Low pressure plasmas and microstructuring technology

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Low Pressure Plasmas and Microstructuring Technology

Gerhard Franz

Low Pressure Plasmas and Microstructuring Technology

123

Prof. Dr. Gerhard Franz Professor for Applied Physics Munich University of Applied Sciences Department of Precision- and Microengineering Engineering Physics 34 Lothstrasse 80335 München Germany [email protected] www.gerhard-franz.org

ISBN 978-3-540-85848-5 e-ISBN 978-3-540-85849-2 DOI 10.1007/978-3-540-85849-2 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009921817 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar S.L. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To my wife

Preface

Over the last forty years, plasma supported processes have attracted ever increasing interest, and now, all modern semiconductor devices undergo at least one plasma-involved processing step, starting from surface cleaning via coating to etching. In total, the range of the treated substrates covers some orders of magnitude: Trenches and linewidths of commercially available devices have already passed the boundary of 100 nm, decorative surface treatment will happen in the mm2 range, and the upper limit is reached with surface protecting layers of windows which are coated with λ/4 layers against IR radiation. The rapid development of the semiconductor industry is inconceivable without the giant progress in the plasma technology. Moore’s law is not carved into stone, and not only the ITRS map1 is subject to change every five years but also new branches develop and others mingle together. Moreover, the quality of conventional materials can be improved by plasma treatment: Cotton becomes more crease-resistant, leather more durable, and the shrinking of wool fibers during the washing process can be significantly reduced. To cut a long story short: More than 150 years after the discovery of the sputtering effect by Grove, plasma-based processes are about to spread out into new fields of research and application [1]—no wonder that the market for etching machines kept growing by an annual rate of 17 % up to the burst of the internet bubble, and it took only some years of recovery to continue the voyage [2]. To realize a multi-color flat panel display measuring 16 square feet, a total of 300 000 LEDs is required. 1994, just in front of the launching of the blue LED, the fraction of LEDs (which effectively emit in the visible range) amounted to about 1/3 of the total compound semiconductor market of $4.4 billion. Four years later, the fraction of so-called high-brightness LEDs had been grown by 64 % (total market 1997: $7.1 billion), and the number of substrates which were subject to epitaxial processes to produce LEDs had been increased by almost 300 % (Fig. 1). This was conservatively updated to an estimated amount of 1

International Technological Roadmap for Semiconductors

VII

VIII

Preface

MBE 5%

LPE 72%

MOVPE 54%

MOVPE VPE 13% 15% 1994: 5550 m

2

LPE 31% VPE 10% 2

1998: 14,190 m

luminescence [lm/W]

InGaAlP/GaAs red/orange

10

1

10-1

AlGaAs/GaAs red

GaP/ZnO/GaP red

GaAsP:N/GaP red/yellow GaP:N/GaP green

GaAsP/GaAs red

GaAs/GaP

1960

1970

1980 year

GaN/InGaN on sapphire

SiC on SiC

1990

2000

Fig. 1. From 1994 to 1998, the area which had been epitaxially grown for LEDs rised by about 300 % from 5 550 m2 to 14 190 m2 (about 11/2 soccer pitches). Simultaneously, the main interest moved from relatively simple techniques (liquid-phase epitaxy) to highly sophisticated gas-phase epitaxy. The etching techniques covered the whole palette from sewing via wet etching up to dry etching methods, and here from hard gunfire by an argon ion beam to subtle etching by generating a reactive plasma of high density but consisting of ions with relatively low energy. This is associated with a steep increase in colorful brillance or efficacy.

$1 billion in 2003 [3]. Despite the deep crisis which was even aggravated by the assassination on Sep 11, the market had recovered. Alone the market for high-brightness LEDs has hit the volume of the total LED market of 1997 [4] (in numbers of LEDs shipped: 20 billion units in 2004, 30 billion units in 2006), and the next landmark has been reached just now: Almost every third LED is used for automotive purposes, most of them belong to the category high-brightness [5]. (Figs. 2). Progress is made most obvious by a flashback. In 1978, the costs for a complete wafer fab amounted to about $20 mio., the narrowest geometries were about 2 and 3 microns, and the yield remained at ridiculous 10 %. Some statements from the first number of Semiconductor International from the year 1979:

Preface

IX 30

7.5

total LED usage [%]

LED market [$ billions]

10.0

5.0

2.5

0.0 2000 2002 2004

2006 year

2008 2010 2012

20

10

0 2000

2002

2004 2006 year

2008

2010

Fig. 2. 2007, the market for LEDs alone has hit the total volume of III/V compound semiconductors of 1997. In particular, the market for automotive LEDs has grown by more than 30 % in the last five years. By the end of 2008, almost every third LED is expected to be part of an automotive illumination with a growing fraction of high-brightness LEDs [4] [5].

• Progress in the area of photoresist and masks provided, deep-UV lithography will be the next technological step [6]. • Plasma etching will become a very important technology for pattern transfer, and plasma deposition shows the same auspicious propects . . . [7]. • Many of the problems what design is concerned can get over provided – a less efficient chip is accepted by the market, – test functions are integrated on the chip, – which is synonymous not to strive for the highest packing density [8]. This was 1978 [9]: • minimum of the structures by about 2.5 μm; • the design was at a maximum of eight masks; • clean room class between 100 and 1 000; • first application of steppers for processing of 4" wafers; • gate electrodes with doped polysilicon; • first operation of lasers for annealing purposes;

X

Preface • minimum gate oxide thickness: 100 ˚ A; • wet etching still dominates, plasma etching as high-end technology is operated only in areas where undercut is not tolerable.

For the public, the computer market was entirely dominated by dinosaur computers, but the first programmable calculators were offered by HewlettPackard and Texas Instruments, the overwhelming part of light was generated by Edison’ evanescent bulb, and “cold light” was just a dream of some scientists. Ion Beam Etching with Ar (DC) anisotropic etching low selectivity poor etch rate sputter yield at ≈ 60 ◦ poor efficiency massive damage

MW-CCP 2.45 GHz ashing with O2 very soft etching very low etch rates no anisotropy

RF-Sputtering 13.56 MHz option w. magnetron with Ar sputtering of dielectrics

downstream

reactive @

@ @

@ capacitive @ coupling @ @

capacitive coupling

MW

@ @ R ? @ reactive sputtering CCP-RF oxides from metals @ capacitive “hot” electrode @ coupling @ R @ CCP-RIE: (ME) “ RIE” 13.56, 27.12 MHz ne ≥ 109 /cm3 antenna anisotropic, selective etch.@ coupling @ prone to high damage @ helicon R discharges @ static 6 13.56 MHz ne ≥ 1012 /cm3 introduction of static resonant excit. magnetic fields

? RF-Ion Etching with Ar for large areas

RF ? Ion Beam Etching with ICP-RF mainly 2 MHz reactive, soft anisotropic processes

RF

@ static @ @ R @

? ICP-RIE 2 or 13.56 MHz non-resonant excit. ne ≥ 1012 /cm3 reactive, soft, anisotropic very high etch rates

? ECR-RIE 2.45 GHz resonant excitat. ne ≥ 1012 /cm3 reactive, soft, anisotropic very high etch rates

Fig. 3. Flow Diagram for Mutual Development of Excitation Methods and Reactive Processes

Preface

XI Sputtering RF DC-Magnetron t

t IBC Dielectrics, Metals Diamond, DLC

Plasma Coating

t PECVD Dielectrics, Metals Diamond, DLC

t

Ion-Plating Dense Metals PECVD, Plasma Enhanced Chemical Vapour Deposition: p ≥ 1000 mTorr (130 Pa): sample on grounded electrode; IBC, Ion Beam Coating; Ion Plating: p < 1 mTorr, evaporation of very dense metal on a sample atop a powered electrode;

Fig. 4. Various dry etching methods. They mainly differ in the excitation method.

Today, the line widths of the most advanced devices are less than 1 000 ˚ A, but the main issues in nanometer devices are still the patterning of the gate (MOS devices) or the mirror or facet (semiconductor lasers). The end of the common planar technology although further miniaturized is predicted to be reached by less than a decade. But perhaps the prognostic power will be comparable with the score of the prognosis for the world-wide oil deposits which should be depleted always “in the next fifteen years”, who knows? Therefore, we consider to address again a broad readership for the next edition of this book. After a short phenomenological introduction, the various methods to generate charged carriers are extensively discussed, keeping in mind that low temperature plasmas have been subject of intense research for now more than a century which comprises the methods to characterize the plasmas. In the second half, the technological techniques of surface refinement are discussed in two lengthy chapters, and we become again aware of the mutual challenges of surface treatment and plasma technology. Detailed derivations are compiled in a special chapter, including the detailed but tedious algebra. It has become customary to spell acronyms which are introduced just recently in capitals which are gradually transformed to small letters, and it is a matter of taste to decide whether this time is already passed by. Since this book is mostly directed to plasma beginners, I preferred to apply capitals throughout.

XII

Preface CCP-IE RF: PE, RIE, MERIE MW: PE t

ICP-IE

t

Plasma Etching

t

IBE

RIBE CAIBE

t

MW-RIE ECR-RIE CCP, Capacitively Coupled Plasma; ECR, Electron Cyclotron Resonance, downstream; ICP, Inductively Coupled Plasma, downstream; MW, Micro Wave (2.45 GHz); PE, Plasma Etching: p > 75 mTorr (10 Pa): sample on grounded electrode; IE, Ion Etching, RIE, Reactive Ion Etching: p with < v > the mean thermal velocity. The mean free path λ itself depends on particle density n and cross section σ according to Eq. (2.3) because it roughly scales with the target area of the molecule. In the simple approach of kinetic gas theory, this cross section is the only one considered and is the cross section for elastic scattering (see [14]). In this approximation, the cross section for elastic scattering is independent of temperature (energy) and is tabulated in units of A), πa20 —the cross section of the hydrogen atom with a0 Bohr’s radius (0.529 ˚ 8.82 × 10−17 cm−2 [sometimes the averaged collision number Pc across 1 cm for A2 ]. a gas at 1 Torr (133 Pa) and 0 ◦ C is found; the relation is σ = 0.283Pc in ˚ A typical value for λ is 4.5 cm for nitrogen at 1 mTorr (0.2 Pa). In fact, the cross section weakly depends on temperature (kinetic energy): With decreasing temperature (reduced molecular speed), it rises slowly due to prolongation of the interaction. For example, by reducing the temperature by 200 ◦ C from +100 ◦ C to −100 ◦ , σel rises by 30 % for nitrogen and oxygen. 2.2.1.2 Cross section and mean free path in plasmas. This temperature dependence is caused by weak intermolecular forces (polarization), which exhibit a weak r−6 dependence for the attractive part of the Lennard-Jones potential ΦLJ . In plasmas, however, we have bare charges which lead to intense, farreaching interactions which can be described analytically only in some simple cases. To begin with, we introduce some experimental details for atoms and electrons: • Atoms – In a sputter system operated with argon at 50 mTorr (7 Pa), the mean free path of the argon atoms is approximately 1.54 mm (25 A2 ). meV Ar, σtot = 26 ˚

8

2 Collisions and cross sections • Ions – The mean free path of ions is considerably smaller. λ for Ar+ ions of medium velocity in argon at 30 mTorr (4 Pa) and 4 eV is about 0.87 mm and rises for 10 eV ions to 1 mm and for 100 eV atoms to 1.4 mm. – In ion beam sources which are driven at significantly lower pressures, they are larger by orders of magnitude: For 400 eV Ar+ ions and a gas pressure of 10−4 Torr (13 mPa), λ equals 51.6 cm. • Electrons – λe of fast electrons in argon [16 eV, which is at the maximum of σ (26 ˚ A2 )] amounts to about 2.5 mm at a pressure of 50 mTorr (7 Pa), corresponding to about 7.5 mm at 20 mTorr (3 Pa). ˚2 ) show a – Slower electrons (Ekin ≈ 4 eV) with a smaller σ (9.7 A longer λe (at 20 mTorr about 13 mm) which applies also to rapid electrons (λe for 100 eV electrons is about 75 mm at 20 mTorr in argon). The frequencies for elastic collisions between electrons and neutrals yield, according to νm = nσv: ∗ for 1 Torr (133 Pa, 3.54 × 1016 cm−3 ) 3.8 × 109 s−1 , ∗ for 100 mTorr (13.3 Pa) 382 × 106 s−1 and ∗ for 10 mTorr (1.3 Pa) 38 × 106 s−1 .

2.2.2 Definitions First, we consider a beam of monoenergetic particles orientated in a parallel direction with respect to the x-axis and exhibiting a flux of N particles per cm2 sec (Fig. 2.1). z dW

Nt

q j

x Np

y

Fig. 2.1. Definition of the differential cross section. Np projectiles collide with Nt targets and are deflected into dΩ, the differential solid angle element.

2.2 Elastic collisions

9

In the case of hitting an annular target with area 2πb db (b: inner diameter, b + db: outer diameter), we define the differential scattering cross section for solid angle dΩ = 2π sin ϑdϑ: dσ(v, ϑ) b db = , dΩ sin ϑ dϑ and the total cross section (“c” for collision) σc (v) = 2π

π dσ(ϑ) 0

dΩ

(2.4)

dΩ,

(2.5)

which, in turn, does not allow any statement of the angular dependence. Furthermore, we define the frequency of collision according to νc (v) = nσc (v)v,

(2.6)

the mean free path again, but decorated with a subscript “c” λ≈

1 nσc (v)

(2.7)

1 . pλ

(2.8)

and the probability of scattering Pm =

The change of momentum is in the center-of-mass system (Fig. 2.2):

q b

CM

r j

Fig. 2.2. Collision between two spheres in the center-of-mass system. b: scattering parameter, θ, ϕ: scattering angle, CM: center of mass.

Δp = mv(1 − cos ϑ) :

(2.9) ◦

Collisions in the forward direction are minimally weighted (cos 0 = 1), those in the backward direction (cos 180◦ = −1), however, are considered most prominent: The diffusion is slowed down most effectively by collisions which are orientated backwards. For the treatment of the transfer of energy and momentum, the most important property is the cross section for momentum transfer, which is σc (v), but weighted by the factor 1 − cos ϑ:

10

2 Collisions and cross sections π dσ(ϑ)

(1 − cos ϑ)dΩ, dΩ along with the frequency of momentum transfer: σm (v, ϑ) = 2π

(2.10)

0

v (2.11) = νc 1 − cos ϑ. λ The cross section of momentum transfer σm can be extracted from the total collision cross section when its angular dependence is known. For a head-on collision (ϑ = 180◦ = π), we calculate, using the laws of conservation for energy and momentum, the maximum amount of transferred energy to (m the mass of the pushing sphere, M the mass of the stroken sphere) νm =

EM,kin =

4mM Em,kin (m + M )2

(2.12)

which is often rewritten using Langevin’s energy loss parameter L=

2m m+M

(2.13)

eventually yielding [15] 2M (2.14) Em,kin . m+M It can easily be shown that the maximum of this function is given for m = M : Collisions between ions with high kinetic energy and slowly moving parent atoms are very effective (cf. Sect. 2.5.2). On the other hand, the energy transfer between an electron and a heavy neutral is negligible (me /mAr ≈ 70 000). EM,kin = L

2.3 Elastic collisions between electrons and neutrals 2.3.1 The total cross section Measurement of the total cross section (sum of all possible scattering processes) was first performed by Ramsauer using the instrument which is sketched in Fig. 2.3. By exposing a plate of zinc sulfide to UV light, electrons are released at point A by the photoelectric effect. Without collisions, they are forced into orbital motions when a perpendicular magnetic field is applied. They pass all the slots from S1 to S5 until they hit the collector C. With elastic collisions, however, electrons are deflected and cannot pass; with inelastic collisions (without deflection), kinetic energy is lost, the orbital diameter increases and these electrons fail to reach the collector as well. Under a certain pressure p, the electron current is measured in B (current i) and C (current I). For known distance s between S4 and S5 , the cross section of absorption can be evaluated from the currents i and I at two different pressures p1 and p2 according to Beer’s law with α = nσ:

2.3 Elastic collisions between electrons and neutrals S2

Fig. 2.3. Ramsauer’s apparatus to measure the total cross section [16]. First, electrons are released from a slab made of ZnS. By applying a magnetic field orientated normally to the plane of the paper the electrons are forced in orbital motions. Their diameter is given by the distance between slots S2 and S5 . Scattering, however, will lead to losses in the Faraday cages B and C.

S1

hn

S3 S4

A S5

B

11

C

i = Ie−αps (p1 − p2 )αs = ln

(2.15)

I1 i 2 , I2 i 1

(2.16)

and for the lightest targets (hydrogen and helium), a hyperbolic dependence as pictured in Fig. 2.4, is obtained. For zero energy, the curve saturates at a certain constant level. 12

2

6

s [10

cm ]

8

-16

10 helium

Fig. 2.4. Experimental cross sections for the elastic collision of electrons through helium as function of electron energy according to [17].

4 2 0 0

10 20 electron energy [eV]

30

2.3.2 Differential cross section We consider now the total cross section and the cross section of momentum transfer, which can be evaluated after having measured the angular depen-

12

2 Collisions and cross sections

dence of the scattering amplitude according to Eq. (2.10). It was again Ramsauer (this time together with Kollath) who applied the measuring principle sketched in Fig. 2.5 [18]. A reactor contains a filament that emits electrons which are accelerated across an electric field. These electrons are deflected by gas molecules and are caught in Faraday cages and eventually measured.

electrometer with Faraday cups

filament grid system

reactor

collector

Fig. 2.5. Sketch of the apparatus developed by Ramsauer and Kollath to determine the angular dependence of the cross section for elastic scattering of electrons after [18].

Some general guidelines are: • At sufficiently high electron energies, we find an intense zero beam: Almost no deflection and then a gradual decline for large scattering angles. • Approaching low electron energies, this picture becomes more complicated. Maxima and minima occur which resemble X-ray diffractograms of fluids (Fig. 2.6). For the first time, this was observed by Bullard and Massey in argon. They explained these patterns by diffraction of electron waves at symmetrically scattering atoms, comparable with the pattern of light waves which are scattered at spheres of similar dimension [19]. However, for the quantitative interpretation of this phenomenon, a quantum mechanical treatise is required. • The complexity of this pattern rises with increasing atomic number of the scattering atom. For elastic scattering of electrons of low energy, a successful collision is assumed to have occurred for a deflecting angle of 90◦ . According to Eq. (2.11), νm then equals νc . For the noble gases, both dependencies (on velocity and on angle) have been measured; hence, σc (v) and σm (v, ϑ) can be compared. For low

2.3 Elastic collisions between electrons and neutrals

13

normalized scattering intensity

1.00

0.75

4.00 eV

0.50 2.80 eV

Fig. 2.6. Scattering of electrons at argon atoms for very c low electron energies [20] ( J. Wiley & Sons, Inc.).

2.00 eV

0.25 1.15 eV

0.00 0

45 90 135 scattering angle [°]

180

energies, the agreement is satisfactory (argon) or almost perfect (helium, Figs. 2.7).

25 argon

6 20

sc sm

4

s [10-16cm2]

s [10-16cm2]

helium

sc sm

15 10

2 5

0

0


60

0 0


60

Fig. 2.7. Comparison between total cross section σc and cross section for momentum transfer σm for the lightest noble gas helium and the medium-light noble gas argon, which behaves entirely differently for low electron energies [20]. For helium, σm exceeds σc for low electron energies: The scattering in the backward direction becomes more important than forward scattering.

2.3.3 Modeling 2.3.3.1 Cross section for the interaction between a point charge and an induced dipole. For potentials which decrease with r−4 (i. e. for forces which decline with r−5 ), there exists a simple correlation between the frequency of collisions and number density, which was pointed out by Maxwell [21].

14


An electron is moving in a field of a potential Φ which declines with r−n (Fig. 2.2). Introducing polar coordinates yields 2 = v 2 = v2 + v⊥

∂r ∂t

2

+ r2

∂φ ∂t

2

,

(2.17)

with v and v⊥ the two components of the velocity v which are orientated in parallel or in perpendicular fashion with respect to the radius vector r, we find for energy and angular momentum m0 m0 v 2 +Φ= · E= 2 2

∂r ∂t

2

+r

2

∂φ 2 ∂t

+ Φ = const,

(2.18)

∂φ = const. (2.19) ∂t With b the scattering parameter, the distance between the two particles under consideration (for lacking interaction, the trajectory of the electron would be a straight line) we get for the angular momentum L = m0 r2

∂φ = m0 bv0 , (2.20) ∂t with the boundary condition for the initial energy E0 (initial velocity v0 ) for vanishing potential Φ = 0 m0 r2

m0 v02 . 2 Since the time-dependent derivations of the coordinates are E0 =

∂φ ∂r ∂φ ∂r ∂r = v0 br2 ∧ = · = · v0 br2 , ∂t ∂t ∂φ ∂t ∂φ

(2.21)

(2.22)

we obtain

∂r ∂φ

2

E0 b2

E0 b2 + Φ = E0 . r4 r2 From Eq. (2.23), the angular dependence of r (∂r/∂φ) yields r2 ∂r =± ∂φ b

+

b2 Φ 1− 2 − r E0

(2.23)

1/2

,

(2.24)

with the minus sign for approaching particles, the plus sign for disappearing ones. This function exhibits a minimum at ∂r/∂φ = 0, when the argument of the square root vanishes. With Φ = ar−n and b = cr, we find for the minimum distance rmin n rmin =

2a m0 v02 (1

− c2 )

.

(2.25)

2 , Applying the hard sphere model, the minimum distance rmin equals to σscatt = πrmin and we can write it down with combined constants:

2.3 Elastic collisions between electrons and neutrals σscatt =

15

A . v 4/n

(2.26)

A , v

(2.27)

For n = 4, this yields the simple form σscatt =

which is the potential which will form between a point charge and an induced dipole with moment μind (with α the polarizability): 1 1 μind = αE ∧ Epot = − μ · E = − αE 2 . 2 2

(2.28)

For a Coulomb potential, we calculate the potential energy to 1 e2 Epot = − α 04 . 2 r

(2.29)

With this derivation, we have obtained the important result that for a force with a declining behavior according to r−5 , e. g. a point charge in a neutral gas which exerts polarization effects, the specific conduct of the EEDF (and hence ve ) does not matter at all. In this very special case, the frequency of elastic collision between electrons and neutrals νm (and also the mean free path λ) will become a√single function of the number density [21].1 σ will then scale with 1/v ∝ 1/ Ekin (Fig. 2.8).

12

s [10-16 cm 2]

10 8

Fig. 2.8. Experimental and calculated cross sections for the elastic collision of electrons through helium as a function of electron kinetic energy according to [17].

6 4 2 0

0

10

20

30

E [eV]

1 For the interaction between point charges Coulomb’s law holds: The potential drops according to 1/rn=1 . In this case, σscatt = A/v 4 , which determines the generalized resistance η in the case of high-density plasmas (cf. Sect. 14.7.2).

16


2.3.3.2 Ramsauer effect. For electron energies larger than some tens V, the cross section decreases gradually; it scales with the inverted ionization potential and is proportional to the polarizability (to a first order approximation: the atomic number) since scattering takes place at the bound electrons of the atoms. Very low values of the scattering cross section are caused by weak interaction with the higher noble gas atoms (Ramsauer effect, Fig. 2.7.2 [22]). This effect was explained for the first time using quantum mechanical methods by Allis and Morse using the partial wave method. According to this theory, the cross section of elastic scattering is composed of partial cross sections, which are denoted s scattering for l = 0, p scattering for l = 1 etc. (ηl is the partial phase shift of the waves with angular momentum L = l(l + 1)¯ h, p=h ¯ k): σ0 =

σl ∧ σl =

l

4π (2l + 1) sin2 ηl . k2

(2.30.1)

The affiliated potentials become deeper but simultaneously more short-ranged— the atomic radius grows by about 150 % when going from helium (0.49 ˚ A) to xenon (1.24 ˚ A) leading to a ratio of atomic volumes of about 16, whereas the number of electrons increases by a factor of 27! For slow electrons (k → 0 ∧ λ → ∞), L becomes very small, and for the noble gas atoms argon and heavier, only the component of zero order, i. e. s scattering, remains of importance: 4π sin2 η0 . (2.30.2) k2 η0 → π at k = 0 ⇒ σ0 = 0. Hence, a head-on collision (ϑ = π) between an electron and an atom cannot be observed. For the heaviest noble gas atoms (Xe and Rn) this effect is much more distinct than for the lighter ones (Ar and Kr) and vanishes for He and Ne. σ0 =

2.3.4 The frequency of elastic collisions between electrons and neutrals For the special cases of hydrogen and helium, the collision frequency simply scales with discharge pressure (number density): νm = const × p0 , which follow the equation with p0 = 273/T × p [p in Torr (in Pa), Figs. 2.9]: • He: νm = 2.31 × 109 p0 (3.07 × 1011 p0 ); • H2 : νm = 5.93 × 109 p0 (7.89 × 1011 p0 ). This does not hold true for the heavier noble gases (at least for neon, we see a fairly good approximation with an asymptotic behavior) and a strange conduct is observed for chlorine. Here, the maximum is foreshadowed by a minimum at very low energies in Cl· and Cl2 (Fig. 2.10). Following Massey, we estimate the mean free paths for electrons λe in argon according to Eq. (2.7) [25]. For typical electron energies of several electronvolts, we calculate a λe of several centimeters (Figs. 2.11).

2.3 Elastic collisions between electrons and neutrals

6

25 H2

n/p [10 9 sec Torr] -1

n/p [10 9 sec Torr] -1

5 4 3

He

2 1 0 0

17

5

10

15 20 Ekin [eV]

25

30

35

Xe Kr

20

Ar

15 10 5 0 0

Ne

5

10

15 20 Ekin [eV]

25

30

35

Fig. 2.9. Frequency of elastic collisions between electrons and neutrals for hydrogen and helium, and the heavier noble gases [23].

nm [sec -1]

108

107

Cl2

Cl

106 10-2

10-1

100 101 Ekin [eV]

102

Fig. 2.10. Frequency of elastic collisions between electrons and neutrals for Cl· and Cl2 after [24]; discharge pressure 10 mTorr.

2.3.5 Cross section and rate constant for argon By Eqs. (2.6) + (2.11), the cross section for a specific scattering process is related to the probability of its occurrence. This frequency ν depends on the density of target molecules, and reducing ν to the unit density yields the rate constant k, k = < ve > σ

(2.31.1)

in particular for the momentum transfer km : km = < ve > σelast .

(2.31.2)

This relation allies atomic properties which must be evaluated by scattering experiments with macroscopic kinetic parameters, with the mean electron velocity being the connecting link. The directed energy E from the scattering experi-

18


100 argon

10 1

1 mTorr 5 mTorr 10 mTorr

0.1 0.01 0

100 mTorr

3

mean free path [cm]

mean free path [cm]

100

argon

10

0.5 eV 1.0 eV 1.6 eV 2.4 eV 4.3 eV 8 eV 12 eV

1

50 mTorr

6 9 12 electron energy [eV]

0.1 0

15

10

20 30 40 pressure [mTorr]

50

60

Fig. 2.11. Calculated mean free paths of low-energy electrons (below the ionization threshold) in argon after [25].

ments has to be replaced by the thermal energy E for a discharge. Most frequently, a Maxwellian distribution is assumed but especially for low plasma densities (E = kB Te ), however, this assumption has turned out to be highly questionable (Sect. 14.1). The connection between σ and k is pictured for argon in Figs. 2.12. The data for σelast have been compiled from Nakanishi and Szmytkowski [26, 27]. 30 100

s

-1

3

10

-1

k

1

-2

10

-6

k

s

k [10 cm sec ]

1 10

selast [10-16 cm2]

20

10

k [10 -6 cm3sec-1]

selast [10-16 cm2]

2

10-3 0.1

0

0

25

50

75

electron energy [eV]

0 100

0.01

0.1

1

10

10-4 100


Fig. 2.12. Argon: total cross section for elastic scattering σelast and the corresponding rate constant k = σelast × < ve > after [25] − [27].

2.4 Elastic collisions between heavy particles The scattering among heavy particles exhibits a remarkably distinct maximum in the forward direction [28]. Measuring such an angular dependence is almost hopeless. For energies of a few electronvolts, the requirements for angular resolu-

2.4 Elastic collisions between heavy particles

19

tion are difficult to meet, and these demands increase with rising momentum of the projectiles. The accuracy necessary to measure the total elastic cross section for argon is for thermal atoms 0.70◦ , those with 1 000 K temperature require an accuracy of 0.30◦ , which is even sharper for atoms of 1 eV kinetic energy (11 600 K, 0.11◦ ). These difficulties are reinforced for the measurement and calculation of the differential cross section which means the determination of the fraction of molecules which hit target molecules and are subsequently scattered into the solid angle dΩ = 2π sin θdθ. In Table 2.1, the results of calculations for the differential cross section of helium against a beam of protons for two different scattering potentials (Coulomb and Hartree, a simple pseudopotential) are compiled [29].

Table 2.1. Differential cross sections dσ(ϑ)/dΩ per unit solid angle for protons of kinetic energy of 110 eV in helium in units of a20 .

0 12 28 34 57 80 114 137 167

dσ(ϑ)/dΩ Hartree potential Coulomb potential 9 × 103 ∞ 7.85 124.0 2.00 6.10 0.72 2.85 0.21 0.40 0.08 0.12 0.04 0.04 0.03 0.02

This forward direction is more distinct for larger kinetic energies of the projectiles (coming from low energies, only elastic scattering processes can occur). Normally, the determination of the angular dependence is confined to the range without the zero beam. One of the pioneering experiments was carried out by Berry and Cramer in neon and argon [30] − [33] (Fig. 2.13). We note that the cross section remains at several ˚ A2 up to kinetic energies of about 500 eV. Hence, an ion beam is almost entirely dissipated after having passed a distance of only 1 cm at high pressures (cf. Sect. 6.8). However, some advanced, indirect methods are available, e. g. measuring the complex conductivity (Sect. 5.6), and the line width of the electron cyclotron resonance, which is mainly determined by the collisions between electrons and neutrals [34] (Sect. 7.5).

20


4 A

s[10-16 cm2]

3

A - 250 eV B - 500 eV C - 750 eV D - 1000 eV E - 1250 eV F - 1500 eV G - 1750 eV

B

C D 2 E F G

1

0 0

45

90 135 scattering angle [°]

180

Fig. 2.13. Cross section of Ar+ ions in argon as a function of scattering angle [30].

2.5 Inelastic collisions 2.5.1 Inelastic collisions between electrons and heavy particles During inelastic collisions, internal grades of freedom are excited by transformation of kinetic energy of the colliding projectiles into internal energy levels of the target. For electron impact, the most important reactions are with A∗ an excited molecule: • Excitation in upper molecular states (vibration levels or electronic states, [Eq. (2.32)]. • Dissociation into radicals (which is very effective in discharges through electronegative gases [35], [Eq. (2.33)]. • Dissociation into ions [Eq. (2.34)]. • α-ionization [Eq. (2.35)]. • Ionizing dissociation [Eq. (2.36)] and • Electron attachment (dissociative or electron attachment, respectively), for electronegative gases [Eq. (2.37)]. e− + A2 −→ A∗2 + e−

(2.32)

e− + A2 −→ 2 A∗ + e−

(2.33)

e− + A2 −→ A+ + A− + e−

(2.34)

2.5 Inelastic collisions

21 − e− + A2 −→ A+ 2 + 2e

(2.35)

e− + A2 −→ A+ + A + 2 e−

(2.36)

e− + A2 −→ A− + A∗ .

(2.37)

2.5.1.1 Experimental methods. In atomic gases, dissociation reactions do not take place; one of the “classical” reaction is the ionization of mercury vapor by electron impact (Franck-Hertz experiment, Figs. 2.14). In a triode tube

anode grid filament 0.5 V

_ 20 V

+

400 14.7 V 9.8 V

200

sion [10-16 cm2]

anode current [a. U.]

30

300

4.9 V

100

0

0

2

4

6 8 10 12 anode voltage [V]

14

16

20

10

0 10


16

Fig. 2.14. Franck-Hertz experiment. Top: experimental setup, bottom: the anodic current (LHS) and the cross section of ionization (RHS) as functions of acceleration voltage (electron energy) in mercury vapor [36]. At the ionization potential of 4.9 eV and its multiples, the characteristic exhibits distinct deviations from the expected behavior, and the cross section does not exhibit a smooth behavior [37].

filled with mercury vapor, the electrons which are emitted by a glowing cathode are accelerated through a uniform field to a grid with the anode plate placed directly behind the grid. Those which have passed it will be slowly decelerated by elastic collisions with gas atoms, but in general, we observe a slight increase of the anodic current with growing anodic voltage. However, when the ionization potential of mercury (4.9 eV) has been reached, the I(V ) characteristic

22


drops sharply. This phenomenon is caused by inelastic collisions of the electrons with mercury atoms which are ionized by electron impact. A small retarding field between grid and anode causes the anodic current to vanish at this value, because the electrons have less than 4.9 eV kinetic energy and are not able to move against this retarding potential: Hgvapor + e− −→ Hg+ + 2 e− .

(2.38)

This setup has been improved by Maier-Leibnitz [38] and further by Schulz and Fox who employed a retarding potential difference method [39]. They concluded that the electron energy distribution function, EEDF, in the original Frank-Hertz experiment was very broad, too broad to obtain an accurate measurement of the excitation probability. For that purpose, the energy of the bombarding electrons as well as the energy of those that have lost energy in inelastic collisions must be known. Therefore, they constructed an energy filter system which made use of also in the determination of the ion energy distribution function IEDF (cf. Chap. 6). By means of this three-grid optics, it is possible to extract a nearly monoenergetic beam of electrons (Figs. 2.15).

filament

F1

F2

F3

0 collision chamber

F [a.u.]

retarding potential aperture F3 F1 F2

collector

x [a. u.]

Fig. 2.15. By employing the retarding potential difference method, Schulz and Fox could significantly improve the accuracy of the determination of the inelastic scattering cross section [39].

The first grid is positively biased by approximately 3 V with respect to the filament and serves to draw electrons. The second grid is slightly negative with respect to the filament, the electrons must run uphill against a retarding potential, and as a result, the EEDF is chopped off. It is only the high-energy fraction of the electrons which is able to pass. The third grid shows the same potential as the first grid. Since the EEDF can be sensitively influenced by variation of Φ2 , Φ2 + ΔΦ2 , we obtain two values for the collector current which can be solely referred to the electrons with energy ΔΦ2 . By variation of the retarding potential Φ2 , grid 2 also varies the voltage to the collision chamber which determines the collector current, i. e. the EEDF.


23

2.5.1.2 Cross section. The cross section for an inelastic scattering process exhibits a threshold which is followed by a steep rise (several orders of magnitude) to reach a maximum. This is sharply peaked for optical transitions at energies which are close to the threshold, and a little broader for ionizations at about 100−150 eV, which is consistent with the de Broglie wavelength of the electrons of approximately 1 ˚ A, the typical diameter of the targets where the electron waves are expected to be deflected most intensely. For higher energies, we find a slow decrease since the time of interaction between the collision partners is gradually shortened (Fig. 2.16 in linear scale, Figs. 2.17 in logarithmic scale).

cross section [10-16cm2]

3

2 argon hydrogen

1

0

10

100

1000

Fig. 2.16. Typical energy dependence of the cross section for inelastic scattering between electrons and molecules for argon and hydrogen.


The cross section of ionization has been comprehensively described by Bethe and Salpeter [40, 41]. According to Born’s approximation, the total cross section of a discrete excitement into state n with energy En of the excited state can be approximated by

1 4πa20 z 2 C A ln(4 B T ) + + O , (2.39) T T T2 with a0 Bohr’s radius, z the charge and T the kinetic energy of the projectile, mostly given in Rydberg (= 13.6 eV), irrespective of the nature of the projectile. A is the squared matrix element of the specific excitation, and the B, C are constants which will be generated during the integration [42], and the factor 4 in the logarithmic summand which is left for historical reasons. The third term is a very small correction, and most frequently, the second term is neglected as well, and we arrive at the following simplified approximation [43, 44], which is known as Bethe’s asymptotic cross section [41]: σn =

4πa20 z 2 [A ln(4 B T )] . (2.40) T Unfortunately, this theory holds best for large kinetic energies of the projectiles. However, for ionization and excitation processes by electron impact, the energy σn =

24


dependence just above threshold is of extreme importance. To describe this regime properly, in particular the steep rise beyond threshold and the maximum, Wannier developed a theory for single ionization which states that in the energy range near threshold, the probability of ionization rises exponentially with the energy in excess of the ionization energy (more correctly: raised to the 1.127th power) [45, 46], which had been improved by Geltman [47]. Lotz introduced an equation with three parameters (a, b and c), which approximates the cross sections of most of the interesting gases within a deviation of less than ±10% to experimental data [44] (Fig. 2.17.2):

1 21P

0.1

1

2S

3

0.01

0.001 10

ln(T /En ) T 1 − b exp −c −1 T En

cross section [10-17 cm 2]

cross section [10-17 cm2]

σn = a

2P

(2.41)

10

1 argon hydrogen

0.1

0.01 10


.

100 electron energy [eV]

1000

Fig. 2.17. Typical dependence of the inelastic cross section of molecules and electrons on the electronic kinetic energy. Note the double logarithmic scale to make the steep increase visible. LHS: energy dependence of the cross section for excitation into various optical levels of helium, according to the formula of Mityureva and Smirnov [48]. RHS: Lotz’s theory for ionization of hydrogen and argon, experimental data for argon are from one of the most cited papers by Rapp and Englander-Golden [49].

The main difference between ionization and excitation into optical levels by electron impact is caused by the electrostatic interaction of three charges after the collision process. For the first time, this problem has been addressed by Wannier ([46], also cf. [50], Fig. 2.17.1). As a main result, the maximum for excitation into optical levels is located very close to the threshold, whereas for ionization, the difference in energy between maximum and threshold will be no less than a factor of 4 in excess of the ionization energy [48]. The graphs of Figs. 2.17.1 are plotted applying an approximation containing three adjustable parameters σ0 , φ, and γ:

σn = σ0

T − En T

−γ

T −1+φ En

.

(2.42)


25

In summary, we find the cross section for ionization by electronic impact to rise sharply from zero at a specific threshold by several orders of magnitude until it reaches a maximum at approximately 100 eV because the de Broglie wavelength of about 1.2 ˚ A fits the atomic dimensions. For further increases of electron energies, the cross section gradually decreases. At 500 eV, it has typically weakened by a factor of about 2 compared with its maximum value (Fig. 2.18).

Hg

sion [10-16 cm2]

10

air Ar Ne

1

H2 He

0.1 10


Fig. 2.18. Cross section for ionization by electron impact for various gases in double-logc Oxford arithmic scale [51] ( University Press).

10000

2.5.1.3 Rate constant for ionization. Employing Eqs. (2.31), we ally the frequency of ionization with the rate constant for this process, pictured in Figs. 2.19 for ionization of argon, with a Maxwellian distribution assumed. In fact, it is the rate constant we are interested in to calculate the carrier generation. Especially at the threshold, the slope of the k(E) curve repeats the steep increase of the cross section. In low temperature plasmas, it is but these electrons in the high-energy tail that are responsible for the maintenance of the discharge. 2.5.1.4 Electron attachment. Electron attachment plays a significant role in discharges through electronegative gases. One of the main processes is dissociative electron attachment, e. g.: O2 + e− −→ O− + O,

(2.43)

SF6 + e− −→ SF− 5 + F,

(2.44)

a two-body collision leading to the formation of a negatively charged oxygen ion and and oxygen atom, or a very special case, the formation of a SF− 5 ion and a fluorine atom.2 2

It is this ion which captures almost all the electrons at very low discharge pressures.

26


3

Lotz Li

2

1 argon

0

0

250

500

750

1000

1000 k s

1

100

0.1

0.01 10


10

argon

100 electron energy [eV]

1000

rate constant [10-11 cm3sec-1]



10

1

Fig. 2.19. Ionization by electron impact for argon. LHS: Two different formulae, according to Lotz and Li, yield an almost indistinguishable result [44, 52]. RHS: For cross section and rate constant, a Maxwellian distribution for the electrons assumed. The calculation is based on the Lotz formula [44].

normalized cross section

1.0 Schulz Craggs et al.

0.8 0.6 0.4 0.2 0.0 4


12

Fig. 2.20. Normalized cross section of electron attachment and formation of O− ions. The asterisks were evaluated by Schulz, the circles are due to measurements by Craggs et al. The maximum (Schulz) has been found at 1.25 × 10−18 c J. Wicm2 and 6.7 eV [53] ( ley & Sons, Inc.).

Atoms are not able to capture electrons in a two-body collision process because the laws of conservation of momentum are violated. Therefore, the cross sections are very low (in the order of 10−18 cm2 ), but exhibit sharp maxima at electron energies of a few eV (Fig. 2.20). The density of negative ions can far surpass that of the electrons which changes the characteristics of the discharge completely. As is evident from Fig. 2.22, the cross section for dissociative attachment exhibits two maxima for oxygen. The first can be referred to reaction (2.43), whereas the maximum at higher energies is due to the reaction e− + O2 −→ O− + O+ + e− .

(2.45)


27

One of the most prominent cases is that of BCl3 as an additive in discharges through chlorine. In principle, three reaction paths are possible [54]: BCl3 + e− −→ BCl∗3 ,

(2.46.1)

BCl3 + e− −→ BCl2 + Cl− ,

(2.46.2)

BCl3 + e− −→ BCl + Cl− 2,

(2.46.3)

BCl∗3

the metastable parent anion, which can be effectively generated by with extremely low energy electrons up to 0.1 eV [54]. Although the pyramidal BCl− 3 ion is expected to be long lived, (Petrovic et al. detected this anion in a mass spectrometer, which indicates a lifetime in the microsecond range [55]), due to its small electron affinity of 0.33 eV, it will be easily discharged by electron impact. All of the discharges which are dealt with in the following chapters exhibit higher mean electron energies. Therefore, BCl− 3 anions are most likely to elude detection.3 Tav et al. have estimated the cross section for dissociative electron attachment to BCl3 to be ≤ 5 × 10−18 cm2 [according to Eqs. (2.46.2) and (2.46.3)] and for electron attachment to BCl3 according to Eq. (2.46.1) to 4.5 × 10−15 cm2 . 2.5.1.5 Total collision cross section. Since the numerous possibilites of excitation exclude each other, their probabilities must be added up, and we can define a total cross section for all possible excitations [57]: σtot =

Pi σi (v),

(2.47)

i

with Pi the probability and σi the differential cross section for reaction i. For example, the maximum of the total cross section for electron impact for argon A2 and can be found just above the ionization amounts to 26 × 10−16 cm2 = 26 ˚ threshold of 15.76 eV. This means an electronic mean free path of 1.5 mm at a discharge pressure of 50 mTorr (7 Pa, n = 2 × 1015 cm−3 ). For low electron energies, the cross section for elastic scattering almost equals the total scattering cross section since the velocities of the electrons do not suffice for atomic or molecular excitation. Especially for the noble gases, numerous measurements have been performed and modeled. A compilation of these measurements is pictured in Fig. 2.21. It was Myers who split up the total cross section σO2 ,tot into its separate parts for oxygen (Fig. 2.22) [58]. For energies up to approximately 50 eV, the 3 However, Gottscho and Gaede succeeded in generating and observing this anion in low-frequency discharges at 50 kHz where the electron density will completely relax during each half cycle leaving ample time for the slow electrons to generate the BCl− 3 anion [56]. Using a non-invasive method (laser photodetachment), they could detect this anion before the electrons were reheated in the next half cycle.

28


40 Xe Kr

s[10-16 cm2]

30 20

Ar

10 Ne

0

0

2

4 6 8 electron velocity [V1/2]

10

Fig. 2.21. Experimental scattering total cross sections of the heavy noble gases for low electron velocities. The almost complete transparency at very low velocites is striking; the distinct maximum corresponds well with the ionizac Oxtion energy Eion [25] ( ford University Press). Compare with Fig. 2.7.2.

cross section for elastic scattering is the most important contribution to the cross section for ionization.

A

-15

10

G

s [cm2]

10-17

C

B

B F

C E

10-19

E

D

D G

C

10-21 0.1

1

10

Fig. 2.22. The single contributions of the total cross section for electronic impact were split up by Myers for O2 [58]. The letters denote the contributions as follows: A: elastic scattering, B: rotatory excitation, C: oscillatory excitation, D: excitation to singlet-oxygen, E: dissociative attachment (two maxima!), F: excitation to upper electronic states, G: ionization.

E [eV]

2.5.2 Inelastic collisions between heavy particles To begin with, the literature covering the low-energy range, which is of paramount importance in glow discharges, is widely scattered, in particular when we compare this range with the countless number of papers dealing with energies exceeding 1 keV. This is mainly caused by the low cross sections for inelastic processes. Hence, elastic scattering will still dominate the interaction between the gas constituents. The main reactions between heavy particles are: • Generation of electrons by ionic impact [so-called β-ionization, Fig. 2.23, Eq. (2.48)].


29

• Excitation to electronic levels [Eq. (2.49)]. • Electron stripping [Eq. (2.50)]. • Simple charge transfer [Figs. 2.25 − 2.28, Eq. (2.51)]. • Double charge transfer [Eq. (2.52)].

+

Ar in Ar

sion [10-16 cm2]

10

H2+ in H 2 Ne+ in Ne

He+ in He

1

0.1 2 10

103 ion energy [eV]

104

Fig. 2.23. β-ionization: cross section of ionization by ions of the parent gas after [59]. The values are not corrected by ionizations caused by secc Oxford ondary electrons ( University Press).

+ + − A+ 2 + B −→ A2 + B + e ,

(2.48)

A+,∗ + B −→ A+ + B∗ ,

(2.49)

A+ + B −→ A2+ + B + e− ,

(2.50)

B+ + A −→ A+ + B,

(2.51)

A+ + B −→ A− + B2+ .

(2.52)

As we can see from Fig. 2.23, the threshold for ionization by ionic impact is relatively low but amounts to about twice the ionization potential. The classical theory of J.J. Thomson yields cross sections equal in size for equal velocities of ions and electrons. The same result is obtained with quantum mechanics with the restriction that the impact of the colliding particles is large compared with their pair potential (Born’s approximation). Choosing the abscissa scale skilfully (in fact, it is the momentum of the colliding particle), we can find a similar behavior between the ionic and the electronic cross section, however, dilated by some orders of magnitude for the heavy particles (Fig. 2.24). The

30


sion [10-16 cm2]

electron energy [eV] 100 1000

Hooper et al., 1962 Smith, 1930 Tozer, Craggs; 1930 Bleakney, 1930 Lampe et al., 1957

1

0.1

1 proton energy [MeV]

Fig. 2.24. Comparison of the ionization cross sections of argon for protons (dashed) and c J. Wielectrons (solid) [60] ( ley & Sons, Inc.).

cross sections for ionization (β-ionization) do not reach hogh values until the ions have been accelerated to velocities which are comparable to those of the electrons for α-ionization. Hence, for energies up to several hundreds of eV, the ionic cross section is smaller by about two orders of magnitude. For the same kinetic energy, the ionic momentum is considerably greater, which leads to very short de-Broglie wavelengths. Hence, even for inelastic collisions, there are hardly any diffraction effects. 2.5.2.1 Charge transfer. For charge transfer, an almost complete exchange of momentum is of importance. Consider a rapidly moving ion colliding with a slow molecule (Fig. 2.25). This collision can result in a slowly moving ion and a rapid neutral, and the components of velocity are conserved. In the case that both colliding particles have the same mass, this process is very efficient, and this collision is denoted (symmetric or resonant charge transfer, Fig. 2.26). We expect the cross section for this process to be comparable with that for elastic collision. To begin with, we assume A = B, i. e. the ionization potentials should be differ by the amount ΔEion : Eion,B = Eion,A + ΔEion .

(2.53)

Massey discussed this charge transfer by means of the correspondence principle [62]. For a particle velocity v smaller than that of the bounded electrons, they will adapt to this disturbance without an electron transfer (so-called adabatic approximation). We find the time of collision τ to scale inversely with the frequency ν of an electron oscillation with τ ≈ b/v (b the so-called adiabatic parameter):


31

15 H+, Kr

s [10-16 cm2]

12 9

H +, Xe

6

Fig. 2.25. Cross section for the asymmetric charge transfer of protons in noble gases after [61].

3 0

0

5

10

15

20

25

V1/2

1 ΔEion aΔEion a = ∧ν = =⇒ v = . (2.54) v ν h h When the velocity of the particles exceeds this threshold, no adiabatic collision will take place but an ionization will occur. The larger the difference in ionization potential ΔEion , the higher the energetic threshold to become a competitive process. On the other hand, the time for interaction declines with rising energy which eventually reduces the probability of ionization: A maximum has been evolved (Fig. 2.26). τ=

s [10-16 cm2]

100

Fig. 2.26. Principal energy dependence of the two mechanisms for charge transfer. in double-logarithmic scale, the resonant charge transfer linearly scales with energy, whereas the asymmetric charge transfer exhibits an energetic maximum.

10

symmetric charge transfer asymmetric charge transfer

1

1

10

100

1000

E [eV]

2.5.2.2 Resonant charge transfer. For a resonant charge transfer with ΔEion = 0, we expect a monotonous increase in the cross section for ionization for lower energies in the adiabatic approximation. Albeit the correspondence principle predicts this conduct qualitatively correctly, more extensive descriptions of the height of the maximum (e. g. ΔEion /Ekin ) are definitely impossible.

32


The principal shape of the energetic dependence of the cross section is depicted in Fig. 2.26. It can be described by the formula σ = (a − b ln v)2

(2.55)

with a, b empirical constants and v the relative velocity of the colliding particles [63]. For a resonant reaction, there exists a linear relation between the cross section and the logarithmic kinetic energy (Fig. 2.27).

Firsow formula Gilbody, Hasted (1956) Flaks, Solovev (1958) Gustafsson, Lindholm (1960)

sCT [10-16 cm2]

60

40

Fig. 2.27. Resonant charge transfer: Ar+ fast + Arslow −→ Ar+ + Ar fast ; comparison of slow experimental data with theory c J. (Firsow formula) [64] ( Wiley & Sons, Inc.).

20

0 0

20

40 v [eV1/2]

60

80

The cross section reaches a maximum for an ion in its parent gas at slow velocities. In this energy range, it can deliver the main contribution to the total cross section (Fig. 2.28). The electron transfer mostly happens without any exchange of momentum: The generated ions exhibit negligible velocities, and the scattered atoms alter their direction only in a very narrow scale; hence, the scattering cone exhibits a very low opening angle [65]. The measuring of the charge transfer was made possible by an apparatus which consists of a magnetic field which deflects the ion beam and focuses it on to the sample. To prevent multiple ionizations, the cross section of the reactor has to be made sufficiently thin. The ions produced are sucked off by an electric field which is directed perpendicular to the reactor. The ions are measured by a Faraday cage which is negatively biased to avoid distortions by secondary electrons (so-called condensor method) [66]. Ternary collisions occur relatively rarely; the ratio of the probabilities of a two-body collision over a three-body collision is about thousand [67], therefore reactions + 2A+ + B −→ A+ 2 +B

(2.56)

(recombination) are not very likely to happen. Since the number of collisions scales with the squared pressure p (at constant temperature: With the squared number density n), the probability for recombination increases with rising pressure (cf. Sect. 4.7).


33

50

50

s [10-16 cm2]

s [10-16 cm2]

s (total) s (resonant charge transfer) s (elastic collision)

helium

25

0 0

5

10 v [eV1/2]

15

neon

25

20

0 0

5

10 v [eV1/2]

15

20


75

s [10-16 cm2]


argon

50

25

0 0

5

10 v [eV1/2]

15

20

Fig. 2.28. Cross sections for the processes of elastic scattering and symmetric (resonant) charge transfer of He+ in He, Ne+ in Ne, Ar+ in Ar [31] − [33]. At low energies, the resonant charge transfer can deliver the main contribution to the total cross section.

2.5.2.3 Penning ionization. Among the numerous reactions which can occur between heavy particles, we should mention the Penning ionization, collisions of second order. This mechanism becomes important in discharges of reactive gases which are doped with noble gases (cf. Sects. 10.4.3, 10.6.4 and 11.8.6) and consists of ionization by metastable species. For ionization by collisional impact, the metastable level must be higher than the ionization potential of the neutral molecule or atom. Since the return to the ground level from a metastable level is forbidden by selection rules, these levels exhibit considerable lifetimes. The metastable levels for neon (16.62 and 16.7 eV) and those of argon (11.55 and 11.72 eV) are higher than the ionization potentials of most of the gaseous elements and gaseous compounds. By doping a discharge of argon with neon, metastable neon atoms can ionize argon

34


atoms. The state of metastability is denoted with an asterisk (cf. also Fig. 4.4, enlargement of α). Ne∗ + Ar −→ Ne + Ar+ + e− .

(2.57)

The cross sections at thermal energies are somewhat larger than the gas kinetic cross sections (≈ 10−15 cm2 ), therefore, its probability approaches unity, and this process becomes dominant very quickly [68, 69] and will be manifested in a rise of the discharge current [70]. Mierdel discussed consecutive staircase processes which cause an additional ionization after a collision of two metastables [71]; this effect would become more probable with rising density of the metastable atoms. Ar∗ + Ar∗ −→ Ar+ + Ar + e− .

(2.58)

Furthermore, ionization of metastables can happen by electron impact: Ar∗ + e− −→ Ar+ + 2 e− .

(2.59)

Since the energy difference from the metastable level to ionization has shrunk from 15.76 eV to only 4.21 eV, considerably more electrons are able to succeed in an ionization. Although the density of the metastable species is very low, Ingold has estimated this mechanism to be the most important ionization source in discharges of Hg/Ar [72]. 2.5.3 Collisions between photons and molecules As final collision processes between individual particles, we must consider impacts of photons (Fig. 2.29). Again, the threshold of ionization has to be exceeded. But, in contrast to the impacts of charged particles, we observe for photons the occurrence of a steep absorption edge of several orders of magnitude at E = Eion to decline precipitously beyond this sharp maximum. For example, the maximum of the photoelectric excitation is found for argon at 15.5 eV and exhibits a value of about 0.36 × 10−16 cm2 . Hence, the maxima for the yield of ionization for electrons, ions and photons are located at very different values.

2.6 Generation of secondary electrons at surfaces 2.6.1 Electrons The impact of electrons incident on a surface causes the following reactions: • Elastic scattering, the energy does not change but the momentum, very high contribution in the spectrum.

2.6 Generation of secondary electrons at surfaces 40

M2 M3 Ar

s [10-18 cm 2]

30

M1

20

10

Ne

L1

K

Fig. 2.29. Cross section for photo absorptions of c Sprinnoble gases [73] ( ger-Verlag).

L2 L 3

He

0 20

35

40

l [nm]

60

80

• Inelastic scattering, both energy and momentum are changed, small contribution in the spectrum. • Towards low energies, the second rise peaking at energies between 5 and 10 eV is caused by secondary electrons (Fig. 2.30).

1.00

f(E)

0.75

0.50

0.25

0.00 0

50

100 Ese [eV]

150

Fig. 2.30. Energy distribution of secondary electrons, f (E)se , for 160 eV electrons incident on a shiny gold surface (normalized to the low-enc Oxergy maximum) [74] ( ford University Press).

The yield of secondary electrons depends sensitively on • The energy of the electrons incident on the surface, which causes a maximum at medium energies. At low energies, the electrons are rather absorbed, at higher energies, they are scattered elastically. • The chemical composition of the topmost layer: It rises with increasing density and work function with the well-known periodic dependencies. The yield increases with growing fill-up of the electronic shells. When the filling of a new shell has begun before an inner shell has been completely filled, we observed discontinuities, in particular for half-occupied shells.

36

2 Collisions and cross sections • The quality of the surface: the shinier, the higher the yield.

For many metals with a clean surface the yield for secondary electrons, δ, equals unity (Fig. 2.31). For insulating layers, the values for δ can reach 15 (to measure δ, momentum methods are applied, however, the errors are rather large). 2.0

1.5

Ag

d

W Cu

1.0

Mo

0.5 Be

0.0

0


1500

Fig. 2.31. Coefficients for secondary emission, δ, for several metals as a function of the energy of the primary electrons incident on the surface.

In insulating materials, the conduction band lacks electrons almost entirely whereas the valence band is nearly filled. Hence, the primary electrons can generate secondary electrons only by interaction with valence band electrons. But these electrons gain a higher amount of kinetic energy than comparable metal electrons, which enhances their probability of reaching the surface and eventually leaving the substrate. For the same reason, the yield increases with flatter angles of incidence. To differentiate between the electrons, the coefficient for backscattering η refers to those electrons with energy between 50 eV and the energy of the electrons incident on the surface, and the coefficient for secondary electrons δ refers to those electrons having energy between 0 and 50 eV. 2.6.2 Ions Ions incident on the surface can release electrons as well, and this mechanism is of considerable importance in DC discharges and also plays a role in RF discharges which are capacitively coupled. The energy distribution of the secondary electrons responds relatively flatly to the kinetic energy of the incident ions and exhibits a broad maximum between 5 and 10 eV [75] (Fig. 2.32). The distributions for Ei ≤ 200 eV are very similar; γi rises steeply for growing energies, and a maximum becomes visible at very low electron energies. Eventually, for the highest ion energies, the energy distribution of the secondaries peaks close to zero.

2.6 Generation of secondary electrons at surfaces 30 25

1000

10

100 200

15 1000

10 5 0 0

Eion - 2 Wa

n(Ei) [10-3]

20

He+

Mo

Fig. 2.32. Energy distribution of electrons which are released from a surface being atomically cleaned by He+ ions with various kinetic energies. The dashed lines without symbols separate the Auger processes from the other mechanisms which release secondary elecc The American trons [75] ( Physical Society).

E(He+) [eV] 10 40 100 200 600 1000

40

600

10


37

20

An electron can be released only when the sum of the kinetic energy of the ion incident on the surface and the ionization potential exceeds twice the value of the work function Wa : For every electron emitted from the surface, a second one is required to neutralize the incident ion. Ekin (e− ) ≥ Eion − 2 Wa .

(2.60)

This condition is remarkably violated by neon which can be attributed to an additional Auger process [76], and which is not observed when the electron is released by Ne ions of 10 eV kinetic energy. This strange phenomenon is explained by the discharging of ions of higher energy in close vicinity to the surface. Afterwards, these atoms can generate rapidly moving electrons in a process denoted Auger relaxation. The yield of this process, however, exceeds that of the Auger neutralization. Typical values for the yield for secondary electrons, γ, vary between 0.05 and 0.1 for the heavy noble gases incident on molybdenum and tungsten (Fig. 2.33). The distinct energetic minimum of the number of secondary electrons for He+ is absolutely striking. This is repeated for the heavier noble gas, however, to a weaker extent, which is due to a larger penetration depth with rising kinetic energy, and the electrons which are now released are attenuated during their course to the surface. This reduction in secondary electrons is overcompensated by the growth of the yield at further growing kinetic energies. For neon, this effect is covered by the above mentioned Auger relaxation.4 Although the curves of yield are very similar, the electron yield is • always higher for molybdenum than for tungsten, 4 This is valid for ion energies which are typical for glow discharges. Positive ions moving very rapidly (α-particles with energies of 1 MeV) can release up to thirty electrons; their energy amounts then to some keV.

38


0.30

Ne+

gi [electron/ion]

0.25

He+

0.20

W Mo

0.15 Ar+

0.10

Kr+

0.05

+

Xe

0.00 0

250

500

750

1000

Fig. 2.33. Comparison of the yield for secondary electrons, γi , for the ions of the heavy noble gases incident on pure molybdenum and tungsten c The American [75, 77] ( Physical Society).

ion energy [eV]

• and depends on the ionization energy of the incident ion: The lower its ionization potential the less the yield of secondary electrons. This is due to the low work function of molybdenum—for constant energy of the incident ions, the energy of the secondary electrons rises and enhances the probability of being released. On the other hand, Fermi energy and density of states at the Fermi level (this equals the number of conduction electrons divided by the Fermi energy in the free-electron model) energy and the number of released electrons: A falling Fermi energy causes a larger kinetic energy of the emitted electrons, simultaneously, the density of state goes up. Furthermore, Hagstrum investigated the influence of the surface quality on the yield for secondary electrons [78]. Fig. 2.34 impressively illustrates the difficulties obtaining reliable data for δ and γ (γ as anisotropic property additionally depends on the crystal orientation). 0.10 W +

Ar

gi [electron/ion]

0.08

N2/W

Ar+

0.06

W +

Kr

0.04 Kr+

0.02 Xe+

0.00 0

N2/W W

250

500 750 ion energy [eV]

N2/W

1000

Fig. 2.34. γi as a function of kinetic energy of the heavy noble gas ions for pure tungsten and tungsten covered with a c mono layer of nitrogen [78] ( The American Physical Society).

2.6 Generation of secondary electrons at surfaces

39

But also the energy distribution is altered by the contaminated surface: The fraction of rapid electrons is reduced. This is not only caused by a change in the work function but also by fundamental alterations of electron generation mechanisms with the exception of Auger processes. 2.6.3 Photons The electron yield caused by photons, triggered by the Einsteinian photo effect, oscillates between 100 and 1000 ppm in the near UV (NUV) (dependent on the work function Wa ) rising to values of up to 0.5 in the vacuum UV (VUV) (Figs. 2.35 and 2.36).

0

10

2015 10

5

E [eV]

Pt

10

potassium on platinum

Au

_ _

g hn

_

_

-3

10-6

100

200

300 400 l [nm]

500

600

Fig. 2.35. Yield for photoelectrons γhν of gold and platinum as functions of wavelength (lower abscissa) and energy (upper abscissa). γhν additionally depends on the polarization of the incoming light [79]. ⊥ and : E of linearly polarized light is orientated in perpendicular or in parallel fashion, respectively, with rec Oxford spect to the surface ( University Press).

E [eV] 30 0,20

25

20

15

Fig. 2.36. The yield for photoelectrons depends sensitively on the pretreatment of the surface [80]: squares: untreated tungsten electrode; triangles: 5 min heated at T > 1 000 ◦ C and 10 Torr; circles: heated up to reproc ducibility at T > 1 000 ◦ C ( J. Wiley & Sons, Inc.).

0,10

g

hn

0,15

0,05

0,00 40

60

l [nm]

80

100

40


Two main mechanisms have been identified which contribute to the steep increase in yield: First, most of the radiation in the long-wavelength range is reflected, and second, due to conservation of momentum, the excitation of free electrons in the valence band by a two-electron process seems very unlikely to happen. Going to short wavelengths, the reflectivities of the metals begin to decline to very low values (with simultaneous rise in transparency), and it becomes more likely to excite valence electrons. All these observations add experimental proof to the suggestion that this process consists of two steps which both belong to the activation type with a certain threshold, generating the electrons as well as their escaping from the solid. Measuring the work function and the various coefficients γ, δ and η remains a great challenge still, since the yield depends strongly on the purity (volume property), but even more sensitively on the contamination (surface property). Theoretically, the main problem is the exact calculation of the barrier which has to be overcome (model potentials etc., cf. [81]). Although the cross sections for photoionization are extremely small at low energies, this does not mean that they are without significance in glow discharges. In DC discharges, the charged carriers exhibit kinetic energies of several thousands of eV, therefore, photons of high energy are generated by Auger processes.

3 The plasma

3.1 Direct current glow discharge In this chapter, we will give a qualitative description of a very simple discharge, will describe some experiments, and will evaluate the main parameters which determine the quality of a low-pressure discharge: plasma density, electron temperature, Debye length and plasma frequency. Eventually, a model is introduced which was originally derived by Langmuir and MottSmith and which is now known as the Global Model.

3.1.1 Phenomenology One of the most prominent appearances of a glow discharge is the glow itself which is created by relaxation processes of atoms which have been excited by inelastic electronic collisions. The zones which emit light at very different intensities reveal that these relaxation processes do not happen with the same intensity across the discharge (Fig. 3.1). Dark areas can be identified adjacent to the electrodes. Their extension differs with pressure, power input and type of gas, and they are called “dark space”. Both cathodic dark spaces belong to the darkest zones of all: Aston’s and Crooke’s dark space, respectively. When Langmuir denoted the different zones in 1928, he applied the word “sheath” for this zone [12]. However, it has become common to use the word “dark space” for DC sheaths solely. The anodic dark space is significantly lighter than the cathodic glow which appears to be light only because it contrasts with the confining dark spaces. The negative glow with its gas specific color (Table 3.1) exhibits its highest intensity a little way behind the very sharp boundary to the cathodic dark spaces and is smeared out to Faraday’s dark space. All these three zones exhibit nearly the same length. This is approximately the length which has to be covered by the electron to gain enough energy of motion to at least equal the ionization energy or ionization potential. Therefore, this ionization length is proportional to the inverted pressure of the discharge as well as the potential difference between G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 3,

41

42

3 The plasma

Table 3.1. Characteristic colors in the negative glow and in the positive column gas hydrogen nitrogen oxygen helium neon argon chlorine bromine iodine HCl CCl4 sodium potassium rubidium cesium mercury

negative glow faint blue red yellowish blue green orange blue purple blue yellowish brownish green faint green whitish faint blue blue light green green

positive column rosa blue faint blue pink brick red dark red greenish reddish violet rosa whitish green yellow green light red yellow brown greenish

the electrodes.1 Faraday’s dark space is again relatively dark compared with its adjoining zones, in particular the positive column between this dark space and the anode. Its color is characteristic for the ambient, but differs not only in brightness but also in wavelength from the light radiated by the negative glow (cf. Table 3.1), in most cases, it is shifted to longer wavelengths. Its brightness is often uniform but sometimes exhibits several zones of different glow intensity, in particular in molecular gases, so called “striations”, which are either spatially stationary or can oscillate.2 In the positive column, the gas is in the ideal plasma state: The number of carriers equal each other, and there is only a very weak longitudinal field. Reducing the distance between the electrodes or decreasing the pressure will affect the nature of the discharge dramatically. First, the positive column will vanish, and afterwards also Faraday’s dark space. When the electrodes are so close that the anode dips into the negative glow, the discharge is quenched, if the voltage is not simultaneously increased. From this behavior, we can conclude 1 As we have seen in Chap. 2, the cross section of ionization increases exponentially just after the ionization threshold. Hence, this ionization length depends on the energy distribution of the electrons. 2 The positive column is used to generate light with countless variations. The gas filling consists of an inert gas and a metal vapor: mercury in fluorescent lamps, sodium in sodium vapor lamps. These exhibit the highest optical efficiency; which exceeds the light output of a normal incandescent bulb by approximately a factor of 10 at 100 W input power.

3.1 Direct current glow discharge

43

cathodic layers negative glow

anodic glow

_

+

positive column Faraday

Aston Crooke

dark spaces

glowing intensity

anode

I

dc potential

Vc

V

electric field

n+ nj- j +

Ex

space charge density j- + j+ current density

Fig. 3.1. Crooke’s glow discharge tube, approximately 50 cm in length at a pressure of about 133 Pa (1 Torr). The glowing areas are shaded. Below, the spatial dependence c J. Wiley of some properties which are characteristic for the discharge is shown [82] ( & Sons, Inc.).

that the most important zones of a discharge are Crooke’s dark space and the negative glow (Fig. 3.1). The potential does not vary linearly across the distance between the electrodes. Due to the different mobility of the positively and negatively charged carriers, space charges are built up which change the spatial shape of the voltage in a characteristic way: A steep decrease across the cathodic dark space in the so-called cathode fall is followed by a region of constant potential and vanishing field in the negative glow. In the positive column, the potential slowly grows — in the form of a staircase in a sandwich column with striations. In the anodic zone, the anodic sheath exhibits a potential increase which is a little steeper than in the positive column, which is of the order of the ionization potential of the gas in the tube. The carrier distribution is intimately correlated with the course of the potential and the electric field via the Poisson equation. Entering the dark space from the quasineutral plasma, both charge densities decline steeply, with a steeper decrease of the electron density to vanish far in front of the cathode (vice versa for the anode). Due to the different mobilities of the two carrier types, not

44

3 The plasma

Vd I V0

R S

_

+

Fig. 3.2. Circuit to generate a glowing discharge with a well defined source voltage V0 which must exceed a certain voltage Vb (b for breakdown). The (overall) voltage of the discharge will be determined according to Eq. (3.1).

only the sheath thicknesses but also the physics in the sheaths are different. For example, the thin anodic sheath is the only region with an excess of negative charge; the ion density dominates in all other parts of the discharge, in particular the region of the cathode fall. The electric field vanishes in the negative glow and the electric carriers move randomly and mutually screen each other (cf. Sect. 3.3), in the positive column, they migrate with medium velocity according to their polarity to their mutual sheaths. To maintain this process, a relatively low electric field is sufficient. Capture of positive ions at the anode would cause a depletion of them, if there were not a sufficient high field to generate additional ionization. 3.1.2 V-I characteristic As a first experiment, we evaluate the (overall) characteristics of the voltage between the electrodes as a function of electric current with external means, in other words, the V-I characteristic or I(V) characteristic of the discharge as a single phenomenon. For this end, we use the discharge tube of Fig. 3.1 and a circuit with voltage source V0 and adjustable resistance R (Fig. 3.2). After having passed a certain threshold, a current I will flow through the tube, and the voltage of the discharge can be simply calculated according to Vd = V0 − IR.

(3.1)

This characteristic depends on a large number of parameters, not only gaseous density and type of the gas, but also the electrode material. The principal characteristic is plotted in Figs. 3.3. At very low current densities of 1μA cm−2 or less, this discharge is denoted as Townsend or dark discharge because very little light or no light is generated. The main feature of this discharge is the linear rise of the potential between the electrodes which is closely connected with a remarkably linear electric field. In the region of about 1 μA cm−2 , the first space charges begin growing. The zone of potential drop is now confined to the cathode fall. The glow becomes gradually lighter with growth of current, but covers only part of the cathode. When the current is further increased, the cross section of

3.2 Temperature distribution in a plasma

45

Vb Townsend's discharge

normal abnormal glow discharge

abnormal

V [V]

potential

Vb

Vn

normal Townsend

Vn 10-9

10-6

10-3

cathode

I [A]

anode distance

Fig. 3.3. V (I) characteristic of a self-sustained discharge without positive column, and the schematic spatial behavior of its potential (Vb is the potential required for c Academic Press). breakdown) [83], [84] (

the glow continues to grow thereby maintaining the current density constant. This is the range of the normal glow discharge or the range of normal cathode fall. After having extended across the whole electrode, the cathodic current density can only rise with increasing current, which, in turn, causes a growth of the cathodic voltage: abnormal glow discharge or abnormal cathode fall. Further growth of current leads to a maximum of the voltage and an eventual breakdown as an arc [85]. Now, the thermal emission of electrons begins to play a decisive part. The main difference between an arc discharge and a glow discharge is the relatively low cathode fall of the former type, which is connected with a negative I(V ) characteristic because of the steep thermal increase in electrons. Due to the occurrence of space charges und the simultaneous generation of different potentials within the discharge at growing discharge current, recording of the spatial resolved I(V ) characteristic has been a challenge since the 1920s. These potentials are measured with a Langmuir probe [86, 87]. In principle, it consists of a small metallic electrode (in the case of a cylindrical system simply a wire) which is immersed in the plasma and connected to a voltage source by which its potential can be changed in relation to the plasma. Thus the relatively small disturbance of the plasma due to the measuring current will be neglected, at least to first order. This is justified for plasma densities larger than 1010 cm−3 for DC discharges [88] (but cf. [89] and Sect. 9.1 for RF discharges).

3.2 Temperature distribution in a plasma As mentioned in Chap. 1, the plasma state is a dynamic equilibrium of generation and loss of electric carriers. The negatively charged carriers are normally

46

3 The plasma

electrons, the positively charged carriers ions of the ambient gas. The energy which is coupled into the plasma, i. e. the work which has been performed by the electric field (W = F x), can be calculated according to W = e0 Ex with x = 1/2 at2 . Since the acceleration a = e0 E/m, we obtain W =

(e0 Et)2 ; 2m

(3.2)

the energy of the external field will be predominantly transferred to the electrons. The efficiency of the transfer of kinetic energy between the plasma constituents depends on • the collision rate between the particles, and • the mean energy transferred during every collision. As we have seen in Chap. 2, the collision rate itself depends on • the cross section of the specific type of collision, • the mean energy of the projectiles and • the density of the molecules. The process most often considered is the elastic collision between electrons of mean energy < ve > and neutrals of density nn , for which we have the cross section of momentum transfer σm : νm = σm nn < ve > .

(3.3)

In low-temperature/low-pressure plasmas, number densities are in the range 1012 − 1014 /cm3 , and typical values of the cross sections for momentum transfer are in the ˚ A2 range, the electrons exhibit kinetic energies of several electronvolts (2 − 8 eV or 20 000 − 80 000 K; 1 eV equals 11 600 K), which leads to values for νm between 104 and 108 sec−1 . Due to the very large ratio of the masses of electrons and other plasma constituents (me mi ), the transfer of kinetic enery via elastic collisions is nearly neglible (the efficiency is approximately 10−5 , since it is proportional to the ratio of the masses me /mi , see Sect. 2.1). Hence, ions and neutrals remain at relatively low temperatures (≤ 0.1 eV; at room temperature, kB T is about 1/40 eV). This means: • Significant deviations from the conditions for thermodynamic equilibrium: Pressure and temperature have to be the same in the whole system— we must distinguish between the temperature of the electrons and the neutrals/ions, and

3.2 Temperature distribution in a plasma

47

Table 3.2. Mean velocities of electrons and argon ions Energy [eV] 0.1 1,0 10 100 1000

v [cm sec−1 ] of electrons argon 1.9 × 107 6.9 × 104 5.9 × 107 2.1 × 105 1.9 × 108 6.9 × 105 5.9 × 108 2.1 × 106 1.9 × 109 6.9 × 106

• Very high mean energy of the electrons, which amounts to approximately 1 % of the speed of light (Table 3.2).3 Provided the electrons are distributed following a Maxwellian distribution function, their mean energy is

< ve > =

8kB Te , πme

(3.4)

which leads to a current density of the electrons (j = 1/4 ρ v with ρ the density of the carriers) a factor of 103 higher than that of the Ar ions. Because of the low number of collision processes with heavy particles, the energy distribution of the electrons remains different from that of the ions and neutrals which remain at a very low level, typically in the upper range of 102 K (Sect. 9.4.2). With growing pressure the number of collisions between electrons and heavy particles will increase dramatically (this number is proportional to the pressure squared) which leads to an effective equalization of the different temperatures (cf. Sects. 3.5 and 14.1 for a critical understanding of distribution functions and the phenomenon of temperature). This is the main difference between low pressure and high pressure plasmas, which start at about 1 bar and are used in quartz burners, for example. And we understand why a low-pressure plasma serves as an efficient light source: The electric energy is mainly transferred to the electrons which effectively excite, in turn, the optical levels of mercury atoms. Therefore, a low pressure plasma is also a cold plasma. The plasma density of a cold plasma is confined to a range between 108 and 1012 cm−3 and is determined by physical limits: At too low a charge density, the electrostatic interaction is very low, which leads to some isolated charges which are separated across a significant distance; when the plasma density exceeds a certain threshold, the gas will heat up significantly by collisions between electrons and neutrals. 3 We realize a resemblance to the free electron Fermi gas in metals. Here, the Fermi energy EF reaches some eV (e. g. Cu: 7.00 and Au: 5.51 eV); as a first consequence, the Fermi velocity, which formally assigns the velocity of the electrons to their kinetic energy reaches very high values.

48

3 The plasma

Direct measurement of the temperature of the heavy particles (ions and neutrals) of the plasma bulk are impossible. One method to determine the speed of the particles, and hence their temperature, is measurement of the width of a spectral line, which is predominantly determined by the Doppler shift [besides radiation damping [90] and lifetime broadening (so called fourth uncertainty relation ΔE · Δt ≥ h ¯ [91])]: An object radiating with a wavelength which moves away from a resting watcher with velocity v, is recognized by him as having the apparent wavelength (1 + v/c)λ, an approaching object, however, will have apparent wavelength (1 − v/c)λ (cf. Chap. 9). Another possibility is the recording of a rotational vibration spectrum. In absorption, the distribution of intensities in a Bjerrum double band is largely determined by the temperature-dependent population of the rotational levels of the vibration ground state [92], cf. Sect. 9.4.

3.3 Neutralization of charges in an undisturbed plasma The discharge potential between the electrodes is mainly characterized by a steep decline across both the dark spaces, whereas the negative glow remains almost free of fields (constant potential). In the positive column, we note a small rise of the potential with constant electric field (Fig. 3.1). The plasma is effectively screened against disturbing fields, which are caused in the simplest case by ions. This is achieved by the formation of electron clouds which are arranged around the ions. By this, the range of the Coulomb potential is reduced to very small values. This shielding effect was first described by Debye and H¨ uckel in their theory on strong electrolytes in aqueous solutions [93]. Due to a mutual orientation of the ions, clouds of the lighter ions will be formed around the more inert ions. The ionic density around a certain ion in a solution is then given for thermal equilibrium by

e0 Φ(r) n(r) = n0 exp − kB T

(3.5.1)

with n0 the ion density in the undisturbed solution, where n+ + n− = n0 , the charge density ρ(r), however, vanishes at large distances from the ion because of the different polarities of the ions (z the valence of the ion) ±zi e0 (n+ + n− ): ρ(r) =

i

e0 Φ(r) zi e0 n0 exp − kB T

(3.5.2)

For small values of the orientating electrostatic potential, Eq. (3.5.2) may be expanded4 4 This linearization is necessary to apply the fundamental principle of superposition. Accounting for members of higher order would violate this principle.

3.3 Neutralization of charges in an undisturbed plasma

ρ(r) ≈

zi e0 n0

i

49

e0 Φ(r) 1− . kB T

(3.5.3)

We demand electroneutrality for the whole solution, hence, the first sum van ishes ( i zi e0 = 0), and we obtain for one sort of carrier (zi = 1)

ρ(r) = e0 n(r) ≈ e0 n0 −

e0 Φ(r) kB T

(3.6)

and likewise for the potential: Φ(r) ≈ −

kB T n(r) . e0 n0

(3.7)

Inserting Eq. (3.7) into the Poisson equation d2 Φ(r) n(r)e0 = dr2 εε0 eventually leads to the Poisson-Boltzmann equation −

d2 Φ(r) e20 n0 = Φ(r). dr2 εε0 kB T

(3.8)

(3.9)

A solution is5

Φ(r) = Φ0 exp −

r d

(3.10)

with

d = λD =

εε0 kB T , n0 e20

(3.11)

which is the Debye length. The equilibrium between thermal and electrostatic energy determines the value of the screening length: Increasing the charge density reduces the screening length. The ion cloud would collapse if there were no thermal movement of the ions. At its edge, where e0 Φ ≥ kB T , e. g. in close vicinity of the electrodes, the ion cloud is severely disturbed. In a similar way, we can argue for a plasma because highly mobile carriers exist as well. However, they differ significantly in their energy [Eq. (3.2)] which has to be taken into account in a comprehensive analysis (cf. Sects. 3.4 and 14.2). For immobile ions, we obtain eq. (3.11), replacing n0 by ne . In fact, there is a slight thermal movement of the ions as well.6 In a low-pressure discharge, however, they are far away from thermal equilibrium. Even small dragging forces 5

For positive values of the exponent, the equation diverges. Provided also the ions are in thermal equilibrium (Te Ti ), the equation for the reduced temperature for both types would read 6

potential energy [a. u.]

50

3 The plasma

Fig. 3.4. The long-range tail of the Coulomb field 1/r is screened by the formation of an ion cloud with the screening length λD (λD is normalized to unity [94]).

screened Coulomb potential Coulomb potential

distance [a. u.]

cause the ion drift velocity to rise beyond its thermal velocity (a potential of already 0.05 V would equal the thermal velocity of Ar+ ions exhibiting a temperature of 600 K). λD ≈ kTe /ρ has a length of about 100 μm for kB Te = 2 eV and ne = 1010 cm−3 . Across a distance of λD , the electrostatic interaction drops to 1/e (37 %), and in 3 λD , to values of less than 1 % (cf. Fig. 3.4). Numerically, we get approximately (for T in K and ne in electrons/cm3 ) for λD in cm:

λD = 6.91

Te [cm], ne

(3.12.1)

if we prefer to express the electron temperature in eV which is not exact but common:

λD = 743

Te [cm]. ne

(3.12.2)

λD decreases with increasing electron density, and increases with rising electron temperature. Only electron temperature and electron density are inserted into the equation to determine the Debye length, because it is only the electrons which cause the screening effect of the ions due to their tremendously high mobility, no matter if a surplus of negative charge is to be equalized by draining or if a positive excess charge has to be compensated by a flow of electrons into this drain. From an external viewpoint, we find this volume containing mobile charges to behave quasineutral since ne = ni . 1 1 1 1 = + ≈ , T Te Ti Ti and λD would be solely determined by the ions.

3.3 Neutralization of charges in an undisturbed plasma

51

The numerical values of the Debye length cover some orders of magnitude; e. g. in the photosphere of our sun with T = 5 600 K and a plasma density of 1012 cm−3 , λD is about 5 μm, in the Orion nebula with T = 10 000 K and a plasma density of 100 cm−3 , λD is approximately 70 cm (Fig. 1.1). In dense and cold plasmas, the quasineutrality can only be disturbed in the interior of relatively small volumes, in a thin and hot plasma, however, the Debye length can exceed the dimensions of the plasma container. In this case, ions and electrons move independently, and there is no correlation mechanism which would compensate or equalize high potentials within this agglomeration of charges. By means of the Debye length, we can sharply distinguish between gases which are ionized, i. e. which contain but some charged carriers (ne is very small), and plasmas. For those “plasmas”, we calculate very large values for the Debye length that are eventually in the order of the dimensions of the reactor. Evidently such a system cannot be termed quasineutral any more. λD has to be small against the spatial dimensions of the plasma (Langmuir’s definition of a plasma). As a second requirement, we demand a large number N of screening electrons. N can be calculated as the product of the electron density of the undisturbed plasma, ne , and the spherical volume with the Debye length as radius: N=

T 3/2 4π ne λ3D ∝ √ ∨ λD > ne−1/3 , 3 ne

(3.13)

N must be very large (> 100). The importance of this parameter is elucidated from its simple linguistic usage as the plasma parameter: The quasineutrality can be violated only within a sphere of radius λD . Only within this volume can the kinetic energy of the electrons exceed the directing effect of electrostatic interaction.7 To balance out deviations of equilibrium, it does not take longer than λD = τ=√ < v2 >

ε0 m e ne e20

(3.14)

with the mean squared velocity (= 3kB Te /me ). This length is the border between the motion of individual particles and the motion of an ensemble. Mathematically speaking, the long-range tail of the Coulomb potential with its singularity at x = 0 is replaced by the Thomas-Fermi potential (Fig. 3.4). Compared with the mean free path of the electrons λe , which is in the order of cm, λD is between 0.1 and 1 % of λe . This means that the properties of a plasma are not, or only very slightly, disturbed by collisions between the electrons and heavy particles. 7 In the case this condition cannot be met, the plasma behaves rather as a liquid than as a gas.

52

3 The plasma

Vp

potential [a. u.]

0 Vf

floating grounded electrode

Vc

Fig. 3.5. Potential distribution of a discharge which lacks a positive column and Faraday’s dark space; the positive electrode is immersed in the negative glow, and the adjoining sheath of the negatively charged electrode separates it from the negative glow.

distance [a. u.]

3.4 Potential variation in the plasma

As we have seen from Eq. (3.4), the velocity of the electrons exceeds that of the ions by some orders of magnitude. Hence, they can easily evade the glowing plasma and are captured by the walls, leaving behind a positive residual charge. Due to the effective Debye screening, this excess charge cannot easily be distributed over the plasma bulk. On the contrary, this potential difference will be built up across a distance of only several Debye lengths. Across this layer (dark space), the carrier density drops to very small values, and therefore, intense fields can occur without generating large currents, which is impossible in the negative glow where the plasma density exceeds the upper limit of a dark Langmuir plasma (np ≥ 107 cm−3 ). The developing potential is called the wall potential [95] or floating potential, Vf , since it is set up at all isolating walls which are in contact with the plasma. This floating potential can be readily measured against ground and can even be calculated in simple cases. The layer itself is denoted sheath; in particular, in DC discharges this layer is commonly referred to as dark space (cf. Sect. 3.1). One of the physical functions of this sheath is the formation of a potential barrier to bound the electrons electrostatically. It adapts to values to ensure equality of the electron and ion currents. The height of this sheath is relatively low, but sheaths in front of electrodes can very significantly exceed kB Te /e0 . Looking from the plasma bulk, it has charged up to a positive value, the plasma potential, Φp , with respect to all the walls confining it, and the potential distribution can be sketched according to Fig. 3.5.


53

3.4.1 The low-voltage plasma sheath To make this quantitative, we investigate an isolated electrode being immersed in a plasma. In thermal equilibrium, for both types of electric carriers, we take the Maxwellian distribution for granted within the negative glow. In this region, n0 = ne = ni [96]. Due to their considerably different mass, the current densities for both carrier types incident on the substrate are initially different: 1 e0 n e je = e0 ne < ve >= 4 4

8kB Te kB Te = e0 ne πme 2 πme

(3.15.1)

and 1 e0 n i ji = e0 ni < vi >= 4 4

8kB Ti kB Ti = e0 ne , πmi 2 πmi

(3.15.2)

so we obtain for their ratio je = ji

Te mi , Ti me

(3.16.1)

and with ne = ni = n0 < ve > = < vi >

Te mi . Ti me

(3.16.2)

The initial current densities of ions and electrons differ by a factor which exceeds the ratio of their velocities! This gives immediately rise to a surplus of negative charge on the substrate, the potential drops from the initial zero to the floating potential Φf , and in the vicinity of the device, a positive space charge surrounds it due to the deficiency of captured negative electrons. Hence, by the creation of such a potential 1. the electronic current will be reduced to reach a level such that 2. the ionic current will equal the electronic current. 3. Looking outward to the undisturbed plasma, the electric field which is connected to the sheath is rapidly damped out. Therefore, the ionic flux incident on the electrode is by no means enlarged, since the ions entering the sheath have left the quasineutral plasma with very low energy. Only those ions which have reached the sheath will be accelerated. 4. The net result is a depletion of charges in the sheath.

54

3 The plasma

3.4.1.1 Approximation of first order. Although we are predominantly interested in the plasma potential Φp , the potential that is readily measured is the floating potential Φf . To calculate its order of magnitude, we use condition (2), the evenness of the currents incident on the floating electrode. The correct electron current density is the mean current density times a retarding Boltzmann factor, which takes into account its drop from the initial large value due to the creation of the retarding potential Φf < 0 (uphill for the electrons); for the ions, there does not exist any barrier which they have to overcome (downhill for the ions):

e0 (−Φf ) . < vi >=< ve > exp − kB Te

(3.17)

Combining Eqs. (3.16.2) + (3.17), we get

e0 (−Φf ) Te mi = exp − Ti me kB Te

⇒ −Φf =

kB Te Te mi ln , 2e0 Ti me

(3.18)

from which follows the floating potential:

Φf =

kB Te Ti me ln . 2e0 Te mi

(3.19)

Φf is negative in all cases. 3.4.1.2 Approximation of second order. In Sect. 14.2, the ion current density at the boundary presheath/sheath is shown to be8

ji = n 0

kB Te mi

(3.20)

from which we derive the Bohm velocity at the sheath edge to be

vB =

kB Te . mi

(3.21)

Solving for the kinetic energy of the ions, we obtain the Bohm potential according to 1 1 kB Te e0 ΦB = mi vB2 ⇒ ΦB = , 2 2 e0

(3.22)

8 The ions are accelerated across the presheath to a velocity that balances the electronic velocity component along the electric field, i. e. 1/3 of their thermal energy: 2 /2 mi vB = 1/3 3/2 kB Te

1

from which Eqs. (3.20) and (3.21) will follow.


55

which represents the plasma potential Φp at the boundary between the presheath and the sheath. The electron current density is the mean current density times a retarding Boltzmann factor which gives rise to its drop from the initial large value, and dividing by n0 , we get

je 1 e0 (−Φf ) = < ve > exp − . n0 4 kB Te Equating Eqs. (3.20) and (3.23), and using < ve >=

Φf =

(3.23)

8kB Te , πme

we obtain

kB Te πme ln , 2e0 2 mi

(3.24)

which is negative in all cases. In argon, the floating potential would reach 16 V for an electron temperature of 3 eV and 11 V for a Te of 2 eV. 3.4.1.3 Approximation of third order. In the vicinity of the sheath and in the sheath itself, the electrostatic energy is much greater than the thermal energy. To calculate the flux of electric carriers incident on the electrode, the Poisson equation has to be supplemented to arrive at the Poisson-Boltzmann equation (cf. Sect. 3.3). d2 Φ = βΦ, dx2

(3.25)

(with Φ = rφ, φ is a potential which must not be linearized because its value is large compared with kB T, cf. Sect. 14.2). Now, we also take account of the fact that, lacking a reference potential, we can measure but the potential difference, and as the result of a tedious calculation, we find that the fraction of electrons with energies greater than e0 (Φp − Φf ) with Φf the floating potential, that can penetrate the sheath entirely to eventually reach the wall, drops to

e0 (Φp − Φf ) , ne (x) = n0 exp − kB Te

(3.26)

with Te , the electron temperature. Their velocities deviate from the mean electron velocity according to Eq. (3.4) to a large extent. At the sheath boundary, where Φp = ΦB , the electron density equals ne = n0 still. Therefore, the electronic current incident on the negatively charged substrate will be je (x) =

1 ne (x)e0 < ve >, 4

(3.27)

the current density of the positive ions, however [Eq. (14.43)]

1 ji = n0 e0 vB exp − 2

= n0 e0

kB Te 1 exp − mi 2

(3.28)

56

3 The plasma

i. e. the condition for equilibrium ji + je = 0 ∨ −je (x) = ji can be written in this approximation [97]: Φp − Φf =

kB Te 2π me kB Te me ln ⇒ Φp − Φf = ln 2.31 , 2e0 emi 2e0 mi

(3.29)

which is the potential of an isolated floating surface in contact with the plasma for which the net current must be zero. Since the logarithm is negative, the floating potential Φf is always numerically larger and opposite in sign to the plasma potential Φp . This difference is the barrier potential which has to be overcome by the electrons to cross the sheath of an isolated electrode, and it equals approximately the logarithmic ratio of the masses between ions and electrons. The ions are accelerated towards the wall by the initial voltage Φf , since they reach the presheath with a “random” motion leaving the plasma bulk with vanishing electric field. Hence, their kinetic energy is thermal and very small. When entering the presheath, they are accelerated by an electric field to enter the sheath with the directed velocity vB . The calculated values for Φf generally have an upward trend. Cox et al. pointed out that these deviations can often be attributed to the wrong energy distribution of the electrons, which is supposed to be Maxwellian [98] (cf. Sect. 14.1). This assumption is questionable, at least in low-density plasmas. A second assumption refers to the temperature of the ions (room temperature assumed). The detailed discussion of high-voltage plasma sheaths are deferred to Chap. 4 for DC discharges and Chap. 6 for RF discharges. On a qualitative level, the potential of the cathode sheath is set to a level of approximately Φc = Φp − Φcathode , and the potential of a grounded electrode is about Φp . The range of potential drop between uniform plasma and the electrode itself, the sheath, can be divided into two subparts. Hence, above an electrode, we can distinguish at least three zones (Fig. 3.6): • Undisturbed plasma, far away from an electrode, no field, the ions move randomly, and the densities of positively and negatively charged carriers equal each other. Disturbances which occur randomly, and deviations from equilibrium are equalized within the Debye length. • Slightly disturbed plasma, the ions will be accelerated by a weak electric field (transition zone). The densities of positive and negative carriers mutually equal still, its thickness is of the order of the ionic mean free path. This directed velocity is due to a potential drop of 1/2 Te which happens to occur over a long distance between electrode sheath and undisturbed plasma (i. e. a very tiny electric field, so called “presheath”). The ions are accelerated by the Bohm potential ΦB , and the Bohmic presheath terminates when the ions have gained the (directed) velocity Bohm velocity vB ; here, their density has been approximately diminished to



Vp VB

57

neutral plasma no field Bohmic presheath weak field

Fig. 3.6. Undisturbed neutral plasma, the quasi-neutral transition zone (Bohmic presheath) and the cathode fall within the positively charged sheath adjacent to the negatively charged surface.

positively charged cathode sheath

Vc

distance [a. u.]

1

/2 mi vB2 ni = n0 exp − kB Te

= n0 exp −

1 2

n0 =√ . e

(3.30)

For a stable sheath with a surplus of positive charges, it is mandatory that the ionic density across the sheath always exceeds the electronic density. Only then ∀x > 0 :

d2 φ . < ε > = kB Te = me < v 2 >⇒ kB Te = 2 2 3 2

(3.33)

The mean squared velocity < v 2 > and the square of the most probable velocity vw2 show a mutual ratio of 3/2 [101]: 2 . (3.34) 3 The plasma is outwardly screened by the sheath; hence, the interior of the plasma, at least the negative glow, becomes almost free of electric fields. As a direct consequence, the thermal velocity of the electrons far surpasses their drift velocity, which is additionally reduced by collisions with neutrals, v = μE with μ the mobility and E the electric field. Hence, the influence of the electric field on the energy of the electrons (and at second order also the energy of the molecules and ions at the border plasma/sheath, the Bohm edge) is small—and this holds true also for the processes of ionization and diffusion. kB Te =

60

3 The plasma

3.5.2 Energy loss When thermalized electrons hit the wall during their chaotic course, they are annihilated, and all their energy gets lost as well. As it is shown in the kinetic theory of gases, the electron flux incident on the wall with positive direction +x is given by 1 (3.35) Γx = ne < ve >, 4 and the energy flow of n electrons with average energy 1/2 me < v 2 > incident on the wall in positive direction +x with mean velocity < vz > yields to Sx = 2kB Te Γx ;

(3.36)

hence the energy loss is 2kB Te . As we have seen in Chap. 2, the processes of excitation are characterized by low efficiency. On the average, the energy which has to be piled up by an electron must exceed the excitation threshold significantly, and after excitation, the abundant energy will mainly get lost in elastic collisions. The lower threshold of this amount of energy can be generally given by Eexc , in particular for ionization, by Eion . Ions which have traversed the Bohmic presheath have gained a certain amount of directed energy (EB = 1/2 kB Te ) and are about to get lost. Just before they hit a surface, they are further accelerated across the sheath, either in front of the wall of the reactor, or in front of the electrode. In either case, they absorb an amount of energy which is large compared with EB and which, in turn, causes a heavy power demand. In total, the energy consumed by lost ions is 1 (3.37) Ei = kB Te + e0 Φs . 2 It is these three contributions which define the lower limit of energy loss per time and area: 1 E ≥ 2kB Te + Eion + kB Te + e0 Φs , 2 and the energy flow into the plasma must exceed

(3.38)

Pabs (3.39) ⇒ Sabs ≥ nB vB ε O where O denotes the surface of the plasma bulk which terminates at the Bohm edge. In equilibrium, the carrier densities are balanced out, and the temperatures of molecules and electrons differ significantly but are separately constant in time. This is the case provided that Sx =

• the rate of formation for charged carriers equals the rate of loss, and if

3.5 Temperature and density of the electrons

61

• the energy losses of the electrons are balanced out by their energy gain. We will first estimate the temperature of the electrons and then take a closer look to their density. 3.5.3 Electron temperature Applying the law of mass action, we can get an idea of some important properties of electropositive tenuous plasmas in which the plasma density np (= ne ) is small compared with the density of neutrals n0 : np n0 . • How large is the plasma density np ? • How large is the rate of charge generation? We assume a single-step ionization by collisions of neutrals and electrons: A + e− −→ A+ + 2 e− and only monovalent ions, the losses should be due to diffusion to the reactor walls, and we disregard all other loss mechanisms such as recombination processes. Since from a neutral A, two charges are generated, one ion (i) and one electron (e), the bimolecular reaction of formation can be formulated A + e− −→ A+ + 2 e− ,

(3.40)

ni = n e ∧ nA = n 0 − ne ≈ n 0 .

(3.41)

and therefore

According to the law of mass action, we can this express as

εion ni n2e = exp − n0 ne kB Te ne =

√

n0 exp −

⇒

(3.42)

εion , 2kB Te

(3.43)

(simplified Saha equation without consideration of potential degeneration of states). The formation rate of the ions is with kion the bimolecular rate constant for the ionization from A to A+ , ∂nA+ (3.44) = kion ne nA ; ∂t it depends linearly on the electron density, and kion depends only on Te . Hence, in the cylindrical volume πr2 L, the formation rate of the ions amounts to ∂NA+ = kion ne nA πr2 L. ∂t

(3.45)

62

3 The plasma

This is countered with losses by diffusion to the walls of the reactor (velocity B Te ; and at sufficiently low pressures, the ionic mean at the Bohm edge vB = km i free path λi exceeds the dimensions of the reactor (λi R ∧ λi L) yielding approximate solutions for the densities at the radial and axial boundaries. The density of charges at the radial surface is slightly different from the axial one [102, 103], but is neglected during this derivation; np is the plasma density in the center of the reactor:

∂NA+ = nA+ vB 2 πr2 + 2πrL . (3.46) ∂t For a stable system, this loss equals the generation according to a continuity equation

(3.47) kion nA+ nA πr2 L = nA+ vB 2 πr2 + 2πrL ⇒

−

vB nA vB = ⇒ kion = Op,spec kion Op,spec nA

(3.48)

with 2(r + L) , (3.49) rL a factor which reflects the geometric ratio volume over surface. The index “spec” denotes the spectroscopically determined electron temperature. Inspecting Eq. (3.48), we note that the quotient on the left-hand side implicitly includes the electron temperature (i. e. the mean electron energy): kion depends strongly, vB weakly on Te (see also Figs. 2.12 and 9.34).9 In particular, the rate coefficient for ionization must decrease with increasing neutral density (growing pressure), hence, the electron temperature Te must go down [eq. (3.48)]. This effect is more pronounced at lower pressures. We must compare the rate coefficient for ionization, which is closely connected with the cross section for ionization, via Op,spec =

kion = < ve > σion

(3.50)

and which can be obtained by scattering experiments with data obtained by plasma diagnostics. The generative and diffusive parameters are analyzed in Figs. 3.7. For a given geometry of the reactor volume (here: electrode gap: 5 cm, reactor diameter: 43 cm), in the range of low pressures (two orders of magnitude), the plasma property nO does not show any tendency to rise sharply beyond a certain pressure (border at about 100 mTorr). According to this simple model, the electron temperature Te rises 9 In particular, this holds true for energies close to the ionization threshold where the characteristic jump in the inelastic cross section by several orders of magnitude is observed (cf. Sect. 2.5).

3.5 Temperature and density of the electrons

1013

1014

n [cm -3]

1015

20

1015

10

1014

0

1

6

1016

nO

10

Te [eV]

electrode distance: 5 cm electrode diameter: 30 cm O

nO [cm-2]

O [cm]

8 1017

40 30

63

4

2 0

2

10

100

4

10 10 vBohm/k ion [1015/cm2]

p [mTorr]

6

10

10 8

Ar gap 5 cm diameter 43 cm

Te [eV]

6 4 2 0 0

25

50 p [mTorr]

75

100

Fig. 3.7. For a given geometry, the parameters for electron generation and electron loss are correlated with electron temperature. The rate coefficient is modeled applying the (energy dependent) ionization cross section and assuming a Maxwellian distribution of electrons [104, 105] (Sect. 9.4). The smaller the range O, the larger the loss rate by diffusion. To sustain a stable plasma, the electron temperature must consequently rise. The electron temperature is calculated considering different ion densities at the boundary sheath/presheath in axial and radial direction according to the approach by Godyak and Lieberman [102, 103].

• with falling particle density nA (for constant gas temperature, this is the discharge pressure), • when the reactor volume gets smaller, and • when the confining walls around the plasma become larger. In particular, the temperature of the electrons, Te , is determined by the density of the neutral molecules (i. e. for constant gas temperature: the discharge pressure), and via the geometry factor by their gain (by collisional impact) and loss (by diffusion), but it definitely does not depend on the plasma density, np and the power (incoming or absorbed). In this model, the thermal velocity of the electrons scales inversely with ln nA [cf. Eq. (3.43)]. This simple dependence

64

3 The plasma

is confirmed by approaches which take the electron energy distribution function (EEDF) into account (cf. Sect. 14.1). 3.5.4 Electron density The temperature of the electrons is determined by the dynamic equilibrium between their generation and loss. According to Ohm’s law, the power absorbed by a current of ions j in an applied field of intensity E is given by dPabs = js Ed3 x ∨ dPabs = σ E 2 d2 x dx.

(3.51)

The current density js has to be measured at the boundary between Bohmic presheath and sheath and is the product of the density of √ the ions, ρA+ = e0 nA+ = e0 NA+ /V , the Bohm velocity vB and the factor 1/ e [Eqs. (14.43/44)]: e0 NA+ vB , js = √ e V

(3.52)

whereas the (averaged) electric field is given by the drop of the electric potential between electrode level and plasma boundary, so we get dPabs =

e0 NA+ vB d2 x √ E dx. e V

(3.53.1)

From this consideration, we see that dPabs =

e0 NA+ vB d2 x √ Edx e V

(3.53.2)

2

where the ratio dV x describes again the ratio between volume and its confining surface which can be associated again with Op,spec introduced by Eq. (3.49), the specific geometry factor. The absorbed power scales with ion density density and field intensity, and, to a weaker extent, with electron temperature (via vB ). Furthermore, we can infer the number of generated ions, NA+ , to depend directly on the absorbed power, but inversely on the effective geometry, and the intensity of the electric field: N A+ ∝

dPabs , vB Op,spec Edx

(3.54)

and comparing this equation with the rough estimation of eq. (3.39), we see that nB × O must equal NA+ × Op,spec . To achieve high plasma densities, the specific area, the electric field and the thickness of the sheaths (electrode sheath and Bohmic presheath) should be kept as low as possible. On the other hand, failing to achieve a high plasma density can be caused by these fundamental parameters.

3.6 Plasma oscillations

65

Furthermore, the dependence of vB on Te determines the ion impact energy, and the rate coefficient of ionization is a single function of Te , which primarily depends on the gas density (at the sheath edge, the ions have gained a directed energy of order 1/2 kB Te [106], cf. Chap. 9 for the difference between generated and absorbed power). Even in the simplest approximation of a DC discharge, Peff scales with the squared electric field (cf. Chap. 5 for the effective electric field in an HF discharge). This derivation is based on some simple assumptions, primarily on the law of mass action for generation and the diffusion as mechanism for loss and should generally be valid. In particular, this model does not depend on the method of plasma excitation, irrespective of whether they are DC based or HF methods.

3.6 Plasma oscillations We encounter two different types of electrical carriers in an electropositive plasma, as in argon, namely ions and electrons. However, the collision frequency is very high even at modest discharge pressures. Hence the mean free path of electrons, λe , and that of the ions, λi , respectively, is sufficiently short to ensure electroneutrality, even if electric currents are flowing under the influence of an external field. At low frequencies or large wavelengths, we can regard the plasma as a system consisting only of one component (cf. Sect. 3.3). With increasing frequency, this model breaks down. Electrons and ions move mutually independently, causing a separation of charges, and accompanying electric fields are set up which generate electrostatic oscillations. Further rising frequency eventually highlights the inertia of the ions since they cannot follow the fluctuations of the field any longer. They become a background of positive discharge to ensure electroneutrality, and the electrons move against this background as in a viscous liquid (jelly). This plasma is termed Lorentz plasma. This disturbation can be described with the Poisson equation, and for a linear field we obtain ρx ρ dE ⇒E= . = dx ε0 ε0

(3.55)

The force which is applied to the charge density ρ and causes its acceleration, is F = −ε0 e0 E = −e20 nx ⇒

d2 x ne0 x + = 0, dt2 ε0 m 0

(3.56)

which is the well-known equation of a harmonic oscillation with eigenfrequency

ωp =

ne20 , ε0 m 0

(3.57)

66

3 The plasma

so so-called plasma frequency (for a more exact treatment of this problem cf. Sect. 14.3). Since Eq. (3.57) contains only constants, except for n, ωp can be written for electrons (m0 = me ) as ωp = 5.64 × 104 10

√

n.

(3.58)

−3

As an example, for n = 10 cm , we obtain ωp = 5.64 GHz. This would be the angular frequency of the plasma electrons around their position of equilibrium without any damping. However, the oscillations are damped by collisions with neutrals and electrically charged carriers, irrespective of whether they are electrons or heavy ions. To excite such oscillations, the collision frequency νm must be small compared to the plasma frequency, the reciprocal of the time τ which we defined in Eq. (3.14): ωp > νm .

(3.59)

As we can see later on (Chap. 9 and Sect. 14.6), this is a very good approximation, especially in high-density plasmas. But already in low-density plasmas, as in capacitively coupled discharges, νm amounts to only a few percent of the value for ωp . The product of Debye length and plasma frequency results in ε0 kB Te ne20 kB Te 1 = = √ vw 2

ne0

ε0 m e

me

2

(3.60)

with vw the most probable velocity [maximum of the Maxwellian distribution, cf. Eq. (3.34)]. That means: In a plasma, carriers are displaced only to an extent which is of the order of magnitude of the Debye length. Hence, the paramount importance of the Debye length for the description of plasmas will be evident from the fact that also dynamically generated deviations from equilibrium are not stable across a range which is larger than this length. The Debye length separates the plasma into two parts of interaction with waves: For disturbances k kD (with kD the inverted Debye length, the Debye wavevector), the plasma behaves as a continuum; it oscillates as a single ensemble (“cooperative effect” [107]); for disturbances k kD , however, Eq. (3.25) is still valid, which describes the behavior as individual particles. We must distinguish between transverse and longitudinal optical oscillations which are not mutually coupled in the case of no external fields. Transverse optical oscillations are excited by electromagnetic waves in the range ω > ωp . For ω < ωp , the dielectric constant becomes negative; which makes the propagation impossible beyond the skin depth, but opens up the possibility to measure the highest charge density by reflection of microwaves (cf. Sects. 14.6 and 14.7) [108]. Longitudinal optical oscillations, however, are excited by inelastic scattering of electrons with energies greater than their mean energy. The plasma waves exhibit an energy of typically 10 eV and more. For wavevectors k kD , we can approximate the phase velocity according to


67

vph ≈ ωp /k.

(3.61)

The oscillations will propagate (i. e. the local oscillation will turn into waves) by thermal motions of the electrons. These waves can propagate almost undamped since a particle moving slowly against the wave feels almost the same field as a resting particle. For further growth in k, a large fraction of the electrons exhibit thermal velocities which are comparable with vph . This opens the window for a very effective, collisionless energy transfer which becomes so efficient at k ≈ kD that it is useless to speak of organized oscillations (Landau damping [109], cf. Sect. 14.3). Now, at latest, we have entered the range which emphasizes the interaction between mutual pairs rather than with ensembles and between them. Ions oscillate in a plasma as well. According to Eq. (3.57), however, they will move at considerably lower velocities and lower frequencies. For argon with a plasma density of 1010 cm−3 , we find ωp,i = ωp,e me /mAr = 1/300 ωp,e ≈ 20 MHz. Some authors make these oscillations responsible for the striations in the positive column (cf. Sects. 3.1, 4.5 and 7.5).

4 DC discharges

The behavior of a DC discharge is first described in a phenomenological sense, which is followed by the avalanche theory of carrier generation. Next, the issues of energy gain in the cathodic zone and charge generation in the negative glow are dealt with. For breakdown, these processes of charge generation must equal the carrier losses which are the focus of the next section. The features of the positive column, in particular diffusive processes, and the anode zone are topics of the following section. Some special DC discharges are briefly mentioned, and eventually, the similarity rules are discussed.

4.1 Introduction Let’s start with secondary electrons which have been released by γ-processes at the cathode. These secondary electrons leave the cathode with very low energies of only several electronvolts, but are considerably excited across the adjacent electric field of the dark spaces. Within a very short distance (Aston’s dark space), they have accumulated enough energy to excite atoms, which causes the sharply defined cathode layer. As expected, the lines of lower exitation energy are closest to the cathode. Farther away from the cathode, the electrons have already gained sufficient energy that they can easily perform multiple ionizations, which causes an avalanche of ionization in this part of the dark space called Crooke’s dark space. Simultaneously, the electric field diminishes steeply until it has almost entirely vanished when the electrons enter the negative glow. The main physical feature of the adjacent positive column is the diffusive loss of the charged particles, therefore, these issues are discussed in this section. Eventually, with a small increase in potential, the anode zone is reached. In the following section, the possibilities of ionization at the cathode and in the cathodic dark space are discussed, applying Townsend’s approximation. According to this theory, at the surface of the cathode electrons cannot be generated by electrons. G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 4,

69

70

4 DC discharges

4.2 Gaseous generation of carriers 4.2.1 Townsend’s equation There remain the two possibilities that • electrons between cathode and anode can generate secondary electrons in the gas phase (so-called α-electrons) and that • ions incident on the cathode can generate γ-electrons by bombardement of the cathode. With the simplifying assumptions that • the electric field drops linearly between cathode and anode, that • the ion flux is mobility-limited and that • secondary electrons generated on the surface of the cathode do not exhibit any energy distribution but show beam-like character, we can apply the ionization theory developed by Townsend [110, 111]. We denote α and γ the first or primary and the second or secondary Townsend’s ionization coefficient, respectively, and Γe (x) the electron flux. Thus, the generated electron flux yields dΓe (x) = αΓe (x)dx ⇒ d ln Γe (x) = αdx ≈ e0 Φn /Eion ,

(4.1)

Γe (x) = Γe (0) eαx ;

(4.2)

hence with Γe (0) the flux of secondary electrons at the cathode, Eion the ionization energy, and Φn the so-called normal cathode fall. At each successful electron impact, a pair of charges, electrons and ions, is generated with current density je (0)(eαx − 1), which are attracted by the cathode where they release γΓe (0)(eαx − 1) electrons of the second generation ad infinitum (Fig. 4.1).1 At the anode, the current of electrons has grown to (with d the distance between the two electrodes, j = e0 Γ): je (d) = je (0)eαd + γje (0)(eαd − 1)eαd + γ 2 je (0)(eαd − 1)2 eαd + · · · ,

(4.3.1)

or 1 This derivation neglects collisions of the ions. In fact, the ions incident on the cathode experience several collisions, since at pressures between 5 and 100 mTorr (1 and 13 Pa), the ionic mean free path λi amounts to values between 1 mm and 5 cm. Although we neglect ionizations by ionic impact in the gas phase (so-called β-processes), the mean energy of the ions is reduced by these collisions, and this reduces the impact at the electrode.

4.2 Gaseous generation of carriers

-

+

71

Fig. 4.1. Schematic representation of the generation of an avalanche of charged carriers by electron impact. LHS: cathode, RHS: anode.

je (d) = je (0)eαd (1 + γ(eαd − 1) + γ 2 (eαd − 1)2 + · · ·),

(4.3.2)

which represents an infinite geometric series, with sum for γ(eαd − 1) < 1, je (d) =

je (0)eαd je (0) =M 1 − γ(eαd − 1) 1 − γ(M − 1)

(4.4)

with M = eαd .2 This is the common expression for the current growth in a Townsend’s discharge, provided the current density is small (≈ 10−10 A cm−2 ) and the diffusion losses remain negligible [112]. For increasing voltage between the electrodes, both the coefficients α and γ will increase, the denominator in eq. (4.4) approaches zero and can even become negative. At any rate, the current rises sharply, and the striking of the plasma is connected with gas breakdown, or simply breakdown. In fact, ignition of the plasma happens at lower voltages since a positively charged layer is generated in front of the cathode, which leads to a steep increase of the electric field. The characteristics of the discharge changes fundamentally: A feature of the Townsend’s discharge is a constant electric field or a linear rise in potential between cathode and anode. Now, we observe the formation of a positively charged layer in front of the cathode. Due to this field, secondary electrons are accelerated away from the cathode, and the electron density falls to very low values but remains finite (Fig. 3.1). After ignition, the production of secondary electrons at the cathode is sufficient to sustain the discharge: The loss of ions is balanced by released secondary electrons which, on their course to the anode, generate this very number of ions. Hence, the current has become independent from its external cause, and we can write down the condition for stationarity of a self-sustained normal discharge in 2 The electron current will be amplified by M times and would alone not satisfy the continuity equation. However, every ionization impact in the plasma creates the same amount of electrons and their counterparts, and therefore, the continuity equation is valid for the total current. Due to their small mobility, the ions arrive the cathode with a certain time lag.

72

4 DC discharges

Townsend’s approximation [vanishing denominator in Eq. (4.4)], the so-called condition for stationarity:

1 αd = ln 1 + . γ

(4.5)

γeαd = γM = 1 :

(4.6)

Since γ 1, this yields

αd

αd

e positive ions release γe secondary electrons, or that number of ions + 1 ion, which are generated by one electron. With Eq. (4.1), we obtain for the normal cathode fall: Eion Φn = e0

1 1+ . γ

(4.7)

4.2.2 The primary ionization coefficient Provided that no pressure dependent phenomena play any role in the ionization, the ratio between electric field and pressure, E/p (which scales at constant gas temperature with the ratio between electric field and number density, E/n), determines the energy gain of the electrons between two collisions. Since the mean free path of the electrons, λe , is inversely proportional to the number density n, we find as an additional condition (in the following, we use ε instead of E for the energy to avoid confusion with the electric field):

< ε >= f

e0 E p

=f

e0 E n

= f (e0 Eλ).

(4.8)

The dependence on α, the number of ionizations per cm in the direction of the field, must be the same; in addition, α depends on the collision rate per cm (at constant temperature, p ∝ n): α α = pf (e0 Eλ) ⇒ = f (e0 Eλ) = f p

e0 E . p

(4.9)

According to Eq. (4.6), the product of γ and the multiplication factor M is constant. Inserting α using Eq. (4.9) yields

γ exp pd f

E p

= γ exp p df

Vb pd

= 1.

(4.10)

This requires for fixed pd the breakdown voltage Vb to remain constant as well: Vb is a single function of the product between the discharge pressure p and the electrode distance d (for Tgas = const : p ∝ n, Paschen’s law). For large values of E/p, the electrons reach the anode without having collided, i. e. without any chance to ionize, whereas for small values of E/p, the

4.2 Gaseous generation of carriers

73

2000 1600

Vb [V]

1200

hydrogen

800 air

400 0 0

Fig. 4.2. Paschen curves for hydrogen and air after [113]. 5

10 15 pd [Torr mm]

20

25

30

distance between two collisions is too short to gain that amount of kinetic energy which is sufficient for ionization. These counteracting dependencies lead to a relatively broad minimum which can be located at d ≈ λe , the so-called Paschen minimum (Fig. 4.2). α is often empirically approximated by

Bp α Ap (4.11) = · exp − E E E with A and B properties which are characteristic for the gas under consideration (Fig. 4.3). η=

ln a/E

A/eB

B

Fig. 4.3. Characteristic of α/E vs. E/p in double-logarithmic scale. The maximum can be found for the argument B and the value A/Be for the condition α/E = Ap/E exp(−Bp/E) (after [114]).

ln E/p

The decrease in α/E for small arguments is closely related with the excitation of higher electronic states (instead of “successful” ionizations); for large values of E/p, however, the increase in kinetic energy leads to a significant reduction of the ionization cross section. Deviations from this conduct can often be referred to as the Penning effect. To make this visible, it is common to plot η = (nσ)/E vs. E/p (Fig. 4.4). Since the electric field decreases almost entirely across the cathode fall, d in Eq. (4.5) equals the thickness of the cathodic dark

74

4 DC discharges

space, and it is assumed to begin where the field vanishes when the electrons enter the negative glow, neglecting the thin Bohmic presheath [115].

Ne+10-2 % Ar

104 a/E [10-6 V]

Ar

Ne+10-3 % Ar

Ne+ 10-4 % Ar

Fig. 4.4. Enlargement of α by collisional impact with metastables: Penning ionization Ne∗ +Ar → Ne+Ar+ +e− c [70] ( Review Modern Physics).

103 Ne

102

Ar

1

10

102

103

E/p [V/Torr cm]

4.3 The normal cathode fall 4.3.1 The secondary ionization coefficient Attraction of the ions by the cathode and mutual repulsive forces of the ions determine the thickness of the cathodic dark space. Therefore, the dark space is expected to shrink with rising pressure (n ∝ p). Expressed with the condition for stationarity, the thickness of a normal cathode fall, dn , increases for rising gaseous ionization or α-ionization or lessening surface ionization or γ-ionization. From Eq. (4.7), we can see that the height of the cathode fall, Φn , increases with rising ionization potential of the plasma-constituting ambient. Furthermore, γ plays an appreciable role in this dynamic equilibrium by which the height of the cathode fall is eventually fixed. Some qualitative approaches exist regarding Φn as a function of the work function of the electrode material for the same plasma ambient, since electrodes with the lowest work function Wa exhibit the lowest cathode falls (which is similar to gases with the lowest ionization potential or largest cross section for ionization by electrons). However, it seems quite likely that Φn cannot show a simple dependence on Wa because Φn is also determined by γ, which depends on the material of the cathode, its geometry and its surface quality and from the gas ions. Equations (4.1) and (4.7) seem at last to supply us with a satisfactory relation for the thickness of the normal cathode fall, which confirms the preceding assumption:

1 1 . · ln 1 + pdn = αp γ

(4.12)

4.3 The normal cathode fall

75

The thickness of the dark space increases in size for rising α-ionization or falling γ-ionization. The breakdown condition opens an experimental window to crosscheck γ. For this end, we take the logarithm of the condition for stationarity γeαd = 1, and express the exponent by αd = α/E Ed or α/E Vb which yields ln

1 1 = ln Vb − ln . γ η

(4.13)

Plotting the difference of the logarithms of Vb and 1/η vs. the logarithm of E/p yields the reciprocal secondary coefficient γ (Fig. 4.5).

air, brass 5

10

4

F [V]

10

1/h

Ne, Fe

Vb

103

Vb

2

10

Fig. 4.5. Values for the breakdown voltage Vb and 1/η for a parallel-plate capacitor for air and neon after [116].

1/h

1

10 0 10

101

102 E/p [V/Torr cm]

103

4.3.2 V-I characteristic After ignition, the initial linear potential drop between the electrodes will be distorted by the occurence of space charges, and the field cannot be regarded as homogeneous. Therefore, the current density between the electrodes cannot be determined with the simple equations

j = ρv = e0 nv ∧ v =

2e0 2e0 ⇒ j = e0 n , m m

(4.14)

because it has become spatially dependent. To describe these new features of the discharge, in particular the cathodic dark space, which happens to become the layer with the steepest potential drop, besides the condition for stationarity [eq. (4.5)], a relation between • current density of the ions • height of the cathode fall, and

76

4 DC discharges • the thickness of the dark space

is required. 4.3.2.1 Matrix sheath. To get an idea what will happen if we neglect the continuity equation, we consider the ion density to remain constant across the dark space. According to the (one-dimensional) Poisson equation, we find e0 n0 x e0 n0 dE ⇒ E(x) = = dx ε0 ε0 and the potential Φ turns out to depend on the distance squared:

(4.15.1)

e0 n 0 2 x, (4.15.2) ε0 which yields a square-root behavior between the thickness of the dark space, dc , and the potential drop Φ: Φ(x) = −

dc =

2ε0 Φ, e0 n0

(4.16.1)

2e0 √ Φ: kB Te

(4.16.2)

or, in terms of the Debye length,

dc = λD

The thickness of this matrix sheath would exhibit an extension of many Debye lengths. 4.3.2.2 Child-Langmuir sheath. To meet the requirements for continuity, the current density of the ions must drop with increasing velocity.3 This can be caused either by space charges in the high-vacuum regime (case 1) or by friction in the high-pressure regime (case 2), depending on the ratio of the mean free path of the ions over the thickness of the dark space: λi < 1 : mobility limited dc λi > 1 : space charge limited dc The relation between these three properties was first derived by Child in the year 1911. Other derivations are due to Langmuir (1912) and Schottky (1914). 3 From the sheath criterion, the electron density should drop even more, with the remaining electrons accelerated to 0.1×c or more. Both facts are responsible for the low level of emissivity. Few but rapidly moving electrons will lose their energy by exciting neutrals to levels which are far beyond the optical levels.


77

The ion current leaving the quasineutral negative glow, which exhibits a vanishing electric field (dE/dx)x=0 = 0) and is incident on the cathode, is given by j = ρv, where v is calculated via the electrode potential e0 Φ = 1/2m0 v 2 . The Poisson equation can be written as d2 Φ C = −√ dx2 Φ

(4.17.1)

with C=

ε0

j , 2e0 /m0

(4.17.2)

and C < 0 since we move from zero potential downward to the cathode potential. If the first derivative of Φ is regarded as a function of Φ (in many textbooks, this is called multiplication by dΦ/dx): d2 Φ dF dΦ dF dΦ = F [Φ(x)] ⇒ =F , = dx dx2 dΦ dx dΦ this equation can easily be separated yielding

F

C dF =√ ⇒ dΦ Φ

F dF =

√ C √ dΦ ⇒ 1/2 F 2 |Φ 0 = 2C Φ + A. Φ

(4.18)

(4.19)

Assuming dΦ/dx to vanish in the negative glow it follows that A = 0, and repeated integration yields √ √ dΦ 4 3 d = 2 Cdx ⇒ Φ /4 |Φ 0 = 2 Cx |0 , 3 Φ1/4

(4.20)

and finally for Φ and j:

3

9 4 j 2e0 Φ /2 Φ /2 = d2 ∧ j = ε0 . 4ε0 2e0 /m0 9 m0 d2 3

(4.21.1)

Following Bohm, the maximum ion current is given by eqs. (14.43/44). Inserting this relation into Eq. (4.21.1), we get finally for the sheath thickness (with n the plasma density): 2 d= 3

ε0 n

4

3 2e Φ /4 . e0 kB Te

(4.21.2)

This is the high vacuum version of Child’s current density equation (Fig. 4.6). As an example, for an electron energy of 4 eV, a plasma density of 1010 cm−3 and a potential Φ of 100 eV we obtain a sheath thickness of d = 2.06 mm. For the same plasma parameters, we get a Debye length of only 0.16 mm. For higher pressures, the current leaving the negative glow is mobility limited: j = ρu = ρμE, and the mobility is supposed to be independent of the electric field (with u the drift velocity).

78

4 DC discharges 300

300 0.1 mA/cm2 1 mA/cm 2 10 mA/cm2

0 mA/cm 2 10 mA/cm2

200 F[V]

F[V]

200

100

100

0 0.00

0.25

0.50

0.75

1.00

0 0.00

0.25

0.50

0.75

1.00

distance [cm]

distance [cm]

Fig. 4.6. Space charge limited current between two plates, one of which acts as electron source. LHS: various current densities, RHS: attenuation of the potential by ions in the sheath adjacent to the electrode in comparison to a linear potential drop.

j d2 Φ ρ j =− . =− =− dx2 ε0 ε0 u ε0 μE The separation is easily performed with E = − dΦ dx and the product rule

(4.22) d dx

dΦ dx

2

=

2 2 ddxΦ2 dΦ dx :

dΦ 1 d 2 dx

2

dΦ j dx ⇒ = = ε0 μ dx

2j √ d, ε0 μ

(4.23)

and finally for Φ and j: 2 Φ= 3

2j 3/2 9 Φ2 d ∧ j = ε0 μ 3 . ε0 μ 8 d

(4.24.1)

This is the mobility limited version, which is known in the solid state (with finite ε) as Mott-Guerney equation. Inserting Eq. (14.44) into Eq. (4.24.1), we get finally for the sheath thickness in the mobility-limited case

d=

9 3 8 ε0 μ

n0 e0

1 / 6 e 2 mi

kB Te

2

Φ /3 .

(4.24.2)

• In the high vacuum version, the potential of the space charge scales with 4 1 3 x /3 , the field with x /3 , and the current density with Φ /2 . • In the high pressure version (drift velocity is limited by collisions) the 3 1 potential is proportional to x /2 , the field proportional to x /2 , and the 2 current density proportional to Φ .


79

1.00 linear rise in potential space charge limited current mobility limited current

1.00

0.75 0.75

V/Vc

jnorm

0.50

0.25

0.00 0.00

space charge limited mobility limited homogeneous

0.50

0.25

0.25

0.50

0.75

0.00 0.00

1.00

0.25

0.50

0.75

1.00

Fnorm

d1/dc

Fig. 4.7. LHS: Spatial dependence of the potential for some different distributions of the space charge: The difference is very minute between a - linear potential drop without any screening and a - potential drop which is attenuated by a current which is either - space charge limited or - mobility limited. RHS: The same holds true for the cathodic ion current density: Compared to the 1 limit of the homogeneous field which assumes a vanishing charge density (j ∝ Φ /2 ), the differences between the two cases are too minute to be detected, here pictured for normalized properties. The Ohmic law simply scales with Φ1 .

In both cases, the dependence between current density and potential considerably differs from the case of a homogeneous field with a very low charge density, where we find the current density j to depend on the square root of the potential Φ. In particular, the charge density (and consequently the current density) are spatially independent. However, it was Ingold [117], who 3 pointed out that the assumed differences between these possibilities (Φ ∝ d /2 4 for mobility limited current, Φ ∝ d /3 for space charge limited current would be far too small to be distinguished experimentally; even a linear field (Φ ∝ d1 ) would be difficult to confirm [118], Fig. 4.7). Furthermore, the mean free path of the ions, λi , is comparable to the thickness of the cathodic dark space, dc , and we eventually conclude that neither of these equations is correct (at least in principle). In the case of the mobility limited current density, the current density at the surface of the cathode (u = μE) would yield j = ε0

9μ Φ2n 8 d3n

(4.25)

with the plasma regarded as emitting electrode and the cathode as collector electrode with vanishing electric field at the boundary of the cathode fall.

80

4 DC discharges

For the thickness of the dark space at an isolating wall which exhibits a floating potential of typically Φf = −10 V, we can calculate for a collisionless dark space dc to be about 0.2 mm: The dark space is invisible! Becoming more precise, we can express the thickness of the dark space in terms of the Debye length with eqs. (3.15) and (4.24.1) for a one-dimensional Maxwellian distribution of the ions (ion current incident on the electrode). We use Child’s equation to calculate the positive ion current neglecting the electron space charge within the sheath:

3

2e0 Φ /2 4 ε0 = e0 ni vB . 9 m0 d2c

(4.26)

Considering the velocity of the ions at the Bohm edge [cf. Eq. (14.30)] this yields √ d2c =

√ 3 3 32 e0 Φ /2 2 2e0 Φ /4 × λ2D ⇒ dc = × λD , 9 kB Te 3 kB Te

(4.27.1)

or

3

Φ /2 λD : (4.27.2) Φf The cathodic sheath thickness must be considerably larger than the Debye length. Furthermore, we draw the following conclusions from this discussion: dc ≈

• For vanishing je (0) also je (d) becomes zero: To trigger an electron avalanche some free electrons are required. To facilitate the ignition, plasma reactors are often equipped with a filament which emits electrons thermally. • For a self-sustaining discharge, a certain initial electron density must be available. Once the discharge is ignited, the ion current drops by diminishing the discharge voltage, and the required ion current density is provided by radial shrinking of the negative glow. The minimum voltage required to release a sufficient amount of γ-electrons is denoted normal cathode potential Φn . In these discharges, current density and height of the cathode fall remain constant even when the current rises by some orders of magnitude. That is accomplished by enlarging the cross section of the current path until the whole electrode is covered. Now, further rising current leads to an avalanche of increasing current density which will trigger a higher efficiency of γ-processes which, in turn, is connected with a further rise in potential. It is these abnormal cathode falls which are used extensively for sputter deposition and plasma etching. Since the thickness of the sheath depends on flux and potential drop, the characteristics of an etching process or a deposition process can gradually or

4.4 The abnormal cathode fall

81

Table 4.1. Calculated values of properties which characterize an abnormal discharge through argon [122]. energy [eV] 15.8 16 20 30

pd1 [mTorr cm] 319 224 61 27

pd2 [mTorr cm] 323 233 75 53

pdc [mTorr cm] 1545 1103 353 218

α/p [cm−1 mTorr−1 ] 34.8 49.2 157 258

Φc [V] 127 128 147 195

suddenly change when we alter one of these determining properties. Although derived for a high-voltage DC sheath, this model is appropriate for employment in the treatment of the RF sheath. This has been extensively shown by Riemann and coworkers during the late 1980s who also incorporated an energy distribution of the ions for his derivation [119, 120]. For an arbitrary ratio between dc and λi , they obtain a formula which unifies the approaches of a space-charge limited current with a mobility-limited current according to

9 12π dc j d2 × 1 + Φ /2 = 4ε0 2e0 /m0 125 λi 3

2

.

(4.28)

4.4 The abnormal cathode fall As we have previously noted, the current density and the thickness of the dark space depend on the height of the cathode fall in this type of discharge. For ionization in the dark space, we require the mean free path of the electrons, to be short compared with its thickness. By this scattering process, the beam-like, monoenergetic characteristic of the γ-electrons is terminated; the electrons exhibit a certain energy distribution, hence, α is not constant any more. Furthermore, conditions are required to calculate the balance for particles, momentum and energy which depend on the number of collisions between electrons and neutrals. As invariant properties, j/p2 and pd exhibit the following dependencies on the abnormal cathode fall [121]: ⎫

j/p2 ∝ Φc3/2 : space charge limited current; ⎪ ⎪ ⎪

j/p2 ∝ Φ2c : mobility limited current; pd ∝

1 αd 1−

Φ Φc

ln 1 . γΦc

⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

(4.29)

Using these equations, the numbers in Table 4.1 are calculated for a collision frequency νm of 6.6 × 109 sec−1 and a γ of 0.1.

82

4 DC discharges

After having fallen across the (reduced) length pd1 , the electrons are accelerated to energies which equal the ionization potential or even larger. This length is insignificantly shorter than pd2 , which defines the upper limit of enery accumulation due to elastic collisions (mobility limit). On the other hand, pd2 is considerably shorter than pdc , about a factor of 5 just beyond the ionization threshold to drop to about 1/10 at twice the ionization potential; all the length which characterizes the dark space declines with rising potential, in particular, the thickness of the dark space [123]. Its thickness roughly equals the ionization length for the γ-electrons to trigger an ionization. This model gives a rather good description of the experimental data up to the Paschen minimum as communicated by v. Engel [124]. Deviations are mainly caused by the approximation for α (as just mentioned) and its energy dependence (in particular for rapidly moving electrons), but also by the heating of the gas, by which the density is reduced; hence, pd (pd ∝ nd) is no longer an invariant property. Furthermore, Φc begins to rise in the negative glow; therefore, the thickness of the dark space is difficult to fix although the borderline for high cathode falls becomes sharper and better defined. Its length varies between 4 and 10 ×λe [125]. Hence, we can clearly distinguish electrodes which sustain a DC discharge from additional surfaces with adjustable potential. The thickness of the dark space of these electrodes can be calculated using the sheath equation (cf. Sect. 14.2) or by Child’s equation. This difference is mainly caused by the release of γ-electrons which have emerged as a constitutive part of a self-sustained DC discharge. Irrespective of whether the electric field does depend on the thickness of the dark space, for rising γ or increasing current density and cathode fall, the sheath thickness becomes thinner, which leads to a striking increase of the field strength (Fig. 4.8).

10

-1

0.5 j/p

2

0.3

-2

10

-3

dp

0.2 0.1 0.0

10

250

500

750 1000 VC [V]

1250

1500

j/p 2 [A cm -2 Torr -2]

dp [cm Torr]

0.4

Fig. 4.8. Reduced thickness of the dark space pd and reduced current density j/p2 as function of the cathode fall in nitrogen (iron cathode). The values for high voltages have not been corrected by the falling c Oxford gas density [126] ( University Press).

4.4 The abnormal cathode fall

Plotting the product pd which follow

83 j/p2 vs. Φc yields the straight lines in Fig. 4.9,

d j = A + B (Φ − Φ0 ),

(4.30)

an equation which was reported for the first time by Aston [127].

d j 1/2 [10-4 A 1/2 cm -1]

5 H2

theoretical experimental

4

He

3 N2

2 He

H2

Kr

N2

1 CO

0

CO

200

400

Kr

600 800 Fc [V]

1000 1200

Fig. 4.9. Reduced thickness of √ the dark space p d j as function of the cathode fall for several gases. j is calculated according to the space charge limited Child’s equation [127] c Springer-Verlag). (

4.4.1 Discussion of Townsend’s approximation Townsend’s theory postulates the existence of two electron sources: By electron impact, the gaseous constituents are ionized (α-electrons), by ionic impact, electrons are released from the surface a solid (γ-electrons). It should be mentioned again that Townsend’s equation has been derived but for a dark discharge. Effects in other discharges may come to be better understood, but this model cannot be strictly applied to other discharges. This is shown by an example by Davis and Vanderslice [128]: Example 4.1 For a DC discharge through argon, DC voltage: 600 V, at a pressure of 60 mTorr (8 Pa), which means a density of the neutrals of 2 × 1015 cm−3 , the ionization cross section at 100 eV is for electron impact 3 ˚ A2 , for ion impact 0.5 ˚ A2 , 2 ˚ and the total cross section equals 30 A , we calculate an electronic mean free path λe of 1.67 mm, and a thickness of the dark space dc of 1.3 cm. This means for the two tracks of ionization: 1. Ionization by electron impact: The upper limit of ionization is then about 1.9, i. e. at a yield for secondary electrons γ ≈ 0.2, the yield of generation equals 0.2 × (1.9 − 1) = 0.18 because it is a two-step process. 2. Ionization by ion impact: yield of 1.12 ⇒ a maximum rate of 1.12 − 1 = 0.12.

84

4 DC discharges

Albeit σi is smaller than σe by approximately one order of magnitude, the yields are almost equal because of the low density of the secondary electrons. With the supposed thickness of 1.3 cm, every electron would be able to ionize just one neutral species, far too low to sustain the discharge.

• Therefore, the thickness of the dark space should be considerably larger since the yield for secondary electrons does not depend strongly on energy. Also the cross section for ionization is considered to have reached its maximum. No doubt that both the mechanisms just discussed are important, however, they are not sufficient to sustain a discharge. • Moreover, the current density in a normal discharge is between 10−5 and 10−2 A cm−2 , which increases to 10−2 A cm−2 to peak at about 0.5 A cm−2 in an abnormal discharge. Thus, there are considerable carrier losses due to ambipolar diffusion (cf. Sect. 4.7). Eventually, the fraction of electrons released by the photoelectric effect is assumed to be of comparable size with that due to ion bombardement [129]: γi ≈ γhν . • At the anode, an additional ionization must take place to maintain the ion current. Hence, an effective ion source is required either in the negative glow or in the anodic dark space. The height of the anode fall remains in the order of the ionization potential—far too low for further discussion concerning this topic. It remains the negative glow that has been the subject of the scientific discussion for decades. For example, Little and v. Engel could not suppose the negative glow to be the region of any carrier generation [129]. This assumption was grounded on the fact that the only collisions which change the velocity of the ions incident on the cathode are due to resonant charge transfer: + A+ rapid + Aslow −→ Aslow + Arapid .

(4.31)

In most cases, elastic collision cause only small-angle scattering, by which the velocity of the ions is not significantly changed. According to a model developed by Scherzer, all electrons are generated in the negative glow [130]. As Druyvesteyn and Penning had pointed out, the equation of stationarity [Eq. (4.5)] had to be extended to

αdc = ln

1 1+ γ

(1 − β)

(4.32)

with β the quotient between the current densities of the ions and the electrons, respectively ji /je [131]. If ji were a considerable fraction of the total current, α could become significantly smaller. They distinguish between a normal and an abnormal discharge. In the latter discharge, most of the ions are generated in the negative glow, whereas in the first discharge, the main area of ion production is shifted into the region of

4.5 Negative glow and positive column

85

energy transfer, i. e. the cathodic dark space. This is mainly caused by its shrinking width compared with the mean free path of the electrons, λe , but it is also due to the smaller ionization cross section at larger energies (cf. Sect. 2.2). Above all, Townsend’s approximation has been derived for a homogeneous field. This is definitely not the case for the cathode fall which, on the contrary, is extremely inhomogeneous [here, values for E/p of up to 105 V/(Torr cm) across 100 μm in length can occur]. Therefore, α does not depend solely on E/p, but also on the thickness of the dark space: E = E(d).

4.5 Negative glow and positive column Although the negative glow, regarded as a whole, is quasineutral, it is by no means uniform; this is mainly caused by the electron bombardement from the cathodic dark space. Traditionally, three classes of electrons are distinguished, in a DC discharge, their ratio amounts to about 1 : 10 : 1 000 [132]: • Primary electrons of high energy (in fact, these are the γ-electrons ejected from the cathode). • Secondary electrons which are generated by α-processes (electrons → neutrals) or β-processes (ions →neutrals) with energies which are considerably lower. • Thermalized electrons which are decelerated down to the plasma temperature, so-called last electrons, they form the greatest part of these three fractions. When entering the negative glow, the first two groups will be considerably slowed down due to multiple excitations (optical levels and ionizations) which causes the intense glowing at the head of this zone. At these high energies (some hundreds or even thousands of eV), however, the cross section for both these excitations has dropped to very low levels. Hence, the intensity of the glow does not peak at the border to Crooke’s dark space but is somewhat shifted into the negative glow, until the kinetic energy has been decreased to values of about 100 − 150 eV. That is why some primary electrons can pass the negative glow and reach the anode (cf. Sect. 10.2). All the qualities which are characteristic for a beam are lost in the negative glow; a small negative space charge is generated at the anode which causes the formation of a weak electric field which, in turn, drags the electrons into Faraday’s dark space. After having been accelerated across a sufficient distance, they have gained enough kinetic energy to excite neutrals to optical levels or to ions once again which is manifested by the head of the positive column. Since the beam qualities were already lost in the negative glow, this head appears washed out. The cathodic dark space and Faraday’s dark space differ mainly

86

4 DC discharges

by the mean energy of the electrons. In the first, the mean energy is far too high to excite molecules to optical levels, in the latter, the mean energy is insufficient for this excitation. The field of the positive column adapts to that value that ensures a constant density of carriers, i. e. the rate of generation equals the rate of loss. Often, zones of different glowing intensity are observed in the positive column, so-called striations which many authors consider repetitions of the ionization processes in the head of the positive column. From their distance, the mean free path for optical excitation can be inferred. Morgan, however, interprets these striations as nodes and crests of ionic oscillations [133]. Both of these zones are scenes of prototypic processes. In the negative glow, most of the ionization will happen, whereas the positive column is best suited to explain diffusion losses.

4.6 Ionization Due to their high kinetic energy, primary electrons can perform multiple ionizations. But even if the cross section for ionization is supposed to be at maximum (for argon about 29 ˚ A2 at 100 eV) and we extend the negative glow to an unusual length of 5 cm, a rough estimation reveals that only 5 % of charged carriers are generated which are required to sustain the discharge [134]. This is due mainly to the low energy of the electrons which are generated by the ionization impact because they are hardly accelerated by the weak electric field in the negative glow by which the avalanche is readily terminated. Since the exponential term is lacking, the ionization is solely determined by the product je × σ × nAr . Assuming a Maxwellian thermalization of the electrons around the temperature Te = 2 eV/kB and the condition for reaching the electrodes • cathode: E ≥ e0 (Φp + Φcathode ), • anode: E ≥ e0 Φp , the probabilty for a thermalized electron to reach the cathode becomes zero whereas the anode will be reached with a probability of 1%; i. e. thermalized electrons are almost completely trapped in the negative glow, unless they do not get lost by wall reactions. 4.6.1 Ionization in the negative glow This energy between 2 and 8 eV is definitely insufficient to ionize argon, for its first ionization potential is 15.7 eV. For a Maxwellian distribution of electrons, we can calculate the fraction of electrons with energies larger than 15.7 to 0.1 % at a mean electron energy of 2 eV, which rises to 28 % for a mean energy of 8 eV. The generation rate GR of ions (in fact the ionization frequency νion )

4.6 Ionization

87

dnA+ = GRion = kion nA dt is proportional to these quantities:

(4.33)

• The ionization cross section σion for electronic impact. • The energy E of the electrons. • The fraction of electrons with energies f (E) larger than 15.7 eV. • The density of the neutrals nA : GRion =

∞ Eion

nA σion (E) E f (E) dE

(4.34)

with f (E) the normalized EEDF. The rate coefficient kion is obtained by dividing GRion by the density of the neutrals (Sect. 2.3.5): kion =

∞ Eion

σion (E) E f (E) dE.

(4.35)

Provided that the cross section for ionization by electron impact can be linearized close beyond the threshold Eion , σion (v) = a(Φ − Φion ),

(4.36)

with a = 4.05/cm mTorr K for argon [135], which is a very good approximation for the negative glow, the generation rate can be calculated. When we apply analytic expressions for f (v), the integration is made possible in closed form (Sect. 14.1). For a plasma density of 1010 cm−3 and a typical pressure of 50 mTorr (7 Pa) we can calculate the yield for electrons as compiled in Table 4.2. Example 4.2 For a cathodic current density of 1mA cm−2 and a γ of 0.2, the electron current at the edge of the negative glow can be calculated to 0.2 mA cm−2 or 3.5×1015 electrons per cm2 sec. For a length of the negative glow of 3 cm, 1015 electrons per cm2 sec have to be generated. This task can be accomplished by Maxwellian electrons exhibiting a mean energy of 2 eV. Electrons with a temperature slightly higher can also compensate the losses which are caused by diffusion and recombination. {A similar result was obtained by Winters et al. for the generation of ions in a discharge through CF4 (ionization potential of CF4 : 16 eV) [136]}.

Although numerous experiments have shown that the electrons in the negative glow are not distributed according to a Maxwellian distribution, inspection of Table 4.2 clearly shows that a pure Druyvesteynian distribution does not describe the facts as well. This is known as Langmuir paradoxon and has been explained by electron trapping and generation of plasma oscillations (Sects. 14.1 and 14.3). Others supposed these differences to be induced by the measuring technique (artifacts of the Langmuir probe technique) [137]. This

88

4 DC discharges

Table 4.2. Ionization of argon by electrons with varying mean energy < E > as obtained by application of two different distribution functions (Maxwell and Druyvesteyn). E [eV]

Eion / < E >

8 6 5.3 4 3 2.5 2 1.5 1

2 2.6 3 4 5.3 6 8 10 15

GR [cm3 sec−1 ] Maxwell Druyvesteyn 2.72 × 1017 9.8 × 1015 17 1.41 × 10 2.0 × 1014 16 9.96 × 10 5.0 × 1013 3.64 × 1016 1.4 × 1010 1.13 × 1016 4.30 × 1015 1.04 × 1015 1.01 × 1014 1.05 × 1012

could be definitely excluded by comparison of data obtained by the Langmuir probe with data gained with the non-invasive optical emission spectroscopy [98]. The main zone of carrier generation is the negative glow, the main zone of energy gain is the cathodic dark space. The energy is shared among ions and electrons. By ionic impact on the cathode, particles are ejected, by γ-processes, electrons will be released and some other processes will occur (cf. Chap. 10). The generated γ-electrons are accelerated by the cathode fall and gain energy until they enter the negative glow.

4.7 Loss of carriers These processes of generation of carriers are exceeded by losses (still no plasma) or just evened out by losses (breakdown or ignition). The main mechanisms for loss are recombination of charged carriers, and diffusion. Although these mechanisms are important, we have already seen that breakdown, for example the Paschen minimum, can be explained without having an idea of the explicit knowledge of how the electrons are lost. Although both mechanisms are of importance, at low pressures, diffusion dominates, and for pressures higher than about 10 Torr (1 333 Pa), recombination takes over [138]. The pure mechanisms can easily be distinguished since recombination scales with carrier density squared, which simply explains its range of validity in the high-pressure range, whereas in the diffusion regime, losses are exponentially proportional to the carrier density. Recombination takes place in the plasma bulk, diffusion means leaving the plasma bulk and reaction with the wall (surface) of the reactor

4.7 Loss of carriers

89

(heterogeneous reaction). In this section, we limit our interest to diffusion with vanishing field. 4.7.1 Free diffusion We consider an electropositive plasma with ni = ne = n. It is only the electrons which can act as ionizing species. As we have seen in Sect. 4.2, ionization by electron impact is an avalanche process, and the number of ionizations, i. e. the number of charges of the next generation n , is proportional to the number of parent charges n, the density of the gas nn , and a bimolecular rate constant k2 which is connected with the frequency of ionization via the generation rate GR GR = k2 ne nn ⇒ k2 =

νion , nn

(4.37.1)

and it is shown here for electrons: dne = νion ne . (4.37.2) dt Losses in the plasma bulk are recombinations and are neglected for the pressure range considered here. For slow changes, we can start with the extended continuity equation [Eqs. (4.38/39)] and Fick’s first law [Eq. (4.40)]: ne = ne eνion t ⇒

∂n ∂n (4.38) + ∇ · Γ = 0 −→ + ∇ · Γ = νion n, ∂t ∂t and for the steady case, all charges that are generated in the volume are lost by diffusion processes through the (imaginary) walls, which confine the plasma, so we can drop the first term on the LHS to obtain ∇ · Γ = νion n.

(4.39)

Γ = −D∇n

(4.40)

Remembering that

we obtain the three-dimensional Helmholtz equation [139] νion n=0: (4.41) D The amount of generated carriers equals the amount of lost carriers, and the generation of new carriers is proportional to their number. We have introduced the three-dimensional diffusion coefficient, D, which is well-known from kinetic gas theory where we have derived a relationship between the mean square distance < x2 >, mean velocity < v >, the mean free path λ, and the reciprocal time between two collisions (1/τ = νm , random walk) ∇2 n +

90

4 DC discharges

1 < x2 > 1 = λ . (4.42) 3 τ 3 It differs for electrons and ions by some orders of magnitude. In a plasma, it is regarded as spatially constant. For a rectangular box, this boundary-value problem yields D=

πx πy πz cos cos , (4.43) I J K with maximum density at the origin [ne (0) = ne,0 ] and vanishing at the walls, i. e. for x: ne = ne,0 cos

I ∂ne = 0 at x = 0 ⇒ ne = 0 at x = ± , ∂x 2 and the diffusion length D 1 ∨ = νion Λ which can be written in an equivalent form

Λ2 =

νion , D

2

π 1 1 1 1 1 = 2 + 2+ 2∨ 2 = Λ2 Λx Λy Λz Λ I

(4.44)

+

(4.45)

π J

2

+

π K

2

.

(4.46)

Further simplifying for a cube with I = J = K, we obtain 1 νion = 2 =3 D Λ

2 π

I

,

(4.47)

and 3D D ∧ Λ2 = . (4.48) νion νion We note that Λ is independent of pressure because both its constituents, D and νion , scale inversely with pressure. Keeping in mind that the net current at the boundaries will vanish, the same consideration for the ions yields an important relationship between the densities of the two different types of charges Λ2x =

De ne . (4.49) Di The high electron flux directed outward leads to a depletion of the negative carriers and to an excess of positive ions, and no stable plasma can be formed. ni =

4.7.2 Ambipolar diffusion coefficient Since the diffusion coefficients for electrons and ions differ by some orders of magnitude, this also holds true for their fluxes. Above all, the few charges move independently and undisturbed. However, after ignition, in plasmas which meet


91

the requirement for quasineutrality, i. e. their dimensions are large compared with the Debye length (Λ λD , cf. Sect. 3.3), the charge density has tremendously increased. Space charges generate electric fields which exert deflecting forces on the charges which happen to come across. In particular, the motions of electrons and ions will be linked together, and the diffusion rates of electrons and ions equalize to one value in the stationary state. This is achieved by the build-up of an electric field which holds the electrons back in the plasma bulk but accelerates the ions out of it. The total fluxes of the two carriers will become equal [validity of Ohm’s law provided and μ the mobility of the carriers (j = ρ v = e0 n v = e0 n μE)] which are composed of a drift term (due to external forces) and a diffusion term (due to internal forces: random walk): j+ = e0 Γi = e0 μi ni − e0 Di ∇ni ∧ j− = −e0 Γe = −e0 μe ne − e0 De ∇ne . (4.50) Assuming that the deviations from equilibrium remain very small, we demand that the densities of ions and electrons be equal (condition of electroneutrality). We obtain a new value for the electric field E=

Di − De ∇n , μe + μi n

(4.51)

and in this case, the densities in the drift terms vanish, and we arrive at a new diffusion coefficient Da =

μ i De + μ e Di . μi + μe

(4.52)

This coupled motion (diffusion and drift) can be regarded as a modified diffusion which is termed ambipolar diffusion. Remembering that μe μi , this expression can be further simplified by neglecting the ion mobility in the denominator, yielding Da ≈

μi De + D i , μe

(4.53)

and applying the Einsteinian relation D = kB T μ

(4.54)

we arrive at

D a = Di 1 +

Te Ti

≈ Di

Te , Ti

(4.55)

and we can write for the second Fick’s law for vanishing drift dn = e0 Da Δn. dt

(4.56)

92

4 DC discharges

The diffusion of the ions will be accelerated, but that of the electrons is slowed down, and the total diffusion velocity is determined mainly by the mobility of the slower species, but enhanced by the ratio of the mean energies of the mutually coupled species. Hence, ambipolar diffusion coefficients are considerably smaller than that for free electrons. As a first consequence, the electron density falls more slowly, and the electric field which is necessary to sustain the discharge decreases from the topmost value required for ignition or breakdown to values which are considerably lower [23]. Example 4.3 From thermodynamics, we remember typical values for D to be around 10−1 cm2 /sec at 1 bar (DO2 = 1.8 × 10−1 cm2 sec at 0 ◦ C), and since D scales with the mean free path (D = 1/3 λ < v >), it will become larger by four orders of magnitude at 100 mTorr (end of the viscous flow regime), say 750 cm2 /sec. Let’s consider an electron temperature of 3 eV and an ion temperature of 600 K, we find for the ratio Te /Ti a value of around 60 which leads to coefficients for the ambipolar diffusivity of 45 × 103 cm2 sec at 100 mTorr. A parallel-plate reactor 20" (50 cm) in diameter and a gap of 7.5 cm between its plates exhibits a volume of 15 l, and for a gas flow of 100 sccm, this leads to a residence time τ of 2.4 sec. Remembering the equation for the random walk [eq. (4.38)], we eventually obtain a value for the diffusion length Λ of 570 cm, large in size compared with all the dimensions of the reactor.

Both carrier types diffuse very rapidly, which has two consequences: • All the problems concerning density gradients can be linearized. This has the advantage that we can treat issues concerning plasma properties separately from those associated with transport behavior. • Of the two mechanism for equalizing density gradients, diffusion is predominant over convection, i. e. real transport of gas volumes different in density (caused by heating). In particular at discharge pressures of some tens of mTorr, which we encounter in most of the RF reactors which are driven capacitively or inductively, the charged carriers diffuse rapidly thereby causing the plasma to extend readily in the reactor. Raising the pressure, however, confines the plasma to the path between the electrodes. The diffusion becomes ambipolar for plasma densities larger than 108 cm−3 . In low-pressure discharges in which loss is dominated by wall reactions, the diffusion time, which is the averaged time to cover the distance from the origin to the wall, is approximately given by the ratio Q/I: τ=

Q ρV = , I jA

(4.57)

which reads for the cylindrical symmetry of a parallel-plate reactor [l: cylinder length, r: cylinder radius, O: surface: 2πr(r + l)], τ=

n0 e0 πr2 l . vI O

(4.58)


93

Since the carriers must pass a dark space (sheath) we have to insert for ji the current density at the Bohm edge [eq. (14.44)] and we obtain for the typical geometry (r l):

τ ≈l

mi . kB Te

(4.59)

For an argon plasma with an electron energy of 2 eV confined within a tube of 10 cm in length, τ becomes ≈ 45 μsec. In fact, the values are higher by a factor of 2 which can be referred to the nonlinear distribution of charged carriers within the dark space of a DC discharge or the sheath of a HF discharge. The conditions become more complex in discharges of electronegative gases with rising pressure. Now, the density of the negatively charged carriers is considerably influenced by the negative ions, which can even lead to a clear predominance in discharges through SF6 .

4.7.3 Modified boundary Reducing Eq. (4.43) to one dimension and considering (4.46), we find

ne = ne,0 cos

x , Λ

(4.60)

and inserting it into Fick’s first law, we obtain for the flux

Γ=D

ne,0 x sin Λ Λ

(4.61)

which propagates with the diffusion velocity of

u=

Γ x D : = tan n Λ Λ

(4.62)

To meet the the conservation laws which are expressed in the continuity equation (4.38), for vanishing number density a steeply rising diffusion velocity is mandatory, which would become singular at the wall! To circumvent this singularity, we must exclude this hard boundary condition and we demand for real systems a finite velocity and a finite residual density. For high discharge pressures operated at some Torr this is no issue at all since D scales with inverted pressure, and we can set n = 0. However, DC discharges as well as RF discharges are driven at significantly lower pressures, and for them, this boundary condition has to be modified by replacing it by a finite density at some distance in front of the wall. This boundary is the Bohm edge (Sects. 14.2 and 14.4) which takes into account the inertia of the ions.

94

4 DC discharges

4.7.4 Diffusion processes in the positive column We have seen in the last section that the positive column is a relatively placid zone. Its field adapts to that value that ensures a constant density of carriers, i. e. the rate of generation equals the rate of loss. According to Schottky’s assumption that the mean free path of the electrons is small compared with the radius of the column, some collisions will take place. Carrier loss is dominated by ambipolar diffusion in a radially outward direction and subsequent capture by the walls of the reactor. Hence, the charge density is at its maximum in the center, declining gradually when going outward and vanishing eventually at the reactor walls. The potential of the column is more positive than the wall potential due to the larger mobility of the electrons compared with the other charges. The current in the positive column is maintained predominantly by the electrons which are delivered from the adjacent Faraday’s dark space. Provided the length of the positive column is large compared with its diameter, its properties can be regarded as radially symmetric. Furthermore, the diffusion laws can be applied when the radial dimensions exceed the mean free path of the electrons, and we assume the densities of both the carrier types to be equal [cf. derivation with Eqs. (4.50) and (4.51), ni = ne = n0 ], which holds then also be true for their radial gradients dn/dr. In the state of equilibrium, positively and negatively charged carriers migrate outward with the same ambipolar diffusion coefficient. A two-dimensional toroidal element with inner diameter r and outer diameter r + dr can absorb

dn dt

= 2πrDa r

dn dr

(4.63) r

carriers, but will lose

dn dt

= 2π(r + dr)Da r+dr

dn dr

(4.64) r+dr

carriers. Thus, its balance can be calculated according to

dn dt

= −2πDa r+dr

dn r dr

dn −r dr r+dr

dn + dr dr r

,

(4.65.1)

r+dr

or

dn dt

= −2πDa r+dr

d2 n dn r dr + dr , dr2 dr

(4.65.2)

provided no recombinations occur. For the stationary state, dn/dt will vanish:

4.8 Anodic region

95

1 d dn νion r + n = 0, r dr dr Da

(4.66)

which is solved by the Bessel function of the first kind and zero order

n(r) = n0 J0 r

νion Da

(4.67)

with n0 the central concentration at r = 0. The Bessel function of first kind and zero order is comparable to a damped cosine function with shifted roots; n(r) = n0 J0 (2.405r/R) with R = rmax (Sect. 14.7.2, Figs. 14.35), and in three dimensions

ne = n 0 J 0 r

νion πz cos Da K

(4.68)

and for the diffusion length

2.405 1 = Λ2 R

2

+

π K

2

.

(4.69)

4.8 Anodic region The description of the various zones is completed with the anodic zone. Here, the current is drawn from the glow discharge to the external circuit. The properties of the anodic zone depend mainly on the position of the anode with respect to • the quasineutral plasma of the positive column or to • the slightly negative space charge of Faraday’s dark space or to • the positively charged negative glow. Is the anode located within the positive column, the density of ions vanishes at the anode, and a small negative space charge is created in front of the anode, the anodic dark space with the anode fall; the anode is considerably more positive than the potential of the negative glow. To sustain equilibrium in the dark space, the electrons which establish the negative space must balance the number of ions entering the boundary between the positive column and Faraday’s dark space. That is why the energy of the electrons (and the potential, resp.) rises to values which amount approximately the ionization potential of the ambient. Would the anode fall rise to higher values, the number of ionizations would be enlarged until the condition of state is met again. Hence, the anode fall cannot significantly surpass the ionization potential [140] (as in the case of the cathode, Fig. 4.10). Neglecting all the secondary processes, v. Engel developed a simple theory for the anode fall, which can be applied at low pressures below 100 mTorr (15

96

4 DC discharges

Pa) and for low current densities (j < 10−3 A/cm2 ) [141]. A parabolic variation of the field assumed, the anode fall yields V (x) = 3Va

2 x

d

1−

2x 3d

(4.70)

with Va the anode fall and d the thickness of the anodic dark space. 15

f(x) [a. u.]

Fig. 4.10. Anode fall in argon after the theory of von Engel, which dives into the positive column in which we observe a constant rise of potential (and a constant electric field). Va is somewhat smaller than the ionization energy of argon [141].

potential electric field

10 5 0 -5 -3

-2

-1

0 x [mm]

1

2

3

This limitation does not apply for very small anodes and is also not valid for discharges through electronegative gases which form negative ions. By this electron capture, electrons are consumed, and the resulting negative ions are significantly slower than electrons. This gives rise to a steep increase of the anode fall, which can reach 1 000 V in discharges through halogenes [142]. From this dicussion it is obvious that the thickness of the anodic dark space is determined chiefly by the condition of a stable space charge in front of the anode, but secondly by the condition of ionization. The ion current density is considerably smaller than the electron current density; it is approximately ji = −

Di je De

(4.71)

with De , Di the diffusion coefficients. To meet the condition for ionization, the thickness of the dark space must be sufficiently large to generate the ion current according to Eq. (4.71), i. e. to accelerate the electrons to energies which are somewhat larger than the ionization potential Ekin ≥ EIon : da

ji =

0

α je dx = −

Di je . De

(4.72)

Since the electron current density is almost constant, we obtain da Di = α dx. De 0

(4.73)

4.9 Hollow cathode discharge

97

According to this condition, we can calculate a value for pd1 (invariant in similar discharges) of ≈ 10−4 mbar cm (0.1 mTorr cm) in helium, which is considerably smaller than the thickness of the anodic dark space; da has been calculated by Ingold to 3 Torr cm employing the mobility-limited Child’s equation [143]. Further reduction of the electrode distance at constant discharge current will reduce the height of the anode fall, which can eventually become negative if the anode reaches the negative glow, because ions and electrons diffuse to the anode meeting the requirement for neutrality of charges [144, 145]. In this case, the potential of the negative glow is the topmost value of the discharge—a normal case in plasma systems for technological applications (Fig. 3.5). Further reduction to values smaller than the thickness of cathodic dark space gives rise to a steep increase of the potential because ionization at the same current density is evidently made more difficult. This is a further hint at the importance of the negative glow to sustain the glow discharge [a similar phenomenon can be obtained by reducing the discharge pressure (similarity rules)]. We speak of an obstructed glow discharge [130].

4.9 Hollow cathode discharge Other obstructed discharges are the spray discharge and the hollow cathode discharge, which can increasingly be found not only in ion plating systems but also in ion beam systems [146]. The electrons which are ejected from the cathode exhibit a beam-like character in directions normal to the cathode. Sharp bending of a cathode that either end will migrate close to its counterpart gives rise to a melting of their glows (the anode can degenerate to a ring of larger diameter). If the acute angle has reached a certain lower limit, the electrons which are emitted from the first cathode into the mutually glowing zone can be reflected from the second cathode (pendulum effect). Ideally, this angle should reach 180◦ . For constant potential, current density and intensity of the glow will increase from this threshold (pa ≈ 4/3 Torr cm) [147]. For small values for pa (with a the distance between the electrodes), the current density in hollow cathode discharges can far surpass the value observed in normal discharges up to factor of 1 000 (Fig. 4.11); the functional relationship can be approximated to j/p2 ∝ 1/(pa)5/2 in the range a ≈ d [129]. This increase in current density is referred to the dramatic reduction of diffusion to the walls with subsequent loss for charged particles (electrons and ions) and excited particles. Shortening the distance between the two electrodes gives rise to a decrease of the thicknesses of the dark space, which leads, in turn, to a rising field strength at constant cathode fall Φc and an increasing ion density. Furthermore, since the radiation of the discharge will become more intense, the electron yield will rise due to the photoeffect. Instead of the application of two

98

4 DC discharges

VC =

1000 He 250 V

He 400 V

100 j/jn

H2 300 V

10

Ar 300 V

1 10-1

Fig. 4.11. Hollow cathode discharge after [148]. The reduced current density j/jn is plotted vs. the reduced distance between the two electrodes for several gases and cathode falls. The material of the cathode c Oxford University was iron ( Press).

1 pa [Torr cm]

plane-parallel plates, in most technological cases a cylindrical hollow cathode is used. A rise in glowing intensity can be observed in normal reactors at small diameter welded flanges; we speak of a special hollow cathode effect only if the reduced length p a exceeds values of 1 Pa m. At a pressure of 375 mTorr (50 Pa), this means a gap between the electrodes of 2 cm.

4.10 Similarity laws As has become clear from the preceding sections, the exact description of a discharge is difficult and extensive because of the mutual dependencies of the plasma properties. The oldest systematization was presented by de la Rue and M¨ uller [149] who first pointed out that the breakdown voltage Vb does not change significantly if the product pd is held constant (with p the pressure and d the gap between two plane-parallel plates, cf. Sect. 4.2). Since number density n scales with pressure p for constant temperature, but the mean free path λe scales with inverted pressure, we find that pd is proportional to the number density between the electrodes. However, the energy which is gained by an electron between two collisions is proportional to 1/pd; hence, the generation rate will remain constant: Vb ∝ pd, which is the Paschen law. This has been investigated most thoroughly by Steenbeck and can be formulated as follows [150]: Discharges are denoted as similar when potential and current are equal at similar positions, and all the linear dimensions will only vary by a factor of a. Assume two discharges of the same gas (same electrode material), in two reactors which are distinguished—or similar—by a factor of a only in their linear dimensions. Provided that

4.10 Similarity laws

99 T1 = T2 ; V1 = V2 ; u1 = u2 ,

(4.74)

in the case of a Maxwellian distribution, we further have < Ekin,1 > =< Ekin,2 >

(4.75)

d1 = a d2 ∧ r1 = a r2

(4.76)

and

which yields λ1 = a λ 2 ⇒ p 1 =

p2 E2 E2 E1 = , ; E1 = ⇒ a a p1 p2

(4.77)

i. e. E/p is an invariant property. Moreover, from E1 = E2 /a, we can derive with Gauß’s law (E = σ/ε0 ): σ1 = σ2 /a ∧ ρ1 = ρ2 /a2 .

(4.78)

Simplified, we can further state I1 = ρ1 vA ⇒ I2 = ρ2 /a2 v a2 A2 = ρ2 vA2 : I1 = I2 4

(4.79)

Inserting this in Eqs. (4.77), j/p2 evolves as an invariant property as well as E/p: j1 = I/A1 ∧ j2 = I/A2 ∧ p1 = p2 /a ∧ p21 = p22 /a2 ⇒ I Ia2 j1 I j2 j1 = = ∧ 2 = = 2. 2 2 2 2 2 p1 A1 p1 a A2 p2 p1 A2 p2 p2

(4.80)

As we have seen in this chapter, it has become common to plot functional relationships against the properties E/p, E/d or j/p2 which are invariant against alterations of the reactor geometry. Deviations from the similarity laws or rules hint at complicated processes. For normal discharges, j/p2 is constant over a wide range. For abnormal discharges, j and Vc will rise which can be approximated most simply by j/p2 = aV b . The most important reason for this behavior is the heating of the cathode by which the gas is heated as well. If the pressure is not corrected by this increase in temperature, the gas density will decline. If this effect is not taken into account, we calculate values for j/p2 which are far too low. A similar argument holds true for the invariant pd. For example, Druyvesteyn and Penning could show that in a discharge through hydrogen a variation of dc by a factor of 100 lead to a variation of pdc by a factor of only 2 [151]. In abnormal discharges, V and j/p2 are connected via a complicated function [152]; the ionization via metastable atoms can also be detected by deviations from this rule. 4

More precise is j = ρi vi + ρe ve .

100

4 DC discharges

8

v [106 cm/sec]

6 hydrogen

4 2 0 0.0

0.5

1.0

1.5

Fig. 4.12. In discharges through hydrogen, the drift velocity of the electrons linearly increases with the c J. Wiley ratio E/p [153] ( & Sons, Inc.).

E/p [mV/(Torr cm)]

For some gases, the similarity rules are followed with an astonishing precision. A linear functional relationship exists between the mean molecular velocity of hydrogen, helium and a mixture between helium (He) and mercury (Hg), socalled Heg gas, and E/p (Fig. 4.12, cf. Sect. 2.3). The physical meaning behind this similarity rule is the fact that E/p determines the mean energy of the electrons. The mean energy < εe > which is gained by the electron between two collisions is < εe >= e0 Eλ,

(4.81)

during a collision, the mean energy Δε = L (< εe > − < εg >)

(4.82)

with L the Langevin’s energy loss parameter L=

2me mi + me

(4.83)

and < εe > and < εg > the mean enery of the electrons and the gas molecules, respectively, will be transferred. Equating (4.81) and (4.82) yields e0 Eλ = L (< εe > − < εg >)

(4.84)

or < εe > − < εg >≈< εe >=

e0 Eλ , L

(4.85)

E . pσ

(4.86)

eventually, we obtain with σ the cross section < εe >∝ Eλ∨ < εe >∝

4.11 Conclusion

101

Within certain limits, the energy which is transferred to the electron scales with E/p. In a similar sense, j/p2 behaves invariant as well, j = σE with σ the electric conductivity which is, according to the model of free electrons, proportional to the electron density which, in turn, scales with the number density n; the free parameter is τ , the time between two collisions.

4.11 Conclusion • In a DC discharge, a DC current flows across the electrodes through the plasma, electrons are accelerated to the anode, ions to the cathode, respectively. • It is therefore impossible to apply isolating substrates as electrodes. • The discharge is ignited after having reached a certain ratio of electric field to number density. The required voltage is denoted breakthrough voltage. • Once the discharge is stable, currents flow through the discharge, and an avalanche is generated which serves for a self-sustained discharge, provided the operating voltage is maintained. This voltage is significantly lower than the voltage for breakthrough because the mobilities of the carriers (in an electropositive discharge: Positive ions and negative electrons) are mutually coupled. • The region for power transfer is the cathodic dark space, the region of carrier generation is the negative glow. • For maintenance of the discharge, the processes at the cathode are of extreme importance (during γ-processes, ions incident on the electrode with high a kinetic energy, often some keV, liberate γ-electrons from the electrode which take part of the avalanche for carrier generation). • Due to the high mobility of the carriers, the potentials of the electrodes are screened by charges of unlike type forming a space charge around the electrodes. The existence of this space charge suppresses a continuous growth of the elecric field: The zone of steep increase is confined to a thickness of several centimeters at maximum (space charge limited current).

5 High-frequency discharges I

The phenomenological difference in breakdown behavior between DC discharges and HF driven plasmas is used to discuss the processes of carrier gain, predominantly Ohmic heating, and carrier loss (diffusion) which are required for the understanding of HF breakdown. The conditions for equilibrium of gain and loss in the plasma bulk are extensively discussed. Next, power coupling is investigated from an electrical viewpoint on two levels. The chapter is finished with a discussion of matching networks and waveguides for the different methods of coupling.

5.1 Phenomenological introduction In a DC discharge, an electrode which is covered by an electrically isolating plate will be charged to the weakly negative floating potential Φf ; the fluxes of both types of carriers (electrons and ions) become equal in magnitude, irrespective of whether the potential is applied at the rear side of the insulator (cf. Sect. 3.4). At its surface, ions and electrons will be terminated or will recombine, and there is no need to draw off an electric current (which would be impossible). At plasma densities of 1010 cm−3 , a voltage of 10 − 20 V develops across the sheath. The sheaths represent capacitancies which can pile up charges. Capacitance is defined as C = Q/V ; since Q ∝ V , and it takes a certain time to charge up the capacitancies (Q = Idt), the voltage does not alter instantaneously, i. e. without any time of retardation. After ignition, both sides of the insulator drop to the (negative) cathode potential. During bombardment with positively charged ions, the potential becomes more positive (less negative) because electrons are consumed to neutralize the ions: The potential at the surface facing the plasma gradually reaches Φf , which is sufficient to generate an ion bombardement which can clear off the surface from weakly bounded contaminations. However, the energy of the ions incident on the surface is too low to make substantial sputtering feasible [154]. To solve this problem, we can use alternating current (AC). During the positive part of the cycle, the negative charging of the electrode will be removed by ion bombardement. The frequency which is required for successful neutralization can be roughly estimated as follows, provided the electrode current is assumed to remain constant (in fact, it will diminish): C = Q/V = I t/V, ⇒ t = C V /I. G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 5,

103

104


p [mTorr]

1

Fig. 5.1. For rising operating frequency, the pressure which is required to ignite a plasma steeply drops up to some hundreds of kHz to settle down for even higher frequencies [155] c Elsevier). (

10-1

10-2

100

1000 f [kHz]

5 4 S [W/cm2]

14 MHz 7.5 MHz

3 2

3.7 MHz

Fig. 5.2. For rising operating frequency, the efficiency of power coupling increases [157].

1 0 1

10

100

1000

p [mTorr]

For a quartz 3 mm in thickness, its specific capacitance amounts to about 1 pF cm−2 ; for V = 1 000 V and j≈ ≈ 1 mA cm−2 (the current density is estimated from DC measurements and sputtering rates) we calculate a rise time of about 1 μsec. Since the electrode does not charge up instantaneously because the current does not remain constant but drops with falling potential, AC discharges with isolating cathodes can already be operated at frequencies beyond some ten kHz. Koenig and Maissel were the first to demonstrate in 1970 that plasmas can be ignited with rising frequency [156]. Norstrøm extended these investigations [157] (Fig. 5.1). It turned out that the coupling of RF energy close to the ignition voltage is almost independent of pressure but rises steeply for larger pressures. The pressure dependence of power coupling is larger for higher frequencies which indicates further ionizations or even the onset of additional ionization mechanisms (Fig. 5.2). However, the slope of the I(V ) curves is almost the same

5.2 Generation of carriers

105

from which we infer similar electron temperatures (ln je = const − e0 V /kB Te ). This effect can be detected up to some MHz, from which we further conclude that the discharge is sustained by electrons which are not generated by electrode processes (for more details see Sects. 6.1 and 14.4)!

5.2 Generation of carriers To explain this fact, we consider the movement of a free, undamped electron which is accelerated by an oscillating electric field. According to Eqs. (5.1) and (5.2) (a property which is multiplied by “i” exhibits a phase shift of π/2 : eiπ/2 = cos π/2 + i sin π/2 = i), the velocity exhibits a shift of π/2, and the amplitude a shift of π against the exerting force: me

due = −e0 E 0 eiωt dt

(5.1)

e0 1 e0 E 0 eiωt . (5.2) E 0 eiωt ∧ xe = m iω mω 2 On the time average, the power absorption is determined by the integral over 2π iωt 0 e dt = 0: Neglecting the losses by radiation damping, the electron cannot absorb any energy! Hence, another mechanism must be responsible for power absorption. This is suggested also by the fact that the maximum amount of energy that can be absorbed by a carrier within one half of a microwave cycle (2.45 GHz), is less than 0.1 eV at the measured field for breakdown of 100 V/cm. This is more than two orders of magnitude less than the ionization potential of argon [Eion (Ar) = 15.76 eV, Table 5.1 and Fig. 5.3]. ⇒ ue = −

Table 5.1. Velocity, kinetic energy, and maximum amplitude of an electron which oscillates freely in an HF field at the operating frequencies of 13.56 MHz and 2.45 GHz. ν 13.56 MHz

2.45 GHz

E [V/cm] 1 10 100 10 100 1000

v [cm/sec] 2.07 × 107 2.07 × 108 2.07 × 109 1.14 × 106 1.14 × 107 1.14 × 108

Ekin [eV] 0.12 12.2 1217 3.6 × 10−4 3.6 × 10−2 3.6

ψ0 [cm] 2.42 24.2 242 7.4 × 10−5 7.4 × 10−4 7.4 × 10−3

We have seen that an electric field exerts an accelerating force on an electron as long as it collides with a second particle. During the impact, the field drift is destroyed and kinetic energy is transferred. This amount is negligible for an

106


elastic collision with a heavy gas constituent. After this incident, the electron will be accelerated again, but on a higher energy level. By this repeatable process, the electron can accumulate a tremendous amount of energy until it is by a wall collision. Hence, the motion of the electron consists of a large random component and a small drift component. As we have extensively discussed (similarity rules, Sect. 4.10), the energy transferred to the electron scales with E/p with E the electric field and p the discharge pressure. Although the operating frequency in an RF discharge is typically lower by a factor of 10 to 50 compared to a microwave discharge, the pressure is lower by about the same factor. Hence, the picture is similar: We observe many collisions during one oscillation period [at a pressure of 100 mTorr (13 Pa), νm = 9 × 108 sec−1 for 12 eV electrons] and a so-called random phase movement. The elastic collisions between electrons and neutral molecules are mandatory for this mechanism of energy transfer to occur. The (directed) energy which is piled up by the electrons during the drift motion exerted by the electric field is transformed into a random phase motion. Albeit in the two half-waves the electron is accelerated to and fro the electrode, the electron can gain energy in both parts of the AC cycle. In a mathematical sense, the absorbed energy scales with the field squared, and it becomes independent of the sign: Ohmic heating, [Eqs. (5.4) and (5.9)]. For the absorption of power and the increase in conductivity, the ratio ω/νm with ω the operating angular frequency and νm the frequency of elastic collisions and σm the cross section of elastic scattering of electrons with neutrals (m for momentum) or the cross section of momentum transfer νm = σm ue ne

(5.3)

is of paramount importance. This ratio expresses how rapidly a concerted electron movement will be damped by elastic collisions with neutrals. The energy gain of an electron amounts to 1 (5.4) Pabs = − e0 E 0 · ue 2 with ue the drift velocity of the electron, which can be derived from the second Newtonian axiom for harmonic distortion:

me

due + νm ue = me (iωue + νm ue ) = −e0 E 0 eiωt dt

(5.5)

yielding e0 1 E 0 eiωt . (5.6) m iω + νm We define the AC mobility as connecting factor between drift velocity and electric field ue = −


107

e0 1 , (5.7) me iω + νm which differs from the DC mobility by the imaginary summand iω in the denominator by which electron inertia a/(due /dt) is accounted for. For sufficiently low frequencies, this term can be neglected against the damping term mνm ue by which the dissipative losses by elastic collisions are taken into consideration; in this range, μ is real and constant.1 Hence, the AC mobility is a complex scalar function, and its real part describes the energy transfer. This can be readily understood by splitting the function into its real and imaginary part, respectively: μAC = −

e0 iω νm μ=− − 2 ; (5.8.1) 2 2 2 m e ω + νm ω + νm e0 ω νm e0 ; (μ) = − ; (5.8.2) (μ) = − 2 2 2 2 m e ω + νm m e ω + νm and inserting into Eq. (5.4). For the power absorption of N electrons in the volume V (n = N/V ), we obtain for one HF cycle 1T P j · E dt = V T 0 e 1 1 P = ne0 ue · E ⇒ ne0 μAC E 2 , V 2 2 which yields, taking the real part of μ [Eq. (5.8)]:

(5.9.1) (5.9.2)

P νm 2 E02 νm 2 E02 ne20 = = σDC 2 , (5.9.3) 2 2 V mνm νm + ω 2 νm + ω 2 2 with σDC the DC conductivity, and further provided that νm is independent of the electron velocity (this holds true only for hydrogen and √ helium and approximately for the high-energy tail of the EEDF), E = E0 / 2 denotes the RMS field. From Eq. (5.9.3), we see that the ratio ω/νm , which is in fact the ratio of the imaginary part over the real part (μ)/(μ), mainly determines the efficiency of power coupling into the discharge. • For small values of ω/νm (low operating frequency ω and/or high pressure causing high νm ), the mean free path of the electrons decreases, and the power gain per one mean free path decreases with the mean free path squared. Vice versa, 1 Free electrons would exhibit a purely imaginary mobility (and because σ = e0 nμ a purely imaginary conductivity as well) due to the phase shift of 90◦ = 1/2 π against the exciting electric field.—To compare with metals: Even against microwaves, the conductivity of metals remains almost completely real, i. e. field and conduction current are in phase. For example, for copper, n = 8 × 1022 electrons/cm3 ; σ = 5 × 1017 sec−1 = 5, 5 × 107 Ω−1 cm−1 and νm —it is denoted as damping constant g or γ—about 3 × 1013 Hz. However, appreciable deviations are observed in the IR or VIS range which occur in the tenuous plasmas treated here some orders of magnitude lower.

108


• with rising operating frequency f = ω/2π, the amplitude of the electric field must be increased to keep the power input constant (cf. Sect. 6.5). Hence, with constant power input and fixed electron density ne , the energy transferred to the electrons would decrease with rising operating frequency −1 rather than ω −1 (Fig. [158], since the time of electron acceleration is νm 5.3). • Dropping the pressure, νm ω, causes the mean free path to grow, and the absorbed power can be approximated by ne20 P ≈ V 2me νm

νm ω

2

E02 :

(5.10)

Higher plasma densities or higher electric fields of some kV are required to maintain the discharge, and below a certain pressure (number density) threshold, the discharge will extinguish. For vanishing collision frequency νm , the absorbed power equals zero.

0 0.8

10

20

p [mTorr] 30 40

50

60

0 0.8

p [mTorr] 2000 3000

4000

5000

0.6

0.4

Eeff /E 0

0.6 Eeff/E 0

1000

hydrogen ½

w = nm

0.2

E 0/2 13.56 MHz 27.12 MHz

0.4

w= nm

hydrogen

0.2

E 0/2½ 2.45 GHz

0.0

0.0 0

50

100

150 200 250 nm [10 6 sec-1]

300

0

350

5

10

15

20

25

30

nm [10 9 sec-1]

Fig. 5.3. In the case of RF discharges through hydrogen, driven at f = 13.56 MHz, the collision frequency νm equals the angular frequency at 85 MHz, in the case of MW discharges, νm equals the angular operating frequency at 15 GHz. In both cases, the effective field is considerably smaller than the RMS field. For vanishing collision frequency, the effective field becomes zero: No power transfer takes place.

These effects are often taken into account by defining an effective field by 2 = Eeff

2 νm E02 ; 2 + ω2 2 νm

(5.11.1)

this field would transfer the same amount of energy as a steady field E0 , and the denominator effectively averages over these two frequencies. Writing


109 2 Eeff =

1 E02 , ω2 2 1 + ν2

(5.11.2)

m

we readily see that the more efficient coupling with rising operating frequency is attained at the expense of the intensity of the oscillating electric field which will in all cases be smaller than the steady field E0 . For νm ω, i. e. for high discharge pressures, the effective field will √eventually turn into the RMS field √ E0 / 2, for νm ω, we obtain Eeff ≈ E0 / 2 νm /ω (Figs. 5.3 and 5.4 showing two different presentations). The evenness ω = νm marks the lower limit for efficient power transfer. In RF driven discharges, the effective field is reduced to only a few percent at 10 mTorr in argon, which increases to some ten percent at 100 mTorr. In microwave discharges, the pressure should exceed 1 000 mTorr for an effective discharge. Striking the discharge becomes difficult below a certain pressure threshold. 100

100

10 Torr 100 mTorr

10-2

argon

Eeff /E0

Eeff/E 0

10-1

13.56 MHz 40.68 MHz

1000 mTorr

10-2

10 mTorr

100 mTorr

argon 2.45 GHz

10-3 0

2

4

6 Ekin [eV]

8

10

10-4 0

2

4

6 Ekin [eV]

8

10

Fig. 5.4. In the case of discharges through argon, driven at two RF frequencies at 13.56 and 40.68 MHz (LHS) and at the MW freqency of 2.45 GHz (RHS), the relative effective field is plotted vs. the mean energy of the electrons. At 10 mTorr, the effective field is reduced to only a few percent, which makes ignition more difficult and eventually prevents striking the discharge; and also MW discharges should be operated at pressures above 1 Torr.

By this derivation, it has been shown that the power transfer per HF cycle depends on the ratio (μ)/(μ). For fixed pressure (fixed number density), the efficiency of energy input should decline with rising frequency. However, this approach does not take into account an important fact: With rising frequency, the efficiency of ionization steeply increases, and this effect more than makes up for the decrease in power transfer. The mechanism of Ohmic heating is the result of interaction between an electron, the force which is exerted by an electric field, and neutrals which collide with the accelerated electron. This mechanism is mainly active in the plasma bulk provided there is a considerable electromagnetic field. But this can happen

110


only at low plasma densities or at operating frequencies which exceed the plasma frequency of the electrons, which is the case in low-density microwave-driven discharges. These discharges are driven “electrodeless” (they can be operated through a so-called “microwave window”, mainly made of quartz) and thus have very low plasma potential. This, in turn, causes the formation of very thin plasma sheaths. In the theory of continua which is extensively dealt with in Sect. 14.6, we see that this absorption of power can be expressed by a dielectric constant which is larger than zero but less than unity [eq. (14.171)]. For rising plasma density, however, we enter the regime of evanescence, which is characterized by a negative dielectric constant (ωp,e > ω = 2πf , this will happen for an operating frequency of 2.45 GHz at 1.54×1010 Hz or a plasma density of 7.45×1010 cm−3 ). In the high-pressure regime (νm > ω), the plasma impedance shows a complex behavior between inductive and capacitive characteristics, in the low-pressure regime (νm < ω), the impedance is inductive (positive imaginary). As main phenomenological result, the electric field of the incoming wave is damped to 1/e within the skin depth, but this volume is not static as in the DC case but instead, it is vigorously moving. We speak of the creation of electron-deficient breathing sheaths, and Ohmic heating in the plasma becomes less effective. These sheaths are responsible for three effects, displacement current heating, Ohmic heating, and stochastic heating. As they are mainly associated with capacitively coupled plasmas, they are discussed in Chap. 6.

5.3 Operating frequency and the EEDF At the end of the last section, we pointed out that this approach does not take into account the frequency dependence of the different excitations. What we are interested in is high an ionization efficiency but not a waste of energy. It was shown by the groups around Moisan and Wertheimer that in fact it is the efficiency of ionization which increases with rising frequency, albeit the power input declines. This can happen because the electron energy distribution function EEDF will change its slope with increasing frequency (see also Sect. 14.1). In particular, the convex shaped EEDF exhibiting a typical Druyvesteynian characteristic with an extremely low portion of high-energy electrons beyond 15 eV which is characteristic for the DC case gradually reinforces this part of the EEDF which is so important for ionization (cf. Sect. 4.6). This is shown in Fig. 5.5 for same mean electron energy of 3.5 eV and rising operating frequency from DC (A) to microwaves (D) [159, 160]. This adjustment of the EEDF, however, is self-limiting, since for high plasma densities, Coulombic collisions between electrons become more probable and eventually become dominant, and every distribution function transforms to a Maxwellian distribution. The distribu√ tion functions are normalized to 0∞ f (E) E dE = 1.

5.4 Loss mechanisms

111

1

f(E)

10-1

DC

DC

10-2

0 MB

1.25

10-3 10-4

2 DC

0


Fig. 5.5. Keeping the total energy constant, the portion of higher-energetic electrons increases with rising operating frequency. Mean energy < E > = 3.5 eV; various νm /ω-ratios: A: → ∞ (DC); B: 2; C: 1.25; D: 0 (microwave plasma). (pa = 0.15 Torr cm (20 Pa cm), after [159, 160].

15

Since the total absorbed power at constant discharge pressure (number density, which means constant νm ) decreases with rising operating frequency, especially in microwave plasmas, the plasma density is larger than the other types operated at DC or RF for the same absorbed power, Moisan and Wertheimer concluded that less energy is spent in atomic excitation. But this means: Since the generation of carriers depends on the operating frequency, also within the same band (e. g. the RF band between 1 and 100 MHz), the plasma density can be increased by raising the operating frequency. This was first pointed out by Surendra and Graves [161] and is the basic principle of dual-frequency operation (Sect. 6.7).

5.4 Loss mechanisms We have already identified two mechanisms of electron loss, diffusion and attachment. 5.4.1 Diffusion The loss mechanism of diffusion is of paramount importance in the low-pressure regime (cf. Sect. 4.7). However, this mechanism is also modified in HF discharges and has to be analyzed precisely. We have noted that the diffusion coefficients of electrons and ions are mutually coupled after ignition of the discharge. As a first result, the electric field required for maintenance is significantly reduced compared to the breakdown field. In discharges of electropositive gases, the ambipolar diffusion coefficient Da happens to be smaller by orders of magnitude compared with that of the electrons (Fig. 5.6). The solutions of the Helmholtz equation [139]

112

5 High-frequency discharges I 7

10

-1

nion/p [s Torr ]

106

-1

105

Fig. 5.6. Field dependence of the diffusion coefficients in an RF discharge (upper value: breakdown, lower value: sustaining) after [58].

104 3

10

2

10

0

10

20 30 40 E/p [V/Torr cm]

50

60

d2 n νion + n = 0, dx2 Da

(5.12)

simplified for a one-dimensional problem, are set by the boundary condition of vanishing plasma density at the wall, which is justified by the fact that the diffusion constant scales inversely with pressure. Since no odd function can be a solution, Eq. (5.12) is simply solved by the cosine function:

n = n0 cos

νion x Da

(5.13)

with the boundary condition n = 0 at x = ±1/2 L the gap between the electrodes. This represents the solution for the dominant (lowermost) diffusion mode. Higher diffusion modes are solved by the higher harmonics of the cosine function. Moreover, this yields for the ionization frequency and the number density:

νion =

π 2 Da νion π ∧ = 2 L Da L

2

=

1 , Λ2

(5.14)

x . (5.15) Λ with Λ the diffusion length of the electrons. Inspecting Eqs. (5.13) − (5.15), we remark that the constant in the spatial-dependent part, 1/Λ, equals the ratio of the two complicated functions D and νion . Whereas the former depends on gaseous pressure (in fact: on number density and gas temperature), the latter depends exponentially on the temperature of the electrons. To obtain a constant result, the two dependencies must cancel exactly. Equation (5.12) reads, in cylindrical coordinates taking the boundary conditions into account (electron density and its derivations vanish at the walls and at the electrodes): n = n0 eνion t cos

5.4 Loss mechanisms

113

d2 n 1 dn νion +n + =0 2 dr r dr Da

(5.16)

which yields

n = n0 e

νion t

J0

νion r. Da

(5.17)

To have n vanish at the wall rmax = R, we demand that R νion /Da = 2.405, which opens the possibility to determine the ionization frequency νion . This represents the solution for the dominant (lowermost) diffusion mode. Higher diffusion modes with several maxima will be described by Bessel function of higher order [162]. Inserting Eq. (5.17) into the diffusion equation yields for a sustaining discharge

νion 2.405 1 = = Λ2 Da R

2

+

π L

2

(5.18)

with R the radius of the cylinder and L the distance between the electrodes. The diffusion to the walls is described by the first part of the RHS, the diffusion to the electrodes is described by the second part of the RHS of Eq. (5.18). For a uniform electric field, breakdown will occur when the losses by diffusion are at least counterbalanced by ionizations within the plasma, i. e. when νion = Da /Λ2 or Λ2 =

Δn . n

(5.19)

5.4.2 Recombination Whereas diffusion losses are heterogeneous reactions which happen at surfaces, other loss mechanisms take place in the gaseous phase of the plasma. In either case (recombination or attachment), a bimolecular reaction is required. With rising pressure, recombination processes become more likely to occur, and the condition for loss has to be extended by a term which takes the diminishing of carriers due to recombination into account. Since this must be a reaction of second order, Eq. (5.12) has to be extended by a term which is of second order in the ion density: d2 n νion + n − kr n2i = 0 : dx2 Da

(5.20)

The negative sign is because of carrier loss, ni squared because we assume an electropositive plasma with electron density and ion density equal in magnitude. From this equation, we see that recombination processes are likely to occur only at high densities where they readily become the dominating loss mechanism.

114


5.4.3 Attachment in electronegative gases In plasmas with electronegative molecules, we observe strong deviations from this conduct. A simple mass balance which takes into account generation (ionization) and loss (electron attachment), both considered as bimolecular reactions under general consideration of losses by diffusion [Eqs. (5.15) and (5.18)], can be described as follows: Da ne = kion ne nn − ka ne nn (5.21) Λ2 with kion and ka the rate coefficients of ionization and electron detachment, respectively, ne and nn the densities of electrons and neutrals, Da the ambipolar diffusion coefficient and Λ the diffusion length (approximate distance between the electrodes). This equation can be simplified to kion − ka =

Da , nn Λ2

(5.22)

provided the mobilities of positive and negative ions are considered equal and are small compared with that of the electrons. Mobility and diffusion coefficient are joint via the Einsteinian relation 2 D = < εk > μ 3

(5.23.1)

where < εk > denotes the averaged electron energy which is known in this connection as characteristic energy [58] and can be expressed for a Maxwellian distribution as D = kB Te μ

(5.23.2)

σDC . ne0

(5.24)

1 + 2α De 1 + 2α + μe /μIon

(5.25)

with μ= Da can be expressed by [163]: Da =

with α = nn /ne (nn : density of the negative ions). Of course, α is zero for argon, but becomes about 100 for SF6 : Hence, the nature of discharges through electronegative gases (i. e. Da ) can be determined predominantly by the density of the negative ions which can far surpass the density of the electrons appreciably, sometimes, by some orders of magnitude [56]. In discharges through electropositive or inert gases, we observe equilibrium between gain (ionization) and loss (ambipolar diffusion). Therefore, the electron

5.4 Loss mechanisms

115

density can be described with cosine functions or Bessel functions (maximum in the middle of the reactor, zero at the walls). In discharges through strongly electronegative gases, equilibrium is achieved by electron detachment, and diffusion can be neglected, and to first order, the electron density becomes spatially independent. From equation kion − ka =

Da , nn Λ2

(5.26)

we can further extract that reducing the discharge pressure (i. e. diminishing the particle density nn ) will lead to a rise in Da since μ scales inversely with the density of the neutrals. • In discharges of electropositive molecules (kion ka ), this means an increase of kion , which can be only realized for rising mean energy of the electrons (electron temperature, cf. Sect. 3.5). • In discharges of electronegative gases, ka is of comparable magnitude of kion ; kion and ka will change in the same sense, and the electron temperature will change considerably less. Additionally, in this equation, neither the densities of the charged carriers neither their energies (temperatures) will directly appear (the densities are hidden in the ambipolar diffusion coefficient); and to first order, these properties are independent of the coupled power. Within certain limits, Te depends only on gas pressure and on the ratio Te /Ti . 5.4.4 Decay The generation of carriers by electronic impact can be described by the ionization rate νion which depends exponentially on the electron temperature. When the power is turned off, the decay rate of the carriers is determined by the ratio of the Bohm velocity over the critical dimensions of the reactor (real volume over effective surface). The Bohm velocity scales with the square root of Te , and the time-dependent part of the continuity equation becomes ∂ ln ne (5.27) = νloss . ∂t The electron density is expected to decay exponentially after having switched off the external power (“afterglow”, Fig. 5.7), which is the main advantage of pulsed plasmas: The mutual coupling of the diffusion coefficients will be lost, the electrons evade the plasma bulk which, in turn, causes an enrichment of positive ions. When the next HF cycle starts by plasma ignition, the reactions at the substrate surface can be tremendously different from the behavior associated during “normal” operation (Chaps. 11 + 12). −

116


n e [10 9cm -3]

3

Fig. 5.7. The electron density in a cavity resonator exponentially decays after having switched off the HF power c J. Wiley & Sons, [164] ( Inc.).

1

2

4

6 8 t [msec]

10

12

5.5 Breakdown Having discussed extensively the mechanisms for charge gain and charge loss, we want to tackle the question of breakdown in HF discharges, when the electric field is still uniform throughout the plasma volume, for example for the operating frequency f = 13.56 MHz (λ ≈ 22 m). A discharge can be ignited when the electron losses by diffusion, recombination or electron attachment can be compensated by ionization mechanisms. If the losses are solely due to diffusion (cf. Sects. 4.7 and 5.4), the diffusion current can be determined by ∂ρ (5.28) ≥ ne0 νion − ∇ · j ∂t with νion the frequency of ionization, and ρ the charge density. Typical for a mechanism which is controlled by diffusion is the dependence of the breakdown voltages on pressure with distinct minima. We can readily understand this conduct by deriving Eq. (5.9.3) yielding 2 ne2 E 2 ν 2 + ω 2 − 2νm ∂P = 0 0 m 2 ∂νm me 2 (νm + ω 2 )2

(5.29)

which peaks at ω = νm (Fig. 5.8). We can readily identify two limiting cases. 2 ω 2 (many collisions during one oscilla1. High discharge pressure νm tion period): The energy which is transferred from the field to the electron will be dissipated into the plasma by elastic collisions between electrons and neutral molecules, the energy loss per collision Δε is defined via Langevin’s energy loss parameter 2me /(mi + me ), approximately 2me /mi :

power coupling efficiency [a.u.]

5.5 Breakdown

117

w = nm

f = 13.56 MHz

0

50

100 nm [106 sec-1]

150

Δε ≈

200

Fig. 5.8. The efficiency of power coupling peaks when the angular operating frequency matches the collision frequency νm .

2 1 e20 Eeff 2e0 me = < εe > | νm | m e ν m mi

(5.30)

with < εe > the mean energy of the electrons [165], which gives for the electric field

Eeff = νm

2me 2 < εe >, e0 mi

(5.31)

the effective field must be linearly increased with rising pressure (νm ∝ p). Therefore, the dependence is similar to the DC case (energy which is piled up is proportional to E/p with E the electric field, cf. Fig. 4.12). 2 ω 2 (many oscillations per collision): Since in this 2. Low gas pressure νm range, λe will increase, the probability of gaining energy from the field will diminish, which will lead to a rise by the amount of the effective field. Simplifying facts by the assumption that all inelastic collisions will cause an ionization,2 the absorbed power becomes Pabs = νion < εion >, which means for the frequency of ionization:

νion =

2 νm Pabs e20 E02 = , 2 + ω2 εion me εIon νm νm

(5.32.1)

Pabs e2 E 2 νm = 0 0 2. εion me εIon ω

(5.32.2)

simplified for νm ω: νion =

2 Again, the model gas is helium with mercury vapor (Heg gas); He exhibits a metastable level at 19.8 eV which has a lifetime of some msec: Almost every collision of a metastable He atom, He∗ , with an Hg atom will lead to an ionization; the effective ionization potential amounts to about 19.8 eV; νm (Hg) then becomes 2.37 × 109 sec−1 (p in Torr).

118

5 High-frequency discharges I The condition for breakdown is given by νion = D/Λ2 . We note that Λ, the diffusion length, solely depends on the geometry of the reactor, and D can be evaluated from kinetic gas theory according to D = λ √ < v > /3 (with λ the mean free path). More precisely, we have to write < v 2 > instead of < v >, this yields νion =

√ λ < v2 > . 3Λ2

(5.32.3)

√ √ Since < v 2 > can be written as < v 2 > = λνm , we eventually obtain for the ratio between the frequencies of collision and ionization: νion λ2 = νm 3Λ2

(5.33.1)

or νion νm =

< v2 > . 3Λ2

(5.33.2)

With < εe >= 1/2 me < ve2 >, the field eventually becomes ω E0 = e0 Λνm

2 εIon < εe >. 3

(5.34)

The minimum of the field can be located where the frequency for momentum transfer νm equals the operating angular frequency ω. The electric field for breakdown scales with • the inverse frequency of elastic collisions νm [and approximately with the inverted discharge pressure (cf. Sect. 14.1)], and with • the operating angular frequency ω (Fig. 5.9). Furthermore, this dependence holds also true for maintenance of the discharge. This causes the time-independent EEDF to vary with the operating frequency [160], and the efficiency of power input becomes dependent on pressure. For example, for microwave discharges through helium (σ ∝ 1/v) the range for largest power input can be confined between 4.5 and 9 Torr (600 and 1 200 Pa), and the maximum can be located at about 6 Torr (900 Pa). The breakdown at high frequencies is solely determined by the (primary) α-ionization. Knowing the ionization coefficient, it is feasible to determine the strength of the electric field required for breakdown. To put in another way, the ionization coefficient α can be evaluated from this breakdown experiment. From the spatial dependent diffusion equation (5.12), it is evident that νion /D is equivalent to the first Townsend’s coefficient α squared (dependent on the gas, its pressure, the electric field, and, additionally, the operating

5.5 Breakdown

119

E [V/cm]

104

103

2 L = 0.159 cm

2 L = 0.475 cm

102

10 -1 10

1

10 p [Torr]

102

103

Fig. 5.9. Microwave breakdown in He/Hg (Heg gas), L: distance between the electrodes after [166]. Raising the pressure and the collision frequency weakens the effective electric field. Hence, the amount of E which is required for breakdown must be increased.

frequency). α describes the growth of the electron current n/n0 as function of the cathodic distance: n = n0 exp(αx), i. e. the number of ionizations per cm which can also expressed by νion < ue > with νion the frequency of ionization and < ue > the drift velocity (< ue > = μE) or

(5.35)

νion . (5.36) μE α, however, refers to an ionization which is caused by a drift in the DC field, whereas νion /D describes a motion in an oscillating HF field, which is significantly smaller than the motion caused by a steady DC field. Alternatively, this can be expressed by an ionization (creation of an electron-ion pair) by an electron that falls across a voltage of 1 V (instead across a distance of 1 cm). Both the coefficients are related by α=

η=

νIon α ⇒ νion = ημE 2 ; [η] = 1/V, [νion /D] = 1/cm2 . = E μE 2

(5.37)

By definition of ζ ζ=

νion 1 , 2 D Eeff

(5.38.1)

the coefficients are related via ζ=η

μ 1 1 = 2 2 . D Λ Eeff

(5.38.2)

120


From the Einsteinian relation, we see that the ratio D/μ measures the mean energy of the electrons and depends on E/p. Therefore, it is possible to determine η from measurements of ζ and vice versa (Fig. 5.10). However, the mean energy of the electrons can be calculated in a different way for the AC case and the DC case, which can lead to difficulties when the mean energies, i. e. the electron temperatures, are compared.

h[ionizations/V]

10-2

Ayers (DC)

Hale (DC)

10-3 Varnerin and Brown (AC)

10-4 10

100 Eeff/p [V Torr-1 cm-1]

Fig. 5.10. Comparison of the coefficients for ionization α and ζ in a discharge through hydrogen after [167].

The increase of the breakdown field with rising pressure occurs for the same reason as in the DC case: λe drops, and the kinetic energy which can be piled up by the electron between two collisions, decreases as well. Therefore, the field must scale inversely to λe , i. e. direct proportional to the discharge pressure. For low pressures, the mechanisms become different. The efficiency of the energy transfer deteriorates, thereby causing an increase of the breakdown field. In the limit of collisionless plasma, no energy can be absorbed at all. Hence, this minimum is no Paschen minimum [168]. In contrast to DC discharges, widening the distance between the electrodes will cause a reduction of the breakdown field: The equilibrium between electron loss and electron generation is achieved more readily (Fig. 5.11). As in DC discharges, the minimum is the result of two competing mechanisms. 5.5.1 Microwave discharges: model for breakdown To shed light on this breakdown problem from another side and to discuss the limits of energy gain, Brown and Bell investigated the influence of all the parameters which are involved in microwave breakdown [58, 169]. Since it is of high heuristic value and conventional and highly sophisticated microwave discharges have regained much interest as plasma sources for ion beam processing (SLAN), it will be presented here.

5.5 Breakdown

121

60

-1

E/p [V Torr cm ]

50

-1

40

Fig. 5.11. Reducing the gap between the electrodes (reduction of Λ) causes higher losses by diffusion, and the effective field which is required for breakdown must rise (H2 , after [58]).

30 20 10 0 0

1

2

3 4 pL [Torr cm]

5

6

7

We will use the property pd (p: gas pressure, d: distance between the electrodes) which behaves as almost invariant against alterations of the dimensions of the reactor [similarity rules (Sec. 4.10)]. As far as the influence of the HF field is concerned, pλ (with λ the wavelength of the operating field) is the wellmatched parameter. The energy is denoted by ε instead of E to avoid confusion with the electric field, and for the mean free path λ, we use l to avoid confusion with the wavelength λ. For all parameters, the influence on the ratio Λ/λ has to be considered. 5.5.1.1 Frequency. As we have stated in the introductory remarks, a breakdown will occur at the state of equilibrium between generation by ionization and loss by diffusion. At low frequencies, d λ: The field across the dimensions of the reactor is uniform, this is the border of uniform field. For high frequencies, however, a threshold is set by the condition d = πΛ ≥ λ/2

(5.39)

with Λ the diffusion length, yielding pλ . (5.40) 2π We note that this restriction comes into play only for microwave discharges where λ ≈ Λ. pλ = 2π(pΛ) ⇒ pΛ =

5.5.1.2 Pressure. Next, we find that the pressure will confine the range of validity of the kinetic gas theory, that is, when the mean free path le becomes equal to the diffusion length Λ. The probability for collision Pm is inversely proportional to the discharge pressure p, and le = Λe , yielding

122

5 High-frequency discharges I pΛ =

1 , Pm

(5.41)

this property does depend on the (mean) energy of the electrons, Te . This defines the border of the mean free path. 5.5.1.3 Oscillation amplitude. If the amplitude of the HF oscillation becomes sufficiently large, the electrons can reach the electrodes at the peak of every HF cycle and are instantaneously annihilated. The amplitude can be inferred from the second Newtonian law: e0 dx d2 x + νm = − E0 eiωt dt2 dt m

(5.42)

yielding x=−

e0 e0 E0 E0 eiωt = eiωt . m iνm ω − ω 2 m −iω(iω + νm )

(5.43)

The amplitude ψ0 of this oscillation is the amount of the complex number e0 E0 mω

−ω iνm − 2 2 2 ω 2 + νm ω + νm

ψ0 =

:

e0 E0 1 , 2 2 mω ω + νm

which can be written with Eqs. (5.11) for the effective field as √ 2e0 Eeff ψ0 = . mωνm

(5.44) (5.45.1)

(5.45.2)

The amplitude cannot exceed half the distance between the electrodes: √ L 2e0 Eeff . (5.46) = 2 mωνm Substituting 1 2πc v ∧ω = ∧ νm = , ple λ le

(5.47)

√ mπc pL Pm v ∨ pλ = 2 . e0 Eeff /p

(5.48)

Pm = we obtain pλ =

Inserting the constant yields (with L = πΛ): pλ = 106

pΛ , Eeff /p

(5.49)

5.5 Breakdown

123

which can be solved for known breakdown field. This is the border of the oscillation amplitude. When entering this regime, all the electrons are lost within one half of a HF cycle by recombination with the wall (electrodes), and the field must be increased tremendously to compensate for these losses (Fig. 5.12). 350 300 400 mTorr

250 E [V/cm]

300 mTorr

200 150 500 mTorr

Fig. 5.12. At the border of oscillation limit, the breakdown field must be increased tremendously (after [170]).

100 50 0 0

50

100 l [cm]

150

200

5.5.1.4 Low gas pressure. Next, there exists the transition from high pressure to low pressure (many collisions per oscillation → many oscillations between two collisions). The borderline can evidently be drawn at √ < v >2 . (5.50) ω = νm = le For hydrogen [νm = 5.9×109 p0 (p in Torr) or 7.89×1011 p0 (p in Pa)], pλ = 4246. This is the border of the transition of the collision frequency. 5.5.1.5 Breakdown. Eventually, we find the optimum border of the breakdown [Eq. (5.34)] with ω = 2πc/λ yielding 2πc E= e0 Λλνm Inserting the values for hydrogen:

2 εIon < εe >. 3

(5.51)

• νm = 7.89 × 1011 p0 [in Pa] • εion = 15.76 eV, assuming < ε >= εion , we obtain pλ = This borderline is drawn in Fig. 5.13.

0.56 . pΛ

(5.52)

124


104

Fig. 5.13. Meeting the borders of stability in a HF discharge through hydrogen, (1) border of the uniform field, (2) border of the mean free path, (3) border of the oscillation amplitude, (4) border of low gaseous pressure, (5) optimum border of breakdown (after [171]).

pL [Pa cm]

5

10

3

10

2

3 1 4

10

2

1 10-1 10

102

103 104 105 pl [Pa cm]

106

107

108

5.5.1.6 Conclusion. For an HF discharge, there exists an island of stability, which is completely confined by diffusion processes with respect to every (primary) parameter. These boundaries encircle this island of stability perfectly for discharges driven by microwaves (300 MHz or higher), whereas for RF discharges, some conditions become obsolete.

5.6 Maintenance 5.6.1 EEDF and the electric field We first note that νion = nkion = nσion < ve >

(5.53)

using σion , the cross section for ionization by electron impact, and < ve >, the mean electron velocity. For athermal plasmas, the mean energy of the electrons is considerably greater than that of the ions. Balancing generation and loss, and remembering that the diffusion coefficients of both carriers are mutually coupled by the ambipolar diffusion coefficient, Da can be expressed approximately as Da ≈ μ i ε k

(5.54)

with μi the ion mobility and εk the so-called “characteristic electron energy” which is obtained via the Einsteinian relation [Eqs. (5.23)] εk =

De . μe

(5.55)

Approximating Da with the diffusion coefficient defined by Eq. (5.54), we eventually obtain

5.6 Maintenance

125

nkion ≈

μi εe kion μi → ≈ : 2 Λ εk n Λ2

(5.56)

Although both quantities on the LHS (n and kion ) depend on the EEDF, its ratio can be expressed as a function of specific constants of the plasma-constituting gas (which determine μi ), its pressure (which mainly determines n) and the reactor geometry (by which Λ is fixed). In particular, this ratio does not depend on the operating frequency. The independent properties which determine the shape of the EEDF adapt to those values to ensure the ratio kion /εk being kept constant. Following Moisan and Wertheimer [160], we can find two limiting cases: • For low plasma densities, the Coulombic collisions between electrons (i. e. the electron density itself) and the stepwise ionization can be neglected in shaping the EEDF, both kion and < εk > remain unaffected and become sole functions of E/n and E/ω. – In particular, the averaged electron energy in ultrahigh frequency plasmas (some hundreds of MHz or microwave) should always by smaller than that in an RF plasma, even for higher intensity of the electric field. – RF discharges follow neither the E/n-dependence (DC discharges) nor the E/ω (microwave) but show a somewhat mixed behavior. Thus Eq. (5.31) defines the magnitude of the discharge maintenance field (which differs from the externally applied field with operating angular frequency ω), and • for sufficiently high plasma densities, the EEDF becomes Maxwellian, the electron temperature is well-defined and is given by kB Te = 2/3 < εe >. For a given operating frequency ω at a certain number density n, the electron temperature Te will determine the electric field required for maintenance. 5.6.2 Collision frequency In a high-frequency discharge, the frequency of elastic collisions between electrons and neutrals, can be obtained by measuring the complex conductivity (which is connected to the mobility via σ = e0 nμ) (Fig. 5.14): νm σr = . σi ω However, two objectives make matters complicated:

(5.57)

126


• In all other gases except hydrogen and helium, the frequency for elastic collisions νm varies with electron energy and consequently, the concept of the effective field becomes questionable, if we are not interested in the high-energy tail of the EEDF. • The EEDF depends complexly on the operating frequency [172]. The power transfer exhibits a maximum at νm = ω which is a manifestation of the frequency dependence of the (time-dependent) EEDF.

10 -1

s [W -1 cm -1]

10 -2

sr sr si si

1 eV

0,05 eV

10 -3

Fig. 5.14. Comparison of the real part and the imaginary part of the electric conductivc ity of a plasma after [173] ( J. Wiley & Sons, Inc.).

10 -4

10 -5

107

10 9

10 11

n e [cm -3]

Therefore, it is worthwhile to get an idea of the deviations from the simple theory. For chlorine, νm rises from 6.3 × 106 sec−1 at about 1 eV to 135 × 106 sec−1 at about 15 eV. The first value is small even against the angular FCC frequency (13.56 MHz) of 85 MHz, and so we expect the characteristic electron energy to depend solely on the square of E0 /ω, whereas the averaged energy of the tail electrons should show some pressure dependence (Sect. 14.1).

5.7 High-frequency coupling: qualitative approach Considering the generation of carriers, the operating frequency should be 1. lower than the plasma frequency to enter the cutoff regime or regime of evanescence: ω < ωp , and should be 2. matched to the discharge pressure: ω = νm . Also to use dielectric electrodes, high operating frequencies would be advantageous. However, inductivities are then severely damped. It is for this reason that the upper operating limit is given by severe grounding problems which are difficult to overcome. Furthermore, below 100 MHz, the reactor can be designed very flexibly; electrodes and coils can be shaped into almost all geometries. The

5.7 High-frequency coupling: qualitative approach

127

most serious disadvantage is the lack of condition (2). That is why the range between 10 and 20 MHz is a good compromise; and in the 1950s, 13.56 MHz was the only frequency in the RF band which was permitted by the Federal Communications Commission (FCC) to avoid interference with the networks used for communication [174]. Non-linear effects, however, cause the generation of numerous strong higher harmonics: The sixth one is located in the VHF band, the seventh and eighth are located in the band used for airspace surveillance. Moreover, there are several frequency-dependent effects which have to be discussed in the following. As a first summary, we have found that the high cathode voltage which is required in a DC discharge to generate secondary electrons can be considerably reduced at the same power input in an HF discharge: • γ-electrons are not necessary any more to sustain the discharge (so-called α-regime [175]). • The discharge pressure can be significantly lowered. As main result, we can get rid of the electrodes as constitutive elements within the discharge, and this discharge is spoken of an electrodeless discharge. The coupling occurs through the dielectric wall of a bell-jar or a window by means of condensor plates (capacitive coupling) or coils (inductive coupling) (Figs. 5.15, cf. Sect. 5.8).

Fig. 5.15. The coupling can happen capacitively, placing the electrodes within the discharge (LHS) or externally (M); alternatively, coupling can occur in an inductive way (electrode always located externally, RHS), here shown for a barrel reactor [176].

Hence, high-frequency discharges can be divided into the following three classes which are characterized by the interaction between the electromagnetic field and the plasma: 1. E-type: Capacitive or voltage coupling between the RF electrodes which are placed within or out of the discharge reactor, respectively: High a capacitance is required between electrode and plasma, and the sheaths resemble the dielectric layer of a capacitor between the two plates, the electrode

128

5 High-frequency discharges I and the plasma. That the electrodes can be placed outside the plasma reactor is a clear manifestation of the fact that emission of γ-electrons is not required for the maintenance of the plasma. The orientation of the exciting electric field is normal to the surfaces of the electrodes, the reactor walls and their sheaths. The degree of ionization depends on the amplitude of the RF voltage. This is the vast area of parallel-plate reactors (Chaps. 6, 10 and 11, Figs. 5.16). With increasing degree of ionization, i. e. larger plasma density, plasma voltage and plasma resistance decrease dramatically, and we observe an abrupt change to the

2. H-type: The time-varying current induces a time-varying magnetic field in the plasma zone which, in turn, generates a second time-varying electric field which is oriented in a parallel direction with respect to the reactor walls and which sustains the plasma. This excitation requires high conductance between the electrode and the plasma. This regime is denoted inductive or current coupling with large RF currents. The electric field loops will close within the plasma which allows an electrodeless operation. Very high plasma densities (≈ 10%) can be achieved. This is the area of the ICP reactors which are now in common use (Chaps. 7, 10 and 11). For types (1) and (2), the wavelength exceeds the dimensions of the reactor appreciably. For example, the corresponding vacuum wavelength of an RF wave with f = 13.56 MHz (ω = 85.20 MHz) amounts to 22.12 m. Hence, we speak of “quasistationary” discharges or steady-state discharges. Albeit the plasma density achieved with the method of inductive coupling is at least higher by one order of magnitude than that obtained with capacitive coupling, it suffers from spatial inhomogenities; therefore, the plasma is applied in the “downstream” mode [95]. RF excitation

RF excitation target + dark space shield

gas shower head

dss

gas shower head

plasma

plasma

plasma

electrode with substrates



dss

dark space shield HF, RF

vacuum system

vacuum system

vacuum system

Fig. 5.16. In a countless number of modified parallel-plate reactors, capacitively coupled plasmas serve to modify surfaces either by diode sputtering (LHS), ion etching (middle) or by plasma etching and deposition (RHS). dss denotes dark space shield.

5.7 High-frequency coupling: qualitative approach

129

3. With further rise of the operating frequency, the wavelengths of the operating field become comparable or even smaller than the dimensions of the plasma apparatus. For example, the corresponding vacuum wavelength for the common microwave frequency of 2.450 GHz is only 12.25 cm. Following our considerations in Sect. 5.5, the topmost efficiency is reached for pressures of about 1500 mTorr (200 Pa) in discharges through argon, but microwave discharges can also be operated at elevated pressures, sometimes even at atmospheric pressure, however, at the expense of spatial homogenity. According to Eqs. (14.170), waves with frequencies exceeding the plasma frequency will propagate through the plasma almost undisturbed, the plasma acts as a low-loss dielectric, and the amount of dissipated power is very small still; for a driving frequency of 2.45 GHz, this will happen at plasma densities below 8 × 1010 cm−3 . Beyond this threshold (cutoff: ω = ωp ), the evanescent or cutoff regime is entered, and the penetration depth of the wave is restricted to a zone which is determined by the skin depth with a thickness a several centimeters [eq. (14.166)], the heating zone. According to this equation, the skin effect is far more pronounced in microwave-driven discharges (cf. also Figs. 14.28). In this frequency regime, plasma generation appyling capacitive or inductive coupling is almost impractible, and instead, waveguides, coaxial cables and antennae are used (cf. Sect. 5.10). In principle, widely spread, simple microwave reactors consist of four parts: the power supply with the waveguiding system (for low powers: coaxial cables, for high powers: waveguides), an applicator for power matching and a circulator to annihilate the reflected power, and the plasma reactor, a geometrically simple device with either cylindrical or rectangular symmetry, constructed of metallic or dielectric walls (Fig. 5.17). magnetron

TE10

dielectric tube with plasma

rectangular waveguide

Fig. 5.17. Simplified schematic of a microwave discharge in a reactor (here: dielectric tube) which is transparent for electromagnetic waves which are generated in a magnetron and guided via a rectangular waveguide. For better coupling, the reactor can be constructed as resonator with a sliding short. The main sources for power loss are waveguide losses, impedance matching, and radiation.

• The power supply itself consists of a microwave source (magnetron or clystron) and an attached waveguide or resonant cavity to allow easy striking of the discharge. Waveguide losses can be minimized

130

5 High-frequency discharges I by highly-reflecting coatings (silver or gold), the impedance matching can be significantly improved by terminating the waveguide with a horn, gradually flared over several wavelengths. This design also enhances the directivity of the radiation pattern. • The applicator, in its simplest form, consists of the microwave window by which the vacuum recipient is sealed against atmosphere. Application of an antenna improves the characteristics of the radiation pattern, however, at the expense of contamination. If the window is fitted with a converging lens, the refraction index of the plasma which is less than unity has to be taken into account (Sect. 14.6). Therefore, they have to be concave to focus. • To prevent local heating due to the above mentioned skin effect behind the microwave window which consists in most cases of quartz, the plasma vessel has to be designed to act as a resonant cavity. Now, a standing wave is generated in the reactor, and the maximum intensity of the field has moved away from the microwave window and can be found somewhere inside the reactor. A combination of both of them has been realized by Engemann et al. with their slotted antenna (SLAN) [177]. The microwave field is coupled into an annular structure via a tuning element (rod antenna) which is located within a conventional rectangular waveguide with perpendicular orientation. To match the varying conditions of the capacitive plasma, not only can this antenna be varied in length but also the rectangular waveguide can be adapted by a sliding short. The microwave power is selectively radiated into the interior of the reactor through slots which are located at the nodes of the electric field. • The circulator serves to annihilate the reflected power by a dummy load [178]. The problems associated with the skin depth and poor performance at low discharge pressures can also be circumvented by application of an intense static magnetic field which guides the wave (plasmas generated by whistler waves, cf. Chap. 7 and Sects. 14.6 + 14.7).

In these high-frequency discharges, the plasma density is considerably higher than in DC discharges for the same power input. For convenience, we compile some basic properties of the most common plasmas in Table 5.2.

5.8 High-frequency coupling: quantitative approach To ensure a comprehensive understanding of the two most important coupling modes (capacitive and inductive), we will focus on some of the electrical considerations. We will begin with the limiting cases of the two resonant circuits,

5.8 High-frequency coupling: quantitative approach

131

Table 5.2. Plasma sources and the most important parameters discharge pressure, plasma density, and electron temperature along with their most common application. type DC glow

p [mTorr] 1 − 102

Te [eV] 0.1 − 10

np [cm−3 ] 108 − 1011

RF CCP, low p RF CCP, high p RF ICP RF helicon MW convent. MW SLAN MW ECR E beam

1 − 102 102 − 103 1 − 103 0.1 − 10 10 − 103 100 − 103 0.1 − 5 0.01 − 0.1

1 − 10 1−5 1−5

(6.5)

with n0 the density of the neutrals and σm the cross section for momentum transfer. • Within the plasma, σi σe and σe → σDC for νm ω in the pressure range down to 50 mTorr (in argon), for the low-pressure region, which is used for reactive ion etching, νm ω (Sect. 2.2), and σe → iε0 ω ε˜ [eq. (14.154)], • but in the sheath because of ni ne : σi σe . 3 Because n = n(x), σ exhibits a spatial dependence (cf. Chap. 7). σ is connected with the mobility μ via σ = ρμ = e0 nμ.

158

6 High-frequency discharges II

Hence, the current density in the plasma can be approximated by jp ≈ (σe + ε0 iω) Ep ,

(6.6.1)

which further simplifies to ⎫ ⎬

νm ω : jp ≈ σe Ep ;

(6.6.2)

νm ω : jp ≈ ε0 iωEp . ⎭

In both cases, the plasma frequency of the electrons exceeds the typical excitation frequency by one or two orders of magnitude. On the other hand, we find in the sheaths σi ε0 ω, which means ⎫

high frequency range js ≈ ε0 ωEs ; ⎬ low frequency range js ≈ σi Es .

(6.6.3)

⎭

Since σi depends on the ionic density which strongly decreases across the sheath in the direction of the electrode due to the enhancement of the ionic drift velocity, the electric field must vary as well, but in an inverted dependence because of the constancy of js . For high frequencies, however, the electric field across the sheaths will be constant (only displacement current). Due to the principle of continuity, the current in the plasma must equal the currents across the sheaths, i. e. Ip = Is or jp Ap = js As , and we get for the currents in the two limiting cases ⎫

High frequency range σe Ep Ap = ε0 ωEs As ; ⎬ ⎭

Low frequency range σe Ep Ap = σi Es As .

(6.7)

The differential resistance in the plasma bulk can be calculated with the resistivity, dp its length and Ap its cross section dRp = ·

ddp ddp = , Ap σe Ap

(6.8)

and we get for the Ohmic heating within the plasma bulk4 dPp = kΩ

Ip2 ddp j 2 Ap j 2 Ap = kΩ p ddp ⇒ Pp ≈ kΩ p dp σe Ap σ σ

(6.9)

kΩ is a constant which contains the time-averaged sheath thickness and the Debye length [187]. Eventually, we obtain the electric fields for the highfrequency and low-frequency range, respectively (ε0 Es,1 As,1 = ε0 Es,2 As,2 ):

High frequency range Es =

1 ε0 ωAs

Low frequency range Es =

1 σ I As

P p σ e Ap ; kΩ dp

P p σ e Ap kΩ dp

⎫ ⎪ ⎪ ⎬

⎪ ⎭ : ⎪

(6.10)

4 Additionally, the electron density is axially and radially dependent, but only to a weak extent.

6.2 Electric fields across the sheaths

159

the electric field across the sheath is inversely proportional to the electrode surface and directly proportional to the plasma current and the square root of the power coupled into the plasma bulk, respectively. The electric fields in the plasma and across the sheaths can be obtained from σe Es ≈ Ep σi

(6.11.1)

for the low-frequency range, and for the high-frequency range from Es iωε ≈ . Ep iωε0

(6.11.2)

According to Eq. (14.171), the collisionless dielectric constant remains negative for ω < ωp,e and ε ε0 : The electric field across the plasma bulk is directed in the opposite direction to the fields across the sheaths and is much smaller. We find the important results: 1. The RF voltage will drop across the sheaths generating an intense electric field, and only a small electric field is set up across the plasma bulk. 2. The electric field in the LF case is much higher than in the HF case.

Example 6.1 With < v >= kB T /mAr and a discharge pressure of 100 mTorr (13 Pa), a plasma density of 109 /cm3 and the values for the temperatures T , the cross sections σm , the collision frequencies νm , the specific conductivity σ and the resulting electric resistivity for the • low-frequency case with Es /Ep = σe /σi = 67; • high-frequency case with Es /Ep = σe /(2π × 13.56 MHz) = 4.7. For the high-frequency case, the sheath field has dropped to about 1/10 of that for the low-frequency case!

Table 6.1. Calculated values for the electric conductivity of electrons and argon ions according to the free electron model.

T [eV] electrons 3 ions 0.1

σm νm σ [10−16 cm2 ] [10−6 Ω−1 cm−1 ] 20 0.7 GHz 400 25 350 kHz 6

[kΩ cm] 2.5 167

Since the sheath field determines the ionic drift across the sheath, and for the high-frequency case, this field is significantly smaller than for the low-frequency case—all other conditions kept constant—this difference must be taken into consideration for the construction of reactors for ion or plasma etching (cf. Sects. 11.2). In the HF case, the relative strength of the electric field is weakened by

160


• rising excitation frequency (Fig. 6.5) and • increasing collision frequency (i. e. growing particle density or rising discharge pressure).

bias potential [V]

400

300

Fig. 6.5. Bias potential as a function of RF power coupled into the plasma bulk at 13.56 MHz and the second harmonic at 27.12 MHz (high frequency case above ωp,i ) after [188]. p = 600 mTorr (80 Pa); flux ratio GeH4 :H2 = 10:90; area ratio of the electrodes (“hot” to “cold” including the quartz cylinder: 0.65.

RF electrode, 13 MHz RF electrode, 27 MHz grounded electrode, 13 MHz grounded electrode, 27 MHz

200

100

0

0

25

50 RF power [W]

75

100

At lower frequencies, the higher electric field leads to further growth of the sheath thickness. This, in turn, leads to a rise in electron temperature [Eqs. (3.48) and (3.49)], since the bulk volume shrinks but the relative plasma surface remains the same. ε0 ∇E s = e0 (ni −ne ) we obtain for the directionz normal to the electrode ε0 dE ≈ e0 ni dz. Integrating and considering that E will vanish at the (with E = E0 at the electrode surface). If we sheath boundary yields ds = ε0eE(0) 0 ni suppose the potential gradient to be linear (or a potential which depends on d2 ), this remains valid also for the voltages (Vs = Es ds ; and with the dependencies discussed in the preceding section with a slightly higher dependence on As ): ε0 Es As =

ε0 Vs As Ip Ip ds ⇒ ε0 Vs = = ds ω ωAs

(6.12)

with ε0 As /ds the sheath capacity [189].

6.3 Current-voltage characteristic at one electrode In Figs. 6.2 we have seen that the I(V ) characteristic of the sheath is similar to that of a rectifying diode. Its Ohmic part has been outlined in Sect. 3.4; the nonlinearities are controlled by the sheath thickness and the (potential dependent) ion distribution across the sheath. Due to the progressive depletion of electrons across the sheath, which starts at equilibrium values at the Bohm edge but quickly falls to zero, at RF frequencies above the plasma frequency of the ions, ωp,i , e. g. at 13.56 MHz

6.3 Current-voltage characteristic at one electrode

161

(ωp,i < ω ωp,e ), the capacitive displacement current across the sheath nearly equals the conduction current of the electrons within the plasma bulk.5 However, not only the electron density is inhomogeneous but also the ion density must drop to smaller values because the ions are accelerated to the RF driven electrode by the negative DC potential. Even if the discharge is excited by a harmonic voltage which is responded by a harmonic displacement current, the dropping charge density across the sheath leads to a nonlinear response of all the other properties. In terms of the equation of continuity, we can express the condition at the migrating Bohm edge, the last point for which evenness between ni and ne is given [with ω the operating frequency and a one-dimensional problem in x-direction with ˜j = j0 cos ϕ(x) with ϕ the spatial-dependent phase angle which peaks at the sheath boundary with ϕ = 0] dϕ ∂ e0 ni = 0 ⇒ j0 sin ϕ = e0 ωni . (6.13) ∂t dx Evidently, the nonlinear response is caused by the spatial dependence of the phase angle. Appropriate models for a symmetric discharge have been established by Godyak and Sternberg [190, 191] and Lieberman [192]; a model for a capacitive/resistive sheath has been introduced by Klick [193, 194]. Provided that the quantities are known which define the character of the capacitively coupled plasma, the plasma density np , ji , ion current density to the electrodes, and ds , the thickness of the sheaths adjacent to the electrodes, it is possible to calculate the efficiency of power coupling. It has turned out that an uncritical application of equations by which the DC discharge can be described (Bohm’s theory, Child’s law and Langmuir theory), is very questionable [191]. Due to the extreme complexity of the exact models describing the sheath dynamics, in this section, we restrict ourselves to a qualitative treatment of the properties which control the capacitive coupling. However, it is possible to understand the development of the high negative potentials at the RF driven electrode with a one-dimensional model. In this approximation, electrons and ions both exhibit zero temperature. The DC fraction of the sheath voltage is caused by the different mobility of the charged carriers which leads to initial current densities which differ by about three orders of magnitude. Equilibrium at the electrode is reached after several RF cycles when the electron current density is reduced to the ion current level, and the condition for equilibrium reads ∇ · j˜ +

1 2π [ji − je (t)]d(ωt) = 0. (6.14) 2π 0 The electron current density, je , instantaneously follows the electrode potential, i. e. the charge density of the electrons is time-dependent, and it drops from the equilibrium value at the Bohm edge to zero above the electrode (cf. Fig. 5 In more exact treatments, the positive conduction current across the sheath must not be neglected.

162


6.6). The current density of the positive carriers, ji , is independent of time and phase above their plasma frequency due to the inertia of them. Across the whole sheath between x = 0 and x = ds , an excess of positive carriers is required for its stability (cf. Sect. 14.2).6 For harmonic excitation, the extension of the “breathing” sheath is given by

electrode

plasma _ ne

Fig. 6.6. Principal structure of the RF sheath with quasi-stationary ions after Godyak and Lieberman [191, 192].

ni

ne(t)

x

se(t)

dS - se(t) dS

0

se (t) = se cos ωt,

(6.15.1)

ds = s0 + se cos ωt

(6.15.2)

which leads to

as the total extension of the sheath. In the direction of the electric field, the (linear) charge density which is piled up in the sheath amounts to ds −se cos ωt 0

e0 nB dx = e0 nB (ds − se cos ωt),

(6.15.3)

with nB the density of the charged particles at the Bohm edge by which the developing electric field [Es Ep (Sect. 6.2)] is given according to Poisson’s law to Es ≈

e0 nB (ds − se cos ωt) , ε0

(6.16)

and the potential Vs = VDC VDC = −

VDC = −

e0 nB [ds − se cos ωt]2 ⇒ 2ε0

(6.17.1)

e0 nB 2 s2 ds − 2se ds cos ωt + e (1 + cos 2ωt) . 2ε0 2

(6.17.2)

If the potential of the DC sheath becomes positive, it will be opened, and in this short part of the RF cycle the same amount of electrons will flow to 6 This assumption is not entirely correct, but simplifies the picture of two telescoped sheaths.

6.4 Sheath potentials

163

the electrode to equalize the ion current over nearly the whole part of the RF cycle. Furthermore, we see that the response leads to the generation of a second harmonic and a constant average value.

6.4 Sheath potentials 6.4.1 Symmetric system A typical symmetric system in a parallel-plate reactor consists of two electrodes of equal size; one of them is grounded, the other one connected to the high-frequency source via a blocking capacitor which acts as a high-pass filter suppressing the DC current.7 In the state of equilibrium, the electron current to the excitation electrode must equal the positive current because the capacitor blocks the conduction current. Since the plasma bulk must lose an equal number of ions and electrons, this very condition applies also to the grounded electrode. Due to its high electric conductivity, the plasma bulk is an equipotential zone, and the high-frequency generator couples to the plasma across both the electron-deficient dark spaces with same geometry, i. e. same thickness d, same electrode area A and same capacitity C (Fig. 6.7).

electrode sheath

Rh

RP

plasma potential VP(t)

Rw wall sheath

Ch

Cw

Fig. 6.7. Electric model of an RF discharge with an excitation frequency ω/2π above the plasma frequency of the ions ωpi . The plasma is regarded as an electric conductor with highly mobile electrons, whereas in the sheaths of the electrodes, the electron density (and hence the electric conductivity) drops to zero which let them act as capacitancies for the RF current. The diodes in the circuit describe an electron flux across the sheaths for the situation Φp < Ve . By this, the uppermost value of the c The American Institute of Physics). sheath potential is fixed at zero [195] ( 7 The area of the grounded electrode encompasses all the surfaces which are not connected with the RF source.

164


To even out the currents of electrons and ions, the excitation electrode must develop a negative bias voltage, thereby creating an uphill potential for the electrons and a downhill potential for the ions. Hence a DC component is added to the (symmetric RF) sheath potential. At the RF driven electrode, the difference between the applied and (self-biased) electrode potential Ve (t) + VDC and the plasma potential Vp (t) yields the sheath potential Vs (t). At the grounded electrode, this difference between zero and the plasma potential Vp (t) equals the sheath potential Vs (t) (Figs. 6.8).

Fdc [a.u.]

p/2, 3p/2

p

F [a.u.]

0 0

Vdc

0

electrode distance [a. u.]


Fig. 6.8. LHS: the spatial dependence of the DC potential across the discharge in a symmetric discharge. The plasma potential exhibits a significant DC component and equals the sheath potential at the grounded electrode. RHS: the electrode potentials at the RF driven electrode for four equidistant moments of the RF cycle.

The lowermost value of the plasma potential is controlled by the high mobility of the electrons. It must not be less than the lowermost potential of any surface in contact with the plasma bulk, the grounded counterelectrode. This condition defines the instantaneous plasma potential according to 1 (6.18) Vp (t) = VRF (1 + sin ωt) . 2 Both electrodes are almost equivalent and are equally bombarded by plasma particles (Figs. 6.8). The counterphase shift with respect to the other electrode is reflected by an equation similar to eq. (6.17): VDC =

e0 nB (ds + se cos ωt)2 2ε0

(6.19)

which yields for the entire voltage drop VΣ across both sheaths [sum of eqs. (6.17) + (6.19)]: VΣ =

2e0 nB ds se cos ωt. ε0

(6.20)


165

The voltage drop depends only on the fundamental (ω = 2πf ) of the operating voltage.

6.4.2 Asymmetric system 6.4.2.1 Sheath potential theorem. In a typical asymmetric system, we choose the magnitude of the blocking capacitor to be much larger than the capacities of the sheaths. Because of the series connection of the capacities, it does not affect the properties of the plasma bulk and the sheaths. In this configuration, the DC component of the sheath potential of the grounded electrode can differ significantly from its counterpart at the RF driven electrode. In the model of a parallel-plate capacitor, the sheaths are assumed to play the role of the capacitive medium between the two electrodes. For a fixed charge, the area of the electrodes mainly determines the capacitance and the potential drop. This was the begin of a long-lasting debate. Koenig and Maissel pointed out that the assumption of a space-charge limited current would lead to an extremely strong dependence of the voltage ratio across the sheaths on the area ratio [156]:

A1 V1 = V2 A2

4

.

(6.21)

The sheath theorem has been subject for numerous investigations. For its validity, the ion current is supposed to be space-charge limited which does not hold true for pressures above 1 mTorr (0.1 Pa). This means in practice, that we have to face a hybrid between a space-charge limited and a mobility-limited discharge, and the exponent drops down to values between 1 and 2 [95, 196]. Evidently, the discharge will contract at higher discharge pressures, and a lower exponent is just pretended [185] (cf. Sec. 11.4). On the contrary, the exponent can be fixed at 2 and a ratio of “active” surfaces can be defined [188]. Eventually, it could be shown that it is the capacitance of the sheaths rather than the area of the electrodes that controls the ratio of voltages. The voltage which is applied to the RF driven electrode will be divided between the two electrodes and the plasma bulk: VRF sin ωt = (Vh − Vc ) sin ωt + E(t) [l − dt (h) − dc (t)] .

(6.22)

l the distance between the electrodes; dh , dc is the sheath thickness of the RF driven electrode (hot) and grounded electrode (cold), resp., E(t) ist the electric field across the plasma bulk. Provided that the voltage drop across the plasma bulk can be neglected, i. e. that the definition of a plasma potential Vp does really make sense and that the applied RF voltage is not severely distorted by electrical nonlinearities, we can calculate the sheath potentials.

166


6.4.2.2 Calculation. For vanishing DC conduction current in the sheaths, i. e. for Rh ∧ Rc → ∞, the sheaths represent two capacitors connected in series which are charged up to charge Q by VRF : V RF = V h + V c = Q ·

1 1 + , Ch Cc

(6.23)

with V h and V c the potential differences between plasma and target and between plasma and wall, respectively—both voltages are difficult to measure—and Ch and Cc the capacities of the two sheaths. Provided that the DC current through the external circuit vanishes which is ensured by the blocking capacitor, the current incident on the RF driven electrode must equal that incident on the grounded electrode Ih = Ic ⇒ QT = QW ⇒ Ch (Vh − Vp ) = Cc (Vc − Vp ),

(6.24)

which defines the capacitive voltage division: Ch Vh − Vp = . Cc Vc − Vp

(6.25)

In the approximation to first order, the magnitude of the capacitance is controlled by the area ratio of the RF driven electrode and the other surfaces exposed to the plasma, morever, it is determined by the voltage drop across the sheaths. This voltage drop, in turn, influences the sheath thickness via Child’s law: C = C(A, V, d), leading to a thick high-voltage sheath across the RF driven electrode, and a thin low-voltage sheath across the other surfaces. This implies that the capacitive impedance of the larger electrode will have a larger capacitance per unit area and rises by a factor which is larger than the pure area ratio. In the following paragraph, we deal with a completely capacitive coupling of the plasma bulk with plasma potential Vp against the RF driven electrode. As has become evident from previous sections, we must distinguish between a DC part and an RF fraction [195]. The electrode potential of the RF driven electrode can be written as Ve (t) = VDC + VRF sin ωt.

(6.26)

Provided that the sheaths act purely capacitively, the instantaneous plasma potential oscillates around a DC value, V p with amplitude aVRF : Vp (t) = V p + aVRF sin ωt,

(6.27)

with a the coupling factor a=

Ch (V ) . Ch (V ) + Cc (V )

(6.28)


167

The DC component of the plasma potential, V p , is controlled by the ratio of the mobilities electron/ion—as in a DC discharge. Vp (t) cannot be less than the potential of any surface exposed to the plasma: • The potential of the grounded electrode, V = 0. • The potential of the RF driven electrode, V (t) = VDC + VRF sin ωt. The maximum of the plasma potential is V p + aVRF ≥ VDC + VRF ,

(6.29.1)

V p − aVRF ≥ 0.

(6.29.2)

and its minimum

Since we discuss a capacitively coupled plasma with a vanishing external current, both types of carriers must succeed in reaching the electrode during a comparatively long part of the RF cycle (ions) or a sufficiently short part (electrons). Hence, its minimum turns out to be zero, and its maximum to be VDC + VRF , i. e. both the inequalities (6.29) become equations: 1 Ch − Cc . (6.30) (VRF + VDC ) ; VDC = VRF 2 Ch + Cc The sheath potential across the RF driven electrode is the difference between the electrode potential (sum of the applied RF voltage and the DC bias, VDC = Vh − Vc ), and the plasma potential, and can be written approximately as Vp =

Vs (t) = VRF

Cc 1 (sin ωt − 1) = VRF (sin ωt − 1) , Ch + Cc 1 + CChc

(6.31)

which can be written with Eqs. (6.28) − (6.30), (VDC ≤ 0): VRF − VDC · (sin ωt − 1) . (6.32) 2 The sheath potential of the grounded electrode is equal in magnitude but opposite in sign from that of the plasma potential (Fig. 6.9) (VDC ≤ 0) : Vs (t) =

Vp (t) =

VRF VRF + VDC · (1 + sin ωt) = · (1 + sin ωt) . 2 1 + CChc

(6.33)

We distinguish between these borderline cases: • Symmetric system (Ch = Cc )VDC = 0, Vp (t) = 1/2 VRF sin ωt. • Asymmetric system (smaller electrode is RF driven): VDC < 0; Vp (t) < 1 /2 VRF sin ωt; Vp,min = 0, Vp,max = VRF .

168


Vp(t)

V [a. u.]

0 Vs(t)

Ve(t)

VDC

0

90

180

270

360

Fig. 6.9. Comparison of the instantaneous potentials of the plasma bulk, the RF driven electrode and its sheath for a capacitively coupled plasma at the RF driven electrode (ωRF > ωp,i , sheath: Vs , RF driven electrode: Ve , plasma potential: Vp ); phase shifts are neglected. Electrons can flow only for Ve > 0.

phase angle [°]

• Asymmetric system (larger electrode is RF driven): VDC > 0; Vp (t) > 1 /2 VRF sin ωt; Vp,min = 0, Vp,max = VRF . Besides the rectification across the sheaths, we observe in asymmetric systems the additional effect of voltage division between the electrodes. The plasma potential oscillates around a positive DC value, and its lowermost value during the negative half of the RF cycle is just zero. On the other hand, the electrode potentials oscillate around a negative DC value, and their uppermost value during the positive half of the RF cycle equals zero as well. The negative DC value of either electrode develops according to the law for voltage division for pure capacitive coupling. Using laboratory slang, this DC value is denoted “self-bias” voltage or simply “DC bias” or “DC voltage”. Hence, the voltage of the sheath Vs (t) is composed of several components: • A time-dependent RF voltage around a negative DC component for the electrode. • A time-dependent RF voltage around a positive DC component in the plasma bulk. They are split according to the ratio of capacitive voltage division between the electrodes (Fig. 6.10). In the case Ch ≈ 1/3 Cc , the DC bias becomes half the peak-to-peak voltage (VDC ≈ −1/2 VRF and V p ≈ 1/4 VRF ). Typical values for parallel-plate reactors are less than 0.05 [195], and VDC → −VRF . These functional relationships are compiled in Fig. 6.11 along with the grounded case, a configuration with several severe disadvantages. To begin with, no DC bias can be developed. If the small electrode is RF driven, the plasma potential will never come close to ground. The energy of the ions incident on the electrode, however, is the sheath potential, a quantity which is composed of several components. Among them, the plasma potential and the electrode potential are the two most important


169 Fig. 6.10. Spatial dependence of the DC component of the potential across an asymmetric discharge. The sheath potential drop across the grounded electrode differs substantially from the sheath voltage at the RF driven electrode. The plasma potential exhibits but a small DC component.

Fdc [a.u.]

0


potentials. Both of them exhibit a DC and an RF component. Therefore, it is a disastrous misapprehension to conclude etch rates or sputter rates from a knowledge of only the DC bias which have been established in different plasma reactors. For the long-term process control in the same reactor, however, this datum is of high heuristic value.8 For this derivation we did neglect the floating potential, which is calculated according to Eqs. (3.24). In particular for small capacitancies, the absolute value of VDC can become approximately VRF , which almost causes the plasma potential to vanish. Hence, small errors in measuring VDC and VRF strongly influence the value of the plasma potential (for further details see Sect. 14.4.2).

6.4.3 Resistive Coupling The main reason for the difference between the measured plasma potential and its derived value is the assumption of a totally capacitively coupled discharge. To put it shortly, the resistivities of the sheaths, Rh and Rc , which have been neglected so far, have to be taken into account. There is not only a displacement current, IRF , across the sheaths, but we must consider also the small positive ion current Ii [197]—this is simply required for a clear definition of the plasma potential. This effect becomes effective at higher plasma densities and leads to a drop of the plasma potential, since the sheath begins leaking (Fig. 6.12; again, we neglect Vf ), and this results in an additional inductive/resistive component. Let us compare the capacitive coupling with the inductive/resistive coupling as borderline cases: 8 To check the asymmetric behavior of the reactor, the etch rate can be checked with substrates which have been mounted to both electrodes.

170

6 High-frequency discharges II capacitive coupling

DC coupling V(t)

V(t)

V(t)

VP(t) asymmetric small electrode powered

0 VDC

0

VP VS 0 VDC

VP VS 0

V(t)

VP(t) symmetric

0

0

VP VS 0

VP VS 0

V(t)

VP(t) asymmetric large electrode powered

VDC

0

0 VP VS 0 VDC

VP VS

0

90

180 270 360 Phase angle [°] Phasenwinkel [°]

450

540

0

90

180 270 360 Phase angle [°] Phasenwinkel [°]

450

540

Fig. 6.11. Schematic representation of the time-dependent variation of the plasma potential Vp (t) (dashed) and the electrode potential Ve (solid) for three different system geometries, totally capacitive conduct of the sheaths for DC coupled and capacitively coupled electrode, respectively. System geometries: (1) sputtering and ion c The American etching, (2) PECVD and plasma etching, (3) hollow cathode [195] ( Institute of Physics).

• Capacitive coupling: Rh , Rc → ∞, Ch , Cc : finite. • Inductive/resistive coupling: Ch , Cc → 0, Rh , Rc : finite. Ions can only respond to the time-averaged DC potential, i. e. for capacitive coupling, there remains a small but continously flowing current, whereas for inductive/resistive coupling, parts of the wave are omitted by clipping (rectifying behavior of the sheath, Figs. 6.1/6.2 + 6.7). Harmonic excitation V (t) = VRF cos ωt provided, the sheath potential crosses zero for


171

Vp(t) for IRF >> Ii

Fig. 6.12. Comparison of the plasma potentials Vp (t) for purely capacitive (IRF Ii ) and purely resistive (IRF Ii ) cases and a case lying in between. Vf has been neglected, i. e. VRF + VDC Vf [195] c The American Institute of ( Physics).

potential [a. u.]

Vp(t) for IRF ~ I i

0

Vp(t) for IRF (nn : density of the neutrals, σ: elastic cross section, < v e >: electron mean velocity) which must be rewritten in the case of multiple gases, taking into account the dependence of the cross section on < v e >, and with the discharge pressure p as the controllable parameter νm =

p pi nn σi (ve ) < ve >, k B Tn i p

(6.51)

where i denotes the index and pi the partial pressure of species i, Tn the temperature of the neutrals in K.12 Especially at low discharge pressures, when the mean free path approaches the characteristic length of the reactor, the transfer of energy from the electric field to the electrons cannot happen by Ohmic heating. Due to the relatively small number of collisions, a collisionless mechanism dominates the heating of electrons. This process is caused by the “breathing” RF sheath. This movement pushes the electrons back and forth and transfers energy to them. This possibility of such a collision-free mechanism for RF discharges, based on the Fermi acceleration [208] was first introduced by Godyak [209, 210] (Fig. 6.15). 9

10

8

-1

neff [s ]

10

experiment theory

107

106 -4 10

10-3

10-2

10-1

Fig. 6.15. The effective electron collision frequency in a RF discharge through mercury at 40.8 MHz (3 × 13.56 MHz), length of the tube: 6 cm, diameter: 10 cm) [210].

discharge pressure [Torr]

11 As experimentally shown by Popov und Godyak, this mechanism is (at least in discharges through metal vapors) not sufficient to explain the high degree of ionization [205]. They concluded that a collisionless mechanism is reponsible for this effect (surfing or wave riding), at least at pressures below 10 mTorr (1 Pa) [206]. 12 In several cases (H2 and He) the dependence of the cross sections of the neutrals σ on < ve > cancels approximately out and then the collision frequency is merely a function of particle density (cf. Sect. 2.3.4).

178


Historical approach. According to kinetic gas theory, we can evaluate the energy gain of the electrons with initial velocity v incident on a breathing sheath with velocity vw . Due to the sheath’s velocity, only electrons with velocities larger than vw can collide with the sheath in its retreating phase: The number of collisions is reduced, whereas in its expansion phase, the number of collisions is enlarged. We consider the electrons at the sheath edge (density n) to be distributed according to n=

∞ −∞

fw (v, t) dv dt,

(6.52)

where fw (v) denotes the electron energy distribution function EEDF at the sheath boundary, and the electrons which will collide with the sheath (expressed as current density) is dj = (v − vw ) dn = (v − vw )fw (v, t) dv dt.

(6.53)

For v ≥ v w : collision, but reduced intensity, for v ≤ v w : no collision, for v s = 0: immobile wall, we obtain the same result as in kinetic gas theory, for v w < 0: collisions, more intense than against an immobile wall: je (v) =

∞ v−vw

(v − vw )fw (v, t) dv dt

(6.54)

and the force per unit area exerted on the electrons can be expressed with vr = −v + 2vw as dp vw = 2m(vw − v) (v − vw )fw (v, t) dv dt dt dx which yields for the transferred power F =

(6.55)

dP = F dv = 2m(vw − v)(v − vw )vw fw (v, t) dv.

(6.56)

In order to determine fw , we consider a Maxwellian distribution in the undisturbed plasma bulk denoted g(v ) and connect it with the distribution within the sheath via v = v − vs with vs the drift velocity of the electrons at the sheath edge [192] to obtain [211] P = −2m

∞

∞

vw −vs

vw −vs

(vw − vs )(v + vs − vw )2 g(v ) dv+

(6.57)

vs (v + vs − vw )2 g(v ) dv .

Conservation of current requires vw = vs at the sheath edge which lets the first integral in Eq. (6.57) vanish: Reflection from the breathing sheath does not change the distribution of thermal electron velocities. It was the second integral which was supposed to be responsible for stochastic heating [192, 212]. However, Gozadinos et al. pointed out that the second integral of Eq. (6.57) describes the maintenance of the electron drift in the field and its time-averaged integral vanishes as well [211].

6.5 Power input

179

Pressure heating. Gozadinos et al. used an extended fluid approach leading to pressure heating [207, 211]. Despite the pressure heating appears to be much more consistent, both approaches represent different points of view to the same mechanism. The usual fluid approach based on the first-order moments of the Boltzmann equation provides for the permittivity ωe2 =1+ , 0 iω (iω + νm )

(6.58)

2 , a series expansion and for the dissipated RF bulk power assuming ω 2 νm 2 considering all terms of the order of ω or lower

PΩ =

m e νm l e20 ne

1+2

ωk2 1 2 j , ωe2 k 2 k

(6.59)

taking into account higher harmonics k > 1 of the discharge current (density) e2 n 2 j. The term containing the plasma eigenfrequency ωp,e = m0e e0 in Eq. (6.59) reflects approaching local plasma oscillations which also cause an increase of RF power dissipation into the plasma bulk due to an increase of the RF electric field through a low electron density. The first term is the ohmic (real) part of the resistivity and l denotes the effective length of the plasma (electrode gap minus the sheath lengths). Strictly speaking, calculation of the RF current within the chamber and a subsequent averaging were required, but for an approximation of first order, it is sufficient to note that the effect has to be taken into account also when the electron density decreases only in the outer regions of the plasma bulk, which can occur in particular in electronegative gases. The collisionless heating is also expected to depend on j 2 , so that it can be also characterized by a “sheath” collision rate νse , and an effective collision rate can be defined as

νeff = νm 1 + 2

ωk2 2 ωp,e

+ νse

(6.60)

replacing now νm and the first-order plasma oscillation term in Eq. (6.59). Combined model. The models known from the literature describe the collisionless heating as stochastic or pressure heating and rely on the questionable assumption of a sinusoidal discharge current. They provide, after some algebra with the assumption of a constant ion density within the sheath (matrix sheath) [206], 2v e , (6.61.1) se with self-consistent modeling of collisionless movement of ions in the sheath but assuming the minimum sheath thickness to be zero [192], νse =

180


1 e0 V , νse = se πme

(6.61.2)

and, using Lieberman’s collisionless sheath model [192] with a new, more realistic and consistent pressure heating approach [207, 211]:

νse =

⎛ 7 e0 V ⎜ ⎝1 +

18se

πme

⎞−1 1 2e0 V ⎟ ⎠ ,

45π

kB Te

(6.61.3)

with V the mean sheath voltage. The stochastic heating approach results in enhanced heating in case of a decrease of the ion density within the sheath. This is based on a higher electron drift velocity in the quasi-neutral sheath region. The pressure heating depends entirely on the gradient of the ion density within the sheath only and disappears for a homogeneous ion density within the (matrix) sheath [211]. But all Eqs. (6.61) indicate the heating to be independent of gaseous pressure and electron density in the case of a collisionless sheath. However, for the pressure range under consideration this is really true only at very low pressures (below 1 Pa, 7.5 mTorr). Compared to the collisionless case, the gradient of the ion density within the sheath is decreased with growing pressure by the lower drift velocity due to energy loss by collisions with neutrals. This can also be concluded from Gozadinos’s pressure heating model [211] where the core model does not depend on the sheath model but depends strongly on the ratio of the electron density at the boundary to the undisturbed plasma bulk and the sheath edge to the “real” sheath nse /nB . This ratio in “quasi-neutral” range of the sheath equals the ratio of the corresponding ion densities. We can therefore expect a pressure-independent νse in the tiny range between the minimum pressure and slightly higher pressure where the free mean path is still well above the sheath thickness. For a medium pressure above 1 Pa (7.5 mTorr), where the (stochastic or pressure) sheath heating is still dominating, increasing pressure results in a lower νse and νeff . Using the general pressure heating approach [211] provides approximately π ve νse ≈ 8 se

'

ns ln e nB

2 (

.

(6.62)

The Bohm criterion, the conservation of mass and energy and estimating the average in Eq. (6.62) provide √ π e0 V νse ≈ , 8 se me

(6.63)

the square root dependence on V in Eqs. (6.61.2) and (6.61.3) is also caused by the dependence of the density of the ions on their energy. At sufficiently low

6.5 Power input

181

pressures, the ion energy is determined by the sheath potential V . For arbitrary pressures, this provides finally the estimation νse ≈

νse |p=0 1+

σ

p

se knB+Tn

,

(6.64)

where se and σn+ denote the mean sheath thickness and the ion-neutral collision cross section, respectively. Thus the pressure dependence described above can be reduced to the ratio of the sheath thickness and the ionic free mean path, i. e. both collisionless and collisional ion motion will contribute to the heating mechanism. With further growing pressure, Ohmic heating increases and will become the important heating mechanism at pressures well above 75 mTorr (10 Pa) [205] which also leads to a significant change of the EEDF [204, 191]. Substituting the term for the frequency according to Eq. (6.60) leads to a slight modification of the formula for stochastic heating as published by Misium et al. [187]:

3

se /2 me νeff 2 se je A, (6.65) Pst = kst λD e20 nB which takes into account the inhomogeneous ion density and the instantaneous electron density across the sheath. The occurence of the sheath thickness in Eq. (6.65) stresses the importance of this heating mechanism especially at low pressures. 6.5.2 Two regimes of power transfer Capacitively coupled plasmas are generated in the evanescent regime in which the operating frequency f falls short of the plasma frequency of the electrons but exceeds the plasma frequency of the ions. As conseqence, the penetrating electric field is reduced by a factor of 3 within 60 cm at topmost plasma densities of 1010 cm−3 (collisional plasma), which is shortened to 30 cm in collisionless plasmas; reducing the plasmas density lengthens these values by a factor of 3 for every decade (cf. Sect. 14.6). No parallel-plate reactor is known to exceed this dimension nor is any likely to be built. But is should be kept in mind that the mode of excitation will gradually change with increasing plasma density. For low-density plasmas, the intensity of the electric field within the plasma bulk Ep is relatively high, Ep =

j0 me νm , e20 np

(6.66)

and the heating of electrons occurs mostly by Ohmic heating in this area, which is large compared with the sheaths. The mean energy of the electrons is relatively low, and therefore, these electrons remain trapped in the plasma bulk which masters the processes in the discharge.

182


For plasmas with higher density, sheath heating becomes predominant. Since the density of the sheath electrons is dramatically lower than their plasma counterpart, we find Es =

j0 m e ν m Ep , e20 ns

(6.67)

and due to this acceleration, their velocity significantly exceeds that of the electrons in the plasma bulk. The intensity of the electric field Es is determined by the balancing of the ionization processes (which determine the ion density in the sheath). Here, the sheaths will dominate the processes in the discharge.

6.6 Spatial distribution of charged carriers In weakly ionized plasmas, the spatial distribution of the charged carriers is determined by the equilibrium between carrier gain and carrier loss. The latter mainly depends on ambipolar diffusion. At pressures of several Torrs which are customary in microwave discharges, the axial diffusion can be approximated by a cosine function, and the radial diffusion by a Bessel-function of the first kind (Schottky profile):

n(x, r) = n0 cos

πx r , J0 2.405 2L R

(6.68)

with L = a − ds half the distance a between the electrodes (plasma half-length), reduced by the sheath thicknesses ds (Sect. 5.4). We have seen in Sect. 4.7.3 that the zero boundary condition which are applied to solve the diffusion equation leads to a singularity in the ambipolar drift velocity. This is no issue in microwave-driven discharges since they are operated at comparably very high pressures for which the diffusion coefficient Da drops to very small values. However, in RF driven discharges, the discharge pressure is lower by at least two orders of magnitude, and in this pressure range, the drift velocity of the ions exceeds their thermal velocity considerably, and the carrier profile is not very well fitted by Schottky’s approach. This problem has been circumvented by Person who took the inertia of the ions into account assuming that the plasma density at the boundaries remains finite [213]. To make this happen, the ambipolar drift velocity must be equal to the Bohm velocity, i. e. the velocity of the ions is on the brink of supersonic behavior. Based on this assumption, Self and Ewald presented a one-dimensional distribution which contained the two cases at the lower and upper pressure limit model λi L and λi L (Langmuir profile vs. Schottky profile). For the lower limit, the ion transport is not mobility-limited any longer but determined by resonant charge transfer, and the drift velocity has to be replaced from ui = μi E to ui = 2Ekin,i /pi which exceeds the thermal ion velocity considerably. During their course across the Bohmic presheath, the ions have been accelerated to the Bohm velocity, which

6.6 Spatial distribution of charged carriers

183

is always large compared with their thermal velocity in the plasma bulk. But this means that the frequency of the ion-atom collisions becomes independent on the ambipolar drift velocity which prevents a smooth joining of the two borderline cases. For a better approximation, a non-linear transport equation has to be solved, which has been worked out for the first time by Valeri Godyak [214] and has been further refined by Lieberman and Lichtenberg [103]. As in the previous model introduced by Self and Ewald, it is closely correlated with the ionic mean free path, λi . Following Godyak, the plasma density at the edges for x = X and r = R amounts to [214]: ns,X 0.86 ns,r 0.80 ≈ ∧ ≈ . X n0 n0 3 + λi 4 + λRi

(6.69)

Both the distribution profiles (radial and axial) resemble a Langmuir profile with a plasma density on the boundary which amounts 1/2 to 1/3 of that along the discharge axis. At the Langmuir limit, the axial value saturates at 0.5 n0 , and its radial counterpart at 0.4 n0 (Fig. 6.16). λi is defined according to the 0.5 0.4

n a,r/n0

0.3 0.2 0.1 0.0 0.00

axial shape radial shape

0.25

0.50

0.75

1.00

Fig. 6.16. Cylindrical plasma: The plasma density at the axial and radial boundary has dropped to less than half its value along the axis after [102].

l i/L,R

kinetic theory of gases 1 ; (6.70) nσi for σi , we have to insert the sum of the cross sections for elastic scattering and the resonant charge transfer, which are in the order of 10−14 cm2 , and we obtain a λi of about 3 mm in argon at 10 mTorr (1.3 Pa). λi =

The integration of the nonlinear differential equation is sketched in [214] and in great detail in [215]. They start with the equation for ion continuity (ρi = e0 ni ) ∇ · ρi + νi ρi = 0, a Maxwellian distribution of the electrons

(6.71.1)

184


ne = n0 exp

e0 Φ , kB Te

(6.71.2)

and a certain drift velocity of the ions

ui =

2λi e0 . πmi μi

(6.71.3)

After having introduced reduced variables which define a variable α, a normalized ionization rate which describes the balance of ionization and loss at the boundary, and which is the product of the reduced ionization length c=

νi d vB

(6.72.1)

and the ratio between the axial dimensions of the reactor and the ion mean free path

b=

πd , 2λi

(6.72.2)

α = cb

(6.72.3)

they consider slab geometry, and inserting (6.72.3) into (6.72.1) yields the nonlinear differential equation with ξ = x/d, y = n/n0

vB

2λi d π dx

−n

dn = νi n dx

(6.73)

which yields a value for the parameter α and for the relative distribution y(ξ) for the boundary conditions v0 = 0 ∧ vi = vB : ⎡

2 α /3 ξ

1

⎤

1 1 1 π 2(y 3 − 1) /3 − 1 ⎦ √ = ln[(1 − y 3 ) /3 + y] + √ arctan ⎣ + √ . 2 3 3 6 3

(6.74)

For λi /d → 0, α reaches its maximum value at

α0 =

2π √ 3 3

3

=

4 3

(6.75)

which can be seen from Fig. 6.17.

From Eq. 6.74, we obtain the general dependence of y. The relative density 2 n/n0 solely depends on the parameter α /3 ξ with α determined by Eq. (6.72.3) and shown in Fig. 6.18. In this figure, also the Schottky profile and the circular equation y 2 + ξ 2 = 1 are shown. For the low-pressure case, the plasma density within the plasma bulk is much flatter (almost constant) than in the highpressure case but drops significantly steeper than we could expect from eq. (6.68).

6.7 Dual-frequency discharges

185

1.4 a0 = 1.33

a

1.3

1.2

Fig. 6.17. Normalized ionization rate as function of the relative ionic mean free path after [102].

1.1

1.0 1E-3

0.01

0.1 l/2d

1.00

Y

0.75

0.50

y(x) x 2 + y2 = 1 cos(x)

0.25

0.00 0.00

0.25

0.50 X

0.75

1.00

Fig. 6.18. The relative plasma distribution between the electrodes for a planar geometry in axial direction after [102].

6.7 Dual-frequency discharges 6.7.1 Frequency dependence of plasma density In Sect. 5.2, we dealt with spatially homogeneous discharges, and found that raising the operating frequency at equal power input leads to an increase in effective field and high-frequency current. This may be explained by electron inertia which will come into play (although at a plasma density of 1010 /cm3 , the angular plasma frequency is 5.64 GHz, i. e. higher by a factor of 66 still than the operating frequency of 13.56 MHz). Since the instantaneous oscillations of the high-frequency field are not exactly followed as for lower frequencies, higher electric fields are required to obey Poisson’s law or to ensure the effective control of space-charge even in small parts of the RF cycle. As a result, electron inertia counteracts the frequency effects discussed in Sects. 6.2 and 6.5. According to Surendra and Graves who were the first to present MonteCarlo simulations, the sheath thickness scales almost inversely with frequency, whereas the plasma density rises with the square of the applied frequency [161]:

186


• The rising sheath field which is mainly due to a decreasing sheath thickness leads to enhanced ionization rate ⇒ • The plasma density in the plasma bulk will be increased (Fig. 6.19). • Since the electrode sheaths shrink in thickness, the plasma bulk will extend in size [216] (Fig. 6.20). This leads to a reduction of the ratio of plasma volume over surfaces confining the plasma [(Eqs. (3.43 and 3.44)]: • Although the plasma density increases with rising operating frequency, the electron mean energy (electron temperature) remains almost unaffected (Fig. 6.19).

100

jRF

10

1010

0.1

<je>

109 50

100

150 200 p [mTorr]

250

150 200 p [mTorr]

P [mW/cm2]

10

1 <Ee>

1 50

100

150 200 p [mTorr]

250

0.1 300

Fig. 6.19. The plasma density scales with the squared operating frequency, whereas the sheath thickness scales nearly inversely with ωRF according to Monte-Carlo simulations by Surendra and Graves. [161]

electrons

100

Ei dsh

0.01 300

ions

50

E [eV]

np

sheath thickness [mm]

e0 Fp

1

j [A/cm2 ]

np [cm -3]

1011

250

6.7.2 Mutual influence of two electrodes Operating at two different frequencies opens the window to steer ion flux and ion bombardement energy almost independently. Ideally, we want to control the plasma density by the high-frequency excitation [Eq. (6.45)] S=

2 Zω 2 VRF P dx. = A σDC d

(6.76)


187

200

<Ez> [V/cm]

100 0 13.56 MHz 27.12 MHz 40.68 MHz 60 MHz

-100 -200 -300 0.0

0.5

1.0

1.5 2.0 z [cm]

2.5

3.0

Fig. 6.20. The plasma bulk extends its volume at the expense of the shrinking sheaths with rising frequency [216]. Shown here is the axial electric field for a discharge through hydrogen at VRF = 100 V and p = 300 mTorr (40 Pa), reactor dimension: diameter: 12.8 cm, electrode distance: 3.0 cm.

Since the power is proportional to the square of the sheath voltage, the amount of the dissipated power scales as the square of the frequency, the fundamental equation of this effect. 6.7.2.1 Narrow gap. When we turn on the low-frequency electrode, a highvoltage sheath will form. Its extension from the surface of the electrode up to the Bohm edge where the plasma condition of time-averaged quasi-neutrality is violated) is governed by the Poisson equation along with Gauss’s law: V

∇ · E d3 x =

A

E · dA =

A

ρ ρ ρ dA ∧ ∇Φ = − ⇒ Φ ∝ d. ε0 ε0 ε0

(6.77)

To ensure the stability of the sheath for rising voltage, the sheath must thicken. This, in turn, leads to a reduction of the plasma volume. Since the effective boundaries have not been changed significantly, the electron temperature must rise according to Eqs. (3.43 + 3.44) which causes an increase in plasma potential [cf. Eq. (9.2)]. Comparing the modeled sheath thickness, the plasma potential and the mean bombardment ion energy 1 (6.78) Ei = VDC + kB Te 2 with 1/2 kB Te the (directed) ion energy at the Bohm edge and considering a collisionless sheath, Boyle et al. found an almost perfectly congruent behavior [217] (Fig. 6.21). The steeper slope of the plasma potential compared with the mean ion energy is due to the enhanced number of collisions when traversing the thickening sheath. 6.7.2.2 Wide gap. Since the electron temperature remains almost constant after having added the low-frequency electrode to the discharge, we must con-

188


6

80

60 F [V]

ds [mm]

5

4

40

3 0

50

100 150 RF voltage [V]

200

20

250

0

50


200

250

0

mean ion energy [eV]

-10

DC bias [V]

-20 -30 -40 -50 -60

0

50


200

250

30

powered electrode

20 grounded electrode

10 0

50


200

250

Fig. 6.21. Using a Monto-Carlo method, Boyle et al. simulated the sheath width, the plasma potential, the self-bias voltage, and the mean bombardment ion energy in a dual-frequency discharge as functions of the low-frequency voltage and found almost congruent behavior [217]. The power at the high-frequency electrode has been kept constant at 1 200 W.

clude that the processes triggered by the two electrodes have become almost mutually independent. Only in the high-energy regime, there remains an influence of the electrode driven with lower frequency [218]. It is these electrons which are important for the operation of the discharge. The characteristic of the EEDF has now almost entirely transformed to a Maxwellian shape, a straight-line behavior in contrast to a convexly shaped slope for a Druyvesteynian distribution (Fig. 6.22). It is noteworthy to mention the resemblance of this transformation from a Druyvesteynian behavior to a Maxwellian conduct when lowering the discharge pressure, as was first noted by Godyak and Piejak [219]. They reduced the operating pressure in a discharge of argon from 3 000 mTorr (400 Pa) to 70 mTorr (10 Pa). The EEDF changed in shape from a convex shaped Druyvesteynian behavior to a straight-line Maxwellian behavior, but with two different temperatures; i. e. straight lines but different slopes (Fig. 6.23).

189

1000

1000

100

100 EEDF [a. u.]

EEDF [a. u.]


10 1 0.1 0.01 0

10

1

Vlf = 0 V Vlf = -280 V

0.1

Vlf = 0 V Vlf = -200 V

5

10

15

20

25

0.01 0

30

5

10

15

20

25

30



Fig. 6.22. Influence of the electrode gap. LHS: Small gap (large a part of the gap consists of sheaths), the convexly shaped Druyvesteynian distribution transforms to a Maxwellian distribution with two temperatures when the voltage at the low-frequency (LF) electrode rises from 0 V to 280 V. RHS: Large gap (sheaths are only a small fraction of the gap), the shape of the EEDF is nearly Maxwellian [218].

3.00

1010

2.00 109

[T or r]

pr es s

0.30 0.20 0.10

ur e

0.50 107

ch ar ge

f [eV-3/2 cm-3 ]

1.00 108

di s

106 105

0

5

10

15

0.07


Fig. 6.23. Lowering the discharge pressure leads to a transformation of the EEDF. At large pressures, a Druyvesteynian behavior is observed which gradually changes into a Maxwellian distribution with two temperatures [219].

190


This behavior was repeated by Abdel-Fattah and Sugai when they enhanced the ωHF /ωLF -ratio (HF means high(er) frequency, LF stands for low(er) frequency) from unity to 4 (from 13.56 MHz to 54.24 MHz) [220] (Fig. 6.24). This transformation is explained by the increased number of collisions between electrons by which every Druyvesteynian distribution will eventually turn into a Maxwellian distribution (cf. Sect. 14.1).

Tbulk Ttail

f [eV -3/2 cm-3]

109

108

50 44 37

107

27 106

0

5

10

2p w/

[M

]

Hz

13

E [eV]

Fig. 6.24. Raising the discharge frequency by a factor of 4 from 13.56 MHz to 54.24 MHz entirely alters the characteristics of the EEDF from a Druyvesteynian distribution into a Maxwellian distribution, but with two temperatures. This behavior indicates the transition to a second heating mechanism from collisional Ohmic heating to collisionless sheath heating [220].

6.8 Collisional sheaths The maximum kinetic energy of the bombarding ions which have traversed the sheath can be calculated as (Fig. 6.9): e0 Vs = e0 · (V p − Ve );

(6.79)

this is the kinetic energy of an ion incident on the electrode which had not suffered any collisions when traversing the sheath. Relatively early, it became obvious that several scattering mechanisms must exist which cause significant broadening of the ion energy distribution function

6.8 Collisional sheaths

191

(IEDF) and the ion angle distribution function (IADF) which leads to a lowering of the mean energy of the ions. Eventually, a state of equilibrium between the heavy constituents of the plasma is achieved [221]. Several questions arise: • How long is the mean free path λ for an ion in a sheath of thickness ds , or how large is the ratio λ/ds , or how many collisions will happen when an ion traverses the sheath? • Is there an influence of ion inertia on the ion velocity, or does there exist an influence of the phase of the RF cycle on the ion velocity, and if yes: Can this amount be quantified? • Does there exist a correlation between the kinetic energy of the ions and its broadening in the spatial (IADF) and energetic domain (momentum, IEDF)? In general, the ions which enter the pulsating sheath need a time which equals some periods of the RF cycle before they have traversed the sheath. In a typical capacitively coupled plasma, the maximum velocity of an Ar+ -ion in a sheath field of 500 V/cm would be 5 × 106 cm/sec; the time required for traversing the sheath would then be 0.2 μsec. The duration τ of a complete RF cycle is 1/13.56 × 10−6 = 0.074 μsec. We infer from Fig. 6.9 an accelerating force on the ions in the negative half-cycle when the sheath extends and the instantaneous plasma potential decreases; in the positive half-cycle, this force inverts its sign. Therefore, ds (t) and V (t) exhibit a mutual phase shift of π. This means that it takes at least three oscillation periods until the argon ion has crossed the sheath and consequently, it will hit the electrode with the time-averaged sheath potential. As a result, we will see but a small width in the ion energy distribution function IEDF around the mean potential. For excitation frequencies below the plasma frequency of the ions, ωRF < ωp,i , however, ion energies can vary from very small values (when the sheath thickness is thin) to energies which equal the peak RF voltage (when the sheath thickness is at its maximum). The sheath thickness itself can be split into two parts, an RF component and a DC component: ds (t) = ds,DC + ds,RF sin ωt.

(6.80)

Kushner estimated the ratio ds,DC /ds,RF to about 10 in argon for perfectly capacitively coupled sheaths, i. e. for ωRF ωp,i [222]. In this case (typically ωp,i /2π ≈ 1 MHz), the ion motion can be regarded as the response to a mean RF field. • For a sheath thickness which significantly exceeds the mean free path of the ions, λi , the mean sheath voltage is large compared with the

192

6 High-frequency discharges II kinetic energy of the ions incident on the electrode (divided by their charge).

• For comparable values of ds and λi , the mean ion energy is expected to equal the mean sheath voltage. A precise approach to solve this problem would require the solution of Boltzmann’s transport equation for ions, electrons and neutral molecules which are coupled with the Poisson equation for a self-consistent field. The solution of the coupled integral-differential equations with correct boundary conditions would yield the corresponding distribution functions [223]. Another track is the Monte-Carlo method or particle-in-cell method (PIC) [189, 221]. A third track has been found by Economou et al. 1988 [224]. Since it is lucid and transparent, this model will be presented here in detail. Without temperature gradients and magnetic fields, we can consider a plasma consisting of neutrals and singly ionized ions which can be described by three mutually coupled differential equations: 1. The continuity equation [thereby the sheath is expected to lack any source (i. e. vanishing divergence)], which is a rather good approximation due to the very low electron density, which eventually vanishes. 2. The equation of motion of ions traversing the sheath with Newtonian friction (in principle, this equation should take into account the density gradient ∂ni /∂dx). 3. The Poisson equation: ∂ni ∂ni ui + = 0; ∂t ∂x

(6.81.1)

∂ui qE Fi ∂ui + ui = + ; ∂t ∂x m m

(6.81.2)

∂E qni . = ∂x ε0

(6.81.3)

To estimate the force of friction—and every scattering can be regarded as friction—all possible collisions of a test ion with an ensemble of neutrals which are Maxwellian distributed are averaged. The result is a quadratic dependence in ui : Fi = −γi u2i with γi the so-called resistance term. For that end, we consider the relative density ni of particles with mass mi after scattering into the solid angle element dΩ = 2π sin ϑdϑ. Considering conservation of energy and momentum yields for the geometric factor 1 − cos ϑ (cf. Sect. 2.2) which promotes large-angle scattering. For a hard sphere potential 2 , the resistance term can be written as σ(ϑ) = πrhs


193

2 γi = mi ni πrhs .

(6.82)

γ scales with discharge pressure. The resonant charge transfer consists in an (almost) entire exchange of momentum between a fast moving ion and a neutral atom with relatively low velocity. The idealized cross section is then approximated with a δ-function according to 2 σct (η) = πrct

δ(π − η) , sin η

(6.83)

which yields for the total cross section 2 2 σtot = π(rhs + rct ).

(6.84)

The boundary conditions are given by the Bohm criterion (cf. Sect. 14.2). Albeit derived for DC sheaths, we can apply it to the quasi-stationary RF sheath. First of all, the three coupled differential equations can be simplified with the time-averaged E-field with E = −∇V. d(ni ui ) = 0; dx ui

dui q dV γi u2i = − ; dx m dx m d2 V qni =− . 2 dx ε0

(6.85.1) (6.85.2) (6.85.3)

vi = kB Te /mi denotes the (maximum) initial velocity at the Bohmic edge. In fact, it is somewhat smaller, since the ions suffer friction losses in the Bohmic presheath, which exhibits a thickness of about one a Debye length. At the boundary presheath/sheath (x = 0), the boundary conditions are approximate ni (0) = n0 ;

ui (0) =

dV dx

kTe ; m i + γi λ D

= x=0

kB Te . qλD

(6.86.1)

(6.86.2) (6.86.3)

Inserting the Bohm velocity into the formula for the collision number per second, which is well-known from kinetic gas theory, Z = /λ = nσ (with the mean free path λ and the mean velocity : → vB ) yields a lower value for the collision number between the ions and the neutrals in the sheath. For an electron temperature of 2 eV in a discharge through argon (σtot : 26 × 10−16 cm2 ), we obtain the following values:

194


• p = 10 mTorr (1.5 Pa), ds = 10 mm: one collision, • p = 50 mTorr (7 Pa), ds = 5 mm: 3.5 collisions, and • p = 500 mTorr (66 Pa), ds = 4 mm: 30 collisions. Already at relatively low discharge pressures, we find equilibrium between the electrostatic force and friction [225]. 6.8.1 Experiments In the following paragraph, we deal with • scattering by elastic collisions, • scattering by elastic collisions and resonant charge transfer, and • effects of the pulsating RF sheath (RF modulation). It is common to apply a series of vertically arranged consecutive grids which are driven at different voltages (Sect. 2.5) [226]. The resolution depends on their distance, their open area (wire diameter) and the aperture A (Fig. 6.25) [227, 228]. Using various grids with different open surface (1−3), it is possible to vary the disc’s section C which is exposed to the beam: between 0.4 and 5 cm2 , i. e. more than order of magnitude. By application of accelerating or retarding potentials (from −125 to +125 V), it is possible to repel one type of carriers completely (so-called retarding field analyzer). ion beam

Fig. 6.25. Retarding field analyzer: Applying various apertures A and grid distances with various open areas (1 − 3) it is possible to vary the current which will caught by the disc C and eventually measured by the detector. By continous sweeping of the DC potential, the IEDF is determined [229].

The aperture A and the first grid are grounded to generate an area free of fields between the neutralization grid of the ion source and the analyzer for the ion energy. To suppress the influence of the electrons on the current signal, the potential of the downmost grid was fixed at −80 V. This is sufficient to


195

block the electrons from the downstream region and secondary electrons from the collector electrode from penetrating the beam. By deriving the curve for ion current with respect to voltage, the ion energy distribution function IEDF is determined (Fig. 6.26). 0.003 ion current density

0.100 ion energy distribution

0.075 0.050

0.002

0.001

dj/dV [a.u.]

ion current density [mA cm-2 ]

0.125

0.025 0.000

0.000 0

50

100

150

200

Fig. 6.26. Ion current density and ion energy distribution function (dI/dV ) of an argon discharge at 10 mTorr, 150 W, boundary voltage: 50 V [229].

retarding potential [V]

6.8.1.1 Elastic scattering and resonant charge-transfer. The resonant charge transfer is the most significant scattering mechanism for an ion in its parent gas, e. g. Ar+ in Ar (so-called “symmetric” charge transfer); its cross section is remarkably large (cf. Sect. 2.5). On the contrary, the “asymmetric” cross section is significantly smaller; this has been shown for discharges through − − SF6 (SF6 → SF+ 5 + F ; also ions as SF5 are detectable in high concentrations which determine the electronegative character of SF6 -discharges) by Brandt and Jungblut [230]. In most cases, these competing dissociations in several fragments reduce the density of the M+ -ion noticeably, and as a consequence, the resonant charge transfer does not play this paramount role in discharges through molecular gases than in ones through inert gases, however, it cannot be simply neglected. According to analyses with the mass spectrometer, the M+ peak does not disappear until the electron energy has passed 50 eV or higher [231]. But the ionization processes mainly occur in the plasma bulk where the mean electron energy amounts to only several electronvolts! DC discharges. The energy spectrum of ions incident on the electrode was measured for the first time by Davis and Vanderslice in a DC discharge for the three lightest inert gases [128]. Their experiment consisted of an electrode with a drilled hole and attached mass spectrometer; the (optical) sheath width could be measured with an accuracy of ±5 %. Provided that • the field declines linearly across the dark space, • all ions are generated in the negative glow,

196


• the only interaction between ions and neutrals is a symmetric charge transfer and • that the cross section is constant within the regarded energy interval, the energy distribution of the ions incident on the electrode IEDF could be described consistently. As can be seen from Fig. 6.27, in argon the IEDF at 60 mTorr (8 Pa) is very broad: Only a vanishingly small fraction of the Ar+ -ions traverses the dark space collisionless (λ/dC = 0.06 with λAr+ in Ar = 0.4 mm).

relative intensity

1.00 Ar+ in Ar VC: 600 V dC: 1.3 cm p: 60 mTorr dC/l = 15 for -16 2 s = 53 * 10 cm

0.75

0.50

Fig. 6.27. Energy distribution of argon ions in a DC discharge c The American Phys[128] ( ical Society).

0.25

0.00 0.00

0.25

0.50

V/VC

0.75

For constant discharge voltage, the IEDF is relatively weakly influenced by the discharge pressure which is a manifestation of the similarity rules (see Sec. 4.10, the product of pressure and sheath thickness is relatively constant). The same holds true for the collision number in the sheath. The functional relationsship can be written as ⎧ ⎨

⎛

⎞⎫

1 d ⎝ V /2 ⎠⎬ 1− 1− f (E) ∝ 1 · exp ⎩− ⎭ λct Vc (1 − VVc ) /2

1

(6.87)

with λct the mean free path for resonant charge transfer. RF discharges. In a capacitively coupled RF discharge, Ingram and Braithwaite measured the EEDF and IEDF of electrons and ions incident on the grounded electrode as function of pressure with a modified Langmuir probe [232]. The glass tube which serves as a shield for the tungsten rod is sealed by grid with fine mesh on ground potential. Since its width is significantly smaller than the Debye length, we obtain an equipotential surface in front of the electrode. A second grid exhibits a retardation potential to let through only one sort of carrier which are incident on a collector with sweeping bias potential; the collector current can be measured with an amplifier which simultaneously generates the first and second derivatives dI/dV and d2 I/dV 2 .


197

By biasing the second grid to sufficiently negative values, the electron current is suppressed, and all the collector current is pure ion current which is subject to an energy-dispersive analysis, i. e. we record the IEDF. The uppermost value of the bias potential with vanishing probe current marks the highest ion velocity. In Fig. 6.28, this strong dependence of the IEDF on pressure or the mean free path, respectively, is displayed. A sharp, almost monochromatic beam of argon ions traversing through argon has become a diluted distribution by raising the pressure by two orders of magnitude. The onset of thermalization has communicated to about 40 mTorr (5 Pa). Of paramount importance, however, is the factof the exponential decrease of f (v) for declining v which is an important part of f (v)dv.

1.0

normalized IEDF [a. u.]


0.8 0.6

30 mTorr 15 mTorr

0.4 7.5 mTorr

0.2 0.0 0

1.5 mTorr

10

20 Ekin [eV]

30

Fig. 6.28. Measured ion energy distribution functions (IEDF) in argon for various pressures. With rising pressure, the maximum shifts to lower energies and diminishes in intensity. The maximum energy corresponds to the free fall from the plasma potenc IOP Publishing tial to the potential of the Langmuir probe Φp − ΦLp after [232] ( Ltd).

A2 Example 6.2 The cross section for argon at 30 eV, σ(30 eV Ar+ ) equaling 38 ˚ 13 −3 yields at a discharge pressure of 20 mTorr (3 Pa, n = 70 × 10 cm ) a mean free path λ of 5.8 mm, at 60 mTorr (8 Pa), λ = 2 mm. For a sheath thickness of 12 mm, this means two collisions at 20 mTorr and six collisions at 60 mTorr, or stated another way: At 20 mTorr, a fraction of e−d/λ = 12.5% traverse the sheath without collisions, which drops to 0.2 % at 60 mTorr.

198


Both the mechanisms of ion energy broadening can be separated only by energy-dispersive mass spectrometry. In fact, even at discharge pressures of 55 mTorr (7 Pa), the plasma potential Φp at 23 eV, measured at full width at half maximum is broadened by only 2 eV when the Ar+ 2 -ion is used for its determination [195, 196]. For Ar+ 2 , Davis and Vanderslice find a sharply-peaked distribution when V /Vc equals unity, i. e. maximum sheath voltage, which resembles Fig. 6.27 at A2 at 500 eV) [128]. The integral below the peak low pressures (σscatt is only 7 ˚ had shrunk to only 12 % of the total area! These measurements clearly indicate that the resonant charge transfer dominates the scattering processes in the discharges of inert gases [201]. 6.8.1.2 Elastic scattering. This assumption is confirmed by measurements of B.E. Thompson et al. in discharges through molecular gases such as SF6 , CH3 Cl and CH3 Br [189]. Their experimental setup was similar to that of Ingram and Braithwaite, however, they investigated the high-pressure range between 190 mTorr and 1 000 mTorr (25 to 130 Pa) [232]. Since in these gases, the resonant charge transfer does not play any role because most of the ions are only fragments of the neutral species, the mean free path λ is significantly increased. For example, at 190 mTorr (25 Pa) λ amounts to about 350 μm at a sheath thickness ds of about 1 mm, and about 1 /4 of the ions traverse the sheath without any collision. For argon, λi of an ion beam with an energy of 500 eV amounts to about 27 cm to 13 m within a pressure range between 70 and 1 mPa, respectively. At the upper pressure limit, we find a significant number of fast neutrals after only a few cm. Since current densities are normally measured with a Faraday cage, calculations for the sputtering yield have to be corrected by these fast neutrals. For an exponential decay of the ion beam flux with respect to distance, resonant charge transfer can be identified as dominating scattering process. 6.8.1.3 Modulation. But even in collisionless sheaths, a broadening of the IEDF is observed which is due to RF modulation effects. This phenomenon is more distinct for lighter ions and is more easily visible at lower excitation frequencies ωRF . Its occurrence is closely related to the ratio τRF /τi with τi the transit time of an ion across the sheath and τRF the time of an RF cycle. When τi falls short of τRF , then the ions become sensitive for the instantaneous sheath potential. In the simplest case, we investigate a collisionless sheath. Provided the spatial variation of the sheath potential can be calculated according to Child’s law, and the ion velocity is monoenergetic (vx = 2e0 Vs /mi ), then [cf. Eqs. (4.24)]


199

4

Vs (x) = A x /3 ∧ A =

9ji 4ε0

2/3

1 mi /3 × , 2e0

(6.88)

with the constant which does not depend on x (x = 0 means wall). It takes the ion (s the time-averaged sheath thickness) τi =

s dx 0

v(x)

⇒ τi = 3s

mi 2e0 Vs

(6.89)

to fall across the sheath, and for the ratio τi /τRF we obtain 3sω τi = τRF 2π

mi . 2e0 Vs

(6.90)

For τi τRF , i. e. in the low-frequency regime, the ions can respond to the instantaneous sheath potential which finally results in a bimodal IEDF with ΔEi the difference between minimum and maximum sheath drop which scales with the ratio τRF /τi . For more realistic sheaths, the modulation effect has been evaluated by Benoit-Cattin and Bernard for the first time [233] (Fig. 6.29):

f(E)

8e0 VRF ΔE = 3ωds

e0Vs

2e0 VDC , mi

(6.91)

Fig. 6.29. Benoit-Cattin and Bernard derived the bimodal IEDF for a monoenergetic initial ion velocity. The two singular peaks are symmetric about e0 Vs pictured here for two different masses [233].

ion energy [a. u.]

with ds the sheath thickness. Their calculation yielded a bimodal IEDF with two peaks which are also symmetric about e0 Vs , and the energy difference ΔE between them is proportional to ωi /ωRF : Rising ωRF and/or enlarging mi will decrease ΔE, and the two peaks will approach each other and will eventually merge to one single peak. Since the IEDF is assumed to be initially monoenergetic, both the peaks are singular.

200


Four years later, Coburn and Kay performed an experiment in the sheath region of a grounded electrode [196]. In fact, they observed a severe broadening in the ion energy distribution, in particular in light gases (Fig. 6.30).

Eu+ +

Fig. 6.30. Energy distribution of several ions very different in mass in an RF discharge at 13.56 MHz through argon, 75 mTorr, electrode distance: 50 mm, measured at the grounded electrode [196] c The American Institute of ( Physics).

IEDF [a. u.]

H2O

H3+ H2O+ DE

50

H3+

75 100 125 ion energy [eV]

150

175

√ Since the spread in energy ΔE/E scales inversely to mi , this means for water (M = 18) at 100 eV a ΔE of 20 eV, but for Eu (M = 151) only 4 eV. Since the quantity that is responsible for this effect is the ion transit frequency which has to be compared with the operating frequency, we obtain nearly the same spread when we investigate same masses at various operating frequencies. Again, we find the maximum more sharply peaked for E = e0 V s , when we have left the regime for almost instantaneous field response (cf. Eq. (11.3) and Fig. 6.31). 104 +

Cl2 ; 13.56 MHz +

Cl2 ; 100 kHz Cl+; 100 kHz Cl+; 13.56 MHz

IEDF [a. u.]

103

102

Fig. 6.31. Energy distribution of Cl+ - and Cl+ 2 -ions at two different operating frequencies (100 kHz and 13.56 MHz) [234] c The American Institute of ( Physics).

101

100

0

100

200 300 ion energy [eV]

400

500

Hence, ion inertia causes three RF modulation regimes:


201

• (Very) light masses (H+ ): The ion transit time is comparable or even less than the operating frequency. Low DC bias values are accompanied with a distinct bimodal energy spectrum with a typical saddle profile around e0 (VDC ± Vs ). • Medium masses (H2 O+ , Ar+ ): The width of the saddle profile shrinks, it is still remarkable that the profile is asymmetric around the value of VDC and the height of the lower peak of the IEDF always exceeds that of the high-energy peak. Coburn and Kay put this down to the resonant charge transfer which cannot be separated from the RF modulation. For example, the ratio ΔE/E for the Ar+ 2 -ion (no resonant charge transfer possible with the mother molecule) is similar in sharpness to that for Eu+ , although the mass ratios amount to mEu /mAr2 = 152/80 ≈ 2. • Heavy masses (Xe+ ): The ions can only respond to the time-averaged potential V s (t), ΔE/E becames progessively sharper for rising mass or operating frequency. 6.8.2 Computer simulations Simultaneously, numerous Monte-Carlo simulations were performed which took account of collisions within the sheath. Solving the equations of motions in the electric field across the sheath yields the trajectories of the ions between collisions. Applying the Monte-Carlo method randomizes the moment of collision. The cross section itself can be expressed as the sum of the three contributions σtot = σelast + σinelast + σct .

(6.92)

In the case of great uncertainties —this is the case for very low energies —model potentials an be applied (model of hard spheres for elastic collisions: σelast → σhs , Lennard-Jones for the inelastic collision between ions and neutral molecules, σinelast → σLJ ). After every collision, the velocity components, i. e. momentum and kinetic energy, are calculated again. Since there are large differences in velocity between ions and thermal molecules (< E > ≈ 1/40 eV) the calculation can be simplified by the assumption of resting molecules, and collisions between ions can be neglected due to their low density. Next, the oscillating sheath has to be taken into account for the specific moments of inertia of ions and electrons which can respond to the instantaneous electric field up to the GHz range (see Sect. 14.4). It is evident that the time interval for the computation should be small compared • to the time between two collisions: Δt < τ ≈

1 ν

=< v > /λ and

• to the reciprocal value of the operating frequency: Δt < 2π/ω,

202


otherwise, we would take into consideration only the time-averaged E-field and a Maxwellian distribution. On its course, the single ion trajectory is calculated until the final impact on the surface of the electrode. This calculation is then repeated for many ions. Under the assumption that the time average equals the ensemble average, this yields simulated IEDFs and IADFs, often displayed as histograms. The accuracy of this simulation depends on the choice of • the initial conditions: The ions enter the sheath with Bohm velocity vB , • the correct sheath potential and its dependence on the thickness, i. e. the electric field across the sheath, • the accuracy of the differential cross section, • the ratio of sheath thickness over the ionic mean free path ds /λi , and • the size of the ensemble (N = 20 000 is required for substantial statistics, whereas trends are already recognizable for N ≈ 6 000 [235]). Kushner calculated the pressure dependence of the IEDF in discharges through argon [221] and fixed the threshold of thermalization at 40 mTorr (5 Pa) which is in accordance with the results of Ingram and Braithwaite. However, he took account only of the resonant charge transfer. Therefore, B.E. Thompson et al. repeated the calculations and added elastic collisions [189] which does not only modifies the IEDF but also broadens the IADF, i. e. the fragmentation and scattering of the ion beam (the IADF cannot be changed by the resonant charge transfer since the transfer of momentum is supposed to happen in only one collision). Their results can be compiled as follows: • If the ions are supposed to act as hard spheres, in the uniform field of a sheath exhibiting a thickness of 3λi , about 80 % of the ions are thermalized. To have a significant fraction of ions traversed without any collision across the sheath, the sheath must significantly shrink (Fig. 6.32, E ∗ denotes the gain of kinetic energy by the time-averaged field). • For fields exhibiting a spatial dependence as strong as across the sheaths, this value can be doubled to reach about 6λi (Fig. 6.33). But even for ds /λi = 1, i. e. after only one collision, a siginificant fraction of the ions exhibit a direction which deviates from normal incidence.13 • Considering only charge transfer yields an exponential decrease of ion energy, as has been described by Davis and Vanderslice [128]. The mean energy has been cut down to half that value obtained by elastic scattering, since the amount of energy transferred in the latter case is 13 The ratio ds /λi does not exactly equal the number of collisions, since by simple occurence of collisions, the effective sheath thickness grows, and therefore, the transit time as well.


203

1.5 5.5

0.06 0.23

1.0

1

IADF

IEDF

0.04 1

3

0.5

3

0.02 5

5

0.0 0

7

7

2

4 * E/E

6

8

0.00 0

20 40 60 angle of incidence [°]

80

Fig. 6.32. Applying a Monte-Carlo method, B.E. Thompson et al. calculated the IEDF and IADF for several ratios of ds /λi (scales approximately with the collision number) during its course across the sheath for hard spheres and a uniform DC field. For better guiding of the eye, the vertical axis is shifted, E ∗ denotes the gain of kinetic c The American Institute of Physics). energy by the time-averaged field [189] (

significantly less (Pex = 0 for elastic scattering). The case of Pex = 0.5 lies between total elastic scattering and total charge transfer (Fig. 6.34). • For higher energies, the IADF becomes more sharply peaked: A larger fraction of ions traverses the sheath without any collision, since the time for interaction declines (the cross section for elastic scattering decreases with increasing energy beyond the maximum at 100 − 150 eV). • Raising the discharge pressure broadens both the functions IEDF and IADF. • Ions with large a deviation from normal incidence exhibit a lower kinetic energy according to tendency. Therefore, for simulations which model etch profiles, the dependence of the ion flux on energy and angle of incidence has to be taken into account. In addition, B.E. Thompson et al. simulated RF effects with parameter M = ωt, the number of oscillations between the collisions. In general, in the upper limit M → ∞, the distribution should be solely determined by the DC field, and in the lower limit M → 0 by the AC field, respectively. For λi /ds ≈ 1, they found a variation in energy of about 30 % and in angle of about 20 %, at λi /ds ≈ 1/3 , however, these values have been reduced to about 5 and 2 %, respectively. In conclusion, for large values of λi /ds , the IEDF and IADF are tremendously affected by the operating frequency, but for values of λ/dS ≤ 5, the influence is negligible.

204


0.15

0.10

2 IADF

IEDF

3

2

1

2 4

0.05

4 6

6

8 12

0 0

2

4 * E/E

8

6

12

0.00 0

8

20 40 60 angle of incidence [°]

80

Fig. 6.33. IEDF and IADF, calculated with the Monte-Carlo method, for several number of collisions of hard spheres on their course across the sheath supposing a linear field. The vertical scales have been shifted to distinguish the various curves. E ∗ c The American denotes the gain of kinetic energy by the time-averaged field [189] ( Institute of Physics).

1.0

0.06

d(Pex=1.0) = 1.00 d(Pex=0.5) = 0.537

0.8

0.04 IADF

IEDF

0.6 Pex = 0.0 Pex = 0.5 Pex = 1.0

0.4

Pex = 0.0

0.02

0.2

Pex = 0.5

0.0

0.00 0

1

2

3 *

E/E

4

5

0

20 40 angle of incidence [°]

60

Fig. 6.34. Applying a Monte-Carlo method, the IEDF (LHS) and IADF (RHS) are calculated for a scattering mixed in character (collisions between hard spheres and symmetric charge transfer) for a linear DC sheath field. Pex denotes the probability for a symmetric charge transfer which does not cause a change of the IADF (δ functions peaking at ϑ = 0◦ with height 1.00 and 0.58), but does influence the IEDF tremendously. E ∗ denotes the gain of kinetic energy by the time-averaged field [189] c The American Institute of Physics). (


205

6.8.3 Hybrid sheath model Brinkmann et al. have developed a (one-dimensional) Monte-Carlo model that couples the equations of motion with the Poisson equation for both types of carriers [cf. Eqs. (6.81)] [236]: ∂ne,i ∂ne,i ui + = 0; ∂t ∂x

(6.93.1)

qE ∂ui Te,i ∂nr,i ∂ue,i + ue,i = − νm(e,i) ue,i ; + ∂t ∂x me,i me,i ne,i ∂x ε0

∂E qi,j ni,j + e0 ne = 0. = ∂x j

0.025

(6.93.3)

0.07 50 mTorr (6.7 Pa) 85 mTorr (11.3 Pa) 120 mTorr (16 Pa)

0.015 0.01

IEDF [a. u.]

0.06

0.02

IEDF [a.u.]

(6.93.2)

109/cm3 10 3 10 /cm 11 3 10 /cm

0.05 0.04 0.03 0.02

0.005

0.01 0

0

50

100 150 200 energy [eV]

250 300

0

0

50

100 150 200 energy [eV]

250

300

Fig. 6.35. IEDF as functions of discharge pressure p (LHS) and plasma density np (RHS). Since the sheath thickness scales inversely with pressure and plasma density, already at np = 1010 /cm3 the energy of the ions incident on the electrode is only marginally influenced by collisions, and it is no longer detectable at 1011 /cm3 [237].

The term for friction in Eq. (6.93.2) depends on Langevin’s energy loss parameter and on the cross sections [as in Eqs. (6.81) and (6.82) − (6.84), the cross section of elastic scattering is denoted by σelast , and that of the resonant charge transfer is σct ]. As emphasized in Sect. 2.5, in particular in discharges of inert gases, the resonant charge transfer dominates the scattering reactions; by this process, fast neutrals are generated which influence the character of the discharge itself but also take a significant part of secondary reactions (surface reactions: sputtering or etching). Hence, in the study of Sabisch et al., a large part is devoted to the contribution of neutral molecules and their distributions in energy (N EDF ) and angle (N ADF ), respectively [237].

206

6 High-frequency discharges II 0.3 0.4 50 mTorr (6.7 Pa) 85 mTorr (11.3 Pa) 120 mTorr (16 Pa)

0.2 0.15

9

IADF [a. u.]

IADF [a. u.]

0.25

normal incidence: 0°

0.1

0

3

10 /cm 10

3

11

3

10 /cm 10 /cm

0.2 0.1

0.05 0

0.3

2

4 6 angle deviation [°]

8

0 0

10

6 4 2 angle deviation [°]

8

10

Fig. 6.36. IADF as function of discharge pressure (LHS) and the plasma density np (RHS). The beam melts away with rising pressure and rising plasma density, but high a plasma density (1011 /cm3 ) guarantees an almost perfect forward direction [237].

0.07

0.12

0.06

0.10

0.05 0.04 0.03

50 mTorr (6.7 Pa) 85 mTorr (11.3 Pa) 120 mTorr (16 Pa)

0.02

NADF [a. u.]

NEDF [a. u.]

6.8.3.1 Ions. For pressures beyond 45 mTorr (6 Pa), no bimodal distributions are calculated any more (cf. Figs. 6.35−6.37), and the mean energy declines with rising pressure; simultaneously, the beam incident on the electrode broadens in energy and angle (normal incidence: 0◦ ).

0.01 0 10 20 30 40 50 60 70 80 90 100 energy [eV]

0.08 0.06 0.04

50 mTorr (6.6 Pa) 85 mTorr (11.3 Pa) 120 mTorr (16 Pa) normal incidence: 0°

0.02 0 0

5 10 15 20 25 30 35 40 45 50 angle deviation [°]

Fig. 6.37. N EDF and N ADF as functions of discharge pressure. The kinetic energy can reach 100 eV, and higher pressures lead to a reduction in mean energy. The angle distribution sharply peaks at normal incidence, but exhibits a second, less pronounced maximum around 20 − 30◦ [237].

6.8.4 Measurements and modellings 6.8.4.1 IEDF in the sheath. These measurements were extended by Liu et al. applying a time-of-flight method in discharges through argon at 13.56 MHz


207

with pressure as parameter (Fig. 6.38) and compared with own Monte-Carlo calculations (Fig. 6.39) [238]. Although the pressure dependence of the double peak which is created by the harmonic excitation of the sheath could not be reproduced, the calculations resulted in fully developed distribution functions at 500 mTorr (67 Pa) of energy (IEDF) and angle of incidence: After having experienced more than thirty collisions during traverse across the sheath, the shape of the IEDF has changed to Maxwellian behavior. 0.10

Fig. 6.38. Measured dependence of the IEDF on pressure in a capacitively coupled discharge through argon at 13.56 MHz. The double peak at very low pressures is well resolved; at 500 mTorr (67 Pa), the IEDF is fully developed exhibiting Maxwellian behavior (after [238]).


0.08

IEDF

0.06 0.04 0.02 0.00 0

10

20

30

40

ion enenergy [eV]

The maximum has retreated to only several electronvolts and has become independent of DC bias. Furthermore, the IADF exhibits isotropic behavior: The ion energy is independent of the angle of incidence. Albeit care has to be taken when interpreting this result —in particular, the large discrepancy between the primary beam and its simulation is striking—this marks the upper limit of an anisotropic angular ionic distribution (Fig. 6.39).14 The occurence of a bimodal IEDF can be observed only at low pressures with low collision numbers. Only in this case can ions traverse the sheath without any collision responding to the instantaneous sheath potential. Hence, we are far away from equilibrium. For higher pressures (and correspondingly higher collision numbers) the ion distribution within the sheath turns to a Maxwellian distribution beyond a collision number which typically exceeds 6, which allows us to define an ion temperature within the sheath (cf. Sect. 14.1). 6.8.4.2 IEDF in the sheath of the powered electrode. By means of an energy dispersive mass spectrometer, Becker et al. investigated the sheath of the powered electrode in discharges of SF6 which exhibits a strongly electronegative behavior even at very low pressures. The ions enter the spectrometer through an aperture with radius 50 μm. Since the floating system is shielded against RF 14 As has been pointed out in Chap. 2, the differential cross section at very low energies is difficult to measure and still remains unknown for most of the reactive gases; the collision frequency, however, does depend on the cross section.

208

6 High-frequency discharges II 0.20

0.10

0.15

IADF

IADF


0.05

10 mTorr (1.3 Pa) 50 mTorr (6.7 Pa) 500 mTorr (67 pa)

0.10

0.05

0.00

0.00 0

10

20

30

0

40

incidence angle [°]

10

20

30

40

incidence angle [°]

Fig. 6.39. IADFs of a discharge through argon at 13.56 MHz and various pressures, LHS: measured, RHS: Monte-Carlo simulation (after [238]).

fields and against the electrode, the potential difference between spectrometer and DC bias can be adjusted to zero. In particular, the aperture is free of DC fields (but is still exposed to the RF field of the powered electrode). In Fig. 6.40, the IEDF for the SF+ 5 ion is depicted at two different pressures. The simulated spcetrum has been calculated according to Wild’s model [240].

count rate [a. u.]

10 mTorr (1.5 Pa)

calculated experimental 1 mTorr (0.15 Pa)

0

50

100

150

200

250

Fig. 6.40. Measured and calculated IEDF of SF+ 5 at 3 and 10 mTorr (0.5 and 1.5 Pa) at VDC = −200 V. At the low pressure, the saddle-shaped profile is very well resolved. The peaks at low energies are caused by additional ionizations in the sheath of the powered electrode [239].

ion energy [eV]

They calculated the sheath thickness to be 3.4 mm at 3 mTorr (0.5 Pa) and 10 mm at 10 mTorr (1.5 Pa). In the range of the saddle profile, the agreement between measured and calculated distribution is relatively poor which is attributed to additional ionizations in the sheath. According to Biehler’s model, these ionizations are also made responsible for the peaks at significantly lower energies [241]. In conclusion, the elastic scattering changes the kinetic energy of the ions with a strongly forward direction into one with a substantial component of the random walk. Since scattering scales with particle density, a rise in pressure

6.9 DC discharges and capacitively coupled RF plasmas

209

lowers the mean energy of the carriers. In discharges of inert gases, resonant charge transfer is the main scattering mechanism.

6.9 DC discharges and capacitively coupled RF plasmas The character of capacitively coupled discharges is dominated by the sheaths in front of the electrodes. Without the RF field, there exists a boundary between the quasi-neutral plasma and the grounded surface, however, this sheath is very thin and the sheath potential drops to several kB Te /e0 [Eq. (3.24)]. In RF discharges, the sheath potential at the grounded electrode can rise to some hundreds of volts, which is joint with an increase in sheath thickness to several tens of a Debye length because of the rectifying response of the plasma to the RF field. This effect is mainly caused by nonlinearities of the sheath impedance (Sects. 14.4.1/2). Across the sheath, the electron density drops and eventually vanishes which causes the displacement current in the sheath to become comparable with the conduction current in the plasma bulk. For a collisionless sheath, both become equal at the plasma frequency of the ions, ωp,i , in the plasma bulk, however, at the plasma frequency of the electrons, ωp,e , with ωp,e ωRF ωp,i (ωRF denotes the operating angular frequency). The RF discharge exhibits the characteristics of two different types: properties of the negative glow and that of a hollow cathode. • Negative glow: Dissipation of electric power into the plasma by accelerated electrons which collide with neutrals elastically and inelastically. This process dominates in the plasma bulk for x > ds . In the plasma bulk, the electric field induced by the electromagnetic field is small, but finite. For example, at a power density of 0.4 W/cm2 , the electric field

E2 =

1 μ0 /2 S, ε0

(6.94)

amounts to about 12.5 V/cm. • Hollow cathode: In this type of discharge, power is transferred by beamlike γ-electrons which are emitted by the electrode. This process can become dominant in RF sheaths (for plasma electrons, the sheath is uphill, seen from the valley; for sheath electrons, the sheath is downhill, seen from the hill). This is the so-called γ-regime [175]. An additional—collisionless—process with high efficiency, stochastic heating by the pulsating sheath, takes over at about 75 mTorr (10 Pa) when the

210


electron mean free path λe becomes larger than the distance between the electrodes. Especially in the high-energy tail of the EEDF, this heating mechanism dominates all other processes. The HF ionization mechanisms are far more effective than in DC discharges (α-ionization); the breakdown voltages are lower at the same pressure, and electrode processes become of secondary importance. Eventually, the electrodes can be located out of the vacuum reactor (α-regime). AC discharges with an operating frequency significantly lower than the ion plasma frequency (ωRF ωp,i ) behave as DC discharges with changing polarity; but it is only in the RF range that these changes become so fast that charging of insulators is effectively suppressed. With rising operating frequency, the sheath impedance declines; and the center of ionization shifts further into the center of the plasma bulk. Albeit in both types of discharges, the ionization processes take place predominantly in the plasma bulk, but to sustain the discharge, electrode processes are of paramount importance in DC discharges. This is quantitatively summarized in a model which has been developed by Graves and Jensen [242]. Provided that the electron mean free path is large compared to the Debye length: λe λD , the electron temperature Te soars across the cathode fall, since freshly generated γ-electrons are accelerated along the electric field across the dark space. With weaker electric field, but still very low electron density, Te begins declining. Simultaneously, the ionization rate goes up (cf. Sect. 3.5). Since the ionization rate is a reaction of second order, it linearly depends on ne , but exponentially on Te according to the simplest approach of Arrhenius):

d[A+ ] Eion . = k2 [e− ][A] ∧ k2 = k0 exp − dt kB Te

(6.95)

Due to the opposite direction of these effects, a maximum in the ionization rate should occur. After having cooled off, the electrons remain at almost the same temperature (energy) across the distance between the electrodes (Fig. 6.41). 4

8 6

ionization rate

2 4 2 0 0.0

electron temperature

0.2

0.4 0.6 relative distance

0.8

0 1.0

ionization rate [a. u.]

electron temperature [eV]

10

Fig. 6.41. Electron temperature and ionization rate in a DC discharge as a function of the relative electrode distance from anode (0.0) to cathode c IEEE). (1.0) [242] (

6.9 DC discharges and capacitively coupled RF plasmas

211


4

normalized ionization rate

5

0° 90° 180° 270°

3 2

90°

1

270°

0 0.00

0.25


0.50

0.25 0° 90° 180° 270°

0.00 0.00

1.00

0.25


1.00

Fig. 6.42. Electron temperature (LHS) and ionization rate (RHS) in an RF driven c discharge as a function of relative electrode distance within one RF cycle [242] ( IEEE).

Whereas in RF discharges the electron temperature exhibits a similar behavior, the ionization rate has shifted farther into the plasma bulk. Although the role of the sheath for ionization cannot be neglected, it is highly modulated and can become almost zero (Fig. 6.42). Compilation of excitation mechanisms: • Excitation of γ-electrons (predominant in DC discharges). • Ohmic heating (very efficient in RF discharges). • Stochastic heating (only in CCP RF discharges).


10

1 DC RF

0.1 -3 10

Fig. 6.43. Electron temperatures in DC and RF discharges through mercury vapor [243]. 10-2

10-1 100 pd [Torr cm]

101

Compilation of ionization mechanisms:

212


• γ-electrons generate an avalanche of charged carriers during their fall across the dark space (sheath); • photoionization (negligible in RF discharges); • α-ionization in the negative glow (plasma bulk); • symmetric charge transfer. In RF discharges, frequency-dependent scattering processes become of great importance; among them, the frequency of elastic collision, νm , the angular operating frequency (ωRF or simply ω), and the plasma frequency of the ions, ωp,i , play a significant role (RF modulation). For both types of excitation, in the pressure regime below 1000 mTorr (130 Pa) the carrier generation is dominated by collisions, the carrier loss by ambipolar diffusion. Hence, the electron temperature, Te , remains almost constant and depends on the product p a with p the gaseous pressure and a the dimensions of the reactor a = 1/(1/R + 1/L) with R the radius and L the length of the plasma, provided the drift velocity is small compared to the thermal velocity of the electrons [243] (Fig. 6.43).

6.10 Summary • In capacitive RF plasmas of electropositive atoms, the discharge impedance is dominated by the sheath capacitancies, and the plasma resistivity can be neglected. • All the applied voltage is dropped across the two sheaths, and the sheaths act as a capacitive divider for the applied voltage. The large differences in electron and ion mobilities cause the plasma to act as a rectifier. • Model: Due to the high electron mobility, the instantaneous plasma potential cannot drop below the instantaneous potential of either electrode. • Model: For symmetric capacitively coupled systems, this leads to a sinusoidal plasma potential with lowermost values of zero, whereas the topmost values of the sheath potentials, which are equal in size, just reach zero. The potential drop across the electrode sheaths which is caused by the flowing displacement current is symmetric in size, and this negative selfbias amounts to about 1/2 VRF . The symmetric system develops a relatively high plasma potential and a very low self-bias voltage. • Reality: During a very short time of the RF cycle, the electrodes become positive with respect to the plasma, and a strong electron conduction current flows.

6.10 Summary

213

• In asymmetric capacitively coupled systems, the negative self-bias at the smaller electrode is larger in size to maintain the continuity condition for current. Simultaneously, the plasma potential adjusts to lower values, to keep the electrons on ground potential for a short time of the RF cycle. To check whether an intended asymmetric system acts properly, an easy, highly recommended procedure is etching of wafers, covered for example with SiO2 , at either electrode. If the etch rates are comparable or, even worse, are almost equal, then the system is symmetric. • The most severe limitation is the impossibility of enlarging the plasma density with increasing power input. Parasitic processes at low pressures are increased ion bombardment which is mainly due to a steep rise of the plasma potential, and generation of hot electrons at high pressures by the heating of plasma constituents (neutrals and ions) because of the strong increase of elastic collisions between electrons and heavy particles.

7 High-frequency discharges III

We start with a comparison of advantages and disadvantages of capacitively coupled discharges, by which it becomes obvious why and how the semiconductor industry has triggered the demand for high-density plasmas. In the first half, this reflection is devoted to inductively coupled plasmas, in the second half to resonant excitation methods. To focus in this chapter on fundamental properties and interrelations, important, but tedious derivations are deferred in Chap. 14 (Sects. 14.5 − 14.7).

7.1 Introduction The steep potential decrease in the sheaths and an extended zone of constant, relatively high charge density, which is closely connected with a uniform, constant potential, is the main advantage of capacitive discharges. This further simplifies the description of the discharge, since it can be confined to only one dimension. In a parallel-plate reactor, we have two adjacent electrodes whose diameter 2r is large compared to their distance d, in its hexagonal counterpart, we have a coaxial configuration of the electrodes (r/d 1). The excitation frequency f = ω/2π, typically 13 or 27 MHz) is far below ωp,e , but significantly above ωp,i which leads to • a quasi-stationary character (λ/r 1), • and a separation into the glowing, quasi-neutral plasma with thermal electric carriers and sheaths across the electrodes. Here, the plasma density decreases by orders of magnitude, whereas the potential drops to very low values. As a direct consequence, the ions will hit the surface of the negatively biased electrode with high a specific kinetic energy of 1/2 VRF in symmetric discharges, and VRF in highly asymmetric discharges. With the wisdom of hindsight, we now see that despite their advantages compared with DC discharges, the capacitively coupled plasmas suffer from the following inherent disadvantages: • The degree of coupling is very low and deteriorates with decreasing pressure which is required for good anisotropy and radial uniformity of etching processes (cf. Chaps. 11 and 12). G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 7,

215

216


• The energy which is transferred to the electrons scales with the ratio of electric field intensity and discharge pressure E/p (cf. Sects. 4.10 and 5.2). Ion flux and accelerating field cannot be varied independently but will be altered in the same direction. For high ionization which leads to a heavy ion flux incident on the hot electrode, intense electric fields are required inherently, for good radial uniformity, the discharge should be kept low. Hence, for an etching process with good performance, the ratio E/p must reach large values. However, with increasing E/p, we have to face severe setbacks: Not only crystal damage, but also the fidelity of pattern transfer will deteriorate, and the Line Edge Roughness, LER, increases [244, 245]. • From first principles, it is the electrons which gain energy from the RF field. However, an amount sufficient for ionization can only be piled up by collisions with heavy particles (conservative field, ∇ · E = 0). Especially at high power input, this leads to significant parasitic heating of the ions which finally defines the upper limit of capacitive coupling.

To overcome the limits of capacitively coupled plasmas, several high-density plasma sources have been invented in the last two decades. To identify the parameters which are responsible for an enlargement of the plasma density, we observe an ion current nB emanating from the Bohmic edge incident on the electrode with initial Bohm velocity vB . The energy of the electromagnetic wave is transferred to the electrons which dissipate the energy via collisions with the heavy plasma constituents. Hence, the absorbed power Pabs will be Pabs ∝ e0 nB vB V

εi

(7.1)

i

with εi the energies of the various excitations for electrons and ions, plasma density nB at the Bohmic edge, which delivers for nB : nB ∝

Pabs . vB V i εi

(7.2)

Hence, nB can be enlarged by these measures: 1. Increasing the power input Pabs . 2. Decreasing vB {only limited application since vB depends only on the square root of the electron temperature (and from the ionic mass) [cf. Eq. (14.30)]}. 3. Reduction of the surface of the plasma by reduction of the volume of the plasma reactor by which carrier losses at the walls are diminished.

7.2 Inductively coupled plasma

217

4. Selectivity of the energy transfer, i. e. excitation of only the electrons and avoidance of ionic excitation by reducing the sheath potential. One possible improvement is the application of a triodic system in which a second carrier electrode serves to separate the processes of plasma generation (first electrode) and control of the ionic flux and ionic energy (second electrode). This is used extensively in so-called dual-frequency systems which have been en vogue since the mid-1990s. Here, a frequency higher than 13.56 MHz is capacitively coupled to the “hot” electrode (the plasma density scales with the operating frequency following a relationship with a slope less than unity) and the sample is placed above the second electrode, which is driven with 13.56 MHz or lower frequencies. Another method is Magnetically Enhanced Reactive Ion Etching, MERIE: By application of a magnetic field, the efficiency of power transfer and plasma confinement are improved (item 2) [246]. For a given cylindrical reactor with radius R and length L, by application of a strong axial magnetic field, the area for possible losses is reduced from 2πR(R + L) to 2πR2 ; and in directions rectangular to the magnetic field, the diffusion length increases (cf. Sect. 14.5). Disadvantages are E×B drifts which cause radial and azimuthal inhomogenities. Therefore, • the magnetic field should be separated from the process chamber or • should be entirely avoided. The first end is pursued in plasmas which are excited by guided whistler waves, the second one in inductively-coupled discharges (cf. Figs. 7.1). In the first mentioned discharges, the coupling of wave fields is enforced through a dielectric window (ECR discharges) or by a λ/4 or U-shaped antenna (helicon discharges).1 For these discharges, an additional static magnetic field is required. The plasma is penetrated by slow whistler waves.2 In these plasmas, the plasma source is spatially separated from the reaction chamber, and we can separately control the density of the ions and their energy and flux. Therefore, we speak of “downstream” or “remote” plasma sources. By renouncing the magnets, the antenna will mutate to a coil, and we will generate an inductively coupled plasma.

7.2 Inductively coupled plasma For inductive coupling, the antenna is wound around the vacuum recipient and is denoted as coil. This is the helical or solenoidal configuration where the turns 1 The application of a conventional RF design with a π-network is advantageous: The antenna itself serves as inductivity L. 2 “Slow” means: The group velocity of these waves is comparable with or equals the thermal velocity of the electrons: vg2 ≈< ve2 > which allows Landau damping (Sect. 14.3).

218

7 High-frequency discharges III RF ICP coil

RF

ICP coil

CCP electrode

CCP electrode

pumps

pumps

microwave n n

dielectric window

solenoids n

solenoids

n

n

RF

n

antenna

CCP electrode CCP electrode

pumps pumps

Fig. 7.1. Various types of reactors for high-density plasmas. Top: two different coils (wound helically or spirally around the reactor) for inductive coupling, bottom: reactor with magnetically guided whistler waves. LHS: helicon wave, RHS: ECR (lower solenoid not necessary).

have constant diameter. In another widely used arrangement, the coil is placed above the cylindrical reactor in a spiral, planar or stovepipe-type configuration. For example, the GEC reactor has this configuration. Both configurations have their merits and disadvantages; one of the most remarkable features of the GEC reactor is the small gap (several centimeters) between the upper dielectric window and the substrate electrode which can be capacitively coupled to the plasma [247, 248]. The principle of excitation is explained using the helical arrangement, the self-screening at high plasma densities and the transitional jump from the E-mode to the H-mode is shown using the planar configuration. The


219

fields are transmitted through a dielectric window which also serves as a sealant against the atmosphere. To reduce capacitive coupling, a metal sheet or, better, a mesh is placed upon the dielectric window, because the losses caused by eddy currents are smaller in the mesh (Fig. 7.2).

RF generator 13.56 MHz

CH4

p-network

BCl3

ICP coil Faraday shield gas ring

SEERS Langmuir OES

substrate electrode with He-backside cooling

Cl2 H2 O2 Ar

Fig. 7.2. Principal setup of an inductively coupled plasma source with Faraday shields and helical arrangement of the RF coil: turns wound around a quartz cylinder to get rotational symmetry with respect to the axis of the cylinder. The frequencies for plasma generation in the ICP source and for the capacitively coupled substrate holder are often different. The diagnostic tools are installed in the downstream region.

RF generator 2 - 13.56 MHz

L-network

7.2.1 Transformer Model As is evident from Fig. 7.3, the RF current in a coil with n turns and length l induces an azimuthal magnetic field which is orientated normal with respect to the coil current and radially outward from the middle axis. This H-field, in turn, induces a second azimuthal electric field by which a plasma current is generated that is directed antiparallel with respect to the coil current (Lenz’s law). Due to their very high carrier density, the plasmas effectively screen themselves against the field across a skin length δ. Since both the “coils” are mutually telescoped and the inner “coil” consists of only one turn, the induced current density in the plasma is extremely high (principle of the induction oven). Inductively Coupled Plasmas (ICPs) are often denoted as Transformer Coupled Plasmas (TCPs). The power is mainly transferred by Ohmic heating from the toroidal induced electric field; due to the small thickness of the sheath, stochastic heating is negligible. In most cases, the plasma potential is significantly lower than 20 V. As can be seen from Fig. 7.4, the plasma flows to a substrate holder which can be capacitively coupled to the plasma. Since it can be steered independently from the plasma source, plasma density and ion energy can be controlled without

220


mutual influence. In contrast to a capacitively coupled sheath with its electric field directed normally to the electrode, this induced electric field is aligned in parallel fashion with respect to this electrode. The advantages of inductively coupled sources are:

.

B

d

H

E

IRF

r R

l

plasma

IP

Fig. 7.3. LHS: The time varying magnetic field generates a time-varying azimuthal electric field which induces a circumferential current in the plasma which is aligned in parallel fashion with respect to the reactor wall. RHS: Principal sketch of the currents in a plasma source (coil and plasma) driven by inductive coupling (helical configuration, turns wound around a quartz cylinder to get rotational symmetry to the axis of the cylinder, electrons are negatively charged).

• Simple configuration of reactor and plasma source. • Spatial separation of ion generation (by Ohmic heating in the inductively coupled plasma source) and ion energy (in the sheath of the capacitively coupled electrode). • No static magnetic fields, but only RF fields. • Compared with capacitively coupled plasmas, a significantly higher plasma density (1 to 11/2 orders of magnitude higher); compared with resonant plasma excitation methods (ECR or helicon), the plasma density is lower by about one order of magnitude. • No resonant coupling, these discharges rather “forgive” deviating primary plasma parameters (pressure, RF power, gas composition) than discharges which rely on resonant energy transfer. • The magnetic field is orientated parallel to the reactor walls, which remain free of fields and currents; hence, the plasma potential Φp drops only in the sheath of the reactor wall and possible sputtering effects remain negligible.


221

In the plasma source itself, we detect spatial inhomogenities in density and temperature of the electrons due to the small skin depth, which is transferred to the heavy ions via the ambipolar plasma properties [249] (Figs. 7.5, 7.12 and 7.14). These inhomogenities become even more pronounced in the downstream direction, by which compensating currents are triggered. The magnitude of this diffusion potential can be calculated using the Einstein-Smoluchowski equation (mobility μ = e0 D/kB T with D the diffusion coefficient, np : plasma density in the plasma source, n: plasma density at the capacitively coupled electrode):

V =

n kB Te ln . e0 np

(7.3)

RF generator 13,56 MHz 2 ERF

p-network 1 ERF

1 ERF

BRF afterglow

EDC

Fig. 7.4. Principal sketch of the experimental set-up and fields and currents in a plasma source driven by inductive coupling (helical configuration, turns wound around a quartz cylinder to give rotational symmetry to the axis of the cylinder as combined from Figs. 7.2 and 7.3).

RF generator 2 - 13.56 MHz

L-network

As a final result, the most important plasma parameters (plasma density, electron temperature) are nearly equalized downstream at wafer level (common distance: 10−15 cm). As prominent evidence, the supreme uniformity in etchrate for most materials and ambient gases has already been mentioned. Today, radial uniformities within ±1% are demanded across diameters between 8 and 12". For too small a gap between window and substrate, these inhomogenities are likely not to be levelled out. Hopwood et. al. swept a Langmuir probe across the plasma at a constant distance z below the window and could clearly judge a distance of 5.7 cm as being too short to have the inhomogenities levelled off [250]. They could significantly improve the radial uniformity by applying a magnetic multipole arrangement (Fig. 7.5).

222


1.00

normalized Is

0.75

Fig. 7.5. Normalized ion saturation current measured across the diagonal of an ICP source with and without magnetic multipole confinement [250].

0.50 0.25 0.00

measured at z = 5.7 cm magnetic confinement no confinement

-20

-10

0 r [cm]

10

20

7.2.2 Power input for inductive coupling In this section, we apply basic electrodynamic principles to calculate the efficiency of power input for inductive coupling (helical coil) [251, 252]. A cylindrical plasma volume V in a cylindrical glass cylinder with inner radius r, wall thickness R − r and length l, is surrounded by a coil with radius R with n turns (Figs. 7.2 − 7.4). In this volume V between r − δp and r across a length l, the absorbed Ohmic power per turn can be evaluated considering harmonic excitation E = E 0 sin ωt and neglecting terms of second order: IRF

LRF

Ip

Lp

Fig. 7.6. Equivalent circuit of an inductively coupled discharge.

VRF RRF

Rp

Pabs =

V

Sohm d3 x = − r

Pabs =

r−δp

1 r ˜ ˜ j · E l 2πrdr, 2 r−δP

j2 j(r)2 lπrdr ≈ lπrδp σ σ

(7.4.1)

(7.4.2)

with ˜j the current density which is induced in the plasma by the RF field, thereby opposed to the direction of the complex coil current density and E˜ , the complex-conjugated field intensity in the sheath.3 3

The energy flow SOhm is given by 1/2 ˜j · E˜ ; Eq. (5.4) is the special case for an electron.


223

7.2.2.1 Plasma resistance and plasma impedance. The complete induced eddy current within the plasma amounts to IP = jlδp ,

(7.5)

1L . σA

(7.6)

with the plasma resistance Rp =

Using L the length of the current path (L = 2πr) and A the penetrated area, or A = lδP , we further obtain 2πr σlδp

(7.7.1)

e20 n . m e νm

(7.7.2)

Rp = with σ the DC conductivity σ=

The inductivity of the plasma, Lp , can easily be calculated via the magnetic flux Φ = Lp Ip = μ0 Ilp πr2 , which is generated by the current within the skin layer of thickness δp (the induced magnetic field H is directed perpendicular to the induced current density)4 Lp =

μ0 πr2 . l

(7.8)

7.2.2.2 Coupling between coil and plasma. For n turns with radius R at a total length l of the coil, the equations for the transformer apply (cf. Sect. 5.8, induction matrix): VRF = iωL11 IRF + iωL12 Ip Vp = iωL21 IRF + iωL22 Ip ,

(7.9)

with the self inductances L11 in the coil, and L22 in the plasma and the mutual inductances L12 = L21 L11 = L22 = Lp = L12 = L21 = 4

μ0 πn2 R2 , l μ0 πr 2 , l μ0 πnr 2 . l

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

See Sect. 14.6 for the skin layer, in particular Eq. (14.166).

(7.10.1)

224


7.2.2.3 Primary circuit. How large is the impedance measured across the whole coil? Considering the inverted direction of the current within the plasma (Vp = −Rp Ip ), and inserting (7.9.2) into (7.9.1) gives −Rp Ip = iωL21 IRF + iωL22 Ip ⇒ Ip (iωL22 + Rp ) = −iωL21 IRF , and we obtain for the total impedance of the source ZS =

(7.10.2)

VRF IRF

VRF = iωL11 −

iωL21 IRF iωL21 iωL22 + Rp

(7.10.3)

ZS = iωL11 −

iωL21 iωL21 iωL22 + Rp

(7.10.4)

ZS = RS + iωL11 = iωL11 +

ω 2 L221 . iωL22 + Rp

(7.11)

For δp r, it follows that Rp iωL p , and we

can expand the denominator of Rp 1 1 ⇒ 1 − , which yields Eq. (7.11) to obtain Rp +iωL iωLp iωLp p

Zs = iωL11 −

iωL221 Rp 1− Lp iωLp

= iωL11 − iω

L212 Lp

+ Rp

L12 Lp

2

, (7.12.1)

shortened to Zs = iω(L11 − nL12 ) + n2 Rp

(7.12.2)

with Rs = n2 Rp and Ls = L11 − nL12 . 7.2.3 Limits of power input From Eqns. (7.4.2) (P ∝ ) and (7.6 and 7.7.1) (Rp ∝ 1/(σ δp ), the Eqns. (7.7.2) for the DC conductivity and (14.166) for the collisional skin depth which e20 n decreases with increasing plasma density (σ = me νm ∧δp = 2/μωσ), it becomes

evident that the power input for fixed current scales with n1p . The RF field penetrates deeper in thinner plasmas, and the electric field does not fade within the skin depth δp but can be transmitted across the whole plasma column for sufficiently low damping (δp ≥ r). In this case, the excited current density is proportional j∝

IRF IRF = = IRF (n0 r), A V /r

(7.13)

and for this case, Eqs. (7.4) must be integrated from r = 0 (center of the plasma filled cylinder) to r = r (reactor wall):

power input [a. u.]


225 Fig. 7.7. Inductive coupling, power input: As competition between linear increase at √ lower densities and 1/ conduct at high densities, we observe a maximum of power input at approximately r = δp with δp the screening length (skin depth) [253].

1/SQRT behavior product linear increase

nP [a. u.]

Pabs =

r j(r)2 0

σ

πrLdr ≈

j2 2 πr L : σ

(7.14)

For fixed current, the power transferred by inductive coupling evidently scales with plasma density: The efficiency at very low plasma densities is poor since there is no means to absorb the energy. The efficiency increases with rising plasma density but eventually saturates at some topmost level because the plasma screens itself against penetrating electric fields by Debye shielding [253]. Since the reasons for low power input at low and high plasma density are entirely different, it is not possible to evaluate the maximum analytically. By graphical analysis, the maximum for power input can be found at r = δp (Fig. 7.7). This means: Inductive discharges only become stable beyond a certain threshold of the induced current. Hence, the ignition process itself and the low-current operation take place via capacitive coupling. Without a plasma, only Ohmic losses occur in the coil, and a large voltage is set up across the coil. Eventually, the breakdown voltage is reached, and the plasma is ignited by capacitive coupling. At lower current values, coupling will remain capacitive across the high potential difference of the terminal turns of the coils. Immediately after ignition, the plasma is always in this mode. With growing power input and higher coil current/voltage, the degree of ionization will increase which, in turn, leads to a decrease of the resistivity (increase of the conductivity) of the plasma column, and we can find a positive slope for the characteristic plasma current vs. coil voltage. Simultaneously, with growing coil current, the magnetic flux density increases and, again, the induced voltage. Hence, the absorbed power will be increased by these two effects which are mutually contingent. Eventually, the change from capacitive to inductive coupling happens out of the sudden: After having passed the threshold, coupling will take place inductively via the induced electric field within each turn of the coil. Passing the upper limit, due to the high plasma density which causes an electronic plasma frequency ωp,e compa-

226


rable to the angular operating excitation frequency 2π f , the plasma screens itself against disturbing HF fields. To avoid excessive shielding, the operating frequency f can be reduced. In most of the reactors which are commercially available, f is fixed to values below 2 MHz by which the skin depth is enlarged by more than a factor of 4 (cf. Sect. 14.6). 7.2.4 Top coil configuration This evanescent behavior has been studied in detail by Hopwood et al. applying a spirally wound coil atop the cylinder [254] who actually found an exponential decay of the azimuthal B field in the axial direction (Fig. 7.8), measured with a magnetic induction probe [255]. Since the electron plasma frequency is higher than 13.56 MHz the electromagnetic wave is cutoff within several centimeters (i. e. within a few percent of its vacuum wavelength) inside the plasma. The curves in Fig. 7.8 are calculated with δp between 2.1 cm for 900 W and 3.7 cm for 300 W. The radial and axial variation of the radial component of B is shown in Fig. 7.9. Its ring-shaped field geometry is the result of the spiral form of the coil in creation of a double-peak design, which is mirrored in the annular form of the induced E field; its magnitude in the z-direction, however, is severely damped in the overdense plasma due to strong absorption (Fig. 7.9). The magnitude of the B field is typically a few Gauss at the window leading to B(r, z) = B(r)e−z/δ .

(7.15)

6

Br(z) [Gauss]

5 4

900 W 700 W 500 W 300 W

3 2

Fig. 7.8. Br (ω) decreases exponentially with distance from the window [254].

1 0

0

2

4

6

distance below window, z [cm]

7.2.4.1 E-mode and H-mode. In the spiral configuration of the coil, this transition from the E-type to the H-type can be excellently found as jump in the plasma potential [256] (Figs. 7.10/7.11).


227

2.50

B(r)

B D F

Fig. 7.9. ICP discharge of argon, 500 W ICP power, 5 mTorr: radial variation of Br (ω) at three distances below the window [254].

1.25

0.00 -20

-10

0

10

20

Titel X-Achse

30

25

plasma potential

1010

20

electron density

plasma potential [V]

electron density [cm-3 ]

1011

Fig. 7.10. E-mode → H-mode I: sudden change in the characteristics for continuous rise of RF power in an ICP discharge of oxygen/argon (80/20 %) at 20 mTorr (2 2/3 Pa) after [256].

15

109 0

50

100 150 ICP power [W]

200

250

This is caused by both the modes: In E-mode, the potential of the coil is topmost at its center. Hence, an ICP discharge is ignited between the center of the coil and the lower electrode—as in a CCP mode. With rising coil current a sharp transition can be observed because the induced electric field is most intense between center and outer diameter of the coil, and the induced current flows in a ring-shaped region between inner and outer reagion of the coil (Fig. 7.10). This behavior is even more pronounced for pulsed plasmas. As can be seen from Fig. 7.11 (electron and ion density vs. time of a complete RF cycle (2000 μsec, RF off time 100 μsec), the densities for both the carrier types decrease after having extinguished the plasma, but they continue falling after the RF power is restored. The electron density limps behind the ion density to explode after 1000 μsec by about two orders of magnitude. Now, the H-mode is brought back into effect, and the densities of ions and electrons remain stable during the rest of the RF cycle.

228

7 High-frequency discharges III Fig. 7.11. E-mode → H-mode II: sudden change in the density of ions (ni ) and electrons (ne ) in a pulse period characteristics for (RF off time: 100 μsec) [256].

100 msec

H mode

-3

ne, ni [cm ]

1011

E mode

10

10

ions 9

10

electrons

0

500

1000

1500

2000

t [msec]

In a helical arrangement, however, the plasmas are generated along the chamber wall in both E-mode and H-mode. Therefore, the transition occurs smoothly, and the mode transition becomes dim. 7.2.5 Modeling of ICP discharges In the interior of the plasma source, inhomogenities in plasma density and electron temperature are caused by the small skin depth: The excited RF field quickly fades within a distance which is small compared to the spatial dimensions of the plasma source. This nonuniformity creates a diffusion potential which serves to equalize this gradient by ambipolar diffusion of the electric carriers. This equalization leads to a very homogeneous radial distribution of plasma density and electron temperature. This can be easily proven by the superior radial uniformity of the etchrate, provided the wafer is placed at a certain distance in the downstream zone of the reactor. We present two modeling studies. Numerical calculations of some primary properties which characterize the plasma: Plasma density, electron temperature and plasma potential have been carried out by Stewart et al. for argon and can be summarized as follows [249]: • For low pressures (5 mTorr, < 1Pa), the radial profile of the carrier density nP does not depend on the coil configuration (top coil or cylindrical coil). • The electron temperture Te peaks at the edge of the cylindrical plasma reactor, exactly at the locus of power input where the electrons gain energy from the RF field. But even in the center of the cylinder, Te decreases by only 10 % (Fig. 7.12.1). np peaks on-axis; it approximately meets the Cosine-Bessel profile (Schottky profile) which is required from eq. (6.46) and which results from the isothermal, ambipolar diffusion equation (Fig. 7.12.2). The plasma potential remains very shallow and


229 1.00

1.00

0.50

0.75

5 mTorr (2/3 Pa) R/L =1

normal. ne

normal. ne

R/L =1 5 mTorr (2/3 Pa)

R/L =2,5 20 mTorr 2 2/3 Pa

0.75

5 mTorr (2/3 Pa) R/L =2.5

0.25

5

10

0.50

R/L =2.5 5 mTorr (2/3 Pa)

0.25

20 mTorr 2 2/3 Pa R/L =1

0.00 0

20 mTorr 2 2/3 Pa R/L =1

R/L =2.5 20 mTorr 2 2/3 Pa

0.00 0

15

5

10

15

radius [cm]

radius [cm]

3.5

3.5 R/L =2.5

R/L=2.5

5 mTorr (2/3 Pa)

3.0

3.0 Te [eV]

Te [eV]

R/L =1

2.5 20 mTorr 2 2/3 Pa

2.0 0

5 mTorr (2/3 Pa) R/L =1

2.5

R/L=2.5

R/L =2.5 20 mTorr (2 2/3 Pa)

R/L =1

5

R/L =1

10

2.0 0

15

5

10

15

radius [cm]

radius [cm]

6 1.0

5 kIon [10-10 cm3/sec]

0.8 normal. ne

non-uniform Te

0.6

uniform Te + substrate (h = 2 cm)

0.4 uniform Te + substrate (h = 1 cm)

0.2

4

R/L = 2.5 5 mTorr (2/3 Pa)

3 2

R/L = 1 5 mTorr (2/3 Pa)

1

R/L = 2.5, 20 mTorr (2 2/3 Pa)

R/L = 1; 20 mTorr (2 2/3 Pa)

uniform Te

0.0 0

5

10 radius [cm]

15

0 0

5

10

15

radius [cm]

Fig. 7.12. Two-dimensional model of the radial distribution of charged carriers in an inductively coupled discharge for a top coil configuration and a laterally fixed coil after [249] (1: top LHS, 2: top RHS, . . . , 6: bottom RHS).

peaks at about 3 1/2 kB Te in the center of the cylinder. For discharge pressures beyond 20 mTorr (3 Pa), the maximum of np can be shifted radially outward, since the diffusion coefficients become smaller (Fig. 7.12.6).

230


• A non-uniform electron temperature is a necessary but not sufficient condition for a maximum of np which does not peak on-axis of the plasma source (Figs. 7.12.2/7.12.4). • For a coil which surrounds the cylinder in a helical configuration (Figs. 7.12.2, 7.12.4, 7.12.5, 7.12.6), Te rises radially outward which enhances the ionization rate and likewise np in the toroidally shaped border region (skin effect). • The modeling can be entirely changed by installation of a substrate holder (Fig. 7.12.6). This change in geometry has consequences for nearly all dependencies. This equalizing behavior has been also modelled by Panagopoulos et al. for an ICP reactor with an asymmetric pumping port (Fig. 7.13) with chlorine as an etchant [257]. In contrast to the normal design, an annular downstream pumping port, the pumping port is located at one lower quadrant of the reactor, thus generating an azimuthal asymmetry. RF ICP coil inlet 1 pumping port

inlet 3

inlet 2 focus ring CCP electrode

Fig. 7.13. Sketch of a cylindrical ICP reactor with top– coil configuration (15 cm from wafer level) and asymmetric pumping port (90◦ 2.6 cm from wafer level). There are four gas inlets, symmetrically located around the periphery of the reactor (12 cm from wafer level) [257].

We expect the plasma density, and in particular the density of Cl+ ions, which are mainly responsible for the significant increase of the etchrate in ICP discharges (cf. Chap. 12), to follow this azimuthally asymmetric design of the reactor. This should be manifested in the most sensitive property, the etchrate, which should exhibit the same shape. However, it was shown by Panagopoulos et al. applying a modified three-dimensional finite element fluid model that the asymmetry of the reactor design is by no means reflected in the etchrate (Fig. 7.14). Furthermore, it was be demonstrated that a focus ring could play an important role in alleviating these azimuthal nonuniformities. Although the power density at the dielectric window radially varies by a factor of 5 from the

231 7


3

7

5

7

79

3 3

3

5

5 7

3

3

3

1

5 3

3

3

3 7

75

7

3

7

8

7 2

7 5

6

5

5

8

9

6

7

9

5

7

8

5

2

10

9

7

9

8

9

12

9

7

6

1

Fig. 7.14. Because of the inhomogeneous distribution of the charged carriers within the plasma source, equalization processes are triggered which lead to a radially more uniform plasma density and electron temperature at the locus of the substrate. This is shown here for a planar coil atop the bell-jar. In particular, the contour lines denote: - Top left: The radial power gradient is steepest at the outmost turn where also the current peaks [about a factor of 3 larger, between 3.44 × 105 (corresponds to 3) and 10.78 × 105 W/cm2 (corresponds to 9), locus: 14 cm above the substrate position. - Top middle: The density of atomic Cl fluctuates only by ±10 % [1.40 × 1014 (corresponds to 1) and 1.67 × 1014 /cm3 (corresponds to 7)], locus: 14 cm above wafer level. - Top right: The density of Cl+ fluctuates stronger [1.60 × 1011 (corresponds to 3) and 2.80 × 1011 /cm3 (corresponds to 7)], locus: 14 cm above wafer level. - Bottom left: On wafer level, the density of atomic Cl fluctuates by about ±15 % [8.33 × 1013 (corresponds to 2) and 11.00 × 1013 /cm3 (corresponds to 10)]. - Bottom middle: On wafer level, the Cl+ -density fluctuates by a factor of 1.4 [2.00 × 1010 (corresponds to 1) and 5.56 × 1010 /cm3 (corresponds to 9)]. - Bottom right: The etchrate itself, however, is but weakly influenced [fluctuation about ±5 %, 0.51 μm/min (corresponds to 7) and 0.57 μm/min (corresponds to 9)] [257].

center to the edge, which is reflected by a poor radial uniformity of both the

232


chlorine radicals, Cl·, and the Cl+ ions, at wafer level these nonuniformities have levelled off (deviation less than 5 %) [257].5 7.2.6 Conclusion Both types of coils, the planar type and its helical counterpart, generate azimuthal fields, in the case of the planar type an additional vertical (usual z) dependence. Small azimuthal irregularities will be generated at the ground end of the coil, where the current is highest, at pumping ports and other prominent loci. In principle, energy from the RF field to the electrons can be transferred without collisions over a whole period, i. e. the time-averaged amount of energy does not vanish, since the magnetic field is not conservative [254]. Therefore, capacitive and inductive coupling are fundamentally different excitation methods.

7.3 Generation of plasmas supported by magnetic fields 7.3.1 R´ esum´ e of the properties of HF discharges Up to now, we considered methods of electronic energy gain which required elastic collisions between electrons and neutrals. After a consecutive series of these collisions, the electrons have piled up a sufficient amount of energy to ionize the neutral constituents of the plasma, and furthermore, this is a very efficient means to dissipate energy into the plasma. In capacitively coupled RF discharges, the plasma potential and the sheath potential can rise to very high values (hundreds of volts). For different electrode areas, the sheath potential at the smaller electrode can additionally exhibit a large DC component which can skim the peak value of the amplitude of the applied RF voltage. The sheath potential of the electrode is the difference between the potential of the electrode and the plasma potential. At the grounded electrode, the sheath potential equals the negative plasma potential, at the RF driven electrode, it is equal to the difference of the sum of applied RF potential and DC self bias, V˜RF + VDC , and the plasma potential V˜p on the other hand. A further small part appears in the plasma bulk as longitudinal electric field. Since the density of the highly reactive species which are generated by collisional impact is relatively low, for high etchrates high sheath potentials are required. The inherent disadvantage of this type of excitation consists in the nature of coupling: The DC bias which develops at the RF driven electrode depends on pressure and RF power, furthermore, it is directed normal to the surface of the 5 Since the etchrate exhibit a similar dependence, the rate-limiting step is supposed an attack of Cl+ ions, cf. Chap. 12.

7.3 Generation of plasmas supported by magnetic fields

233

electrode (on which the wafer is to be placed). This enhances the anisotropic component of the process of removal, but inevitably causes severe damage to the remaining topmost layers. Therefore, the strategy for high, anisotropic etchrates with combined low damage is obvious: • High plasma density. • Low electron temperature. • But above all: spatial separation of plasma generation and ion acceleration. The first two steps have been made in changing from DC excitation to RF excitation by capacitive coupling. In fact, higher plasma densities are associated with lower bias values at the RF driven electrode. To further enhance the energy transfer, magnetron-supported methods were applied. Among them, MERIE, Magnetron Enhanced Reactive Ion Etching, has become most popular. As was shown by M¨ uller et al., the application of a static magnetic field (130 Gauss or 13 mT) which was orientated in parallel fashion to the surface of the electrode (and the wafer), could enhance the etchrate of silicon by a factor of 7 and the etchrate of Si3 N4 by a factor of 2.5. Simultaneously, the DC bias dropped by a factor of 6 for Si and 5 for Si3 N4 , respectively, all the other parameters left unchanged [246]. All progress considered, the excitation of the plasma remains a capacitively coupled method with the disadvantages just mentioned. Inductively coupled plasmas clearly avoid these setbacks, but, due to the skin effect, they are limited to plasma densities to upper values between 1012 cm−3 in the plasma source, but by about one order of magnitude at wafer level. A qualitative leap forward is achieved by resonant coupling methods. The penetration depth of the wave fields is tremendously enlarged, elegantly avoiding all the problems and setbacks associated with the limitations caused by the skin effect (cf. Sects. 14.6 and 14.7). These experiments with whistler waves can be performed either in the microwave range (ICC frequency: 2.450 GHz); socalled ECR excitation or ECR heating, or in the RF range at typically 10 − 100 MHz (for technological purposes: still at 13.56 MHz): so-called regime of helicon waves. Two effects are benefitted from: the resonance with transverse electromagnetic waves and the enhancement of the diffusion coefficient by the magnetic field. Hence, comprehensive knowledge of the following topics is demanded: • Which waves propagate in the plasma, which will be absorbed? • What is the polarization of these waves? • ⇒ In short: What is the quality of the dispersion in a magnetized plasma?

234


First, we will recapitulate some characteristic phenomena of RF discharges. The AC field is weaker than the DC field at same amplitude. In the plasma bulk, the current is almost completely electronic conduction current, in the sheaths, it is ionic conduction current (small contribution) but mainly displacement current. j p ≈ (σe + iε0 ω)E p ;

(7.16.1)

j s ≈ (σi + iε0 ω)E s ;

(7.16.2)

whereas the proportionality factor between j and E is given by the conductivity (σ = e0 np μ) and can be calculated after Eqs. (5.7) and (6.4) to yield ε0 ωp2 1 np e20 ⇒σ= . (7.17) me iω + νm iω + νm np = np (x) is a spatial dependent function: Starting from topmost values in the center of the discharge, it falls down in outward direction to vanish at the walls. In a cylindrical plasma, this radial decline follows a Bessel function. For a capacitively coupled plasma, the degree of ionization exhibits a typical value between 100 ppm to 0.1 %, that means plasma densities of up to some 1010 /cm3 , and the angular plasma frequency ωp,e exceeds the operating angular frequency by more than one order of magnitude (5.64 GHz vs. 85 MHz), and the RF power can be readily absorbed by the plasma. For further rising plasma density, for example in inductively coupled plasmas, a sharp cutoff will occur in a cold plasma (νm = 0) at ωp,e = ω, and the RF wave will be reflected (Sect. 14.6). At finite temperature with the possibility of elastic collisions, this sharp transition is softened. σ=

7.3.2 Whistler waves 7.3.2.1 Phenomenology. Below the plasma frequency, waves exhibit an evanescent propagation behavior: Within the skin depth, the wave is damped to about one third of its initial value which leads to the phenomenons of cutoff and reflection. However, by application of a strong magnetic field, the sharp border between dielectric matter and metall is softened (cf. Sect. 14.7): Transmission becomes possible in a frequency band below the plasma frequency which is otherwise “forbidden.” The propagation behavior is determined by the dispersion relations and confines the upper boundary to ωc,e : ωc,e ≥ ω. As result, we observe slow-propagating [phase velocity vph is smaller than √ or equals to about 1 % of the TEM value (c = 1/ ε0 μ0 )] whistler or helicon waves. The name “helicon” was first used by Agrain for waves propagating in a high-conductive metal at very low temperatures [258]. “Helicon” is derived from the helical guiding lines of force which cause the electrons to follow. The inherent property of these waves is their almost exclusive coupling of their energy

7.3 Generation of plasmas supported by magnetic fields

235

density to the magnetic field [cf. Sect. 14.6, Eq. (14.164)]. In the gaseous plasma, these waves are normally denoted “whistler waves”; their propagation band is located between the cyclotron resonance frequencies of ions and electrons: ωc,i ≤ ω ≤ ωc,e with ωc = e0 B/m. It was Barkhausen who was the first to note the sound of atmospheric whistler waves [259]: Since it is a sensation which lies within the audible range, sometimes we hear a descending glissando starting at very high pitches when we listen to the radio in the short-wave band. It is caused by lightnings which generate a broad band wave signal. Among them are the whistler waves which propagate along the lines of√the earth’s magnetic field; and since their dispersion behavior follows ∂ω/∂k ∝ ω for ω < 1/2 ωc,e , the higher pitches arrive in front of the lower pitches which explains the audible sensation (cf. Sec. 14.7.1, Fig. 7.15). However, these whistler waves can not only be observed at ground level but also from spacecraft [260]. Vice versa, these waves can generate a plasma. Fig. 7.15. Spectrogram of a whistler recorded in northern Nevada, 1999. The tone, starting from the highest pitches of the piano, descends to the pitch of high a female speech voice, c exhibiting a steep glissando [261] www.auroralchorus.com.

7.3.2.2 Dispersion and absorption. The magnetic field causes the electromagnetic wave to propagate with anisotropic behavior. When viewed along the static magnetic field, we can distinguish between a right-hand circularly polarized wave and a left-hand circularly polarized wave. The first are denoted R-waves, the latter are called L-waves. First, the threshold frequency for propagation is different for both the types. For the R-waves —and it is but this type of waves which can transfer resonant energy to the electrons —this interaction causes a rise in the resonance frequency from ωp,e to ωr , and ωr is a function of both frequencies, ωp,e and ωc,e (cf. Table 14.2). In addition, a band of propagation at low frequencies will arise (n > 1): the regime of slow-propagating whistler waves. Its upper boundary is set by ωc,e . At its lower limit, it is operated with radio frequency, at its upper limit with microwaves. We can chose the intensity of the static magnetic field to match the electron cyclotron frequency with the operating frequency yielding a resonant excitation: so-called “ECR heating”. This holds true for the frequency range6 6 Waves with larger wavelength (lower frequency) are below the cutoff; they exhibit evanescent behavior and will be reflected at the sheath. This mechanism is comparable to the conduct of short waves at the Heavyside layer of the atmosphere.

236

7 High-frequency discharges III ωl ω ≤ ωc,e ,

(7.18)

whereas the plasma frequency of the electrons, ωp,e , should be high against the product of operating frequency ω and cyclotron resonance frequency of the electrons ωc,e : 2 ωp,e ωωc,e

(7.19)

with ωl the lower hybrid resonance, which is defined by 1 1 1 = 2 + . 2 ωl ωp,i ωc,i ωc,e

(7.20)

In this equation, the subscript “i” refers to the ions and their response to the fields which becomes important at very low frequencies. For the conditions (7.18/19), we can neglect ionic movements and electronic gyromotions, and following Stix, we denote this excitation coupled resonance [262]. The range for ECR operation is confined by ω = ωc,e < ωp,e . Besides simple Ohmic heating, another heating mechanism is supposed to become important in this case: Landau damping by helicons, low-frequent whistler waves which slowly move in a narrow tube in a certain angle with respect to the static magnetic field B 0 . Fast electrons can extract energy from the wave field with high an efficiency when their thermal velocity equals the phase velocity of the helicon: vph =

ω , k

(7.21)

which happens to be even lower than the thermal velocity of the electrons in the vicinity of resonance: The phase velocity vph and the wavelength drop to very small values (ideally, they would vanish), and the wavevector k becomes infinite. In contrast to the R-wave, which can push the electrons exclusively, this damping can also happen to L-waves when they interact with electrons of the high-energy tail of the EEDF. It was Frank Chen who suspected Landau damping as the main absorption mechanism in discharges excited by helicons. Besides stochastic heating, this is the second collision-free mechanism for energy transfer. However, the excitation of electrons by Landau damping is restricted to a very narrow area adjacent to vph . At a ratio of ve /vph = 0.1 the efficiency of energy transfer ξ almost vanishes (ξ ≈ 10−51 ), whereas ξ has risen to 0.17 at ve /vph = 0.5 (cf. Sect. 14.3). 7 Landau damping would lead to high a portion of electrons in the tail of the EEDF which could not be proven [264]. On the contrary, the damping of helicon 7 That this mechanism must play a significant role has made obvious by experiments with double Langmuir probes [263]. As Popov has found, the absorption of R-waves increases in the vicinity of the resonance (as it is expected since their phase velocity should decrease to very low values), but he succeeded in separating the two effects by measuring two electron temperatures far away from the ECR layer in the vicinity of the vacuum window.

7.4 Helicons in a bounded plasma

237

waves could be explained by Coulomb collisions [265] since at these high plasma densities of almost 100 %, every EEDF, no matter which is its initial character, turns into a Maxwellian distribution simply by Coulomb collisions. Although the fate of the L-wave during the process of absorption is still unclear, this explanation could shed light onto this open question. In particular, in plasma-filled cylinders at such frequencies far below the Larmor frequency, also the L-wave should contribute to plasma heating by evanescent absorption leading to an enhanced movement, albeit chaotic in character. It is but the R-wave which can solely rise the electron energy. After a long discussion over more than two decades, an additional heating, namely the strong damping of the waves named after Trivelpiece and Gould has evolved the main mechanism of absorption (cf. Sec. 14.7.2). In free space, these are the aforementioned whistlers.

7.4 Helicons in a bounded plasma 7.4.1 Introduction Which waves can propagate in a bounded plasma? As shown in Sect. 14.7.1, the regime of the R-waves encircles two frequency ranges, which are separated by ωc,e . Whereas the upper band exhibits the normal evanescent behavior, the lower band with ω < ωc,e is denoted the range of whistler waves which can propagate through the plasma, albeit with severe damping. Since cylindrical symmetry is provided, the waves can be described with Bessel functions, and the solutions for the wave fields can be obtained in rising accuracy: • Cold plasma: neglect of pressure gradients, • Lossless plasma: neglect of the ηj term (cf. [266] − [269] and Sect. 14.7.2). The propagation of these waves in a hollow wave guide is described by means of the Maxwell equations along with the dispersion relation. In contrast to the free-space behavior, the confinement restricts the propagation, and only certain modes can be transmitted through the plasma. Propagation of waves in free space can be modeled by a transverse wave with E- and B-vectors, respectively, orientated perpendicular to the direction of propagation, which is denoted by k, and all ω(k) values are possible for positive index of refraction n. In contrast to this simple conduct, the modes propagating through a waveguide exhibit field components which are orientated in parallel fashion to the wave vector k. It is possible to have either the complete electric field or the complete magnetic field orientated perpendicular to the k vector, but not both of them. In the former case, we speak of TE-modes, in the latter case, we denote these modes as TM-modes. These modes are caused by currents flowing in the walls of

238


the wave guides which generate a magnetic field in the direction of propagation k. The mode patterns are displayed in Fig. 7.16 for the m = 1 mode and in Sect. 14.7.2 also for the m = 0 mode. It has to be kept in mind, however, that the modes in a plasma-filled cylinder only resemble the modes obtained by application of classical electrodynamics. In contrast to the propagation of cylindric modes in a hollow waveguide which are characterized by vanishing divergence, we must take into account the conductive medium (occurrence of free charges) which causes the electric field to possess a finite divergence (open E-field lines), and the anisotropy which is caused by the static magnetic field. 7.4.2 Dispersion and wave fields The dispersion relation for helicons and whistler waves is derived in Sects. 14.7.1/2. For whistler waves, we can easily calculate the dispersion relation when we neglect electron inertia via Eq. (14.223) according to Eq. (7.22). For bounded helicons, we adopt this approach [Eq. (14.239.1)], and the equation to determine the axial wave vector yields kk =

μ0 e0 ωnp , B0

(7.22)

2 which can be solved along with k 2 = k2 + k⊥ . In combination with the dispersion relation (the wave vector consists of two components which are orientated in parallel and normal directions with respect to the static magnetic field), we obtain solutions of coupled differential equations of second order which further depend on the conditions of the reactor (perfectly isolating or ideally conducting walls and conditions for the equation of motion, see Sect. 14.7.2). As result, the fields of the helicons exhibit hybrid character between pure electrostatic (∇ × E = 0) and pure electromagnetic (∇ · E = 0) behavior. For example, the pattern for m = 1 in Fig. 7.16 resembles that of the TM11 mode for a circular waveguide, and especially the phase-sensitive patterns for the m = 0 exhibit a close relationship between the borderline cases (Figs. 14.39, 14.43 and 14.44).

7.4.3 Antenna coupling To couple energy into a bounded magnetoplasma and to transfer energy to the plasma constituents, the plasma modes should be excited which have their maximum on-axis or between edge and center. Ideally, they should not exhibit any azimuthal dependence to ensure a maximum of radial and azimuthal uniformity. In Figs. 7.17, the radial variations of the first two azimuthal modes with m = 0 and m = 1 for radial wave number n = 1 are displayed along with the squared current density for the first three modes. Since only the first two modes


239

Fig. 7.16. Simplified pattern of the electric field (solid) and magnetic field (dashed) in a helicon wave (m = 1) after [270]. Note the open field lines for E which are caused by the free charges in the plasma– filled waveguide.

peak in the center, we are restricted to apply one of them for plasma generation. The specific feature of the mode with m = 0 is its axial symmetry, therefore, this wave should be radiated from a coaxial antenna in the center of the plasma. Due to reasons of feasibility and contamination, in most cases the wave with m = 1 is chosen and this mode is radiated from an externally wound antenna. The whistler waves are tuned to resonance by variation of • the length of the antenna La , • magnetic induction B0 and • excitation frequency ω; their axial wavelength (λ = 2π/k ) is calculated according to Eq. (14.223) (cos Θ = 1) 2πc λ= ωp

ωc ε0 B0 B0 = 2πc ≈ 5 × 109 [cm] ω ne e0 ω ne ω/2π

(7.23)

with B0 in Gauss, n in cm−3 and ω in Hz. The wavelength is proportional to the square root of the ratio of the static magnetic field over the electron density; this resonance condition is good for weak magnetic fields ≤100 Gauss (10 mT) and = 5 × 1011 cm−3 plasma densities ≤ 1011 cm−3 : for B0 = 100 Gauss (10 mT), n 8 and ω/2π =13.56 MHz we obtain λ ≈ 60 cm. λ scales with 1/np , and vice versa. This resonance condition imposes restrictive conditions on the excitation frequency and the antenna wavelength which are limited through the dispersion 8 We obtain the same result when inserting for the dielectric constant ε = n2 (Maxwell’s relation). Using Eqs. (14.208/209), we find an equation between k, np , and B0 : μ0 e0 np ω B0 2π 5 ≈ 5.6 × 10 [m] k≈ ⇒λ= B0 k np ω

with B0 in T, np in cm−3 and ω in Hz.

240

7 High-frequency discharges III 1.0

1.0 Bz

Br

0.5

Bq

B [r. u.]

B [r. u.]

0.5

0.0 Br

m=1 b=2

0.0 Bz

-0.5

-0.5 0.00

Bq

0.25

0.50 r/a

0.75

1.00

-1.0 0.00

0.25

0.50 r/a

0.75

1.00

1.00 m=0 m=1 m=2

I2

0.75 0.50 0.25 0.00 0.00

0.25

0.50 r/a

0.75

1.00

Fig. 7.17. Radial dependence of the magnetic wave fields and the current density squared (bottom) for the azimuthal modes m = 0 (LHS) and m = 1 (RHS) after [271]. k /k = 1/3 , radial wave number: n = 1.

relation: In a bounded system, λ and k = 2π/λ are free parameters no longer. It turns out that there is a set of eigenvalues of permitted angles which are determined by the radial modes. Hence, only a set of wavelengths N λ /2 fit the length of antenna La , beginning with N = 1 leading to k1, = π/La [270]. According to Eq. (7.22), at fixed ω, k or vph = ω/k , the phase velocity along the tube, the plasma density np should be proportional to the intensity of the magnetic field B0 or vice versa, np is expected to rise linearly with increasing B 0 . However, density jumps will occur when the static magnetic field is enlarged, which leads to the feared mode jumping (see Fig. 7.18): The wavelength does not remain constant along the axis. For the m = 0 mode, k⊥ is fixed to 3.83/a with a the tube radius (Fig. 7.19) [272]. Remembering that k ≈ π/La with La the length of the antenna, we see that for a given radius a, the density np scales with B0 , or vice versa, for a given np , the required field intensity B0 should vary linearly with respect to the tube radius a. To heat electrons specifically by Landau damping, antennas have been constructed according to the condition


241

plasma density [1011 /cm3]

15

Fig. 7.18. Mode jumping occurs when the intensity of the static magnetic field is smoothly increased. The thick line rising from the origin is the theoretically expected dispersion for a helicon wave excited with a wavelength that doubles the length of the antenna [271].

10

5

0 0

500 Bz [Gauss]

1000

antenna length [cm]

60

Fig. 7.19. The axial wavelength as function of np /B [Eq. (7.23)] for an m = 0 mode after [272] for two different tube radii in an approximation neglecting electron inertia yielding typically 2/3 of the wave length applying full theory.

a = 5 cm

40

20

0

a = 10 cm

0

1

2 B0/np [10

3 -11

4

5

3

Gauss cm ]

k =

2π π ≈N λ La

(7.24)

with La the length of the antenna to obtain [273] me E= 2

ω k

2

(7.25)

with E the electron energy. Another effect which is encountered with changes in plasma density or magnetic field, is the sudden change from resonance (propagating and counterpropagating wave act to form a standing wave) to propagation, also in various directions [272]. These observations shed light on the heating mechanism which is explained for the radial component: To begin with low magnetic field intensities, the radial electric field is inductively coupled, i. e. the amplitude evanescently drops during penetration into the plasma, and power input is limited by the conventional skin effect. Hence, the maximum of the electric field is located

242


close to the coils, its local minimum is on-axis (center of the tube). But intensifying the magnetic field results in further penetration of the radial electric field. Eventually, we reach the first onset of a standing wave in radial direction. Simultaneously, the axial plasma density increases due to the improved radial penetration, and the phase velocity of the azimuthal electric field begins to drop. When the azimuthal field either encounters a boundary or can interfere with a counterpropagating wave, a standing wave of the azimuthal field in axial direction begins to form, and the characteristics of propagation changes from radially dominated to axially dominated with an increase of wavelength for both the radial and axial directions according to Eq. (7.22). As result, we find: • The power input through radial electric fields rises with increasing radial mode number, or the other way round: • Standing wave patterns with higher order will appear as eigenfunctions for the radial electric field at higher magnetic fields (Fig. 7.20).

amplitude [a. u.]

30

20

Fig. 7.20. The axial variation of Br and Bz for B0 ≈ 100 Gauss clearly shows a standing wave pattern: At each minimum of the wave fields, there is a phase change of π [271].

Br

Br

10

Bz

Bz

length of antenna

0

20

40 60 axial distance [cm]

80

To ensure a significant energy transfer to the electrons which should be in the order of the ionization maximum of argon (approximately 50 eV), the antenna has to meet the condition La a, a typical value would be about 120 cm. Following Eq. (7.23), however, a short antenna is required with La R. Plasmas in thin, long pipes are difficult to be operated because the electron temperature rises sharply with decreasing cross section of the reactor (cf. Sect. 3.5, also [272]). Therefore, an increase of the static magnetic field is required to enclose the plasma. Provided that Landau damping plays the significant role in exciting these helicons, Chen has compiled the achievable plasma densities along with the conditions for a ratio of L/λ = 7 and an electron temperature of 4 eV (Table 7.1) [274]. These values have to be slightly modified for a different choice of parameters (Te , L/λ and a). For more precise calculations, the dependence of the wave vector k on temperature and the resistance term should be taken into account.


243

Table 7.1. Resonant tuning of the helicon waves to experimental parameters [a: diameter of the cylinder with perfectly conducting walls, λ: 2π/(k), L: 1/(k)], Lp : plasma length. n [cm−3 ] 1011 1012 1013 1014 1015 1016

a [cm] 25.1 7.9 2.5 0.79 0.25 0.08

B [T] 0.004 0.013 0.040 0.126 0.40 1.26

ω [MHz] 0.288 0.912 2.88 9.12 28.8 91.2

λ [cm] 500 159 50 15.9 5.0 1.6

Lp [cm] 3500 1100 350 111 35 11

7.4.4 Operation The discharge is operated in the radiofrequency range (5 − 50 MHz) with the excitation frequency of 13.56 MHz for technical purposes. The antenna is placed upon a plasma cylinder whose walls consist either of a conducting or isolating material. Across its length La , the antenna radiates a time-dependent B-field that couples with the transversal magnetic field of the helicon mode. Furthermore, an electric current is induced directly beneath the antenna wires which is opposite in direction to the antenna current. This current, in turn, excites a transverse electric field across the antenna which can couple with the transverse electric field of the helicon mode (cf. Fig. 7.21) [270]. This leads to a rotation of the wave fields in space and time. A significant assumption of this model is a very high plasma density which justifies the neglect of displacement currents within the plasma, and we regard the discharge as a plasma filled circular hollow waveguide with either isolating (glass) or perfectly conducting walls (metal). The plasma densities are among the highest which ever have been achieved in low-temperature plasmas (up to 100 % which is only possible with high an amount of doubly-ionized species) in conjunction with very low plasma potentials Vp (typically 10 − 15 V). The magnetic fields which are required for this performance are remarkably low (50 − 150 Gauss or 5 − 15 mT, Fig. 7.22). As real challenge, however, remains the matching. In the first matching networks, the π-matching consisted of two variable capacitors and the antenna serves as inductive device. In retrospect—the inventor Boswell himself investigated the matching network for more than two years—it was mainly this difficulty which prevented a commercial breakthrough [273]. Boswell eventually succeeded with his Alcatel reactor mainly constructed by himself [278] (Fig. 7.23).

244


Fig. 7.21. Power coupling of helicon waves (m = 1) via a Boswell-type antenna after [275]. The antenna matches the condition that the currents should alternate in direction every half wavelength. In fact, the solenoid consists of a series of small magnets, and the glass tube is very tall. The ratio height over diameter often exceeds 20 [271, 276].

antenna

RF

downstream applications

30

20 Vp

20

15

np

10 15

5 0

10

150

300

450

11

25

25

plasma density [10

plasma potential [V]

-3

cm ]

30

Fig. 7.22. The sharp drop in the plasma potential at an antenna power of about 350 W indicates the onset of helicon generation. This is reflected in the simultaneous steep increase in plasma density. Below that threshold, the discharge remains in the capacitively coupled modus [277].

600

antenna power [W]

7.4.5 Experiments Boswell described the first experiment in argon at a pressure of 1.5 mTorr (0.2 Pa) in 1970 [273]. The experimental setup consisted of two coaxial cylinders: a source made of glass, and a reactor of glass or metal. The solenoid at the source chamber is required to couple the RF power into the plasma, and the solenoid at the processing chamber confines the plasma. An axial static magnetic field with an intensity of 1.5 kGauss (150 mT) surrounds the tube. The axially orientated antenna generates a standing helicon wave by coupling the transverse antenna fields to the radial plasma fields (Ez = 0). With a double Langmuir probe, Boswell et al. measured plasma densities between 1.5×1011 and 3.8 × 1012 cm−3 , i. e. a degree of ionization between 0.4 and 9 %, dependent on B0 . This causes an index of refraction of up to 1012 by which the classification as resonance phenomenon is justified. Boswell coined the acronym RIPE for etching in these discharges (Resonant Inductive Ion Etching) [278].


245

matching network

RF

antenna

source solenoid

substrate table (isolated or ccp) reactor solenoid

vacuum system

substrate cooling matching network

Fig. 7.23. The first reactor which was commercially available was equipped with an antenna of the Boswell-type c Societe (Alcatel) [278] ( Fran¸caise du Vide).

RF

Later on, Boswell et al. reported on a degree of ionization almost 100 %, measured with a microwave interferometer at electron energies of about 3 eV [275].9 Even when we take into account that highly ionized plasmas consist of a large fraction of double-ionized species which distort the degree of ionization to higher values, this result is very impressive.10 At low static magnetic fields, the color of an argon plasma is white reddish which is due to the lines of atomic argon [Ar(I)]. Increasing B0 to values of 500 Gauss (50 mT), the color suddenly turns to blue (Ar lines at 480.6 and 488 nm); the plasma density has increased by a factor of 5. Additionally, a sharper confinement of the radial profile is combined with the doubling of the axial frequency. Even further rise of the magnetic field to 75 mT causes another turn-over which is associated with a doubling of the plasma density [271]. Boswell himself conducted some RIPE etching runs in the silicon system applying the reactive gases SF6 and Cl2 [278, 279]. At a DC bias of −100 V and a radiated RF power of 500 W (13.56 MHz), very high etchrates could be found [0.4 μm/min at 0.5 mTorr (0.07 Pa) and 1.5 μm/min at 3.75 mTorr (0.5 Pa)]; 9 In most cases, the cutoff will be measured, i. e. the frequency for which the measuring frequency equals the plasma frequency (ω = ωp ), and the wave will be reflected. 10 From the two branches of the Langmuir curve, we obtain values for ni und ne . For single-ionized plasmas holds ne = ni , from the deviations, we can conclude the fraction of multiple ionization with reserve.

246


the anisotropy, defined as A = tan α = lv /lh with lh the horizontal and lv the vertical etchrate, however, dropped by nearly 50 % from 1.0 to 0.6. The systems with coupled resonance are distinguished by an excellent radial uniformity (±1 % across 4"), which can be attributed to the high spatial plasma homogenity, combined with extremely low bias potentials. Furthermore, the competion between etching and deposition in carbon-containing gases can be decided in favor of etching even at very low bias values [280]. 7.4.6 Summary Helicons are superpositions of bounded low-frequency whistler waves which propagate at a common but fixed angle with respect to an axial static magnetic field. From their dispersive behavior, they are mixtures of electromagnetic (∇ · E = 0) and electrostatic (∇ × E = 0) fields. Their field pattern can be described by E ∝ exp [i(ωt − kz − mΘ)]

(7.26)

with Θ the angle between the wave vector and the static magnetic field, and m the number of the the azimuthal mode. In reactors for helicon wave excitation, an RF driven antenna is mounted directly on the reactor wall. Combined with relatively low static magnetic fields, excitation to very high plasma degrees is made feasible. The waves transmit their energy by collisions or a mechanism which is closely connected to Landau damping, interaction with the Trivelpiece-Gould mode. Due to high plasma densities, the sheaths are considerably thinn and completely collisionless. For vanishing magnetic field, a smooth transition to inductively coupled plasmas will happen.

7.5 Electron cyclotron resonance 7.5.1 The electric field and the diffusion length The introduction of plasma sources which were driven with electron cyclotron resonance marked a significant progress in low pressure plasma technology, a spin-off process of projects of developments of nuclear fusion and thrusters.11 Energy of a microwave field is coupled into a plasma by means of a static divergent or radially inhomogeneous magnetic field (quadrupole, octopole or even higher order) at a certain frequency, the Electron Cyclotron Resonance frequency (ECR). As a simple example, we start with the plasma-filled cylindrical tube which is penetrated by an axialsymmetric, static magnetic field. Enforced by the 11 A “spin-off” process denotes the transfer of production processes, materials and know-how from the military/governmental sector into the civil/private sector.

7.5 Electron cyclotron resonance

247

Lorentz force, the electrons will gyrate on orbits and helices with the Larmor radius perpendicular to the lines of this field (Fig. 7.24).

B

F v

Fig. 7.24. Charges are forced to gyrate around the lines of a magnetic field. Due to its small mass, the radius of the electron is smaller by orders of magnitude compared to the ion radius. Changes in the electron velocity (electron energy) alter the radius of the orbit but not the angular frequency. The electrons all rotate with the same Larmor frequency.

At sufficiently low pressures the electron mean free path becomes farther than at least one orbit. If this tube is exposed to a microwave field in the axial direction, a synchronization of these movements can happen at ωc,e = ωop : The electron cylotron frequency equals the angular frequency of the microwave, and the electrons can pile up energy from the wave field which causes the orbits and helices to degenerate to spirals. This resonant enlargement of the energy transfer from the wavefield to the electrons is denoted as ECR heating. For convenience, the incoming microwaves are supposed to be linearly polarized, and they can easily be decomposed into two wave components which are polarized circularly, and L-wave and an R-wave (Figs. 7.25). +E

CCW: R

CW: L

Fig. 7.25. A linearly polarized wave can be decomposed into two circularly polarized components with vanishing phase angle. For positive z, the CW rotating component is the L-wave, the CCW rotating one is the R-wave.

-E

Due to the negative charge of the electron, it is only the R-wave which can enhance the tangential component of the orbital motion of the gyrating electron (Figs. 7.26). Neutral atoms and molecules are excited and ionized by inelastic collisions with the heated electrons. Both the types of electric carriers which are coupled by electric forces experience a slow drift out of the divergent or radially inhomogeneous magnetic field. As a matter of fact, a beam of electron ion pairs leaves the plasma source. This movement is caused by a conversion process of transverse kinetic energy to longitudinal kinetic energy (cf. Sect. 14.5). By steering the degree of divergence, the longitudinal drift velocity component can be

248


L

B

F v

R 0°

90°

180°

270°

Fig. 7.26. From the two circularly polarized waves, it is only the electric field of the R-wave which can continuously transfer energy to the gyrating electron, shown her for four distinct phase angles. For the same phase angles for the L-wave, only two of them happen to push the electron (0◦ and 180◦ ), the remaining two will retard the orbital component, so there is not net result any.

adjusted over a large range. Provided ions moving downstream do not suffer collisions, their trajectories should remain in parallel direction with respect to the static magnetic field, and they eventually approach the substrate with normal incidence and small velocity. Hence, the crystal damage can be kept low — at the expense of a low etchrate. By the spatial separation of plasma source (generation of carriers) and substrate immersed by the plasma (definition of the kinetic energy of the impinging particles), it was possible for the first time to control density and energy of the ions independently. The absorption of energy of the electrons as a whole is shown in the appendix (Sec. 14.7.1). Here, we want to investigate the behavior of a single electron in an electromagnetic field which is perpendicularly penetrated by a static magnetic ˜ field B 0 , which should be large against the fluctuating magnetic wave field B: ˜ ∧ B0 z k B0 B

(7.27)

with k the wave vector of the incident wave, we start with e0 (7.28) B 0 ∧ E = E 0 eiωt ∧ v = v G eiωt m for the equation of motion of the gyration center G [Drude or Langevin equation, Eq. (14.172)] ω=−

m

dv = −e0 E(t) − e0 v × B 0 − νm v, dt

(7.29.1)


249

which yields in components ⎫

(iω + νm )vx + e0mB0 vy = − em0 Ex ⎪ ⎬ − e0mB0 vx + (iω + νm )vy = − em0 Ey ⎪ (iω + νm )vz = − em0 Ez ⎭

(7.29.2)

or as matrix with ωp2 = ne20 /mε0 ⎛

1 ⎜ ⎝ ε0 ωp2

⎞⎛

⎞

⎛

⎞

iω + νm ωc 0 jx Ex ⎟⎜ ⎟ ⎜ ⎟ −ωc iω + νm 0 j = ⎠⎝ y ⎠ ⎝ Ey ⎠ . jz Ex 0 0 iω + νm

(7.29.3)

This is the matrix equation σ −1 ij · j = E;

(7.29.4)

but we are interested in the solution j = σ ij E ⇒ ρ v = ρ μij E ⇒ v = μij · E.

(7.30)

For this end, we must invert Eq. (7.29.3) and obtain the doubly inverted matrix −1 μ−1 with ij ) A=

1 nε0 ωp2

iω + νm [(iω + νm )2 + ωc2 ]

:

(7.31.1)

⎛

⎞

(iω + νm )2 −ωc (iω + νm ) 0 ⎜ ⎟ (iω + νm )2 0 A × ⎝ ωc (iω + νm ) ⎠. 2 2 0 0 (iω + νm ) + ωc

(7.31.2)

μij can then be expressed as ⎛

⎞

μ⊥ iμ× 0 0 ⎟ μij = ⎜ ⎝ −iμ× μ⊥ ⎠, 0 0 μ

(7.32)

which is denoted as usual [281]: μ⊥ : μtransverse : μ× : μnormal : μ :

μparallel :

e20 iω+νm ; m (iω+νm )2 +ωc2 e20 −ωc ; m (iω+νm )2 +ωc2 e20 1 . m iω+νm

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(7.33)

If the operating frequency equals the cyclotron frequency, a singularity will occur. The electron velocity perpendicular to the lines of the static magnetic field diverges, and Eqs. (7.29.2/3) are not valid any longer: At resonance, the

250


spherical movement degenerates into a spiral movement. The resonance is very sharp and only effective within a frequency interval of about ±5 %. As a typical resonance phenomenon, a strong energy absorption occurs even for small fields at ω = ωc , which is but weakly damped at low discharge pressures. For pressures between 10 and 75 mTorr (1 to 10 Pa), the breakdown field is significantly smaller than without a magnetic field (cf. Fig. 7.27. For 2.450 GHz, the magnetic induction for resonance amounts to 875 Gauss or 87.5 mT). The magnetron radiates unpolarized wave fields. Hence, the radial component which is normal with respect to direction of propagation can be split into a left and right circularly polarized fraction, resp. Helicon waves of both polarizations cannot propagate in the plasma-filled cylinder if the refraction index sinks below unity, i. e. at a certain diameter less than λcutoff , for magnetic fields B < Bc,e , the waves will be reflected. However, R-waves can propagate in that part of the plasma column where B > Bc,e . It is but this wave that can push the gyrating electrons (cf. Sect. 14.7.1) [263]. Walking on the resonance ridge (n → ∞, vph → 0), it should be considered that this condition for strong absorption is only valid for the R-wave and can be narrowly missed. The L-wave is likely to be reflected when the cutoff is about to be reached (n → 0, vph → ∞), and perhaps mode hopping is caused at this threshold [282]. Although the fate of the L-wave during the process of absorption still remains unclear, Popov’s proposal of Landau damping of the L-wave by highvelocity electrons could shed light on this open question [263].

EB [V/cm]

100

10 0.0

30 Torr (4 kPa) 18 Torr (2.4 kPa) 8 Torr (1.05 kPa) 4 Torr (530 Pa) 2 Torr (270 Pa) 1 Torr (130 Pa)

0.1

B [T]

0.2

Fig. 7.27. Already in the 1950s, the dependence of the breakdown voltage on the magnetic flux density was subc ject to research [283] ( The American Institute of Physics).

0.3

The resonance is definitely not caused by collisions of electrons, on the contrary: The energy absorption will be limited by collisions of the electrons with heavy particles. Lower damping will increase the Q-factor of the resonance (Fig. 7.28. Hence, the lower condition for resonance should be formulated as


251 ω ≥ νm ,

(7.34)

which means discharge pressures in argon of significantly less than 10 mTorr (less than 1 Pa) for the typical excitation frequency of 2.45 GHz. The effective electric field for a motion perpendicular with respect to the ˜ ⊥ B 0 ) will change as well. To show this, we take the static magnetic field (E real part of Eq. (7.29.3), take the square, average over time and obtain 2 Eeff

ν2 = m 4

1 1 + 2 E02 , 2 2 νm + (ω − ωc ) νm + (ω + ωc )2

(7.35)

2 (ω + ωc,e )2 ). which turns into the RMS field at high discharge pressures (νm At the singularity, the second summand vanishes, and the shape of the effective field will become Lorentzian. For this case, the FWHM equals the frequency of elastic collisions:

νm =

√

Δω,

(7.36)

which facilitates a direct measurement of νm . The absorbed kinetic energy is given by 1 1 2 Ekin = mv 2 = ma2 /νm , 2 2

(7.37)

or

Ekin

1 1 e2 E 2 1 e20 1 = m 0 2 eff2 = + 2 E02 . · 2 + (ω − ω )2 2 m νm 4m νm ν + (ω + ωc,e )2 c,e m

(7.38)

For high discharge pressures, the kinetic energy is approximately (ω νm ): Ekin ≈

e20 E02 . 2 2mνm

(7.39)

This equation is very important for hydrogen and helium, since in these gases, νm does not depend on the particle’s velocity. Only in this case can the absorbed energy be described as a function of frequency and magnetic field. For lower discharge pressures, the resonance at ω = ωc,e will become sharper and the power absorption can be approximated by P (r) ≈

ne (r)e20 2me νm

νm ω − ωc

2

E(r)2 .

(7.40)

Additionally, the diffusion length enlarges, and in cylindrical geometry we obtain (B 0 L, R: radius, h: height of the cylinder): 2 νm 1 = 2 2 Λ νm + (ω + ωc,e )2

2

2.405 π + R h

.

(7.41)

252


nm = 2 wc nm = wc n m = w c/2 n m = w c/4

Eeff/E0

0,6

0,4

Fig. 7.28. For lower discharge pressures, the Q-factor of the c resonance increases [283] ( The American Institute of Physics).

0,2

0,0

0

1

w/w c

2

3

Just at very low pressures the energy transfer is extremely efficient. The magnetic field causes an effective enlargement of the plasma volume in all those directions which are perpendicular with respect to the magnetic field, by the factor 2 νm + (ωc,e + ω)2 . 2 νm

(7.42)

For too low a pressure, the mean free path λe and the closely connected energy absorption will steeply increase, but the ionization drops due to • the density reduction of neutral molecules, and • the gradual decrease of the energy dependent cross section of ionization with increasing energy. Since by increasing the pressure, the efficiency of power absorption will again deteriorate, the working window for ECR is relatively narrow, and the discharge pressures should be kept below a few mTorr [284]. Further increase of the discharge pressure leads to the condition νm ω. When entering this regime, the mean power absorption per collision will decrease, the voltages for breakdown will grow and become proportional to the number density (Fig. 7.27). These competing mechanisms directly lead to a maximum of ionization in the pressure range between 1 and 0.1 mTorr [0.1 and 0.01 Pa, Eq. (7.34)] [285]. Measurements with Langmuir probes in argon revealed a monotonous drop of the electron temperature with rising pressure which has been referred to the increasing collision frequency (cf. Sect. 14.1) [286]. These competing processes cause a maximum in the saturation current measured by a Langmuir probes at a pressure of about 0.075 mTorr (10 mPa): To lower pressures, the number of collision partners decreases, to higher pressures, the electron temperature goes down. Furthermore, due to the low collision number number between the


253

charged carriers and the resonant excitation, it is questionable to speak of an electron temperature at all. At least Asmussen reported on strong deviations from a Maxwellian behavior [284]. In the resonance zone, the losses, predominantly by diffusion, are exactly balanced by carrier generation. Albeit the solenoidal magnetic field minimizes these radial diffusion losses, the very high “electron temperature” effectively counteracts this radial confinement of carriers.12 Temperatures of up to 8 eV have been measured at pressures below 1 mTorr [287]. On the other hand, electrons are guided by the axial field into the downstream region, thereby conserving the structure of the radial electron temperature distribution function and density, and, via ambipolar coupling, also the ionic parameters as well [288]. The electrons that are accelerated in the source volume will ionize the gas molecules to a comparable extent as on the low-frequency border of the band of whistler waves (ω ωc,e ), depending on discharge pressure, flow rates and the microwave power.13 7.5.2 Coupling of microwaves The models which describe the coupling of microwaves assume the microwaves to propagate in the plasma bulk against the magnetic beach of the ECR layer where they are resonantly absorbed. Therefore, the microwave window and the region adjacent to it are the zones which will decisively influence the quality of coupling. Stevens et al. demonstrated with conventional transmission line calculations that for a given plasma density, the optical thickness (product of geometrical thickness and refraction index) will reduce the reflection of the microwave at the surface of the window, thereby stabilizing the preferred mode [290] (cf. Figs. 7.31 + 7.34). 7.5.3 Electron cyclotron resonance heating The electric field of a linearly polarized wave (sum of a circularly right-hand polarized (RHP) wave and a circularly left-hand polarized (LHP) wave of same amplitude), which may propagate in z-direction, is given by (i: unit vector in x-direction) 12 It should be kept in mind that the statistic definition of temperature requires processes of equalization, most simply realized by collisions between like particles (electrons) or collisions between unlike particles (electrons and neutrals). At these low discharge pressures, both processes are not very likely to happen, despite the high degree of ionization. 13 In retrospect, this type of discharge was used to generate the first high-density plasmas. Since this discharge can be operated without metallic electrodes, a contamination with easily sputtered electrode material could be avoided. For the first time, discharges with reactive gases were operated over longer periods (several hundreds of hours [289]) under stable conditions which led to their application as plasma source for reactive ion beam etching (RIBE, cf. Chap. 8).

Intensität [a. u.]

254


Fig. 7.29. The wave should enter the plasma at the highest value of the magnetic field and will transmit its energy to the electrons within the ECR layer that is sharply defined at low discharge pressures.

RHS-Welle in der Resonanzzone Restwelle nach Absorption Begrenzung der Resonanzzone

x [a. u.]

E(r, t) = [iEx (r)]eiωt .

(7.43)

The electric field of the RHP wave rotates around B 0 in counterclockwise direction with frequency ω and transfers energy to the electrons. Secondly, the magnetic field B 0 exerts the Lorentz force −e0 v × B on the electrons which are forced to gyrate with the Larmor frequency ωc,e around B 0 . By these two interlinked processes, a continuous transfer of energy from the wave field to the electrons is ensured.14

Pabs =

V

Sohm d3 x = n0 e0 (x − iy)ERHP

(7.44)

taking account of the radial inhomogenity of the magnetic field which we assume as locally linear, which allows an expansion according to

B(z) = B0 +

∂B ∂z

z = B0 + B z = B0 (1 + αz),

(7.45)

ω=ωc,e

with z the distance between the zone of exact resonance (z = 0), and for the cyclotron frequence, we obtain in an analogous manner: z = ωres (1 + αz), ωc,e (z) = ωres + ωres

(7.46)

which gives the gain in kinetic energy: 1 e2 E 2 t2 Ekin = me v 2 = 0 RHP . 2 2me

(7.47)

For t, we insert the time of gyration in the resonance layer. We start with a thermal electron which will pile up energy in this zone, will further diffuse and 14

On the other hand, the ions are almost not deflected by the LHP wave.


255

1.0

intensity

0.5

0.0

Fig. 7.30. The absorption of the RHS wave by the plasma can be described by an Airy function.

-0.5

-1.0 0.0

0.1

0.2

0.3

0.4

0.5

B/Bc

will be reflected at the peak of the static field. On its course back, it will pass the ECR layer for a second time (Figs. 7.29/7.30). Its initial velocity is 2 v02 = v2 + v⊥

(7.48)

with v the component orientated in parallel fashion and v⊥ the component directed in perpendicular fashion with respect to the static magnetic field B 0 . The equations for the velocity components orientated in parallel and perpendicular direction with respect to B 0 are dv dt dv⊥ dt

≈ −

2 αv⊥,0 , 2 ∼

+ iω(1 + αz)v⊥ ≈ − em0 Ee ,

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(7.49)

with the approximative oscillatory solution for t [291] t=

8 2 π sin x3/2 + , παv ω 3 4

(7.50)

with

2ω x=− αv⊥,0

2/3

αΔz.

(7.51)

The oscillation of t comes about because the resonance zone is passed twice (Fig. 7.30). From Eq. (7.51), we can further see: • t scales inversely with v⊥,0 : the higher the transverse energy of the electron, the less effective the ECR heating. • Additionally, t scales inversely to v : the slower the velocity directed in parallel fashion to the static magnetic field, the longer the residence time

256

7 High-frequency discharges III within the ECR layer, the higher the amount of energy which can be piled up.

7.5.4 Electron cyclotron resonance reactors In the past, different types of ECR reactors have been offered on the market (now, it is cut back to the Hitachi/NTT variant). They mainly differed by the method of microwave coupling into the zone of resonance (Fig. 7.1). The technical realization is displayed in Figs. 7.31 and 7.34/7.35, respectively, for the two prototypes: waveguide or cavity applicator. The microwave energy is coupled into the recipient via a coaxial input probe (cavity applicator) or via a waveguide (waveguide applicator). In this case, the static magnetic field can be oriented in parallel or perpendicular fashion with respect to the direction of the wave with wave vector k (helicon waves and TG waves): • k B 0 or • k ⊥ B 0 15 In both cases, the metal housing confines the discharge, however, often a quartz tube is placed into the interior of the waveguide to avoid severe deterioration of the waveguide’s surface. The microwave field (so-called TE-wave) is generated in a microwave oscillator at constant frequency but variable cw power output (magnetron) and is guided into a tightly flanged rectangular waveguide (waveguide applicator) or a coaxial input probe (cavity applicator). Just at its entrance, the power is rectified and measured by a detector diode. The power coupled into the plasma column is the difference between the forward power and the reflected power. An orthogonal circulator serves to protect the magnetron by guiding the reflected ground wave into a water-cooled dummy load. 7.5.4.1 Waveguide applicator. The microwave energy is coupled into the recipient via a hollow waveguide. This waveguide can come either as radial waveguide (Figs. 7.31 + 7.34) or cavity resonator (Fig. 7.35) with the case as outward boundary (either made of metal or protected by a cylinder of quartz to avoid contamination what has consequences for the dispersion of the helicons, Sect. 14.7.2). To optimize the transmittance, the waveguide is equipped with a three-stub tuner. It consists of three stubs, mostly made of brass, which serve to match the wave to the highly inductive load of the plasma-filled hollow waveguide. With these stubs (section of coaxial line 1/4 λ long and short-circuited at one end), dielectric losses can be fought. 15 This case is realized in so-called distributed ECR systems (DECR). Here, the X-wave with a resonance at the upper hybrid frequency [Eq. (14.207) and Table 14.2] takes over the role of the R-wave.


257

Because of the cylindrical symmetry, their solutions are Bessel functions. Only whistler waves can propagate which are guided by the static magnetic field B 0 (q B 0 , for details see Sects. 14.7.1/2). dummy load three-stub tuner

orthogonal circulator magnetron

coupler

detector diode

l/4-window solenoides

ECR plasma

ion beam grid optics

Fig. 7.31. Schematic view of a divergent field ECR waveguide applicator with opc Oxford Instruments 1993). tional ion beam grid optics used for etching purposes (

By guiding the microwaves through waveguides, their “freedom of movement” is severely confined, and the waves answer by generation of rectangular and cylindrical waves in the waveguide and by cylindrical whistler waves in the plasma-filled tube. In vacuum, the electric and magnetic fields are always rectangular to the wave vector k. For guided waves, however, this holds only true for TE waves (for the electric field vector), or TM waves (for the magnetic field vector). The amplitudes of these waves exhibit a radial and azimuthal dependence and vary also in time which can be described by the general wave equation ψ1,2 = A1,2 Jm (γ1,2 r)eikz eimϑ e−iωt

(7.52)

with Ai amplitude constants and Jm the Bessel function of order m. Via a mirror tilted by 45◦ or a taper, the wave is deflected by 90◦ . Just in front of the dielectric window made of quartz or sapphire, a waveguide transformer transforms the rectangular TE wave into a cylindrical TM wave. The simplest mode converter were a cuboid that generated a circular TE11 wave from a rectangular TE10 wave (Fig. 7.32 M, λcutoff = 6.12 cm). At a frequency of 2.45 GHz, this mode required a plasma tube radius of 3.59 cm (λcutoff = 7.19 cm) [292] (Fig. 7.32 LHS). Albeit for this mode, the electric field peaks on-axis, it is azimuthally asymmetric. This could lead to etch profiles which are non-radially symmetric. Hence, a cylindrical converter is applied

258


which generates the TM01 cylindrical wave. To propagate, this mode requires a minimum radius of 4.67 cm [293] (Fig. 7.32 RHS, λcutoff = 9.34 cm). Its profile is concentric with outward-bound growing or fading magnetic field.

H

E

E

E

H

Fig. 7.32. LHS: The TE11 wave is the dominant wave with a cutoff-wavelength of 3.59 cm at 2.45 GHz. Albeit this mode peaks on-axis, it exhibits an azimuthal dependence. Therefore, the TM01 wave is generated applying a cylindrical converter (RHS) [294]. M: the rectangular TE10 wave.

By application of two magnets, the resonance condition can be met at three positions (Fig. 7.33). At the upper magnet [coming from higher a magnetic field than required for resonance (875 Gauss)] and above and below the lower magnet, so that the residue of the helicon wave can be absorbed. But even for lower magnetic field intensities at the lower magnet, the electrons will be mirrored between the two magnets (“magnetic bottle”, cf. Sect. 14.5). At a certain lower threshold, the magnetic bottle will leak, and electrons will flow downstream. Although the Larmor radius of the ions is larger by orders of magnitude, so they should be not affected, Coulombic interaction will force the ions to a collimated, equal flow of electron ion pairs out of the magnetic bottle. Provided they do not suffer collisions, their trajectories should remain parallel to the static magnetic field, and they eventually impinge on the substrate with normal incidence (so-called “plasma stream” mode). A third magnet at wafer level is scarcely used. The static magnetic field is divergent in radial and axial directions: • Radial direction: The intensity of the magnetic field rises in outward direction from the center to the edge of the discharge. As result, carriers drift to the discharge center increasing the charge density. This counteracts the steep gradient which has been created by the strong ionization in the annular, very narrow shaped ECR layer. By varying the intensity of the magnetic field, the zone of resonant energy absorption can be radially shifted. • Axial direction: The topmost value of the magnetic field should be found at the microwave window. The field strength must exceed the value which


259

1250

B [Gauss]

1000

125 A 875 Gauss

120 A 110 A

750

90 A 70 A

500

substrate level

50 A

250

30 A 10 A

0 0

15

30 d [cm]

45

60

Fig. 7.33. Principal sketch (“brontosaurus” shape) of a typical waveguide system with its magnetic field gradient for two magnets [295]. The efficiency of ionization is higher for the two-magnet system since the magnetic bottle mirrors the electrons back and forth, however, also the diffusion losses are higher than in a one-magnet system.

is necessary for resonance (cf. Fig. 7.33, cutoff), otherwise, the wave will be reflected rather than absorbed. Since the intensity of the magnetic field gradually decreases in the downstream direction, a magnetic bottle can be formed which leaks a little in the downstream direction. At the microwave window, a strong magnetic field serves to have the axial component of the electron velocity vanish, at the other end (downstream), B 0 should exhibit its lowest value which exerts a force downstream on the plasma (cf. Sect. 14.5.1). In thermodynamic equilibrium, a potential difference ΔΦ is generated which causes a flow of charged particles: ΔΦ = −

μ ΔB , e0 kB Te

(7.53)

and by this variation in plasma density, an additional potential difference is generated: (np : plasma density in the plasma source, n: plasma density at the capacitively driven RF electrode):

kB Te n V = ln . e0 np

(7.54)

For an excellent performance of the ECR source, this divergent magnetic field has evolved to be extremely important. Its tasks are twofold: • The ECR layer is confined to a very narrow annular zone where the condition for resonance is met. By means of the divergent magnetic field, the

260


microwave magnetron inert gas feed symmetric plasma coupler

rectangular waveguide

mirror

l/4 quartz window magnet 1

monochromator photomultiplier or photodiode array

magnet 2 reactive gas feed

mass spectrometer for RGA

wafer HR grid

RF

substrate electrode RF generator + matching network

turbo pump mechanical pump computer computer

Fig. 7.34. Besides the application of two magnets to effectively screen electrons from evading the plasma source, the separation into inert and reactive gases is strongly recommended. In particular, film-generating gases will belong to this group. Some of c Oxford Instruments them polymerize in the plasma rather than on the surface ( 1993).

zone of equal ion density is extended a little bit, above all, the density will be reduced. • As main result, a small ambipolar field is generated which exerts a net force which is directed outward, and the ions are forced to leave the weakening field. 7.5.4.2 Cavity applicator. Alternatively, the resonant excitation of a plasma volume can be achieved by coupling microwave power into a system which consists of a tunable microwave cavity and a radially inhomogeneous static magnetic field (Fig. 7.35). Only a fraction of the cavity volume is occupied by the plasma source which is sealed “upstream” by a quartz window. The plasma source can be optionally confined by a screen or grid in downstream direction to operate as an ion beam source (“ion beam mode”) as well. Typically, it consists of N pairs of permanent magnets with a pole-face maximum field of about 3000 Gauss, assembled as an octagon with eight magnets [284] or as a circle with twelve magnets [296] (two of them are shown in Fig. 7.35) to form a cylindrical resonant zone in the plasma source which exhibits a diameter of 12.5 cm to ensure the condition of resonance, approximately 1 cm inside the discharge zone. The process gas is introduced via pinholes (not shown) in the annular ring below the magnets [284]. The microwave power is coupled into the cavity via a coaxial


261

input port, and the cavity is tuned to resonance via the sliding short [297]. The volume of the resonant cavity is defined by the

2r sliding short

coaxial

input probe

h

viewing port

input port quartz window solenoid

12.5 cm ECR plasma to process chamber

solenoid

Fig. 7.35. Cross sectional view of a cavity multicusp ECR applicator after [284].

• diameter of the cylinder (2R), and the • height (h) of the remaining cylinder between sliding short (upper level), and the end plate (lower level) for excitation, and for operation by the volume denoted as “plasma”. Hence, various single modes can be excited (common are the TE111 or TE211 ). The modes of the fields vary in axial and radial directions, the modes with numbers greater than zero also in the azimuthal direction. For example, the TM-mode for a cylinder with inner radius R and height h has the solution ψ(r, θ) = Jm (γmn r)e±imθ

(7.55)

with γmn = xmn /R with xmn the nth roots of the Bessel functions Jm (x) = 0 (m is the azimuthal mode number), and the electric field Eθ must vanish at r = R. Hence, the resonance frequencies of the TM-mode are given by

ωmnp

c =√ εμ

x2mn p2 π 2 c 1 p2 π 2 R 2 + 2 ⇒ ωmnp = √ x2mn + 2 R h R εμ h2

(7.56)

with m, p = 0, 1, 2 . . . , but n = 1, 2, 3 . . . (Table 7.2 and Figs. 14.36). The resonance frequency of the lowermost mode is given by

262


2.405 c , ω010 = √ εμ R

(7.57)

Table 7.2. Roots of the Bessel functions which are important for the design of cavity resonators. order 0

1

2

Jm x01 x02 x03 x11 x12 x13 x21 x22 x23

= 2.405 = 5.520 = 8.654 = 3.832 = 7.016 = 10.174 = 5.136 = 8.417 = 11.620

Jm x01 x02 x03 x11 x12 x13 x21 x22 x23

= 3.832 = 7.016 = 10.174 = 1.814 = 5.331 = 8.536 = 3.054 = 6.706 = 9.970

and is independent of h! Hence, simple tuning by changing the geometry of the reactor is impossible [298]. The relations for the fields are obtained by Ez = E0 J0

2.405r −iωt ; e R

(7.58.1)

2.405r −iωt √ Bθ = −i εμJ1 . (7.58.2) e R For TE-modes, Eq. (7.55) is valid in principle, but due to the boundary condition for Bz

∂ψ ∂r

= 0,

(7.59)

r=R

we have the condition γmn = xmn /R where xmn equals the nth root of the first derivative of the Bessel function Jm (x) = 0, and we obtain for the resonance frequencies

ωmnp

c =√ εμ

2 p2 π 2 c 1 p2 π 2 R 2 xmn 2 + xmn + 2 ⇒ ωmnp = √ 2 R h R εμ h2

(7.60)

with m = 0, 1, 2 . . . , but n, p = 1, 2, 3 . . . The lowermost TE-mode with m = n = p = 1 exhibits a resonance frequency of

ω111

1.841 c R2 = √ 1 + 2.912 2 . εμ R h

(7.61)


263

For h > 2.03 R, this mode dominates. Since the frequency depends on the ratio h/R, a simple matching is made possible by variation of the length of the cavity resonator. Since the Q-factor just for this mode is extremely low, in most cases the TE011 -mode is used for matching. After ignition, the waves will be absorbed and the E-field breaks down, the Q-factor reaches unity, and no intensity pattern can be detected any more. The lowermost TM-mode which will be solved by a Bessel function of zero order exhibits a resonance frequency which is independent of the axial distance. Therefore, no tuning with this mode is possible, and the lowermost TE-mode is used (TE111 ). For h > 2.03 R, its resonance frequency is below the lowermost TM-mode, and the TE111 -mode will become the fundamental mode; its eigenfrequency does depend on the h/R ratio and the process of tuning is carried out with ease. The efficiency of coupling is mainly set by the damping of the whistler waves which is expressed by its inverted property, the Q-factor. Besides the conductivity of the plasma, the external parameters which influence the Q-factor are • the operating frequency [the ratio between the plasma frequency (ωp ∝ n2e ) and operating frequency determines the regime of propagation and the skin depth δ, Sect. 14.6], and • the dimensions of the cavity resonator which both determine the • special mode which matches these conditions and will dominate the oscillating behavior of the plasma. The Q-factors peak around unity, and the resonance curves are found to be relatively broad. Referring the skin depth δ to the wavelength of the operating mode which scales with the Q-factor and plotting this ratio vs. the geometric ratio 2R/h for a cylindrical cavity resonator, Q for TE-modes peaks for 2R = h (Fig. 7.36). After having struck the plasma which causes the electric resistance to decline precipitously, the electric field collapses. The advantages over the waveguide applicator are: • No external matching stubs are required; this is perfectly accomplished by the sliding short. • A standing wave is impressed against the plasma source sealed by the quartz window without the difficulties caused by the occurence of reflected powers. • It can easily be operated in the two modes: plasma stream mode and ion beam mode.

264


0.8 TE012

Q [d/l]

0.6

TE011 TE112

0.4

TM012 TE111

TE211 TM011

0.2

TM010

0.0

0

1

2

3

Fig. 7.36. Q as function of the dimension (cross section/height) of a cylindrical cavity resonator for different modes after [299]. Albeit the Q-factors peak at 2r/h = 1, this maximum is not very distinct.

2R/h

7.5.4.3 Conclusion. In ECR discharges, the electron temperatures Te are typically between 2 and 6 eV, but peaking at 8 eV at discharge pressures below 1 mTorr, and the plasma densities are in the range between 1011 cm−3 and 1012 cm−3 , i. e. higher by up to three orders of magnitude, compared with capacitively coupled discharges, and about one order of magnitude higher than inductively coupled discharges [282]. This leads to significantly shorter Debye lengths [λD ≈ 6.91× nTPe with np in cm−3 and Te in K, Eq. (3.11)]. The pressures should be less than 2 mTorr to avoid severe damping of the resonance (collisional damping), but higher than 0.5 mTorr to ensure an effective ionization: a classic example of a ridge walk. Due to the resonant power transfer which happens at pressures which are significantly lower than in other discharge types, the degree of ionization is higher than for most of the other types, but at the expense of higher electron temperatures [300]: A larger fraction of them can overcome the retarding potential of an unbiased substrate [282]. Evidently, this stress would lead to higher substrate temperatures (and eventually to damage) than in comparable capacitively coupled discharges but this constructive feature can be effectively fought by installation of a second magnet and the configuration of a magnetic bottle. Applying a certain substrate bias causes the etchrate to rise and improves the degree of anisotropy. But most important is the fact that ion generation in the plasma source is uncoupled from ion acceleration at the substrate. To enlarge the plasma density, high substrate potentials are no longer mandatory. This uncoupling is carried out by electric means and offers the possibility to separate these two processes also spatially apart (distance typically 5 − 30 cm), the plasma processes at the substrate happen “downstream”, but the discharge pressure remains still the same. This is the plasma beam mode. Installing a grid optics which has been optimized according to the laws of Paschen and Child the ECR source is turned into an electrodeless broad beam source (“ion beam mode”, Chap. 8, Fig. 7.31).


265

The essential parts of an ECR source are: • A stable microwave source. • The application of a circulator, to transmit the reflected power in a dummy load where it is annihilated. • A rectifying detector diode to measure forward and reflected power. • A three-stub tuner to match the source with the inductive load of the plasma including the λ/4-window. • A mode converter that transforms the rectangular wave into a circular wave. • Finally, the correct dimensioning of the divergent magnetic field: topmost value at the microwave window, lowermost value at the plasma exit (Sects. 14.5 and 14.7.1). ECR plasmas offer the opportunity to study the term and the conditions for quasineutrality quite easily. The neutral gas dissociates into charges carriers with very different mobilities: 5 eV-electrons at 875 Gauss exhibit a Larmor radius of only 86 μm, whereas this radius amounts to 2.15 mm for thermal (500 K) Ar ions (ωc,i is 2.1 × 105 Hz at 875 Gauss). By the order of magnitude, it reaches the dimensions of the reactor. Therefore, the IEDF is not expected to show any dependence on the intensity of the magnetic field. The strong magnetization, however, causes an ambipolar electric field in both directions parallel and normal with respect to the magnetic field which influences and guides the ions as well! The electrons which have gained velocity (are accelerated) in the ECR layer immediately experience a longitudinal force due to the diverging magnetic field. Thus its transverse velocity component is converted into a longitudinal velocity component. This causes a drain of electrons in the excitation zone which is instantaneously counterbalanced by Coulombic forces between electrons and ions. As result, also the ions are longitudinally guided (i. e. accelerated) by the magnetic field, and a neutralized beam of electron-ion pairs leaves the diverging magnetic field and impinges on the substrate [284, 301]. The plasma flickering which can also be observed in discharges excited by helicon waves at 13.56 MHz can be caused by several phenomena [302]: • Oscillations which are fed back between the discharge and the matching network. • Mode hopping. • Instabilities within the discharge caused by the collisions of gas flows with strongly different temperatures (hot gas current leaving the plasma source, cold gas flow which comes from the nozzles of the gas ring just above the RF electrode).

266


7.6 Comparison of high-density discharges For all the methods introduced, plasma generation is spatially separated from its application, and this is called downstream configuration. Two of the methods discussed are distinguished by resonant absorption of whistler waves; and collisionless mechanisms for power transfer are effective. In ECR discharges, the RHP whistler waves are resonant at the cyclotron frequency; another possibility for power transfer is the absorption by Landau damping [263]; this would include also the absorption of LHP whistler waves. To match the condition of resonance, a certain intensity of the magnetic field is mandatory for a given frequency: For 2.45 GHz, the intensity of the magnetic field is 875 Gauss. The microwaves penetrate the plasma via a λ/4 window; for reasons of process stability, its optical quality must remain constant. Employing more than one magnet, the condition of resonance can be met at different positions within the plasma source which opens the possibility to construct a magnetic bottle with two magnetic mirrors which are more or less transparent, causing the electrons to evade the plasma with a relatively strong longitudinal component. Since the electrons are partly mirrored back into the plasma source, they can trigger further ionizations, and these systems are characterized by an even higher plasma density despite the inevitable diffusion losses which are caused by the geometrical construction of this source. However, the lower the pressure the sharper the resonance; and high-efficiency processes are often connected to razor-edge situations and results—not very desirable for a production-line process. In the case of helicon-driven discharges, the transverse modes of an RF driven external antenna couple with a standing internal helicon wave. The πmatching network can be configured with the antenna itself as inductance, and the plasma densities can exceed 100 %, neglecting multiple ionizations. Its big disadvantage, however, is the phenomenon of mode hopping which is caused by the quasi-resonant coupling method. To avoid mode hopping, the design of the reactor has to be shaped very carefully. The magnetic fields exceed the intensity of the earth’s magnetic field (0.2 Gauss) by a factor of 50 − 100 but are less intense than in ECR-driven discharges (875 Gauss for 2.45 GHz). Interfering E×B drift which would cause radial and azimuthal inhomogenities are considerably lower (they can be observed for oblique mutual orientation of the two fields). For vanishing magnetic field, the helicon-driven discharges gradually becomes a simple inductively coupled discharge without any resonant coupling behavior. Although the plasma density is lower by one order of magnitude compared to the resonant methods which declines further at wafer level, it is more than sufficient for most applications. In contrast to parallel-plate reactors which can be considered as “fat” systems with a ratio radius/length much larger than unity, all these high-density plasma systems exhibit geometric configurations with a ratio radius/length approaching unity. This causes radial inhomogenities in the plasma source which

7.6 Comparison of high-density discharges

267

have not yet been treated mathematically. The very high plasma densities (between 1011 and 1013 cm−3 in the plasma source) are leveled down by more than one order of magnitude at wafer level, which is closely connected with a radial homogenization. Due to the resonant excitation, the electron temperatures in ECR discharges exceed values of 5 eV with ease to peak at values of 10 eV for very low pressures. Furthermore, high plasma densities cause very thin, collisionless sheaths. The resonant coupling leads to very high plasma densities and the coupling itself has been subject to many experimental investigations and theoretical derivations, but for production plants, it has been the inductively coupled discharges which have won the competition for a high-quality, most reproducible tool. Even in these discharges, the etchrates remain very high, and most frequently the additional application of moderating gases is recommended.

8 Ion beam systems

8.1 Introduction The reactors considered so far consist of a plasma source which is either spatially connected with the processing chamber (DC reactors and CCP reactors) or partially separated as in the high-density reactors, which are spoken of “downstream”reactors. Spatial separation between plasma source and processing chamber leads to ion beam processing which was initially pushed by NASA at the end of the 1950s when they dreamt of interplanetary missions which chould be made feasible with ion thrusters [303, 304]. For optimized thrust, the fuels should combine high atomic weight with low ionization potential to facilitate high ion densities and high a momentum at the same power input. Simultaneously, corrosion should be kept at low a level. Therefore, most experiments were performed with mercury (instead of cesium). This research remained but a dream, but as a typical spin-off process, these systems were identified as plasma sources for inert gas processing. Systems which were progressively improved are now known under the names ion milling system or microetch system [305]. These systems consist of an ion beam source and a spatially separated processing chamber, i. e. a vacuum reactor with numerous manipulating systems (Fig. 8.1). The plasma source consists of the plasma generation system itself which is confined against the other part of the reactor by the grid optics which serves in a first step to extract ions out of the plasma source and in a consecutive step to accelerate them. Plasmas can be generated by all possible methods. We can distinguish between DC excitation using a Kaufman source or a hollow cathode and HF excitation: RF or ECR. But we can also oppose the Kaufman source against filamentless sources which allow the generation of an ion beam free of contamination—at least in principle (Fig. 8.2). Although the extracted ion beam is denoted broad beam, it consists of a bundle of single beamlets (Fig. 8.3). The optics itself consists of a system of grids (1 − 4) which are aligned in “eclipsed” position and all exhibit the same diameter of the holes, at least to first order. In fact, they are delicately constructed, not only as far as their slightly different diameter is concerned (radially outward on the same grid and between different grids), but also the G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 8,

269

270

8 Ion beam systems inlet for reactive gases (CAIBE) ion beam source

ion current aperture (pivotable)

inlet for reactive gases (RIBE) load-lock valve process chamber substrate table (rotating, water cooled,tiltable)

ion beam

argon

anode

acceleration grid

turbomolecular pump

substrates current probe (pivotable)

process chamber in opened position

roughing pump

Fig. 8.1. Sketch of a reactive ion beam etching system (RIBE system with CAIBE option, equipped with a Kaufman source). By means of a rake containing a chain of Faraday cups, the radial characteristic of the ion beam can be measured (IEDF and IADF).

ratio hole diameter/thickness, distance of the grids and shape (dished or plane grids). Since the grids are always set to different potentials, this optics is denoted acceleration deceleration system. The core of the processing chamber is the substrate holder. This system is a machine by itself, since it can be tilted against the beam, it can be cooled or heated and to cap it all, it can even be rotated. On top of this substrate holder, the sample is placed. In the case of a wafer, it must be fixed mechanically with cramps when a coolant will flow below the wafer (gaseous or liquid). From this sample, material can be removed (ion milling), but it is also possible to fix this system with respect to a target from which material is sputtered away by the ion beam. This sputtered material will eventually deposit on the substrate. Then, this system acts in the mode of ion beam deposition or secondary ion beam deposition [306], although the atoms which will deposit onto the substrate are uncharged. These two chambers are totally separated in an electrical sense, but not as far as the vacuum is concerned. To meet technological challenges, i. e. high current density along with high radial uniformity of the beam, the latter must be guaranteed at the entrance of the grid optics, whereas the first can be influenced by the plasma source and by the grid optics. By means of the grid system, the vacuum conductivity is severely deteriorated, which leads to a pressure rise of

8.2 Plasma sources

271

about one order of magnitude above the vacuum in the processing chamber. By this configuration, very low pressures are possible (typically ≤ 10−4 Torr or 0.1 Pa) in the plasma source at pressures in the processing chamber of 10−5 Torr (10 mPa) or even less. Therefore, of all the plasma processing methods available, cross contamination is kept at lowest levels.

8.2 Plasma sources 8.2.1 Kaufman source In Fig. 8.2, a short historical sketch of ion beam technology is presented. The oldest plasma source is the Kaufman source [307] (see also Fig. 8.1). It consists of grounded or isolated cylindrical walls and a glowing wire which emits electrons thermally (cathode). Adjacent to the reactor wall, the cylindrical anode is installed which is set to the topmost potential within the discharge. The discharge voltage is defined as the difference between cathode potential and anode potential. By this configuration, the ions are accelerated towards the grid optics, whereas the electrons remain confined in the plasma source. The working gas, most frequently argon, is introduced into the plasma source through a port in the grounded wall. Electrons are emitted from the glowing wire and on their course to the anode, they strike the neutrals and ionize by collisional impact. Since the mean free path of the electrons becomes very long at these low pressures, which would increase the required volume for a stable discharge up to several m3 , the electrons must be contained by magnetic fields (electron confinement by magnetrons). By this configuration, not only the pressure which is required to sustain the discharge can be lowered, but also the discharge voltage between cathode and anode can be fixed at very low values. Whereas a low pressure also contributes to a lean beam due to the fact that collisions between the ions are suppressed, the latter effect serves to keep the plasma potential at low levels. Since the second ionization potential of argon is 27.6 eV (two-step process) or 43.4 eV (one-step process: 15.8 eV + 27.6 eV), only very few double-charged ions are generated, which leads not only to poor performance of the ion beam due to inevitable effects in the acceleration system of the grid optics but also to lattice damage. Its main advantages are the ease of use and simplicity, however, it suffers from a limited cathode lifetime (cathode wear by evaporation processes) which is further reduced when reactive gases are applied. Hence, a good performance with ever-increasing demands for radial uniformities becomes more and more difficult to meet.

272

8 Ion beam systems

RF (13.56 MHz)

1995

PBN coil

MW 2.45 GHz

PBN

1985

ECR SLAN ion optics

1975

permanent magnets filament DC

neutralization filament

Fig. 8.2. The progress in the ion beam technology mainly consists in the development of new plasma sources [from the Kaufman source to MW sources (ECR and SLAN) and RF sources], and in the application of refined neutralizing elements (from simple tungsten wires in the oldest Kaufman sources to so-called plasma-bridge neutralizers). In contrast to the Kaufman source, with the newly developed sources, almost c Oxford Instruments 1993). no contamination is detectable (

8.2.2 HF sources The experiments with HF sources can be traced back to the 1960s, mainly to the group of H.W. Loeb at Justus-Liebig University in Gießen, which favored RF sources, mainly coupled inductively. Others experimented with capacitive coupling and ECR sources [308, 309]; comprehensive treatments have been reported by Jongen and Lyneis and Wolf [310, 311]. In all cases, the use of a dielectric window is required to bring the RF energy into the chamber, or RF electrodes have to be applied within the chamber which itself is isolated from ground. If capacitively coupled, they are subject to ion bombardement (and source of contamination), if a dielectric window for inductive coupling is the favored solution, this requires maintenance. But no matter which method has been applied, the plasma density is significantly higher than in DC discharges, although the upper plasma density in ICP discharges is limited to about 1011 cm−3 due to the

8.2 Plasma sources

273

screening which is denoted the skin effect (cf. Sect. 14.6). But most technicians voted for ICP instead of ECR, simply because of practical considerations (too bulky and expensive, Fig. 8.2). But it should be kept in mind that due to the strong magnetic field in ECR discharges, the electrons are effectively confined within the plasma bulk and therefore very seldom hit the walls, by which, in turn, also ion bombardement is avoided. But also in ICP discharges, the plasma potential is kept at low voltages, and since the walls are on floating potential, the ions approach wall and grid optics by ambipolar diffusion. ICP driven ion beam sources dominate the market, therefore, we want to focus on this method of plasma generation.1 8.2.2.1 Design of a grid optics with RF source. Excitation by inductive ˜ induces an electric field coupling means that an oscillating magnetic field B ˜ field. Hence, we have ˜ ind which is orientated perpendicular to the exciting B E two possibilities to wind a coil around the vacuum cylinder which has to be made of quartz or alumina (so-called axial arrangement or radial arrangement). Mounting the coil in a planar arrangement at the top of the plasma chamber is the third possibility. This configuration comes very close to the standard of the GEC cell. Remembering the demands for low density of contaminants and a homogeneous ion beam with high ion flux, we see that only the radial arrangement meets these requirements. • Only when the axes of the plasma source and the coil coincide, is the in˜ ind which guides electrons and ions orientated parallel duced electric field E to the chamber walls, so that they do not intersect them, which avoids ion bombardement. This is only possible with the radial arrangement. • The ion beam density ji is the product of ion density ni and their (mean) velocity < v >. The ion density peaks at the axial center and decays in outward direction due to diffusive loss. Fortunately, the mean ion energy drops from its topmost values at the wall to lower values in the axial center, but only for the radial configuration: These two counteracting effects can effectively cancel out, and the radial uniformity of a high-density beam is ensured. Again, this is only possible for the radial arrangement. To make this quantitative, we find that the RF coil generates an axial magnetic field BRF : n (8.1) jRF,0 sin ωt, l (n: number of turns, l: length of the coil), which is inhomogeneous for l r, i. e. in the axial center, the radial intensity B(r) is lower than at the chamber ˜ induces an azimuthally orientated electric vortex field wall. This fluctuating B BRF = μ0

1

The SLAN source has been introduced in Sect. 5.6.

274

8 Ion beam systems

1 (8.2) rω BRF,0 cos ωt. 2 Because of the skin effect2 and the inhomogenity of BRF , this is no more than a rough estimation.3 This radial rise of electron temperature Te which causes an increase in the velocity of the ions via the Bohm velocity vB is characteristic for RF sources and cannot be found in Kaufman sources. Following Bohm, the extractable current density depends on the electron temperature Te according to [Eqs. (14.43/44)] Eind =

ji = 0.6065 e0 n0

kB Te . mi

(8.3)

Because of the counteracting effects which determine ni and vi , we obtain the flat response for the extractable current density ji = ji (r), and we can simply calculate the total extractable current emerging of N aperture holes with diameter ρ according to Ii = Aji = N πρ2 ji .

(8.4)

The radial dependence of the ion flux Γi (r) which is measured just below the grid optics of a 14" source exhibits an almost flat response within the inner 10", a slight increase which is caused by the rising field which is followed by the steep drop at the edge. The densities in the center and at the maximum (1" from the edge) differ by no more than a factor of 2 [313].

The voltage peaks between the terminals of the coil, and we have found that an inductively coupled discharge is ignited in the CCP mode (cf. Sect. 7.2.3). Capacitive coupling can be minimized by lowering the operating frequency. If it is lowered between the expected plasma frequency, no capacitive coupling can occur. In this case, problems to strike the discharge are likely to occur. And since we are involved with inductive coupling, a π-network must be applied to match the plasma impedance to the DC output of the RF generator (50 Ω). 8.2.2.2 Boundary voltage. Installation of a circuit which would push the ions towards the grid optics and would simultaneously retain the electrons in the ion source would improve the process of beam generation. For this purpose, Ranjan et al. installed an accelerator circuit, consisting of a coil and a tuneable capacitor in series, which taps a fraction of the power from the plasma to the RF accelerator electrode and generates a boundary voltage Vb (t) (Fig. 8.3) [229]. This voltage is the sum of a DC offset VDC and RF voltage with a peak-to-peak voltage Vpp (cf. Sect. 6.4): 2 At 13 MHz, the skin depth can be calculated to 7 mm in a cold (collisionless) plasma at a plasma density of 5 × 1011 /cm3 [cf. Eq. (14.167)]. This value is enlarged to several centimeters at lower operating frequencies. 3 Due to the cylindrical geometry, E(r) rises as a Bessel function of first order [312].

8.3 Grid optics cooling water

to oscilloscope

275 gas inlet 6” flange

beam acceleration electrode inductive coil

RF power supply

quartz tube

Fig. 8.3. Sketch of an ion beam source with accelerator electrode [229].

neutralizer grid

Vb (t) = VDC + Vpp sin(ωt).

(8.5)

The acceleration electrode is connected over a high voltage connector to a high-frequency digitizing oscilloscope. The shape of the boundary voltage can be explained by the capacitive coupling of the electrode to the plasma. Because the areas of both acceleration electrode and walls with ground potential are different, a DC potential can be formed. The discharge is asymmetric and VDC ≈ Vpp . Vpp scales with VDC and both can not be tuned independently. In this case, VDC was always set and Vpp followed. Because of their mass, the ions in the plasma can not follow the sinusoidal potential Vpp (typically at the radiofrequency of 13.56 MHz), but they are repelled by VDC by the acceleration electrode and are accelerated towards the neutralizer grid. The distance between acceleration electrode and grid optics is several centimeters and much longer than the M F P , so the ions can thermalize.

8.3 Grid optics 8.3.1 Configuration and potential adjustment The ions which are generated by collisional impact in the plasma source are now extracted and subsequently accelerated by the grid optics in order to generate a collimated beam which should simultaneously exhibit high intensity (ion flux) and high radial uniformity. To realize these opposing requirements, at least partly, highly sophisticated extraction devices have been fabricated which consist of at least one grid but which have recently been extended to a maximum of four grids. Each of them consists of hundreds of holes which are drilled into a wafer of molybdenum or graphite in a specially calculated design. Eventually, they are carefully aligned in the eclipsed configuration. Out of each hole, a beamlet will be radiated which is divergent in shape, but its angle of divergence can be adjusted by several parameters (angle of divergence α (Fig. 8.4, rB : radius of the broad beam at the target, rG : radius of the broad beam at the grid optics, s: distance between grid optics and target):

276

8 Ion beam systems

a rB

rG

s Fig. 8.4. The broad beam consists of a bundle of single beams, so-called beamlets, which are extracted from the plasma source by means of a grid optics (ion optics is also in common use). Depending on the variation of the potential, we obtain a focused (α < 0◦ ), collimated (α = 0◦ ) or defocused (α > 0◦ ) beam. Under rG , we understand the radius of the broad beam at the grid optics level, rB is the radius at target level which are separated by the distance s which determines the angle of divergence [eq. (8.6)]. The target itself can consist of a substrate to be etched, or a target from which material is sputtered on a substrate, or a rake containing a chain of Faraday cups for purposes of optimization (beam current, ion angle distribution function IADF, ion energy distribution function IEDF).

r B − rG . (8.6) s For a beamlet which is principally divergent in nature (although the broad beam itself can be parallel, convergent or divergent following this definition), we use the similar equation to define the angle of microdivergence as the truncated cone angle which encloses 95 % of the beamlet current [314], Fig. 8.5. arctan α =

ρB − ρ G . (8.7) s However, ρB and ρG are now the radii of the beamlet at grid optics level and target level, respectively, and s the distance between lowest grid and target. For optimization of the grid optics, measurements are made of the voltages which are applied to the grids and the currents which are picked up by them, the ion-beam current itself and its radial dependence by application of a chain of Faraday cups. The grid currents increase for poor focusing of the beamlets and have to be tuned to zero for optimized focusing.4 arctan α =

4 Since the ion beam current scales with the emission current in the Kaufman source and the RF current which is transmitted by the RF generator in the RF case, respectively, only these currents are subject to measurement when the ion beam source is operated in the “normal” mode. Therefore, the parameter of second order (etch rate or sputter rate), will depend on the state of the individual machine.

8.3 Grid optics

277 a

2rg

s

2rb

Fig. 8.5. For a diameter of a beamlet of 2ρg = 2 mm at the ion optics, a divergence of only 5◦ will cause a widening of the beamlet cone 2ρb to more than double its original value on its course of about 300 mm in length. The angle of divergence of a beamlet will follow the equation arctan α = (ρb − ρg )/s.

8.3.2 Screen grid 8.3.2.1 Kaufman source. The task of the inner grid is not only to suppress evading of electrons out of the plasma chamber but also to focus the ions into small beamlets, and is is denoted screen grid or electron suppressor. This grid is usually set at cathode potential, i. e. a little less than the plasma potential Φp , and the discharge voltage is the difference between cathode and anode (inner skin of the reactor or separate electrode). This additional discharge voltage is small compared to the potential of the inner grid; and it is often referred to as plasma potential, and its absolute value is set to the topmost value within the discharge which leads to an effective confinement of the electrons but repels the ions. Even further containment of these primary electrons within the plasma chamber is accomplished by means of an axial magnetic field which additionally serves to prolong their lifetime. It must be at least as intense as to prevent a primary electron from reaching the anode without have experienced a collisional impact. On their enlarged spiral course, they can additionally contribute to a higher degree of ionization by impact. Albeit the ions are influenced to but significantly lower extent, the counteracting electrostatic force which is exerted by the electrons will also keep the ions confined, and the inner skin of the plasma source is almost completely prevented from ion bombardement since the plasma potential which determines the sputtering power is only slightly higher by some volts than the discharge voltage. 8.3.2.2 RF source. The ions pass the screen grid which is set to high a posiB Te , which is considerably higher tive potential with the Bohm velocity vB = km i than the mean molecular velocity; the other walls of the plasma source are isolated from ground. Therefore, the walls of the plasma source are only subject to the bombardement of ions which are accelerated to no more than the Bohm velocity [Eqs. (3.21/3.22)]. Despite the low plasma density (a few percent at maximum), the flux of the ion current incident on the screen grid is dramatically higher than • the flux of the neutral molecules incident on the grid optics and

278

8 Ion beam systems

• the flux of the ions to the other surfaces which confine the plasma source since the fluctuating magnetic field reduces diffusive losses. This behavior justifies neglect of other losses and taking into account only the losses which happen at the screen grid—and which are intended.

8.3.3 Accelerator grid

potential [a. u.]

The potential of the outer grid or accelerator grid is fixed to negative values with respect to the potential of the screen grid. Typically, its absolute value is set to 1/10 up to 1/4 of the plasma potential which generates the required electric field to extract the ions out of the plasma source. Furthermore, the electrons are prevented from flowing back into the plasma source, which can be misinterpreted as an increasing ion current. The ions are generated at Φp , are accelerated to Φa − Φp = Vt and decelerated to ground potential within the ion beam, and their final energy is then e0 (Vp − Vg ) = e0 Vn (Fig. 8.6).

Vn Vt

ground aperture potential

screen grid

grid potential acceleration grid

Fig. 8.6. Variation of potential in the acceleration system of a two-grid optics. The plasma potential is set approximately to the anode potential, the potential of the beam which has passed the neutralization system is set to ground potential. The focusing is mainly determined by c R = Vn /Vt [315] (The American Institute of Physics).

distance [a. u.]

8.4 Qualitative treatment of beam extraction The ion trajectories are mainly determined by the potential distribution in the system of the ion optics which consists of two or three grids in customary designs (cf. Figs. 8.5 and 8.6, also for the used nomenclature); in simple extraction systems, also single-grid optics are in use, and in the most advanced system, a four-grid optics serves to optimize the beam performance.

8.4 Qualitative treatment of beam extraction

279

8.4.1 Extraction without a grid In principle, it is possible to extract ions by means of a divergent magnetic field. This principle is realized in the mirror system of ECR reactors where the magnetic bottle is adjusted to leak at one mirror into the processing chamber (see Sects. 7.5 and 14.5). In thermodynamic equilibrium, a potential difference ΔΦ is generated which causes a flow of charged particles: μ ΔB . (8.8) e0 kB Te Despite its ease of application, the main severe setback is the relatively high discharge pressure which is required to strike the discharge, but then also to sustain it. Since the mean free path scales inversely with pressure, the beam readily diverges at higher pressures, and furthermore, the energy of the ions is difficult to change. Since a grid exhibits a poor conductance, its application within one reactor wich is evacuated with one pumping system makes a mode of operation possible, in which we simultaneously observe two different pressures in two compartements. We get benefit from the low conductance of the grid: high vacuum in the processing chamber, poor vacuum (sometimes more than an order of magnitude less) in the plasma source. ΔΦ = −

8.4.2 Extraction with one grid A plasma sheath will be formed around each hole (aperture) which serves to screen the grid potential against the presheath of the plasma. As we have seen in Sect. 14.2, this presheath is necessary to build a stable sheath. In this presheath, the ions are accelerated to the Bohm velocity, in the sheath, the ions are accelerated further and focused to form a single beamlet. The dimensions of the aperture are solely determined by the mechanical stability and manufacturability. Therefore, they are fabricated of molybdenum or graphite. This sheath consists of a space-charge limited layer of positive ions; its thickness scales inversely with the square root of the plasma density. Using a single grid, we easily see that the hole diameter must be small against the sheath thickness as calculated using Eq. (4.21.2). Otherwise, the electrons would not be influenced by the grid and its electric field would flow unaffected through the (intended) apertures. For example, for an ion density of ni = 1011 cm−3 , an electron energy of 5 eV, and an extraction voltage of 25 V, the diameter of the holes must not exceed 0.2 mm! From this condition, we can also deduce that the thickness of the grid also must shrink; otherwise, we would face large losses in beam intensity due to the deflection of the ions which would impinge the grid within the acceleration “tunnel”, and so they get lost. Since the single-grid extraction lacks an effective protection against the ionic attack, it is subject to intense erosion.

280

8 Ion beam systems

8.4.3 Extraction with two or more grids The screen grid determines the absolute magnitude of the plasma potential Φp , by the potential of the acceleration grid, which is always negatively biased against the screen grid, the electric field is set up which exerts a force on the ion beam flux by which the beamlet is focused. En route this distance, the velocity of the ions is dramatically enhanced, and the current density increases. According to continuity of the current, the penetrated area must shrink. The narrowest diameter of the beam is the meniscus. The third grid is set to ground potential and retards the beam effectively to the beam potential. Moreover, it effectively prevents the sputtering of the acceleration grid by ions which would be directed to its downstream side. Poor screening of the middle grid mainly leads to contamination. The wear of the grid is almost unmeasurable. But above all, it improves the performance: better ion beam divergence at higher ratios of R = Vn /Vt [313].

8.5 Quantitative treatment of beam extraction In the approximation to first order, the ions arrive the sheath at the screen grid at the Bohm velocity; they are accelerated by the strong field between the two grids which causes a focusing of the beam. In the case of correct focusing, it will pass the aperture of the acceleration grid.

8.5.1 Current density 8.5.1.1 Derivations from Child’s law. Since the ion beam is space-charge limited, it can be determined applying Child’s law: 3 / 2e0 Vt 2 4 , ji = ε 0 9 mi le2

(8.9)

with

le =

lg2 +

d2s . 4

(8.10)

Although derived for the one-dimensional case, this equation can be applied to calculate the extractable current density of a broad-beam ion source. Vt denotes the potential difference between the two grids, and for l, the diagonal le is inserted l instead of lg , the distance between the two grids. Hence, we have to consider the thicknesses of the two grids, their distance and the acceleration

8.5 Quantitative treatment of beam extraction

281

voltage between them, and eventually the radius of screen holes (cf. Fig. 8.7) and the ratio mass to charge.5 screen grid

accelerator grid

discharge chamber plasma

Fig. 8.7. Beam divergence in a two-grid system as function of the normalized perveance. The ions are extracted from the volume with diameter le [cf. Eq. (8.9)] [315] c The American Institute of ( Physics).

le ds

da a/2

ts

lg

ta

3 • I varies with V /2 . This is why the ratio of these properties, the so-called perveance, is regarded rather than currents for a given voltage [316];

4 2e0 Ii ; P = 3 = ε0 9 mi V /2

(8.11)

it is constant for a given optics and gas composition; it rises for a given extraction voltage when the ion current is increased or when the extraction voltage is reduced at a given current density. • From Eq. (8.11), it becomes also evident that the upper limit of the perveance depends only on the ionic mass mi . • Furthermore, we infer from Eqs. (8.9) and (8.10) that the current density scales inversely with l2 which represents the extraction area. This yields, for the ion current itself that can be drawn through an aperture with diameter d:

2e0 3/2 4 V : Ii = ji le2 = ε0 9 mi

(8.12)

5 This derivation is based on the assumption that the ratio of the hole diameter of the accelerator grid, da , and that of the screen grid, ds equals unity. For different hole diameters, in particular in the case da < ds the extractable current is smaller than the maximum current calculated with Eq. (8.9). Most grid designs are constructed this way as it has turned out that the best collimation is obtained at current densities which are just a little short of the possible maximum.

282

8 Ion beam systems it is independent on the aperture da itself because le and da both scale similarly, and their variations cancel out.

Assuming that the source area of ion emission for one beamlet equals the area of one hole screen grid, 1/4 πd2s , we define the normalized perveance per hole Π: ji Π= 3 V /2

le ds

2

4ε0 = 9

2e0 . mi

(8.13)

The upper limit for current drawing is given by the breakdown field which is roughly 20 kV/cm. But above all, with increasing acceleration voltage Vt , the divergence of the ion beam eventually exceeds any tolerable mark. 8.5.1.2 Grid transparency. The extractable current can also be enhanced by raising the grid transparency η. For a hexagonal configuration of spherical apertures, the well-known formula for hexagonal close-packed structures yields η=

1 πrs2 2 √ 3 2 d 4

(8.14)

with d the distance between the centers of the adjacent holes (d = rs +x, x being the width of the bridge between adjacent holes with radius rs ). The transparency cannot be raised beyond η = 0.907, since this is the limiting value for vanishing bridges between the holes. By raising the hole diameter from 1.5 to 3.5 mm, the transparency increases by a factor of 5.5, and due to the reduced surface of the grids, issues of contamination also become less serious.6 The focusing of the ion beamlets, however, deteriorates with rising perveance. Furthermore, issues of fabrication and stabilization, respectively, limit the lower length of the aperture tunnel: Grids which are thinner than 250 μm are hardly stable across a larger diameter of roughly 8". Since the sheaths extend into the plasma, the ions are focused when they pass the apertures of the acceleration grid. Therefore, the diameters of the accelerator holes can be significantly smaller than those of the screen holes without sacrificying much of the beam current. Values down to 2/3 for da /ds have been reported. In particular, this is advantageous for larger ion beam sources, since this feature reduces the conductance of the grid optics by which the consumption of the process gases will go down. For smaller sources or for maximum current, however, the diameters of the apertures of all grids should be equal. For an optics with a two-grid design with a screen hole diameter ds of 2 mm and a grid spacing lg of 1 mm, at a total voltage of 500 V a single aperture should 6 It has to be mentioned that the extractable current density drops below the value calculated by eq. (E.9) when the apertures of the screen grid are smaller than 2 mm in diameter. The reason for this behavior is unclear [303].


283

extract a maximum of 0.1 mA of singly ionized Ar ions. A typical hexagonal aperture configuration exhibits then a density of 18.5 holes/cm2 with a center-to-center hole spacing of 2.5 mm. The average current density is then 1.85 mA/cm2 . This dependence on Vt should hold up to the breakdown limit of roughly 20 kV/cm across the gap lg [317].

For hole diameters of less than 2 mm, the plasma is almost shielded from the grid optics. If the ratio hole diameter/grid spacing exceeds unity, aberrations begin deteriorating the beam collimation.

8.5.2 Focusing and divergence From first principles, it is evident that a grid with holes that first accelerates the ions and then decelerates them can never generate a collimated beam, a beam without any divergence, since acceleration means focusing and deceleration is synonymous with defocusing. In fact, the broad beam consists of hundreds of beamlets [cf. Eq. (8.7) and Figs. 8.4 and 8.7]. We find the lowermost angle of defocusing to be no less than 5◦ . In contrast to the method to calculate and optimize the current density via Child’s law, the methods to determine the parameters which influence the collimation of the beam are still empirical. The ion beam divergence depends most strongly on the ratio R = Vn /Vt , the distance between all the grids, and the normalized perveance per hole. The focusing deteriorates with increasing perveance (rising current) which is caused by the growing constriction of the ion beam current density. Simultaneously, the electron current also increases which shallows the constriction of the ion current at the outer boundary of the accelerator grid (Fig. 8.7). Further increasing perveance enlarges the diameter of the beamlet by electrostatic repulsion: The divergence and the grid current increase once more which is mainly caused by the thinning of the sheaths. For excessive extraction, the outer part of the ion-beam current impinges the edges of the aperture tunnel. These ions contribute to the grid current (which is, in fact, loss current) and cause substantial sputtering. These effects determine the upper limit of the extractable ion current. The lower limit is determined by the backstreaming of the electrons which are necessary for the neutralization of the ion beam. To suppress this flow, a sufficiently negative potential is required (see Sect. 8.6). We insert Vt = Vs + Va into Eq. (8.9) and see that the absolute value of Vt can be influenced by both grid potentials. A highly negative value for Va is responsible for a high current density, whereas the ion energy is determined by the value for Vs . On the other hand, the electric field of this grid also prevents neutralization of the space charge below this grid, which causes a widening of the beam (deterioration of the collimation). Therefore, this effect can only by circumvented by lowering the absolute value of Va , which mainly determines

284

8 Ion beam systems

Vt . But this enhances the ratio R = Vn /Vt and, in turn, lowers the extractable ion current. This beam widening, the performance of the ion current can be evaluated by plotting the angle of divergence vs. the perveance (Fig. 8.8) [315].

relative beam expansion

1.00

0.75 Vn /Vt = 0.5

0.50

0.9

0.25

0.00 0.00

Fig. 8.8. Beam divergence in a two-grid system as function of the normalized perveance c The American Insti[315] ( tute of Physics).

0.7

0.25

0.50 0.75 normalized perveance

1.00

Although this sketch is the shape of a broad beam, we can use this pattern to describe the divergence of a beamlet as a function of the I(V ) characteristic. With rising normalized perveance, the positive space charge of the beamlet exerts a stronger electrostatic force on the diffuse electrons. With deeper penetration of the electron cloud, however, the deflection of the beamlet becomes shallower which causes, in turn, a smaller angle of divergence. Further increase of the normalized perveance, however, gradually deteriorates the collimation of the beamlet: First of all, a rising ion current will lead to enhanced charge transfer between ions and neutrals (cf. Sect. 8.7.3), and although the slow ions are readily captured by the accelerator grid, the generated rapidly moving neutrals are not affected by the potential at all. Furthermore, ions get increasingly lost due to impingement at the orifice of the accelerator grid (the aperture of the accelerator grid is smaller by typically 20 % compared with that of the screen grid). A good collimation is considerably influenced by a low ratio for R. This ratio can be varied over a limited range only and has to be kept above at least 0.5 to prevent backstreaming of the electrons. Example 8.1 To enhance the focusing, the ratio lg /ds can be increased. Consider doubling this value from 0.5 to 1.0 at a value of R of 0.7. This change in the grid optics reduces the angle of divergence α from 26◦ to about 20◦ (the half-angle α1/2 is reduced from 13◦ to 10◦ , respectively) [317]. Since R can be enhanced to 0.9 without fearing the danger of electron backstreaming, the divergence can be further improved to only 12◦ .

The divergence is mainly dependent on the normalized perveance, the distance between the two grids and the ratio R. The first parameter most strongly depends on the ratio screen hole diameter over grid thickness of the accelerator grid, but relatively weakly [318].


285

8.5.3 Conclusion The requirements for collimation and current density are a classical example for mutually opposing targets: • High collimation requires high R. • High beam current density requires low R. That is why in Kaufman sources, this ratio can be varied between 0.5 and 0.9, or to put it another way: Within certain limits, ion energy and ion flux are mutually independent. At relatively high ion energies of 0.5 − 3 keV which are necessary for etching and secondary ion beam deposition, the ion beam can be sufficiently collimated (several ten degrees at a FWHM of 10 eV). For surfaces which should be kept free of lattice damage, the ion energy must not exceed 100 eV. For an R of 0.5, this would mean an acceleration voltage Vt of approximately 200 V, which causes a steep decrease of the ion flux (0.5 mA cm−2 at 1 000 eV represents about 0.015 mA cm−2 or a bombardment of less than a monolayer in 10 sec at 100 V [319]). To solve this dilemma, we have several possibilities: • The single electrode [319]: When using only one electrode, we can get rid off the space-charge limitation. This allows a significantly higher ion beam current density because the course of ion acceleration is now determined by the sheath thickness of the single apertures. By application of micromesh grids with mesh diameters of 100 − 300 μm which are a little smaller than the sheath thickness or just equal it, the ion current density can be considerably enlarged [> 10 mA/cm2 at very low energy (10 − 200 eV)] [314]. But operating with ions of very low energy is mandatory to avoid severe grid erosion; otherwise, the lifetime is restricted to hours or days. • The configuration with three grids by which the divergence is reduced at low values for R; at high values for R, the improvement is vanishingly small (Figs. 8.9 and 8.10).

8.5.4 Three-grid ion beam source The higher expenditure for three-grid design is justified only by a performance which is significantly improved compared with two-grid optics (Fig. 8.9). In particular, the collimation is better, especially at low values for R (cf. Fig. 8.10). In a two-grid optics, the ions are focused to the smallest beam diameter when they have just passed the holes of the accelerator grid. At this boundary, they leave the zone of well-defined potential and emerge into a region with poorly defined, but rising potential (cf. Fig. 8.6) where they are decelerated

286

8 Ion beam systems screen grid

accelerator grid

decelerator grid


screen grid

accelerator grid

decelerator grid


neutralization surface


le ds

da

dd a/2

ts

lg

ta

ld

td

Fig. 8.9. The beam divergence can be effectively reduced by adding a third grid: c The American Institute of Physics). three-grid optics [315] (

and simultaneously defocused according to basic laws (Poisson equation). The zone of retardation and defocusing can be clearly defined by application of a third grid which is set to ground potential and positioned downstream of the accelerator grid, and which is called the decelerator grid. Fixing the surface of neutralization at this grid results in flatter equipotential surfaces between this grid and the accelerator grid. And flatter equipotential surfaces cause less off-axis deflection of the emerging ions [313].

beam divergence [°]

30

electron backstreaming limit two-grid optics

20

three-grid optics

10

0 0.0

0.2

0.4

0.6 R

0.8

1.0

Fig. 8.10. Reducing the beam divergence by installation of a third grid downstream of the accelerator grid and set to ground potential, here shown as function of R = Vn /Vt which is called the decelerator grid; data are from [313] and [320]. Triangles down: ld /ds = 1.0, triangles up: ld /ds = 0.123.

A reduction of the hole diameter in the decelerating grid (referring to the hole diameter in the screen grid) serves to further improve the collimation of the ion beam. A value of 0.83 for dd /ds was found as the result of an optimization procedure [316]. With similar parameters (hole diameter, grid distances),


287

potential, energy [a.u.]


a three-grid optics can deliver an ion beam of double the intensity of a two-grid optics [321]. The principal shape of the ion energy distribution function (IEDF) is sketched in Figs. 8.11.

potential ion energy

0


0


x [a. u.]

x [a. u.]


0

x [a. u.]

Fig. 8.11. Ion optics: variation of the potential for a grid optics with one, two and three grids (solid) and its influence on the ion energy (dotted): (1) One grid: large divergence but high current density, operation possible only at low energies due to severe grid erosion. (2) Two grids: low divergence and large ion-beam current density. (3) Three grids: lowest divergence at medium ion-beam current density, but complic Oxford Instruments 1993). cated set-up (

Especially at low ion energies which should be favored to avoid lattice damage, this ratio can be extended by a factor of 100 [319, 322]. Studies have shown that the three-grid optics is superior to the two-grid optics, especially what the divergence of the ion beam is concerned. For best results, the three grids should be spaced as closely as possible. However, d, the grid distance, is mainly limited by • the intensity of electric fields, which is mainly determined by the acceleration voltage which can cause arcing, but in almost all cases by the

288

8 Ion beam systems

• (failing) mechanical stability of the grids due to thermal stress and their mutual displacement by thermal expansion. That is why most grids are made of molybdenum or graphite (another reason is their low sputtering yield which reduces the risk of contamination).

8.5.5 Four-grid ion beam source Focusing is achieved by subjecting the ion beam to high potential differences which limits the extractable current by excessive ion impingement to the grid optics. To combat this upper limit, an additional grid could serve to split this voltage into two separated regions. The fourth grid then electrostatically focuses the ions after they are extracted from the plasma and are accelerated through a set of focusing and defocusing grids (Fig. 8.12) [323]. As has been shown by screen grid

accelerator grid

focus grid

decelerator grid



Fig. 8.12. By installing a fourth grid, the beam divergence can be further reduced [323, 324]. 250 V

-470 V

-1650 V

0V

Hayes et al., the collimation of the beam can be effectively improved by adding a fourth grid (Figs. 8.13). Over a wide range of ion energies, the divergence can be suppressed to values smaller than 4◦ .

8.6 Neutralization 8.6.1 Principle of operation A beam of ions, all positively charged, would readily disrupt due to mutual repulsion. What is really achieved within the grid optics is a tremendous transfer of energy from the electrons (which carry the predominant part of the energy in all non-equilibrium plasmas) to the ions. Not only is the charge of the entire ion beam balanced by a stationary electron cloud but in each small volume in the ion beam, the number of negative charges equals the number of positive charges. At operating energies of some hundreds of eV, however, the cross sections of the

8.6 Neutralization

289

10 8 6 4

beam divergence angle [degs]

beam divergence angle [degs]

250 V Vt: 470 V

3-grid, 98 % 3-grid, 95 % 4-grid, 98 % 4-grid, 95 %

2 0 0

1

2

3

4

5

8 750 V Vt: 1050 V

6 3-grid, 98 % 3-grid, 95 % 4-grid, 98 % 4-grid, 95 %

4 2 0 0.5

P [10-9 A/V3/2]

1.0

1.5

2.0

2.5

P [10 -9 A/V3/2]

Fig. 8.13. The angle of divergence can be further reduced for the broad beam by application of a fourth grid (LHS: 250 V beam energy, RHS: 750 V) [324].

ions for electron attachment have already dropped to very low values; therefore, the ions still remain “alive”. This type of neutralization is called charge neutralization. One main source for charge neutralizing electrons is a neutralizer, which serves to compensate the charge of the ion beam completely. In the case of its malfunctioning, mainly because the coupling between ion beam and neutralizer is not working properly (either distance too far or adjustment problems between ion beam and neutralizer), the required negative charge is drawn from every grounded surface, which can be observed as arcing. If the charge neutralizing electrons travel with the same average velocity towards the target as the ions, then the beam itself is neutralized: current neutralization. The origin of these electrons is the target surface (cf. Chap. 2), where the electrons are generated by • γ-process: generation of secondary electrons due to collisional impact by ions, • δ-process: generation of secondary electrons due to collisional impact by electrons, • γhν -process: generation of photoelectrons, which all together play a significant role at voltages of some hundreds of eV. When the ion beam impinges on the grounded surface of a metal, ions will recombine with electrons at the surface, and the electric current that will flow from the DC source to the target can easily be measured and is used to adapt the conditions at the neutralization element dynamically, because it equals the instantaneous ion current. Problems can arise at the surface of insulators or floating surfaces of a metallic conductor since the lack of neutralizing electrons

290

8 Ion beam systems

will reduce the velocity of the ion beam incident on the surface by which the efficiency of the electron-generating processes severely deteriorates. A deficient beam neutralization manifests again by arcing and dropping the etchrate (sputtering rate).

8.6.2 Neutralization elements To avoid this defocusing, we can use a simple filament which has to be located a short distance downstream with respect to the ion optics to guarantee effective coupling to the extracted ion beam. The potential of the hot filament is set to values between 20 and 100 V below the potential of the beam plasma,indexbeam plasma by which the filament current density becomes space-charge limited: The emitted current cannot be increased by either raising the potential (referred to the potential of the beam plasma) or by raising the temperature of the filament but only by increasing the length of the rod which is immersed in the ion beam plasma. This leads into an insoluble dilemma: To increase the lifetime of the filament to tolerable values, this must exhibit a certain thickness which requires a relatively high operating voltage. On the other hand, the potential with respect to the potential of the beam plasma should be kept as similar as possible to avoid sputtering. No matter how this optimization problem is solved, it has become evident that contamination by the filament is the main concern of this technique. This problem cannot be circumvented by other geometries. Because of the space-charge limitation of the electron current, the efficiency of a toroidal element which encircles only the outer regions of the ion beam is considerably lower [325]. Applying this method, it is possible to sputter from isolating targets that will not become charged during operation. Widening of the beam to angles of up to 30◦ , however, shows that the neutralization often does not work properly. In most of these cases, it is recommanded to set the emission current of the neutralizer slightly above the ion-beam current. To avoid these problems of contamination and short lifetime, filamentless sources have become popular. They make use of small discharges which are driven by RF, mostly in the ICP mode; but also DC hollow cathode discharges are applied. In both cases, the main difference to open sources consist in the presence of an aperture in the keeper electrode by which effective penetration of the ion beam is ensured at low voltage difference and large electron current density, thereby providing a conductive plasma bridge between ion beam and neutralizer whose current density is controlled by means of the bias voltage (Fig. 8.14) [326, 327]. To avoid electron backstreaming into the ion optics (which could be confused with the same amount of ions leaving the ion optics), the ion optics have to be protected by a highly retarding potential. Hence the effective neutralization can be achieved by

8.7 Process optimization

291

keeper electrode

bias voltage

keeper voltage

Fig. 8.14. One of the niches of hollow-cathode discharges are neutralizers in ion beam systems which are driven separately from the main source and electrically connected to the ion beam by means of a plasma bridge.

• adjusting the filament current until the emission current evens up the ion beam current measured at the surface of the grounded target, i. e. until the net current to the grounded target will vanish, or by • adjusting the filament current until the electron current which is incident on the floating target can even up the potential of the floating target to ground. At zero voltage, • adjusting the power of the plasma source in order to bring the coupling voltage between ion beam and plasma source down to zero.

8.7 Process optimization A process should be optimized taking at least these parameters into account 1. acceleration voltage and substrate damage, 2. plasma potential and substrate damage, 3. ion-beam current density and the resulting loss current at the accelerator grid, and 4. electron backstreaming. 8.7.1 Maximum power and substrate damage As we have derived in Sect. 8.5, the maximum ion-beam current density can be calculated according to the space-charge limited Child equation [eq. (8.9)]. Since substrate damage is mainly caused by high energies of the projectiles incident on the surface, and this energy is determined by Φa , the potential of the accelerator grid, Φa cannot rise beyond −750 V, and the current is in fact

292

8 Ion beam systems

limited by the source volume lg .7 It turns out that an upper value of 0.5 W cm−2 for a substrate with good thermal contact at its backside (He backside cooling preferred) must not be exceeded; at 500 V acceleration voltage8 this means an upper current density limit of 1 mA cm−2 .

8.7.2 Discharge voltage and substrate damage The density of double-ionized species steeply rises with increasing plasma potential Vp which is closely connected to the discharge voltage. For example, Kaufman estimated the fraction of double-ionized argon ions at 20 % at discharge voltages Vd between 60 and 70 V, whereas it is negligibly small at 35V ≤ Vd ≤ 40 V [328]. But double-ionized species generate severe substrate damage due to the larger penetration depth. Since all these processes are connected with damage of chemical bonds, or jumping to crystal sites etc., they exhibit a certain threshold (activation energy). After having passed this threshold, the rate constant for this process rises exponentially, and since the dose for the generation of lattice damage is relatively low (1 − 2 mA sec/cm2 ), the actual fraction of double-ionized species is almost insignificant.

8.7.3 Power and grid current Evidently, excessive beam currents cause rapid wear of the grid optics which causes, in turn, contamination problems. But if the beam currents drop below a certain threshold, process times become intolerably long. The loss current which can be measured at the acceleration grid consists of two parts: the direct ion current generated in the plasma source and the charge transfer current (Sect. 2.5.2). On their course to the ion optics, rapidly moving ions will hit slowly moving neutrals and will mutually change their velocity (or their charge). These rapidly moving neutrals will cause the same collisional impact than an ion would do. But a slowly moving ion happens to be captured more readily by the grid than a rapidly moving ion. Therefore, the grid current predominantly consists of these slow ions; and they will impinge on the upstream side of the grid optics. However, at higher acceleration voltages, two other effects will come into play: 1. Rapidly moving ions from the downstream side of the grid optics will take over the importance of the just mentioned effect on the upstream side of the grid optics. 7 Substrate damage can be caused by physical and chemical effects, from the degradation of photoresists to the complete amorphization (“atomization”) of the surface. 8 At this acceleration voltage, the Ar+ ions exhibit velocities of 50 km/sec or 180 000 km/h (the second cosmic velocity to leave the gravitational field of the earth is about 40 000 km/h).

8.7 Process optimization screen grid

293 Fig. 8.15. The characteristics of the ion trajectories will change for rising ion-beam current. More and more ions will impinge the grid during their course through the grid optics which increases the loss current. This current is measured at the accelerator grid. Schematically shown here are trajectories for normal current (dotted), maximum current (broken), and excessive current (solid, after [328]). This also causes contamination problems.

accelerator grid


accelerator current [a.u.]

2. The number of collisions caused by ions which are on course to or within the aperture tunnel will increase steeply (Fig. 8.15). Beyond a certain threshold, this current dominates all other contributions and can be recognized by plotting grid current versus beam current. Deviations from linearity which are due to the charge transfer current indicate the optimized operating point (Fig. 8.16).

impingement current maximum beam current

charge exchange current

Fig. 8.16. At larger beam currents, the onset of direct impingement of the ions on course through the grid optics is indicated by a steep rise in the loss current measured at the accelerator grid. It will become the main mechanism for ion loss (after [328]).

beam current [a.u.]

8.7.4 Electron backstreaming In the case of keeping the accelerator voltage at too low an (absolute) value, electrons will flow back into the plasma source. This movement cannot be elec-

294

8 Ion beam systems

trically distinguished by a forward motion of the ions in the outward direction. This effect can effectively be suppressed by tuning the acceleration voltage to sufficiently negative values (larger numbers). Plotting the I(V ) curve, the onset of electron backstreaming is indicated by a steep increase at very low values of Va . Backstreaming of the electrons into the plasma source will be effectively prevented for Va beyond this point (Fig. 8.17).

ion-beam current [a.u.]

electron backstreaming current

ion-beam current

Fig. 8.17. For low accelerator voltages, electrons are likely to stream back into the plasma source, thereby simulating an increasing ion-beam current (after [328]).

accelerator voltage [a.u.]

8.7.5 Current density of the ion beam For sputtering, damage is of no concern, but the ion flux should be as high as possible to shorten the process time. For etching, issues of beam divergence and the voltage of the screen grid, Vn , come to the fore, since a high-quality collimation allows larger distances between the ion optics and the substrate which reduces contamination problems. The beamlets superimpose eventually to form a homogeneous broad beam with a relatively flat beam profile. We can influence the flux of this broad beam at a certain distance by focusing or defocusing. This can be most readily achieved by a systematic displacement of the holes in one grid with respect to the holes of the other grid. This can be further improved with dished grids (Figs. 8.18). 8.7.6 Uniformity Even provided the divergence of the beamlet is no larger than 5◦ , this means a widening to more than double its initial diameter over a distance s of 300 mm (Fig. 8.5). For a one-dimensional array of holes 2 mm in diameter at a distance of 7 mm, the pattern of the grid optics remains clearly visible for a distance between


295

Fig. 8.18. By measuring the ion-beam current density across the target, we obtain evidence of its radial uniformity. If the broad beam appears focused at target level, it is spoken of as a convergent beam (top), if it is defocused, the beam is denoted divergent (bottom). The radial uniformity can be controlled not only by a systematic mutual displacement of the grids but also by shaping into the third dimension (dishing, RHS). A high ion flux is advantageous for sputter processes, whereas for etching processes, c Oxford Instruments 1993). the broad beam should be radially uniform (

grid optics and substrate of 300 mm and an angle of divergence of 5◦ . At a ratio of 1 : 1.2 (diameter/distance of the centers of adjacent holes), however, the pattern is already blurred (Figs. 8.19). In reality, these are two-dimensional grids with a ratio between aperture diameter and distance of the centers of considerably larger than unity in a hexagonal configuration. Inspecting the radial uniformity of the broad beam measured with a rake containing a chain of Faraday cups, this is sufficiently homogeneous (Fig. 8.20). However, this measurement does not allow a statement concerning the collimation of the beamlets. On the other hand, we can qualify the radial uniformity of the broad beam with respect to the current density from this measurement. The homogenity does slightly deteriorate with sinking current density (at 70 mA/cm2 , the radial uniformity deviates by ±1.8 % and is improved to ±1.04 % at 80 mA/cm2 ). This is advantageous for ion beam coating processes (cf. Sect. 10.9), but has some disadvantages as far as etching is concerned; the IADF is always uncertain (cf. Sect. 11.6). From this, it is evident that homogenity of the beam consisting of hundreds of overlapping beamlets, measured in mA/cm2 , is not at all an issue. Therefore, no beam pattern is visible at the target. The distribution of the atoms sputtered from the target mainly follows a cosine distribution, since at these low pressures of 0.1 mTorr or less, there is no scattering of the sputtered atoms flying to the

296

8 Ion beam systems 150

current density [a. u.]

current density [a. u.]

150

100

50

0

0

25

50

-250 a. u.

100

50

0

75

d [mm]

0

25

50

75

d [mm]

Fig. 8.19. LHS: For a diameter of a single beamlet at the accelerator orifice of 2 mm, the pattern of the grid optics (linear chain of holes 7 mm in diameter) remains clearly visible for a distance between grid optics and substrate of 300 mm and an angle of divergence of 5◦ Lower: two adjacent beamlets, upper: resulting pattern of the current density. RHS: With the same diameter of the aperture hole, but at the very close distance of only 2.4 mm between centers, the beamlets smear out to a homogeneous linear beam profile. Lower: six adjacent beamlets, upper: resulting profile of the linear beam, which is nearly homogeneous (±5% between mean value and extreme values. In reality, the grids show a two-dimensional, hexagonal configuration with a ratio between aperture diameter and distance of the centers of considerably larger than unity.

relative beam intensity

1.00 0.95

Fig. 8.20. The radial uniformity of the curent density of the ion beam, measured with a rake containing a chain of Faraday cups at target level, is prerequisite for a homogec Veeco Instr. neous coating 2002.

0.90 0.85 0.80 0.75 0.70 -15

70 mA/cm 2

-10

-5

75 mA/cm 2

0

80 mA/cm 2

5

10

15

position [cm]

substrate by ambient molecules (as is the case for RF diode sputtering). This is shown as a conclusion in Chap. 10. What is really interesting, however, is the deviation from normal incidence. Widening the beam by only 5◦ has dramatic consequences for the etching angle (Chap. 11). This etching angle is a function of the angle of incidence of the ions


297

and the distribution function of their velocity (denoted as ion angle distribution function IEDF and ion angle distribution function IADF) [329].

9 Plasma diagnostics

This chapter is divided into two parts: In the first part, several techniques are introduced and discussed from the very beginning to a level of being able to read the original literature. The second part is devoted to some results which are presented for high-frequency plasmas (CCPs, ICPs and whistler wave generated plasmas), especially discharges through electronegative gases.

The various plasmas can be analyzed in several ways to obtain the parameters which determine their qualities: • Plasma density; more precise carrier density (ni ). • Electron temperature. • Temperature of molecules and ions. • Frequency of elastic collision between electrons and neutrals. • Chemical and electrical composition of the plasma. • Chemical and energy resolved composition of the plasma sheaths. • Absolute determination of the densities of reactive species. • Determination of electrical data. Since the methods are different, the results are different as well, even when we consider the same property. Prominent examples are the electron temperature, measured with optical emission spectroscopy (OES) or with a Langmuir probe, or the plasma density (measured with electron spectroscopy (SEERS), microwave interferometry, or with a Langmuir probe, cf. Table 9.1). In Chap. 6, we have already discussed plasma analysis, applying energy resolved mass spectrometry to determine the IEDF. Since no method can give answers to all the questions, some methods are introduced and critically compared. It must be admitted that a definitive method for determining the plasma density is still lacking, which explains the numerous different approaches. G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 9,

299

300


Table 9.1. Advanced methods to determine important plasma parameters. AA: Advanced Actinometry; SEERS: Self-Excited Electron Resonance Spectroscopy; Z: analysis of the impedance Z; μ: microwave interferometry. Physical property VP nP Te Tbulk PRF,eff νm cos ϕ Vrms Irms

Method of determination Langmuir AA SEERS Z √ √ √ √ √ √ √ √ ? √ √ √ √

μ √

9.1 Langmuir probe 9.1.1 Introduction A Langmuir probe consists of a spherical or cylindrical electrode which is immersed in the plasma. By applying a potential difference which is referred to a system immersed in the plasma as well, the so-called reference electrode, we can record a current voltage characteristic (Fig. 9.1). ¿From its shape (Fig. 9.2), important plasma parameters can be deduced. The theory was established by Langmuir and Mott-Smith in a series of papers beginning in the year 1926 for DC discharges through argon [330]. If not explicitely stated, the following remarks refer to such an electropositive gas without any negative ions. We first start with an inductive approach of the phenomenology of the I(V ) characteristic, will continue with a detailed explanation of grounding problems and the definition of the reference potential, and the later sections are devoted to certain details of the catching of carriers by the probe (absorption of electrons and ions).

9.1.2 Conditions for performance Planar probe theory applies for the following conditions: • The dimensions of the plasma should be large compared to the Debye length.

9.1 Langmuir probe

301

aluminum wings KF 40 flange vacuum feedthrough tungsten rod aluminum tube

clamping ring glass tube

Fig. 9.1. Setup of a Langmuir probe. Vacuum part with a tube made of aluminum to protect the glas capillary which contains the probe rod made of refractory metal with large-area stainless steel wings which serve as high frequency compensating electrode and a vacuum flange includc [331]. ing electric feedthroughs This system is attached to a linear translator which allows spatial measurements along the axis of the reactor.

• The temperature of the electrons Te should be much higher than the temperature of the ions Ti . • The discharge pressure should be so low that the mean free paths of both carriers should be large compared to the sheath thickness of the probe: collision free sheath. • The sheath thickness rs should be small compared to the radius rp of the tip of the Langmuir probe.1 In these cases, the area of the sheath edge is comparable (only slightly larger) than the area of the surface of the rod). Otherwise, the effective area of the probe is difficult to define. The length of the rod lp which is not protected by the glass sleeve and protrudes into the plasma should be large compared to its diameter to meet the requirements for a cylindrical probe. • The material of the electrode should be chemically inert, exhibit a low sputtering yield and a high work function: Its conduct should be neutral. This can be best fulfilled using refractory metals as tungsten, rhenium or rhodium. • The Debye length of the plasma should be much smaller than the radius of the probe to keep the disturbance of the charges as small as possible [250]. The area of the reference electrode should be large. Especially in reactors with electrically insulated walls, or when their electric state will change during a process [e. g. processes with coating plasmas (CF4 , CH4 1

See Chap. 4 for the relationship sheath thickness and Debye length.

302

9 Plasma diagnostics etc.)], its application is recommended [332]. Moreover, its additional advantage is the straightforward determination of the floating potential Vf (cf. Sects. 9.1.5/6).

• The inevitable disturbance of the charge distribution should not alter the properties of the plasma significantly. • In the (weak) field of the probe, there should be no temporary variation of the spatial distribution of the carriers. • The probe should be an absolute drain for the carriers. • The geometry of the probe should be spherical or cylindrical; in the following, we confine to the cylindrical type. ¿From these conditions, it is evident that a Langmuir probe must not be used in zones with low electron density and hot electron temperature, especially not in the sheaths. For this end, Gottscho, Zarowin and Czarnetzki have proposed (and brought to perfection) the application of the Stark effect, i. e. the splitting of originally degenerate lines in the strong electric field of a sheath [333] − [335]. With this method, a spatial resolution of about 0.1 mm3 in volume has been demonstrated. However, these measurements require a homogeneous field to avoid Stark broadening, which does not show any temperature dependence — in contrast to e. g. the Doppler effect [336]. As result, with this unique method the determination of the electric field of a sheath has been made possible — and by this, the independent control of the sheath thickness which is electrically active (in contrast to the sheath thickness which is determined by optical investigations). A biased electrode will collect electrically charged carriers which are accelerated by the electric field according to Coulomb’s law. This loss disturbs the quasi-neutrality of the plasma which responds by equalization processes: a flow of electrons into the (positively charged) drain or a flow of electrons out of the (negatively charged) source. This process leads to the build-up of a selfconsistent sheath in front of the electrode. To act as plasma probe, its electric field must not exceed the threshold of electron generation by secondary processes (thermal excitation or γ-processes). Likewise, the electrode current must be kept below a disturbing level (cf. Sect. 9.1.12 for determining the plasma potential). • During operation, the potential distribution is disturbed within the Debye length. • This leads to a flow of charged particles and, furthermore, to the built-up of space charges. Due to the strong asymmetry of the I(V ) characteristic, which is caused by the large mass ratio between electrons and ions, the initial electron current far surpasses the ion current by some orders of magnitude (cf. Sect. 3.4).

9.1 Langmuir probe

303

ji ∝ je

me . mi

(9.1)

Hence, the negatively charged space charge is much more important. Since the ion current is very small in magnitude, its measurement is very delicate, and the determination or at least estimation of the two properties electron density ne and electron temperature Te constitutes a significant part of the theory of the Langmuir probe. 9.1.3 Characteristic of the Langmuir probe

electron-saturation current

3

ln IS [mA]

VP

2

retarding-field regime ln j = const - e0 /kTe * V

1

Fig. 9.2. Scheme of an idealized I(V ) characteristic of a Langmuir probe to evaluate the most important plasma parameters electron density, ion density, electron temperature, and plasma potential [337].

VP (d2j/dV2 ) = 0 VP = VS

VF

0 ion-saturation current B A -20

0

20

40

DF[V]

9.1.3.1 Principle of the measuring technique. We regard the plasma bulk as a homogeneous zone with constant plasma density which lacks an electric field. It is confined by sheaths which are terminated by conducting reactor walls to ensure a proper circuit. In this case, a current from the Langmuir probe to the reactor walls will flow through plasma and sheaths if an electric potential is applied to the Langmuir probe. The principle can be seen from Fig. 9.2 [337]. The potential of the cylindrical rod is compared with that of the anode or with ground potential. Applying highly negative potentials will retard the electrons but accelerate ions which create a positively charged space layer around this negative potential. Therefore, this ion current quickly saturates (so-called space charge limited current, points left from A). With growing potential (i. e. less negative values), fast electrons can be absorbed by which the net current of of positive ions is diminished (condition: 1/2me ve2 ≥ e0 V , Point B). At a voltage which is still negative, 0 > V = Vf , both the fluxes are equal in magnitude, and there is no net current. This potential is

304


denoted the floating potential, because it is the potential of a conductive system immersed in the plasma but not connected to a reference potential, in particular not to ground. Further increase of the probe potential leads to weakening of the ion current, but to a steep growth of the negative current (which consists of electrons). This region is called the electron collection region, transition region or retarding-field region. Eventually, the potential of the plasma equals the probe potential. Only at this point will an electric field not repel or accelerate either carrier type, and they move to the probe according to their thermal velocities. According to Poisson, this is synonymous with a lack of space charges. Here, at the plasma potential, the retarding-field regime turns into the electron saturation regime, more or less sharply. To determine Φp exactly, we plot the logarithm of the probe current density vs. probe voltage. Due to the asymmetrical mobilities of the electric carriers, the V (I) characteristic is asymmetrical to the origin and increases steeply in the retarding-field region. Thanks to the logarithmic suppression, it seems to be a straight line in most cases. In principle, this steep increase is terminated at the plasma potential, and the slope of the characteristic should become significantly shallower and can be approximated by a second straight line. Following this definition, the plasma potential is determined by the point of inflection of the I(V ) characteristic (d2 I/dV 2 = 0). Another common definition is the intersection of both of the extended lines of electron saturation current and retarding-field region (Fig. 9.2). The concordance between these two approaches will become better for steeper slopes of the I(V ) characteristic. Only electrons which have invaded the sheath can traverse the sheath and can be eventually collected as probe current. If the thickness of the sheath (and hence its volume) were independent of the applied potential, the electron current could not grow even with further increasing potential but saturates at a real constant value. In fact, not only the area of the probe is subject to change but also the thickness of the negative space charge. Since this sheath consists of higly mobile electrons, it is significantly more extended than the ionic sheath in the regimes of the ion saturation current and the retarding-field regime, respectively. This is the reason why the electron current further increases even beyond the plasma potential, however, with a smaller slope. 9.1.3.2 Extraction of plasma parameters. According to the equation 1 (9.2) me ve2 ≥ e0 (Φp − ΦLp ) = e0 VLp 2 with Φp the plasma potential, ΦLp the probe potential and VLp the probe voltage, respectively, the energy distribution of the electrons EEDF can be determined by variation of the electron current with respect to the potential of the Langmuir probe ΦLp . The mean value of the energy is straightforwardly obtained by application of a normalized distribution function f on the measured energy values according to the relation

9.1 Langmuir probe

305

<E > =

∞

E f (E) dE.

(9.3)

0

For a Maxwellian distribution of the electrons, the fraction of the electrons which have a higher energy than e0 VLp and are able to overcome the sheath, is given by

Ne = N0 exp −

e0 VLp , kB Te

(9.4.1)

which yields for the current density

e0 Ne < v > e0 VLp je = exp − V 4 kB Te

⇒ je = e0 n

e0 VLp kB Te exp − , (9.4.2) 2πme kB Te

from which follows ln je = const −

e0 VLp . kB Te

(9.5)

¿From this equation, two properties can be derived: • Equation (9.5) is a straight line with a slope of e0 /kB Te (since d/dVLp ln je = 1/je × dje /dVLp = −e0 /kB Te ). Provided the effective area of the probe is exactly known, we can extract the electron temperature.2 • For VLp = 0, i. e. at Φp = ΦLp , the electron density n0 is given from the current density:

je = e0 n0

kB Te . 2πme

(9.6)

¿From a linear increase of the logarithmic current density in the fieldretarding regime, we can conclude that the electrons are distributed Maxwellian. In all other cases, the energy distribution of the electrons must be described by other functions. A very popular function is the Druyvesteynian distribution function [339], which has been derived for distributions in electric fields. Its application is superior for higher electron energies and low(er) electron densities (cf. Sect. 14.1).3 2 Numerous trials are reported to apply this equation to the ions which led to ridiculously high values for Ti . The reason is a change in the energy distribution of the ions IEDF caused by the sheath formation [338]. 3 This description is valid only for the ideal case. As a matter of fact, in most cases, no sharp bend in the I(V ) characteristic is found. Then two or even more slopes must be determined — with a necessary spread in Φp (i. e. Maxwellian distribution of the electrons with two temperatures).

306


9.1.4 Plasma potential We have already realized that “plasma potential” is not only difficult to measure but also complicated to define. The potential of the Langmuir probe or the potential of the reference electrode can be absolutely determined. The plasma potential Φp is defined as that potential of the probe where no space charges (and electric fields) around the probe are present. Hence, the carriers move to the probe solely with their thermal velocity, experiencing no drift in an accelerating or retarding field. This voltage can only be evaluated from the I(V ) characteristic, and it is the value of its abscissa of the point of inflection, which coincides with the straight lines which approximate the electron saturation current and the electron retarding zone. The point which can be directly extracted from the I(V ) characteristic, however, is the floating potential Φf , which exhibits the additional advantage that it can be theoretically determined (Eq. 2.21) when the current of positive carriers is neglected, e. g. in discharges of a electropositive gas (inert gas or oxygen at low pressures). In the context with the electron temperature Te , we have to discuss (Fig. 9.3): • The plasma potential Φp , referred to the potential of the reference electrode. • The potential of the probe ΦLp , referred to the plasma potential, i. e. the sheath potential of the Langmuir probe. • VLp , the potential difference measured with the Langmuir probe.

potential

Fs

Fp Vp

Fig. 9.3. Going from the plasma potential Φp to more negative values for the voltage, the probe is surrounded by a cloud of positive ions creating a positively charged sheath potential which has to be corrected for.

distance

We measure VLp which deviates from Φp by the sheath potential Φs . Only when the potential ΦLp equals the plasma potential Φp , is no sheath present, and only here it holds true that

9.1 Langmuir probe

307 Fd probe 2

probe 1

relative potential

Fb

Fg

Fa

Fig. 9.4. Sketch of the potentials across two electrodes which are immersed in the plasma: Steep sheaths and a voltage difference between the electrodes which is rather caused by space charges than by Ohmic resistance. Φγ − Φα is the external voltage after [340].

Fa distance

ΦLp = Φp .

(9.7)

To evaluate the I(V ) characteristic in order to obtain the significant plasma parameters, ΦLp is the highest necessary abscissa value. For even more positive potentials, the probe is surrounded by a cloud of negative carriers, i. e. electrons; consequently the measured value for Φp has to be corrected by a negative value. For even more negative potentials, however, the probe is surrounded by a cloud of positive carriers, i. e. ions, consequently the measured value for VLp has to be corrected by a positive value (Fig. 9.3): Φs = ΦLp − Φp .

(9.8)

In the following sections, we will focus on this principal problem of determining the plasma potential and the floating potential. 9.1.5 Principle of the double probe We have dealt with the potential of the Langmuir probe in an absolute way insofar as we considered the floating potential as easily measurable (in a floating system, the net current drawn from the plasma is zero). However, it is not that easy. In Fig. 9.4, we have sketched the voltages between the potentials Φ of points β and α and δ and γ, respectively. These denote the sheath voltages, and the voltage between the potentials Φβ and Φδ is the potential drop in the neutral, undisturbed Ar+ plasma, consisting only of (fast) electrons and (slow) Ar+ ions. This potential difference is caused by differences of space charges in the plasma rather than by the existence of an Ohmic resistance and is small compared to Vβ,α and Vδ,γ , further simplied: This potential difference vanishes.

308


Provided that the sheath voltages do not depend on the external voltage, we can start with the condition of current continuity in this circuit and the definition of the potentials [340] (Ip denotes the current in the circuit of the two electrodes defining the Langmuir probe): Ip = I1− + I1+ ∧ Ip = −(I2− + I2+ );

(9.9.1)

(Φβ − Φα) + (Φδ − Φβ ) = (Φγ − Φα) + (Φδ − Φγ )∨ Vβα + Vδβ = Vγα + Vδγ .

(9.9.2)

Since the currents of ions and electrons both depend on the potential difference between plasma and the probes, both the sheath potential drops will adjust to values that condition (9.9.1) is fulfilled. For equal or comparable probe areas, we denote this setup a double probe system. For a floating probe system, which does not draw any net current from the plasma, it is symmetric with reference to the floating potential Φf . In its simplest form, this system consists of two identical probes which are connected to a voltage source. The current flowing in the circuit can be measured as a function of the voltage between these two probes. The inherent advantage over a simple Langmuir probe is twofold: • The electrodes are on floating potential Φf and are not connected to the electrodes of the discharge tube. Fluctuations of the plasma potential Φp will cause a similar fluctuation of the floating potential Φf which leads to constancy of the sheath potential Φs : Φs = Φp − Φf . • The probe current can be kept at very low values and hence the (negative) influence on the plasma should be small [341]. Discharges which are lacking in clearly defined electrode potentials, in particular electrodeless high-frequency discharges, are the main candidates for their application. The symmetrical setup leads to a I(V ) characteristic which is symmetrical to the origin for vanishing potential difference in the plasma (i. e. for constant plasma potential). Otherwise, the I(V ) characteristic is shifted by this difference (Fig. 9.5). The main advantage of the double probe system is the smaller disturbance of the plasma compared with a single probe measurement [341]: For equal areas of both the electrodes, the electron current which flows to the positively biased electrode is limited by the flow of positive ions to the other electrode, the socalled reference elecetrode, and not by reaching the plasma potential. Hence, the assumption that the sheath potentials are not influenced by the externally applied voltage, is justified to an extent.

9.1 Langmuir probe

309

1.0

Is [mA]

0.5

II,1 Rs = 1/(D Is/DV)

0.0 II,2

-0.5

-1.0 -20

-10

0 DF [V]

10

20

Fig. 9.5. I(V ) of an idealized double probe with a characteristic symmetrical to the origin (no difference in plasma potential for both the electrodes) [341].

However, this system lacks the possibility of measuring either a plasmapotential or a floating potential. But for a Maxwellian distribution of the electrons, the two common parameters, i. e. electron temperature and plasma density can be evaluated from the I(V ) characteristic. To avoid this dilemma, at least two possibilities do exist: The application of a large-area electrode or the addition of a third electrode whose potential with reference to the plasma is kept constant under all circumstances. 9.1.6 Principle of the asymmetrical double probe By choosing a strong asymmetrical ratio, we can heighten the ion current at the large area electrode (1) to values which are large compared to the electron current at the small area electrode (2) even at the plasma potential (Vδγ = 0): I2− I1+ .

(9.10.1)

I1− = −I+,1 ∧ Ip ≈ 0,

(9.10.2)

Then eq. (9.9.1) simplifies to

because of I1+ I2+ ∧ I1− I2− : The resulting small current only consists of an ionic and electronic contribution at the large electrode which mutually cancel out. This can happen only at high a negative potential. Now, even strong variations of the external voltage Vγα at the small area electrode influence the current at the large area electrode only to a very small extent. Thus, we can vary the probe potential by variation of Vγα without affecting the voltage between the large area electrode and the plasma. Henceforth, we refer to the large area electrode as “reference electrode”, exhibiting a strong negative potential with respect to the plasma potential and to the smaller probe. Remembering our

310


definition of the floating potential Φf , we realize the accordance of the reference potential with the floating potential [342]). Decreasing of the external potential Vγα from Φp to smaller values at the smaller electrode (going left in the I(V ) characteristic), leads to strong variations of the electron current with the ion current remaining nearly constant: free fall in the electron retarding regime. This effect is accompanied by relatively small variations of either currents to the reference electrode, and it can be made more distinct by increasing the ratio of the probe areas: the transformation ratio.4 By increasing the potential to values beyond the plasma potential Φp , however, only little changes are expected because the range of saturation of the electron current is quickly reached. Setting the potential of the reference electrode to zero leads to a positive shift for the I(V ) characteristic. In particular, the value for Φf shifts to positive values. If the electrons are assumed to be Maxwellian distributed, we can connect the probe current with the voltage between Langmuir probe and reference electrode by ln Ip = ln Ip,s −

e0 (Φp − Vb ) kB Te

(9.11)

with Ip the probe current, Ip,s the probe current in the saturation regime, Φp the plasma potential, referred to the reference electrode, and Vb the voltage between probe and reference electrode. As result, we have obtained the characteristic of a single probe, but with clearly defined reference potential. 9.1.7 Determination of potentials in high-frequency discharges If a Langmuir probe lacks a reference electrode with constant potential, we are faced with severe difficulties of interpretation. Additionally, plasma fluctuations have a strong influence on the characteristic of a Langmuir probe [183]. Boschi and Magistrelli were the first to point out the difficulties in interpreting the Langmuir probe data “polluted” with RF noise [343], and afterwards, the HF modulation of the EEDF and various plasma properties, e. g. the electron density, respectively, has been extensively discussed by Winkler et al. [344, 345], Godyak [346] and Wiesemann [347], just to mention a few. As could be shown by Flender et al. [347], the second derivation of the Langmuir characteristic which serves to evaluate the EEDF, very sensitively reacts to an insufficient RF compensation, whereas the characteristic itself remains almost unaffected (Fig. 9.6). Hence, the plasma density np and the plasma potential Φp can be evaluated from the Langmuir characteristic, but the EEDF can be not. 4 In the best case, the inner surface of the plasma reactor serves as reference electrode. To ensure constant surface conductivity, reactive gases must not be processed which could contaminate the reactor walls.

9.1 Langmuir probe

311 15

0.7

0.5

I [mA]

0.4

B D

10 5 I [mA]

0.6

B D V(t) H

0.3 0.2

0

0.1

-5

0.0 -0.1 15

20

25

30

35

-10 15

40

20

25

30

35

40

V [V]

V [V]

Fig. 9.6. I(V ) characteristic and its second derivation, as measured with good RF compensation (solid) and the original signal convoluted with a sinusoidal voltage of V0 = 5V (dashed) after [347].

Winkler et al. showed that in capacitively coupled plasmas driven at 13 MHz, i. e. significantly beyond ωp,i , a threshold of approximately 75 mTorr (10 Pa) can be drawn which has to be exceeded to find the derived plasma properties significantly frequency-modulated. Hence, at discharge pressures which are common for reactive ion etching (RIE), no HF modulation is detectable: Only the HF component of the plasma potential has to be considered because of an additional potential drop across the forming sheath, the RF self-bias [183]. Hence, we expect the probe voltage to be modulated by a DC component VDC and an AC component V˜ , respectively [348]. For sinusoidal variation, the voltage in Eq. (9.5) is modified from5 Vp −→ VDC + V˜ sin ωt.

(9.12)

Averaging over an RF cycle of duration T , i. e. e0 V˜ 1T exp[iV˜ sin(ωt)] dt = J0 − T 0 kB Te

(9.13)

yields the Bessel function of zero order with pure imaginary argument [349], and we obtain for the current density ln je = je,0 −

e0 (VDC + ΔV ) kB Te

(9.14)

with [343]

kB Te e0 V˜ ΔV = − ln J0 − e0 kB Te

:

(9.15)

5 Being more precise, there is an additional voltage drop across the sheath caused by the electron current of the order of R A je (t) with A the effective area of the tip.

312


rounded "knee" I (sat) e

3

VP

ln Ip [mA]

2 D Vdc

1 VF

VP (d 2I/dV2) = 0

VF

0 Ii (sat)

-1 -40

-20

0

20

Fig. 9.7. High frequency discharge: The characteristics is shifted by the self biasing process, the electron current across the sheath distorts its upper part [rounded “knee”, (after [348])].

V [V]

Due to the balancing of electron and ion current, the ln I(V ) characteristic of the probe is shifted by ΔV with respect to the applied bias. For small ratios of e0 V˜ /kB Te which are smaller than or equal unity, only a negative shift is imposed on the ln I(V ) characteristic, for large ratios, however, a significant rounding will occur at the “knee” induced by the Ohmic term R A je (t) (cf. Fig. 9.7) [337]. Careful attention has to be paid to filter the signal from high-frequency noise. This can be accomplished by a transmission line beginning with an inductor which terminates the tungsten rod of the Langmuir probe. This transmission line contains a series of circuits with resonance frequencies which equal the fundamental and its overtones of the excitation frequency. To avoid stray capacitancies of the probe wire to the outer ground, the mechanics of the Langmuir probe often comes along in triaxial technique. For this end, an additional inner shielding is used which is capacitively coupled to the plasma by means of large-area compensating electrodes, for example metal wings, which are attached to the Langmuir probe (cf. Fig. 9.1). Since the probe impedance is mainly determined by the capacitance between Langmuir probe and plasma, an additional shunting capacitance which is exposed to the same plasma potential Φp , will reduce the impedance of the Langmuir probe to low values, which would be unachievable by other means. Now, a requirement established by Godyak et al. can be readily met [346]: 1 Te Zs ≤ Zext 2 VRF

(9.16)

with Zs and Zext the impedances of the Langmuir probe (and plasma) and the external probe circuit, mainly the DC resistance of the RF driven electrode sheath (which are comparable without this shunting impedance). It should be kept in mind that this requirement has to be followed by all the Fourier

9.1 Langmuir probe

313

components which are generated by the rectification of the sinusoidal voltage applied to the RF driven electrode. Hence, it can follow, at least in principle, the high-frequency amplitude of the plasma potential without any phase shift. In this inner shielding, no currents would flow except the inevitable capacitive coupling to the outer shielding. Another possibility is the application of a double probe or a triple probe, respectively [350]. With this probe, it is possible to create a reference potential even in electrodeless discharges [351]. On the other hand, the oscillations can be investigated with a carefully designed circuit. Wood et al. connected the rod to a series of parallel LC elements within the probe shaft [352]. If they were tuned to resonate at 13.56 MHz they would follow the oscillations of the plasma potential, and the typical parameters (ne , Te ) could be extracted as if they were taken in a DC discharge. They could verify the self-consistent model with a highly nonlinear sheath motion which was established by Lieberman in 1988 [192] (cf. Sect. 14.4). This model improved the picture of a sinusoidal sheath oscillation which had been introduced by Godyak ten years earlier [206]. Later on, Lieberman’s model was brilliantly confirmed by Vender and Boswell with particle-in-cell (PIC) simulations in 1990 [353], and by Gozadinos et al. with their considerations for stochastic heating [211]. 9.1.8 Details We will now focus on some details of the collecting process of carriers which arrive at the Bohmic presheath of the Langmuir probe. We will briefly discuss the limits of the theory for a thin and a thick sheath, respectively, and will finally focus on operating a Langmuir probe in a magnetic field. 9.1.9 Probe radius The geometry of the probe and the density of the plasma (not only the density of electrically charged carriers but also the number density) mutually influence the regimes in which an adequate evaluation of the probe measurements is possible without a strong simplification. In principle, we distinguish between these regimes: 1. A regime for which the radius of the probe, rLp , is large compared to the thickness of the sheath, rs − rLp , around the probe (Fig. 9.8) [354]. This is a layer of positive ions which begins on the surface of the probe and extends to a plane where the finite electron density must be considered. 2. A regime for which the radius of the probe, rLp , is very small or comparable to the Debye length and to the thickness of the sheath, rs − rLp , around the probe (cf. Sect. 3.3, Fig. 9.8).

314


3. Eventually, also the mean free path of the ions and its relation to the Debye length plays an important role in discharges at higher pressures → (1). In the first case, the surface of the cylinder with radius rLp is comparable to the surface of the cylinder with radius rLp , whereas in the latter case, the surface of the rod is small compared to the radius of the sheath. Since the radius of the sheath edge is a property which evolves from a model, the area of this cylinder depends on its accuracy. In particular, the current density becomes a questionable quantity.

rs rp li rs rp Fig. 9.8. Top, simplest assumption: Probe radius rLp and radius of the sheath rs are both comparable, but small against the mean free path of the ions λi : collisionless sheath. For rLp rs , every electron which has arrived at the Bohm edge will be collected. Bottom, next assumption: Probe radius rLp is small compared to the sheath radius rs . Carriers are influenced (from slight deflection to collection) up to the sheath edge.

Since the metallic rod represents an equipotential surface, the upper limit for the probe radius should be the Debye length. As outlined in Chap. 4, the sheath can be estimated to be several Debye lengths in thickness.6 For a ratio rLp ≤ 1, (9.17) λD the motions of the carriers can and must be taken into consideration. This is possible with an analytical solution (OML). In low-pressure plasmas and in 6

This assumption is questionable for large negative values of the bias voltage.

9.1 Langmuir probe

315

high-density plasmas, respectively, this condition can easily be fulfilled. In these discharges, the mean free path of both carrier types is large compared to the sheath thickness rs (collisionless sheath); at higher discharge pressures, however, the probe itself alters the gradient of carrier distribution. In this pressure regime, further approximations are required, which were first numerical in nature [355], but recently, some parametrized solutions have been developed [356, 357]. The electron density around a spherical probe deviates from the value for infinite distance from the probe by [358]

n = n∞ 1 −

rs rLp

(9.18)

with rs the radius of the negative sheath whose thickness rs − rLp equals several Debye lengths [359]. At large number densities, the number of collisions for ions and electrons within the space charge becomes finite. This leads to a weakening of the steep increase of the electron current, and is very complicated to treat. Therefore, we limit our discussion to the collisionless sheath with the condition λe,i (rLp , rs ). 9.1.10 Thin sheath: space charge limited current 9.1.10.1 Positive ions. For strongly negative probe potentials, a positively charged space charge will form around the electrode which outwardly screens B Te . Although this voltage is small comthe probe potential by the fraction k2e 0 pared with the total plasma potential, Φp , the leaking field associated with this potential leads to a considerable increase of the ion flux, and the ions have attained a component of (directed) kinetic energy which vastly exceeds their thermal energy. At the boundary between Bohmic presheath and probe sheath, they are accelerated to the Bohm velocity vB . In the sheath, the electron density drops sharply to very low values and eventually vanishes (Fig. 9.9), and we get for the ion flux and the ion current density [Sect. 14.2, Eqs. (14.43/44)]:

ji,max = e0 Γi,max =

kB Te n0 e0 = 0.606 n0 e0 emi

kB Te , mi

(9.19)

it is considerably enlarged (the square root is the sound velocity of the ions!): The ions are accelerated from the very low thermal energy Ei ≈ 0.1 eV to approximately the mean electron energy (several electronvolts)! It has turned out in many cases that the probe current may be calculated applying the area of the rod of the Langmuir probe [360]. 9.1.10.2 Electrons. The region of the linear increase of the probe current for which not only the name retarding zone is customary but which is also denoted the electron repelling current, starts at negative values of the probe potential. In this regime, a change of the characteristic of the electron current will occur.

316


Vp VB


neutral plasma no field Bohmic presheath weak field positively charged cathode sheath

Fig. 9.9. The three zones in front of an electrode: undisturbed quasi-neutral plasma, quasi-neutral Bohmic presheath, positively charged sheath with cathode fall.

Vc

distance [a. u.]

At its beginning, it is determined by a retarding potential which will be replaced by a diffusion-limited behavior at less negative potentials. The current density can be calculated [with ΦLp the probe potential (Sect. 3.4, Eqs. (3.20) − (3.23))]

1 1 e0 ΦLp kB Te je = ne (x)e0 < ve >= n0 e0 · exp − . 4 4 2πme kB Te

(9.20)

Plotting the logarithmic probe current versus probe voltage should result in a straight line, and from its slope, we can evaluate the electron temperature Te , provided the energy of the ensemble is distributed Maxwellian. For rs −rLp rLp , all electrons will reach the probe. In that case, the current for both carrier types can be calculated according to I± = j± As

(9.21)

with As the surface of the sheath. In principle, the plasma density can be extracted from the ion current. Since it is smaller by orders of magnitude than the electron current, the errors are always very large [361]. A theory for the case of very small probe radii and collisionless sheaths has been established by Allen, Boyd and Reynolds [362], the limiting case of a finite sheath was thoroughly treated by Bernstein and Rabinowitz [363] and Laframboise [355] also for higher number densities with a collisional sheath. 9.1.10.3 Finite electron temperature. For finite electron temperature, the electrons can leave the surface of the probe without an extracting field by which a retarding potential is caused (V < 0 V) which suppresses any further emission. Hence, the V (x) characteristic drops across its whole course: For small distances, the potential values are below zero, and also at the anode, we do not reach the values which we are accustomed to when we apply the Child-Langmuir

9.1 Langmuir probe

317

equation (Sect. 4.3, Fig. 9.10). Here, Vm denotes a potential which has been established by Langmuir [86]. For a value of 3 Vm2 , the V (x) characteristic exhibits its minimum. 20

300

200

0 mA/cm 10 mA/cm 2 100 mA/cm2

10 F[V+Vm]

h = 0.1

F[V]

Vm: 0 V Vm: -15 V Vm: -30 V Vm: -50 V

15

2

100

5 0 -5 -10

0 0.00

-15 0.25

0.50 d [cm]

0.75

0.00

1.00

0.05

0.10

0.15

0.20

d [cm]

Fig. 9.10. For finite electron temperature, the potential in the vicinity of the probe drops to values below 0 V, and the minimum of the potential dependence is located in front of the electrode; this is shown here for a ratio η = keB0 VTpe of 0.1 for a sheath thickness of 400 μm and various values for Vm and an η = 0.1 and electron current densities je of 10 mA/cm2 . Note the larger scale of the abscissa at the RHS figure.

For thin sheaths (rs rLp ), the current is space-charge limited and can be expressed by Eq. (9.22). Its value depends on potential via the the changing thickness of the sheath. Provided that ΦLp kB Te /e0 , we can apply the diode equation (4.21), and for a planar probe with a one-dimensional sheath, the current is given by

Ie = je ALp

3/2

ΦLp −4 2e0 = ALp . ε0 9 me (rs − rLp )2

(9.22)

¿From the I(V ) characteristic, we can eventually calculate the sheath thickness rs . For finite electron temperature and/or small Langmuir probe potential ΦLp , Eq. (9.22) has to be extended by a correction term [364], and we define xs = rs − rLp :

Ie = je ALp

3/2

⎡

⎤

kB Te −4 2e0 ΦLp ⎣ ⎦ ALp . = 1 + 2.66 ε0 2 9 m e xs e0 ΦLp

(9.23.1)

Further growth of the sheath thickness requires an additional correction for cylindrical and spherical geometry (Fig. 9.11) [365]:

318


6 5

Fig. 9.11. With a geometrical correction of the Child Langmuir equation, it is possible to calculate the probe current for sheath thicknesses between “very thin” and “thick”, α2 for spherical geometry, and β 2 for cylindrical geometry.

a2, b 2

4 a2

3

b

2

2 1 0 1

2

3

4

rs/rp

Ie = je ALp

3/2

⎡

⎤

kB Te 2e0 ΦLp ⎣ −4 ⎦ ALp , = 1 + 2.66 ε0 9 me β 2 x2s e0 ΦLp

(9.23.2)

with

rLp . (9.24) rs The factors α (for spherical probes) and β (for cylindrical probes) equal unity for planar probes but are only approximately one for the other geometries when the sheath thickness equals the probe diameter, but they increase sharply with growing ratio of sheath thickness to probe diameter (Fig. 9.11). Consider a rod being 250 μm in thickness; this would mean for a capacitively coupled plasma with Te = 3 eV and ne = 1010 /cm3 a Debye length of about 130 μm. Turning to inductively coupled plasmas with their higher densities (say: Te = 2 eV and ne = 1011 /cm3 ), this would mean λD ≈ 27 μm. For these plasmas, we would characterize this rod as being relatively large in size. β = γ − 0.4γ 2 + . . . ∧ γ = ln

9.1.11 Thick sheath: Orbital Motion Theory (OML Theory) Since the electron current always exceeds the ion current considerably (with the exception of electronegative plasmas and even at high negative values of the probe potential), we are going to scrutinize the situation in front of the rod with respect to applications in reactive, high-frequency plasmas. As in Chap. 3, we start from the assumption that we can distinguish three different zones. In contrast to that situation, we approach now a positively biased rod (Fig. 9.9): • Undisturbed, quasi-neutral plasma. • This region is followed by a transition zone with finite field but equal density of positively and negatively charged carriers.

9.1 Langmuir probe

319

• Eventually, we enter the sheath with steeply dropping potential; the density of positively charged carriers is minute compared to that of the negatively charged carriers (electrons). Whether a charge really hits the electrode or not does not depend only on the sheath thickness but also on the amount of the initial velocity in the presheath (i. e. the electron temperature). Furthermore, the composition of its velocity (it it composed by a radial and tangential component) has to be taken into account which further influences the spatial distribution of the carriers across the electric field. In the case that tangential components exist, the carrier exhibits a certain angular momentum, and its motion can considerably deviate from the direction which is determined by the central field. If orbital motions are possible, the Boltzmann equation can be solved only numerically (cf. Fig. 9.12, OML theory from Orbital Motions Limited theory). vt,s

Fig. 9.12. An electron arrives at the Bohm edge with thermal velocity kB Te . Shown its velocity components (radial and tangential) during its fall to the rod after Swift and Schwar [366].

vr,s plasma plasma sheath

vt,p probe rp vr,p

rs

¿From Fig. 9.13, we can address the energetic problem: After having passed the boundary between Bohmic presheath and sheath, an ion with vanishing angular momentum L will fall in a straight line to the electrode (potential negative); for finite values of L, the potential can take all possible values. For positive values of V (r), no trapping is feasible. Hence, the probe current will become dependent on the potentials of the plasmas and the Langmuir probe, but also on the electron temperature in a complicated manner [363, 367]. In the case of TTei 1 and for small positively biased probes, Lam showed that the diameter of the probe, rLp , coincides with the radius of the sheath, rs . The voltage drop across the Bohmic presheath can then be calculated according to ΦB = 0.69

kB Ti , e0

(9.25)

with ΦLp ΦB [367], and we obtain in the next higher approximation

ΦLp Ti Ti 1, 45 ∧ 0.69 . Te Ve Te

(9.26)

320

9 Plasma diagnostics L2max /2 Mr 2 > V(r)

2

L2 between 0 and L max 2

L =0



2

L max /2 Mr > V(r)

L2 between L min and L max

L2 between L min and L max

radius [a. u.]

radius [a. u.]

Fig. 9.13. The effective potential of an ion into the electric field of a Langmuir probe after Bernstein [363]. For vanishing L, the ion falls straight into the potential minimum, for large values of L without a potential minimum, the ion cannot be captured. LHS, small values; RHS, large values of L.

For a capacitively coupled plasma with an electron energy of 3 eV and an ion temperature of 600 K, we find TTei ≈ 0.02. The advantage of this approximation can be found in the clear definition of the sheath, and from this boundary, the properties of the plasma (density and temperature of the carriers) equal those of the undisturbed plasma. The total probe potential drops from this very borderline. 9.1.11.1 Electron saturation current. An electron which has arrived at the boundary r = rs and moves further in the direction of the electrode with r = rLp , exhibits three components of velocity with respect to the electrode of length l (l rLp ): • A parallel component (v ). • A radial component (vr ). • A tangential component (vt ). energy E and angular momentum L are conserved according to 1 1 2 2 2 2 2 2 + vt,s + v,s ) − e0 Φs = me (vr,Lp + vt,Lp + v,Lp ) − e0 ΦLp (9.27) E = me (vr,s 2 2

L = me vrs = me vrLp ⇒ vLp = vs

rs . rLp

(9.28)

Since we are not interested in the direction parallel to l, we can fix v = 0. An electron only contributes to the probe current if

9.1 Langmuir probe

321

1 1 2 2 2 2 0 ≤ me vr,Lp = me (vr,s + vt,s − vt,Lp ) + e0 ΦLp , (9.29.1) 2 2 and from that, we find for the radial component of the velocity at r the probe radius rLp (cf. Fig. 9.12): ⎡

0≤

2 vr,Lp

=

2 vr,s

+

2 vt,s

rs ⎣1 − rLp

2 ⎤ ⎦+

2e0 ΦLp , me

(9.29.2)

2 : resolved for vt,s

2 = vt,s

2 rLp 2e0 VLp 2 2 + vr,s − vr,Lp . 2 me rs2 − rLp

(9.30.1)

When the radial component of velocity must exceed zero at the boundary, it follows that the squared tangential component is at least zero: 2 vt,s ≤

2 rLp 2e0 VLp 2 + vr,s . 2 2 me rs − rLp

(9.30.2)

• From Eqs. (9.29) we can see that for positive ΦLp , the radial component of the velocity can just vanish. • For negative values of VLp , we can extract from Eq. (9.30.2) that for vanishing tangential component, the radial component can be calculated ac 2e0 VLp cording to − me . The electron flux incident on the electrode with area 2πrs , length l and a radial velocity of vr,s is dΓe = 2πrs Lvr,s f (vr,s , vt,s , v,s )dvr,s dvt,s dv,s .

(9.31)

The limits are for: • vr : 0 and ∞, • vt : −vt and +vt , where 2 2e0 VLp rLp 2 + vr,s vt = 2 me rs2 − rLp

(9.32)

• v : −∞ and ∞: Γe = 2πrs l

∞ ∞ vt 0

−∞

−vt

vr,s f (vr,s , vt,s , v,s )dvr,s dvt,s dv,s .

(9.33)

Starting with a Maxwellian distribution for f (v) at the sheath boundary,

322


dn = n0

me 2πkB Te

3

exp −

me 2 2 2 vr,s + vt,s + v,s dvr,s dvt,s dv,s 2kB Te

(9.34)

and inserting this in Eq. (9.33), we obtain with erf(x) the error function and erfc(x) the complementary error function 2 x −y2 2 ∞ −y2 erf(x) = √ e dy ∧ erfc(x) = √ e dy, π 0 π x

(9.35)

and for the flux incident on the electrode ⎧

⎛

⎞

2 rLp kB Te ⎨ rs ⎝ e0 ΦLp ⎠ 1 − erfc + Γe = 2πrLp n0 l 2 2 ⎩ 2πme rLp rs − rLp kB Te ⎫

e0 ΦLp r2 e0 VLp ⎬ erfc 2 s 2 ; + exp kB Te rs − rLp kB Te ⎭

(9.36)

a complicated function which depends on the thickness of the sheath, the diameter of the probe, and on the probe potential; to cap it all, the (relatively unknown) relation between the extension of the sheath and the probe potential are required. Hence, we must carefully choose the initial conditions either to determine exactly this latter mentioned relation or to make the electron current independent of the probe potential, e. g. at very low plasma densities. Therefore, we focus on the two limiting cases: 1. Thick sheath:

rLp rs

2. Thin sheath:

rs −rLp rs

→ 0. 1.

Thick sheath. For this limit, Eq. (9.36) simplifies to

Γe = 2πrLp n0 l

rs rLp × 1 − erfc rLp rs

e0 ΦLp kB Te

kB Te × 2πme

e0 ΦLp + exp erfc kB Te

e0 ΦLp kB Te

.

(9.37.1)

Since 1 - erfc(x) = erf(x), we further obtain

Γe = 2πrLp n0 l

rs rLp erf × rLp rs

e0 ΦLp kB Te

kB Te × 2πme

e0 ΦLp + exp erfc kB Te

e0 ΦLp kB Te

,

(9.37.2)

9.1 Langmuir probe

323

2.5

Fig. 9.14. Comparison of the functions according to eqs. (9.37.3) and (9.38), resp. For e Φ > 2, both values of x = k0B TLp e the functions become indistinguishable and explain the square-root behavior of the electron saturation current for cylindrical probes.

f(x)

2.0

eq. (9.37.3) eq. (9.38)

1.5

1.0 0

1

2 x = eV0/kBTe

3

4

which becomes for small arguments of the error function, i. e. for which erf(x) → √2π x, will transform to

kB Te Γe = 2πrLp n0 l 2πme

2 √ π

e0 ΦLp e0 ΦLp + exp erfc kB Te kB Te

rLp rs

e0 ΦLp kB Te

e0 ΦLp kB Te

→ 0,

,

(9.37.3)

As Langmuir and Compton showed [368], for probe potentials 2kB Te , e0

ΦLp ≥

(9.37.4)

i. e. erfc(2) ≈ 0, the expression in parentheses in Eq. (9.37.3) becomes congruent with the significantly simpler function (expanding the exponential term)

2 √ 1+ π

e0 ΦLp , kB Te

(9.37.5)

so we eventually obtain the function (Fig. 9.14) √ Γe = 4 πrLp n0 l r

kB Te 1+ 2πme

e0 ΦLp kB Te

if VLp ≥

2kB Te . e0

(9.38)

Provided that rLps → 0, the electron flux becomes independent of the diameter of the sheath for probe potentials which are large compared to the thermal energy of the electron (kB Te /e0 ) which applies to the region of electron saturation current. From Eq. (9.38), the square-root behavior of the I(V ) characteristic becomes evident.

324

9 Plasma diagnostics r −r

Thin sheath. For the other limiting case, i. e. s rs Lp 1, both the complementary error functions in Eq. (9.36) vanish, and the calculation of the electron flux simplifies to

Γe = 2πrs n0 l

1 kB Te = As n0 < ve > 2πme 4

(9.39)

where As equals the area of the probe ALp which is, in fact, the same result as previously obtained [Eqs. (9.20/21)]. Since rLp = rs , every electron which has reached the edge of the sheath will also reach the electrode itself. Hence, the diameter of the sheath can be calculated using the space-charge limited version of the Child-Langmuir equation [Eqs. (4.21)]. 9.1.11.2 Electron current in the retarding-field region. In this regime, the probe potential ΦLp becomes negative, the electrons must run against a retarding potential and for small thermal energy or sufficiently negative probe potential ΦLp , they will not be collected by the probe. From Eq. (9.30.2), it is evident that for negative ΦLp , the smallestradial component of the electron 2e0 ΦLp to reach the surface of the velocity at the sheath boundary must equal me probe. This leads to the final condition Γe = −2πrs l

∞

∞ vt

2e0 Φ 1 ( m Lp ) /2 e

−∞

−vt

vr,s f (vr,s , vt,s , v,s )dvr,s dvt,s dv,s ,

(9.40)

yielding the well-known exponential relation

Ie = Ie,0 exp

e0 ΦLp , kB Te

(9.41.1)

with the electron current in the undisturbed plasma 1 Ie,0 = e0 Γe = e0 ALp n0 < ve > . 4 < v > is the mean thermal energy

(9.41.2)

8kB Te . πme

9.1.11.3 Current transition at plasma potential. According to our definition of the plasma potential, the electron retarding regime terminates at this point and will pass into the electron saturation regime. The sheath thickness r −r but vanishes at this point, i. e. s rs Lp → ∞. Inserting this condition into Eq. (9.36) leads to the demanded concordance:

Ie = 2πrLp e0 n0 l

kB Te . 2πme

(9.42)

9.1 Langmuir probe

325

5

ln Ip [mA]

4 3 2 1

sum of both currents

electron current

0 -1 -40

ion current

-20

0

20

40

Fig. 9.15. By extrapolation of the ion saturation current up to the plasma potential and subtracting it from the probe current, we can determine the pure electron current (ion current not to scale).

DF [V]

9.1.11.4 Ion current. In principle, we can use the negative branch of the I(V ) characteristic to evaluate the ion current which becomes more and more dominant for strong negatively biased values of the probe potential. Due to their small mobility, the current due to the ions is smaller by orders of magnitude, compared with the electron current. Therefore, a correct derivation for this current should be based on the fact of the presence of both types of carriers, a high density of low-mobile ions, and a small density of high-mobile electrons which are distributed Maxwellian. A derivation which takes all these issues into account has been given by Allen et al. [362]. However, the approximate solutions of this differential equation yield only unsatisfactory results, and the easiest (and most frequently applied) method is to extrapolate the ion current up to values of the plasma potential from the ion saturation current (Fig. 9.15). For values less than the plasma potential, the probe potential is negative referred to the plasma potential, and ions are accelerated to the Langmuir probe. Orbital motion limited theory (OML) does not take into consideration either the collisions between ions and neutrals or the mutual collisions between ions, which leads to an overestimation of the density of the positive ions [369]. The discrepancies in plasma densities provided by microwave interferometry and Langmuir probes are mainly due to this neglect [370].

9.1.11.5 Summary. For a thin sheath (rs −rLp rLp ), the electron current is given by the simple Eq. (9.20), and thermal corrections can be considered using Eqs. (9.22)−(9.24). For a thick sheath (rs −rLp rLp ), the current is limited by orbital motions (OML) and is given by Eq. (9.36) with the two approximations Eq. (9.38) for the thick sheath and Eq. (9.39) for the thin sheath, respectively, which is self-consistent with Eqs. (9.19/20) for the thin sheath and a spacecharge limited current.

326


9.1.12 Electron temperature and plasma potential In the simple model introduced in Sect. 9.1 [linear behavior of the I(V ) characteristic in the electron retarding regime], we can easily calculate the electron temperature according to Eq. (9.5). This conduct is only justified for thermal equilibrium which is characterized by a Maxwellian distribution (Sect. 14.1). For a non-Maxwellian behavior of the EEDF, the I(V ) characteristic, however, is nonlinear. In particular, this behavior can be observed at higher electron energies and lower electron densities. The low number of collisions prevents an effective thermalization; moreover, these high-energy electrons lose their energy during inelastic collisions and get lost by various processes (diffusion and recombination). For a bent I(V ) characteristic, it is common to apply the Druyvesteynian distribution which can be extracted from the second current derivative: d2 I 4me dN V dv = 2 N Ae0 dV 2

(9.43)

with A the effective probe surface, V = Φp −ΦLp , dN the number of electrons in the interval dv. To summerize, the plasma potential Φp can be evaluated from: • V → ln je : intersection of the approximating lines for electron retarding regime and the electron saturation current, respectively; • V → d2 je /dV 2 : maximum of the first derivative [339, 371]; • V → dI/dV : maximum of the Maxwellian distribution. The second method normally provides slightly smaller values of Φp than those determined with the first one. The smaller values of Φp provide, in turn, smaller values of the probe current and the electron density ne . 9.1.13 Influence of a magnetic field In a magnetic field, we have to consider the Larmor radii of electrons and ions and their ratio to the diameter of the rod and the Debye length. Considering electron energies of several electronvolts and ion energies of several hundreds in ◦ C, we expect Larmor radii of several hundreds of micrometers for the electrons and several centimeters for the ions in a magnetic field with B < 100 Gauss, which are both significantly larger than the corresponding rLp and λD . Therefore, they will not affect the probe characteristics, in particular in helicon discharges. This can be quite different in ECR discharges with magnetic fields which are higher by more than one order of magnitude. For a rough estimation, we consider a plasma density in the ECR region (which peaks at 1012 /cm3 ), and an electron temperature of about 8 eV. This would lead to a Larmor radius of 700 μm; so they become comparable with the thickness of the rod.

9.1 Langmuir probe

327

4 with grounded electrode Te = 3.3 eV

60 40

without grounded electrode

ln (Isat - I)

probe current [mA]

80

2

Te = 6.7 eV

0

20 with electrode floating

Te = 3.1 eV

0 0

20 40 60 probe voltage[V]

80

-2

0

25 50 probe voltage [V]

75

Fig. 9.16. The measurement without a grounded electrode yields an erronoeus high electron temperature [369].

9.1.14 Measurements With the theory considered so far, we want now to address some special problems in reactive plasmas which are driven by high-frequency fields. 9.1.14.1 Grounding problems. In contrast to DC discharges of electropositive gases (mainly argon), in RF discharges signals are distorted by capacitive pick-up (cf. Sect. 9.1.7). Especially in capacitively coupled discharges, this problem has been addressed by several authors (cf. [348]). However, the reactor walls are grounded, at least in principle, and reliable measuring should require a single RF compensated probe, since fluctuations of the plasma potential vary the current to the wall but this charge flux is instantaneously equalized. High-density plasma reactors with electron cyclotron resonance sources or inductively coupled sources, or even reactors with anodized walls lack these grounded areas. Especially when high electron currents are drawn, the plasma immediately reacts with a reduced electron flow (corresponding a positive offset current) to the walls to maintain its quasineutrality. Since this positive excess charge cannot be conducted away, the wall potential will rise during the measuring process (cf. Fig. 9.16) [369]. Hence, the measurements with single Langmuir probes have to be carefully checked to avoid misleading results. The catalogue of measures consists of large reference electrodes and double and triple electrodes. The application of the latter type, however, excludes the measurement of the spatially resolved plasma potential. 9.1.14.2 Determination of the characteristic. In Fig. 9.17, the probe characteristics through HF discharges of reactive gases are shown. The main problem is a noisy original signal which has to be properly smoothed to obtain a sig-

328


nal which can be interpreted. From the squared characteristic, the logarithm is taken which is subjected to a fast Fourier transformation (Fig. 9.17.1). Whereas the high-frequency part of the spectrum is caused by noise, its lowfrequency section is due to the experiment. Chosing a certain cutoff frequency (COF) which is marked in Fig. 9.17.1, it is this part of the frequency spectrum which is convoluted with appropriate filter functions (e. g. Gauss, sometimes also a square function), but mostly the Blackman function

nπ 2nπ + 0.08 sin (9.44) COF COF with n = 1 the fundamental, and COF the uppermost harmonic which is taken into account: 1 ≤ n ≤ COF. The next step is the retransformation to the time domain, and the second derivative is taken (Fig. 9.17.2) which yields the plasma potential, Φp . The final result will be represented as in Fig. 9.17.3 and Fig. 9.17.4. F (f ) = 0.42 + 0.5 cos

6

log [I(n)2]

2 0

1.0

0.1 x d2I/dF2

Ffl: 22.7 V Fp: 30.2 V ne: 1.0 x 1010 cm-3 COF: 112

4

Ffl: 22.7 V Fp: 30.2 V 10 -3 ne: 1.0 x 10 cm

0.5

0.0

-2 0

50

100

150

-0.5 -40

200

-20

-3

20

40

20

40

3

-4

Ffl: 22.7 V Fp: 30.2 V 10 -3 ne: 1.0 x 10 cm <Ee>: 2.3 eV lD: 92mm

2

-5

Ffl: 22.7 V Fp: 30.2 V 10 -3 ne: 1.0 x 10 cm

-6

I [mA]

log [I(F)]

0 F [V]

n

1

0

-7 15

20

25 F [V]

30

-40

-20

0 F [V]

Fig. 9.17. The main problem in HF discharges is the noisy signal which requires convolution with a smoothing function. LHS, top: logarithm of the squared Fourier spectrum of the real V-I characteristic of a Langmuir probe (cutoff frequency: vertical rod), RHS, top: second derivation of the electron current with respect to voltage, RHS, bottom: final characteristic, smoothed using a filter function at a certain cutoff frequency, LHS, bottom: logarithm of this smoothed V-I characteristic.

9.1 Langmuir probe

329

25 experimental EEDF: MB Te = 2.7 eV

15

f(E) [108/(eV cm3)]

f(E) [108/(eV cm3)]

20

10

5

0 0

20

experimental EEDF: MB Te = 3.8 eV

15 10 5

10

E [eV]

20

30

0 0

5

E [eV]

10

15

20

Fig. 9.18. Electron energy distribution functions (EEDFs), measured with a Langmuir probe (dotted: experimental, solid: calculated under the assumption of a Maxwellian distribution of the electrons for a mean electron energy of 2.7 eV for 22 c IOP Publishing mTorr (3 Pa, LHS) and 3.8 eV for 100 mTorr (14 Pa, RHS) [98] ( Ltd.).

9.1.14.3 Electron temperature and plasma potential. To calculate the electron temperature from the second current derivative according to Eq. (9.43), the measurement has to be deeply extended into the regime of the electron saturation current, since the energy of the collected electrons increases with growing voltage. The second prerequisite is the exact determination of the ion current in order to subtract its (small amount) from the total current. There are innumerable hints for a non-Maxwellian behavior of the electrons, especially in capacitively coupled discharges, particular beyond the energy of the first inelastic threshold (for argon: metastable states at 11.55 and 11.72 eV, [372]): Here, the EEDF resembles a modified Druyvesteynian distribution rather than a Maxwellian distribution (Figs. 9.18 + 9.19) [98, 373, 374]. Deviations from this simple logI-V characteristic are typical for RF discharges through electronegative ambients [137]. Sometimes, the calculated electron temperature are higher by a factor of 2 to 3 than in discharges through inert gases for comparable conditions; furthermore, ni can exceed ne significantly. This is easily explained with the functionality of the electronegative gas which acts as a trap for electrons: − SF6 + 2 e− −→ SF− 5 +F .

(9.45)

For example, CCl4 exhibits one of the largest cross sections for electron absorption [375]. Since it is well known that the cross section is larger for slow-moving electrons than that for rapid electrons, slow electrons are be captured more easily, and the mean energy of the remaining electrons will grow.

330

9 Plasma diagnostics 1 2 eV -1

probe current [A]

10

argon

10-2

10-3

10-4 0

4.7 eV

10

20

30

Fig. 9.19. Semilogarithmic plot of the electron current vs. the retarding probe potential for a pressure of 15 mTorr (2 Pa) in an argon discharge. The different slopes of the lines which approximate the measured curve intersect at the ionization potential [232] c IOP Publishing Ltd.). (

probe potential [V]

All these experimental facts indicate that the electrons have neither Maxwellian nor Druyvesteynian distribution. But we have to keep in mind that both distributions are based on the assumption that solely elastic collisions are admitted and will occur. For that derivation, electrons with mass me and temperature Te are in mutual thermal equilibrium with molecules with mass MM and temperature TM . This assumption may hold for small electric fields. Here, energy losses by inelastic collisions can be kept low since only very few electrons are accelerated to sufficiently high energies. If the intensity of the electric field reaches such a value that the thermal energy can be neglected, it is mainly the functional dependence of the cross section that matters (cf. Sect. 14.1).

9.1.15 Conclusion In principle, the Langmuir probe draws current from the plasma which causes inevitable feedback, in particular, this is the case for high potentials. Therefore, the determination of the plasma potential should take place at the point of inflection and should be avoided when entering the regime of electron saturation current. For sufficiently steep increase of the retarding current, the error should be kept below 10 %. A singular property of the measuring principle of the Langmuir probe is the spatial resolution in vertical and radial direction, at least of the methods discussed here (another technique is laser-induced fluorescence, LIF). This opens the window for measuring space charges. Also the plasma potential Φp can only be obtained with this technique. It is important for the determination of the total energy of projectile ions which are incident on surfaces and are accelerated in the field of the sheath.

9.2 Self-Excited Electron Resonance Spectroscopy

331

In simple DC discharges of inert gases, we really find I(V) characteristic which can easily be interpreted following Eqs. (9.5) and (9.6) to extract the electron temperature and plasma density. The application of Langmuir probes in high-frequency discharges through reactive gases still remains a great challenge although by effective filtering the rough signal can be widely cleared of the RF noise. For drawing plasma parameters of inert gases, it still represents the standard. The technique of the Langmuir probe has been developed for the evaluation of parameters in DC discharges through inert gases. Provided that the higher harmonics in the probe circuit are perfectly compensated, the Langmuir probe can also be employed in HF discharges, retaining its reputation as standard for plasma diagnostics. In discharges of reactive gases, however, the measuring tip is readily attacked which causes severe difficulties in evaluating the spectra. Moreover, measurements with a Langmuir probe answer only some questions. Therefore, other methods have been introduced which are non-invasive.

9.2 Self-Excited Electron Resonance Spectroscopy 9.2.1 Non-linear response between voltage and current In the simplest case, a capacitively coupled discharge consists of two parts: • The plasma sheaths in front of the two electrodes. • The plasma bulk between the confining layers. The variations of the voltage across the sheath enforce the charges to an oscillating movement. In the case of a linear response the eigenfrequency would be the operating frequency. Due to their inertness, the ions do not respond to this force, and also the electrons follow the field with a certain retardation. This causes a nonlinear term to the electron current, and higher harmonics far from the operating frequency are generated. It is these overtones which are taken into account for SEERS. Since most of the discharge current consists of displacement current, this method is extremely robust to surface contaminations no matter non-conducting layers have been deposited on the sensor or on the reactor wall, and the measurement itself does not influence the plasma. For an asymmetric HF discharge, we find for ωHF ωp,i the following electrical properties (Fig. 9.20): • Plasma: Because of the high density of charged carriers, only small fields can be set up. Although the plasma behaves quasi-neutral, the conductivity is dominated by the electrons due to their small inertia which is not zero. Therefore, the plasma impedance is inductive in character with a large Ohmic fraction.

332


electrode

plasma _ ne

ni

ne(t)

0

se(t)

dS - se(t)

se dS

x

Fig. 9.20. Model of the rectifying sheath after Godyak and Lieberman [102, 192]: ne vanishes for x < se , the width of steep drop being only several Debye lengths. The slightly curved decrease should go over the calculated drop. ds −se (t) is the (electrically defined) sheath thickness (Sect. 14.4).

• Plasma sheath at the powered electrode: The thickness scarcely depends on discharge pressure and sheath potential, very thick compared with the Debye length; the ions are accelerated in response to the electric field which has been set up. • Plasma sheath at the grounded electrode: Very low thickness which can be calculated according to Child-Langmuir in the collisionless case, very small sheath potential. • The time-averaged electron density at the electrodes is almost zero. • The ion density across the sheath always exceeds the time-averaged electron density but is always smaller than the ion density in the plasma bulk: ji = e0 ni (s)vi (s) ⇒ • Z: almost no Ohmic part (R), only capacitive components (C). • Φ: large drop across the powered electrode (cathode fall). • I: very low ion current, predominantly displacement current, extremely nonlinear response of the resulting current with respect to the exciting potential with rectifying, diode-like characteristic (Sect. 14.4). In a simple model, the plasma responds with a harmonic displacement current across the sheaths and a harmonic conduction current with an inductive component in the bulk. As we have seen in Sects. 6.3, 6.5 and 14.4., these capacitive and inductive sources cause nonlinear components, i. e. a spectrum of higher harmonics. Concentrating first on the plasma bulk, we find that deviations of the Ohmiic law j = σE are due to this response, and applying Eq. (6.4) for the DC conductivity yields


333

dj e2 np = 0 E − νm j. dt me

(9.46)

For a one-dimensional plasma, the length of the plasma bulk l is the geometric length (distance between the electrodes − thicknesses of both the sheaths: l ≈ d − se ),7 and A0 equals the (circular) electrode surface [π/4 (2R)2 ]. Assuming a linear potential drop across the plasma bulk, we find next that E = −Vbulk /l. Inserting this in Eq. (9.46) and resolving for Vbulk , we obtain

me l dj Vbulk + νm j . = 2 − A0 e0 np dt

(9.47.1)

The first term on the RHS can be easily identified as the inductive part, the second term on the RHS as the Ohmic part: −

Vbulk dj = Lbulk + Rbulk j. A0 dt

(9.47.2)

The displacement current across the sheaths is given by j = ε0

∂E ∂t

(9.48.1)

which can be rearranged to j = −ε0

∂ ∂Vs ∂x ∂t

(9.48.2)

leading to

d

∂Vs ∂t

=−

j dx ε0

(9.48.3)

with Vs the voltage drop across the sheath. Integrating, we further note that the (electronic) sheath thickness se is a complicated function of the instantaneous sheath voltage, which depends on the time-dependent phase se = se {V [ϕ(t)]},

(9.49)

and we eventually obtain −

dV j = se {V [ϕ(t)]}. dt ε0

(9.50)

Combining Eqs. (9.47.1) and (9.50), we find for the temporal variation of the discharge voltage according to Klick [202] 7

For a two-dimensional model (R d): l≈

1 +

1 d−se

2 R

.

334


me l 1 dV se dj d2 j νm + 2 . − = j+ 2 A0 dt ε0 np e0 dt dt

(9.51)

Eventually, we have to consider the stochastic heating by an effective frequency of momentum transfer which equals the sum of the collision frequency between electrons and neutrals (νm ) and the stochastic frequency between electrons and the breathing sheath (νstoch ) by νeff = νm + νstoch .

(9.52)

Equation (9.51) describes an oscillating circuit which is excited by an external voltage Vs (t) which harmonically follows with a certain damping expressed by νeff : • Polarizable space charge in the sheaths: capacitance C. • Inert mass of the electrons [derivations of j with respect to t in Eq. (9.51)]: inductance L. • Power absorption and power dissipation in the plasma (Ohmic heating) for resistance R; for pressures lower than approximately 100 mTorr (15 Pa): Onset of additional stochastic heating which does depend on pressure only marginally, but will become more and more significant for further dropping pressure. The equivalent circuit is pictured in Fig. 9.21 (cf. Sect. 5.8).

Rp

Lp

Zp

plasma bulk Cs,1 V1

Cp

Cs,2 Vp

V2

Fig. 9.21. Equivalent circuit of a capacitive discharge. The (complex) plasma impedance is split into several contributions. Sheaths and plasma bulk set up a damped oscillation circuit with series resonance and parallel resonance [376].

Due to the non-linear response of the space charge in front of the electrode, not only is the first harmonic generated, which is responsible for the lowermost resonances (parallel and series; the series resonance represents the maximum of the HF current in the Fourier spectrum), but also their higher harmonics [in an algebraic sense, by the second term on the RHS of Eq. (9.51)].


335

If the damping is supposed to remain small (by neglecting the Ohmic resistance of the plasma bulk), there will be two resonances, a parallel resonance (maximum in voltage) and a series resonance [maximum in current, cf. Eq. (5.97)] [376]: 2 = ωe,p

1 ne20 , = LC ε0 m e

(9.53.1)

se , se + s1 + l

(9.53.2)

2 ωg2 = ωe,p

• se : sheath thickness in front of the “hot” electrode (in argon about 10 mm for several tens of mTorr and several hundreds of volts); • s1 : sheath thickness in front of the “cold” electrode; • l: (effective) geometric length of the plasma bulk. A confined discharge glows within cylindrical volume whose circumference is given by the “hot” electrode and its length l equals the gap between the ‘ ‘hot” electrode and its counterpart (up to densities of aproximately some 109 /cm3 );8 • ωe,p : electronic (parallel) plasma frequency; • ωg : geometric (serial) plasma frequency (always lower than ωe !). ¿From Eqs. (9.53), we see that the resonant frequencies are closely connected with the electronic plasma density ne ; and this property plays a decisive role for the electric conductivity in the extended free-electron model [cf. also Eq. (6.4)]: ε0 . (9.54) iω + νeff The time distance between the highest signals is the inverted excitation frequency, whereas the geometric resonance will be given by the distance in time between adjacent peaks. From this difference, the spatially integrated plasma density can be evaluated, from the damping of this series, the effective collision frequency νeff is obtained which equals the sum of the contributions of collisions between electrons and neutrals, νm , and between electrons and the breathing sheath, νstoch [Eq. (9.52)], respectively [377] (Fig. 9.22, discharge through oxygen). To determine the plasma parameters plasma density ne and effective collision frequency νeff , exactly those nonlinearities are made use of which complicate matters with Langmuir probes, and we obtain: σ = ωp2

8 This does not hold for high-density plasmas (starting at about 1010 /cm3 at wafer level); additional parasitic paths between plasma and the confining reactor walls make the modeling more difficult since the skin effect effectively screens the plasma from penetrating electric fields. These high-density plasmas can be generated in capacitively coupled plasmas as well, however at higher operating frequencies (typically 3 × 13.56 MHz).

336


0.020 f

sensor current [mA]

0.015

ng

0.010 0.005 0.000 -0.005 0

50

100

150

200

Fig. 9.22. SEERS: A time-dependent discharge voltage is picked up at the antenna of the sensor. The induced current is made visible. The harmonic excitation at f = 13.56 MHz (74 nsec) is modulated with highly damped oscillations of the SEERS resonance νg ; 12.08 nsec (83 MHz). [378].

time [nsec]

• A volume-averaged electron density ne from ωg . • A volume-averaged effective collision frequency of the electrons via the damping of the Fourier components of the resonance. • The electrical conductivity σ of the plasma bulk according to the freeelectron model [hydrodynamic approach, Eq. (6.4)].

9.2.2 Technical realization To get a comprehensive idea of the processes at the electrode, a direct measurement at the RF driven electrode would be the best and most straightforward way. Due to inevitable stray capacitancies, this path is blocked. Therefore, the detour consists of mounting the sensor into the grounded reactor wall to become an integral part of it. The sensor itself consists of a coaxial pick-up (50 Ω) used as antenna which is seated into an insulating disk most frequently made of teflon which measures about 10 mm in diameter. This device is fitted into a metallic disk which is mounted into a flange (typically 40 mm in diameter) of the reactor wall, thereby taking care of setting its surface in-line with the reactor wall (Fig. 9.23). To ensure good electric contact between reactor wall and this disk, a steel spring is wound around it (Fig. 9.24). The ground potential of the sensor should be on general ground, and the sensor has now become an integral part of the reactor wall. The signal of the discharge current is picked up and the induced voltage is measured in a fast digital oscilloscope with a bandwidth of 500 MHz. This is shown schematically in Fig. 9.25. With some defaults which take into account the electrical nature of the gas and the dimensions of the reactor, Eq. (9.51) is solved by an iterative procedure to obtain the plasma parameters ωg and νeff .


teflon isolation

electrical sensor

337

Fig. 9.23. Vacuum part of the SEERS sensor, here displayed for the ICP reactor DPS 300 of Applied Materials. The electric sensor is insulated by a ring made of teflon. Sensors with an optical window for simultaneous OES are also available [194].

Fig. 9.24. SEERS sensor (exploded view). By proper seating, the insulated sensor should become an integral part of the grounded electrode [194].

9.2.3 Inherent properties 9.2.3.1 Electronic plasma density. The Fourier components of the induced voltage are the basis for an iterative calculation which results in a value for the volume-averaged electron density ne . For this self-consistent loop, the discharge is supposed to behave entirely capacitively, and the DC part of the sheath impedance should be small against its RF part. However, at least three issues have to be taken into account: • This model cannot be strictly applied since a small resistive part should be taken into account which is due to power transfer to the ions (cf. Sect. 6.5). This part is completely neglected as well as the small (influence of the) floating potential Φf . • The rigid correlation of the geometric factor in Eqs. (9.53) with the parallel resonance ωg → ωe,p is only valid for an entirely capacitive, symmetric coupling [cf. Eqs. (5.95) for the symmetric case of an HF discharge]. For an asymmetric discharge, the geometry factor goes to unity [Eqs. (5.96) and

338

9 Plasma diagnostics (5.98)], the two resonant frequencies become similar and their difference will eventually vanish.

• In some cases, neglecting the DC (real) part is not allowed. For 10 mTorr (1.3 Pa), an operating frequency of 13.56 MHz and an electron tempera≈ 2 in a discharge through argon; according ture of 4 eV, the ratio νωm = 86 38 to Eq. (5.57), this equals the ratio between real part and imaginary part (L/R). Damping by elastic collisions eventually confines the range for measurement in an upward direction.

plasma

RF sensor

HERCULES RF

RF current

RF voltage

Fast Fourier transformation SEERS model electronic collision rate electronic plasma density DC conductivity

Fig. 9.25. Scheme of the SEERS sensor. The discharge current is picked up by a floating antenna and will induce a small voltage in the oscilloscope. By means of a fast Fourier transformation (FFT), various plasma parameters are calculated and displayed graphically [193].

It should be noted further that Eq. (9.51) requires a mean value for the (electrically defined) sheath thickness se which will be generated during the evaluation procedure. It was Godyak who first pointed out that the sheath thickness determined by optical methods exceeds the capacitive sheath thickness considerably [379]. The visual contrast peaks at the border of the oscillating sheath, compared with the mean position of the sheath (Fig. 9.20) [379]. Comparing electron densities recorded by Langmuir probes and by SEERS sensors, the lower values obtained by SEERS, about one order of magnitude, catch the eye. This can be referred to the spatial averaging (Fig. 9.26). 9.2.3.2 Frequency of momentum transfer. ¿From the damping of the Fourier components, we get an idea of the importance of the heating mechanisms and their pressure dependence. From Eqs. (2.6) and (2.11), we can infer the frequency of momentum transfer to be given to νm =< ve > σn

(9.55)

which indicates that Ohmic heating becomes increasingly insignificant for lowering the number density n [which should be synonymous for lowering the discharge pressure p, see also Eq. (5.3)]. That it is possible to strike RF driven


339

10

Fig. 9.26. Comparison of the electron densities ne of argon, measured by a SEERS sensor and by a Langmuir probe. SEERS averages over the plasma volume, whereas Langmuir is a spatially-resolving method ⇒ ne , measured by the latter method, exceeds the SEERS data by almost one order of magnitude [380].

ne [109/cm3]

Ar, Langmuir Ar, SEERS

Langmuir vs. SEERS emitted RF power: 150 W

1

0

20

40

60 80 p [mTorr]

100

120

discharges even at pressures below 100 mTorr has decisively fueled the search for another heating mechanism which has been turned out to be stochastic the root mean square of velocity scales with temperature √ heating. Since √ ( < v 2 > ∝ Te ), and according to the global model, Te ∝ 1/ ln n (cf. Sect. 3.5), the collision frequency as defined by Eq. (9.52) is expected to follow νm ∝

10.0

neff [108 sec-1 ]

5.0

2.5

0.0

25

50

75

(9.56)

Fig. 9.27. In the pressure range below 100 mTorr, the two mechanisms for power transfer, Ohmic heating and stochastic heating, counteract in opposed direction. As main result, the effective collision frequency remains almost constant [381].

Ar/Kr [DC bias] 300 500 700

7.5

n , ln n

100

p [mTorr]

and the drop in νm for lowering the pressure is partially compensated for by the increasing importance of νstoch (Fig. 9.27; note that at very low bias voltages, significant deviations from the capacitive character are detected. This is reflected by a decreasing power factor [380]). This behavior has also been reported for discharges through mercury [210]. It should be explicitely noted that all the equations involved contain the number density n but not the pressure p. However, pressure is used as one of

340


the most prominent independent plasma parameters. Most frequently, pressure is measured downstream at room temperature. This can lead to erroneous values which cause misleading consequences for rising gas temperature in the plasma source or at wafer level (ideal gas behavior provided, p ∝ n, but n ∝ 1/T ). The temperatures of the plasma bulk have been measured recently and have been found to exceed room temperature significantly even at low or moderate power levels (Sect. 9.4.2). 9.2.4 Conclusion In conclusion, the data picked up by the SEERS sensor and evaluated by considering an entirely capacitive behavior deliver similar dependencies as they can be obtained by Langmuir probes. Due to its principle of volume averaging, the values for the electron density must fall short of those measured by a Langmuir probe in the glowing plasma bulk. For absolute measurements and comparisons, the data should be calibrated by several spatially resolved measurements with a Langmuir probe or verified by a means of a microwave reflectometer. Its domain is the capacitively coupled discharge but the system has been recently applied also to inductively coupled plasmas. Since it is a non-invasive method, it can be installed with ease, being instantaneously ready to use. Because of the capacitive coupling of its measuring tip, even thick coatings of more than 1 mm do not hamper the measuring (which makes Langmuir probe measurements in coating plasmas nearly impossible, and also non-invasive optical measurements of these plasmas are not really recommended).

9.3 Impedance analysis An electric measurement in the RF circuit itself is very complicated because of the requirement to take into account several complex properties V = ZI. Displacement currents, but also eddy currents have to be considered, and the electric field is the resultant of two potentials E = −∇V −

∂A ∂t

(9.57)

and must not be evaluated by simple integration of a field equation. Instead of this, the RF fields of propagating waves are measured, in fact those of the forward and reflected wave, respectively, and referred to the usual terminal resistance of Rref = 50 Ω (cf. Chap. 5). To accomplish this task, this device which is called V (I) probe or Z-scan is flanged into the circuit between matching network and driven electrode (Fig. 9.28). In principle, it is a two-port with four gates, which contains transducers for current and voltage, and it opens the possibility to measure actual values

9.3 Impedance analysis

341

for current, voltage and the phase angle between them—decisive properties to define the character of the discharge and to check its models [198, 382, 383]: • Phase angle ϕ between current and voltage −90◦ : entirely capacitive. • Phase angle ϕ between current and voltage 0◦ : ideally resistive. • Phase angle ϕ between current and voltage 90◦ : entirely inductive. The most general question is the evaluation of the impedance matrix (SMatrix) with the properties S11 and S22 , respectively, the reflectivity coefficients (input and output) and S12 and S21 , respectively, the gain (forward and reflected) with r and f the amplitudes of the reflected and incident wave, respectively, referred to Rref .

r1 r2

=

S11 S12 S21 S22

f1 f2

(9.58.1)

resolved in two equations r1 = S11 f1 + S12 f2 r2 = S21 f1 + S22 f2 ,

(9.58.2)

whose values are referred to the terminal resistance of Rref = 50 Ω, and the Ohmic law can be written according to U =ZI ⇒u=zi⇒ √

Z U = I Rref , Rref Rref

(9.59)

which leads to the well-known equation for the reflectivity coefficient ρ: ρ=

z−1 1+ρ ⇔z= . z+1 1−ρ

(9.60)

This system can be significantly simplified after adjustment since the two-port + plasma mutates to a one-port with vanishing reflection coefficient ρ. The plasma impedance will become ZP = 50

RF source

matching network

Z-scan

1+ρ Ω. 1−ρ

(9.61)

plasma

Fig. 9.28. The instrument to measure the impedance is inserted in the circuit between matching network and driven electrode and records data up to the fifth harmonic.

342


The V (I) probe is specially designed not to induce shifts in key process parameters or have its measurement influenced by the hostile conditions in the current path. From this extremely carefully designed housing, shielded cables transmit the current and voltage signals to the processing unit, in which the raw signal is filtered and analyzed in terms of the first five harmonics in voltage, current, including the phase angle ϕ of the fundamental, from which significant plasma properties can be deduced which define the characteristics of the discharge. Among them are the effective power input, the complex reactance Z and its components and the resulting self-bias voltage VDC . We have seen that the behavior of highly capacitive, asymmetric discharges is characterized by extreme nonlinearity of the sheath current with respect to the exciting sinusoidal voltage, most frequently at 13.56 MHz. With this instrument, these nonlinearities (i. e. higher harmonics of 13.56 MHz) are subject to further investigation (Figs. 9.29). Of particular interest is the change in characteristic when primary parameters are changed. From these graphs, it will become evident that the model of an entirely capacitive discharge is almost perfectly realized in discharges through a gas with high ionization potential (such as argon) at very low discharge pressures and high power (high bias voltage) [380]. 200

0.4

150

argon

75 mTorr 38 mTorr 15 mTorr 4 mTorr

0.3

100

cos f

absorbed RF power [W]

argon

50

0 0

4 mTorr 15 mTorr 38 mTorr 75 mTorr 112.5 mTorr

0.2 0.1

100

200

300

400

radiated RF power [W]

500

0.0

0

200

400 600 bias voltage [V]

800

1000

Fig. 9.29. The electrical behavior of an electropositive inert gas argon, shown for the ratio of absorbed power with respect to emitted power and its power factor [380].

However, it has to be taken into account that the overwhelming part of the current does not flow through the discharge but through the L-matching network, since the current intensity in each is in inverse proportion to their respective resistances. Example 9.1 Consider a typical value for the sheath capacitance of 60 pF (1 cm sheath thickness, area of the electrode 750 cm2 ) which leads to an impedance of 197 Ω at f = 13.56 MHz. Typical values of the capacitance in the matching network are

9.4 Optical emission spectroscopy (OES)

343

in the order of 300 pF which leads to Z = 40 Ω, and since the current intensity in each is in inverse proportion to their respective impedances, only 17 % of the voltage decrease are due to the discharge current.

But that means: If the fraction of the measured current is that small, the V (I) probe is not very sensitive to plasma parameters and even worse, to changes of them.

9.4 Optical emission spectroscopy (OES) It has turned out that in low-pressure low-temperature plasmas, the EEDF deviates from the Maxwellian prototype to a certain extent. Since this distribution rigidly holds only for non-charged particles, this behavior is intelligible and could be expected. When inspecting the retardation zone of a Langmuir characteristics, the deviations from the straight-line behavior are a first hint of this conduct. Due to the strong asymmetry of the V (I) line, the high-energy electrons are hidden behind the background of the small ion current and are nearly undetectable. Since these electrons cause all the processes that are necessary to ignite and sustain a plasma with its characteristic colors, their density and energy distribution can be determined by a quantitative evaluation of the optical emission spectrum.9 This method was invented by Malyshev and Donnelly in the late 1990s and has been named Trace Rare Gas Optical Emission Spectroscopy TRG-OES [384, 385]. An experimental setup is sketched in Fig. 9.30. Mirrors MFCs

Gasinlet One Hole Shower Head Langmuir Wafer SEERS

RIE-Chamber Pumping System

Ar Cl2 Kr BCl Xe 3 A CH Cl24 H BCl 2 3 O22 O Anode Optical 2 Lens Fiber System

Monochromator Photomultiplier + Photodiode Array

HR-Grating

Powered Cathode RF Z-Scan RF RF-Generator + Matching Unit Computer

Fig. 9.30. Experimental setup of an optical spectrometer flanged to a reactor for optical emission spectroscopy (OES).

9 All the atoms and molecules which are generated during processing are subject to an analysis (cf. Sect. 11.8).

344


9.4.1 Electron temperature with OES Upper electronic states will relax by emitting light of characteristic wavelength [386] which is accessible by optical grid spectrographs within the range between 200 nm and 1000 nm. Typical mechanisms are compiled in Eqs. (9.62) and sketched in the corresponding Fig. 9.31 (the asterisk denotes an excited or metastable state): Ai + e− −→ Aj + e−

(9.62.1)

Ai + e− −→ Ak + e−

(9.62.2)

e−

Aj −→ Al + e− e−

Ak −→ Am + e−

(9.62.3) (9.62.4)

upper levels j,k

excitation by electron collision

cascading @ hν ? relaxation J @ @ by J @ @ collisions of 2nd R @ J @ @ optical J @ R @ relaxation J hν metastable levels l,m J J J ^ J

kind

lower level i

Fig. 9.31. Sketch of the electronic excitation and subsequent relaxation into two states of medium energy: The population of the excited level does depend sensitively on the EEDF.

Since the intensity of these spectral lines which are observed in emission sensitively depends on the population of the upper level, the simplifying assumption will be made in the approximation of first order that the equilibrium of occupation results from the excitation from the ground level by electron impact and relaxation by spontaneous emission. Since these conditions resemble those of the solar corona, this model is denoted corona model. 9.4.1.1 Corona model and its validity. Applying this model, the electron temperature Te can be directly determined by measuring the intensity of optical emission lines. When a species A with density nA is excited by electron impact from level Ai with energy Ei to level Aj with energy Ej by a reaction of second order and subsequently will relax to level Ak with energy Ek , the measured intensity of the emitted line IAi,j ,k should be [387]: IAi,j ,k = kij ne nAi bj,k

(9.63)


345

provided the spectral sensitivity of the spectrometer is constant over this specific spectral range. The (constant) branching ratio bj,k = Ajk /

Aji

(9.64)

i<j

can easily be determined with the data collected in the Wiese tables [388]. To fit the rate coefficient kij =

∞ Ej

Ee σ(E) EEDF dE,

(9.65)

we must choose an electron energy distribution function (EEDF). TeOES is then defined as the inverted slope of the function ln Fe (E) vs. E for E > 10 eV with fe (E) the EEDF [389]. For a Maxwellian or Druyvesteynian distribution, respectively, the EEDF is represented by an analytical expression. It is evident from this definition that the so-defined electron temperature is a characteristic property of the high-energy tail of the EEDF and may not uncritically extrapolated to its low-energy range. Only for cases in which the energy of the electrons is distributed according to the laws of a canonical ensemble, the Maxwellian approach can be adopted, and in this case TeOES would coincide with the Te value obtained with Langmuir probes. The energy-dependent cross section σ which exponentially depends on temperature just above the threshold, will be parametrized applying the Bethe formula [43]. We obtain an integral which depends only on the electron temperature Te . Since we may calculate with the ideal gas law, the densities nA,B scale with their partial pressures pA,B . Hence, for the intensity I of two spectral lines, we can state IA = kA bA ne nA ∧ IB = kB bB ne nB .

(9.66)

In a capacitively coupled discharge, the degree of ionization is low enough (≈ 0.1 %) to consider nA,B being constant. Calculating the quotient, the unknown electron density cancels, and this ratio is going to be compared with the measured intensities, and we obtain Te : kA bA ne nA IA kA bA IA = ⇒ = . IB kB bB ne nB IB kB bB

(9.67)

The validity of this model mainly depends on three assumptions: • The excitation from the ground state to the upper state is caused by electron impact [384, 390]. • The energy dependence of the cross section for this electronic excitation is similar [385, 391].

346


• There is corona equilibrium between excitation by electron impact and spontaneous relaxation by radiation (this is no problem for low-density plasmas which are optically thin, in particular, for capacitively coupled plasmas) [361]. For plasmas with higher electron temperatures (such as ECR discharges) and higher electron densities (such as inductively coupled plasmas and plamas excited by helicons and ECR), respectively, the paths of excitation has to be considered very carefully; most frequently, metastable states and cascading effects from higher levels are involved [387]. This method is known as Advanced Actinometry10 and has been extended by Donnelly et al. to TRG-OES and requires a simultaneous non-consecutive control of two spectral lines. A grid of high resolution (better than 1 ˚ A) exhibits only a small spectral range, therefore, the use of two inert gases is recommended, the gauge components. A line can be employed if its Einsteinian coefficient and its branching ratio is known; the increase of the spectral line should scale with rising power. Table 9.2. Important lines for Advanced Actinometry wavelength [˚ A] 4379.9 4526.2 7256.65 7414.12 7547.09 7744.94 4200.67 7503.87 7514.65 7635.11 7723.76 7587.41 7601.54 7685.25 7694.54

assignment

Cl(I): 5p4 D03/2 → 4s4 P3/2 Cl(I): 5p2 P03/2 → 4s2 P3/2 Cl(I): 4p4 S03/2 → 4s4 P5/2 Cl(I): 4p2 P3/2 → 4s4 P5/2 Cl(I): 4p4 S3/2 → 4s4 P3/2 Cl(I): 4p4 S03/2 → 4s4 P01/2 Ar(I): 5p[2 12 ] → 4s[1 12 ]0 Ar(I): 4p [ 12 ] → 4s [ 12 ]0 Ar(I): 4p[ 12 ] → 4s[1 12 ]0 Ar(I): 4p[1 12 ] → 4s[1 12 ]0 Ar(I): 4p[1 12 ] → 4s[1 12 ]0 Kr(I): 5p[ 12 ] → 5s[ 32 ]0 Kr(I): 5p[ 32 ] → 5s[ 32 ]0 Kr(I): 5p [ 12 ] → 5s [ 12 ]0 Kr(I): 5p[ 32 ] → 5s[ 32 ]0

branching ratio bki not determined not determined 0.32 0.10 0.22 0.11 not determined 1.00 1.00 0.41 0.127 1.00 0.57 1.00 0.13

energy of the upper level [eV] 11.82 11.94 10,63 10.59 10.63 10.63 14.50 13.48 13.27 13.17 13.15 11.67 11.55 12.26 11.53

9.4.1.2 Direct electronic excitation. At least two different paths of excitation have to be taken into account: either a transition from the ground state with 10 This method has been introduced by Coburn und Chen to determine the density of excited species by adding just one gas, the gauge gas (ακτ ις is Greek for beam) [392].


347

relatively small cross section or excitation by metastables with large cross section (Chap. 2). In capacitively coupled plasmas with their low plasma density, the excitation by metastables can be neglected, but in high-density plasmas, this bypassing is quite likely to become a serious competitor. Fortunately, this excitation can be excluded for some upper levels, and by comparison with a suspected two-level process, the inclusion of metastables can be definitely ruled out (or not) [390]. To decide whether metastables are unimportant the ratio of the two intensities under inspection is plotted vs. the absorbed power. If this ratio remains constant the influence of metastables can be neglected. In Fig. 9.32, this is shown for the two argon lines at 750 nm and 751 nm, and the three krypton lines at 758 nm, 760 nm, and 768 nm. This conduct is supposed to be due to the low plasma density, at least in the range between 100 and 400 W which corresponds to a power density of 0.1 to 0.3 W/cm2 . Having this fact in mind, the rate coefficients for a one-step process can be calculated. 1.00

intensity ratio

0.75 Kr 758/760 Kr 768/758 Ar 751/750

0.50

0.25

0.00 0

100

200

300

400

500

Fig. 9.32. OES: background corrected lines of argon and krypton at 15 mTorr (2 Pa) as a function of the absorbed RF power.

RF power [W]

9.4.1.3 Parametrization of the cross section. Cross sections in the energy range of interest (several eV) are measured but for some inert gases. These lines are effectively parametrized applying a simplified Bethe formula which has been improved by Lotz [43, 44, 104] [cf. Eq. (2.41)]: σ = const

ln(T /Ej ) Ej

(9.68)

with T the energy of the electrons and Ej the energy of the upper level which is reached by a one-step excitation by electron impact (Table 9.3). Because of the large deviations between the measurements of different authors, for reasons of consistency the application of only one source is strongly recommended. In Table 9.3, the data communicated by Feltsan and Zape-

348


Table 9.3. Constants of the simplified Bethe formula for cross sections of the inert gases argon and krypton after Feltsan and Zapesochnyi [393] − [395].

# wavelength upper level constant [˚ A] Ej [eV] [10−19 cm2 ] 1 Ar 7503.9 13.48 5418 2 Ar 7514.7 13.27 2016 3 Kr 7587.4 11.67 3283 4 Kr 7601.5 11.55 5783 5 Kr 7685.3 12.26 803 6 Kr 7694.5 11.5 4397

sochnyi are listed for transitions in the UV/VIS range [393], [394].11 They were fitted within ±3 % to the maximum of the cross section which goes perfectly over the steep rise beyond the threshold energy. It must be noted that the data are not corrected for optical cascading. 9.4.1.4 Details of the evaluation. Provided that the corona model is valid, the actual electron temperature is obtained by comparing the ratio of the calculated spectral lines with that of the experimental ones [387]. To keep the method simple and in order to reduce possible sources of experimental error, the application of only one inert gas is recommended which should be excited to different ionization levels. For argon, this should be the neutral gas, Ar(I), and single-ionized gas, Ar(II) [398]. Most frequently, this approach turns out to be scarcely executable since the intensity of the Ar(II) line is far too feeble, in particular in discharges through electronegative gases such as SF6 , Cl2 or BCl3 . Therefore, a second inert gas has to be employed; the combinations Ar/Kr or Ar/Xe have been proven to be most appropriate whereas the upper levels of neon are already too high (Tables 9.2 + 9.3).12 In a capacitively coupled plasma through electronegative gases such as chlorine, the lowest doping levels amount to about 10 % of the flow of the main gas to obtain reliable line intensities. This error may appear very large but is inevitable since in chlorine, the spectral intensity of the Ar line at 7511 ˚ A is attenuated more severely than in neon (by a factor of 3); in BCl3 , this factor is 2 at 75 mTorr (10 Pa), but rises to 10 at 7.5 mTorr (1 Pa). These rate coefficients serve to calculate the electron temperature Te,MB for a Maxwellian electron distribution. In our case, the equally weighted ratios of the line intensities # 3/1, 4/2, 3/2 with 1.6 eV ≤ ΔE ≤ 1.8 eV are used with ΔE denoting the difference between the upper levels of Ar and Kr. 11 Many authors prefer the data reported by Mityureva [48, 396] or those compiled by Szmytkowski et al. [397]. 12 Further doping leads to an increasing dilution of the inspected gas.


349

As can be seen from Fig. 9.33, raising the discharge pressure generally causes the electron temperature to drop, and the composition of the atmosphere can add some subtle modifications to this trend. 10 8

Ar gap 5 cm diameter 43 cm

Te [eV]

6 4

Fig. 9.33. Electron temperatures in argon, experimental data and modeling the plasma employing the global model.

2 0 0

25

50 p [mTorr]

75

100

One of the most critical issues is the cross section. Systematic errors can be caused by the necessity of applying the values of several authors or the deficient accuracy of the analytical formula which can easily lead to different results deviating by a factor of 2 or so. This is pictured in Fig. 9.34 for two different formulae to fit the rate coefficient of the first ionization of argon. In Fig. 9.34, 1x10-6

kIon [cm3/sec]

1x10-9

Fig. 9.34. The rate coefficient for the first ionization of argon calculated with two different formulae shows a significant variation, especially in the important energy range below 10 eV.

-12

1x10

Lotz formula Bethe formula

1x10-15 1x10-18

100

101 Te [eV]

the (simplified) Bethe formula is used according to ln Te /Eion (9.69) Eion with T the energy of the electrons; next in order follows the Lotz formula which goes over the experimental shape much more precisely (cf. Sect. 2.5): σion = C1

σion = C1

ln Te /Eion Te 1 − C2 exp −C3 −1 Eion Eion

.

(9.70)

350


9.4.1.5 Chosing the right electron distribution. Lowering the plasma density, the application of a Maxwellian distribution for the EEDF becomes questionable to an increasing degree. As shown in Sect. 14.1, it is bizarre to expect electrons to act as uncharged particles (for which the Maxwellian distribution holds); only a sufficient number of collisions between charged particles can force the EEDF into the desired direction which happens, in fact, in highdensity plasmas [389] − [391]. The most striking experimental argument is the growing gap between the electron temperatures which are obtained by two independent methods, the measurement with a Langmuir probe and that with OES. For comparison, the rate coefficients of two lines (Ar 7511 ˚ A and Kr 7580 ˚ A) are modeled not only with the approach of Druyvesteyn but also with that of Maxwell and Boltzmann [339, 399] (Figs. 9.35 − 9.38, “D” denotes Druyvesteyn, “MB”denotes Maxwell-Boltzmann). 0.06

1.00 0.75

Maxwell-Boltzmann Druyvesteyn

f(E)

f(E)

0.50

0.04

= 2.5 eV

Maxwell-Boltzmann Druyvesteyn = 10 eV

0.02 0.25

0.00 0.0

0.00 2.5

5.0 Te [eV]

7.5

0

10.0

10

20 30 Te [eV]

40

50

40

50

0

10

-2

10 -1

f(E)

10 f(E)

= 2.5 eV -2

10

= 10 eV

-3

10



10-4 10-3 0.0

2.5

5.0 Te [eV]

7.5

10.0

0

10

20 30 Te [eV]

Fig. 9.35. Both expressions which describe the EEDF analytically: Maxwell-Boltzmann and Dryvesteyn, exhibit a different behavior. For the same mean energy, the latter is broader, its maximum is shifted towards higher energies and is less intense than the first function. In the high-energy range which is required to excite optical levels, the Maxwellian distribution decreases more slowly than the Druyvesteynian distribution. Note the different scale in the linear plots and the convex shape for the Druyvesteynian distribution in the logarithmic plots.


351

Comparing the pressure dependence of the electron temperature which has been calculated with the two different distribution functions, the habitude is almost the same for both of them, however, the electron temperature is simply shifted to higher values—about 2 eV—for the Druyvesteynian distribution (Fig. 9.38). The real EEDF, however, is supposed to be found rather between these extreme distributions and the absolute values obtained for Te are quite worth discussing [400]. 1.0

1.0

f(E), s(E), normalized

f(E), s(E), normalized

D, 2.5 eV

0.8 0.6 0.4

3

2S

0.2 MB, 2.5 eV

0.0 0

25

50 energy [eV]

75

100

0.8 0.6 0.4 D, 10 eV

0.2

3

2S

MB, 10 eV

0.0 0

25

50

75

100

energy [eV]

Fig. 9.36. Comparison of the two energy distribution functions (MB or D) for the electrons with the cross section for a specific optical transition, linear scale. The difficulty of fitting the “right” electronic temperature is evident: For mean energy of 2.5 eV, there are almost no electrons at 20 eV!

9.4.1.6 RF power. According to the global model introduced in Sect. 3.5, Te should scale inversely with the logarithm of the discharge pressure, but should be almost independent on the absorbed power. By and large, this will be observed, in fact. Deviations from this behavior, for example for BCl3 , should be mainly due to the strange conduct of the cross section rather than to chemical effects (electron attachment). 9.4.1.7 Corona model: limits of applicability. • The ratio of the rate coefficients is not constant. • The densities of the gauge components are not constant. • There are several (also non-radiative) paths to relax from the upper level. • There are different paths to reach a certain excited level.

352


3

Ar 750.4 nm D Ar 750.4 nm MB

k [10-9 cm3/sec]

k [10-9 cm3/sec]

3

2

1

0

Kr 758.7 nm D Kr 758.7 nm MB

2

1

0 2

4

6

8 10 Te [eV]

12

14

2

4

6

8 10 Te [eV]

12

14

Fig. 9.37. Rate coefficients calculated with Eq. (9.57) for two lines which are used for OES most frequently. In the low-energy range below 10 eV, electrons distributed Druyvesteynian must rise their “temperature” by about 1/2 eV over the Maxwellian “temperature” to make up for the lower occupation in the high-energy tail of the distribution.

• Provided the cross section for the gauge component deviates significantly from that of the working gas (for example an etchant), changes in the EEDF can influence the ratio kA /kAr considerably. • Non-radiative relaxation mechanisms influence both the gauge components (inert gases) and the gas under inspection, most frequently, in the same direction, therefore, their impact is not very large. For the molecular etchant, several mechanisms can be considered [401]. In chlorine plasmas, for example, atomic Cl can be generated on several tracks. The main mechanisms are reactions by electron impact and electron attachment: Cl2 + e− −→ 2 Cl· +e− ,

(9.71)

Cl2 + e− −→ Cl· +Cl− .

(9.72)

For both these reactions, the cross section amounts to about 10−16 cm2 [35]. Chlorine atoms are excited by consecutive electron impact as well: Cl + e− −→ Cl∗ + e−

(9.73)

and a cross section one order less in magnitude (about 10−17 cm2 ) [402]. Assuming a ratio between nCl /nCl2 ≈ 0.1 which is the largest ratio measured with IR methods, the rate of formation for atomic Cl· by dissociation

9.4 Optical emission spectroscopy (OES) 7


6 5

D, 2 Pa

4 D, 10 Pa MB, 2 Pa

3 2

MB, 10 Pa

1 0 0.00

0.25

0.50

0.75

1.00

353

Fig. 9.38. Electron temperature as a function of composition (molar fraction) for two different pressures in discharges of BCl3 /Cl2 , calculated for a D-distribution and a MB one (absorbed RF power: 75 W). The habitude of the matching curves is almost the same except a constant upwarded shift for the D-distribution [399].

molar fraction BCl 3

and electron attachment is larger by two orders of magnitude than the excitation rate of this very atom. The chlorine radicals generated by the reactions (9.71/72) are very likely to be in excited states, and the intensity of the chlorine emission line rather describes the rate of formation of Cl· than its density: d[Cl] = (kD + kA )[Cl2 ][e− ], dt

(9.74)

keeping in mind that the atomic chlorine is excited to various upper levels. 9.4.2 Plasma gas temperature Since temperature is statistically correlated to the mean energy of the ensemble under consideration which follows a certain distribution (for neutrals: Maxwellian, for electrons: Maxwellian or Dryvesteynian, but both questionable), the most straightforward method to measure gaseous temperatures uses the broadening of highly resolved spectral lines in emission or absorption. The width of a spectral line is determined by radiation damping [90] and lifetime broadening (so-called 4th uncertainty relation ΔE Δt ≥ h ¯ [91]), but predominantly by the Doppler shift: an object radiation with wavelength λ receding (approaching) with velocity v from the observer is seen by him with the shifted wavelength [the factor is (1 + v/c)λ in the first case, but (1 − v/c)λ for the latter situation]. In a gaseous discharge, the spectral lines are broadened due to the chaotic motion of its constituting particles. From the shape of the spectral line the temperature can be evaluated according to

354


λ Δλ = 2 c

2kB T m

(9.75)

at FWHM (full width at half maximum) [403] (Fig. 9.39).

normalized intensity

1.00 T/3 T 3T

0.75

0.50

Fig. 9.39. Doppler broadening of a spectral line at various temperatures. The intensity in arbitrary units is plotc Oxford ted vs. Δλ/λ [404] ( University Press).

0.25

0.00

-3

-2

-1

0 Dl/l

1

2

3

The Doppler shift leads to a considerable broadening of the line; and for a given temperature, the broadening is more distinct for large wavelengths and small masses. In the UV/VIS spectrometers which are in common use, however, the resolution is not sufficient to resolve this fine structure. Another possibility is the measurement of the rotational fine structure of vibration spectra of the excited state. The intensity distribution of a Bjerrum’s double band is predominantly determined by the occupation of the rotational levels of the ground state, and the probability of occupation of a single state is given by

J(J + 1)h2 NJ = (2J + 1) N0 exp − 8πΘkB Trot

(9.76)

with Θ the moment of inertia, J the rotational quantum number, and Trot the rotational temperature [405]. Therefore, Eq. (9.76) determines the intensity distribution of the rotational vibration band [92], Fig. 9.40. The term for this double band is y = |x|e−x

2

(9.77)

with

x=

A ωR ∧ ωR = 2kB Trot

by which the maximum is located at

kB Trot , A

(9.78)

9.4 Optical emission spectroscopy (OES) 1 x= √ . 2

355

(9.79)

intensity [a. u.]

0.50

Fig. 9.40. Bjerrum’s double band of a rotational vibration spectrum which occurs by coupling of the vibrational transition with the rotational transic Sprintions ωv ± ωR,i [406] ( ger-Verlag).

0.25

0.00

-2

-1

0 J

1

2

To observe this spectrum, the presence of a molecule is mandatory whose dipole moment is subject to change during the transition (case of the rotational vibration spectrum) or which are Raman active. Most frequently, the excitation of rotational levels of nitrogen is employed which has been added to the ambient of interest [407] − [411]. But above all, it has to be ensured that the rotational temperature Trot equals the translational temperature T ; it is only in this case that the hyperfine structure of electronic transitions can be employed to evaluate gas temperatures. It is well known from kinetic gas theory and molecular spectroscopy that by twobody collisions, a highly frequent exchange between translational and rotational motions takes place resulting in equilibrium. Hence, rotational spectroscopy is a very powerful means to evaluate the translational energy/temperature of the neutrals. In plasmas, higher electronic states are occupied by electron impact excitation, which does not affect the ground state distribution provided the plasma density is sufficiently low. The gas temperature will then be evaluated by comparing the experimental and calculated spectrum. Why is nitrogen the favorite for this type of diagnostics? In plasmas, lines are investigated in emission i. e., transitions from higher electronic energy level to lower electronic energy level. This transition is connected with a series of rotational vibrational bands (E , v , J → E , v , J ). In emission, the gas temperature can be determined spectroscopically from the rotational fine structure of the vibration spectrum of excited electronic levels which is abbreviated to vibronic levels. The accuracy of the measurement is the better the closer the minima of the potential curves coincide. According to the Born-Oppenheimer approximation, alterations within the electronic structure happen instantaneously (within

356


10−15 sec), whereas alterations within the molecular skeleton constituted by the individual arrangement of the nuclei happen within one period of a molecular vibration—slower by orders of magnitude. This is why transitions within the potential diagram occur vertically thereby following the Franck-Condon principle (Fig. 9.41). 25

potential energy [eV]

20

15

10

B3Pg+5

10

A3Su+

15

N( 4S0) + N+ (3P)

5

15

N( 4S0) + N+ (2P0)

3

4

C Pu-

10

5

5

15

4 0 + 2 0 3 B Pg N( S ) + N ( D )

10 1

3

B Pg

10 5

0 1.0

3

-

C Pu

+

X Sg+

4 0 + 4 0 N( S ) + N ( S )

3

+ A Su first negative 3 B Pg second positive

1.5 2.0 2.5 internuclear distance [Å]

system system

3.0

Fig. 9.41. N2 : For electronic excitation from the ground state, the minima of several potential curves coincide almost exactly. By evaluating the vibronic transitions measured in emission, the gas temperature can be inferred [412].

In most cases, however, the potential curves are mutually shifted, at least slightly. This causes the electronic excitation not to lead to the occupation of the vibrational ground state of the upper electronic level but to an upper vibrational level with an upper rotational level. To infer the temperatures in the ground state from emission spectra out of the upper state via Eqs. (9.76) − (9.79), the minima of the potential curves should coincide. This condition is met for the 1 + 3 − excitation from the ground state X 1 Σ+ g for the systems X Σg → C Πu and 3 + X 1 Σ+ g → B Πg . These upper levels can relax into the first positive system 2 + → X Σg (between 540 and 1000 nm) or in the second positive system B 2 Σ+ u 3 − → B Π C 3 Π− u g (between 300 and 460 nm) [407, 413]. Since the excited state is a triplet system, each band consists of three subbands denoted P, Q, and R. The gas temperature is determined by simulation of the spectra and best fit with experimental data [414]. Most sensitive to this modeling is the violetdegraded tail which should be perfectly penciled over for the complete system under inspection. As cross check, Davis and Gottscho found that in discharges of nitrogen, this value equaled the result obtained with laser-induced fluorescence (LIF) [415]. However, from this first experiment in 1983, it lasted for another twenty years until the huge potential of this method was really acknowledged. Recently, Donnelly et al. and Sawin et al. published papers which dealt with inductively coupled discharges (ICP) of chlorine [408, 409, 411]. 9.4.2.1 Features in noble and inert gases. As we have seen in Chap. 2, metastables play a significant role in discharges through noble gases. In fact, their content remains in the lower percent region, but their temperature can


357

significantly exceed the temperature of the overwhelming majority of atoms. It was shown recently by Wang et al. that their contribution has to be taken into account to have the simulated spectrum matched properly to the experimental data [412], especially for high pressures (Fig. 9.42).

experimental spectrum synthetic spectrum (one-temperature fit)

intensity [a. u.]

intensity [a. u.]

experimental spectrum synthetic spectra (two-tempeature fit)

3300

3300

3315

3315

3330 l [Å]

best fit: TrH = 1750 K TrL = 960 K (R = 0.35)

3300 3345

3360

3375

3300

3315 3315

3330 l [Å]

3345

3360

3375

Fig. 9.42. In discharges through noble gases, the influence of metastables has to be taken into account to have the simulate spectrum matched properly to the experimental spectrum, shown here for gas temperature measurements of micro-discharges through argon [412].

In a capacitively coupled plasma for same power input, hydrogen remains relatively cool compared with the heavy noble gas argon (Fig. 9.43). For typical power input between 100 and 400 W (S ≈ 1/8 − 1/2 W/cm2 ), the gas temperature rises by a factor of 2. Raising the pressure by a factor of 4 is mirrored by a rise in temperature by a factor of 4 as well. But even for moderate power input, the gas temperature quickly rises to 600 K. This different behavior of hydrogen is caused by the additional mechanism of heat conduction. Whereas the mechanisms for heat dissipation in heavy molecular gases are predominantly radiation and convection, these are supplemented by resonant charge-transfer in inert gases; however, its contribution is difficult to quantify. But in lighter gases, heat conduction will replace convection as the main mechanism which turns out to be very effective. In ICP discharges, the limiting factor for power coupling is the skin effect which confines the plasma density to values of about 1011 /cm3 [253]. In CCP discharges, excessive electron heating is limited by the simultaneous power input into the ambient due to the change from pure capacitive coupling to capacitive/resistive coupling: In the thick plasma sheath, also ions are heated. Since the mass ratio between ions and neutrals equals unity, it is evident that the energy transfer into neutrals is very efficient in capacitively coupled plasmas. Moreover, the effective plasma volume in “simple” parallel-plate reactors is small compared to ICP discharges which are mainly driven in the downstream mode,

358


1500 pressure [mTorr]

113

Ar

pressure [mTorr]

500

38

1200

15

113 23

4 4

T [K]

T [K]

900 600

400

H2

150 W line

300 0 0

300

250

500 DC bias [V]

750

1000

0

200

400 600 DC bias [V]

800

1000

Fig. 9.43. Using OES with detector gas nitrogen, the rotational gas temperature has been evaluated in discharges through argon and the molecular gas hydrogen [381].

the effect of gas heating should be very pronounced in CCPs (although, in most practical cases, the absolute power input in ICPs is higher).

9.5 R´ esum´ e • Langmuir probe – Advantages: Recording of important plasma parameters which are spatially resolved: electron density ne , ion density ni , EEDF which can be approximated to the electron temperature Te of the plasma bulk provided several prereqisites are met, determination of the plasma potential Φp . – Disadvantages: Difficult to operate in production reactors, very sensitive tip, low signal/noise ratio in high-frequency discharges which complicates the evaluation. Severe grounding problems: When the interior of the reactor is coated with insulating layers, the return current path is blocked which leads to an oscillating plasma potential Φp of a single Langmuir probe. Double probes and probes with reference electrodes can overcome this setback but face other complications. • Advanced actinometry – Advantage: Non-intrusive, easy to apply and can be employed in production tools. In contrast to the measurement with Langmuir probes, the EEDF in the high-energy tail is measured. It is these electrons which are responsible for sustaining the plasma.

9.6 Properties of Electronegative Plasmas

359

– Disadvantage: Very complicated evaluation procedure, and only one plasma parameter is obtained, the method integrates and does not deliver a spatially resolved signal. • SEERS – Advantage: Skilful employment of just the electrical noise, no problems in HF discharges to evaluate mean values for the electron density ne , the effective collision frequency νeff and the absorbed power. Easy to apply, very robust, which predestine this method for production tools. Some production issues, for example the first wafer effect, have been quantified for the first time employing this method. ∗ Electronic plasma density: For argon, fairly good agreement has been found between SEERS and Langmuir [380]. ∗ Effective collision frequency: This property has evolved to be the criterion of reproducibility. To make effective use, the energy dependence of the cross section at very low energies is mandatory. – Disadvantage: No spatially resolved signal is obtained, no value for Te . • V (I) probe – Advantage: Direct measurement of complex electric properties is feasible. – Disadvantage: The chamber capacitance must be small compared to the impedance of the plasma. Due to the unfavorable ratio of the capacitances under consideration, only a small fraction of the measured current flows into the discharge which reduces the sensitivity of this device.

9.6 Properties of Electronegative Plasmas In the following, we will deal with chlorine-containing, electronegative plasmas which are widely used for anisotropic etching. Their properties are extracted applying the methods described in the first part of this chapter (np , Te , νm ), and they are used to discuss some mechanistic aspects of surface reactions in Chap. 12 in capacitively driven discharges (CCP), inductively coupled plasmas (ICP), and ECR driven discharges. The properties are the density of neutrals [parent molecules vs. neutral fragments (radicals)], plasma density: electron density and the density of positive ions, electron temperature and the density of negative ions.

360


9.7 Capacitively coupled plasmas 9.7.1 Electrical considerations Employing V (I) probes for different gases, the significant losses catch the eye. Only 50 % of the radiated RF power are absorbed by argon; in molecular gases (chlorine and boron trichloride) these losses remain a little bit smaller (Fig. 9.44). 200

350

150

300 75 mTorr 37.5 mTorr 15 mTorr 3.75 mTorr



Ar

100

50

250 200

Cl2 75 mTorr 37.5 mTorr 15 mTorr 3.75 mTorr

150 100 50

0 0

100

200 300 400 radiated RF power [W]

0 0

500

BCl 3

250

75 mTorr 37.5 mTorr 15 mTorr 3.75 mTorr



200 150 100 50 0 0

200 300 400 radiated RF power [W]

500

300

300 250

100

100

200

300

400


500

200

Cl 2:BCl3 (10:10) 75 mTorr 37.5 mTorr 15 mTorr 3.75 mTorr

150 100 50 0 0

100

200

300

400

500


Fig. 9.44. Comparing four gases (one atomic, three molecular), we see that the power absorption is quite similar. The power absorption for the molecular gases is higher than for argon, however, this does not lead to a higher degree of ionization but is consumed by parasitic reactions [380].

Simultaneously, the power factor also increases for higher pressures in chlorine to equal unity for very high discharge pressures (112.5 mTorr) and very low power input ( 0.5, high temperature zone: Onset of recrystallization, massive growth of the grains due to steep increase of the surface mobility, since diffusion is an activation-type process, some columnar structure remains visible.

zone 2

zone 1

zone 3

Fig. 10.12. Microstructure diagram according to Movchan and Demchisin for physical vapor deposition [469]. Ts : substrate temperature, Tm : melting temperature.

Ts/Tm

As far as the number of independent parameters is concerned, sputtering is superior to evaporation techniques. The influence of the discharge pressure has been the subject of many treatises. As we have seen in Chap. 6, raising the pressure will change the IEDF, and the mean ion energy is reduced. Thornton has reasoned this additional degree of freedom (Fig. 10.13). He established a model which extends the considerations of Movchan and Demchisin to sputtering mechanisms by considering two counteracting mechanisms. Collisional impact by the inert gas atoms reduces the normal component in the deposition flux but enhances the oblique component which leads to an equalization of topographical differences. Rising pressure is associated with increasing thermalization of the impinging atoms which simultaneously causes a reduction of the residual energy of the adatoms. As it has turned out the first effect is superior to the second one, and we observe the columnar grains to grow at lower substrate temperatures

390

10 Plasma deposition processes

with rising discharge pressure. On the other hand, best adhesion is expected at low discharge pressures. Due to ion bombardment and substrate temperature, the microstructure can vary from completely amorphous up to polycrystalline. For example, a lower substrate temperature can be compensated by an increase in ion energy. Thornton distinguishes four ranges:

zone 3 zone 2

zone 1

1.0

zone T

0.8 30 20 argon 10 pressure [mTorr]

0.4

0.6 Ts/Tm

Fig. 10.13. Microstructure diagram according to Thornton who identified four different regimes of film growth [469]. Ts : substrate temperature, Tm : melting temperature.

0.2 1

1. Zone 1: Columnar growth predominantly determines the rise in film thickness, however, the columns are loosely packed and separated by voids. For low substrate temperatures, the pores cannot be filled out. 2. Zone T: The columnar growth becomes less important; simultaneously, the porosity decreases since the voids are filled by surface diffusions leading to densely packed grains. Additionally, ion bombardment assists in smoothing the surface. 3. Zone 2: Further rise of diffusion processes will lead to the formation of crystalline columnar grains which, in turn, causes surface roughening. 4. Zone 3: The diffusion has reached its maximum intensity. A polycrystalline layer with a rough surface has been formed. Up to the state of coalescence, ideal growth of the islands is assumed, which is justified by the small extension of the nuclei. So, dislocations can easily be equalized by transport processes. After having reached the state of coalescence, this is probably not possible any more. Earlier or later, these point defects and dislocations are incorporated, even by intended epitaxial growth. These dislocations are supposed to be responsible for the voids which will prevent a perfect match of the columns during coalescence. The thermodynamic end of film formation is simply the attainment of the minimum in surface energy. The mechanisms are evaporation and condensation and diffusion at the surface and in the volume as well. Impurities of the sputter ambient or of the residual gas also contribute to the build-up of tensions. Since they can trigger both types (compressive or

10.2 Sputtering kinetics

391

tensile stress), skilful doping of the sputter atmosphere can sometimes generate a stress-free range [470]. The two different limiting cases of adhesion lead to the following morphological classification. 1. Interfacial cohesion: Two different layers stick together but will not mutually diffuse nor form a chemical compound. 2. Cohesion by interdiffusion: gradual transition from one layer to the other. 3. Cohesion by interdiffusion and the formation of an (oxidic) interlayer. 4. Additional mechanical interlock by Kirkendall pores. Provided the substrate surface is not contaminated by organic films (grease and oil), the oxide layer can be removed by backsputtering. But even at a very low pressure of less than 10−6 Torr (0.1 mPa), the coverage time to form a single atomic layer will be less than one second. By raising the temperature, the time interval over which the surface remains free of contamination can be prolonged. The phenomenon of interfacial cohesion will occur for low nucleation densities. One of the lowest values is observed for gold: Since almost no bonding will happen between the adatoms, the surface energy is superior to nucleation and small droplets will form rather than a coherent film, the paradigm for poor adhesion.6 The two other mechanisms intermingle since they differ just by the formation of a chemical compound as interfacial layer. Hudson and Somekh conducted stress experiments in systems which involved niobium as counterpart of nickel, silicon, and tantalum [475]. All these deposits come along as amorphous film and do not form chemical compounds but they differ decisively with respect to their short-range order. For example, the niobium-rich compositions have metallic structure, whereas the silicon-rich compositions exhibit an amorphous tetrahedral structure. The third case serves as the basis for the connection of ceramics with metals: Consider an oxide (SiO2 ) which is coated with a metal whose oxide exhibits a very high formation energy. Adhesion is then achieved by the following reaction (molybdenum on top of glass): Mo + SiO2 −→ MoO2 + Si.

(10.9)

To generate an Ohmic contact, the metal should form several oxides in different oxidation states. It is mandatory that one of these oxides exhibits a 6 It is customary to deposit a glue layer of titanium as a first step since titanium can reduce nearly all oxides by forming an oxide with large phase width. To avoid interdiffusion between titanium and gold, a stopping layer (platinum or molybdenum) follows next to form the Bell contact [471] − [473]. Another example which has been treated by Mattox and McDonald is the system cadmium/iron which are also mutually insoluble [474].

392


metallic conductivity (better than 3×102 Ω−1 cm−1 ). In the case of molybdenum, these are the molybdenum bronzes [476]. As far as adhesion is concerned, sputtered films are superior to evaporated films. Although the initial high energy of the sputtered atoms is thermalized to a great extent en route to the substrate, the mean energy of the impinging atoms is still a multiple of the lattice energy. An additional amount of energy is provided by the rapid electrons from the plasma bulk which can support interface reactions and interdiffusions as well as additional nucleations which, in turn, supports the interlinking of the two layers at the interface. Therefore, a subsequent sintering step improves the adhesion rather than facilitates the separation of the coating from the surface. As an example, films of sputtered Ta/Si up to 2 μm in thickness exhibited perfect adhesion even after having been sintered for 60 minutes in argon at a temperature of 900 ◦ C (Fig. 10.14).

Fig. 10.14. SEM micrographs of a split wafer after deposition (single crystal Si, 30 nm gate oxide, 300 nm poly-Si, 2000 nm TaSi2 , LHS) and after sintering at 900 ◦ C in Ar (RHS) [477].

To enhance cohesion, several techniques have evolved to be straightforward: • By presputtering, the purification of a target just before a sputtering process is termed. Simultaneously, the system is heated and brought into a state of equilibrium. • For backsputtering or sputter cleaning, the substrates are placed on top of an RF driven electrode, and the “target” is subjected to the ions incident on it (in the RF case also subjected to low-energy electrons). In the opposite to a glow discharge cleaning process, the surface is gently removed. Applying a DC discharge to metals with a native oxide gives rise to a very low negative bias voltage between 10 and 20 V by which the native oxide can be effectively removed in situ, without being exposed to oxygen-containing atmosphere prior to the real sputtering process (Sect. 5.1). Otherwise, a wet-chemical pretreatment is strictly required for further processing.

10.3 Target topography

393

10.3 Target topography In the preceding section, we have discussed kinetic aspects. Considering rates, one main feature is the transformation of the surface by ionic impact. Real planes which have been proven to be originally smooth on an atomic scale will change appearance in a delicate process which is mainly characterized by redeposition of removed atoms and the so-called cone formation. Causing a high scattering effect, the surface is made to appear dull, and can even look like black velvet; and a shiny target (Fig. 10.2) which is proudly installed in a reactor will soon exhibit a tarnished surface. These secondary effects are an inevitable phenomenon of surface modifications which are caused by ionic bombardment [478]. Most frequently are observed stairs, deep holes which look like crystallographically etched, rounded and shallow pits, and cones and pyramids which are located in the middle of a pit; their formation has been thoroughly investigated. Furthermore, the formation of bubbles is observed at high ionic doses and ion energies (> 10 keV), as has been generated for the first time as blistering in fusion reactors. Figures 10.15 show typical examples of the morphology of a surface, which has been exposed to perpendicular ion bombardment. But roughness can also arise from plasma etching with no physical component at all. It is a well-known fact that treatment of silicon with molecular fluorine which has been further activated in a high-frequency, high-pressure plasma (barrel reactor with or without a Faraday cage) leads to tarnished surfaces. This feature prevented molecular fluorine from becoming technologically applicable, selective etchant for silicon over silicon dioxide (selectivity is greater than 100:1). In both cases, roughness increases with pressure and temperature. From a fundamental point of view which is described by the so-called roughness-induced mechanism, three states are distinguished (Figs. 10.16) [479]: • Primary effects: Creation of a patterned structure by primary particles leads to the formation of secondary particles. • Secondary effects: Bombardment of removed secondary particles leads to the formation of cones and pyramids. • Tertiary effects: Cones and pyramids are standing within pits, trenches, and holes (Figs. 10.15, bottom). This state can be left, and the surface will be planarized almost completely (Fig. 10.17). However, the contribution of the primary and secondary particles during this process of planarization still remains unclear [482].

394


Fig. 10.15. Formation of a bizarre surface topography (acute pyramids in SiC (top, LHS), somewhat less acute pyramids in AlGaAs (top, RHS), cones and truncated cones within pits of GaAs (bottom) [480, 481].

10.3.1 Historical review 10.3.1.1 Roughness-induced mechanism. One of the first investigations of cone formation was carried out by Wehner who correctly recognized the interrelation between the transformation of originally smooth polycrystalline surfaces into conical structures with the angular dependence of the sputtering yield [483]. Additional investigations made by Wilson and Kidd at polycrystalline wet etched surfaces of gold revealed the formation of cones which exhibited a faceted surface (pyramids). This indicated the importance of sputtering effects at crystalline surfaces; in the fcc system, the (100) and (111) surfaces, resp., are most stable against ionic bombardment [484].


A

395

B

C

E

D

Fig. 10.16. The various states of pyramid formation according to the roughness-induced mechanism after [482]: a convexly shaped structure (A) transforms to a faceted pyramid (B). After formation of a trench (C) (secondary effects), the shape of the c pyramid will be defined sharper (D), until it is planarized by tertiary effects (E). ( Gordon & Breach Science Publishers, Inc.)

Fig. 10.17. Even on a surface with an initially perfectly smooth appearance the tertiary structure will evolve after some transformations. This state is characterized by a contour of a propagating wave, here shown for a GaAs surface before (LHS) and after (RHS) [485] − [487].

These observations were confirmed by Sigmund with a first model [479]. Cones were expected to be torn down more likely than valleys; but he showed that small irregularities on top of a relatively smooth surface show the tendency to grow during ion bombardment provided this process is supreme over

396


atomic migration; and he called his theory roughness-induced mechanism (Fig. 10.16). The first process should be temperature-independent; but the second one belongs to the thermally activated mechanisms. Therefore, a discrimination between these two possibilities should be made possible by a temperature change: Lowering the temperature should cause rougher surfaces and vice versa. Cones should be planarized more slowly on a smooth surface provided that their dimensions are smaller or at least in the order of magnitude of the penetration depth of the ions. 10.3.1.2 Contamination-induced mechanism. Simultaneously, surfaces were subjected to the mutual contamination of particles which were ejected by two opposing targets, one made of copper, the other made of molybdenum [488, 489]. The sputter rate of copper vastly exceeds that of molybdenum, and the cone formation on the copper surface has been explained by particles of molybdenum which mask the copper surface preventing further attack which would eventually cause the growth of cones. As expected, the formation of cones declined when the molybdenum target was removed. Five years later, Hudson found just the opposite behavior: Formation of pyramids can also be caused by particles exhibiting a higher sputtering yield which have been deposited on a surface with lower sputtering yield [490]. Eventually, Auciello noted that two conditions have to be met to have the secondary structures described by Wehner and Hajicek occur: The surface must exhibit a certain roughness, and the substrate must be kept at relatively high temperature [491]. Taking these facts into account, Kaufman and Robinson introduced a model that included the surface diffusion of these contaminating particles, socalled seed atoms, and their agglomeration to larger clusters which they denoted seed clustering [492]. This model resembles the nucleation theory which has been described in the preceding section.7 The occurrence of an activation energy in their kinetic equation indicates the existence of a certain threshold which has to be exceeded to have the secondary structure developed (diffusion coefficients of the contaminating atoms are proportional to exp(−Eact /kB T ) with T the surface temperature). This also holds true for the density of cones itself. Remembering the equation for the random walk: < r >2 ∝ Dt,

(10.10)

we note that < r > scales according to ∝

√

t exp

−Eact , 2kB T

(10.11)

7 They considered the occurrence of shock waves [493]; because of its adiabatic character, no thermodynamic equilibrium is required. The generation of shock waves should scale with the ion flux which should, in turn, enhance the diffusion rate.


397

and we can explain the dependence of the (mean half) distance between the single clusters ln < r > = −Eact /2kB T + const.

(10.12)

The activation energy can be evaluated by plotting the mean half distance (≈ 200 nm) vs. absolute temperature (Fig. 10.18) and proves to be approximately 1 eV for aluminum that has been contaminated by molybdenum or gold.

[nm]

1000

100

Fig. 10.18. Mean half distance < r > of aluminum cones on gold as a function of the reciprocal absolute temperature c The American Insti([492] tute of Physics).

Eact = 1.04 eV

10 1.2

1.3 1/T [10-3/K]

1.4

However, this model is applicable for low ion doses only, i. e. for current densities below 0.5 mA cm−2 or fluxes of 3 × 1015 cm−2 , and this mechanism is called the contamination-induced mechanism. For increased ion flux, diffusion processes will become more important, last but not least because of heat dissipation problems (Fig. 10.19).

[mm]

15

10

5

0.0

0.5

1.0 j [mA/cm2 ]

1.5

2.0

Fig. 10.19. Dependence of the mean distance < r > of cones on a copper surface contaminated by molybdenum as a function of the current density c Elsevier of Ar+ -ions [494] ( Science Publishers B.V.).

Several premises of this theory can be questioned, e. g. clusters and bombarding ions are considered to be in a state of equilibrium, but this model could solve two difficult issues:

398


• Secondary structures occur on a substrate by bombardment of every material irrespective of whether the sputtering yield of the contaminating species is higher or lower than that of the substrate. • In most experiments, the temperature of the substrate was supposed to exceed the melting temperature; for a short pulse (10 − 100 psec) after the collisional impact, the local temperature has been estimated to peak between 1 000 and 3 000 ◦ C, depending on the ion flux. This would mean that phase transitions should be included into this model, by which the formation of thermal spikes and whiskers can be explained easily [495]. These contradictory observations and their interpretation, mainly by the demand for the occurrence of a critical temperature, became subject to a detailed systematic series of experiments. 10.3.2 Comparison of topographical mechanisms From the aforementioned, it seemed quite likely that defects should play a significant role in the process of surface modification, and this assumption has been confirmed in a series of elegant experiments, conducted by Auciello and Kelly as well as Robinson and Rossnagel during the 1970s and 1980s [496]. The secondary structure is supposed to be caused by these mechanisms: • Sputtering. • Reflexion of the ions. • Redeposition. • Surface diffusion. All these processes lead to the formation of ridges of quadratic or tringular cross section, resp., pits and cones (round or with sharp pyramidal surfaces) and take into account the described mechanisms of cone formation, i. e. defects and inclusions at grain boundaries, surface contamination and preferential removal in distinct crystallographic directions and planes, and eventually, the surface defects which are caused by different mechanisms were contrasted for the first time [494]. The properties of impurity-seeded defects exhibit the following properties: • A certain threshold density of contaminating particles is required for their formation (≈ 0.1 %). • Spots of refractive material tend to cause defects with more ease than low-melting materials. • The density of cones does not depend on the time of exposure.


399

• The contaminants must be continuously delivered. • Temperature and substrate bias mainly determine the structure and the density of the cones. • For rising ion flux, complicated structures (whiskers) will dominate the defects. Some features of the roughness-induced cones and pyramids are: • They occur independent of impurities and contaminations. • The form is subject to change from regular or crystalline structures at high discharge pressures to broader, rounded shapes at lower pressures. • They preferentially occur in materials of high sputtering yield (aluminum and the coinage metals). • In most cases, their density is low. • Single, almost instantaneous formation at the surface which can be removed by a very high ion dose (≈ 1019 − 1020 cm−2 ) and high energies (>10 keV). • Bombardment of highly-energetic ions seems to favor or to induce rough surfaces, sometimes generating a uniform texture. Both the mechanisms have in common: • Reflection of high-energetic ions by the surfaces of the cone. • Redeposition of sputtered material. • Durability of shape once a cone or a pyramid has been formed. • Angular dependent sputtering effects. • Pits out of which pyramids and cones will grow; by increased erosion of primary ions and sputtered particles, these pits will develop. Cones and pyramids mainly differ by a crystallographic effect: The facets of a pyramid are connected with the structure of the substrate by means of the facets of the pit. The various states of surface modification have been described in a simple model by Auciello [491] (Fig. 10.16): • Due to the angular dependence of the sputtering yield, S(ϑ), convex shaped structures will be formed preferentially until facets occur which exhibit an inclination angle of ϑ: A pyramid or a cone are generated. By means of a Monte-Carlo simulation, Rossnagel showed that convex shaped structures can give birth not only to cones but also to pyramids [497].

400


• Scattered ions of the primary beam and neutrals which have been sputtered from the pyramids dig a trench around the pyramid which enhances its spatial definition in the first step [498]. • However, exactly this feature causes erosion that is more intense than in the neighborhood, and the pyramid is subject to rapid planarization, leaving a pit as remnant. Three mechanisms have evolved which cause the formation of a microstructure out of an originally plane surface [482]. • Erosion by physical sputtering, which is connected with the presence of defects which are either original or caused by sputtering. • Contaminating particles on the surface. • Deposited or released contaminating atoms can cause the formation of small crystallites by radiation-induced migration. The secondary surface which has been generated by contamination-induced processes can be subsumed as a special case of the roughness-induced mechanism: The different atoms must segregate at uneven sites, inclusions, or grain boundaries etc. which have been caused by ion bombardment, thereby causing the formation of cones and pyramids. Otherwise, the structure will be planarized [499]. Before the process is started, the degree of smoothness has to be measured and this value has to be taken into account for nucleation and pyramid formation. Out of surfaces with a smooth surface, only few pyramids will grow, but even on an amorphous surface, a secondary structure will evolve [485]. For sufficiently high ion dose,, pyramids disappear to leave pits behind. This is described by Carter’s theory: The contour of the surface is altered following the form of a propagating wave [485] − [487] (Figs. 10.16 and 10.17).

10.4 Sputtering conditions Since the gas ions should react neither with the target nor with the atoms which are about to constitute the depositing layer, inert gases are applied as main component, and among them, argon. For special purposes, sometimes also krypton and xenon are in use. The borderline at low pressures is drawn by the number of ionizations by collisional impact, which scales with number density and the discharge pressure. Below a pressure of 5 mTorr or about 0.5 Pa, the sputtering rates drop significantly, most pronounced in DC discharges. The upper limit is determined by the reduced substrate bias at the same power input. Since the sputtered atoms are subject to more intense scattering on their course between target and substrate, and the sputtering rate is the result of ejection (at the

10.4 Sputtering conditions

401

150

60

copper target ER [nm/min]

ER [nm/min]

titanium target

100 Au

40

20

Ti Si

Cu

50 0

5

10

15 20 pAr [mTorr]

25

30

0 0

5

10 15 pAr [mTorr]

20

25

Fig. 10.20. Etchrates (ER) of substrates consisting of layers of gold and copper, both mounted on a copper target (LHS) and of layers of titanium and silicon, both mounted on a target of titanium (RHS), as function of argon pressure at constant ion flux. The drop in the etchrate, a result of backdiffusion induced by scattering of argon atoms, is significant. This effect is overshadowed by a second one in the case of the titanium target. By backdiffusion and redeposition of silicon atoms atop the titanium target, compounds are formed which exhibit a sputtering rate which is considerably less than that of silicon. Hence, the selectivitity between silicon and titanium is inverted c Philips). by varying the pressure (after [500]

target) and transport (across the gap), also this fact contributes to a decline of the sputtering rate with rising pressure. Therefore, the drop in the sputtering rate for copper and gold by a factor of 3 comes about because the pressure is varied by a factor of 4 [from 7.5 Torr (1 Pa) to 30 mTorr (4 Pa)]; for both these coinage metals, the properties pressure and sputtering yield almost scale inversely (Fig. 10.20.1). This phenomenon becomes an increasingly serious problem if the material which is just sputtered away will diffuse back and will condense atop the substrate, however, forming a compound which exhibits a lower sputtering rate than the initial component (Fig. 10.20.2). At 100 mTorr (15 Pa), the mean free path for nitrogen is in the order of several millimeters, and this marks the upper limit for efficient diode sputtering. The lower threshold is given by sustaining requirements and should not fall short of 10 mTorr (1.5 Pa, Fig. 10.21). This is further confirmed by the analogous Figs. 6.27 + 6.28. The significant reduction of the sputtering rate beyond 100 mTorr is well modeled by “first principles” calculations.

402


deposition rate [nm/min]

50 40

Fig. 10.21. The pressure range for effective RF diode sputtering is set between 10 and 100 mTorr (1.5 and 15 Pa) [501].

30 20 10 0

0

20

40 60 p [mTorr]

80

100

10.4.1 Electrical properties The lower limit is simply given by stability requirements and can be set to several tens of electronvolts. The upper limit of the electrode voltage is set not only by safety requirements (significant generation of X-rays beyond energies of approximately 10 kV) but also by a decline of sputtering yield due to the rising competition of implantation. At too high a degree of ionization, however, the sputtered neutrals themselves were the subject of ionization and would be accelerated back to the electrode target. But within this energy range, the I(V) characteristic can be continuously varied by sweeping the discharge pressure. Therefore, the actual voltage range is set between 500 and 5 000 V. Since the power loss in the matching network can easily exceed several ten percent, the voltage is often measured directly at the electrodes. Applying a low-pass filter, the DC component can be measured; the HF component is made visible by an oscilloscope after attenuation (Fig. 10.1).

10.4.2 Temperature control of the substrate As we have seen in Sect. 10.2.3, the process of film formation sensitively depends on the temperature of the substrate. Already in the simplest kinetic theory, the rate constant of a reaction depends exponentially on the inverted temperature: ln k = A − Eact /kB T, the Arrhenius equation. This is valid in principle not only for recrystallization reactions during sputtering but also for etch reactions during plasma etching. Even the anisotropy of the etching process can decisively be influenced by the temperature of the substrate [502] (Figs. 11.14). The problem to control the temperature of the substrate is at least threefold: Mechanical considerations meet those of vacuum technology and the methods


403

of temperature measurement itself. Due to the different technological state of the substrate holders, various paths can be followed. 10.4.2.1 Temperature measurement. Hussla et al. made use of the socalled fluoroptical thermometry [503]: A small tablet made of europium lanthanium oxysulfide is mounted on a glass fiber which is subsequently fixed to the sample. The oxysulfide is excited in the UV range, and the fluorescence signal which strongly depends on temperature, is transmitted through the very same glass fiber and processed in a UV/VIS spectrometer [504]. This method allows a temperature measurement in-situ of the wafer surface but is invasive in nature with all the problems involved. Best of all is an indicator which is built in the sample. As has been shown by Mitchell and Gottscho in the case of III/V-semiconductors, the recording of the photoluminescence signal (PL signal) provides exact information of the temperature in the sample [505]. The photoluminescence is excited by a pulsed laser with locked detection by which the background of the glowing plasma is considerably reduced; furthermore, the temperature stress due to the measurement is reduced. Especially in the temperature range between −100 and 200 ◦ C, this method is superior to pyrometric methods, which display good performance beyond 300 ◦ C. Using pyrometry, the absolute temperature can be measured in principle provided a blackbody radiation behavior is followed. Calibration is required at least at one point, but better are two or three points [melting points of lead (327.3 ◦ C), tin (505.1 ◦ C), and aluminum (660.1 ◦ C)]. Unfortunately, the sensitivity is extremely low especially in the range of interest where photoresist degrades (temperatures must kept below 180 ◦ C even for somewhat robust resists)—and chemical reactions with an activation energy of about 1/2 eV double their rate every 10 K. It was in the late 1990s when Donnelly et al. revitalized a technique introduced by Hacman in 1968 and which is based on (infrared) laser interferometry [506] − [508]. In principle, a laser spot is directed at a wafer which must be polished on both sides to ensure the reflection of large portions of incoming light either at the front surface or at the back surface. This condition restricts the operating frequency of the laser to f < Egap . The interference pattern of the reflected beams is caused by their different optical paths, which depends for the beam reflected at the back surface on wafer thickness d and refraction index n. The latter is a complicated function on the real part of the dielectric constant εr , the DC conductivity σ and the frequency f of the operating laser [cf. Eqs. (14.160)]. Since εr and σ both sensitively depend on temperature, we have a means to measure this important parameter provided the functions ε(T ) and σ(T ) are known. Along with the (smaller) correction for thermal expansion, the interference pattern can be correlated with temperature. As Donnelly et al. have

404


found, for typical semiconductor wafers 500 μm in thickness (Si, GaAs, InP), a full interference cycle λ/2n is reached for a rise in temperature of only 3 K. Some obstacles have to be thoroughly circumvented: • The frequency of the laser should be carefully chosen since in GaAs, very strong absorption bands happen to be adjacent to Egap . • Albeit for semiconductors with their positive temperature coefficient with respect to electric conductivity, dσ/dT , an almost similar dependence follows for the temperature dependence of the charge density ρ = e0 n, the rise in εi is relatively small [509], a considerable high-energy shift of the absorption edge has to be taken into account which leads to a decline of dρ/dT , especially for n-type semiconductors. 10.4.2.2 Temperature control. The substrates are placed on top of a substrate platen which is transported from the load-lock to the reactor chamber to be mounted on the substrate holder (electrode). For certain standardized substrates, for example wafers, we can get rid off this platen, and the wafer is fixed by clamping jaws and will be directly mounted on top of the substrate holder, thereby saving one point of contact between substrate holder and substrate platen. This device is perforated by a system of pipes through which can flow a thermostating liquid or gas. The thermostating fluid (glycerine, glycol, iso-propanole) is chosen either to realize lower temperatures (down to −50 ◦ C) or high temperatures (up to 200 ◦ C). Even higher temperatures are achieved using small rapid thermal annealing systems. In the simplest model, both the surfaces which are in mutual contact are supposed to be flat and smooth. In fact, however, they always exhibit a microscopical roughness which is large as against molecular dimensions, and the actual contact takes place only between the peaks of these surfaces which is termed a three point contact. As a consequence, heat transfer is massively disturbed by the gap between the substrate and its holder. Furthermore, the substrate has a past which can be excellently observed when wafers are inspected: The various technological steps, in particular coatings with metals or dielectric layers lead to a severe mechanical tension which gives rise to bending of the wafer by which the contacting peaks between wafer and plate are further reduced. To cap it all, most plasma processes take place at pressures below 1 Torr (130 Pa). Here, the heat conductivity becomes pressure dependent and severely deteriorates. To enhance the thermal conductance between wafer and substrate platen, the mechanical contact between them has to be improved. First amateurish attempts involved fixation by an adhesive tape or a spin-coated wax. By employing this method one got an idea of what was happening, but the substrate is prone to contamination even after a thorough cleaning procedure. As investigations of Egerton et al. have shown, one of the most reliable methods is gluing with


405

guidance silver which remains reserved, however, for scientific and development purposes [510] (Fig. 10.22).

200

200

three point contact elastomer helium flow, water helium flow, kryo

150

150 100 T [°C]

100 T [°C]

three point contact elastomer helium flow, water helium flow, kryo

50

50 0

0

-50

-50 0

15

30 t [min]

45

60

0

15

30 t [min]

45

60

Fig. 10.22. Depending on the quality of the heat transfer between wafer and wafer platen, the temperature of a substrate can show very different values [4"-wafer (Si)]; c Oxford Instruments 1993). LHS: medium, RHS: low power density

This setback of three point contact can be effectively tackled by direct and extensive coupling of the substrate’s backside by a gas. For efficient cooling (by heat conductance or convection), the gas should exhibit a high coefficient of heat conductivity. Due to its inertness, helium is employed most frequently. Typically, the pressure must be kept within the range between 1 and 10 Torr (133 to 1330 Pa) which is large as against the operating pressure. Since an upward force is exerted on the wafer, it has to be mounted reliably, which is accomplished by fastening with mechanical clamps. In the first versions, the wafer was lifted against an O-ring to avoid escape of the gas. The coolant gas would be introduced subsequently into the small gap (50 − 300 μm in depth), thereby avoiding a complicated system of tubes. In the majority of modern versions, the O-ring is got rid of, and the wasted helium is introduced into the reactor. Due to its high ionization potential, the characteristics of the discharge is changed but slightly (Fig. 10.23). The clamps have the advantage of simple mounting but are an inevitable source of contamination and particles, especially in deposition processes where not only the wafer is subject to the coating but also all other parts which are exposed to the process. Each time when the clamps are lifted to release the wafer, the coat will be shattered and particles are spread over the wafer edge. Moreover, the clamps mechanically support the wafer edge only which can cause wafer bowing. For a perfect process, the mechanical stability of the substrate is taken for granted, but evidently, in a lot of cases, it is not. For example, wafers

406

10 Plasma deposition processes clamping ring wafer pins

electrode

clamping ring wafer

gap O-ring

electrode

pins

gap O-ring

(optional)

helium pipe

(optional)

helium pipe

Fig. 10.23. The wafer is mounted on the electrode by means of a narrow annular support and is kept in place by a clamping ring which is vertically adjustable. Helium flows through the gap between electrode and wafer which ensures very effective cooling. The temperature is controlled by means of a thermocouple fixed in the electrode or directly at the backside of the wafer.

of Si are second to none, especially superior to GaAs, but InP being the most brittle of all, causing wafer fractures. Because this method evidently causes problems by wear and generation of particles, the next generation employs an electrostatic system which seems to be superior and is now in widespread use. Interest in this system is also fueled by other applications since more and more processes take place on thinned wafers. Several systems are in use, but the simplest system, which explains the principle, resembles a capacitor with two plates whose gap is filled by a high resistance dielectric. Whereas the substrate (wafer) is the upper plate, the lower plate is the cooled substrate holder or electrode which is charged to a high voltage (Fig. 10.24.1). Appyling a voltage to the electrode causes charge separation (polarization) in the counterelectrode (i. e. the backside of the wafer) intermediated by the dielectric, but clamping is not effective until the circuit is completed, which requires ignition of a plasma which in fact establishes the counter-electrode. This setup caused handling problems, and therefore, a practical system uses several electrodes. Each of them is charged to a different voltage causing effective charge separation without the presence of a plasma (Fig. 10.24.2). To avoid the generation of permanent space charges, low-frequency pole reversal is applied. The adhesion forces are not restricted to the small wafer edge any more, but the wafer is subjected to the electrostatic force across its whole area which improves the heat contact and dramatically the risk of wafer fracture according to F = 1/2 εE 2 A = 1/2 εU 2 /d2 A with U typically between 200 V and 2 kV. However, the perfect adhesion deteriorates the possibilities of flow. If the coolant gas were introduced only from one orifice it could not effectively be dispensed

10.4 Sputtering conditions wafer _ _ _ _ _ _ _ _ + + + + + + + + electrode (unipolar)

407 wafer _ _ _ _ _ + + + + + +

_ _ _

electrode (bipolar)

Fig. 10.24. Principle of electrostatic clamping: A capacitor is formed by the wafer (upper electrode) and substrate holder (lower electrode), separated by a high resistance dielectric. By application of a high voltage at the substrate holder (electrode), an areal force is subjected to the wafer which reduces the hazard of wafer fracture. Helium is introduced into the gap to conduct away the generated heat. The clamping is improved by employing a divided electrode, each part charged to a different voltage (after Oxford Plasma Technology 2007).

below the wafer. Hence it is necessary to carve out a system of trenches into the dielectric which effectively carries the wafer, and now, helium is introduced into the dielectric to conduct away the generated heat. In contrast to mechanical clamping, the waste helium will afterwards stream out into the processing chamber.

10.4.3 Contamination 10.4.3.1 Target and purity requirements. Dependent on its application, various purities of the target are sufficient or necessary. Highest purity is demanded by processes in the technology of semiconductors; background contaminations even in the ppm range can deteriorate the quality of devices, their yield and long-term stability. Whereas it is relatively simple to fabricate metals and alloys of high purity, this can become a serious problem for non-mixable and refractory metals but also for substances which easily sublimate. In these cases, the methods of powder metallurgy are applied: The materials are carefully mixed and subsequently compressed applying high pressures and high temperatures (isostatic pressing) and finally sintered. Protecting layers do not require these standards. They serve to protect layers and materials underneath (for example coating of CDs and DVDs with aluminum) or they simply meet decorative purposes [coating of organic polymers with highly reflecting metals (chromium) for radiator grills, bumpers etc.], which has been accomplished in former times by the wet-chemical treatment of the polymer. Mostly, ABS, a copolymerisate consisting of acryl nitrile, butadiene, styrene has been exposed to chromium sulfuric acid to roughen the surface and to generate electrically active sites which has been subsequently galvanized; a process with diverse pollutants with serious environmental issues.

408


To meet the requirements for the coating of optical surfaces, e. g. to reach a certain degree of extinction across a broader optical band, the purity of the target must be chosen between these two borderline cases. 10.4.3.2 Contamination by argon. During the sputtering process, the substrate is subject to a continuous flux of particles which come not only from the target (mainly atoms or small molecular units) but are constituents of the process gas (mainly argon with its dopants). The ratio of the fluxes amounts to about 104 in the case of Ar/Al at 20 mTorr (3 Pa). More than forty years ago, Winters and Kay suspected incorporation of processing gases into sputtered films would affect not only primary properties such as nucleation and growth but would also influence derived properties, e. g. resistivity and magnetic properties [511]. Therefore, it would be not very surprising to detect imbedded argon in the films. To confirm this assumption, nickel films which were deposited by DC sputtering were vaporized by a resistance furnace or by laser-induced flash and subsequently analyzed by mass spectrometry. Whereas direct bombardment of the growing layer leads to an increased incorporation with rising bias which peaks at 7 % for −500 V VDC (Fig. 10.25), the ratio Ar/Ni remained below 1 % without bias. This value even dropped with growing pressure.

Fig. 10.25. If the growing film is subjected to the bombardment of Ar+ ions, the argon content can reach values up to 7 % [511]. The very low content for values below 200 V is attributed to favored sputtering of previously physisorbed argon.

Ar/Ni [at %]

1

0.1

0.01 0

100

200 300 VDC [-V]

400

500

Winters et al. presented the following model [512, 513]. Raising the deposition temperature causes the argon content to drop because argon is exclusively bound by physisorptive mechanisms which exhibit an energy of formation of less than about 0.5 eV. But raising the discharge pressure also leads to a decreasing content of argon. This is due to a reduced implantation of rapid argon neutrals into the topmost layers of the target. After being discharged, they are bounced back from the target and are consequently unaffected by electric forces, which facilitates their incorporation in the film which just forms on the substrate. Since scattering by collisional impact is more likely at higher pressures, the incorporation declines with rising pressure (Fig. 10.26).


409

argon content [atomic %]

1.00 before annealing after annealing

0.75

Fig. 10.26. Argon content in layers deposited by sputtering in argon as a function of discharge pressure as deposited and after annealing for 60 min at 900◦ C in argon [514].

0.50

0.25

0.00 0

20

40 60 p [mTorr]

80

100

10.4.3.3 Contamination by other gases. Almost all metals form stable oxides with very high lattice energies, and this is why the most sensitive case is sputtering of metals. For example, at medium power, a single atomic layer of tantalum is deposited in five seconds, which means a layer thickness of about 50 ˚ A in one minute. At a typical argon pressure of 20 mTorr (3 Pa), the resistivity already rises beyond a partial pressure of oxygen of 10 μPa (ratio 50 ppm, Fig. 10.27). If the oxygen comes from the system (prominent water peak in the mass spectrum), this can be overcome by heating, increasing the pumping speed or other methods. If is is a gaseous component of the sputter gas, however, every further efforts are futile: The sputtering gas must exhibit a purity of at least 99.9995 %.

r [mW cm]

1000

100

10-7

10-6 1x10-5 1x10-4 partial pressure O2 [Torr]

10-3

Fig. 10.27. Double logarithmic plot of the electric resistivity of layers of Ta/Si sputtered in argon vs. the partial pressure of oxygen [515].

For a perfect sputtering experiment, a surface is required which is free of contamination. That is why the particle density of the ion beam must exceed the “beam density” of the contaminating gases. That flux has been estimated by Yonts and Harrison to exceed 0.1 mA cm−2 at a partial pressure of 1 × 10−8

410


Torr to avoid any negative influence on the sputtering yield [516]. This can be easily achieved with today’s sputter powers (and correspondingly high currents). Problems can arise for RIE processes. 10.4.3.4 Reactive sputtering. On the other hand, this very fact can be made use of to dope a coating by adding a certain reactive gas to the sputter gas (usually argon). This is denoted reactive sputtering. The whole palette from the pure metal to the pure salt (insulator) can be covered by this method. The flows of the dopants are separately adjusted by needle valves or mass flow controllers under the assumption that the pumping speed of the pump is not changed when the total flow into the chamber is only slightly different [517]; applying optical spectrometers or mass spectrometers, the reactive dissociation products of these gases are traced. Of high interest are the conditions at the target which will react with the added gas. However, the degree of interaction is still unclear. Several authors have proposed chemisorption, i. e. a weak chemical reaction with the topmost target atoms. As shown by de Gryse et al., ions of the reactive gas are implanted as a first step [518] exhibiting a certain depth-dependent concentration maximum. After having reached a certain threshold concentration, they are consumed by chemical reactions, forming compounds whose bond strength is higher than that between the host lattice atoms. Now, the sputter rate will start declining, which, in turn, enhances the concentration of the implanted species (avalanche), and the mechanism gradually changes from metallic sputtering to reactive mode sputtering. During this course, uncontrolled flashovers or arcing can happen provided the charge in this layer exceeds its dielectric strength. During this punctual arc discharge, all the material adjacent to the flare is subject to evaporation. Furthermore, a DC discharge will eventually be terminated, since both the electrodes, the anode and the cathode, are subject to chemical attack by the reactive gas, which can lead to coverage with an isolating film. This effect is referred to as “disappearing anode” because this electrode will disappear from the electrons [519], and the process itself is denoted poisoning [518]. One solution to this problem is dual magnetron sputtering (cf. Sect. 10.7), another is pulsed reactive sputtering at duty cycles up to 350 kHz. By this process, the target is discharged, which prevents arcing. Despite these difficulties, numerous stable processes have been established in the last decade. We mention here the fabrication of four coatings for very different applications: • Transparent and electrically conducting films [the gap energy of these materials is shifted into the UV range; for a carrier density in the range between 1020 and 1021 cm−3 , the plasma frequency is still in the IR, so they become transparent for visible light (Sect. 3.6)]; very popular are targets made of indium/tin by which ITO layers [indium oxide (In2 O3 ) and tin oxide (SnO2 , doping level typically 5 − 6 %)] are fabricated [520].

10.5 Sputtering with bias techniques

411

• λ/4-layers to coat optical surfaces. • Blue electrochromic films made of WOx (so-called “tungsten blue”) by reactive sputtering of tungsten in an ambient of argon and oxygen [521], whereas also tungsten bronze with metallic conductivities is accessible at very low dopant levels of oxygen. • Thin isolating layers of SiO2 and Si3 N4 which exhibit perfect adhesion to the substrate and exhibit very good performance even at high temperatures. What are the requirements that have to be met by a λ/4-layer? To realize an amplification of 25 dB in the case of an optical semiconductor amplifier, the residual reflection R must fall short of 10−4 . Simultaneously, the deviations of the index of refraction must not exceed Δn = ±0.009; that means for the thickness a deviation of Δd = ±1.2 ˚ A, i. e. one atomic layer [522, 523]. The main setback for reactive sputtering is the relatively low deposition rate which is even not counterbalanced by the utmost attainable purity of the deposited films. Therefore, this method has been almost entirely replaced by (PE)CVD (cf. Sect. 10.8). 10.4.3.5 Bombardment with other particles. The substrate is also subject to the bombardment of positive ions. The discusssion is deferred to Sect. 10.5. Their flux is considerably lower than the electron flux. The largest electron fraction has its origin in the plasma bulk; their mean energy is considerably lower than 10 eV and is given by kB Te . Rapid electrons are generated by γprocesses in the target and are subsequently accelerated by the sheath field of the target. Since at these energies, the cross section for collisional impact steeply declines, these electrons will reach the substrate almost unaffected, and, although small in number, they are, almost alone, responsible for the power which is transferred and dissipated into the substrate [156, 524]; this has been proven by the retarded potential technique (in principle, a triode which measures the V-I characteristic or electron currents with varying energy) [525]. Photons are generated by collisional impact of ions or electrons with a surface. Since the voltages in a sputtering system do not exceed 5 kV, the most energetic radiation is in the band of soft X-rays. By collisional impact, they can release electrons which is made use of for analytic purposes (ESCA).

10.5 Sputtering with bias techniques An additional bias voltage at the substrate can also influence charged particles. Especially in discharges through electronegative gases (e. g. oxygen or iodine), a positively biased electrode exerts a force on the negative ions which subsequently

412


− react at the surface of the substrate.8 With ions of oxygen (mainly O− 2 , O ), a plasma oxidation is carried out [526], which is an interesting alternative to other selective oxidation processes (e. g. anodic oxidation [527]). Depositing an isolating film automatically causes a certain bias potential which is difficult to control. In order to make this bias voltage an adjustable parameter, RF diode sputter reactors are equipped with a power splitter.

10.5.1 Deposition rate and film composition Is the isolated substrate subjected to a negative potential Vbias the plasma potential Vp will remain unaffected in the approximation of first order [528]. The electrical conducting walls of the sputtering system are still grounded; the potential of the sheath, referred to the substrate is given by Vs = Φp − Φbias (because of the difficulties measuring Φp , in most cases the bias potential Φbias is measured with respect to ground); positive ions of the plasma bulk are accelerated towards the substrate which becomes a second target. In fact, it is the topmost layer which is just growing. Since the layer is still amorphous and the atoms have not reached their equilibrium positions, relatively low values of the substrate bias already reduce the layer growth strongly which indicates that sputtering of amorphous layers is determined by other mechanisms than those aforementioned for the sputtering of a crystalline layer (Fig. 10.28.1). In fact, it 70 tantalum content [atomic %]


45 40 35 30 25 20 15

0

50

100 150 200 substrate bias [-V]

250

60 50 40 30 20

0

75

150

225

300

substrate bias [-V]

Fig. 10.28. The deposition rate and composition of the film are strongly affected by the applied substrate bias, especially in the range beyond 100 V (backsputtering) [529].

was found that the composition of a growing film is strongly affected by biasing the substrate (Fig. 10.28.2), and it is exactly the interval of strongest change in composition where the deposition rate sharply goes down (Fig. 10.29). 8 Even in this case, Φp is more positive than the sum of the sheath potential and the bias potential Φs + Φbias , and electrons and negative ions must drift against a retarding potential.


413


40

35

30

25

20

30

40 50 Ta content [atomic %]

60

Fig. 10.29. Crossplot of Figs. 10.27: The deposition rate suddenly declines sharply in the range where the layer composition is subject to change due to the backsputtering effect [530].

10.5.2 Further film properties Almost all properties, including resistivity, hardness and purity, can be improved by the application of a bias voltage very often. In particular, the gas incorporation (mainly argon) can be controlled more efficiently. At bias voltages of absolutely less than 100 V, this incorporation often shows a minimum: • Decline to minimum: The ions which are incident on the substrate have gained enough energy to remove the loosely-bound argon atoms. • Rise beyond minimum: The kinetic energy of the argon ions is high enough to be implanted into the growing film. At these energies (>100 eV), the film itself becomes subject to violent sputtering attacks, and its composition can be influenced (Figs. 10.25 and 10.28). 10.5.3 Mechanisms of bias sputtering To start with, we must distinguish between several bond types between the metal and the dopant [511]. 1. There are metals which exhibit almost no affinity to any of the potential contaminants (gold, but not platinum). 2. Another group can absorb contaminants not only by chemisorption but also by formation of a chemical bond (tungsten in relation to nitrogen). 3. A third group only forms chemical bonds (nickel in the case of nitrogen). In the first case, substrate biasing will facilitate the incorporation of energetic ions which are not constituents of the film. This effect will be often overshadowed in the other two cases by backsputtering effects of atoms which are already

414


incorporated but exhibit lower energies of formation than the atoms which constitute the film [eq. (10.1)]. For dopant atoms, however, which spontaneously form a chemical bond with an atom of the host lattice which exceeds the lattice energy of the host lattice considerably, significant removal by biasing cannot be expected; in fact, most of the sensitive properties (electrical resistivity) often deteriorate severely. Removal of atoms which constitute the film but happen to be located at other sites than a “normal” crystal site can enhance the density; this often explains the improved conductivity of gold layers which are sputtered with a supporting bias voltage. As a second benefit, the threshold for electron bombardment is raised which is mainly responsible for the heating of the sample. And higher temperatures favor recrystallization processes: Increased grain growth leads to a rough, tarnished surface which is often unwelcome due to decorative reasons. During cosputtering of Nb/Cr layers, Mawella and Sheward observed columnar structures with separating voids which will already disappear at bias voltages of no more than −50 V [531]. By rising the bias to about −100 V, the columns are replaced by a densely-packed, micro-crystalline structure; and even higher bias voltages lead to striations which are not characterized by material contrast, i. e. varying local composition, but are rather attributed to locally different sputtering behavior.

10.5.4 Homogenity of coating at rectangular steps Applying a substrate bias voltage, the coating of acute angles and steps can be further improved. From first principles, this feature is one of the most significant distinctions from evaporating methods, since the beam of evaporated atoms remains its character at pressures which are lower by three or more orders of magnitude compared to sputtering methods. This does not allow conformal coating, a layer homogeneous in thickness irrespective of whether the angle between beam and surface is zero or 90◦ . For the depositing process, the overwhelming fraction of layer constituents consists of almost or completely thermalized neutrals which are unaffected by electric forces. This advantage over evaporation can be further consolidated by biasing the substrate. Material which has been deposited perpendicularly to normal incidence is then removed with more ease than material deposited at the sidewalls (Fig. 10.30). The removed atoms will deposit on these sidewalls to a large extent thereby evening out the anisotropic coating behavior. This equalizing in thickness is further facilitated by the conduct of the sputtered atoms, which leave the surface in an approximate cosine distribution rather than in directions normal to it, as is the case at higher energies [532].


100 mm

100 mm 200 mm 400 mm

415

Fig. 10.30. Biasing the substrate will lead to rounded shapes due to enhanced ionic attack at positions with higher field intensity.

Directly at the corners and angles, the deposited layers must become rounded off by biasing because the sharp angles develop more intense fields, which cause higher ion bombardment and preferred removal.

10.5.5 Mechanical tension and substrate bias Layers deposited by evaporation or sputtering techniques both show a dilatory or tensile stress, whereas layers deposited by galvanic methods generate a compressive stress [533] (Sect. 10.2.3). This mechanical property can easily be affected by additional biasing. As prototype, it was shown by Mansour et al. that in sputtered layers of Si3 N4 , the mechanical tension can be influenced over a wide range [534]. A layer free of tension can be deposited at very low values at the substrate bias (≤25 V); although the deposition rate is large, the surface exhibits a smooth character with a high degree of hardness; with rising bias, the values go negative (tensile stress) to show a second zero crossover of the stress-bias function beyond −300 V at exceedingly low deposition rates (cf. Fig. 10.28.1). These observations agree with the investigations of Hoffman and Gaerttner, who measured a dramatic reduction of the mechanical tension of evaporated chromium layers when it was subject to bombardment with highlyenergetic ions of Xe+ [535]. Cuomo et al. reduced the tensile stress of evaporated films of niob to almost zero by low-energy bombardment of ions of Ar+ [536]. By biasing the substrate, the change in layer composition intensely influences the inner tension of the layer. In most cases, the stress is tensile in nature and will be bent concavely on the coated side. This is attributed to a reduction in columnar growth and subsequent loose packing between the columns [534, 537]. The function stress vs. bias shows a corresponding behavior with a zero crossover at relatively low bias values. Knoll et al. have found it at about

416


−50 V for ZrO2 /Y2 O3 layers, Mansour et al. detected the zero crossover at a bias of about −25 V for layers of Si3 N4 . In the end, layers deposited by substrate bias are very densely packed. Self diffusion energy and melting point are connected via the Flynn formula Esd = const Tm ,

(10.13)

with Esd the self diffusion energy and Tm the melting temperature [538]: The degree of amorphization scales with the melting temperature. Sputtered layers of aluminum are microcrystalline, but considering the amorphous state of asdeposited layers of refractory metals for “normal” RF-diode sputtering with a relatively low packing density, the progress can be attributed to the removal of atoms which have occupied a “wrong” site (lattice point or lattice defect). In conclusion, the method of substrate biasing opens a wide process window to influence the composition, crystallinity and topography of the growing layer. In principle, the deposited layers must become smoother by biasing because in rough surfaces with peaks and valleys, the peaks develop more intense fields which cause higher ion bombardment and preferred removal. The method of biasing can be regarded as in-situ healing by ionic bombardment [539]. It should be kept in mind that the substrate platen or the liner is subject to ion bombardment. As result, the whole reactor, but in particular the target and the deposited layer, will be contaminated by the constituents of the substrate platen.

10.6 Sputter deposition of multicomponent films In principle, films consisting of more than one component can be obtained by the following methods: • Sputtering from a single target (sintered or mosaic target) [540, 541]. • Sputtering from several single targets which is denoted cosputtering: The substrate is mounted on top of a carousel and is rotating below the targets; to achieve a homogenous layer in thickness and composition, complicated movement (planetary drive) is necessary [542]. • Reactive sputtering, at least one more gas is let into the reactor. • Furthermore, these methods can be combined. Since the composition of the layer depends on the composition of the target(s), the sputtering rates of the components (depends on power and pressure), and the etchrates (for bias sputtering), and the cross section for momentum transfer σm (depends on energy and pressure), and the coefficient for condensation, it can deviate considerably from the composition of the target. How vapor pressure,

10.6 Sputter deposition of multicomponent films

417

sputtering yield, atomic weight and the coefficient for condensation influence the composition of the layer has been the subject of a systematic treatise by Winters et al. [512]. They showed that the calculated compositions of the layers deposited by sputtering were in good (qualitative) agreement with experiment. 10.6.1 Target processes Sputtering from a metallic target begins by removal of the native oxide. In the case of DC operation, the discharge current drops when the metal is uncovered due to the difference in the yield of γ-electrons (cf. Sect. 2.6). In the case of a compound target, we have to consider different sputtering rates. The metal with the higher sputtering yield will become enriched in the film, but will become depleted at the target surface. Its sputtering rate declines until the initial target composition is reached again. Starting from the stoichiometry AB with x the sputtering yield of component A and y the sputtering yield of component B, the initial stoichiometry at the surface will be shifted to Ay Bx , and eventually material is sputtered from the compound target with the composition AB, generalized xAy × yBx . As an example, if the sputtering yield of A doubles the sputtering yield of B in the target AB, we start with the stoichiometry AB2 . Albeit the concentration of B now doubles that of A, but A is sputtered away with the sputtering rate of B doubled, the final stoichiometry of the sputtering process will reach 2 A1/2 1/2 B2 = AB, and as result of this dynamic equilibrium, the sputtering yields of the different metals equal the stoichiometry of the target in the stationary state. The method of choice, however, is cosputtering with the simultaneous operation of at least two targets. Since sandwiched layers are obtained, all possible layer compositions can be tailored in a gradual or abrupt manner. Since this sandwiched composition is obtained in-situ, the interdiffusion of the very thin individual layers by a subsequent thermal process is an important issue in cosputtering techniques. But one of the most significant assets is the purity of those one-component targets, which exceeds the purity of sintered targets by orders of magnitude since the materials do not represent chemical compounds but only mixtures. 10.6.2 Between target and substrate During the flight towards the substrate, the cross section for momentum transfer σm between the heavy plasma constituents is of importance. σm determines not only the amount of dissipated energy but also the angle distribution of the neutral layer constituents (in most cases atoms). The impact of the scattering processes can influence the layer composition. A nice example for the pressure dependence of the coating rate which is synonymous for the film composition is shown in Fig. 10.31. The collisional impact between argon and the light silicon

418


atoms leads to considerable scattering by which the content of silicon in the layer is reduced. On the other hand, tantalum atoms with the same kinetic energy are scattered to a significantly less extent.

Ta content [atomic %]

50

45

40

35

30 0

25

50 75 p [mTorr]

100

Fig. 10.31. Composition of Ta/Si films as a function of discharge pressure [543].

10.6.3 Substrate processes Furthermore, the layer growth depends on the number of striking atoms which will not evaporate again. This ratio is denoted condensation coefficient; and the layer composition will be subject to its elemental variation. By application of biasing techniques, a certain amount of the film can be eventually removed (Sec. 10.5). Since this selection process happens in-situ, no equalization mechanisms are possible, substrate biasing does change the layer composition. Several tens of volts are sufficient to vary the stoichiometry even in dielectric compounds with higher lattice energies. For example, the index of refraction very sensitively reacts on changes of the composition. It could be controlled in the range 1.53 ≤ n ≤ 1.58 in the case of ZnO [544]. 10.6.4 Preparative aspects Applying sputtering methods, single-phase multi-element layers are accessible which are thermodynamically instable because they crystallize in several phases out of the melt of that very composition. For example, amorphous films of lanthanoid alloys have been deposited by sputtering from a multi-phase target (Gd/Co and Gd/Co with Au or Mo) which contained at least one component far beyond its solubility limit [545].9 Sputtering of dielectric compounds such 9 In the course of these investigations, magneto-optic memories were subject to characterization. Gd is a very interesting candidate since it shows seven unpaired 4f-electrons and the magnetization scales approximately to the product of unpaired spins and density. The

10.6 Sputter deposition of multicomponent films

419

as SiO2 or Si3 N4 leads to a depletion of oxygen or nitrogen in the deposited layer. This indicates the mechanism of sputtering as process which partly or completely atomizes the lattice structure. Considering that one of the most stable compounds, Al2 O3 , exhibits a lattice energy of a little less than 20 eV (−380 kcal/mol or 16.5 eV), this is small compared with the energy of the incident ions. Coburn et al. could offer evidence that the stoichiometric ratio metal oxide/oxide does not exceed the value of 0.4 even for the most stable oxides such as Ta2 O5 or TiO2 [547]. To compensate for the loss, the method of choice is reactive sputtering: Argon is doped with oxygen (or nitrogen in the case of the nitrides) to preserve the stoichiometry. On the other hand, this inherently deficient sputtering process offers numerous possibilities to study metal-nonmetal transitions which are caused by understoichiometric ratios [476, 548]. A striking phenomenon is the steep decline of the sputtering rate even when only small amounts of a reactive gas are added to the inert gas. This cannot be caused by an additional oxide formation, since the topmost surface of the target has been oxidized anyway (Sect. 10.4.3). Oxygen acts as electron trap rather and by this process, the efficiency of the ionization process is considerably reduced. Furthermore, oxygen is preferently oxidized by the Penning process − Ar∗ + O2 −→ O+ 2 + Ar + e ;

(10.14)

and compared with argon, its efficiency for sputtering is considerably reduced; and finally the density of metastable argon atoms will be reduced (which causes a reduction in the density of Ar+ ions, Sect. 2.5) [549]. The doping of oxygen in discharges through argon does change its visible characteristics; it appears that the total pressure would rise, which is indicated by the contraction and the sharp boundary of the sheath as well as by the reduction of the impedance of the discharge and the DC bias. This is also caused by the fact that the mean drift velocity of the negative carriers towards the electrode is now reduced due − to the increasing fraction of inert ions such as O− 2 and O . Considering the substrate (which is normally grounded), this causes the formation of a sheath with negative ions, which has not been subject to research yet—without the observation that in DC discharges the anode fall can increase considerably (Sect. 4.8) [550]. Jones et al. pointed out that the tear down of a crystalline target will happen layer by layer [551], i. e. after having removed a layer of silicon in a quartz target, the next layer which is subject to the ionic attack would be a layer of oxygen which, however, will be formed again immediately. That would really mean that no more than one atomic layer of silicon could ever be removed. following effect is made use of: For writing, the magnetization (vector orientated perpendicular to the storage medium, rotated by 90◦ with respect to the recording tape) is altered by increasing the temperature beyond the Curie point by means of a laser beam. For reading, a low-power laser made of AlGaAs/GaAs is used. The plane of the (polarized) laser beam is turned by some minutes, dependent on the magnetization (Kerr effect) [546].

420


In fact, they found saturation at a partial pressure of oxygen of about 5 μTorr (0.07 Pa). Compared to the sputtering yield without doping with oxygen, it has dropped by a factor of about 2. This would mean that the removed oxygen will be replaced by about 50 % from deeper strata.

10.7 Sputtering systems with increased plasma density 10.7.1 Magnetically improved sputtering systems Magnetically enhanced systems have been constructed to enhance the sputtering rate and to enlarge the usable pressure range. Since electrons are trapped more efficiently in the plasma bulk at the same plasma potential the electron bombardment of the substrate is reduced. The main feature of these systems is a device which is called magnetron. It is mounted directly on the backside of the target.10 A magnetic field which is orientated parallel to the surface of the target enhances the plasma density in front of it, and in turn, the sputtering rate will rise. In principle, the Lorentz force FL = Qv × B exerts a force on moving electric charges. Albeit only the electrons are forced on spirals since the ions remain nearly unaffected by magnetic fields of only 10 mT (mAr /me ≈ 70 000), the principle of electroneutrality has to be met, which leads to an enhancement of the ion density as well. 10.7.1.1 Theory. The fraction of electrons which possess a velocity component perpendicular to the magnetic field move in response to the Lorentz force, performing circular motions (centrifugal force = Lorentz force). According to v = ω × r (with v ⊥ B), we obtain the Larmor equation: 2 m e v⊥ e0 B me v⊥ ⇒ ωc,e = (10.15) = e0 Bv⊥ ⇒ r = r Be0 me with ωc,e the electron cyclotron frequency; the trajectory is for constant electric field a helix. By the way, at a plasma density of 1010 cm−3 , the plasma frequency equals the cyclotron frequency for a magnetic field of 330 Gauss (33 mT). The ratio of these two properties is [552] 2 ωp,e nmc2 material energy density = = . 2 ωc,e H2 2 × magnetic energy density

(10.16)

To increase the fraction of electrons which exhibit a large velocity component perpendicular to the magnetic field, the electric field (E v) is directed normal to the magnetic field. Using Eq. (10.15), we see that the radius of the helix will shrink with rising magnetic field. Electrons with an energy of 5 eV are constricted to helices with a diameter of about 3/4 cm. Therefore, the magnetic 10

This term was originally used for tubes which generate microwaves.

10.7 Sputtering systems with increased plasma density

421

field has an effect comparable to an increase in pressure, however, perpendicular to its direction (Fig. 10.32). The consequences for the discharge can be described as follows: • The sheath thickness will shrink, and since the electrons in the sheath exhibit beam-like character, this effect is very effective. • The discharge is constricted in the middle, which increases the intensity of the glow. • The I(V) characteristic is considerably changed: The electrons are forced to move on helical orbits which not only enlarges their course through the discharge but also causes the velocity component directed to the wall to vanish. Electrons which would otherwise get lost by wall reactions can act more effectively. As main technological result, magnetron-supported DC discharges can undercut RF discharges in the area of metal coating, and that is why they have remained an important element in production lines. The discussion of the equations of motions will be given in detail for a DC discharge. anode

target

S

N N

S

electron cycloides

S N race track

ions

S

Fig. 10.32. Mode of action of a planar magnetron. The LHS pictures the cross section of the field lines of the magnetic field. The electron density is raised in the target zone which, in turn, leads to a locally improved sputtering rate. RHS: The ions move in response to the electrostatic force to which they are subjected, which explains the c Cambridge University Press). development of race tracks (after [553]

To win the equation for electronic movement in the constant electric field of a cathodic sheath with E = const, we start from the following assumption that the static magnetic field B 0 is orientated parallel with respect to the z-axis, and the yz-plane is extended by B 0 and E. For a collisionless plasma, the equation of motion reads: e0 dv = − (E + v 0 × B 0 ) dt m

(10.17)

422


and in components dx2 dt2

= ωc,e dy dt

dy 2 dt2

=

dz 2 dt2

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

− ωc,e dx dt ⎪ ⎪ ⎪ ⎪ e0 ⎭ = me Ez . e0 E me y

(10.18)

From the third equation, we see that the charge is moving uniformly accelerated parallel with respect to the magnetic field: z=

e0 Ez 2 t + v0,z t. 2me

(10.19)

Combining the first and second equation along with the definition X = dx/dt + idy/dt, dX ie0 + iωX = Ey dt m

(10.20)

X = ae−iωt ,

(10.21)

yields

the general solution of the homogeneous equation. With e0 Ey , mω a specific solution of the inhomogeneous equation yields a=

(10.22)

e0 Ey . (10.23) mω Splitting into real and imaginary parts (e−iωt = cos ωt − i sin ωt) yields ae−iωt +

dx e0 Ey dy = a cos ωt + ∧ = −a sin ωt. (10.24) dt mω dt At t = 0, the velocity possesses only one x-component. The second integration yields with the initial conditions x = y = 0 at t = 0 : a e0 Ey a sin ωt + t ∧ y = (cos ωt − 1). (10.25) ω mω ω For a = −e0 Ey /mω, we obtain under consideration of ωc,e = e0 B/m a cycloid as projection of the trajectory into the xy-plane: x=

Ey Ey (ωt − 2 sin ωt) ∧ y = (1 − cos ωt). B B which is pictured in Fig. 10.33 as cycloid 1 along with x=

x=

Ey Ey (ωt − sin ωt) ∧ y = (1 − cos ωt), B B

(10.26.1)

(10.26.2)

10.7 Sputtering systems with increased plasma density

423

0.4

0.3

f(x)

cycloid 1 cycloid 2

Fig. 10.33. The electronic trajectories are cycloids of type 1 as calculated with Eq. (10.26.1). For comparison, the “introductory” cycloid following Eq. (10.26.2) is also shown as cycloid 2.

0.2

0.1 0.0

0

1

2 x

denoted cycloid 2, but with half the amplitude of cycloid 1. In fact, the sheath field shows a linear decrease rather than being constant, and we improve this approach to second order:

y dc

E = 2E 0 1 −

(10.27)

with dc the sheath thickness; and we fix the boundary at vanishing field E(dc ) = 0 and define a magnetic field which is orientated perpendicular with respect to the sheath field. The field at the surface of the target is given by

Vc =

Edy =

dc E0 y 0

dc

dy =

E0 dc , 2

(10.28)

with

E0 = 2

Vc 2Vc y ∧E = 1− . dc dc dc

(10.29)

The equations of motion can be written according to

d2 x e0 2Vc y dy d2 y = ωc,e = 1− ∧ 2 dt dt dt2 me dc dc

− ωc,e

dx . dt

(10.30)

With the boundary condition of vanishing dy/dt at the target surface (y = 0), we obtain for dx/dt: dx = ωc,e y, dt

(10.31)

and for the equation of motion with α = 2e0 Vc /(me d2c ): d2 y 2 + [α + ωc,e ]y = αdc . dt2

(10.32)

424


At the surface of the electrode, y and dy/dt will vanish, and y is given by the Lagrangeian method: y=

2Vc dc 2Vc 2 + me0e ωc,e dc

(1 − cos ωt)

(10.33)

with ω2 =

e0 E0 2 + ωc,e , me dc

(10.34)

again the equation of a cycloid which turns into the Larmor equation ωc,e = e0 B/me for E0 = 0. With this condition, the distance of an electron from the target is ymax =

4Vc dc 2Vc me 2 + ω 2 dc e0 c,e

(10.35)

at maximum. For a sheath thickness smaller than ymax , the course of the electrons is just insignificantly enlarged. Eventually for a critical value of the magnetic field, the curvature of the trajectory becomes so narrow that the electron will be trapped. Thereby, the magnetic field lets the sheath shrink considerably, and we find as limiting condition: ymax < dc ⇒ Vc
[eV]

H2 H 2/GeH4 9:1

1

0

25

50 75 RF power [W]

100

Fig. 10.37. Mean energy of the electrons in a discharge of GeH4 /H2 as function of absorbed RF power [p = 7.5 mTorr (1 Pa)] [567].

The main advantage over “simple” CVD reactors is the reduction in process temperature, which has enabled the deposition of homogeneous layers consisting of Si3 N4 or SiO2 with low inner tension and simultaneously high radial uniformity, which serve as passivation layers covering traces of aluminum on a printed circuit board (SiO2 is conventionally deposited out of a mixture of SiH4 and N2 O at temperatures between 400 and 450 ◦ C in low-pressure CVD reactors). But even more delicate organic polymers have come within reach. Since the monomeric species are prone to thermal decomposition, they have to be handled with great care. Otherwise, we observe the onset of volume polymerization instead of the intended polymerization on the surface of the substrate. Moreover, the plasma will enhance not only the density of reactive species in the gas phase but also acts on the surface. Active sites are created and unintended contaminants are removed. Hence, layers deposited by PECVD are often superior to CVD films. With increasing operating frequency, the tension of the deposited layer often changes its sign from compressive to tensile stress when the ionic plasma frequency ωp,i /2π is passed. Therefore, a small window can be opened for depositing layers nearly free of stress [570]. This indicates the influence of the energy of bombarding ions: Below ωp,i /2π, the energy of the ions which traverse the sheath can vary between zero and the maximum of the sheath potential; beyond this value, the distribution becomes sharper around the mean value VDC (Sect. 6.8). 10.8.1 Instantaneous mass spectrometry These coating plasmas can be instantaneously analyzed by differentially pumped mass spectrometers. The graphs of Fig. 10.38 are an impressive example of the numerous different species which are generated in a CCP discharge of SiH4 at a pressure of 75 mTorr (10 Pa) and which eventually form silicium clusters [571].

432


The operating frequency can be read from the modulation of the intensities of the MS spectrum (2.5 kHz). 500 2000 Si2H5 Si3H7

1000

0 0

200

400

600

800

1000

Si5H11

200 Si6H13

100 0

0

200

400

600

800

1000

250 Si20H26

Si40H52

200

15

150

10

100

5

50

200

400

600

800

1000

count rate dimer [cps]

count rate monomer [cps/1000]

25

0 0

300

t [msec]

t [msec]

20

Si4H9

400 count rate [cps/1000]

count rate [cps/1000]

SiH3

0

Fig. 10.38. Mass spectrometric analysis of a CCP discharge through SiH4 which is used to deposit polycrystalline silicon [571].

t [msec]

10.8.2 Diamond-like coatings (DLCs) Not only diamonds arouse strange feelings, also ordinary graphite is distinguished from all other elements by its high melting point and boiling point which is topped only by the other modification of carbon. Although it is possible to evaporate graphite by e-beam,12 several experimental issues have to be faced. Among them, heat conduction is the most prominent problem, but also the fact that graphite sublimates at these low pressures. This solid-state evaporation reduces not only the quality of the deposited layers but also leads to low reproducibility. Therefore, sputtering had been the method of choice to coat hard, wear-resistant layers for a long time. But today, this has been completely replaced by PECVD. Hard carbon layers were first deposited more than thirty years ago. Aisenberg and Chabot were the first to report the fabrication of diamond-like coatings (DLCs) applying an ion beam method [572], [573]. During the following years, numerous methods have been developed which are characterized by their chemical approach: A hydrocarboneous gas is decomposed in a plasma 12 It is also a common technique to coat ultra-thin layers by flash evaporation, e. g. for preparing SEM samples. However, these films are of no technical use.

10.8 Plasma Enhanced Chemical Vapour Deposition (PECVD)

433

which is operated in the RF range or MW range, respectively, whereas simple sputtering, even when supported by a magnetron, does not lead to this goal. Diamond consists of tetravalent carbon atoms which are configured in a tetrahedral environment. Referred to a certain carbon atom, all neighbors are carbon atoms as well. To start with the right molecule and to facilitate the progress of gentle diamond-making, a usual suspect would be a molecule with a lot of tetrahedral bonds between carbon atoms such as adamantane. However, the “chemical nose” quickly leads one astray, since the most successful approach has been the use of simple methane [574]. On the contrary, as has been proven by Spitsyn, Derjaguin and Kobayashi et al., hydrogen has to be added to methane to shift the equilibrium Cdiamond Camorphous Cgraphite

(10.39)

to the left-hand side [575] − [577]. In these plasmas, molecular hydrogen is dissociated into atoms to a large extent, and this atomic hydrogen preferentially reacts with graphitic carbon (which has been generated in a parasitic reaction) to volatile compounds.13 By this reaction, the phase purity of the diamond layer is significantly improved. As Bachmann has pointed out, atomic hydrogen could react with a terminal hydrogen atom to molecular hydrogen, thereby generating an active site which would facilitate the docking of another active carbon atom [578]. There is little hope to sell these diamonds on Fifth Avenue,

Fig. 10.39. Crystals of diamond, fabricated in a plasma c Astex, Inc. 1990). (

but they are of interest as industrial diamonds (Fig. 10.39).14 13 In the West, these fundamental observations and investigations were only paid scornful attention, since the very same Derjaguin brought the discovery of polywater before the scientific public five years earlier, a new modification of H2 O which simply turned out to be sweat of russian scientists. 14 In fact, gems such as the Star of Africa or the Koh-i-nor, the Mountain of Light, were never sold—it adorns the crown of the Queen since it would bring disaster to the kings. Can this be regarded as a fable told by superstitious people, this is fact: The owners of the blue

434


10.8.2.1 Applications. These films have been applied for optical purposes (coating of optical surfaces in the near UV and the far IR as well), to solve tribological issues (coating of wear-resistant thin layers on three-dimensional substrates, formed ad libitum, even with deep grooves, and, last but not the least, microelectronic and optoelectronic devices. Thanks to its Young’s modulus, which is second to none, and the highest atomic density of all elements, diamond exhibits the best heat conductivity of all known materials, diamond is predestined to serve as an efficient heat spreader on top of heat-generating semiconductor chips. Diamond should replace sapphire as robust window in measuring cells which contain highly aggressive materials. Submarine glass fiber cables are equipped with photonic amplifiers which could be coated with AR coatings. Due to its robustness against intense light, diamond should be the first choice for a mirror coating for high-power lasers. It should be emphasized again that this low-pressure synthesis of extremely hard and wear-resistant materials—the deposition of diamond-like (BN)∞ has been reported as well [579]—has extended the potential of surface finishing tremendously. In the public, this has been similarly judged. On September 14, 1986, the New York Times headlined: “New Era of Technology Seen in Diamond Coating Process” [580]. Diamonds have been applied in the area of high-end private reproduction of music (HiFi). HiFi enthusiasts will remember the moving-coil cartridges DV Karat and DV Karat Diamond of Dynavector which created a furor by featuring a cantilever made of sapphire and diamond, respectively. In the house-internal publications which accompanied these systems, the application of these materials was motivated by the very low dispersion of the bending vibrations and the very high sound velocity which is caused by its outstanding Young’s modulus; employing a very short cantilever, the resonance frequency could be shifted far beyond the audible range with ease (ν ≥ 50 kHz).15 To compile the attraction of diamond for all the sorts of applications is hardly appropriate to its majesty (Table 10.1). Its purity is checked by Raman spectroscopy: A first-class diamond is distinguished by one single sharp peak at 1 332 cm−1 , the symmetric C-C valence vibration (symmetric vibrations can only be observed in the Raman spectrum),whereas the lattice vibration can be located at very low wave numbers (ω = k/m with k the force constant of the bond which is very large because of the very high bond energy and the low Hope diamond all died unnatural deaths. With its last owner, it has lain at the bottom of the sea now for almost a century, aboard the Titanic. 15 However, one must be wary of an uncritical admiration of this material. It has been reported that Sumitomo and JVC would employ membranes of diamond as material for loadspeakers [578]. It is evident that only the high-pitch range can be considered. Experiences with plasma speakers constructed by Magnat have shown that a steep rise time is but one criterion to judge the quality of a loadspeaker which has to be met certainly but recedes behind matching problems and discolorations of multiway loadspeakers. Again and again surprising is the neutral reproduction of sound by broadband electrostatic speakers which just image the range of the fundamental and first overtone of the human voice with extremely low distortion.

10.8 Plasma Enhanced Chemical Vapour Deposition (PECVD)

435

Table 10.1. Diamond: important properties and areas of applications. property hardest material highest thermal conductivity resistant against UV-radiation, X-rays and γ-rays smallest molar volume despite loose-packed structure large band gap electron/hole mobility high transparency across a wide spectral range highest sound velocity highest Young’s modulus electr. resistivity

value ≈ 104 kg/mm2 (Vickers) 20 – 21 W/cm K (5 × better than Cu)

3.417 cm3 (Au: 10.21 cm3 ) 5.45 eV 1 900/1 600 cm2 /Vs

18.2 km/sec (2 ×vs of Al) 1.2 × 1012 N/m2 1013 Ωcm

application coating for cutting tools heat sink for electron. devices replacement of silicon diode arrays because of better S/N ratio

electronic highspeed devices for very high T window material for measuring cells membranes replacement of glass for lithog. masks

mass) which explains the singular applicability as window material in high-end measuring cells.

10.8.2.2 Diamond electronics. The large band gap of more than 5.5 eV has aroused great interest for the founding of a new electronics. Very low parasitic contributions to photo-induced currents should allow the fabrication of devices with very high signal/noise ratios. An effective p-doping is carried out with boron. To raise an electron, the energy of 0.35 eV (approximately 14 × kB Te ) must be expended. At room temperature, less than 1 % holes are available. The real working range of a diamond electronics can be found at elevated temperatures (500 − 700 ◦ C). An n-doping with nitrogen, however, has turned out to be much more difficult since the energy of the electron bond amounts to about 1.7 eV; phosphorous, on the other hand, does not fit into the diamond lattice. That is why p-FETs have been fabricated as a first step. A big advantage is that diamond can be directly metallized since no troublesome native oxide prevents a good Ohmic contact. On the contrary, it has turned out that etching with pure oxygen is relatively difficult. NO2 , which forms a surface film with better sticking qualities has been

436


proven to be much better suited; its etchrate exceeds that of oxygen by about one order of magnitude [581]. Due to the severe setback of not being able to realize reliable n-doping, the potential of diamond electronics is difficult to estimate. This may be justified by the merging of aesthetic and scientific aspects in an absolutely brilliant way. From history it is well-known that this confluence has sometimes triggered sensational progress. However, the euphoria of the time of discovery (the early 1990s) has faded away.

10.9 Ion beam deposition (IBD) In capacitively coupled plasmas and DC driven plasmas, the electrode (target) is a constitutive part of the discharge, perhaps the most essential of all. Energy and density of ions depend mainly on conditions (applied voltage, surface conditions . . . ). Ion beam methods are regarded as the oldest method which allows the separate tuning of ion energy and ion density. As has been explained in Chap. 8, one of the most prominent advantages consists in the very low pressure within the processing chamber, which is typically lower by one order of magnitude than the pressure in the plasma chamber which is caused by the poor conductance of the grid system which separates both chambers. Working in the deposition mode, the ion beam is directed towards the target. But on their course to the substrate, the ejected particles are no subject to additional scattering processes (Fig. 10.40). Due to the deficient thermalization of the particles, the beam which will impinge on the substrate—in modern systems at least 50 cm apart—exhibits a Knudsen characteristic, similar to that of a cudgel of evaporated atoms, and its constituents exhibit a kinetic energy similar to that of the atoms ejected out from an RF diode sputtering target. As main technological result, the quality of film adhesion obtained with these methods is also comparable. Coating to reduce the reflectance (AR coating) of photonic devices is preferentially carried out in ion beam reactors.

substrate plasma source p 1 mTorr

p

0.1 mTorr sputtered atoms

ion beam

target

Fig. 10.40. Because of the (intended) poor conductance of the grid system, the pressure in the plasma source is higher by one order of magnitude than the pressure in the processing chamber by which scattering processes of ejected atoms are effectively reduced.

10.9 Ion beam deposition (IBD)

437

98

0.02

96

0.04

94

0.06

92 90

normal incidence angle = 90° Ta2O5 Al2O3 SiO2 TiO2

-7.5 -5.0

-2.5 0.0 2.5 5.0 radial position [cm]

0.08 7.5

0.10

100

0.00

99

0.01

98

0.02

97 96 95

0.03

oblique incidence angle = 40° Ta2O5 Al2O3 SiO2 TiO2

-7.5 -5.0

-2.5 0.0 2.5 5.0 radial position [cm]

rel. deviation

0.00 deposition rate [nm/min]

100

rel. deviation


A prominent parameter is the radial uniformity of the beam of bombarding Ar+ ions which could be significantly improved by the operation with ICP sources (Figs. 10.41, Sect. 8.7).

0.04 7.5

0.05

Fig. 10.41. The radial uniformity of the deposited layer does significantly depend on the angle of incidence of sputtering ions at the target. LHS: normal incidence, RHS: c Veeco (2002) [582]. tilted by 50◦

Many properties of the deposited film can be influenced in-situ. For example, the angle of incidence at which the Ar+ ions bombard the target can be varied. But by this tilting, the angle between target and substrate is also changed. Caused by the broader sputtering angle, however, this effect is not as prominent as in the former case. At least, no further collisions will occur on the course between target and substrate, which facilitates the observation of the dependence of the radial uniformity on the mutual tilting, and the subsequent optimization. It has turned out that the radial uniformity can be considerably improved by tilting the target with respect to the ion beam (Figs. 10.40). Simultaneously, the compressive stress will be reduced since atoms which occupy defects (lattice defects, interstitials . . . ) will be kicked out more frequently for oblique incidence. DLC coatings can be fabricated applying a graphite target in discharges of an inert gas (N2 or Ar) doped with CH4 . Compared with CVD deposits, these IBD layers exhibit a finer grain and are extremely smooth which favors tribological applications. The roughness of a DLC film 100 nm in thickness has been measured to be 1 − 2 ˚ A using an AFM [582]. At the very beginning of diamond coatings, we can find the amorphous diamond. First regarded as an accident, it has evolved as a material with the lowest friction coefficient ever found (< 0.1, lowest values reported are between 0.05 and 0.07 for films deposited by magnetron sputter ion plating [583, 584]),

438


together with a very high hardness, approaching 40 kbar (GPa), however, restricted to an operating temperature range below 220 ◦ C.16

16 For comparison: TiCN, another very popular wear-resistant coating, exhibits a friction coefficient of 0.45 and a hardness of approximately 35 kbar (GPa), and can be employed at temperatures up to 400 ◦ C [585].

11 Plasma etch processes

First we discuss the dependence of primary etch parameters (etchrate, uniformity) on independent primary parameters (pressure, gas composition. . . ), and continue with the microfeatures which depend mainly on secondary parameters (such as electron density, ion density, electron temperature. . . ) in all the various types of discharges, avoiding going too deeply into chemical details, which are deferred to Chap. 12. A short introduction into damage caused by physical and chemical impact concludes this chapter.

11.1 Introduction Sputtering is a process which makes use of the momentum transfer of ions which are incident on a negatively biased electrode, the target. By this process, constituents of the target will be ejected from the surface. Since the remaining surface of the target does not matter, the sputter rate can be simply increased with growing self-bias of the capacitively coupled target. Typical sheath voltages are 1 kV or even higher, i. e. typical values for lattice energies are surpassed by more than two orders of magnitude. We expect the isotropic distribution of ion velocities within the plasma bulk, the IEDF to be transformed to a typical beam characteristics which can be softened by collisions across the sheath, mainly depending on the discharge pressure which determines the sheath thickness for constant bias voltage. For a collisionless sheath, the ions will hit the surface with normal incidence which results in an anisotropic tearing down which remain unnoticed at a target which is used for sputtering. However, if the “target” is patterned, this anisotropic attack can be measured: This phenomenon is used for lithography, which is Greek for “to write into stone”, and this is the domain of dry etching. To enhance the etchrate, it is common to add reactive gases to the inert gas, mostly argon (Reactive Ion Etching, RIE). As it has turned out, the term RIE is actually a misnomer, since high etching yields can be obtained, irrespective of whether the ions are chemically inert or chemically reactive, while the composition of neutral impinging species dominates etchrates and selectivity (cf. Chap. 12). And this holds also true for the use of the various methods. In the literature, we find the terms sputter etching, ion etching, and plasma etching. In the following, we will refer to the critical review articles of Bollinger et al. [586, 587] and Coburn and Winters [588]. G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 11,

439

440


As far as the consumption of the etchant gas is concerned, we can distinguish between these methods. During plasma etching, chemical reactions take place between etchant species and the substrate to form volatile species. The role of the plasma is simply the generation of the etchant. Provided no crystallographic effects occur, this type of etching should result in an isotropic etch profile. The etchrate during (ion enhanced) plasma etching is altered by at least three effects: • Ions can sputter away absorbed and low-volatile reaction products (e. g. InCl/InCl3 at the surface of InP). • Impinging ions produce lattice damage thereby creating active sites extending several monolayers below the surface. • Sidewall passivation by plasma generation of polymer precursor radicals. This layer prevents chemical etching leading to an isotropic profile unless ions with a high velocity component normal to the electric field destroy this coverage. An ideal etching method should facilitate (see Fig. 11.1 for some definitions): • A precise control of the geometry which has been transferred into the substrate. • Low density of surface defects. • An anisotropic etch: The vertical etch depth should exceed the horizontal etch width (Fig. 11.1). • The chemical selectivity S (differences in etch rate) of the etchant against mask material and substrate supports the possibility to dig deep holes and trenches. • The selectivity against the layers in the substrate should equal either unity or ∞ (Fig. 11.1). S → ∞ allows long times of overetch, and a simultaneous control of the progress of etching can be avoided, S → 1 does not generate protruding balconies or other inevitable microfeatures. • The anisotropic character has to be controlled. An (x, y, z)-reference standard is required which is naturally delivered with the mask. Ideally, it should not have suffered from the ion impact (Fig. 11.2). In the past three decades, numerous processes have been developed to meet these requirements in different ways. They can roughly be distinguished into four different categories: • Purely chemical processes. The substrate is located within a barrel reactor and is shielded by a Faraday cage to prevent a plasma attack (barrel reactor etching, Fig. 11.4). The reaction partners are only activated neutrals, mostly radicals, but no ions and electrons, respectively; the reaction can be supported by radiation emanated from the plasma.

11.1 Introduction

441 mask layer to be etched

WF W0

substrate

isotropic profile

M0 q

W0 dV

d h MF q

conic profile

MF

M0 W0

anisotropic profile

Fig. 11.1. Profiles for a purely isotropic, purely anisotropic etching and a characteristic profile lying in between these two borderline cases, a conical etching with either positive or negative gradient [here, a positive gradient is displayed since negative gradients occur only during wet etching with solutions which are sensitive to certain crystallographic planes, typically (111) or (111)]. w is the distance between the edges of the mask, m the width of the mask, the index “0” before, the index f after the etching. dh is the horizontal under etch, dv the vertical removal. The ratio dv /dh or the difference 1 − dh /dv is denoted anisotropy ratio or aspect ratio. A purely anisotropical etching exhibits an aspect ratio A of infinity or unity: A = ∞ ∨ 1, a purely isotropical etching is characterized by an aspect ratio A of unity or zero: A = 1 ∨ 0. The cone c Academic Press). angle Θ is defined as arctan dv /dh [589, 590] ( x mask 1 1.5 2

2.5 substrate

h

Fig. 11.2. Isotropic etching does not necessarily lead to rounded etch profiles (cf. Fig. 11.1). For severe underetch, the inclination angle does increase. Underetching the mask by 100 % (x/h = 1) up to 250 % (x/h = 2, 5), only simulates an anisotropic characteristic. c The American Chemical Society). [591] (

• Plasma processes which are carried out in an inverted sputtering reactor, which is called a parallel-plate reactor. The method of excitation is capacitive coupling (CCP). The reaction occurs directly in the glowing plasma and is influenced by ions, radiation and electric fields (ion etching, plasma etching, Fig. 11.3); not only physical, but also chemical influences and processes. In this case, the name of the process is prefixed by “reactive”. • The substrate is placed outside of the plasma source, in the so-called “downstream” zone. The excitation takes place by inductive coupling (ICP), or by whistler waves (ECR) or helicons; and due to intended diffusion losses and equalization, the plasma density at substrate level is lower typically by one order of magnitude compared with the plasma density in the plasma source. Chemical and physical influences.

442


• If the plasma source is separated from the downstream zone by a grid or a system of several grids, the method is called ion beam etching (Fig. 8.1). The substrate is subject to an ion beam which has to be neutralized at wafer level at the latest. In the best case, the neutrals will be formed in the plasma source. Etching by an inert gas: Only physical influences have to be considered, etching in reactive gases adds a chemical component. Since we deal with processes which are in part physical or chemical in nature, sometimes easy to separate, sometimes with a synergistic overlap, the discussion is divided into the Chaps. 11 and 12.

11.2 Sputter etching The oldest dry etching technique is sputter etching. Under this method, we understand the removal of material by sputtering with an inert gas; therefore, it is a purely physical process which is dominated by momentum transfer. The surface is partly covered by a mask, the removal is selective, and the surface is patterned as treated with a positive photoresist (lithography). Mirrors MFCs

Gasinlet One Hole Shower Head Langmuir Wafer SEERS

RIE-Chamber Pumping System

Ar Cl2 Kr BCl Xe 3 A CH Cl24 H BCl 2 3 O22 O Anode Optical 2 Lens Fiber System

Monochromator Photomultiplier + Photodiode Array

HR-Grating

Powered Cathode RF Z-Scan RF RF-Generator + Matching Unit Computer

Fig. 11.3. Parallel-plate reactor with extensive (but never complete) plasma diagnostics: Langmuir probe, SEERS sensor, optical emission spectroskopy with grid and multichannel analyzer (OMA). In most cases, a mass spectrometer is flanged via a KF 40 port to control the residual gas (RGA) . . . and to combat inevitable leaks.

This can be carried out in a discharge of an inverted sputtering reactor in which the substrate is subjected to RF power. This arrangement is a denoted parallel-plate reactor (Fig. 11.3), or in an ion beam system. In the parallel-plate reactor, the gases are fed via a central pipe or a gas shower head, which is part

11.3 Reactive etch processes

443

plasma chamber cylinder (quartz) magnetron

Faraday cage

waveguide or resonant cavity microwave window

wafer on wafer sledge

rotary pump

Fig. 11.4. In a barrel reactor, the ideal of isotropic and soft plasma etching is realized, c Technics Plasma GmbH 1990). especially with application of a Faraday cage (

of the grounded electrode; exhaustion occurs through an annular gap between the RF driven electrode and reactor wall. Every material can be sputtered away, however, this advantage limits this method because of the poor selectivity (ratio of etchrates between aluminum and aluminum oxide is about a factor of 50) which requires relatively high energies of the ions incident on the surface. That is why sputter etching has never reached the level of a technological method. However, it is applied to clean a surface in situ just prior to sputter deposition.

11.3 Reactive etch processes During the first half of the 1960s, plasma ashing of photoresist (PR) was the first process to be carried out in barrel reactors [592, 593]. The barrel reactor system consisted of an evacuated “fat” tube, the barrel, within a microwave resonator (Fig. 11.4). The coupling takes place by radiating via a waveguide, sometimes equipped with a flared horn to match the wave impedance of the waveguide to that of free space. The size of the cavity must be adjusted to match the resonance condition at the operating frequency which facilitates ignition. For small plasma volumes or low plasma densities, the resonance frequency is only modestly put out of tune, and we can calculate the required dimensions from their unperturbed values, normally the TE111 mode [cf. eq. (7.60)]. Since the discharge is electrodeless, and the discharge pressure typically reaches 1 Torr (130 Pa) or even more, the plasma potential can be kept very low, and the substrates experience but a soft ionic attack. Additionally, the samples can be separated from the plasma by a metallic net which serves as Faraday cage. By this configuration, the plasma range is confined to the annular gap between tunnel and reactor wall, and the mesh can only be passed by uncharged

444


particles. Hence, the substrate is subjected to particles which are randomly incident on its surface, and the etching is isotropic. This configuration effectively prevents an ionic bombardment, the selectivity is second to none, and crystal damage can be almost completely avoided, however, at the expense of very low etchrates.1 One of the most common processes is plasma ashing. Often, the plasma is generated by microwaves at medium pressures (more than 750 mTorr or 100 Pa) and serves to create reactive oxygen atoms by collisional impact: O2 + e− −→ 2 O : + e− .

(11.1)

The etching process consists in a chemical reaction of neutral radicals (species with unpaired electrons, so-called open-shell systems) with the substrate (in this case: with organic compounds to volatile carbon dioxide and water) and can be carried out in the plasma bulk but also in a screened barrel reactor. A customary process consists of the etching of SiO2 or Si3 N4 with CF4 . The first step of this reaction is the formation of free fluorine radicals: CF4 + e− −→ ·CF3 + F· + e−

(11.2)

(and other species, cf. Chap. 12), that react, in turn, with the silicon-containing compounds to volatile SiF4 . The plasma can even generate highly reactive species (radicals and radical ions) out of relatively inert molecules which can be formed otherwise only by thermic dissociation at very high temperatures, e. g. CF4 . Needless to say that the reactivity of already reactive molecules, e. g. the halogenes, is further enhanced to unprecedented levels, not only with respect to increased etchrates but also with respect to corrosion and environmental issues. These species can react at the surface of a solid to volatile components {rule of thumb: vapor pressure exceeds 10−5 Torr (1 mPa) at 20 ◦ C [594]}, which are removed by the pumping system.2 Due to the high operating frequency, even at plasma densities of 1010 /cm3 , the skin depth declines to only some cm [eqs. (14.166/167)]. Therefore, we can distinguish between an operating plasma adjacent to the magnetron and the remote plasma in the interior of the reactor behind the high-density plasma “wall”. Plasmas operated in bigger reactors suffer from spatial inhomogenities 1 This process has still survived as so-called “glow discharge cleaning”. The substrates are placed on a substrate holder which is located in the glow discharge of a barrel reactor driven at 13.56 MHZ or 2.45 GHz. The substrates are attacked by low-energy ions and electrons; contaminants will desorb by soft sputtering or by evaporation after the substrate has warmed up. Organic contaminants are ashed coldly in pure oxygen. Subsequently, a native oxide will be formed on the surface. Applying a Faraday cage will even prevent the attack of soft ions and electrons. With mass spectrometry, it has been proven that no species with masses larger than 44 (CO2 ) are formed during the ashing of polyimides. With OES, reaction products as CH, H2 O, CO and H2 could be detected. 2 But also in a soft plasma, the reaction must be thermodynamically allowed; a reaction of alumina (Al2 O3 ) with chlorine to volatile Al2 Cl6 is not possible in a barrel reactor.

11.3 Reactive etch processes

445

in plasma density and reactive species which are partly evened out by diffusion processes.3 Simultaneously, the sputter etching technique with an inert gas (argon), carried out in a parallel-plate reactor, had been improved. By hybridization of these methods, two new techniques were born: Reactive Ion Etching (RIE) and Plasma Etching (PE). The method has been adopted from the anisotropic etching in a parallelplate reactor, the chemistry (etching with reactive gases) from the barrel reactor, respectively. The two methods mainly differ in the following features: • Plasma etching: The substrates are placed on a grounded electrode, and the sheath potential equals the (negative) plasma potential. • Ion etching: The substrates are placed on an RF driven electrode, and the sheath potential is high with an added DC component for asymmetric area ratio between the electrodes; simultaneously, the plasma potential drops to fairly low values (cf. Sect. 6.3). Both methods exhibit inherent—but complementary—advantages and disadvantages [594]. The high plasma potential causes an ionic attack at all surfaces which are exposed to the plasma. By this unselective process, non-volatile compounds and atoms are ejected which will subsequently deposit in an indiscriminate manner at all surfaces . . . and eventually on the surface of the sample. Furthermore, the plasma potential is much more difficult to measure than the negative DC bias which also suppresses the attack of electrons. On the other hand, wafer handling systems, cooling systems etc. have to be installed at the RF driven electrode which requires greater expenditure on construction and operation details than it would be necessary at the grounded electrode of a PE system. With this configuration, patterns can be transferred exhibiting a vertical etch depth which is significantly larger than the horizontal pattern (so-called anisotropy), in contrast to plasma ashing and isotropic wet etching. With these methods, the horizontal and vertical etchrates are often very similar or even equal. The anisotropic behavior can additionally be supported by the choice of the discharge pressure (for ion etching, it is significantly lower (< 150 mTorr or 25 Pa) than for plasma etching (> 450 mTorr or 60 Pa). The higher discharge pressure does not inevitably cause a lower anisotropy. Processing with excitation frequencies lying below the plasma frequency of the ions (Sect. 6.2) leads to the RF modulation of the sheath voltage, und the topmost value is significantly higher than the time-averaged DC potential Vs = 3 Recently, some barrel reactors have been constructed which are operated with RF at 13.56 or 27.12 MHz. The power is coupled capacitively by an external electrode wound around the tube.

446


VDC [eq. (6.32)], provided the time to traverse the sheath falls short of a quarter period of the RF cycle, or put it positively: the ion transit frequency is higher than the excitation frequency. In the case of a collisionless sheath, this yields for VDC = −1/2 VRF [234] VDC 2π VDC sin ωt d(ωt) = . (11.3) 2π 0 π Because of the increase in the electric field, the discharge pressure can be increased without loss of anisotropy, since in this frequency regime, the acceleration of the ions is determined by the ratio E/p (cf. Sect. 4.10) [334] (Figs. 6.29 − 6.31). By the reduction of the ionic mean free path, the risk for surface contamination is reduced, but simultaneously, the desorption of non-volatile (reaction) products is made more difficult. Moreover, the shrinkage of the sheath thickness with rising pressure and with rising frequency has to be taken into account (d ∝ 1/ν, [595]). The ion etching and plasma etching regimes can be distinguished more straightforwardly according to the dominating role played by neutrals (plasma etching) or ions (ion etching). The upper pressure limit can be drawn approximately when the mean free path of the ions equals the sheath thickness. Beyond this border, one of the most important advantages, the possibility of anisotropic pattern transfer, is considerably impeded. To sum up, over the last forty years processes have been developed which facilitate patterning from hard to very soft, from completely physical in character to almost completely chemical in character, and they often consider environmental issues. Compared to the first method, sputter etching, the most recent processes have become very selective not only as far as the selectivity mask material vs. single layer is concerned but even stacks of different, but closerelated layers can be etched with ever improving selectivity and anisotropy. Simultaneously, the etchrates can be enhanced without losing radial uniformity by applying high-density plasmas. With this benefit, even processes with a very low etchrate can be envisaged. Applying large diameter substrate holders, numerous substrates can be simultaneously etched. Although with an etchrate which is comparatively low, the throughput can be kept high, and bottlenecks can be avoided, at least in principle. Vmax =

11.4 General dependence on independent properties As in the case of wet etching, we have to optimize the conditions of primary (independent) parameters with respect to the dependent parameters. However, most of the relationships have evolved to be at least bi-directional, i. e. one parameter influences more than two command variables. For example, the substrate temperature influences the chemical part of the etch rate. The etchrate is

11.4 General dependence on independent properties

447

an anisotropic property, hence its horizontal part will alter differently from the vertical part: The aspect ratio (which refers to one material) will be changed. But also the etchrates of two different materials are not supposed to be changed synchronously by the same factor. Therefore, altering the temperature necessarily causes a change of selectivity (ratio of the etchrates of two different materials) in vertical and horizontal direction, respectively. Another example, more physical in nature, is the gas pressure, which influences not only the number density and the density of the reactive species but also the electron temperature (to approximately first order, Sect. 3.5) and the frequency of elastic collisions between the electrons and the neutrals. The point of view can be that of the independent parameters (substrate temperature, gas pressure . . . ) as well as the dependent parameters (etchrate, selectivity, aspect ratio . . . ). First, the methods to measure the independent parameters are introduced and exemplified by simple cases concerning the etchrate, and in the next section, we consider the same problems, but from the point of view of the dependent parameter.

11.4.1 Substrate temperature 11.4.1.1 Introduction. One of the most important parameters, the temperature of the substrate, however, is still lacking in reliable and systematic investigation. This is mainly caused by the difficulties in measuring the temperature of a wafer which is going to be etched. Measuring and controlling the temperature at the topside without disturbance of the etching process is difficult to realize, but also measuring at the backside can lead to erroneous results and conclusions because of the poor thermal conductivity of most of the semiconductors, it takes time for temperature change to make its way through it. Albeit most of the investigations of the time-dependence of the etchrate showed an increase in the first minutes of the process, this has been referred to all other possible reasons. However, suspicion was not cast on the rise in substrate temperature. The deterioration of photoresist clearly proved the higher temperatures, but on the other hand, the ion-induced reactions which are responsible for the anisotropic etching should not significantly depend on temperature, since these reactions were supposed to belong to the type of ion etching for which the activation energy is small compared with abundance of kinetic energy delivered by the ions incident on the surface. Only the “parasitic” chemical reactions which are processes of the activation type and which cause underetching were suspected to be dependent on temperature. To suppress these reactions and also to avoid the degradation of the photoresist, all reactors are equipped with an effective cooling system rather than a heating system. A preliminary end of the reactor development is evidently reached with the system of helium backside cooling, which gets round the use of a transfer plate and applies the coolant directly to the wafer (cf. Sect. 10.4.2).

448


Two different techniques have evolved: Either the wafer is clamped mechanically to the substrate electrode, in the early stage of development supported by an O-ring (Fig. 10.23), or the wafer is clamped electrostatically (Figs. 10.24). The former technique was limited to a maximum temperature of about 150 ◦ C, and it was realized very soon that one could get rid off the O-ring because the wasted helium only raised the discharge pressure but had hardly any effect due to its inertness. Electrostatic bonding can be used at temperatures even higher than 400 ◦ C for processing times up to several hours, in particular in production lines. In both cases, the small volume underneath this system is flooded with helium (up to a pressure of about 10 Torr, cf. Sect. 10.4.2). 11.4.1.2 Etchrate and its temperature dependence. The temperature dependence of the etchrate has been the subject of research since Mullius and Coburn inspected this feature for silicon in fluorine-containing gases between −196 ◦ C and room temperature [596]. For this interval, they did not find any temperature dependence. Hence, even at “liquid nitrogen”, etching is already spontaneous (which explains the isotropic profile of the structures). A different behavior has been found for the III/V compounds GaAs and InP. Investigations by Donnelly et al. in capacitively coupled discharges revealed an activation energy of 0.46 eV [597]. McNevin investigated the etchrate of InP in a capacitively coupled chlorine plasma as a function of the substrate temperature and found a value for the activation energy which is close to the energy of evaporation of InCl3 [598] from which she concluded that the etchrate of InP in Cl2 is mainly determined by the desorption of the etch products (InCl and InCl3 ) which are low-volatile in nature (Sect. 12.6). Anisotropy and etchrate have been the subject of research also for ion beam etching (Fig. 11.14, Sect. 11.6) [599]. For etching of GaAs in chlorine, the value for the activation energy was confirmed by Donnelly et al. again. 11.4.2 Gas composition From wet etching, it is well known that a highly reactive etchant exhibits a high etch rate at the expense of high a selectivity and vice versa. For example, the etchrate is slowed down by dilution, on the other hand, the selectivity can be improved. Additives as complex-building reagents (F− -ions in the case of Si) can increase the etchrate dramatically, whereas moderators can slow down the etchrate. This behavior is also found for dry etching. The degree of aggressivity of etchants can be steered over a wide range either by physical dilution: Adding a second, less reactive gas, or by chemical dilution: Adding of gases with a lesser content of the reactive species. In the first case, it has to be additionally investigated whether the diluting gas exhibits an etching effect, simply by sputtering.


449

Since for the same kinetic energy, the sputtering power comes about because of a high specific mass of the projectiles, it can be advantageous to apply helium instead of argon [600]. In the case of chemical dilution, the progress of the reaction can be altered over a wide range, from PECVD to RIE (cf. Sect. 12.4, Fig. 12.8).

11.4.3 Gas pressure and RF power As we have discussed in Chap. 6, the DC bias potential is mainly determined by pressure and RF power which, in turn, dominates the technological parameters etchrate and anisotropy. The higher the pressure, the higher the number density, and the greater the generation rate of the active species and the flux of these species incident on the electrode. Consider the simplest case of a bimolecular reaction as rate-limiting step: e− + A −→ A+ + 2 e− .

(11.4)

Since the electron density scales with the number density of A, i. e. with the discharge pressure (properly measured), and the rate of ionization of species A is additionally proportional to its density, the flux of A+ incident on the substrate will become proportional to p2 , a well-known experimental fact which can also be derived from the similarity rules (cf. Sect. 4.10). According to Eqs. (4.21) and (4.24), the sheath thicknesses d2c and d3c scale inversely with ion current density for space charge limited current or mobility limited current, which gives rise to a drop of the DC bias potential. Furthermore, the number of collisions between heavy particles (neutrals and ions) will increase, which deteriorates the directed bombardment and the anisotropy. These competing mechanisms cause one maximum in the etchrate as well as another maximum in anisotropy as a function of discharge pressure. The DC bias potential will principally increase with rising power input, however, its upper border is set by radiation damage in the mask as well as in the substrate. In capacitively coupled discharges, its dependence on pressure and power is approximately given by [352] VDC = A

√

P exp(Bp),

(11.5)

with A and B, B < 0 constants (Fig. 11.5). If the ion energy can be separately controlled (as in the case of ECR discharges), the relationship between capacitively coupled power and DC bias becomes linear. The counteracting dependencies of the DC bias voltage on RF power and pressure and the electron density on pressure give rise to a double maximum in the etchrate with respect to DC bias and pressure (Fig. 11.6). To conclude this paragraph, two points should be emphasized:

450


microwave power 50 W 300 W 350 W 500 W 600 W

DC bias [-V]

300

200

100

0

100 RF power [W]

200

ER [nm/min]

0

0 p

50

rr] To [m

500 100 250

375 V] ias [DC b

Fig. 11.5. Dependence of DC bias on RF power at the capacitively driven electrode with additional plasma generation in a spatially separated plasma source of an ECR source located upstream. For increasing microwave power, the square– root dependence between DC bias and RF power will transform to a straight-line behavior [601].

Fig. 11.6. The etchrate as a function of substrate bias (DC bias) and discharge pressure at a constant gas flow (20 sccm BCl3 , 5 sccm Ar), exhibits a typical saddle-shaped design (double maximum in pressure and dc bias) [602].

• Raising the pressure reduces the mean free path (of the ions); on the other hand, not only the density of reactive species will increase but also the rate of recombination, which can cause depositing of polymers. Moreover, the DC bias drops for constant power input due to the rising plasma density (and higher ion flux) which further supports the formation of polymers, since the ions incident on the substrate do not have enough kinetic energy to sputter away the polymer which has just grown (see Sect. 12.4). • At low frequencies (from DC to several tens of kHz), the discharge is mainly sustained by γ-electrons which are ejected out of the cathode by ion impact. γ mainly depends on the ionic species and on the surface quality of the electrode, but is insensitive to changes of the ion energy. Hence, raising the acceleration voltage, i. e. the DC bias, does not cause the electron generation to increase significantly, however, the γ-electrons are further accelerated and enter the plasma bulk with a higher kinetic


451

energy which, in turn, enhances the plasma density, and the ratio of the densities between ions and radicals (reactive neutrals) is subject to change.

11.4.4 Electrode geometry In order to obtain a large dynamical range of the negative DC bias voltage, the area ratio of the electrodes, i. e. the RF driven electrode and its counteracting grounded electrode, should be kept as low as possible — at least according to the potential theorem of Koenig and Maissel (cf. also Sect. 6.4). Horwitz pointed out that only surfaces which conduct the RF current can be included in the equation of the sheath theorem of Koenig and Maissel [199]: Vh /Vc = (Ac /Ah )4 [156]. For a linear enlargement of the reactor, the ratio Ac /Ah considerably decreases. Example 11.1 For an electrode diameter of 15 cm and an annular gap between electrode and reactor wall of 5 cm, we obtain a ratio Ac /Ah = 6.8. By doubling the diameter of the electrode and leaving all other parameters constant, the area ratio drops to 3.45.

This reduction also influences the sheath thicknesses and, in turn, the plasma potential which goes up because the system becomes more symmetrical, and “hot” and “cold” electrode gradually converge, which would mean that both the electrodes become subject to almost equal ion bombardment. To increase the asymmetric behavior by increasing the area of the cold (grounded) electrode, Maissel recommended the application of so-called “catcher anodes” which consist of metallic stripes fixed as numerous concentric fences on the cold electrode [603]. The distance between the electrodes defines the gap, d. We must address this issue from the point of electron and ion view, respectively in terms of the ratio gap over mean free path. The mean free path of the electrons λe and the sheath thickness ds amount for low-energetic electrons (1 − 10 eV) at pressures for 10 to 50 mTorr (1 to 7 Pa) to several millimeters, whereas the gap is larger by at least one order of magnitude. The frequency of momentum transfer, νm , exceeds the operating frequency ωRF considerably (50 − 100 MHz vs. 2π × 13.56 MHz). The energy which can be piled up by an electron scales with < E 2 > /νm with < E 2 > the averaged electric field squared, and the electron velocity is mobility-limited. For λ d, the gained energy would scale with E × ωRF , and only in this case would the energy depend on the gap. Narrowing the gap to very small values or decreasing the discharge pressure will cause a steep rise in VDC (steeper for gap reduction than for pressure reduction) since the plasma electrons are trapped more easily by the electrodes (hollow cathode discharge, Fig. 11.7, Sect. 4.9). To ensure a stable process, the gap must necessarily exceed the added sheath thicknesses to avoid their merging:

452


The electrons would gain energy via the very efficient process of wave surfing which is difficult to handle. For the ions, their mean free path varies typically between 5 mm at 10 mTorr (1 Pa) and 5 μm at 10 Torr (13 mbar). In conventional parallel-plate reactors, the gap equals several tens of the mean free path of ions and neutrals in the range up to 100 mTorr (15 Pa). For 100 mTorr, the mean free path of nitrogen is about 0.5 mm which yields a ratio of 100 for a gap of 50 mm. It is evident that gaps of less than 1 cm for a diameter of 30 cm or even more cannot be easily handled. When the gap has shrunk to 5 mm for a pressure of 10 Torr (1 500 Pa), the ratio has then increased by a factor of 10! Because of the small mean free path λ of the heavy constituents of the plasma, variations of the gap do not influence the etching mechanisms considerably [185]. The DC bias voltage can be influenced by the electrode distance only to a very small extent since both carrier types are thermalized in the plasma bulk. 250 constant electrode distance constant discharge presssure

VDC [-V]

225 200 175 150 125 0

50

100 150 pd [mTorr cm]

200

Fig. 11.7. Calculated DC bias voltage for discharges in carbon monoxide as a function of the invariant property pd with p the gaseous pressure and d the gap width. The squares are calculated for constant pressure, the circles for constant gap width after [604] c The American Institute of ( Physics).

Both effects are explained by the global model introduced in Sect. 3.5: Reducing the volume deteriorates the ratio volume over effective surface to smaller amounts, and this gives rise to higher electron temperatures, an increase in the plasma potential as well as a higher self-bias potential. 11.4.5 Gas flow effects and the loading effect 11.4.5.1 Gas flow. In these reactors, the discharge pressure covers several orders of magnitude from somewhat below 1 mTorr to several hundreds of mTorr. This pressure is the result of a dynamic equilibrium: Above a certain base pressure, a certain gas flow will generate a certain leak rate (upstream control) which is controlled by a comparable pumping speed (downstream) resulting in a constant residence time of the species, which is normally in the range between one second and one minute. The gas flow is mea-


453

Table 11.1. Discharge pressure, mean free path λ at a diameter of d = 50 cm, the Knudsen number K = λ/d, and type of gas flow.

pressure mean free path K [Torr] λ [cm] 10−3 5 0.1 10−2 0.5 0.01 10−1 0.05 10−3 −3 1 5 × 10 10−4

gas flow Knudsen Knudsen viscous viscous

sured in sccm, where 1 sccm equals 1 cm3 of an ideal gas under standard conditions of temperature and pressure (298 K und 1 bar): 1 sccm contains 6.0223 × 1023 molecules/22 414 cm3 min = 2.69×1019 molecules/cm3 min or 1.69 l Pa/sec or 12.7×10−3 l Torr/sec. Higher discharge pressures are associated with higher gas flows and vice versa. For correct scaling, the following relation has to be considered (Table 11.1, Knudsen flow regime: 0.5 ≥ K ≥ 0.01): τ = V /S ∧ τ =

pV , Q

(11.6)

with τ the residence time in the reactor, V the reactor volume [l], S the pumping power [l/sec] and Q the gas flow (1 Torr l/sec = 79 sccm), and the influences of pressure and gas composition, respectively, on the pumping power are neglected in an approximation of first order. Example 11.2 Provided the reactants are not consumed, residence times of several seconds can be achieved in a parallel-plate reactor of 50 l volume and 20 mTorr and 50 sccm gas flow: τ = 1.6 sec (RIE mode), or in the same reactor, but inversely driven in the PECVD mode at 1 000 mTorr, 50 l and 300 sccm, this time increasing by almost a factor of 10 (13 sec).

In nearly all of the cases considered, the gas flow will remain laminar (viscous),4 which is ensured by a Reynold’s number R smaller than approximately thousand: R∝

Q ηd

(11.7.1)

with η the dynamic viscosity and d the diameter of the pipe. R depends on the ratio of the velocity and the kinematic viscosity (η/ρ) and is nearly independent of pressure: Since the dynamic viscosity is roughly independent of pressure, and 4 We distinguish between the range of viscous flow (down to about p × d > 1 Pa cm), next in line the Knudsen regime (down to about 10−2 Pa cm) which is followed by the molecular regime.

454


the density scales linearly with pressure, the kinematic viscosity scales with 1/p, ≈ const. and thus R ∝ 1/p 1/p Whereas the upstream gas flow is controlled by mass flow controllers, the downstream gas flow is determined by the throughput (pumping speed at chamber pressure, Q = pS). From this, we can calculate the actual flow velocity according to Γ Q = nA n with A the cross section of the valve between reactor and pump. v=

(11.7.2)

Example 11.3 For typical RIE conditions (CCP) of Q (50 sccm), A (80 cm2 for a VAT 100 valve) and a number density n (6.5 × 1014 /cm3 at 20 mTorr), we obtain v = 2.6 × 104 cm/sec which is in the order of magnitude of sound velocities, whereas for RIE conditions in an ICP reactor (lower pressure but also slightly lower gas flow), v will become a little smaller.

Since the gas transport is dominated by diffusion τ is difficult to be estimated. Comparing the etchrates for similar but not identical systems leads to severe discrepancies between them which is often a result of gas flow effects: For steady parameters, the etchrate increases steeply with rising flow, then saturates for the moment and finally decreases for even larger flows. Calculating the number of entering molecules, we can define a utilization factor from the number of molecules leaving, which have reacted or not (for Si: CF4 → SiF4 ). For a 5" wafer of silicon which will be etched with an etchrate of 25 nm/min at a CF4 flow of 20 sccm, the utilization factor equals ≈ 1/2 , i. e. every second CF4 molecule will form SiF4 . For that large an utilization factor, the gas flow is far too low, and raising the gas flow will consequently improve the etchrate until it saturates. The decrease for very high gas flows can be explained by the larger gas throughput which is required to maintain the same discharge pressure: The active species are pumped away before they reach the active sites.

etch rate [a. u.]

pumping rate limited etching diffusion-rate limited etching

effective etch rate

flow rate [a. u.]

Fig. 11.8. The etchrate at constant pressure exhibits a maximum at medium flow rates as the result of two competing mechanisms: too low a flow depletes the supply of reactive species; at too high a flow, they are pumped away before they can make it to the active site (after [605].


455

Chapman and Minkiewicz analyzed the dependence of the etchrate on these parameters [605] (Fig. 11.8): • Generation of reactive species. • Etchrate limitation due to depletion of reactants at low gas flows. • Etchrate limitation due to pumping loss of reactants. 11.4.5.2 Loading effect. To put it another way: Looking from the substrate, for a certain area of open, etchable surface a sufficiently large number of etching species must be provided. For too large a surface, the etchrate will depend on the size of the open, etchable surface which is called loading effect, Fig. 11.9. 250

ER [nm/min]

200 1000 W

150 500 W

100 300 W

50 150 W

0

0

100

200

300

400

Fig. 11.9. The etchrate can significantly depend on the open, etchable surface or the load: etchrate of silicon in a discharge of CF4 /O2 at 50 mTorr (7 Pa) and a flow of 11 l/min [606].

2

A [cm ]

For reactive processes, we have to consider physical sputtering effects and chemical etching effects, and these contributions may be separated by means of the loading effect. Since chemically reactive species are consumed during the etching process, the etchrate will saturate for increased loading, and further enhancement of the etchrate can be realized only by raising the bias voltage. As this process is the result of two opposite trends, a maximum can be observed at medium flow rates. Since the rise to the maximum is due to the increasing supply of chemically reactive species, the loading effect can be most easily detected for high etchrates, for example gold or aluminum in chlorine-containing gases. In principle, it is comparable with the slowing down of the etchrate in liquid media which is due to the progressive dilution of the etchant. Mogab showed that for a reaction of second order, a loading effect can be observed if in the equation ERN =

kab τ kgr 1 + constkab τ N FVA nA

(11.8)

with • ERN : etchrate of A when N wafers are subject to etching simultaneously

456


• kab : rate constant of the reaction aA + bB → Aa Bb , (b = 1) • τ and kgr : lifetime and rate of generation of the reactive species • FA : area fraction which is occupied by a wafer with material A to be etched • nA number density of material A: nA =

ρNA M

(11.9)

with NA the Avogadro constant and M the molecular weight • V the volume of the plasma reactor the second summand becomes equal, or better, exceeds unity [607].5 In this case, the etchrate yields approximately kgr V , (11.10) nA N FA and the inverted etchrate linearly depends on the area of the substrate [587] [608]. The maximum etchrate can be found for vanishing area fraction FA → 0. Two experiments with different loading will then reveal the loading-dependent etch rate. A loading effect is expected to occur rather for fast-etching materials (large rate constant) and for durable reactive species which can be compensated by a large plasma volume (a large number of active species) or by lowering the gas temperature. In the simplest case, the etchrate depends exponentially on the absolute temperature according to Arrhenius’s law: k = k0 exp(−Eact /RT ). Therefore, the lifetime τ of the reactive species should be as short as possible. It depends on several processes: absorption, convection, recombination in the gas and at surfaces, and eventually on various system parameters: gas temperature and pumping speed. τ can be approximated by ERN ≈

−1 + τc−1 τ −1 = τflow

(11.11)

with τflow the flowrate limited lifetime and τc the lifetime which considers all other processes except the etch process itself. For τflow τC the etchrate is flowrate limited (τflow ∝ 1/Q) : the flow velocity of the reactive species through the reactor is definitely too high to get involved in a surface reaction. Even at very high flow rates, however, τflow of the parent molecules is in the order of some seconds and hence larger than τc : a flowrate limited etchrate is not likely to happen even under these circumstances.6 5 This equation can easily be derived from the equation of continuity and Fick’s first law for a stationary state. 6 It was reported that the ashing of photoresist should be controlled by convective processes at high oxygen flows [609].


457

For vanishing layer thickness at the end of a process, the etchrate will dramatically rise, especially if the etchrate is composed of a larger horizontal (chemical) component and a smaller vertical (physical) one. Hence, the vertical component of the etchrate suddenly declines precipitously, and a process which could be performed anisotropically sharply turns to an isotropic characteristic. The vertical component of the etchrate is dominated by ion bombardment and exhibits but a loading effect of second order since it mainly depends on the DC bias (and not on the supply of etchant). As we have seen the etching reaction must not dominate the gas consumption. This can achieved by choosing a substrate plate which consists of the same material as the sample which is going to be etched. Now, the etched surface remains almost constant during the whole process [594]. That the substrate plate should be chemically related with the substrate itself is recommended for another important reason. At higher ion energies, the electrode is subject to sputtering. If it consists of a material which does not readily form a volatile compound with the etchant, the substrate is prone to contamination with sputtered material, which has consequences not only for the morphology of the etching but also for the electrical conduction of the substrate [610]. The loading effect cannot be avoided by simply raising the RF power and, in turn, the DC bias since it is mainly a consumption effect. Since the loading effect is the result of competition between the etching process and other loss mechanisms for the etchant and its reactive component, the etching process must lag behind the other processes. In the case of a loading effect, the etchrate for vanishing area (FA → 0) must be noted and the fraction of the open area which has been really exposed to the plasma (corrected by the area covered with photoresist).

11.4.6 Transport effects and reactor design The gas flow effects as considered so far represent just the (phenomenological) peak of a new iceberg. Problems of transport of reacting species and products will become evident which manisfest in a spatially dependent etch characteristic, and which are predominantly influenced not only by the type of discharge but also by the design of the reactor. This had been the topic of intense research especially at the blossoming of capacitively coupled discharges in the early 1980s where specially shaped reactors for reactive ion etching, so-called hexode reactors, were in widespread use. They were equipped with specially shaped wafer holders and could contain some tens of wafers which had to be mounted by hand but could be processed simultaneously. With rising diameter of the wafers and handling systems by which the wafers could be automatically handled, however, they lost ground against their toughest rivals, the parallel-plate reactors which had to face competition with the next generation of plasma reactors with independently driven plasma sources. The parallel-plate reactor is characterized by

458


two opposing circularly shaped electrode plates (Fig. 11.3). In most cases, the gas shower head is integrated into the grounded electrode. The pumping system is flanged via an annularly shaped port to the reactor, but also systems with a central pumping port are known [611]; the radial and axial plasma density can be described by a Schottky profile at higher pressures and a Langmuir profile at lower pressures (Sect. 6.6). We know that silicon is etched spontaneously by fluorine atoms or fluorine radicals which are formed by decomposing CF4 (for details see Sect. 12.1). A radial flow assumed, the equation of continuity can be formulated with D the diffusion coefficient [612], because of the cylindrical symmetry in cylindrical coordinates: vr

∂c(F) = D∇2 c(F) + Δ; ∂r

(11.12)

with c(F) the concentration of free fluorine atoms, the Laplacian operator in cylindrical coordinates

1 ∂ ∂ r ∇ = r ∂r ∂r 2

+

1 ∂2 ∂2 + r2 ∂ϑ2 ∂z 2

(11.13)

and Δ is composed of the rate equations for formation (kgr ) and loss mechanisms different from diffusion loss (klr ):

−

CF4 + e− −→ CF− 3 +F

(11.14)

dc(CF4 ) = kb · c(CF4 ) · ne dt

(11.15)

with c(CF4 ) the concentration of CF4 , kb the rate constant of second order for the formation of fluorine radicals and ne the electronen density (“b” for bimolecular). To balance the radicals of fluorine, we have to extend this equation by two terms to account for the loss by etching and recombination: Δ = kgr − klr = kb a c(CF4 ) ne − ke c(F) − kr c(F)2

(11.16)

with ke the rate constant for etching and kr that of the recombination. a is the stoichiometric coefficient (in this case: a = 1). Since the etching reaction occurs at the surface and competes with losses by diffusion and recombination, we consider the ratio surface/volume to (2πr2 + 4πL r)/(2πr2 l) which is set to 1/2L for r 2L with 2L the electrode distance; the flux of F-radicals incident on the wall is 1/4 < v > c(F), and the mean velocity of the gas flow can be written as ur =

Qr 2 4πLrmax

(11.17)


459

with Q the gas flow in cm3 sec−1 and rmax the reactor radius. For r L, ur is approximately independent of L, the radial velocity in the middle of the reactor vanishes and linearly rises in the outward direction (this remains valid not only for a shower head but also for a central gas inlet), and axial gradients in concentration can be neglected. This yields for the balance of the F-radicals:

d c(F) 1 d Qr d c(F) r =D 2 4πLrmax dr r dr dt

R w − c(F)vmax 2L 8L (11.18) with R and w area-dependent rate constants. For the boundary conditions at r = 0 and r = rmax , this yields + kb ne c(CF4 ) −

dc(F) = 0 ∧ c(F) = cmax . dr c(CF4 ) can be estimated with the ideal gas law to

(11.19)

p − m c(F). (11.20) RT Recombinations in the gas volume strongly depend on pressure (k ∝ p2 ) and can be neglected for pressures below 10 Torr (1 300 Pa) (Sect. 4.7). It is customary to introduce dimensionless variables to obtain c(CF4 ) =

• Θ1 = • ζ=

c(F ) p/RT

r rmax

• Θe =

ne

• Pe =

1 4πLD

• Th =

1 2LD

• Da =

1 2 m kb rmax D

• γ=

= 2, 316J0 (2, 405ζ) ·Q

2 · kE rmax

1 v r2 8LD max max

< ne > w

to yield [613] d2 1 − Peζ 2 d Θ + Θ1 + DaΘe (1 − mΘ1 ) − Th Θ1 − γ Θ1 = 0. 1 dζ 2 ζ dζ

(11.21)

In these equations, the characteristic length is the radius of the reactor rmax which often occurs squared. The solutions of the differential equation can be obtained with the boundary conditions cex dΘ1 = 0 ∧ Θmax = dζ p/RT

(11.22)

460


for ζ = 0 and Θ1 = Θex at ζ = 1; furthermore, the equation for mass balance is required. 1 0

(Da Θe (1 − Θ1 ) − γΘ1 − Th · Θ1 )ζdζ =

m 0

Th(Θ1 )ζdζ + Pe Θmax (11.23)

with m = rwafer ./.rmax . Definitions of some important properties (since these properties are dimensionless to have them easily compared for different reactor sizes, they can differ per definitionem). For a comprehensive treatment see [614]: • Peclet number, Pe, describes the ratio between convection (in the gravitational field) or drift (in an E field) and diffusion; in most cases, this can simply be expressed by the ratio of the two lengths reactor length 2L or diameter 2rmax and diffusion length (Sect. 4.7), and a correlation between the gas flow rates Q (upstream) and the pumping speed S (downstream) is drawn which refers to the residence time and the fluid velocity u, and we can calculate the Peclet number according to Pe =

2L × u . D

(11.24)

For high flow rates and/or small diffusion coefficients (large Pe), convection dominates the reaction, and intense density gradients will be built up. For small Pe, however, density gradients decline, diffusion dominates the transport, and reaction rates will become linear. Typical values in parallel-plate reactors are below P e ≤ 0.1 (Figs. 11.10 and 11.11). Since the diffusion coefficient scales with inverted pressure or inverted number density (D = 1/3 λ), and the residence time τ comes out to be proportional to pressure, √ the diffusion length Λ itself which contains the product of both (Λ = Dτ , random walk) does not show a pressure dependence which should be valid for the Peclet number as well. • Two numbers which relate the rates for generation and loss by surface reactions, respectively, to diffusion. – With the Damk¨ ohler number of second order, Da, we calculate the ratio of the generation rate of the etchant over its diffusion velocity. For large values of Da, the etchant is generated faster than it disappears, which immediately leads to unequal diffusion profiles. – The Thiele module, Th, describes the ratio of the chemical etchrate to diffusion. For large rate constants or low diffusion velocities at high pressures, large density gradients are created (we use Fick’s second law)


normal. concentration

0.40 0.35 0.30

Fig. 11.10. Normalized concentration Θ of the reactive species as a function of the normalized ratio ζ for various values of the Peclet number Pe (Da = Th). One quarter of the substrate area is covered with the (ζ = 0 to ζ = 1/2 ) c The Electrochemical [615] ( Society).

0.5 1 2

0.25 0.20 0.15 0.10 0.00

open area (to be etched)

0.25

0.50 0.75 normal. radius

1.00

Fig. 11.11. Normalized concentration Θ of the reactive species as a function of normalized radius ζ for various values of the Peclet number Pe (Da = 4,3; Th/Da = 1,023) for a parallel– plate reactor. The gradient of concentration grows for rising c The Electrogas flow [613] ( chemical Society).


1.00

0.75

Da = 4.3 Th/Da = 1.023

0.50 0.02

0.25

1.00

0.00 0.00

0.25


461

1.00

d2 c dc = kc = D 2 dt dx

(11.25)

c = c0 exp [−T h x]

(11.26)

which yields

with

Th =

k . D

(11.27)

For T h ≥ 1, the etching reaction is rapid compared with diffusion processes, and as the main result, density gradients will be created which cause a depletion of the etchant either in the wafer center (for a single-wafer process) or towards the end of the wafer row (for a multiwafer process. Large values of the Thiele module (k → ∞) indicate

462

11 Plasma etch processes vanishing of the etchant: The etching reaction will become diffusioncontrolled. For T h ≤ 1, the diffusion is more rapid compared with the etch reaction and the etch profiles and etchrates become more uniform (Fig. 11.12). Since the ratio Th/Da describes the characteristic of consumption (by etching) and generation, the Thiele module should always be kept smaller or equal than the Damk¨ ohler number. For given generation rate Da, the gradient of concentration will decline for lowered etchrate (for a chemical reaction, the concentration of the reactive species scales with the etchrate, Fig. 11.13). From Eq. (11.27), we see that the Thiele module contains the ratio of the rate constant of the surface reaction (which should not depend on pressure) to the diffusion constant with D ∝ 1/p. Hence, with falling pressure, the Thiele module will rise, and improved uniformity is attained at the expense of reduced absolute deposition rates, since the absolute concentrations are lower.

• γ, the coefficient for recombination at the wall, mainly depends on the surface conditions. Its mathematical dependence is comparable to that of the Thiele module.

0.3 normal. concentration

0.1 1

0.2 10

0.1

100 open area (surface to be etched)

0.0 0.00

0.25


1.00

Fig. 11.12. Normalized concentration Θ of the reactive species as a function of normalized radius ζ for various values of the Thiele module Th. For very high etchrates, the bull’s eye will become a distinct feature of the etching characteristic (radially uniform electron distribuc The tion supposed) [615] ( Electrochemical Society).

As a result, the concentration of the reactant can be calculated employing Bessel functions of higher order. For radial uniformity of the etching reaction, the concentration of the reactant is required to be independent of the radius. This will not not happen, in fact, we observe a radial dependence according to c(ζ) ∝ ζ P e/2 .

(11.28)

Pe, should exhibit very small values. Rough estimates of the order of magnitude (Pe = rmax v/D) show that even for a gas flow which is directed in parallel fashion to the wafers the Peclet number is smaller than 0.05. Since in most reactors, the gas flow is directed normal with respect to the surface of the wafer,


463

the actual Peclet numbers are even smaller [616]. Small values for Pe are synonymous with low gas flows, low pressures and small wafer diameters or low throughput (Fig. 11.13). Fig. 11.13. Normalized concentration Θ of the reactive species as a function of normalized radius ζ for various values Th/Da, the ratio of generation rate to etchrate for Pe = 0.02 and Da = 4.3 for a parallel-plate reacc The Electrochemitor [613] ( cal Society).


1.00 Pe = 0.02 Da = 4.3

0.75

0.50 1.023

0.25 2.33

0.00 0.00

0.25


1.00

In some cases, it has been proven that the radial dependence of the etch rate strongly depends on the radial electron profile which normally deviates from a behavior which could be denoted radially constant (Sect. 6.6). At higher discharge pressures, it follows a Bessel function of first order as was shown by Bigio in a discharge through Hg/Ar by measuring the excited levels of mercury (Hg, 63 P0,1,2 ) [617]. For further discrimination between chemical etchrate and ion-assisted etchrate, these effects can be separated in some cases. Chemical etching is a thermally activated process which is started by the adsorption of radicals and a subsequent reaction between these radicals and active sites. Both steps can be improved by ion bombardment, and the vertical component of the etchrate rises more steeply than its horizontal component, which causes the anisotropic characteristic, and this is taken account of by splitting the rate constant of the etching reaction into two parts [615]: ktot = kn + k+

(11.29.1)

with kn the rate constant of the chemical etching by a spontaneous reaction of the fluorine radicals with the surface sites: RGn = kn c1

(11.29.2)

and k+ the rate constant of the ion-induced reaction RG+ = k+ I+ EI .

(11.29.3)

The second term comprises the voltage drop across the sheath and the entrance velocity of the ions, vB . Although the chemical component at the wafer edge is

464


higher than in its center, the total etchrate in the center can exceed that at the edge (Sects. 12.1 and 12.3). In principle, the etchrate must exhibit a radial non-uniformity irrespective of the special geometry of the reactor, no matter if a central or annular gas withdrawal is installed or a gas shower head or a central gas feed is employed [618].

11.5 Characteristics of dry etching The pattern transfer does not happen at a 1:1 scale with rectangular sidewalls of the remaining structure and perfect line edge resolution (LER, Figs. 11.23 + 11.24). This is caused mainly by the competition between chemical and physical effects (Fig. 11.14) and further due to secondary effects at the mask edges that include these features: • Mask erosion (principal sketches: Figs. 11.15 and 11.16, SEM micrographs: Figs. 11.17 and 11.18). • Subsequent faceting and cone formation (Figs. 11.15 − 11.18). • Redeposition at the mask edges (Figs. 11.25 − 11.27). • This feature includes the phenomenon of selectivity. Furthermore, numerous microfeatures have been identified in the last two decades. Among them are: • Trenching (Fig. 11.28). • Foot formation by shadowing (Fig. 11.29). • Microloading (RIE lag, Figs. 11.30 − 11.35), grouped into: – ARDE (Area Ratio Dependent Etching). – Sidewall bowing. – Notching. • Other mechanisms which are mainly caused by surface charging (Figs. 11.36 − 11.38). The associated electric fields in these narrow spaces (1 μm or less in width) cause the ion trajectories are deflected from the angle of normal incidence.

11.5 Characteristics of dry etching

465

11.5.1 Anisotropy One of the most prominent features of dry etching is the anisotropic behavior of the etchrate. As has been evolved in the preceding sections, it depends mainly on the substrate bias. Detailed investigations in ICP discharges have been carried out by numerous researchers; one of the most recent studies is due to Choi et al. for the etching angle in GaN [619]. Anisotropic etching is the result of forces which act simultaneously and in an synergistic behavior (see Figs. 11.14 for the dependence of etchrate and etching angle as functions of chlorine flow and temperature). It is evident that this feature can best be judged just after dry etch processing before having removed the mask. 400

300 10 sccm 7 sccm 4 sccm 2 sccm Ar IBE

200

300 ER [nm/min]

ER [nm/min]

250

150 100

125 °C 100 °C 75 °C 50 °C 25 °C -25 °C

200

100

50 -25

0

25

50 T [°C]

75

100

0

125

0

2

4 6 8 chlorine flow [sccm]

10

80

T [°C]

50 °C

-25 °C

negative slope of laser facets

60

positive slope of laser facets

40

125 °C

20

0

2

4 6 8 chlorine flow [sccm]

2 mm

10

Fig. 11.14. Ion beam etching: Etchrate and etching angle as function of chlorine flow c P. Unger. and temperature in GaAs after [620]

11.5.2 Selectivity The ratio of etchrates of various layers/materials is one of the most prominent topics in reactive ion (plasma) etching. This ratio is denoted selectivity and encompasses both physical and chemical issues. To the first class we subsume all

466


the problems associated with momentum transfer (cf. Chap. 10), whereas problems of reaction energy and its temperature dependence, sidewall passivation . . . can be regarded as a chemical or thermodynamic issue (cf. Chap. 12). In contrast to wet etch processes, where the selectivity can reach very high values (often more than several hundred), in dry etch processes this value seldom exceeds 50 [621]. This causes difficulties for a sandwich which consists of at least two layers with very different thicknesses. To prevent etching of a thin layer lying underneath a thick layer, the etchrate must be uniform across the whole substrate and its value (and dependence on temperature, initial effects, etc.) must be exactly known. To get round at least the latter problem, an endpoint detector should be used. Dimigen and L¨ uthje pointed out that the patterning process into the substrate is also influenced by the microstructure of the substrate [500]. The enhancement of the etchrate by making use of a chemical component means that even minute distinctions in the substrate material will delicately respond to the etching attack and can cause considerable distinctions in the exactness of the cut. Not only the edge roughness of the mask of photoresist, refractory metal (Ti) or dielectric material (Al2 O3 or Si3 N4 ) will be transferred but the edge of the freshly etched substrate has also to be considered as part of the mask which blankets the lower lying parts. The attack is impeded at the grain boundaries which are reproduced in the next layer. In principle, a polycrystalline layer causes a rougher surface than an amorphous film.

11.5.3 Mask effects 11.5.3.1 Erosion. By erosion of the mask edges, the distances between the structures will grow during the etching process. Sharp angles will round, and the structure will first become fuzzy and eventually, also larger details will be blurred and get lost. The degree of the erosion predominantly depends on the resistance against chemical and physical attack at the tolerable thickness which is given by the lateral resolution of the pattern to be transferred (Fig. 11.15). 11.5.3.2 Faceting. The intensity of mask erosion mainly depends on the angle of incidence. For a soft mask (photoresist), the top zone exhibits rounded corners; if it is heated beyond the flow point, the top zone is rounded as a whole, and all angles will occur. Corresponding to the angle with highest etchrate, facets are going to be formed (Fig. 11.16). The onset of this process takes place at the rounded corners which happen to exhibit the highest etchrate and continues progressively until the mask edge retreats, exposing the surface of the substrate to ionic attacks. Since the etchrate in this material is higher than in the mask, more substrate material is removed and a truncated cone with positive inclination angle will be formed.


467

Fig. 11.15. Schematic representation of mask erosion and subsequent faceting of c IEEE). surfaces which were initially protected by the mask (after [622],

To suppress this faceting, thin metallic or dielectric masks with nearly vertical flanks are employed which exhibit a very low etchrate compared with the substrate material: The selectivity should be as high as possible, and the mask should never retreat behind its initial lateral boundaries (Figs. 11.17 + 11.18). 11.5.3.3 Metal masks and trilevel photoresist. For discharge pressures between 5 mTorr and 100 mTorr, a perfect control of the inclination angle is impossible, and faceting remains a critical issue in ion etching still.7 To prevent sloped sidewalls which generate tapering in the substrate, thin masks with high a resistance against etching are employed. They consist either of metals which are chemically resistant or of dielectric layers (oxidic or nitridic in nature) with high lattice energies which makes them resistant against sputtering. In some cases (etching of InP-based III/V semiconductors in mixtures of methane/hydrogen), also masks of photoresist are still used still [623]. The angular dependence of the etchrate is relatively low; their operating time can be often prolonged in reactive gases due to the formation of dielectric top layers with very high lattice energies (oxides, carbides, nitrides, Fig. 11.19). The patterning of these layers is carried out either in aqueous solution, e. g. sputtered Al2 O3 is etched in diluted phosphoric acid (H3 PO4 ) which has been patterned and defined by a “normal” mask of photoresist, or by dry etching, e. g. Si3 N4 which has been deposited by (PE)-CVD is etched in CF4 /O2 . To ensure lateral and vertical structure fidelity, the most effective tool are thick masks of inert material which can be removed easily after the process. The best way to achieve this goal is to establish a highly sophisticated photoresist 7 Cf. Chap. 10 for ion beam techniques; due to the angle distribution of the beamlets which exhibits a maximum at about 20◦ with respect to normal incidence, a slightly enhanced etchrate is also observed for ion beam etching.

468


1.00

normal. ER

0.75

q1

0.50

q2 0.25

q3 0.00 0

30

60

90

q [°]

Fig. 11.16. The etchrate (ER) predominantly depends on the angle of incidence θ of the impinging ions. The fact that surface and sidewall of the mask do not meet an a right angle in most cases causes a higher etchrate in the top region which further reduces he steepness of the mask during the etch process: faceting with subsequent c Philips). formation of cones (after [500]

Fig. 11.17. Due to faceting, the inclination angle of the etched sidewalls is positive: Tapering of InP in a mixture of ethane/hydrogen [10:40, 40 mTorr (5 Pa), 0,25 W cm−2 ]. Note the rippled sidewall which is the result of the perfect lithographic transfer of the mask edge [624].

technique. To make this evident, in Fig. 11.18 two similar structures with different masks (photoresist) are shown. The mask in the LHS micrograph has been eroded, which caused faceting and tapering in the substrate; the thick mask in the RHS micrograph, however, has withstood the plasma and has transferred its structure into the substrate perfectly. This principle is realized in the trilevel technique, which is started by spinning a very thick photoresist which is subsequently heated at very high temperatures. This first layer is called bottom resist.8 This layer is subject to a coating process: A thin dielectric layer of SiO2 or Si3 N4 is sputtered, and a 8

For the type AZ 4562, thicknesses of 8 − 10 μm and 180 ◦ C are common.


469

Fig. 11.18. Topmost prerequisite for a perfect pattern transfer is the etch resistance of the mask. The incidence angle of the (residual) mask determines the initial incidence angle in the substrate which is here shown for a sandwiched (truncated) cone in AlGaAs/GaAs and its perfectly etched cylindrical counterpart (photoresist mask).

2.5

ER(q)/ER(0)

2.0

Ti

1.5 Mo

1.0

0.5

0

20

40 q [°]

60

Fig. 11.19. Various refractory metals exhibit a very low angular dependence of their etchrate and are well suited to employment as mask material for preparations which require steep sidewalls (Eion = 1 keV) c The American Insti[625] ( tute of Physics).

second photoresist, the structure photoresist, is patterned. The transformation of its pattern into the dielectric layers is carried out reactively by an RIE step in CF4 /O2 . In the last patterning step, not only the bottom resist is patterned but also the structure photoresist is removed quantitatively. This mask served to transfer patterns into GaN and even into SiC down to depths of several micrometers [418] [626] (Figs. 11.20 − 11.22).

11.5.3.4 LER and CD. When the lateral dimensions of the structures to be transferred reached 200 nm, the roughness of the lateral structure itself became increasingly an issue, and the photoresist, the chromium mask and its fabrication

470

11 Plasma etch processes structure resist dielectric layer bottom resist substrate

layer sequence

after PR lithography

after CF4/O2 RIE

after O2 RIE

Fig. 11.20. Principal sketch for the three layer technique [418, 601, 626].

Fig. 11.21. The mask, hard-baked and with a topmost layer of Si3 N4 which is still visible, after the O2 etch treatment, ready for use [515, 520].

Fig. 11.22. Employing a trilevel photoresist (top, dark) with a special technique for smoothing the sidewalls of the photoresist, it was possible to etch facets into GaN/InGaN (bottom, light) with vertical, extremely smooth sidewalls [418, 601, 626].


471

came into focus. It’s true that structure fidelity is the perfect transformation of the lateral dimensions into the substrate, but transformation of what? The light-sensitive polymers which are used as photoresists (PR) are deteriorated with relative ease and become fuzzy, moreover, they flow at elevated temperatures. Masks which remain unaffected to the physical attack, as oxides, titanium or aluminum (in O2 doped atmosphere) have to be patterned in a preceding step with PR. This patterning can be performed in a wet etching procedure (for example: SiO2 with aqueous HF/NH4 ), or by a dry etching process (for example: SiO2 with CF4 /O2 ). Often rough mask edges can be observed, especially at very high magnitudes. This is often due to poor performance of the patterning. However, critical analysis of the process of mask fabrication itself revealed a certain roughness of the chromium masks themselves, and these fringes are transferred by a properly performed photolithographic process first into the photoresist and afterwards into the etched substrate. The issues have been coined LER (Line Edge Roughness) [245] and CD (Critical Dimension), which is given in relation to the pitch of the photoresist, sometimes also in absolute numbers [244]). CD. The resolution rises with decreasing thickness of the resist, but to combat pinholes and voids, the thickness of the photoresist must not be less than about 500 nm. It is evident that a parasitic exposure of an unwritten area by scattered light (or electrons for a resist which is sensitive to an electron beam) will degrade the resolution. This effect is not only dependent on light scattering but also on exposure and development conditions, and, above all, on the backscatter coefficient of the substrate. Metals with high reflection coefficients scatter significantly more efficiently than surfaces of semiconductors. Critical dimension is also an issue for postbaking procedures (Fig. 11.23).

1.0

height [mm]

0.8 0.6

contrast = 100 1

0.4 0.2 0.0

-20

-10

0 CD [nm]

10

20

Fig. 11.23. The critical dimension can be significantly influenced by the contrast of the photoresist and by postbaking.

LER. Among others, Kaindl et al. have addressed the issue of LER by realizing that the fringes in the chromium mask could cause a self-affine fractal roughness

472


in the photoresist which is further transferred into the substrate [601]. With a gentle etching step, they succeeded in smoothing the edges of the photoresist without giving up the lateral control of the photoresist structures (Fig. 11.24).

Fig. 11.24. A mask of photoresist, as developed (LHS) and after having smoothed the fringes of the mask edge (RHS) [601].

As far as the issue of nanopatterning is concerned, this discussion is continued in Sect. 11.5.8. 11.5.4 Redeposition and sidewall passivation Besides this effect which is caused by the mask-induced faceting, we have to consider the fate of atoms and molecular fragments which have been removed from the bottom of the hole or trench and have not left the scene of action completely but have been deposited at adjacent walls. This effect is called redeposition and contributes to the phenomenon of cone formation [627]. If the material is assumed to be ejected following a cosinoidal distribution, we can express the particle flux into the integrated solid angle by

Γ(Ω) = Γ0

h 1− √ 2 d + h2

(11.30)

with Γ0 the flux normal with respect to the surface, d the distance to the adjacent wall and h its height. Consulting Eq. (11.30), we see that only for very flat structures standing wide apart does the flux not depend on their height (or depth, Fig. 11.25). By deposition of layers at the sidewalls which offer higher resistance to etching than the original substrate, the initial structure will congeal; and this


473

1.00 hole diameter: 1mm hole diameter: 5mm

relative flux

0.75

0.50

0.25

0.00 0

1

2 3 4 hole depth [mm]

5

6

Fig. 11.25. Particle flux into half the solid angle originating from the bottom of a hole for two different diameters.

process is called sidewall passivation. These films can emerge from the etchant gas (plasma-enhanced polymerization at the surface) or can originate from cross reactions between etched products (or its decomposed radicals) and the etchant. These films were observed during etching of deep trenches into silicon with halogens (mixture of BCl3 /O2 /HCl) [628, 629]; and in an XPS study, Oehrlein et al. pointed out that this passivation layer consisted of boron oxide which is likely to be generated between the gaseous components BCl3 and O2 [630]. In this case, the oxygen can also originate from adsorbed water films or from small leaks. But other passivating films are known which can only be generated by reactions of the etchant gas alone, e. g. during etching of GaAs with BCl3 /Ar a polymeric subchloride is formed which prevents a horizontal attack [631] (Sects. 12.4 and 12.6 for the competition between etching and deposition). It is well known that chlorine atoms cannot attack SiO2 without ionic bombardment [632]; therefore, an SiOx layer at the sidewalls prevents effectively a horizontal (chemical) attack and significantly contributes to conserve the structure. But by varying the content between fluorine and chlorine in the system (C2 F6 /Cl2 ) ⇔ (CF4 /O2 ), the degree of anisotropy can be adjusted from overhanging to isotropic [632]. The positive contribution for the stabilization of the intended vertical structure can turn into the opposite if the deposited film does not exhibit any volatility. In inert atmospheres, backsputtered material which will be redeposited at the flanks of the mask will cause a devastating change of the intended pattern (Fig. 11.26). After having removed the mask, a wall of redeposited material will be left behind. This redeposition can be circumvented by employing special masks: • Thin masks consisting of wear-resistant material such as oxides, refractory metals in reactive ambients.

474


Fig. 11.26. GaAs has been etched in argon. After having removed the mask of photoresist, excellently formed hare’s ears are left behind [633].

• Convex shaped masks made of photoresist which are heated just below the flowing temperature (which, however, causes a slight loss of resolution, Fig. 11.27).

11.5.5 Microfeatures 11.5.5.1 Trenching. The aforementioned effects can be attributed to a synergistic interaction between mask and substrate. We now want to discuss phenomena that will occur in the substrate itself. Among them, trenching is the most prominent feature. The enhanced erosion at the foot of the etched structure by which an adjacent trench is formed is mainly caused by an increased ion flux to this area [306, 634]. We distinguish between isolated and joint structures (nests), and in the latter case, again between trenches and bridges [635, 636] (Fig. 11.28). The trench which will be formed during trench etchings is consequently called a microtrench. Its origin is caused by reflections of ions at oblique flanks of the attacked photoresist, charged photoresist, vertical sidewalls of the freshly etched substrate, and perhaps by sputtered material from the flanks of the structure which will condense in the center of the floor rather than at its edges which causes the formation of a ground calotte (Fig. 11.28). Ion beam etching is prone to cause this effect (Sect. 11.6). Sopori and Chang could show that the enhancement in the etchrate can be described by a formula which is similar to that introduced for redeposition:

d , ER = ERbulk − const 1 − √ 2 d + h2

(11.31)


475

(1) substrate with patterened mask

(2) after etching

(3) after mask removal rectangular

rounded

metal/oxide

Fig. 11.27. To reduce redeposition effects or to prevent them, the mask can be either convex shaped (by heating) or can be made very thin (dielectric or metallic layers) c The American Institute of Physics). [637] (

with h denoting the thickness of the mask and d the lateral distance from the mask [461]: The thicker the mask the larger the influence of the subtrahend. Lane et al. have modeled various influences in ICP discharges [635, 636]. In particular, the effect of the DC bias could be identified: Rising bias causes broadening of the microtrench and a flattening of the ground calotte which can be completely avoided for very large bias values. They refer this to a narrowing of the IADF with rising bias voltage. Bogart et al. proved that charging the mask can by no means explain the various effects in trenching. Irrespective of whether they used a mask of SiO2 or tungsten, they could not observe any difference when they etched Si(100) or poly-Si in inductively coupled discharges through Cl2 [639]. 11.5.5.2 Shadowing. The initial trajectories of the ions across the sheath are straight lines parallel to the electric field. However, some ions are deflected by collisions with neutrals and charged carriers, and especially for low bias voltages and high plasma densities, the IADF broadens, which causes shadowing of the mask, so-called proximity effects when ridges and other isolated devices (cylinders) are generated (Fig. 11.29).

476


mask

mask

Fig. 11.28. The formation of a microtrench is caused by an increased ion flux to the foot of the structure, shown in a nested system of trenches (top, LHS) and in isolated trenches and bridges (bottom, LHS) and a micrograph of an isolated bridge in InP which has been generated in a reactive plasma [CH3 Cl/H2 (ratio 10:30 sccm), 30 mTorr (4 Pa), 0.3 W/cm2 ] [638].

This foot is typically 10 % of the total structure in height. Coming from low values for the DC bias, shadowing is the first structural effect which is followed by trenching with rising bias voltages. Since the effects which are caused by shadowing are independent of the conductivity of the mask material (tungsten vs. silicon oxide) this effect again cannot be explained by charging and subsequent deflection of the ions [639]. 11.5.5.3 Microloading. Under microloading, we subsume all effects by which the etchrate will vary for similar structures, i. e. structures which rather show the same pattern but in different scale by which the structures are farther apart for lower packing density or vice versa. Structures which are less densely packed (large part of the substrate is covered by masks) will be attacked more rapidly. For a diffusion controlled reaction, this is often attributed to a depletion of the etchant at locations of high consumption (Fig. 11.30). For a reaction which is characterized by a high consumption of the etchant and controlled by diffusion, this conduct can be attributed to a depletion of the etchant. It should be remembered, however, that the mean free path of the ions, λi , is large compared with the lateral dimensions which are now in common use, sometimes by orders of magnitude. On this scale, the trajectories of the ions


477

mask

Fig. 11.29. During the preparation of a trench, the ion-induced anisotropic character of the process will become feebler, which is caused by shadowing of the ions incident on the bottom of the structure. This shadowing causes the generation of a broad foot, here pictured for the ion etching of a vertical cavity laser made of AlAs/GaAlAs/GaAs, consisting of the upper mirror (17 triples totalling 2.88 μm), the upper spacer (AlGaAs, 122 nm), the active zone [InGaAs, single quantum well (SQW) 8 nm], the lower spacer (AlGaAs, 122 nm) and a fraction of the lower mirror [again triples (AlAs/AlGaAs/GaAs, 20 − 60 nm in thickness each]. The contour lines of the foot can be excellently seen, differing in height by only 20 − 60 nm each. The mask which serves as metallization is the so-called Bell contact [471] − [473, 602].

PR

PR

PR

substrate substrate

PR substrate

substrate

Fig. 11.30. Microloading (sketch): Although the same structure, due to the different packing density the RHS pattern is etched more rapidly than the LHS one.

are straight lines. This holds true for the regime of (reactive) ion etching for pressures between 5 and 50 mTorr (1 and 7 Pa, RIE with CCPs and ICPs) and is even more valid for high-density applications (ECR, helicon) and ion beam etching [(R)IBE], which are operated between half an order and an order lower in pressure.

478


That is why this effect is more difficult to deal with than the first-order loading effect, which is manifested in a bull’s eye across large fractions of the wafer and which is due to a depletion of neutral species. To account for this effect, a depletion of ions within the sheath must also occur. Due to reasons of continuity, the ion flux at the Bohm edge must equal that at the electrode surface. On their course across the sheath, the ions are accelerated in response to the sheath field, and according to ji = ρi vi , their density must drop. As we know, the ions alone are not responsible for the removal, but they trigger the process. This is illustrated by the following example. Example 11.4 The typical density of capacitively coupled plasmas is in the order of 1010 /cm3 which causes an ion flux of about 1015 /cm2 sec. This is countered by a flux of neutrals which is higher by about 3 to 4 orders of magnitude. Already at medium etchrates of 100 nm/min, the flux of species leaving the surface exceeds the ion flux by more than a factor of 10: Provided every ionic impact is successful (which means a sputtering rate of unity), even moderate etchrates cannot be explained sufficiently.

From this estimation, the enormous leverage effect of the ions should become evident: Already small alterations of the ion density significantly influence the etchrate. If ions are really consumed during an etching process, a double gradient in concentration is established: 1. A chemically nonselective drop in ion density across the sheath which is demanded by continuity, starting at the plasma boundary and terminating at the surface of the electrode. 2. A chemically selective drop caused by real consumption adjacent to the surface of the electrode. The loss due to chemical consumption cannot be compensated by a nonselective equalization of charges across the sheath. In the case of depletion of a specific sort of ion by selective consumption, this is due to deficient diffusion and/or formation of the specific ionic species which is responsible for the etching reaction, which, in turn, is caused by too high a reaction rate of the etching process itself. Since only the surface reaction is controllable, this must become again the rate-limiting step. This can be achieved most easily by cooling down the substrate electrode [640]. To take this effect into account, all other possibilities involve alteration of the pattern. 11.5.5.4 Aspect-Ratio Dependent Etching (ARDE). Probably the first to discover a dependence of the etchrate on the geometry of the pattern were Bruce and Reinberg [641]. Since then, it has been found that narrow trenches and holes are etched more slowly than larger ones and that the outermost flank of a periodic structure is etched more rapidly than interior slots. And almost always, the sidewalls are slightly distorted or some other microfeatures appear [Figs. 11.31 (sketch) + 11.32 (micrograph)].


mask _ _ _ _ _ + _ _ _ _ _ _ _

479

mask

mask

+ _

_

_

Fig. 11.31. Sketches of unwanted microfeatures which are generated during the etching of slots and trenches, caused by charging or deficient sidewall passivation and transport problems. LHS, sidewall bowing caused by charging of the sidewalls; middle, RIE lag due to transport problems; RHS, notching at a vertical etch stop when a layer is reached which offers high a resistance against further etching.

This effect can be investigated systematically for structures which are laterally apart by less than 1 μm (Fig. 11.33). This phenomenon is called RIE lag [642], and in nested structures of trenches or ridges, this effect is mostly denoted ARDE (Area Ratio Dependent Etching).

Fig. 11.32. ARDE: The lateral dimension determines the vertical etchrate below a certain threshold. If the ratio of window width over etch depth falls below unity, RIE lag will probably occur. This is shown here for slot etching into GaAs (the left trench c A. Goodyear Oxford Plasma exhibits a depth of 77 μm, Cl2 /BCl3 -Plasma) Technology, 2003.

Four different mechanisms are currently discussed: Knudsen transport, ion shadowing, neutral shadowing, and different insulator charging [643]. Since this ratio deteriorates during etching deeper, this leads to a time-dependent etch

480


rate. The RIE lag is less distinct at lower pressures which suggests a transport problem as possible cause (Sect. 11.4). • The anisotropic component is determined by the development of the IADF which is farther spread at higher pressures; in particular the “beam” component at normal incidence is reduced. • Although the number density increases with rising pressure, evening out of concentration gradients will become more difficult; and in this case, a gradient exists within the slot.

1.0

ER/ER(0)

0.9 0.8 0.7 0.6 0.5 0

hole diameter 0.9 mm 0.7 mm 0.45mm 0.3 mm 0.25mm

5

10

15

20

25

Fig. 11.33. For apertures less than 1 μm, the normalized etchrate depends linearly on the aspect ratio Δ (normalized etchrate referred to the initial etchrate for the first μm, Δ = depth/width, after [644]).

D

Coburn and Winters established a model which is based on considerations of flow mechanics to describe the transport in narrow slots which is valid in the Knudsen regime [645] (Fig. 11.34). The etchrate is assumed to vanish at the sidewalls (rate constant k = 0). The fluxes at the aperture and at the bottom are Γ0 and Γb , respectively, corresponding to a rate constant kb at the bottom. Provided there are no losses due to scattering, reflexions (coefficient R) or consumption (reactive etching), we can write down the equation of continuity Γ0 − (1 − R)Γ0 = kb Γb + R(1 − kb )Γb .

(11.32.1)

The difference on the LHS equals the fraction of the flow which reaches the bottom after having experienced losses caused by reflection or scattering, and the first term on the RHS represents the fraction of the flow which has reached the bottom but is lost by reflection, and we find for the ratio of the fluxes at the aperture and at the bottom of the slot R Γb = . (11.32.2) Γ0 R + kb − R kb To a first oder approximation, the etchrate is proportional to the flow of the etchant, and this must be the ratio of the etchrates for a certain aspect ratio


481

1.00 k= 0.1

ER/ER0

0.75

0.25

0.50 0.5 1.0

0.25

0.00 0

2

4

6

8

10

D

Fig. 11.34. Coburn and Winters established a model based on flow mechanics which is valid in the Knudsen regime. The etchrates are shown as a function of the aspect ratio for various rate constants of the etching reaction [645]. The etchrates are normalized to the initial etchrate at the surface (Δ = 0).

Δ =depth/diameter compared to that at the aperture for vanishing depth (Δ = 0): ERΔ Γb = . ER0 Γ0

(11.33)

• For very low aspect ratios, the etchrate is expected to depend on the transport behavior. • Only for very low rate constants is the influence of transport on the rate constant is negligible (very low ratio between the densities of ions and neutrals for low DC bias or very high pressures). Reducing the reactivity of the etchant would improve the evenness of the etchrate, i. e. the RIE lag is pushed back. But it has to be kept in mind that transport to the local site must not be the rate-limiting step but the etching reaction itself. This has been observed by Fujiwara et al. who carried out trench etchings in Cl2 , HCl, HBr, and HI [646], Fig. 11.35. The RIE lag is caused by several (at least two) different mechanisms: With progressive etch depth, transport of the neutral etchant is continuously hampered more violently. Hence, this effect is more pronounced for highly reactive species. Additionally, the shadowing of ions which are not exactly aligned in parallel fashion with the electric field becomes more severe with increasing depth [647]. 11.5.6 Charging effects Since etching reactors are operated at radio frequency, charging of the mask has ceased to be an issue. For a mask with its lateral dimensions large compared with

482


HBr, HI

ER/ER(HBr,HI)

1.00

0.98

HCl

0.96

Cl2

0.94

0.5

1.0 1.5 pattern diameter [mm]

2.0

Fig. 11.35. The reactivity of the etchant determines the microscopic uniformity. In the order Cl2 , HCl, HBr, and HI, the influence on ARDE is most intense for the very reactive Cl2 , whereas for the gases HBr and HI, which are very slow to react, no influence is detectable (after [646]).

its height, Economou and Alkire estimated the current flowing across this mask to be almost completely displacement current at an operating frequency of 13.56 MHz: The mask acts as simple capacitor and deviations in potential of less than 1 V are typical, which is completely negligible for the energies which are typical for RIE processes—the intensity of the electric field, however, is comparable with that of the sheath [648].9 It was Ingram who first noted that during digging a trench or a hole in an isolating substrate the electric field within the hole will be bent due to different charging of the walls with respect to the bottom. Since the angle distributions of electrons and ions differ significantly [649], at the beginning of the digging process, ions collide with the bottom of the hole rather than with the sidewalls, which will, in turn, cause the sidewalls to be charged negatively with respect to the bottom [650]. This causes deflections of the ions from their initial course which marks the onset of sidewall bowing. Since these processes differ with time and aspect ratio, modeling has turned out to be very complicated. Arnold and Sawin have estimated such potential differences to reach approximately 3kB Te /e0 for low aspect ratios (2/1 for height to width) [651] (Figs. 11.36 + 11.37). These charging issues become serious with increasing aspect ratios since the rising potential retards the impinging ions. This charging explains several phenomena: • Sidewall bowing or barrelling, which describes the outward-directed shaping of the sidewalls of trenches and holes. • Formation of dovetails at the bottom of trenches and holes, sometimes termed notching (Fig. 11.38). 9 For operating frequencies below ωp,i , charging cannot be neglected any longer. For the same geometry, the capacitor will charge to 10 V at 100 kHz.

11.5 Characteristics of dry etching electrons

ions

483 electrons

_ _ _

ions incident on the walls

+

+

+

charge buildup

Fig. 11.36. Electrons exhibit a large angle distribution and cause negative charging of the sidewalls which gradually deflects ions from its initial course leading to sidewall bowing (after [651]).

11.5.7 High-end etching processes with high density plasmas The difficulties which capacitively coupled plasmas have to face are twofold. • The pressure is in the order of several tens of mTorr which does not admit collisionless traversing of the ions across the sheath. Furthermore, the transport of less volatile compounds is hampered. Reactions which exhibit no activation energies are controlled by diffusion, and lower pressures would favor larger diffusion coefficients and, in turn, better radial uniformity of etching in the macroscopic range (loading effect of first order) and in the microscopic range: In the lateral dimension (microloading) and in the vertical dimension (ARDE). Aspect ratios which exceed ten are very difficult to achieve. • Plasma density and ion energy cannot be controlled independently, and for acceptable etchrates in low-etching materials (GaN, AlN, SiC . . . ) or for shifting the plasma-induced equilibrium deposition removal to the right-hand side, bias potentials of several hundreds of volts are required. By the application of high-density plasmas which circumvent these difficulties, great progress has been made over the whole range of applications. Not only have low-damage etchings of only several nm in depth been realized but also real cuts through thinned wafers of more than 100 μm across. In silicon, aspect ratios of 25 are easily accessible, and aspect ratios > 100 have been reported. This allows the fabrication of very narrow trenches and pilasters, which can be called a system of slots (Figs. 11.39). Via holes have been bored in semiconductors, extremely finely resolved with very smooth sidewalls (Fig. 11.40). In this case, the depth is 60 μm, no trenching is visible which proves the significance of the chemical part of this process. Real deep etching processes with a depth in the order of 100 μm and high aspect ratios are now possible in high-density plasmas (ECR and ICP). The limiting factor is no longer the etchrate itself for which the limitations have been extensively discussed in the preceding sections but the temperature stress

484


2.0

normal. flow

1.5 A

1.0

A

0.5

Fig. 11.37. Since the fluxes of ions (dashed) and electrons (solid) to the walls and to the bottom of a trench of isolating material are different at the start of the “digging”, the potential becomes locally different very quickly: At the walls will be negatively charged, and the bottom becomes positively charged which causes sidewall bowing (after [651]).

B

normal. potential

0.0

0.8

0.4

0.0

-0.4 -4

-2

0

2

4

normal. radius

A

C

B

C

Fig. 11.38. Modeling of a trench with sidewall bowing (A); RIE lag (B); local dove tailing (C). Solid line: after PR processing, dashed line: after a short etching process (after [651]).

of the photoresist and other materials which are temperature-sensitive. For a long time, it has remained a challenge to replace the process of chip separation which has been conventionally carried out by sawing. With etchrates which are equal or even exceed 1 μm/min across a large wafer, dry etching can compete with convential separation. Concerning the smoothness of the sidewalls, the requirements for chip separation are less severe than for etching via holes (Fig. 11.41). The action items which have to be faced during chip separation encompass mechanical problems and issues of vacuum technology. Albeit the progress in radial uniformity is evident compared with etching in capacitively coupled discharges, it is not at all perfect. During the course of the etching, slight de-


485

Fig. 11.39. Comb structure in silicon exhibiting an aspect ratio of more than 25, c University of etched in a system Plasmalab 100 of Oxford Plasma Technology ( Twente, 2002).

Fig. 11.40. Via hole in silicon, very fine resolution but with a depth of 60 μm, soft-shaped foot, realized in a system Plasmalab 100 of Oxford c A. Plasma Technology ( Goodyear, Oxford Plasma Technology, 2003).

viations in non-uniformity do not matter, however, at the end, the first holes are created. Now, the helium which serves as coolant could invade the reactor. To avoid such an accident the following procedure has been worked out during the processing of so-called high brightness diodes. In the last frontend step, the wafer is glued onto a temperature-resistant foil which restricts the upper limit of the separation rate to about 1 μm/min, and is followed by a polished flip-chip technique which includes removal of the foil. However, not only one-step patterning is considered and realized. The spacer in Fig. 11.42 has been etched by a two-step procedure. First, a trench is dug into GaAs and overgrown with Si3 N4 , and in the second step, the length of the gate is reduced by an anisotropic etch step [ECR, Ar/CF4 /O2 at 0.2 mTorr (30 mPa)] which is carried out in two sub-steps. At a microwave power of 200 W, a bias voltage of −55 V is applied which is added to the plasma potential of +15 V resulting in a total voltage of 70 V. When the endpoint is reached, the bias

486


Fig. 11.41. LHS: Separation of high brightness diodes in a high-density plasma through BCl3 /Cl2 driven by ECR [418]; RHS: holes in silicon, ICP (University of c 2003, both structures realized in reactors Plasmalab System 100 of Oxford Twente Plasma Technology).

voltage is shut off. Using this procedure, gate lengths can be realized which are not achievable with conventional technology.

Fig. 11.42. Etching of a spacer in Si3 N4 to reduce the gate length (ECR, Ar/CF4 /O2 ) [652].

It should be mentioned that not only vertical sidewalls are striven for but also slanted flanks can become an interesting structural feature. Etching contact holes into Si3 N4 on GaAs, for example, can reduce the current density in a metallic layer that will be deposited after having removed the photoresist by a gentle ashing step in oxygen in-situ. All in all, the main advantage of high-density plasmas is the spatial separation between ion generation (in the plasma source) and ion acceleration (at


487

the electrode). Capacitively driven discharges occupy an intermediate position between DC discharges and high-density plasmas. • Very gentle depositing and etching heave become available, which is impossible with capacitively coupled discharges where energies typically peak at 300 − 450 eV although the medium enery is far less. Devices can be fabricated whose patterned surface must not show lattice damage and other traps which localize the highly mobile carriers in MESFETs (metalsemiconductor field effect transistors) or HEMTs (high electron mobility transistors). • At discharge pressures below 1 mTorr (0.1 Pa, approximately one order and a half in magnitude below the pressure applied for ion beam etching), most solid reaction products are more volatile which has made feasible reproducible etching processes in InP applying chlorine (Sect. 12.6). • The effect of ARDE can be suppressed almost completely in ECR discharges. Investigations by Nojiri et al. suggest the overwhelming influence of discharge pressure on the feature of ARDE [653] (Fig. 11.43).

ECR-RIE (3 mTorr)

ER/ER0

1.00

CCP-RIE (49 mTorr)

0.75 CCP-PE (2025 mTorr)

0.50 0

1

2 3 hole diameter [ mm]

4

Fig. 11.43. Etch rates of holes as a function of their diameter, compiled for three different methods. The strong dependence of ARDE on discharge pressure is evident. The data are reduced to ER0 which denotes the topmost etchrate in a high-density plasma (after [653]).

• These observations fit in with the reduced shadowing at lower pressures. As most prominent indicator, we have made out the formation of a foot. At pressures below 1.5 mTorr which are typically for ECR discharges, the foot is pushed back almost completely. The main reason is not only the beam characteristic of the ions impinging on the surface but also the very thin collisionless sheath which is the result of the high plasma density and the low sheath potential.

488


11.5.8 Towards nanostructures CD and LER are a continuing issue in the sub-μm region. We have already indicated that LER is caused by fringed chromium masks [601]; moreover, it is triggered by some other parameters during conventional photoresist processing [654] − [656], and it can even be influenced by the molecular weight of the photoresist [657]. For a rough estimation, we remember that photosensitive molecules of the resin exhibit molecular diameters in the order of 1 or 2 nm. For linewidths of 2 μm (2 000 nm), the typical thickness of the photoresist is around 0.5 to 1 μm depending on the application. Processing of a positive photoresist consists of exposure (breaking of bonds and forming of an acid), postexposure baking and subsequent dilution of the acid (as anionic species) which must lead to a granular structure of the developed photoresist which goes unnoticed in this example since LER induced by photoresist processing is in the order of 0.1 %. It is evident that in the sub-μm range, not only the patterning techniques but also the tools and materials came into focus again. Will features scale down with the dimensions of the devices or will they remain the same? In the latter case, their importance would increase, and strategies had to be developed to combat this behavior. For inductively coupled discharges, these issues have been investigated recently by Rangelow et al. [643]. In features below 50 nm, increasing pressure leads to positively sloped sidewalls and increased microtrenching. This can be mainly attributed to a broadening of the IADF. Reducing the pressures sharpens the IADF which leads to steeper sidewalls but simultaneously, the ionic attack at the mask edges leads to mask faceting which can cause CD loss. Moreover, also ARDE becomes a serious issue for decreasing pressures. Since the IADF is shaped more beamlike for lower pressures this indicates that shadowing of neutrals is mainly responsible for the RIE lag (Fig. 11.44).

11.6 Special features of ion beam etching As has been shown in Sect. 10.1, the sputtering rate depends on the energy of the impinging ions and on their angle of incidence [483]. The latter feature can be subjected to an analysis only in ion beam systems. Qualitatively, a strong dependence on the angle of incidence indicates a physically dominated etching process; the less its dependence the higher the chemical influence [658]. In contrast to reactive ion etching, with ion beam etching faceting problems can be tackled. Faceting can be suppressed by tilting the sample with respect to the beam so that Θtilt = 90◦ − Θmax ,

(11.34)

with Θtilt the tilting angle and Θmax the angle of maximum etchrate. Since the latter exhibits a relatively broad maximum, Eq. (11.35) is only approximate

11.6 Special features of ion beam etching

489

half pitch [nm] 100

60

50

40

30

10

pressure [mTorr]

etchrate dominated by microtrenching

pinch-off

7 sidewall bowing

Fig. 11.44. Influence of pitch and pressure on the microfeatures in a typical periodic structure after Rangelow [643].

5 mask faceting

RIE lag

2

(Fig. 11.45; the sample is subjected to rotation below the beam to avoid unbalanced removal).

1.00

rel. ER

0.75

0.50

0.00 0

10°

90° 80° 70°

0.25

30

60 Q [°]

90 20°

Fig. 11.45. The most effective strategy to avoid faceting and trenching consists in c The American tilting the sample with respect to the incident ion beam [637] ( Institute of Physics).

According to Eq. (11.34), having tilted the sample properly the maximum etch rate occurs approximately at normal incidence, simultaneously, the etched structures exhibit rectangular profiles. Reducing the angle of incidence (the ions impinge normal to the surface if the angle of incidence of is 90◦ ) should cause the sputter rate to increase because the energy of a single sputtering ion will be spread over a larger area. On the other hand, the beam intensity decreases. For very small angles of incidence,

490


the ions just skim along parallel to the surface, only very small fractions of energy can be transmitted, and the influence of sputtering will be reduced. This comes about because the sputter rate increases with rising angle (up to values of about 45◦ ) to drop more gently at higher angles [659]. For some materials, in particular for refractory metals of high atomic order, the sputter rate does not depend on the angle of incidence since the collision cascade in the topmost region is very shallow. Here, the beam intensity dominates other factors (Fig. 11.46).

ER [nm/min]

40

30

300 V ~ 0.32 mA/cm2 GaAs Au Al SiO2 PR Ti

GaAs

Au Al

20

SiO2 PR

10

Ti

0 0

30

60

90

Fig. 11.46. Etch rates of some materials as a function of the angle of incidence of the ion beam (0◦ : parallel, 90◦ : normal to the surfacee [637] [660] c The American Institute of ( Physics).

Q [°]

The first to fit the angular dependence of the etch rate was Rangelow [660]. He found a polynomial dependence on trigonometrical functions; the etchrate of silicon bombarded by argon ions with an energy of 600 eV follows the equation ER(Θ) = 0.03925 cos Θ + 0.051 cos2 Θ − 0.0373 cos4 Θ.

(11.35)

Therefore, the inclination of the sidewall (due to physical momentum transfer) is mainly determined by the angle of the sputtering yield. However, close inspection reveals a slight bending, either concavely or convexly shaped which is due to the unevitable divergence of the beam (Fig. 11.47). In particular, this holds true for IBE and RIBE [661]. For CAIBE with a strong chemical component, this effected may be levelled out. 11.6.1 Applications Etching is carried out by direct bombardment of a target and coating by a two-step process. First, a target is bombarded by ions from which secondary


491 Fig. 11.47. The smearing out of the beamlets leads necessarily to a certain undercut (negative angle, light grey) below the masks (dark grey) which deteriorates the lateral fidelity inevitably. This negative angle can also be observed in argon and is therefore not due to a chemical attack of any reactive reagent. This is an effect of second order, since the angle of the sidewall is mainly determined by the angle of the sputtering yield.

atoms are ejected. Second, a substrate is subjected to bombardment by these non-thermalized atoms and will be coated. Compared with processes carried out in RF diode sputter systems, the main advantages are for coating: • Reduction of the bombardment of rapid electrons since the substrate is located outside of the plasma chamber. • Since the working pressure is significantly lower: – The atoms incident on the substrate exhibit a kinetic energy which is higher and more uniform because energy-consuming collisions happen less frequently. The higher energy of the landed atoms causes higher mobility of the adatoms, especially during the process of nucleation and other steps. – Lower contamination. • Furthermore, a system to cool or to heat the substrate can be installed with ease since the substrate is no part of the electrical system. For etching, the possibility of angle-dependent etching is considered the most prominent advantage—an additional degree of freedom by which every sidewall angle can be accessed simply by choosing the tilting angle ϑtilt with respect to the oncoming ions.10 10 For RIE, the substrate is placed upon the “hot” electrode and the motion of the ions follows the line of the electric field which is in a direction normal to the surface. Exceptions are narrow holes and trenches in isolating substrates in which the electric field is prone to bending effects. To prove this assumption, samples whose dimensions were small compared with the sheath thickness have been fixed onto inclined planes which were located on the electrode. In fact, etched laser ridges exhibited another inclination angle, however, not only that given by Eq. (11.35) but the complete sequence of angles between these borderline cases [662].

492


Fig. 11.48. A system of laser facets at exactly 90◦ and exactly 45◦ , etched into an AlGaAs/GaAs sandwich layer, realized in an IBE system of Veeco [664].

It is very difficult to obtain inclined planes exhibiting an inclination angle of exactly 45◦ by wet etching methods. The most straightforward method has evolved in catching the product which is kinetically controlled before the thermodynamically controlled product has been formed to a significant extent [663]. By ion beam etching, this angle is easily accessible, not only in inert gases [664] but also in their reactive counterparts [665, 666]. Typical micrographs of a laser facet in GaAs/AlGaAs etched with IBE are shown in Figs. 11.48.

Beam current parameters, in particular ion energy and ion flux, can be controlled independently of target processes, and due to the progress which has taken place in the area of neutralization systems, also insulators can be subjected to an ion beam without being faced with charging effects. This has been a huge disadvantage in production lines for long a time since large beam diameters could only be realized (and kept constant in performance) with tremendous expenditure. In 1978, Robinson reported on the development of a multipole magnetron which creates a beam 30 cm in diameter [667] and could be kept constant across ±10 cm with respect to its center by ±5%—at a current density of 0.5 − 0.75 mA cm−2 at 750 V acceleration voltage. In the meantime, 10" guns have become commercially available, and the trend to develop even larger systems lasts undiminished. Extrapolating the time lag from Robinson’s publication to the commercial availability of a system with 50 cm diameter and a current density of 1 mA/cm2 at a deviation of ±4 % which was presented by Hitachi in 1989, these systems will be available by the end of this decade [668]. Due to the low number density ion beam etching was considered to allow only low etch rates. Applying reactive gases, this disadvantage could be dispelled. Optical endpoint control must make use of the UV range (for example the 252 nm line of silicon) since the filaments emit a continous spectrum in the


493

visible range [659] which can be circumvented by application of plasma sources which operate in the HF range (RF or MW). 11.6.2 Ion beam assisted etching: IBAE or CAIBE Since capacitively coupled plasma sources and ion beam systems were developed simultaneously, the disadvantage of the first could be described from the beginning. Ions and chemically reactive species (radicals) are generated in the same plasma, and their concentrations and energies cannot be controlled independently. Ion beam etching does not exhibit this disadavantageous feature but is prone to contaminations which are caused by sputtering of the grids. A combination of both methods was developed by Geis and Lincoln in the early 1980s [669, 670]. The ions were generated in a Kaufman source and the parent molecules of the reactive species were delivered by a gas shower ring which was located above the substrate (Fig. 8.1). With this system, the energy of the ions can be controlled within several electronvolts, i. e. one part in a hundred. The first system which was subjected to this concerted action of ions (Ar) and aggressive neutrals (Cl2 ) was GaAs at pressures of less than 0.1 mTorr (10 mPa). Besides the normal dependence of the etchrate at the same ion energy,11 the etchrate of silicon rose by a factor of 15 when Cl2 was added (Sects. 12.1 and 12.2). Additionally, the surface remained considerably smoother compared to an attack with a pure inert gas, which emphasizes the importance of the chemical component. This method is advanced to a customary etching method under the name CAIBE (acronym from Chemical Assisted Ion Beam Etching). A very prominent area of research of CAIBE is the etching of smooth facets of Fabry-Perot lasers. A conventional semiconductor laser consists of a ridge waveguide of III/V semiconductors, about 2 μm in height and several micrometers in width; its lengths varies typically between 200 and 400 μm (so-called MCRW laser).12 In the conventional fabrication method, the mirrors are generated by cleaving. To ensure high performance and high production yield, especially for the cleaving procedure, the laser ridge has to be defined with respect to certain orientations of the crystal within ±1◦ , in vertical and lateral direction, respectively. With wet etching methods rectangular facets cannot be realized since the [111] planes with an inclination angle of 54 3/4 ◦ determine the geometry as the most slowly reacting planes. Conventional dry etching in parallel-plate reactors suffers most frequently from a selectivity between InP and the quaternary alloy InGaAsP so large that protruding “noses” and other features are caused which reduce the intended smoothness considerably. Here, CAIBE evolves as method of choice: A high nonselective fraction (Ar+ ) leads 11 The etchrate of He+ ions is lower by more than one order of magnitude compared with Ar+ ions, which stresses the importance of physical sputtering for the total etching result. 12 Metal Clad Ridge Waveguide laser

494


to a uniform removal, chemically highly reactive species which attack the semiconductor without any discrimination as well as ensuring a rapid process [671] − [673] (Figs. 11.49 and 11.50).13

optical output power [W]

2.0 cleaved etched

1.5

18 °C, cw

1.0

0.5

0.0 0

1

2

3

4

5

6

7

operating current [A]

Fig. 11.49. A laser facet, approximately 8 μm in depth, carved out of an InGaAlAs sandwich applying CAIBE (Cl2 /Ar). The inclination angle of the facet is exactly 90◦ with respect to ground (Tsubstrate : 75 ◦ C, Eion : 400 eV). The residual roughness amounts to only 3 − 5 nm (AFM) which does not influence the optical properties of the laser diode at all, here shown for a broad-area laser (area of the waveguide: 1000 μm × 100 μm) [672].

After having carved the facet out of the semiconducting substrate, an insitu surface passivation can be carried out. This is strongly recommended especially for materials such as AlAs which are oxidized even in a dry but oxygencontaining ambient [674]. To improve the long-term stability of these lasers, Deichsel and Franz employed a discharge through H2 S which served to sulfidize the topmost layer which is otherwise prone to rapid degradation during operation (catastrophic optical destruction, COD) [675].

11.7 Damage With the wisdom of hindsight, it appears very strange that damage in the remaining substrate has been no issue for years since material removal itself, although intended, is the most serious damage. The first systematic investigations featuring not only the topmost layer but also the adjacent volume were published by the end of the 1980s. We can distinguish between the sources of the damage: • UV and X-rays. 13 Employing chlorine in the ion gun itself (real RIBE) will cause a high amount of Cl+ 2 ions which leads to very high etchrates which are difficult to control.

11.7 Damage

495

optical output power [mW]

300 SC 2000mm 1000mm 500mm 250mm

200

21 °C, cw

100

0 0

200

400

600

800

operating current [mA]

Fig. 11.50. A bent laser facet of unstable resonators, carved out of an AlGaAs/GaAs structure employing CAIBE (Cl2 /Ar), area of the active zone: 500 μm × 100 μm. In the diagram are noted the radii of the curvature, for comparison also the data of a laser with two cleaved facets (SC) [673].

• Ions. • Unintended influences of the sheath field. • Unintended reactions within the plasma bulk. • Unintended reactions at the surface and in the adjacent volume (reactions to non-volatile films, chemical passivation (most prominent is hydrogen). But we can also focus on their influence on the surface: • Displacement of atoms which constitute the crystal or the topmost film, i. e. lattice defects by ion bombardment. • Surface contamination by etch products (polymers), sputtered and redeposited material of the mask or of the freshly etched sidewalls, or etchants which have not reacted completely, or alterations in stoichiometry by intermixing of individual layers. • Implantation of ions with higher energy in subcutaneous zones. • Short-circuits or degradation of electric properties by the sheath field. • Promotion of electrons into the conduction band of insulators by mediumenergy photons, thereby creating trapped holes (E¯hω ≤ 10 eV). • Damage of metal-covered gate oxides by high-energy photons (E¯hω ≤ 15 eV).

496


Surface contamination is very difficult or even impossible to detect if an etched surface is subsequently coated in-situ. Diffusion processes out of this overlaying layer into deeper zones can cause uncontrolled doping. Very serious is electrostatic charging of a dielectric layer, which has to be opened during the etching, for example a gate oxide which is liberated from a wrapping of electrically conducting poly-silicon. In the meantime, this layer has been thinned to so low a value that it withstands voltages of no more than 15 V. This is no issue during etching itself, which is carried out at 13.56 MHz (or multiples of this frequency) or at 2.45 GHz, however, the extinguishing process requires serious attention.14 Fortunately, a broad palette of probes is at hand to observe and to evaluate the damage [676]. Among them are (Table 11.2): 1. Scanning electron microscopy (SEM). 2. Auger electron spectroscopy (AES). 3. Atomic force microscopy (AFM). 4. Fourier transformed infrared spectroscopy (FTIRS) for hydrogen [677]. 5. Time-of-flight-SIMS (TOF-SIMS). 6. Scanning transmission electron microscopy (STEM).

Table 11.2. Surface-sensitive methods to investigate and evaluate damage caused by dry etching.

method SEM Auger STEM PL SIMS Hall C(V ) Fabry-Perot Damping

qualitatively quantitatively depth profile √ √ √ √ √ √ √ √ √ √ √ √ √

14 During the switching off, the potential of an electrically conducting electrode approaches the level of the counterelectrode and sets up a substantial voltage between gate oxide and electrode. If the electrode itself is made of a dielectric material, gate oxide and electrode will be charged to similar levels, and the voltage burst is split between gate and substrate electrode according to their capacitancies which significantly reduces breakdown hazard.

11.7 Damage

497

In a certain sense, all these methods count atoms rather than effects. For problems which are more physical in nature, another system of electrical and optical methods can be applied as a simple test device. In most cases, one focuses on a property which suddenly declines precipitously: 1. Schottky diodes [678]: • Heights of Schottky barriers. • Ideality factors. • V-I-forward characteristic. • inverse voltage and inverse current [679, 680]. 2. Transistors (HBTs) [681]: • Collector current vs. voltages of emitter or collector, resp. • Leak current vs. voltages of emitter or collector, resp. • Gate oxide charging and measurement of the surface potential [682, 683]. 3. Laser diodes, semiconducting waveguides: • Loss of intensity in waveguides. • Loss of photoluminescence which is caused by damage in the active zone. • C(V) measurements [684]. • Hall measurements [684]. These methods differ not only in the intensity of the measured effect but are also sensitive to the depth of the damage. Electrical measurements are supposed to detect damage most sensitively [685], for example evaluating the characteristics of Schottky barriers [C(V ) and I(V )] [686]:

1 I = AT exp − kB T 2

e0 φB −

e0 VF 4πdεi

,

(11.36)

with A the (effective) Richardson constant, φB the height of the barrier, d the thickness and εi the dielectric constant of the insulator. For a relative comparison of the etching processes, εi is set equal ε0 . Applying Eq. (11.36), a depth-resolved profile of the damage within 100 nm or even more can be obtained whereas an amorphization of the lattice will happen in most cases only in the topmost layers (typical depths between 5 and 30 nm, measured with STEM or AES) [687, 688].

498


Hetero-bipolar transistors (HBTs). For a gradual change in the quality of etching from low bias to high bias voltages, the layer resistance will increase continuously which causes a fatal influence on the current amplification. The collector current is expected to decline with a simultaneous rise of the leakage current which is mainly caused by the generation of recombination centers. This is an effect caused by amorphization of the lattice whereas hydrogen-containing etchants lead to passivation of holes. III-V semiconductors. Surfaces of InP are more sensitive and are affected more severely than GaAs and its related compounds. This is mainly due to a higher recombination rate and also to intensified Fermi level pinning [689]. In InP, three zones can be distinguished after plasma treatment of CH4 /H2 [690]: 1. Topmost region, approximately 150 ˚ A in thickness: a high degree of disorder and deficient stoichiometry, enhancement of indium. The degradation of electrical and structural properties can cause a complete failure of the rectifying conduct of Schottky diodes. 2. The second zone, beneath, approximately 150 to 400 ˚ A in thickness, no stoichiometric deviations, is characterized by these features: • Significant lattice disordering. • Aggregated defects which can be detected by ion channeling and point defects which cause a compensation of shallow doping levels in this zone. 3. This layer blankets isolated point defects which can be detected down to a depth of about 1000 ˚ A. The ubiqity of hydrogen leads to an almost inevitable contamination during the various states of processing: • Growing processes with metal-organic chemical vapor deposition [MOCVD, carrier gas or source gas, this does not apply for molecular beam epitaxy (MBE)]. • Hydrogen-containing reagent for etching (wet or dry). • Annealing processes in hydrogen. In III/V semiconductors, the p-dopant substitutes a lattice site (Mg occupies a Ga site in GaN, Zn a III site in GaAs or InP). Hydrogen passivates by formation of a strong V-H bond [691] which is made use of in a quantitative determination of the passivation by correlating an electrical measurement with the IR spectrum [692]. The position of the absorption peak defines the strength of the bond between hydrogen and the group-V element, by the intensity, the degree of damage is calculated.

11.8 Process control

499

In GaP, GaAs, and InP, this passivation can be almost completely reverted by annealing at 400 ◦ C for 1 min [693]; after 20 min, absolutely no passivation is detectable [694]. This is not valid for GaN; here, much higher temperatures are required. Low-damage etchings. Low-damage etching is one of the main advantages of etching with high-density plasmas. One of the first proofs was presented by Constantine et al. and Pearton et al. [678, 695]. Introducing a patterning process of high an anisotropy with an aspect ratio of more than ten without any substrate bias, GaAs and InP were etched in an ECR generated plasma with very low damage. This has been proven by several physical methods: • The I(V) characteristic of Schottky diodes has been compared with their counterparts which had been patterned by wet etching and by ion beam etching. The inverse voltage of the latter exceeded that of the other two by about 10 %, and there was scarcely any difference between the wet etched sample and the ECR etched sample, driven at zero bias voltage. • An independent proof could be submitted by the reduction of the PL signal in GaAs for increasing the DC bias by more than 50 V. The curve at 100 V contrasts sharply with the array of curves for 0 V, 50 V and wet eched sample which are almost indistinguishable.

11.8 Process control The superior performance of a dry etching process is mainly based upon the mechanism of ion-induced etching, of a synergistic interplay between chemical interaction and physical influence of a plasma (momentum transfer of ions and molecular excitement by electrons and photons). But here is also a weak point: Looking at the sputtering yields for argon, the values only differ by one order of magnitude between an element with low yield such as carbon (graphite) or a dielectric as sapphire (Al2 O3 ) and an element with high sputtering yield, for example gold. In other words: The selectivity S, the difference between the removal rates, does not exceed the value of 20 very frequently, even with chemical support. Turned into a positive context, the outcome parameters, most prominently the etchrate, do not depend very sensitively on the primary parameters, resulting in a certain robustness with respect to accidental deviations of pressure, gas flow and power. This dilemma sheds light on the efforts to develop reliable methods for process control, i. e. methods to detect an endpoint instantaneously. Considering an accuracy of ±25 nm in depth at a total etch depth between 3 and 5 μm, a resolution in length has to be achieved within ±1 %. For etchrates of only 50 nm/min, changes of a signal have to be qualified as endpoint signal within ±15 sec.

500


As we have seen in the preceding chapters, plasma processes depend on several parameters which can be evaluated by suitable probes, and the effect of the influencing properties, either solely or concerted, on the conduct of the process has to be made transparent by an in-situ measurement. Irrespective of whether deposition or etching is concerned, the control has to be carried out by measuring the deposition rate or etchrate or, even better, by tracing a physical property that suddenly changes, i. e. real end point detection. Both requirements show a certain overlap and are discussed together, replacing removal by growth in the continuous text. For on-line process control, two classes of methods can be applied in principle. They mainly differ in the kind of data collection: either gathering of signals from the sample or from the plasma bulk. Only methods which belong to the first class allow the absolute measurement of different levels. • Surface-sensitive class: – Ellipsometry. – Interferometric methods with lasers (LI). • Global class: – Impedance of the discharge. – Optical emission spectroscopy (OES). – Mass spectrometry (MS). 11.8.1 Impedance of a discharge Since the impedance of a discharge is strongly influenced by the parameters plasma density, surface quality (morphology, electric conductivity . . . ), and the composition of the ambient, variations of the substrate bias can be monitored for endpoint detection. In the case of ashing of photoresist, this value has been observed to suddenly decline precipitously by a factor of 2, but most frequently the change remains within several percent, dependent on the load (Fig. 11.51). From its characteristic, it has to be counted to the integrating methods which make use of a relatively weak discharge signal, but the other side of the coin is its relative robustness . . . and it is easily applicable. In semiconductor processing, however, this method has been replaced by the other methods which will be described in the following sections. 11.8.2 Ellipsometry Ellipsometry makes use of the fact that the degree of polarization is sensitively changed by reflection. The coating of a surface with only one atomic layer causes a measurable change in the degree of polarization. As measured quantities, the optical constants n and κ (refraction index and absorption coefficient) of the


501

0.75 lAl

signal [a. u.]

start

Fig. 11.51. Time-dependent courses of two signals: The optical emission and the impedance during etching of aluminum show a synchronous signal. The relative change in impedance is better defined than the optical signal [696] c The American Institute of ( Physics).

endpoint

0.50 VDC

0.25

0.00 0

ts ~ 7 sec

2

ts ~ 9 sec

4

6

8

10

12

t [min]

substrate and of the film are used to evaluate a complicated relation between film thickness and the optical constants, which includes measured (1 − 3) and calculated (4) quantities: ∼

1. The optical constants n and κ (n= n − iκ). ∼

∼

2. The relation between the (complex) reflected amplitudes E r / E r⊥ , most frequently measured at the angle of main incidence φ = Φ (at this angle, ∼

∼

the phase difference between E and E ⊥ amounts to exactly 90◦ ). 3. The phase difference δ. 4. Fresnel’s reflection coefficients for the ratio of the amplitudes of the reflected and the incident beam for weakly absorbing media (for strong absorbtion in the extended Beer’s formula): ∼

R⊥ =

E r⊥ ∼

=−

E e⊥ ∼

R =

E r ∼

E e

=

sin(φ − χ) sin(φ + χ)

tan(φ − χ) tan(φ + χ)

(11.37.1)

(11.37.2)

with φ the angle of the incident beam and χ that of the refracted beam, and the final equation reads ∼

E r ∼

E r⊥ with

= tan Ψeiδ =

(R1 + R2 e−2ix )(1 + R⊥1 R⊥2 e−2ix ) (1 + R1 R2 e−2ix )(R⊥1 + R⊥2 e−2ix )

(11.38)

502


reflected beam

wafer

incident beam

ion beam

sender

analyzerr

Fig. 11.52. In-situ measurement during an etching process in an ion beam system c Veeco Instruments 2002. employing ellipsometry

x=

2π 2 d n1 − n20 sin2 Φ, λ

(11.39)

and R1 , R2 Beer’s or Fresnel’s coefficients for the interfaces ambient/film and film/substrate, resp.; n0 and n1 are the refraction indices of the ambient and the film, resp., and d denotes the film thickness and Φ the angle of main incidence. The measurement is carried out with a laser beam which is elliptically polarized by a λ/4 sheet and is reflected at the sample. As a function of film thickness and its optical constants, the polarization of the beam is altered. After having passed an analysator sheet, the beam intensity is detected by a photodiode and evaluated. In the past, ellipsometry was a valuable tool to control the quality of grown films ex-post. For a long-wavelength laser, the layers to be characterized are transparent, and employing modern computers, this method has evolved to be a valuable tool to control layer growth in-situ (Fig. 11.52) [697]. 11.8.3 Optical emission spectroscopy The third method is based upon the frequency selective compilation of emission lines in the NUV/VIS/NIR range of the plasma (OES). That emission lines are observed which are located in the visible range with their adjacent rims in the near UV/IR range is motivated by two facts: • The etchant gases are transparent. • Strong lines simplify the tracing even for very low signals when the end point is approached (Table 11.3). The light, which is subjected to spectroscopic analysis, is transmitted via a quartz window and a glass fiber to an optical grid. The window can be orientated in parallel fashion with respect to the probe (one looks directly upon the surface


503

Table 11.3. Common lines for endpoint detection employing OES. film to be etched PR Si Si3 N4 Si3 N4 SiO2 Al Al GaAs GaAs GaAs InP InP

volatile component CO SiF∗ N∗ CN∗ CO AlCl∗ Al∗ As∗ Ga∗ GaCl∗ In∗ InCl∗

wavelength [nm] 297.7; 483.5; 519.5 777.0 674.0 388.3 297.7; 458.3; 519.5 261.4; 279.0 396.2 228.9; 235.0; 245.7; 278.0; 286.0 287.4; 403.3; 417.2 249.1; 334.8; 338.5 325.6; 410.1; 451.1 (fluorescence) 267.3; 350.0

2 convex lenses glass fiber to spectrograph (OMA) small sample in the center of the “hot” electrode

Fig. 11.53. Since the area of the sample can be very small with respect to the area of the electrode, the application of a focusing optics is strongly recommended to couple the light of the sample directly into the glass fiber [698].

of the wafer through the plasma bulk and the sheath); other systems employ a focusing lens system which is orientated normal to the wafer (Fig. 11.53). The grid is often coated with a metal (blazing); for certain values of the angle of incidence we observe highlighted specular reflection. The spectrum is either amplified by a multiplier or analyzed by an Optical Multichannel Analyzer (OMA). For endpoint detection, a certain frequency band and its temporal variation is traced. OES as introduced so far is a real instantaneous method and does not interfere with processes since it is non-invasive. Due to the richness of lines in this range, OES requires a well-resolving grid with 1 200, better 2 400, grooves per mm which ensures a spectral resolution of 2.5 × 10−3 or better. As a typical example, an overlook spectrum is recorded during etching of GaAs in a discharge of Cl2 /Ar (Fig. 11.54). We recognize the broad band of Cl+ 2 which is centered at 454.9 nm, but effectively covers the range between 375 nm and 550 nm topped by several lines of atomic chlorine, Cl(I), but also a very prominent “fingerprint”

504


region which is well-known as GaCl hand which tops another broad molecular band, here caused by Cl2 [418]. 1500

Fig. 11.54. Overlook spectrum recorded during the etching of GaAs in Cl2 /Ar. Very prominent is the GaCl hand with 5 fingers which tops a broad Cl2 band [418].

1250 intensity [a. u.]

GaCl hand

1000 750 500 329.5 nm: Cl2: 1 P 1u

250 0

X1Sg+

454.9 nm: Cl2+: A 2P u X2 Pg 438.8 nm: Cl(I): upper state: 11.74 eV 452.2 nm: Cl(I): upper state: 11.94 eV

300

400

500

l [nm]

A typical case of monitoring, the fabrication of a VCSEL (Vertical Cavity Surface Emitting Laser) with incorporated endpoint structure is shown in Fig. 11.55. At least two lines of gallium are subjected for tracing which are due to Cap-Layer cap layer

p-doped p-doped top mirror Top Mirror Active activeRegion zone

n-doped n-doped bottom mirror Bottom Mirror

Substrat Substrate

AlAs GaAs/AlAs-Superlattice AlAs/GaAs superlattice

GaAs GaAs

Fig. 11.55. Fundamental structure of a VCSEL (LHS) and a real structure, carved out of GaAs/AlGaAs (RHS). A typical VCSEL consists of an active layer of quantum wells (totaling 0.2 μm in thickness) whose dimensions and material will determine the emitted wavelength with adjoining Bragg mirrors in upward and downward directions. Either of them is piled up of about 20 pairs of λ/4 layers of AlAs/GaAs.

the transitions at • 403.3 nm (52 S1/2 → 42 P1/2 ), and • 417.2 nm (52 S1/2 → 42 P3/2 ).


505

As can be deduced from Fig. 11.56, the upper stack of consecutive pairs of AlGaAs/GaAs can be perfectly resolved and marks a good contrast with the adjoining spacer. In most cases, the dynamics deteriorates during the course of the process, in particular for small samples [(about 1/4 2"), the larger the open surface exposed to the plasma, the smaller the loading effect of first order].

relative intensity [a. u.]

1.00

0.75

0.50

0.25

0.00 0

10

20 t [min]

30

Fig. 11.56. OES spectrum (Ga line at 403.3 nm) of a VCSEL structure with upper mirror consisting of a stack of consecutive pairs of AlGaAs/AlAs layers, a triple sandwich (spacer) which is adjoined by a double sandwich in front of the active zone. Etching must be terminated in the double sandwich at latest.

The spatial dependence of excitation is the result of two opposing effects. • Approaching the electrode from the Bohm edge, the electric field steeply rises, and the energy of the electrons is so large that non-radiative transitions are most likely to be excited. • Going in an outward direction from the substrate, the density of removed layer constituents drops with the inverted distance squared. According to Herzberg, the lifetime of the excited states lies within the span 1 to 10 nsec [699]. Based on this assumption, it may be concluded that the removed atoms have passed only a fraction of their mean free path (several hundreds of mikrons) between excitation and emission, and the probability of excitation can be regarded as constant; in particular, it is independent of discharge pressure and voltage drop. Consequently, the line intensity scales with the etchrate [700] and is suitable as method for endpoint detection with two main possible tracks: 1. The line(s) of a volatile reaction product are traced temporarily. The endpoint is indicated by a steep decrease in intensity. 2. Provided the lines of the etchant are known, these are subject to the spactral analysis. Approaching the endpoint is manifested in a rise in intensity since the etchant is not consumed any longer, and its concentration increases.

506


In the first case, the intensity of a spectral line of a species with a very low concentration is traced, whereas a main component is measured in the latter case. The quality of endpoint detection depends mainly on the detector (resolution power, which should be better than 1 ˚ A, and sensitivity), and on the loading effect of first order, in particular in the second case. To attain a certain detectable limit, a certain load is required certainly, but on the other hand, care has to be taken to avoid an overload: Since the etchrate rises steeply for reducing load, the flux of the reaction product remains nearly constant just when the endpoint is approached. Consequently, lines of the etchant and the product should be traced simultaneously. 11.8.4 Interferometric methods Laser reflectometry or laser interferometry (LI) uses temporal changes in light intensity which are caused by reflections at surfaces or interfaces. We can distinguish three cases: • Metal layers. • Dielectric layers. • Layers of semiconductors. 11.8.4.1 Metals and dielectrics. For normal incidence, the reflectivity of a film on top of a substrate can be determined according to Beer’s formula (Fig. 11.57): 0 0 0 E 02 κ21 + (n1 − 1)2 0 r0 0 0 =R= 2 0 Ee 0 κ1 + (n1 + 1)2

(11.40)

with n1 and κ1 the refraction index and the absorption index, resp., of this layer ∼ (n= n − iκ) and E r as well as E e the amplitudes of the reflected and incident wave, resp. Metals are characterized by very high reflection coefficients R which means κ n ⇒ R ≈ 1. During the course of the etching of a metal layer which has been deposited on a low-reflecting substrate, R remains constant and suddenly drops precipitously when the metal layer becomes transparent. In the case of a stack made of several metal layers with comparable κ, however, their difference goes to zero, which excludes this method from successful application. Only a large difference in n allows reliable tracing. To make matters worse, chemical reactions can cause surface tarnishing by which the reflectivity can be reduced drastically. For dielectric layers, κ n, and the beam reflected at the surface will interfere with the beam which is reflected at the lower interface of the film. If a laser is directed to a stack of various layers, an interference pattern is observed, and an extremum occurs for the condition

11.8 Process control beam splitter AlGaAs laser

507

optical fiber MFCs x-y table Ar BCl3 CH gas inlet H2 4

mirrors monochromator photomultiplier or photodiode array

anode wafer

HR grating

powered electrode RIE chamber RF generator + pumping system

RF

matching network

computer

Fig. 11.57. In-situ control with laser interferometry (LI). Whereas the temporal dependence of the emission lines is traced by optical emission spectroscopy (OES) [e. g. A (52 S1/2 → 42 P3/2 )] [701], for Ga the lines at 4033 ˚ A (52 S1/2 → 42 P1/2 ) and 4172 ˚ LI uses the variations in intensity of the reflected light of a solid state laser which are caused by various refraction indices.

λ Δd =N (11.41) n 2 with Δd the residual geometric thickness (multiplied by n to obtain the optical thickness) and the RHS denotes multiples of 1/2 λ with N a whole number: The intensity of the reflected beams varies sinusoidally with the residual thickness and is a periodic function of n1 Δd/λ. For known dispersion curve of the substrates, the signal can be calculated previously and can be compared on-line with the experimental interferogram [Figs. 11.58, Eq. (11.42) [702]]:

R=

n21 (1 − n2 )2 + (n2 − n21 )2 + (1 − n21 )(n21 − n22 ) · cos n1 4πd λ , n21 (1 + n2 )2 + (n2 + n21 )2 + (1 − n21 )(n21 − n22 ) · cos n1 4πd λ

(11.42)

with n2 the refraction index of the substrate. The etchrate ER and the growth rate GR, can easily be determined by measuring the time between two extrema to get a value in real time. 11.8.4.2 Semiconductors. This method can be applied for semiconductor etchings as well, with one restriction. Since Eqs. (11.41) are applicable only for transparent films (κ n), and semiconductors become transparent at near infrared has been reached (λ ≥ 1 μm), infrared light has to be applied. The application of a wavelength of 1 μm leads to an uncertainty of about 1000 ˚ A (n = 3.5) or one part in ten, and for even longer wavelengths, the absolute uncertainty will further deteriorate (at 5 μm Δλ/λ ≥ 0.5 μm).

508

11 Plasma etch processes 1.0 0.8 rel. intensity

calc. intensity [a. u.]

40

30

0.6 0.4 0.2

20

0

5

10

0.0

t [ a. u.]

0

5

10 15 t [min]

20

25

Fig. 11.58. An endpoint detection employing a laser interferometer (top, RHS) and the comparison with its interferogram which has been calculated previously (top, LHS) and the result as SEM micrograph (bottom) [601]. The large amplitudes are caused by the considerable differences in the refraction indices between AlAs and GaAs.

In conclusion, punctate measurements are possible with the laser interferometer, which are required for a sharp interferogram.

11.8.5 CCD controlled laser interferometry Employing a beam divider opens the window for two-beam interferometry. For comparison, the measuring beam is directed at the shiny area of the substrate and is analyzed by the reference beam which is reflected by a non-tilted mirror, for example, in a CCD camera [703] (Fig. 11.59) and the measuring of the depth is traced back to the counting of the number of maxima of wave trains. To measure the absolute depth, the measuring beam is split and is simultaneously directed to a reference zone with a lateral dimension of typical 300 × 150 μm2 , and to the etching area. Due to the growing difference in height between etching level and reference level, the divided beams get out of step, and the phase shift is measured electronically by means of the reference beam, which is reflected by the slightly tilted mirror into the optics (Fig. 11.60).


509

Frame grabber

CCD - Camera PC

optical system

Wafer area

Shutter for mode selection:

Interferometer

• shutter open:

1 mm²

depth measurement

Monitor Beam splitter Laser

Collimator

Intensity tuner

• shutter closed:

a

Intensity tuner

endpoint detection

Shutter

150 mm

Reference mirror

XY-table (± 25 mm)

Tilted reference mirror to produce interference pattern on CCD matrix

Wafer

Plasma Chamber

50 mm

Fig. 11.59. Experimental set-up of a modified laser interferometer after John et al. [703].

masked surface

> 300 µm

etched surface

On chip regions to be analyzed interference pattern

> 150 µm

Layout phase shift (d)

d1

reference area

phase shift = f (etch depth) pixels on camera chip

d2

d = 4pd/l d : etch depth

measurement area

l : wave length

Fig. 11.60. By scanning two areas which differ in their height, the two fractions of the measuring beam develop a phase shift which is made optically visible by interference with the reference beam, thereby allowing a real and direct measurement of the etched depth or height [703].

510


This configuration is superior to “simple” interferometry since measurements in real space are now feasible. This method can not only be applied for pure semiconductor etching but also for the etching of metallic films on top of semiconductors where the vast difference in reflectivities between metals and semiconductors is made use of (Sect. 14.6, Figs. 11.61). Reflectivites can be measured simply by removing the reference mirror from the optical path, keeping in mind that the area of the laser spot determines the lateral resolution in the conventional measurement. John et al. employ a CCD matrix as receiving element, and the laser illuminates a surface region measuring 3 mm2 at maximum. Since each picture element (pixel) can be read out individually, the area which is subjected to simultaneous data recording is limited by the lower value of 1 μm2 and its upper counterpart of about 3 mm2 . The spots which are to be inspected can be searched and subsequently fixed by a optics equipped with a macro lens optics. 1.0

etch depth etchrate

200 rel. intensity [a. u.]

rel. reflectivity

0.9 start

end

0.8

0.7 0.6

0

60

120 180 t [sec]

240

300

end begin InGaAs GaAs etching

150 100 50 0 -60

end GaAs etching

begin InGaAs etching

0

60

120 180 240 300 360 420 t [sec]

Fig. 11.61. LHS.: Temporal resolved reflectivity of a spot of WSiNx (area: 1×1μm2 ), reference mirror not in optical path. RHS: Absolute depth and the calculated etchrate in a sandwiched layer system of InGaAs/GaAs (Cl2 , capacitively coupled plasma, reference mirror in optical path) [703].

In conclusion, it is possible to record laterally resolved measurements of the reflectivity and in-situ depth-resolved measurements in one analytic instrument.

11.8.6 Mass spectrometry For mass spectrometric diagnosis, pressures of less than 0.1 mTorr ( . (12.42) 4 ∗ Bohm velocity (continuity equation at the border between sheath and Bohmic presheath): ΓCl2 =

vB =

nCl+2 vB kB Te ∧ ΓCl2 + = . mi 4

(12.43)

– Distribution functions of the electron energy EEDF or, even better but rarely well defined, the electron temperature Te . – Density of the charged carriers (positive and negative), but at least the electron density ne . • Surface – Sheath potential Φs (sum of plasma potential and electrode potential; either of them has a DC-component and an RF-component) since in high-density plasmas additional etching occurs triggered by ions: ER+ = k+ ECl+2 ΓCl+2 (1 − Θ),

(12.44)

k+ ∝ (Vp + VDC ).

(12.45)

with

– Rate constants of the reactions which are, in turn, complicated functions of the involved plasma parameters and the gaseous temperature Tbulk . – Sidewall passivation. – IEDF and IADF which mirror the collisions with neutrals on the course of the ions across the sheath. Exemplary for the various interdependences, we should consider qualitatively the decline of the discharge pressure in high-density plasmas with simultaneous increase of the plasma density. The sheath thickness will shrink to such low values that the sheath can be considered collisionless. The ion current incident on the electrode becomes more beam-like: Not only the distribution of the kinetic energy of the ions (IEDF) but also their angle distribution IADF) becomes sharper. In particular, this leads to an increase of the parallel component of the velocity with respect to the electric field [764]. To sum up, a multitude of processes has to be taken into account, and information can and must be obtained even from observations farther away from this focus. For example, an observed damage feature can be caused by a certain

12.9 Simulation of dry etching processes

555

surface reaction [765]. The simulation of these profiles was and has remained a real challenge which has been mastered by several authors and groups in the last decade. Among them are the groups of Sawin at MIT [238], Alkire at the University of Illinois [648], Rangelow at the University of Kassel (now at Technical University of Ilmenau) [766], McVittie at Stanford University [767], Jurgensen at AT & T [768]. All these methods have in common that the distribution functions IEDF and IADF which have been obtained by MonteCarlo calculations introduced in Sect. 6.8 are combined with the surface kinetics of the specific reaction. The simulation programs are composed of several modules to take various aspects and requirements into account (ion etching, plasma etching, pressure dependence, power absorption . . . ). Besides the modeling of plasma reactions, this is synonymous with the inclusion of problems concerning the kinetics at the surface, the specific reaction of interest, but also a possible split into a physical fraction and a chemical one to describe the row of secondary effects such as influences caused by the mask and its material: Redeposition, sidewall passivation, shadowing, micromasking, trenching; concerning depositions, also surface diffusion and bias effects which influence the coating over sharply edged corners should be considered. Though the modularization into single algorithms facilitates the application, it should be stressed that a multitude of factors has not been satisfactorily dealt with. Among them are: • Influence of ultraviolet radiation and rapid electrons on surface reactions. • Synergies between “physical” and “chemical” processes. • Alteration of mechanisms until a stable state has been reached (constant temperature, constant plasma parameter). The ions entering the sheath of the RF driven electrode are first considered beam-like. This beam will broaden in energy and angle by collisions with neutrals. No matter how large the discharge pressure: The mean free path of the ions, λi , is always large compared with the dimensions of the structure. Consequently, the ion trajectories are straight lines which are subject to a distribution with respect to the angle of incidence. But this distribution, the IADF, is an energy function again: Ions which have only a component parallel with respect to the electric sheath field exhibit the topmost energy, whereas ions with a component normal with respect to the electric field show the lowest values of this distribution function. As result, the sidewalls are subjected to a bombardment of ions with medium energy, and the resulting etchrate is definitely lower than in the vertical direction. If the sidewall is protected by a polymeric layer, the sidewalls can be expected to show a vertical slope, in the case it is not, other forms will be found (sidewall bowing when the walls are charged). For a purely ion-induced mechanism, the yield

556

12 Etch Mechanisms

Al mask

Al mask

Si

Al mask

Si

Al mask

Al mask

Si

Al mask

Si Si

Si

Fig. 12.20. A via-hole dug into silicon in a plasma through SF6 /O2 (top, LHS) and its simulation under consideration of a fully developed IADF and sidewall passivation (top, RHS), neglecting sidewall passivation (bottom, LHS) and neglecting of any IADF (bottom, RHS) [766].

scales with the energy of the bombarding ions, and the result is the well-known graph for the angle dependence of the sputtering rate which can suppressed only by perfect sidewall passivation. The energy flux has been modeled as double integral over the energyweighted IADF by Shaqfeh and Jurgensen [768]. Assuming the yield per bombarding particle scales with its energy, the etchrate should be proportional to the energy flux incident on the surface (in spherical coordinates): ER ∝

2π π ϕ=0

Γ(ϑ)E(ϑ) sin2 ϑdϑ dϕ

(12.46)

ϑ=0

with Γ(ϑ) the ion flux and E(ϑ) its mean energy—we know that the heavy ions see but the mean value of the sheath potential. On a boundless expanse, the flux of the ions is symmetric to the azimuth which is orientated normal with respect to the electric field; consequently, the energy flux does not show any angular dependence. Approaching the mask, the flux of ions is reduced by shadowing, and it is those energy-deficient ions and the neutrals which have been generated by symmetric charge transfer which have got lost. The main result is the foot at the basis of the etched structure. Comparing the various programs, the resulting shapes and patterns differ but in nuances. This should be demonstrated by a via-hole dug into silicon

12.9 Simulation of dry etching processes

557

employing a capacitively coupled plasma through CF4 /O2 and its simulation with sidewall passivation and fully developed IADF (Fig. 12.20) [766, 769]. Also in plasma physics, many roads lead to Rome.

13 Outlook

In an approximation to zero order, the plasmas differ with respect to their discharge pressures [ion beam methods (IB) which can be operated down to 10−4 Torr or 0.1 Pa to microwave operation which work even at atmospheric pressure] and operating frequencies [DC to MW operation (technical frequency: 2.45 GHz)]. Finer tuning leads to resonant methods (ECR and helicon) and non-resonant techniques (CCP and ICP). Concerning the type of etching, we discriminate between ion beam etching [(IBE), pressure below 10−4 Torr], ion etching (at least one order of magnitude higher), and plasma etching (another order of magnitude higher). Resonant methods are operated in the intermediate range between ion beam etching and ion etching. For resonant methods, the etch rates are second to none, next in line is ICP which is superior to CCP. We can also make a distinction between etching reactors which are operated downstream (ECR, helicon, ICP, and IB) and others with a parallel-plate reactor (DC and LF, CCP). The etch rates are steadily correlated with the plasma density but no longer to the applied voltage which was necessary in the stone age of plasma etching when the plasma was excited by a capacitively coupled electrode. The term ion etching was introduced when CCPs were in infancy, but we have seen that in these discharges, the ratio of highly excited neutrals to ions exceeds a factor of thousand with great ease. Therefore, ion etching in parallel-plate reactors is a hybrid of some “thoroughbred” processes: Ions are responsible for the anisotropic character, but high etch rates are made possible by the reactive neutrals. This happens also in ICP reactors where the majority of reactive species remains reactive neutrals still; but mainly caused by the higher plasma density, the issues radial uniformity and low etch rates have ceased to arouse interest. Due to the higher density, lower DC bias voltages are required to achieve high anisotropic etching characteristics, the sheaths can be thinner and are most frequently collisionless, which improves the controllability of the etching process in terms of precise endpoint, selectivity, and damage. The latter issue is inevitably allied with ion beam etching which has made use of high acceleration voltages until reactive processes were introduced. On the other hand, we do not want to forget the merits of this simple process since ion milling was the first method which even diamond could not withstand. Since then, more delicate forms of damage have come into the foreground which are more chemical in nature, and G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 13,

559

560

13 Outlook

a large palette of chemical methods to “count” atoms as well as physical and electrotechnical techniques has been developed to detect and quantify this damage.

Fig. 13.1. A shuffle wheel which has been carved out of a monocrystalline silicon membrane subsequently fixed on a dry-etched axis by a flip-chip technique to form a turbine [770].

Since optimum operating frequency and discharge are closely related, pure microwave discharge are not recommended for operation in the low mTorr range. This is the domain of capacitively coupled radio frequency discharges. Its lower pressure limit is given by the collapse of the ionization avalanche. At higher pressures, the number of collisions between electrons and neutrals rises steeply, and this deteriorates the efficiency of the intended power input to the electrons. As a direct consequence of this dissipation process, the gas is heated up, by which, in turn, the collision number is again enhanced, and the upper limit of power transfer is quickly reached. But these discharges can be driven at higher pressures, the requirements for a low base vacuum are not very critical, and microwave discharges act as workhorses for photoresist ashing and for degreasing purposes. In these discharges, the substrate is immersed by the glowing plasma, and the steering procedures to influence the energy of the impinging ions and

13 Outlook

561

their density are mutually coupled. This alliance can be split by discharges which are operated downstream, ICP being the most popular due to the non-resonant heating process which can easily be applied to reactors with large diameters.

Fig. 13.2. Cantilever with doped silicon tips for scanning tunneling microscopy [top: bar represents 1.2 mm (overview) or 0.12 mm (detail); bottom: bar represents 1 mm (overview, LHS), needles, RHS: vertical bar represents 10 μm for the right needle] [771].

Resonant techniques lack radial homogenity and their razor-sharp processes aggravate process stability, and fortunately, a lot of questions remain still open. For example, experimental proof should be added to the suggestion that the L-wave can transfer its energy by a Landau damping mechanism, the heating mechanism of the helicons has been elucidated just recently, but even lower azimuthal modes are difficult to grasp. Modeling of these discharges has not reached the state of CCPs yet, but for these discharges, stochastic heating is still an open issue. Deposition techniques have become commercially available by which not only deposition rates were boosted to unprecedented highs but which also opened the window to generate materials which are located far away from thermodynamic equilibrium. But one of the most spectacular developments has been the deposition of diamondlike films. With simple sputtering, it is possible to obtain carbonaceous layers which are similar to graphite, but reactive processes have opened the window for depositing low-pressure diamonds. As far as the role of chemistry for etching is concerned, reactive processes have increased etchrates and selectivities, the anisotropic character of etch processes could be dramatically enhanced by preferred etching in the vertical direction by ionic impact and by the mechanism of sidewall passivation. Isolating

562

13 Outlook

films (oxides, nitrides, carbides with very high evaporation energies) can be coated by RF sputtering in reactive ambients. By plasma treatment on molecular gases, a number of reactive intermediates is quickly generated which can form stable compounds in the plasma bulk or on surfaces, respectively. These deposits can be very sensitive to numerous reagents, and that is why a growing number of compounds has been successfully prepared in a glow discharge—as every chemist learns in his freshman lecture. It is modern myth to believe in high energies in a glow discharge: It is only the electrons which are heated up to energies of several electronvolts, but these electrons generate reactive intermediates which float in a sea of cold neutrals and can now react consecutively on unprecedented tracks. It is the nature of these intermediates to react sensitively on temperature which blocks the conventional chemical way of long refluxing procedures. The trend for high-end plasmas is low bias voltage combined with high plasma density in order to realize high processing rates with low surface damage which most frequently requires specular surface quality. After some years of falling behind, capacitively coupled plasmas have caught up to the inductively coupled plasmas again. After having achieved this progress, a basic principle could be reanimated: The lower the bias voltage the higher the selectivity of the process, the easier the definition of etch stop, especially where etching of chemically related layers is concerned with or without highly sophisticated methods for endpoint detection. But various pressure ranges and possiblities of plasma generation distinguish their complementary utilizability. For example, the low selectivity which is caused by the relatively high acceleration voltages of ion beam processing can be approved for dry etched facets of Fabry-Perot lasers. This progess in technology would not have been possible by the mutual driving of plasma processing and plasma technology. Numerous effects came to be better understood by employing diagnostic techniques which analyze the plasma parameters of second order [self-excited electron resonance spectroscopy (SEERS), energy-dispersive mass spectrometry (ED-MS) . . . ] which opens the possibility of finding the chemical parameters for plasma-enhanced processes by rational findings rather than empirical ones. These experimental and diagnostic efforts, in conjunction with the advanced theory, seem at last to supply us with a satisfactory understanding of most of the issues of low-temperature plasmas, although heritage from ancient times is still present; among them we find the word “ion etching” in thin plasmas which is rather a euphemism. Applied uncritically, it would mean that the etching is determined by particles which appear in the kinetical rate equation. However, ions will trigger the etching process and are mainly responsible for the anisotropic character of the process. When so-called ion etching was blossoming, real highdensity plasmas with ionization degrees of up to 100 % were developed. In the case of the helicons, details stood in the way of making these discharges a commercial success, however, it was a great time to pursue the development of the theory of helicon waves. The next generation of discharges, the inductively cou-

13 Outlook

563

pled plasmas, met the requirements of the semiconductor industry for stable processes with high etchrates of more than 1 μm/min even in “stubborn” materials. The availability of these processes opened the window for microtechnology, since it has become possible to realize anisotropic structures in the range of mm. The combination of these methods: Shallow but fine structures, combined with rougher, relief printing methods with LIGA-techniques1 or flip-chip techniques has led to microelectromechanical systems (MEMS). Here, the micro-turbine should be mentioned (Fig. 13.1), or ink-jet heads and pressure sensors; tips for scanning tunneling microscopy have been carved out of single-crystalline silicon (Figs. 13.2) [772]. The last case represents a monolithic integration of tip, and cantilever, pressure sensor and electronic control system. Plasma science is a typical cross-section science: Inputs from scattering experiments are required to improve the calculations for the IEDF, IADF, and EEDF; plasmas modify surfaces, so we need badly the input from surface science what sticking coefficients, desorption constants . . . is concerned. Completely new systems become reality, and such a potential is fueling growth in this young market, although a lot of problems still remain unsolved, e. g. the issue of reliability of micromirror-based switches. These three-dimensional devices have the micromirrors tilted freely on one or even two planes to route the light and could be the optical switches of the future [773]. As an example, the MEMS micromirror is displayed in Fig. 13.3. Capping of MEMS structure

Unrestricted MEMS post

Vacuum sealed cavity

All silicon package

c Fig. 13.3. MEMS micromirror as designed and fabricated by Silex Microsystems ( Silex Microsystems, 2007).

The synergistic acting in combination of numerous technologies has solved problems by unconventional paths which have been open questions for decades. The straightforward synthesis of diamonds is a prominent example which stunned the public. Cross-section technology—with these steps, plasma technology links microtechnology and information technology via hybrid transition forms up to monolithic integration [771] − [776]. The merging technologies have triggered a project funded by the European Commission to bridge the gap between photonics and nanotechnologies, being 1

German acronym for lithographic electroforming with subsequent (plastic) moulding.

564

13 Outlook

aware of the big chances when technologies of different maturity are brought together. Some of the ideas are displayed in Fig. 13.4 [777]. 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 silicon laser with silicon nanocrystals on silicon quantum dots silicon laser with erbium-doped silicon quantum dots improved semiconductor lasers based on III/V quantum dots microstructured fibers for long-haul transmission based on photonic crystals microstructured fibers for fiber-to-the home

white (LED) microstructured fibers for long-haul transmission based on photonic crystals hybrid solar cells based on II/VI semiconductors

basic R&D

applied R&D

first applications

mass production

Fig. 13.4. Intense European efforts to challenge the American leadership are bundled in the European technological roadmap for photonics and nanotechnologies [777].

The environmental tolerance of the plasma is one of the assets which cannot be underestimated. • For coatings, we can most frequently get rid of the highly toxic heavy metal compounds. • For etching processes, less toxic gaeseous products are generated, the amount of waste water is significantly reduced, and pollution abatement is no longer an issue. In the 1980s still, degreasing of surfaces most frequently happened by application of freons, even with a large content of chlorine; today, plasma ashers with a reactor volume of some m3 do their job (Fig. 13.5). Simultaneously, the plastic surfaces can be made hydrophilic by the implantation of polar groups2 which makes a finish with a permanent water-soluble laquer feasible [779]; the plastic surfaces can be subjected to glue, ink or metals as well. Thin foils are coated by so-called roll-to-roll coaters, employing ion-plating methods, very short process cycles can be realized [780], Fig. 13.6. 2 This is measured by the significant reduction of the contact angle against water which has been shown to drop from values of around 150 ◦ to values of about 10 ◦ , see [778].

13 Outlook

565 Fig. 13.5. Microwave asher which was used to activate the crash barriers which were mounted at the lower part of the doors of the W 126 of Daimler-Benz, the first automobile with plastic parts sprayed in car color. The dimensions of the reactor were (W/D/H): 2260 x 2000 x c Technics Plasma 2290 mm ( 1992).

The investment in a plasma reactor clearly exceeds the costs which are required to purchase a water bath with toxic solvents by orders of magnitude. This setback will be compensated by its economical long-term advantages, not to forget environmetal issues, taking into account the unpredictable prerequisties set by our governments.

pay-off coil

take-up coil

foil foil

in-situ thickness measurement to the pumping system of the tape chamber

plasma treatment

gas feed

to the pump system of the coating chamber

cooling roller

plasma evaporation source

Fig. 13.6. Scheme of a roll– to-roll coater with ion plating system to coat thin foils [780].

Plasma-enhanced methods opened the royal way to solve numerous issues of surface modification.

14 Advanced Topics

14.1 Electron Energy Distribution Functions (EEDF s) 14.1.1 Boltzmann Equation The energy distribution function of the electrons, EEDF, is the solution of the Boltzmann equation in plasmas which are dominated by binary collisions between electrons and neutrals, elastic and inelastic ones. Since it describes the density of the electrons in the six-dimensional phase space of space and momentum, it is an equation of continuity of gain and loss and can be written as f (r, v). Its product f (r, v) dr dv equals the number N of electrons within the volume element d3 r d3 v. Integration over momentum yields the electron density ne at point r: ne =

∞ −∞

f d3 v.

(14.1)

The distribution of velocities within an element of phase space is the result of gain and loss and can be described by e0 E ∂f ∇v f = + v ∇r f + ∂t me

∂f ∂t

.

(14.2)

bc

The first term describes the change in time, the second term reflects the current behavior of the electrons by diffusion, and the third term takes the drift behavior by an external electric field into account. By these three terms, the net current to and fro the considered volume element is encompassed. Collisions between electrons and neutrals are weighted by the Langevin energy loss parameter L = 2m/M ; therefore, their amount is but minute. In contrast, the collisions between electrons shape the EEDF considerably. 14.1.2 External field as small disturbance We start our analysis with a homogeneous plasma which is anisotropically distorted but by a small leaking electric field E. According to Winkler et al., F G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 14,

567

568

14 Advanced Topics

can be split into two terms [344]: a large amount, isotropic in character, f0 , and a small anisotropic contribution, φ: f = f0 + φ(v).

(14.3)

Since the anisotropic contribution peaks for v pointing in the direction of ∇v φ, this part can be written as v ∇φ(v). (14.4) v Substituting Eqs. (14.3) and (14.4) into Eq. (14.1) leads, after cumbersome algebra, to [781] f 1 (v) =

∂f0 v e0 E ∂v 2 f 1 + ∇r ·f 1 − = ∂t 3 3me v 2 ∂v

e0 E ∂f0 ∂f 1 + v∇r f0 − = ∂t me ∂v

∂f0 ∂t

∂f 1 ∂t

∂f0 + ∂t elast

+ elast

∂f 1 ∂t

∂f0 + ∂t inelast

+ inelast

∂f 1 ∂t

, (14.5.1) Coul

; (14.5.2) Coul

the three summands on the RHS describe the three types of binary collisions, elastic, inelastic, and Coulombic in character. 14.1.2.1 Elastic collisions.

∂f0 ∂t

me 1 ∂ 1 ∂ kB Tg ∂f0 = νm v 2 ; (νm v 3 f0 ) + 2 2 M v ∂v v ∂v M ∂v

elast

∂f 1 ∂t

(14.6.1)

= −νm f 1 ,

(14.6.2)

elast

with me the electron mass which collides with the neutral with mass M , and Tg the temperature of the heavy plasma constituents (gas temperature). 14.1.2.2 Inelastic collisions.

∂f0 ∂t

= inelast

v j

v

f0 (v )νi,j (v ) − f0 (v)νi,j (v) .

∂f 1 ∂t

(14.7.1)

= 0,

(14.7.2)

inelast

where Ekin = Ekin + Ei , and it will be summed over all possible processes j. The frequency of every inelastic collision j is approximated by νi = N/V vσi (v) with N the number of molecules within the volume V .

14.1 Electron Energy Distribution Functions (EEDFs)

569

Coulombic interaction can be neglected in low-density plasmas (i. e. up to a degree of ionization of approximately 10−4 ). This still remains valid in most capacitively coupled plasmas, but does not hold true for inductively coupled plasmas and state-of-the-art plasmas driven by ECR and helicons. In this approximation, the final form of Eqs. (14.5) reads

e0 E ∂ 2 ∂f0 v + ∇r f 1 − (v f 1 ) = ∂t 3 3me v 2 ∂v

νm kB Tg ∂f0 me 1 ∂ νm v 3 f0 + M v 2 ∂v me ∂v

+

j

v f ν − f0 νi,j ; v 0 i,j

(14.8.1)

∂f 1 e0 E ∂f0 + v∇r f0 − = −νm f 1 . ∂t me ∂v

(14.8.2)

14.1.3 Approximate solutions of the Boltzmann equation 14.1.3.1 High frequencies. For a high-frequency field E 0 eiωt , Eq. (14.8.1) can be simplified by neglecting gradients and the temporal dependence of f0 ,1 Eq. (14.8.1) will vanish for

v + f ν − f0 νi,j . v 0 i,j j (14.9) If the temporal dependence of f 1 varies harmonically with eiωt as well, a solution for f 1 can easily given with e0 E ∂ 2 me 1 ∂ νm kB Tg ∂f0 νm v 3 f0 + (v f 1 ) + 3me v 2 ∂v M v 2 ∂v me ∂v

f1 =

∂f0 e0 E , me (νm − iω) ∂v

(14.10)

which yields 1 ∂ 2v 2 ∂v

kB Tg ∂f0 e0 E02 νm v 2 ∂f0 2m + νm v 2 vf0 + 3m2e (νm + ω)2 ∂v M M ∂v +

j

v f ν − f0 νi,j v 0 i,j

+

= 0.

(14.11.1)

by inserting Eq. (14.10) into Eq. (14.9), provided the scalar product between f 1 and E can be regarded as the time-averaged mean value of the real parts of the two vectors (cf. Chap. 5).2 This equation equation describes the transfer of electric energy to the electrons. This will become evident by multiplying each term of Eq. (14.11.1) with ∂f0 /∂t is small against f0 for ω Lνm . 2 2 In Eq. (14.11), we recognize the effective field Eeff in the term 1/2 E02 νm /(νm + ω 2 ) of Eqs. (5.11). Therefore, Eq. (14.11) remains valid for a DC field as well Eeff = EDC . 1 2

570

14 Advanced Topics

the term for the kinetic energy and by subsequent integration over all velocities. Provided that νm does not depend on v (over the whole energy range, this is valid only for H2 and He, however, for most gases also for the high-energy tail of f ), this yields [23, 58] (e0 Eeff )2 ∞ f0 4πv 2 dv = m e νm 0 ∞ ∞ 2me me v 2 2me 3kB Tg f0 4πv 2 dv− νm f0 4πv 2 dv − νm M 2 M 2 0 0

−

∞ j

0

v f ν − f0 νi,j v 0 i,j

mv 2 4πv 2 dv. 2

(14.11.2)

• The term on the LHS describes the rate for transferring the energy of the electric field to the electrons. In particular, for very high operating frequencies (microwaves), the dependence of the Boltzmann equation on the two parameters E0 /n and E0 /ω is reduced to the second parameter [782]. • The three terms on the RHS represent: – The dissipation of electronic energy by elastic collisions. – The energy gain of the (slow) electrons by collisions with (rapid) neutrals. – The energy loss of the electrons by Coulombic collisions. 14.1.3.2 Maxwellian distribution. For vanishing electric field and neglect of inelastic collisions, we get the approximation of first order from Eq. (14.11.1): vf0 +

kB Tg ∂f0 = 0. me ∂v

(14.12.1)

In this case, the distribution function f0 is given by f0 = C e−me v

2 /2k

B Tg

,

(14.12.2)

and the normalization constant C will be determined by the normalization condition n = 4π

∞ 0

f0 v 2 dv.

(14.13)

According to this approach, the electrons are in thermal equilibrium with the gas molecules with temperature Tg .


571

14.1.3.3 Margenau distribution. For small electric fields, the term for the inelastic collisions in Eqs. (14.11) will vanish due to the very low electron density: 1 ∂ v 2 ∂v

2 2 kB Tg ∂f0 e20 Eeff v ∂f0 me + νm v 2 vf0 + 3m2e νm ∂v M M ∂v

= 0;

(14.14.1)

for vanishing electric field, f0 becomes Maxwellian distributed, and taking Eq. (14.12.2) into account, the integration yields

v

2

2 kB Tg ∂f0 e20 Eeff ∂f0 me + νm vf0 + 2 3me νm ∂v M me ∂v

= 0,

(14.14.2)

from which we obtain

2 e20 Eeff M ∂f0 + kB Tg + me vf0 = 0, 2 2 3me νm ∂v

(14.14.3)

which gives as solution along with the normalization condition for the constant [Eq. (14.13)]

f0 = C exp −

v

0

me vdv 2 2 kB Tg + e20 Eeff M/3m2e νm

,

(14.15)

the Margenau distribution. 14.1.3.4 Druyvesteynian distribution. Does the second term containing the effective electric field in Eq. (14.15) exceed the first containing kB Tg consid2 (DC case) erably, then we can solve the Margenau distribution for ω 2 νm 2 2 and for ω νm (microwave case) and obtain an analytic solution remembering 2 lim Eeff =

ω→0

E02 E 2 νm 2 = 0 2, ∧ lim Eeff νm →0 2 2 ω

(14.16)

which yields in the DC case

f0 = C exp −

v

0

me vdv 2 e20 E02 M/6m2e νm

,

(14.17.1)

.

(14.17.2)

and for the microwave case

f0 = C exp −

v 0

me vdv e20 E02 M/6m2e ω 2

f0 can be solved analytically for two special cases: • Provided that the cross section σ(ve ) scales with the inverted velocity of the electrons ve , the frequency for momentum transfer, νm = N/V σ(ve )ve will become independent of ve since both the terms are neutralized (see

572

14 Advanced Topics the discussion in Sect. 2.3). In this special case (H2 und He), νm only depends on the number density n = N/V , or n ∝ p ∧ n ∝ 1/Tg :

me v 2 f0 = C exp − 2) 2e20 E02 M/(6m2e νm

,

(14.18.1)

which is the same algebraic form as Eq. (14.12.2) for kB Te =

e20 E02 M , 2 6m2e νm

(14.18.2)

and the distribution becomes Maxwellian again. • In the other borderline case, the cross section does not depend on the electron velocity at all, and νm = σm N/V v becomes a linear function in velocity and pressure (provided that Tg = const), and we can write for the Margenau distribution:

f0 = C exp −

0

v

2 3 6m3e N 2 σm v dv e20 E02 M

,

(14.18.3)

,

(14.18.4)

which yields

(me v 2 /2)2 f0 = C exp − 2 2 2 e0 E0 M/6me N 2 σm

the Druyvesteynian distribution. We can infer from Eq. (14.18.4) a 4 variation with e−Bv , and electrons distributed Druyvesteynian distribution exhibit a lower tail of high-energy electrons than those distributed according to Maxwell-Boltzmann for same mean energy < E > (Fig. 14.1). The Druyvesteynian distribution describes the conduct of electrons in low-frequency electric field which exchange energy with neutrals by elastic collisions assuming the mean free path λe or the collision cross section for momentum transfer σ to be independent of energy. Translated into energies, this formula reads for argon √

E n(E) = C π exp −0.54 <E >

2

,

(14.19)

For the integration constant C, see for example [783]. Plotting the logarithm of the distribution functions vs. energy, we find a descending straight line for Maxwellian distributed electrons, but for Druyvesteynian distributed electrons an inverted parabola (Fig. 14.2). In both cases, the electron temperature depends on E0 /p:3 3 The electron density, however, does influence the electron temperature with rising degree of ionization.


573

0.2 f(E)

= 2.5 eV

Fig. 14.1. Comparison of the distributions according to Maxwell-Boltzmann (MB) and Druyvesteyn (D) for two mean energies < E > c Review Modern (after [339] Physics).


0.1

= 10 eV

0.0 0

10

20 30 Te [eV]

40

50

0 -2

D MB

Fig. 14.2. Caused by collisions between the electrons, the Druyvesteynian distribution which is valid for weak electric fields will be transferred to a Maxwellian disc Pergamon tribution [784] ( Press PLC).

ln f

-4 -6

MB D

-8

Z=5 Z=50

-10 0

2

4 Ekin/kBTe

6

8

• σm (v) ∝ 1/v: νm ∝ p as well as n ∝ p/Tg ⇒ Eq. (14.18.2). It should be kept in mind that even for the higher inert gases, the functional behavior beyond the peak follows 1/v; and the peak itself is caused by quntum mechnanical effects of higher order (Figs. 2.9 + 2.10 with the Ramsauer minimum).

• σm = const: In the denominator of Eq. (14.18.4), we recognize the mean electron energy squared. Since the gas density remains constant for Tg = const, the mean energy of the electrons depends only on E0 /n or E0 /p. The latter is an extraordinary physical property since it constitutes the circle of similarity rules (cf. Sect. 4.10, where we have found the transferred energy to the electrons to scale with E0 /p).

574

14 Advanced Topics

For the microwave case [Eqs. (14.16) and (14.17.2)], a similar derivation yields a dependence on E0 /ω. Between these limits (ω νm or ω νm ), the electron temperature depends on both parameters, on E0 /n and E0 /ω.4 For these two borderline cases, we obtain analytic solutions of the Boltzmann equation. We have neglected Coulombic collisions between electrons and stepwise, multiple ionization. Taking account inelastic collisions as well results in distributions which can be solved only numerically and which lie in between a Maxwellian distribution and a Druyvesteynian distribution. As Rundle et al. have pointed out, both the distribution functions can be extracted from x √ E , f (v) = a E exp −b <E >

(14.20)

with b = 3/2 and x = 1 for a Maxwellian distribution and b = 0.54 und x = 2 for a Druyvesteynian distribution [374]. They fit the exponent to 1.6 ± 0.1 for a DC O2 discharge. This was suggested by the nonlinear behavior but opposite curvature of the distribution functions. For heavy molecules with internal degrees of freedom (in particular vibration states with energies of less than 1 eV which can be significantly occupied at slightly elevated gas temperatures), inelastic collisions come to the fore. In conjunction with these observations, model calculations by Moisan et al. [160] but also by Ferreira and Loureiro [785] seem at last to supply us with a satisfactory understanding of the dissipation of electronic energy via elastic and inelastic collisions (Fig. 14.3). Due to its higher fraction of rapid electrons, the Maxwellian distribution is superior to the Druyvesteynian distribution, thereby stabilizing the high-density discharge. 100

power transfer [%]

80

MW

DC

MB

60 MB

40 20 0 0.01

MW

MB

DC

MW

0.1

1

DC

10

Fig. 14.3. Mechanisms to dissipate electronic energy. Parameter: νm /2πf (either DC, microwave (MW), or Maxwell-Boltzmann (MB) [160].

pd [Torr cm]

An electric field causes a transformation of the Maxwellian shaped EEDF to a Druyvesteynian distribution. Inelastic collisions and electron electron 4 At low operating frequencies, this model breaks down since it assumes the EEDF to be time-independent which is strictly true only for ω = 0 [160].


575

collisions can roll back this shift; for the latter process, the outstanding efficiency to transfer energy must be noted. As can be seen from Fig. 14.2, for collision numbers which exceed 50 (ionization degree = plasma density/number density > 10−3 ), the EEDF has turned back to Maxwellian shape, whereas for collision numbers smaller than 5, the EEDF remains Druyvesteynian [786]. It cannot have escaped the reader that this is exactly the borderline between low-density plasmas, most prominently represented by the capacitively coupled discharge (CCP), and the high-density plasmas. The discharge that would fit best in here is the inductively coupled one (ICP). Eventually, there remains the possibility of a collisionless energy transfer by plasma waves (Landau damping, Sect. 14.3). 14.1.4 Frequency effects 2 → 0, i. Deriving the Margenau distribution, we assumed the condition ω 2 /νm e. either very low operating frequencies or high discharge pressures, respectively. Moving to higher frequencies, the EEDF is going to be modulated. The temporal development of f0 is determined by the frequency of energy transfer νe in its relation to the operating frequency, that of f 1 , however, by the ratio between the frequency of momentum transfer νm and the operating frequency; νe and νm are connected via the Langevin energy loss parameter (νj denotes the frequency for atomic or molecular excitation to level j):

νe =

2me νm + νj = Lνm + νj . mi + me j j

(14.21)

In the isotropic bulk plasma, the energy transfer is predominant, but momentum transfer dominates across the anisotropic sheath. For ω νe , the electrons see the time-averaged potential, for ω νe , they are subjected rather to the instantaneous potential, and the energy which is to be transferred is closely correlated to the time-dependent electric field. Over the whole energy range, f0 is modulated with its second harmonic 2ω since the power transfer depends on the squared intensity of the electric field (Sect. 5.2). For rising operating frequency ω, however, this modulation becomes less pronounced, first visible in the plasma bulk rather than in the plasma tail, adding experimental proof to Winkler’s suggestion that νe is stabilized by inelastic energetic collisions in the sheath which shape the EEDF of the plasma tail considerably [344]. In the range νe < ω, the EEDF is solved by the quasi-stationary Boltzmann equation (for 50 kHz); for 13.56 MHz, the time-dependent variant must be employed; the cross section is parametrized and the electric field within the plasma bulk is assumed to scale with the measured sheath potential. Using the quasi-stationary Boltzmann equation is justified by the experimental evidence of perfect modulation of the EEDF. As shown by Seeb¨ ock and W.E. K¨ ohler, this condition is met in argon at 225 mTorr (30 Pa). They found the ratio of the collision frequencies νm /νe to exceed 100 (109 /107 ) at field intensi-

576

14 Advanced Topics

4

DC

Te [eV]

w/nm=0.425

MB

w/nm=0.8

1

MB

0

Fig. 14.4. For the same effective electric field Eeff and same number density, the electron temperature additionally depends on the operating frequency [782].

MW

5 Eeff /n [10-16 V cm2]

10

ties between 1 and 20 V/cm, values which are typical for the plasma bulk [787]. For the limiting case νe (E) → ω, this modulation will vanish, and the EEDF has become time-independent. For an inert gas (neon) this borderline will be passed beyond 200 mTorr (25 Pa) and 13 MHz, for molecular gases, these limits are shifted to slighty higher values (approximately 40 MHz for hydrogen).

14.2 Sheath and presheath 14.2.1 Conditions In the late 1940s, Bohm showed that the two distinct zones of a discharge, the plasma bulk and the sheath, must be separated by a third dividing zone which exhibits the properties of the adjoining zones. It is quasineutral (as in the plasma bulk) with a superimposed field (as across the sheath). We start with the following assumptions: • Electrons and ions constitute an athermal plasma. A temperature for each component can be defined, but the values diverge by more than two orders of magnitude, so that the ion temperature is set to zero:

Ti = Te ; vi =

2e0 V0 ; ve = mi

2e0 V0 ; me

(14.22)

i. e. a separated Maxwellian distribution for the electrons [788]: • The ion current density is constant in space and time (continuity): ji = ρi v i . • The potential decreases monotonously.

14.2 Sheath and presheath

577

• The sheath is space-charged limited and collisionless. 14.2.2 Derivation WithEq. (14.22), the ion current in the undisturbed plasma is given by ji = e0 n0 (2e0 V0 )/(mi ) with n0 = ni = ne . Approaching the sheath, the kinetic energy of the ions rises according to 1/2 mv(x)2 = 1/2 mv02 − e0 V (x) and the velocity becomes 2e0 (V (x) − V0 ) , v(x) = v0 1 − 2

(14.23)

mv0

which modifies the ion density according to the condition of continuity ji = const = n0 v0 = n(x)v(x) to: n(x) =

n0 1−

2e0 (V (x)−V0 ) mv02

.

(14.24)

Entering the sheath from negligible fields, the ions are accelerated by the shield field, and simultaneously, their density drops. Inserting into the PoissonBoltzmann equation [Eqs. (3.9) and (3.25)], this yields the so-called sheath equation: ⎛ ⎜ dV ε0 2 = −n0 e0 ⎜ ⎝ dx 2

1 1−

⎞ e0 (V (x) − V0 ) ⎟ ⎟. − ⎠ k T

2e0 (V (x)−V0 ) mi v02

− exp

(14.25)

B e

Regarding dV /dx as function of V : dV /dx = F (U (x)) ⇒ d2 V /dx2 = dF/dV dV /dx, a separation and straightforward integration is possible, and we obtain for ε0 (dV /dx)2 : ⎛

⎞

e0 (V (x) − V0 ) mv02 kB Te 1 − 2e0 (V (x) − V0 ) ⎠ + C. n0 e0 ⎝ exp − + e0 kB Te e0 mv02 (14.26) The constant C vanishes for dV /dx at V = V0 which equals, for ε0 (dV /dx)2 with

A = 2n0 e0 ∧ B = ⎡

mv02 , e0 ⎛

(14.27.1) ⎞⎤

2e0 (V (x) − V0 ) kB Te e0 (V (x) − V0 ) exp − 1 + B ⎝1 − − 1⎠⎦ . A⎣ e0 kB Te mv02 (14.27.2))

578

14 Advanced Topics Fig. 14.5. Integrated sheath equation for several plasma densities. Electron temperature: 2 eV, sheath voltage: 450 V. For entry into the plasma bulk, the potential equals zero, its topmost value is reached on the electrode (d = 0 cm). λD is 333, 105 and 33.5 μm, resp.

0 450 V 2 eV

-150 10

F [V]

10 cm

-3

109 cm -3

-300 11

-3

10 cm

-450 0.0

0.5

1.0 d [cm]

1.5

2.0

At the border of the sheath, V (x) ≈ V0 , and both the functions can be expanded obtaining:

2n0 e0

e0 ΔV 1 e0 ΔV kB Te 1+ + e0 kB Te 2 kB Te

⎡

+2n0 e0 ⎣

⎛

2e0 ΔV 1 mv02 ⎝ 1− − 2 e0 2mv0 2

2

e0 ΔV mv02

−1

2

+ ⎞⎤

− 1⎠⎦ ,

(14.28)

with ΔV = V (x) − V0 for ε0 (dV /dx)2 from which follows immediately

n0 e20 dV = dx ε0

1 1 − kB Te mv02

ΔV.

(14.29)

To ensure a positive radicand, 1/kB Te must exceed 1/mi v02 = 1/(2 Ekin ) or

kB Te = vB (14.30) mI which is denoted Bohm velocity. This is Bohm’s sheath criterion. The ions entering the sheath have to be accelerated to drift velocities which are determined by the thermal electron speed or the electron temperature. This ion drift velocity is significantly larger than the mean thermal ionic speed and nearly equals the sound velocity of ion waves which can propagate even at Ti = 0.5 As result, the ionic flux can even exceed the flux of the thermal neutrals. Albeit the absolute value of the potential has steeply dropped from its initial value at the electrode to low a value, it is just as intense as to disturb the plasma; its magnitude is calculated according to e0 ΦB = 1/2 mi vi2 = kB Ti ≥ kB Te /2 v0 ≥

5 The sound velocity depends on the electron temperature since the ions are subjected to an electric field which scales with Te and on the inertial ion mass. From Eq. (14.30), no dispersion behavior can be deduced, and the ω(k) relation remains constant. Ion waves are waves of constant velocity for small k (vph = vg ); just for large k, they become waves of constant frequency. Because of high damping, ∂ω/∂k ≈ 0.


579

ΦB ≥

1 kB Te . 2 e0

(14.31)

14.2.3 Presheath Ions randomly penetrate the presheath and are accelerated, in response to the leaking electric field, in the direction of that field towards the Bohm edge. By this process, the density of both types of charges is reduced but quasineutrality is maintained. We will now inspect this process starting with the entering condition: n(x) = ne (x) ⇒ ln ni (x) = ln ne (x),

(14.32)

and for the dynamics across the presheath, their drop should be in step: d 1 d 1 d d ln ni (x) = ni (x) ∧ ni (x) = ln ne (x). dx ni (x) dx ni (x) dx dx

(14.33)

Noting that ji (x) = e0 vi (x), we obtain for the variation of the ion density 1 d e0 vi (x) d ji (x) vi (x) d ji (x) ni (x) = = ni (x) dx ji (x) dx e0 vi (x) ji (x) dx vi (x)

(14.34.1)

which can easily be transformed to 1 d d d ni (x) = ln ji (x) − vi (x). ni (x) dx dx dx

(14.34.2)

Focusing now on the dropping electron density, we first note that

e0 Φ(x) ne (x) = ne (0) exp − kB Te

⇒ ln ne (x) − ln ne (0) = −

e0 Φ(x) kB Te

(14.35.1)

from which its spatial variation is derived to be (negative sign of e0 ): d e0 Φ(x) . ln ne (x) = dx kB Te

(14.35.2)

Equating Eqs. (14.34.2) and (14.35.2) yields d d e0 Φ(x) ln ji (x) = ln vi (x) + dx dx kB Te

(14.36.1)

1 d 1 d e0 Φ(x) . ji (x) = vi (x) + ji (x) dx vi (x) dx kB Te

(14.36.2)

or

580

14 Advanced Topics

In the presheath, the spatial dependent ion velocity exceeds the thermal ion velocity but remains always below the Bohm velocity vB , and for the ion current, the inequation holds 1 d e0 Φ(x) 1 d . ji (x) > vi (x) + ji (x) dx vB dx kB Te

(14.37)

1 d j (x) to Neglecting ionizations across the sheath which would cause ji (x) dx i become larger than zero, the LHS must always equal zero which means that the RHS must always be negative. By considering collisions across their traverse, the effective acceleration of the ions will fall short of the expected increase in velocity.

14.2.4 Charge density across the sheath At the border of the sheath with ΦB , the directed energy of the ions (one degree of freedom with respect to the electrode) is closely allied with the mean thermal energy of the electrons. In the following, this potential is denoted as Bohm potential VB . If the potential exceeds this value, the condition for linearization of the Poisson-Boltzmann equation, e0 ΔV kB Te , does not hold any more, and the ions will fall into the sheath which exhibits the spatial dependence of 4 x /3 according to Child’s law. To see what will happen if V < (kB Te )/(2e0 ), Eq. (14.25) is expanded to the first (linear) term, taking into account VB = mv02 /2e0 . ⎛

d2 V e0 ΔV ε0 2 = n0 e0 ⎝exp dx kB Te ⇒ ε0

⎞

−

d2 V ≈ n0 e0 ΔV dx2

1 1 − ΔV /VB

⎠

(14.38.1)

1 e0 − . kB Te 2VB

(14.38.2)

For Φ > kB Te /2e0 , the solution for Φ is exponential, for Φ < kB Te /2e0 but oscillatory, and the sheath will collapse. To see how this happens, we expand the term for the charge density for small Vx − VB : ne ≈ n 0

1−

e0 ΔV kB Te

;

ΔV ; 2VB e0 1 ni − ne ≈ n0 ΔV − . kB Te 2VB ni ≈ n 0

1−

(14.39.1) (14.39.2) (14.39.3)

As the ions are accelerated by the field to a kinetic energy e0 ΔV , their density must decline to ensure a constant current density [see conditions in Eqs. (14.22)]. If this drop across the sheath happens too fast, the highly mobile


581 1 kB Te/1.5 kB Te/2 kB Te/4 kB Te/12 n(e)

0.75

rel. charge density n(V)/n0

rel. charge density n(V)/n0

1.00

0.50

0.25

0.00 0

1

2

3 F [V]

4

5

0.1

0

kBTe/1.5 kBTe/2 kBTe/4 kBTe/12 n(e)

2

4 F [V]

6

8

Fig. 14.6. Start of the Bohmic transition zone: The Bohm potential must exceed 1/ k T to ensure an excess of the ion density over the electron density across the 2 B e whole sheath which is prerequisite for a stable sheath. In semilogarithmic scale (RHS), the electron density decreases linearly.

electrons—which will always leak into the region of positive potential to some extent—will create a net negative space charge just in front of the electrode which, in turn, enhances the ion velocity again. This avalanche of ions would eventually result in a collapse of the sheath. • For VB < kB Te /2e0 , the ion density drops faster than the electron density which quickly exceeds the force which is exerted on the ions at the sheath border (this happens in the plasma bulk when fluctuations occur or shock waves propagate [789]). • For VB > kB Te /2e0 , i. e. for sufficiently high initial ion velocity at the borderline of the sheath, the ion density declines less slowly than the electron density, and it is ensured that the time-averaged sheath potential remains positive (cf. Figs. 14.6). Although the sheath happens to be very thin it screens the plasma almost entirely from disturbing potentials. Since the electrons have thermal energies of several electronvolts, a certain fraction (which can be calculated by a Boltzmann term) of them can overcome such a barrier; to put it another way, residual fields can leak into the plasma which accelerates all the ions in response to this force to velocities which considerably exceed their thermal velocity. Assuming the potential and the field to vanish at the sheath boundary is justified only by the tremendous rise of both of them across the sheath. In fact, the potential at the sheath boundary has already reached the value kB Te /2e0 which causes a steep rise of the ion current incident on the substrate.

582

14 Advanced Topics

14.2.5 Approximations The maximum current can easily be worked out taking into account the component of the ion velocity in a direction parallel to the sheath field which is 1/3 of the most probable velocity (maximum of the EEDF):

< vi >MB +vi,B =

2 mi

e0 VS +

kB Ti . 3

(14.40)

At the boundary of the sheath, the ion density can be calculated taking into account the equilibrium ne ≈ ni :

ni = n0 exp −

e0 VS , kB Te

(14.41)

from which follows, for the ion current density,

ji = e0 ni vi ⇒ ji =

2 mi

e0 VS +

kB Te e0 VS . · e0 n0 exp − 3 kB Te

(14.42)

The maximum current which is mandatory for the preservation of the sheath is given by the derivation of this equation with eVs = kB Te /2 − kB Ti /3 to

ji,max =

600 10 9 cm -3 2 eV

450

F [V]

450 V

300 600 V

150 0 0

300 V

1 d [cm]

1 Ti kB Te e0 n0 exp − + . mi 2 3Te

(14.43)

Fig. 14.7. Integrated sheath equation for some sheath voltages, electron temperature: 2.0 eV, plasma density: 109 cm−3 . For entry into the plasma bulk at x = 0 cm, the potential vanishes, its topmost absolute value is reached on the electrode, i V below zero level at 0 cm. λD is 333 μm.

2

So long as Ti is small compared with Te , the ion current does not depend considerably on Ti . Even for hot plasmas for which Ti ≈ Te , this is no larger than a factor of 1.4. For the phenomena we are interested in, the maximum ion current can be approximated to


583

ji,max =

kB Te n0 e0 = 0.606 n0 e0 · emi

kB Te . mi

(14.44)

(A more precise calculation yields a factor of 0.566 for Ti = 0.01 Te and a factor of 0.537 (instead of 0.606) for Ti = 0.5 Te [790]). For several typical conditions, the integrated sheath equation (14.23) has 1 dy been numerically integrated according to y = h(y) ∧ y = y(x) ⇒ x = h(y) (Figs. 14.5, 14.7, and 14.8). Entering the sheath (low sheath voltage), the field 4 slowly rises to reach eventually the precipitious fall according to Child’s x /3 law. For sheath voltages of several hundreds of volts and typical plasma densities of several 1010 cm−3 , the sheath thickness amounts to several millimeters, causing dramatic consequences for the distribution functions of the impinging ions with respect to angle and energy, respectively [IEDF and IADF (Sect. 6.8)]. 14.2.6 Conclusion Sheaths which can be subjected to the sheath equation (14.25) are called Langmuir sheaths. 0 9

10 cm 450 V

F [V]

-100

-3

-200 -300

0.5 eV 2 eV

-400

1 eV

4 eV

-500 0.0

0.5

1.0

1.5 d [cm]

2.0

2.5

Fig. 14.8. Integrated sheath equation for some electron temperatures. plasma density: 1010 cm−3 , sheath voltage: 450 V, for four electron temperatures. For entry into the plasma bulk, the potential vanishes, its topmost absolute value is reached on the electrode.

The derivation is exact for a DC sheath. In a capacitively coupled RF sheath with leaking currents, this is approximatively valid, at least for ωp,i ωRF .

14.3 Plasma oscillations The theory of plasma oscillations has been developed by Bohm and Gross [791, 792] as well as by Landau (compiled in [793]) and has been verified by Barrett et al. in several experiments [794]. By interacting with these oscillations, electrons can draw energy from this plasma motion.

584

14 Advanced Topics

By entirely neglecting the dynamic conduct of the ions so that they can be considered as stationary background, the charge density of the electrons can be described by n(x, t) which moves with a constant velocity v(x, t). The equilibrium density is then given by ±e0 n0 . In the absence of a magnetic field B 0 , the equation of continuity reads ∂n + ∇ · (nv) = 0, (14.45) ∂t and the equation of motion taking into account the convective derivation ∂ d = +∇ dt ∂t

(14.46)

as well as F = −∇E ⇒

m 0 ne

∂v F 1 1 = = − · ∇E = − · ∇p : ∂t m m mn

(14.47.1)

∂v + (v · ∇)v = −e0 ne (E + v × B) − ∇p. ∂t

(14.47.2)

With this set, the required Maxwell equations can be written as ε0 ∇ · E = e0 (ni − ne )

(14.48.1)

∇ · B = 0,

(14.48.2)

∇×E =−

∂B , ∂t

(14.48.3)

1 ∂E e0 nv + . (14.48.4) ε0 ∂t Considering just small changes, the equations can be linearized. The properties which characterize the plasma state can be represented by a sum of its equilibrium value and by a (comparatively small) value which is due to the disturbance caused by oscillations and waves, and no terms higher in order than unity will enter the initial equations of the perturbation theory of first order. With v 0 = E 0 = ∇n0 and its vanishing temporal derivations, we get: (1) The equation of continuity: c2 ∇ × B =

∂(n0 + n) + ∇ · [(n0 + n)(v 0 + v)] = 0. (14.49.1) ∂t We suppose the terms n0 v 0 , nv 0 and the therm nv to become zero by linearization, and it remains: ∂n + n0 ∇ · v = 0. ∂t

(14.49.2)


585

(2) The equation of motion:

m 0 ne

∂v + (v ∇)v = −e0 n0 (E + v × B) − ∇p; ∂t

(14.50.1)

LHS after linearization: m0 ne ∂v/∂t; RHS first term after linearization: −e0 n0 E; for the second term, we have to take into account that the density fluctuations will happen out of the sudden. Hence, we cannot apply the isothermal equation of state ∇p = kB Te ∇n but rather the adiabatic ∇p = γkB Te ∇n. In the kinetic gas theory, γ represents the ratio of the specific heats cp /cV . This taking in mind, the second term becomes ∇p = γkB Te ∇n, and the equation of motions reads m 0 n0

∂v = −e0 n0 E − γkB Te ∇n. ∂t

(14.50.2)

(3) The Poisson equation: ε0 ∇ · E = e0 ,

(14.51)

since we provided stationarity of the ions. `re’s law: (4) Ampe c2 ∇ × B =

1 ∂E e0 n0 v + . ε0 ∂t

(14.52)

The electric forces will couple the motions of the particles resulting in organized oscillations, and the wave equation for the periodic fluctuations in density is obtained by inserting the equation of motion into the equation of continuity; `re’s law: for the field variations, the equation of motion is inserted into Ampe (1) Equation of continuity: ∂n + n0 ∇ · v = 0; ∂t

(14.53)

(2) Equation of motion: m 0 n0

∂v = −e0 n0 E − γkB Te ∇n; ∂t

(14.54)

`re’s law: (3) Ampe c2 ∇ × B =

1 ∂E . e0 n0 + ε0 ∂t

(14.55)

Taking the derivation of Eq. (14.53) with respect to t and inserting it into Eq. (14.54) yields e20 n0 γkB Te 2 ∂2n − n− ∇ n = 0. ∂t2 ε0 m 0 m0

(14.56)

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14 Advanced Topics

By taking the derivation of Eq. (14.55) with respect to t and inserting Eq. (14.54) considering ∇n = ε0 ∇(∇ · E)/e0 , we obtain ∂2E e2 n0 γkB Te ∂B − 0 E− ∇(∇ · E) = c2 ∇ × . 2 ∂t ε0 m 0 m0 ∂t

(14.57)

Inspecting Eqs. (14.56) and (14.57), we realize the structural identity of the LHS of both these equations, and consequently, this must be valid for the RHS as well. Provided that ∂B/∂ vanishes, this would immediately cause ∇ × E to become zero according to Faraday’s law. This, in turn, is equivalent to the statement that E is the gradient of a scalar electrostatic potential since B and its derivation with respect to t are zero. Neglecting the thermal energy of the electrons yields the wave equation e20 n0 ∂n2 − n = 0; ∂t2 ε0 m 0

(14.58)

same derivation for the field variation. Assuming a plane wave, the plasma frequency ωp turns out to be

ωp =

e20 n0 . ε0 m 0

(14.59)

According to this derivation, a plasma oscillation is longitudinal and electrostatic in character; and its eigenfrequency does solely depend on the plasma density. From the condition E k, we infer k × E = 0. But for a plane wave, Faraday’s law states that ∇ × E = ik × E = 0 which equals dB/dt = 0. 14.3.1 Dispersion relation From Eq. (14.59) we conclude that there is no correlation between ω and k, i. e. the oscillation lacks any dispersion behavior and cannot propagate, its group velocity dω/dk equals zero. But taking into account the term for the kinetic energy of the electrons which has been neglected, we see for E: −ω 2 E −

e20 n0 γkB Te 2 E+ k E = 0, ε0 m 0 m0

(14.60)

from which we can extract the dispersion relation of the longitudinal wave: ωk2 = ωp2 +

γkB Te 2 k . m0

(14.61)

This equation is but valid for small wave vectors k or large wavelengths (for large k, quadratic effects come into play). The coefficient of k 2 describes the propagation behavior. We can exclude the value γ = cp /cV = (N + 2)/N with N the number of degrees of freedom which holds true for an acoustic wave where atoms mutually collide. Since the frequency of the variations in density


587

and field is considerably larger, the electrons are not subjected to collisions over an oscillation period 2π/ω. Since a longitudinal wave in only one direction is concerned, its only degree of translational freedom N is unity and we find γ = (1 + 2)/1 = 3, which yields for the coefficient of k 2 : 3kB Te /m0 . But this is equal to the well-known mean squared velocity of the electrons; and the dispersion relation of the longitudinal wave for a one-dimensional Maxwellian distribution can be expressed as6 _ _ _ _ _ _ _ _ _ _

+ + + + + + + + +

plasma

+ + + + + + + + +

_ _ _ _ _ _ _ _ _

+

d

Fig. 14.9. The longitudinal electrostatic wave can penetrate only several Debye c Academic lengths [795] ( Press).

ωk2 = ωp2 + 3 < v 2 > k 2 .

(14.62.1)

Since < v 2 >= kB Te /me and vw2 = 2kB Te /me , we see that 3 ωk2 = ωp2 + vw2 k 2 . (14.62.2) 2 Plasma waves resemble acoustic waves by causing periodic compressions and rarefactions. However, their dispersion relation and the mechanism of generation is entirely different. The particles in a gas or in a liquid oscillate in response to short-range forces to which they are subjected. The density is relatively high which causes many collisions over one period. In a plasma, however, the particles respond to long-range forces by slow motions, and their mean distance changes but scarcely over one oscillation period. As direct consequence of this behavior, Vlasov defined a mean space-charge potential and applied it those longitudinal oscillations in density (loc. cit. in [791]). Due to the long-range tail 6

Although derived according to the principles of kinetic gas theory, this equation is also valid for a degenerated free electron Fermi gas; since their mean kinetic energy equals 3/5 EF at T = 0 K, we find with vF = ωp /kD 3 ωk2 = ωp2 + vF2 k 2 . 5 (This is an approximation for small values of k; for a more detailed analysis cf. [796] − [798], where the same result is obtained via Lindhard’s dielectric function ε = ε(k, ω), but with outstanding elegance!)

588

14 Advanced Topics

of the Coulombic interaction (a Debye length of 100 μm corresponds to a chain of 300 000 atoms), small deviations from the equilibrium value of density can cause large changes in potential. Due to the chaotic thermal motion of the particles, this disturbance will propagate by an oscillatory motion; a localization of the disturbance is only possible at reduced temperatures. First, we use the dispersion relation [Eqs. (14.62)] to estimate the skin depth of this type of waves into the plasma. Writing

ω2 ωp − 1, k= √ 3 < v 2 > ωp2

(14.63)

ωp k= √ i 3 < v2 >

(14.64)

yields

for ωp2 ω 2 for k; and the skin depth reads 1 δ= =3 k

kB Te me

ε0 m e = 3 λD , ne20

(14.65)

and remains at a very low level (cf. Sect. 14.6 for the skin depth of electromagnetic waves, Fig. 14.9). Second, we investigate the propagation behavior for the limiting cases. For that end, we define the Debye wave vector by kD = 2π/λD and insert this term into the dispersion relation which then reads

ω = ωp 1 +

12π 2

k kD

2

.

(14.66)

For large wavelengths (k 1 ∨ k kD ), ωk ≈ ωp (Fig. 14.10), and the phase velocity of the longitudinal wave equals vph = ωp /k, the group velocity yields to

vg = 2ωdω = 3 < v 2 > 2kdk ⇒ dω/dk = 3

< v2 > 3 vw2 = , vph 2 vph

(14.67)

√ √ a very small value. For k kD , we find vph < v 2 > and vg √ < v 2 >. Only for wavelengths comparable with the Debye length λD = 1/ 2vw /ωph , ω deviates significantly √ from ωph . For k → kD , vg becomes larger and larger but falls short of the limit < v 2 >, the phase velocity of an acoustic wave with γ = 3 in an electron gas. For large k (small wavelengths), the group velocity of the electrostatic plasma waves equals the thermal velocity of the electrons. For small k (large wavelengths), the thermal velocity of the electrons significantly exceeds the


589

6

3

w k2/w P2 = 1 + 3k2/w P2 group velocity

wk2/w P2 = 1 + 3k2/wP2 group velocity phase velocity (3 ) 1/2

4 wk2/wP2

wk2/wP2

2

2

1

0

-2

-2

0 k/k D

2

0 -2

-1

0 k/k D

1

2

2 + 3 < v2 > k2 Fig. 14.10. Brillouin diagrams of the dispersion relation ωk2 = ωph covering a large range (LHS) and the detailed middle section (RHS) for k. - Phase velocity vph = ωk : for every point a straight line from origin. : for every point a tangent. - Group velocity vg = dω √ dk 2 >: For large k (small λ) v - Thermal velocity 3 < v 2 > = 3/2 < vw ph and vg are very √ 2 similar. For small k, vg is considerably smaller than 3 < v > although vph exceeds √ 3 < v 2 > to a large extent.

group velocity of these waves, although their phase velocity does show opposite behavior. For the range of k considered, the waves travel with constant frequency. 14.3.2 Landau damping From Figs. 14.10 and 14.11 we learn that the phase velocity can be found at very high values for small wave vectors. For k → kD , vph moves to the left and will become comparable to the mean thermal velocity of the electrons. As a result of this deceleration, the wave can trap slow moving electrons with velocities of vph ± Δv and will guide them subsequently. Since the Maxwellian distribution is characterized by a disproportion between slow and rapid electrons, this trapping eventually causes a damping of the wave. In turn, the electron distribution is definitely disturbed at v = vph , and the fraction of electrons moving more rapidly is increased (dotted line at v = vph in Fig. 14.11). On the other hand, this damping can be neglected for low wave vectors. In this dispersion relation, the energy distribution function coincides with the temperature defined by statistical mechanics in the case of a Maxwellian distribution of the electrons. Bohm and Gross [791, 793] and, following another path, Landau showed that this dispersion relation represents the limiting case (real part) of a relationship which can be solely solved by complex frequencies,

590

14 Advanced Topics

0.75 n0(v) 1/2

f(v)

0.50 Dv

0.25

0.00 -3

-2

-1

0

1 v 2 ph

Fig. 14.11. One-dimensional energy distribution of the elecrons according to Maxwell-Boltzmann [800].

3

v [a. u.]

and this means: The real part describes the oscillatory effect, the imaginary part describes the damping (or generation) of a plasma wave, which is known as Landau damping (a very elegant derivation is due to Frank Chen [799]). The imaginary part of the dispersion relation is given for k kD according to (ω) = π/2ωph vF2 (df /dv)v=vph ,

(14.68.1)

with df /dv the derivation of a distribution function with respect to velocity. For (ω) > 0, a disturbance is generated, for (ω) < 0, the disturbance is damped. The sign is fixed by the slope of df /dv at the position of the phase velocity. For a Maxwellian distribution, this yields

k2 27π k3 (ω) = − ωph D2 exp − D2 , 8 2k 2k

(14.68.2)

and the intensity of the damping process is determined by the ratio between (ω) and ωph or the Q-factor, the change of amplitude within one oscillation period 2π/ωph . For small k, the damping remains small as well, k/kD = 0.1, for example, (ω)/ωph equals 10−51 , but for k/kD = 0.5, the ratio has already risen to 0.17. For large k, the exponential factor saturates at unity, but 1/k 2 will vanish and the damping will become again very small. In between, however, we can identify a maximum which can be found at k = kD by differentiating Eq. (14.68.2); for k > kD , the damping is already more intense than outlined by Eq. (14.68.2) and even exceeds the real part of the dispersion relation as calculated according to Eq. (14.62.1) [801]. For a Maxwellian distribution, we find a dependence df /dv < 0 beyond the maximum which has been identified as the condition for damping. To change the sign of df /dv, the density of high-energy electrons has to be increased

14.4 Capacitive coupling for collisionless sheaths

591

significantly as will happen for an electron beam which is shot into the plasma. This beam is provided by the electrons which are accelerated by the processes in the sheath. If the current density passes a certain critical limit, df /dv > 0, and a second maximum at high velocities can occur. Growing oscillations can develop at the expense of the high-energy electrons by a collisionless mechanism which is known as waveriding resonance. At a critical density, the damping of the wave turns first into an instability and eventually into an enlargement of the wave, and this phenomenon can be described by [802]:

Tb vb mvb2 nb = const exp − , np Tp up 2kB Tp

(14.69)

with b, p characterizing the properties of the beam and the undisturbed plasma, respectively, and up the most probable electron velocity in the plasma (maximum of the Maxwell-Boltzmann distribution). These electrons might be responsible for the Maxwellization of the electrons of the high-energy tail of the EEDF (E < E >) at elevated electron densities. Kennedy and Fridman have found the ratio a to exceed unity significantly a=

νee 1 νc L

(14.70)

to make this happen (νee : frequency of electron electron collisions; νc : frequency of electron neutral collisions, L: Langevin’s energy loss parameter) [137], [803] − [805]. Landau damping is also suspected to play a significant role in helicon discharges as well as for ECR heating as far as the fate of the L-wave is concerned [263, 289].

14.4 Capacitive coupling for collisionless sheaths 14.4.1 The symmetric case 14.4.1.1 Introduction. It is evident that a model for a homogeneous plasma is not the appropriate approach to calculate the carrier densities and potentials across the sheaths. Although the cause of their inhomogenous conduct can be expressed concisely as acceleration of the carriers, a mathematically satisfying explanation is extremely difficult. Within the limits of a collisionless sheath, however, this problem could be solved analytically [192, 806].7 In 1976, Godyak started modeling of capacitive sheaths, and he supposed the sheath oscillation to move sinusoidally, assuming two telescoped sheaths, one of them stationary, the other one oscillating [206]. This model was improved significantly by Lieberman in the late 1980s. As a main result, he obtained a self-consistent model 7 Godyak and Sternburg solved the problem for the low-potential collisional sheath for the first time [214]. Another approach is due to Goedheer and Meijer, who presented a numerical solution of the Boltzmann transport equation [807, 808].

592

14 Advanced Topics

with a highly nonlinear sheath motion [192]. Later on, Lieberman’s model was brilliantly confirmed by Vender and Boswell with particle-in-cell (PIC) simulations in 1990 [353], and by Gozadinos et al. with their considerations for stochastic heating [211]. Wood et al. could experimentally verify this model with Langmuir probes [352]. Here, we present a hybrid model which is based on Godyak’s and Lieberman’s models as well. 14.4.1.2 Assumptions. We start with the following assumptions: • The plasma frequency of the ions, ωp,i , is small compared to the angular frequency of the excitation, ωRF , which is small against the plasma frequency of the electrons, ωp,e . • To generate a stable sheath, ni must always exceed ne , and the ions must arriveat the Bohm edge x = se (t) with a directed velocity of magnitude vB = kB Te /mi . • Field – The RF field of the sheath is large compared to the longitudinal field in the plasma bulk which is often approximated to zero (constant plasma potential): Es Ep . – The electrons are in equilibrium within the field of the sheath. – The sheath thickness is large compared with the Debye length, (λD dS ), which is met for VRF kB Te /e0 . • Ions – The ions form a stationary ionic part of width ds . Descending from the plasma bulk (x = ds ) to the electrode (x = 0), the ion density drops according to the condition of current continuity, but saturates at a finite level at x = 0. – In the plasma: conduction current of the electrons, the displacement current is very small: ε0 dEp /dt σEp . – The boundary ionic sheath/plasma is stationary, the ions enter the sheath with the Bohm velocity vB = kB Te /mi , and the DC ionic current can be calculated applying Child’s law. – The spatial displacement of the ions in the sheath is small against its thickness. The sheath is collisionless, for the quasi-stationary case (ωRF ωp,i ), the ions feel only the time-averaged field. – In the quasi-neutral Bohmic presheath: conduction current of the electrons, just in front of the powered electrode: displacement current which is large compared to the time-averaged conduction current of the ions: e0 ni dx/dt = ε0 dES /dt ji , je .


593

• Electrons – The electrons instantaneously respond to the RF field, i. e. electron inertia is not considered. – The electron sheath oscillates between x = 0 (electrode surface) and x = ds (border between sheath and presheath), its time dependence is denoted se (t), and its actual thickness equals ds − se (t). Its instantaneous density ne [ϕ(x)] can equal ni (x) but will never exceed it. The spatial-dependent phase ϕ(x) is sensitive to the time interval during which the electron density is dried out. For large x (close to the plasma), this time is short, for small x (close to the electrode), this time extends over nearly the whole RF cycle. For x > se (t), ne = ni , and the field vanishes. For x < se (t), ne becomes zero within a few Debye lengths [807], and the electron sheath terminates. Now, a residual charge of (positive) ions remains in front of the electrode which seems to oscillate with the operating frequency. Its thickness equals se (t). This charge surplus gives rise to an instantaneous field within this sheath. Phenomenologically, we can compare this breathing with the propagation of a damped wave incident on the electrode but never reaching it. ∗ At x = ds , we cross the boundary to the quasi-neutral Bohmic presheath. For a short moment, the electron sheath extends between 0 and ds , and the HF current is conduction current of the electrons. ∗ For x < se (t), ne has vanished, and the remaining sheath contains only ions with low mobility; hence the HF current becomes pure displacement current. ∗ In the DC limit, ω → 0, se becomes zero as well, for ωRF νm , se ≈ ds (Fig. 14.12). So we have to evaluate the boundary se (t), the density ne [se (t)], and the potential V (t). 14.4.1.3 Spatial sheath structure. As we know, the time-averaged potential V (x), the ionic density ni (x) and the velocity of the ions, vi (x), are spatial functions; they are reduced to one dimension: n0 v B = ni v i ∧

1 1 mi vi2 = mi vB2 + e0 V (x), 2 2

(14.71)

and we obtain for the ion density ni =

n0 1 − 2 ek0 VB T(x) e

.

(14.72)

594

14 Advanced Topics

electrode

plasma _ ne

ni

ne(t)

0

se(t)

dS - se(t)

se dS

x

Fig. 14.12. Principal structure of a capacitively coupled high-potential sheath. se (t) is the instantaneous boundary of the electron sheath. se is the maximum thickness of the electron sheath and equals ds for a very short part of the RF cycle.

The Poisson equation gives with se (t) the distance from the electrode (x = 0) to the boundary of the pulsating electron sheath: x < se (t) :

dE e0 se (t) e0 ni (x)dx = ni (x) ⇒ E = dx ε0 ε0 x

(14.73)

(electron density vanishes) and x > se (t) :

dE =0⇒E=0 dx

(14.74)

(quasi-neutrality).8 Moreover, a harmonically excited displacement current is assumed to flow across the sheath: jRF (t) = j0 sin ϕ = j0 sin ωt,

(14.75)

which equals the electron conduction current at the pulsating sheath boundary se (t) at phase angle ϕ, (Fig. 14.12). To take into account the boundary motion, we have to consider the phase angle ϕ(x) = ϕ[x(t)] considering the principle of continuity [Eq. (6.13)]: dϕ (14.76) = −e0 ωni (x) ⇒ j0 d(cos ϕ) = e0 ωni (x)dx. dx After integration within the limits s(t) = x and s(t) = se , whereas s(t) = se at ϕ=0: j0 sin ϕ

e0 x j0 ni (x)dx, (1 − cos ϕ) = ε0 ω ε 0 se

(14.77)

we obtain the electric field of the ion sheath for values x < se (t) with Eqs. (14.73) and (14.77) to 8 This assumption is questionable since the electron density decreases with a Boltzmann term. To meet the condition ni = ne , a further field is set up [809].

14.4 Capacitive coupling for collisionless sheaths E(x, ϕ) =

595

e0 se (t) e0 se (t) e0 x ni (x)dx = ni (x)dx − ni (x)dx ε0 x ε 0 se ε 0 se

(14.78.1)

e0 se (t) j0 ni (x)dx − (1 − cos ωt); ε 0 se ε0 ω

(14.78.2)

or E(x, ωt) =

the time-averaged mean value of the electric field is given by: E(x) =

1 ϕ E(x, ωt)d(ωt); 2π −ϕ

(14.79)

since only within the phase angle ωt = ±ϕ, s(t) > x, hence we obtain E(x) =

e0 ϕ x j0 ni (x)dx + (sin ϕ − ϕ). ε 0 π se ε0 ωπ

(14.80)

With Eq. (14.78.1), and considering ϕ = ωt at se across the whole sheath: e0 ϕ se ϕ j0 ni (x)dx = (1 − cos ϕ), ε0 π 0 π ωε0

(14.81)

and we obtain for the time-averaged field E(x) =

j0 [sin ϕ − ϕ cos ϕ]. ε0 πω

(14.82)

This equation determines the time-independent component of the electric field, and, in turn, the dropping density of the ions across the sheath. Both functions are closely allied according to the principle of continuity: The sheath boundary moves more slowly in areas of elevated ion density and vice versa, and we can define three different regimes. • the collisionless case:

vi =

2e0 Φ , mi

(14.83.1)

• the mobility-limited case: vi (x) = μi E(x),

(14.83.2)

• and the case observed most frequently where the drift velocity scales with the square root of the electric field: 1 mi vi2 = e0 λi E(x) ⇒ vi = 2

√ e0 λi 2 E(x). mi

From Eq. (14.82), we can further derive the following:

(14.83.3)

596

14 Advanced Topics

1. The time averaged field strength scales inversely with the operating frequency ω which has to be taken into account in high-frequency systems (cf. Sects. 6.2 and 6.5) [420] and which is made use of in dual-frequency systems (Sect. 6.7). Since the voltage drop across the sheath which is operated at higher frequency remains small, the energy of the ions is scarcely influenced. 2. The intensity of the averaged field scales linearly with the ion density at the Bohm edge, the boundary at which ni equals the plasma density ne . Higher plasma densities cause shrinking sheath widths and dropping sheath potentials, hence the intensity of the electric field remains almost constant or is reduced more slowly. Since j0 sin ωt = e0 ni dx/dt at x = se ∧ ωt = ϕ : dϕ ω e0 n0 dϕ ω · = ⇒ = e0 ni . dx j sin ϕ dx j sin ϕ 0 1 − 2e0 V (x)/kB Te 0

(14.84)

Eventually, we have obtained two gradients: One of them is the potential V (x), and the other represents the phase angle ϕ, which determines the instantaneous boundary of the electron sheath. Separation of variables and integration over the limits V (x) = 0 and V (x) and ϕ = 0 and ϕ, respectively, yields after some algebra

1−

2e0 V (x) ϕ 3 ϕ =1+H · cos 2ϕ + − sin 2ϕ kB Te 4 2 8

(14.85)

with Lieberman’s factor H, which describes the quadratic expansion ratio of the sheath thickness in terms of the Debye length λD :

1 se 2 j02 = (14.86) 2 kB Te ε0 e0 n0 πω π λD with λD the Debye length at the Bohmic presheath. The rectifying behavior in the capacitive sheath leads to a large expansion of the sheath along with a steep rise of the self-biased DC sheath voltage. For large RF voltages, the sheath becomes several tens of Debye length in thickness. On the LHS of Eq. (14.85), we identify the factor that connects ni and n0 , and we finally get H=

n0 ϕ 3 ϕ =1+H cos 2ϕ + − sin 2ϕ ni 4 2 8 for the ion density and for the time-averaged potential: kB Te V (x) = 2e0

ϕ 3 ϕ 1− 1+H cos 2ϕ + − sin 2ϕ 4 2 8

(14.87)

2

.

(14.88)


597

normal. sheath thickness

3

Fig. 14.13. A sinusoidally excited sheath exhibits a strong nonlinear behavior. This is obvious for small values of the displacement current (small H which is equivalent to e0 VRF /kB Te 1). Only for values of H > 100 is the conduction current negligible against the displacement current.

H=1 H = 10 H = 100 H = 10000

2

1

0

-3

-2

-1

0

1

2

3

phase angle [rad]

The sheath thickness can be evaluated by inserting of Eq. (14.85) into eq. (14.84) and integrating over the limits of V (x) = 0 at ϕ = 0 and V (x) at ϕ to

x=

j0 H · 1 − cos ϕ + e0 n0 ω 8

1 11 3 . sin ϕ + sin 3ϕ − ϕ 3 cos ϕ + cos 3ϕ 2 18 3 (14.89.2) For x = se (t) and ϕ = ωt, we obtain a nonlinear equation for the pulsating electron sheath. x(t) is an even function of ωt. Hence, the terms which cause an antisymmetric (odd) behavior, i. e. terms which contain phase and sine, respectively, have to be replaced by −ωt for −π ≤ ωt ≤ 0. For large H, i. e. e0 VRF /kB Te 1, neglecting the first term on the RHS of Eq. (14.89) does not introduce a significant error, and we obtain for the thickness of the self-consistent RF sheath at ϕ = π ∨ x = se : x=

πAλD · 1 − cos ϕ +

se =

H 8

1 11 3 ∧ sin ϕ + sin 3ϕ − ϕ 3 cos ϕ + cos 3ϕ 2 18 3 (14.89.1)

j0 5π 12 e0 n0 ω se , H ∨H = e0 n0 ω 12 5π j0

(14.90)

and its potential V (x) =

kB Te 9π 2 2 H . e0 32

(14.91)

Inserting Eq. (14.86) into Eq. (14.90) yields 1 H= π

12 se 5 λD

2/3

,

(14.92)

598

14 Advanced Topics

which gives direct insight into the blowing-up process of the sheath by the rectification (Fig. 14.14). The normalized electron sheath thickness is obtained by inserting Eq. (14.90) into Eq. (14.89.2) 0

0

0 0 3 3 11 12 1 x (1 − cos ϕ) + sin ϕ + sin 3ϕ − ϕ 3 cos ϕ + cos 3ϕ 00 . = 01 − se 0 5πH 10π 2 18 3 (14.93) which is shown in Fig. 14.14. Since this is an odd function in ϕ, the absolute values have to be taken. The origin is defined as the sharp drop of the electron density during the descent to the surface of the electrode. As a result of the rectification across the sheath, the sheath thickness is deformed as a function of time or phase angle. During only a small part of the RF cycle, the boundary of the actual electron sheath is close to the electrode. 1.0

cos

0.5

(14.92)

0.0

-4

-3

-2

-1 0 1 2 phase angle [rad]

3

4

normal. sheath thickness

normal. electron sheath thickness

1.0

cos

(14.92)

0.5

0.0

-4

-3

-2

-1 0 1 2 phase angle [rad]

3

4

Fig. 14.14. The rectification across the sheath deforms the sinusoidal variation of the sheath thickness as a function of time or phase angle. LHS: electron sheath: Zero denotes vanishing of the electron sheath, the whole sheath consists of positive charges. Unity denotes reaching the electrode at x = 0 and phase ϕ = π, the positive and negative charges mutually cancel out. RHS: The surplus of the positive charge is the complementary picture. At ϕ = π, the electron sheath is congruent to the ion sheath and the surplus vanishes, at ϕ = 0 (almost for the fourth and the first quadrants), there is no electron sheath, and the positive sheath extends from the surface of the electrode (x = 0) to the boundary presheath/sheath (x = ds ).

14.4.1.4 Carrier density and sheath potential. Using Lieberman’s model for a capacitive sheath, we have calculated the instantaneous boundary of the electron sheath and look now for the time-averaged electron density. If the time-averaged electron density is calculated applying a very simple approach [192] (at the boundary presheath/sheath: ne /ni = 1, on the electrode: ne /ni = 0), we can roughly evaluate the time-averaged difference of the carrier densities. In Fig. 14.15, the crossplot of Eqs. (14.87) and (14.89.2) shows the


599

density drop of the ions and the difference of the densities of ions and electrons (time-averaged). It can be seen that across the whole sheath, the electron density remains smaller than the ion density, which is the criterion of sheath stability (cf. Sect. 14.2).

normalized carrier density

1.00

0.75

Fig. 14.15. The drop of carrier densities across the sheath which develops a DC bias potential of approximately 20 × ΦB . The time-averaged electron density remains below the ion density.

ni ne ni-ne

0.50

0.25

0.00 0.00

0.25 0.50 0.75 normalized sheath thickness

1.00

Due to their inertia, the ions do not respond to the instantaneous field, and the ion current and the thickness of the collisionless sheath are assumed to 3 depend on the sheath potential via the Child-Langmuir (ji ∝ V /2 /d2 with constant 4/9 = 0.44). By inserting se and V (x), Lieberman’s approach yields nearly twice this value, namely 0.82. For the thickness of the self-consistent sheath, however, we obtain for given ion current density and voltage a value which exceeds the thickness of a DC sheath by more than 1/3 . This is attributed to the sharply decreasing electron density across the sheath which necessarily draws a reduced ion density. Integrating yields the time-dependent sheath potential: V (t) =

πkB Te H{4 cos ωt + cos 2ωt + 3+ 4e0

1 15 5 3 1 (14.94.1) + cos ωt + ωt + ωt cos 2ωt + cos 4ωt − 16π 3π 8 3 16 1 238 sin 4ωt − 5 sin 2ωt + . 18 64 Within the limits −π und π and for large H, we find a time-dependent potential as depicted in Fig. 14.16. The maximum can be found at ϕ = 0 und amounts to +H

125π πkB Te H 8+ (14.94.2) H . 4e0 48 Expanding the time-dependent potential V (t) into a Fourier series yields the averaged value V = V0 : V (0) =

600

14 Advanced Topics

3πkB Te 3π A , H 1+ (14.94.3) 4e0 8 whereas the first two overtones for H 1 shrink to only 12 and 4 % of the fundamental, respectively. In particular, the ratio of the voltages V /V (0) = 54/25 ≈ 2.15. For a symmetrically driven RF discharge, the two RF sheaths are connected in series. At the hot electrode, we find a sheath voltage of Vt (t), at the cold electrode, a sheath voltage of Vw (t) = Vt (ωt − π). Expanding again in a Fourier series yields in a drop out of all the even higher harmonics including the DC component V0 . All the higher harmonics become very small, for example, Lieberman finds only 4.15 % of the fundamental [192], whereas Godyak calculates the third overtone to 6.4 % of it [810]. The mean current across the electrodes vanishes. V0 =

normal. sheath potential

1.00

0.75

0.50 0.25

0.00 -3

-2

-1

0

1

2

3

Fig. 14.16. For large displacement currents (large H), the maximum of the time-dependent potentials can be located at ϕ = 0.

phase angle [rad]

This model is valid for capacitive sheaths and high potentials. However, the beginning γ-processes at the electrodes will significantly change the distribution of the electric carriers across the sheath, not only where their density is concerned but also their energy distribution [811]. In weakly ionized RF plasmas, the sheaths are the only nonlinear elements and generate a significant DC component and a strong first overtone. In a planar symmetric RF discharge, the higher harmonics contribute to only a few percent. Hence, this type can be described as linear phenomenon. In a geometrically symmetric RF discharge, the DC component does not take place; however, this is not necessarily the case in an electrodynamic sense. In particular, if the capacitancies of plasma and sheath become comparable, the displacement current can cause a sigificant difference between cold (grounded) and hot (excited) electrode. 14.4.1.5 Sheath dynamics. Applying Eq. (14.93) for an estimation of sheath velocity and acceleration of the sheath, we obtain for the relative velocity


601

v 3ω 3 3 = − cos ϕ + cos 3ϕ + ϕ (3 sin ϕ + sin 3ϕ) , se 10π 2 2

(14.95)

and for the relative acceleration a 3ω = se 10π

7 9 sin ϕ − sin 3ϕ + 3ϕ (cos ϕ + cos 3ϕ) . 2 2 1.0 normal. sheath acceleration

normal. sheath velocity

1.0 sheath thickness

0.5

0.0 sheath velocity

-0.5

-1.0

(14.96)

-3

-2

-1 0 1 phase angle [rad]

2

3

0.5 sheath velocity

sheath thickness

0.0 sheath acceleration

-0.5

-1.0

-3

-2

-1 0 1 phase angle [rad]

2

3

Fig. 14.17. Extending Lieberman’s model of a nonlinear high-voltage capacitive sheath, we can derive the sheath dynamics (after [353]). According to the principle of continuity, an elevated ion density is related to a slower motion of the boundary and vice versa.

Whereas the sheath attains its topmost acceleration at the expected value ϕ = ±π (i. e. when touching the wall), its topmost velocity is reached at ϕ = 2.405 radians: the sheath velocity reaches its maximum at a time of the RF cycle when the sheath edge is still quite close to the wall [in Figs. 14.17, the representations are shifted by π to account for the evenness of Eq. (14.95) and the oddness of Eq. (14.96), respectively]. 14.4.2 The asymmetric case Asymmetric discharges are nonlinear phenomena to a high degree. The higher harmonics of the current can become as strong as the exciting RF current, in particular at low discharge pressures. In the case of solving the equations for the sheaths and the plasma bulk separately and further combining them under consideration of the continuity condition (the total currents incident on both the electrodes a and b must be equal: Ia = Ib ), we obtain a ratio of integrals for the voltage ratio between hot and cold electrode. This ratio is definitely not determined by the ratio of

602

14 Advanced Topics

the electrode areas as assumed in the approximation of “zero order” which was established by Koenig and Maissel with their sheath theorem [156], but by Va = Vb

nb (x)d2 x 2 A,a na (x)d x

q

A,b

.

(14.97)

q can take several values: 4 for the collisionless fall, 3 for the elastic collision, /2 for the resonant charge transfer, n is the plasma density at the Bohm edge and takes different values for the various transport mechanisms. Moreover, calculations by Lieberman and Savas revealed an increase of the ion current density about 40 % at the smaller electrode compared with the larger one [812]. In particular, the ratio of the sheath voltages does not depend on the exciting RF voltage or on the electron temperature and does depend on the ratio of the electrode areas but via the plasma density at the Bohm edge. As we have seen the connection between the exciting instantaneous amplitude VRF and the voltage drop across the sheath VDC is very complicated and contains an insoluble integral [Eq. (14.97)]. For the asymmetric case, Garscadden and Emeleus [183] and Butler and Kino [813] started with the following simplifications: 5

• The negative current density equals the positive one when one electrode is connected to a sinusoidal RF voltage VRF sin ωt and this electrode will be biased with a negative voltage VDC (cf. Sect. 6.5). • The ion current density is independent of the sheath voltage (cf. Sect. 14.2). • The sinusoidal voltage will not be distorted. It follows [cf. Eqs. (3.27), (3.28), (6.30), (14.44)] with VDC = V p − VDC ∧ Vs = VRF

Cc : Ch + Cc

e0 (VDC + VRF sin ωt) je = e0 n0 exp kB Te

(14.98) kB Te 2πme

(14.99)

and

ji = e0 n0

kB Te . emi

(14.100)

The time averaged electron current density, je , i. e. the integral 1T e0 exp VRF sin ωt dt T 0 kB Te

(14.101)

yields the Bessel function of zero order with purely imaginary argument [349]:


J0 −i

603

e0 VRF . kB Te

(14.102)

For zero current across the sheath follows VDC

kB Te =− e0

1 e0 VRF emi ln + ln J0 2 2πme kB Te

(14.103)

with the term for the floating potential Vf =

kB Te emi ln 2e0 2πme

(14.104)

from Sect. 3.4. There are two limiting cases: e0 VRF kB Te : VDC = −Vf .

(14.105.1)

e0 VRF >≈ 10 × kB Te :

(14.105.2)

−VDC = Vf + VRF −

kB Te e0 ln 2π VRF . 2e0 kB Te

(14.105.3)

For large RF voltages, the developing DC bias equals the sum of floating potential and the amplitude of the RF voltage, corrected by a logarithmic term. Plotting the DC fraction from Eq. (14.105.3) vs. the RF fraction Vs (t) yields a straight line with unity slope which confirms this approach. For small voltages, the limit Vf is reached (Fig. 14.18) which causes severe distortions of the RF signal. In Fig. 14.19, this is shown for the boundary voltage of a capacitively

DC voltage [V]

100

VRF

10 Vf

0.1

1 10 RF voltage [V]

100

Fig. 14.18. Sheath voltages: ordinate: DC fraction, abscissa: RF fraction. The straight line is calculated according to eq. (14.105) after c The American [119] [195] Institute of Physics.

coupled electrode in an ion beam system for four different positive values for VDC (16 V, 30 V, 50 V, and 72.2 V). It can be seen that at low voltages, the response is far from being sinusoidal.

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14 Advanced Topics

boundary voltage [V]

100 75

70 V

Fig. 14.19. Boundary voltage on the acceleration electrode for several DC settings. Note the deviation from harmonic behavior for low voltages [229].

50 V

50

30 V

25 10 V

0

0.0

0.5

1.0

1.5

2.0

2.5

time [msec]

14.5 Motion in a magnetic field 14.5.1 The magnetic bottle The general equation of motion of an electron in a a static magnetic field reads dv ; (14.106) dt its solution is a superposition of a circular motion v c with respect to a center of gyration Z which moves with velocity v z (v z B): F = −e0 (v × B) = m

v = vc + vz = ωc × rc + vz .

(14.107)

The acceleration by the magnetic field can be expressed by the Grassmann identity: dv = ω c × v c = −ωc2 · r c , dt and inserting this into the equation of motion yields e0 (v z × B + v c × B) = −m (ω c × v c ) .

(14.108)

(14.109)

The motion is composed of two components: a circular motion around B with the cyclotron frequency ω c , combined with a second motion along B. Since the first vector product vanishes because the vectors are assumed to be collinear, the Larmor relation reads e0 B . (14.110) m The amount of the cyclotron frequency depends only on the mass of the gyrating particle, its direction on the charge. With a static electric field imposed on a magnetic field B, the Lorentz force is modified to F = mdv/dt = −e0 [E + (v d × B)]. An approach similar to that of Eq. (14.107) yields ωc =

14.5 Motion in a magnetic field

605

E + v d × B = 0,

(14.111)

E = B × vd,

(14.112)

or

from which we get E × B = B × v d × B ⇒ E × B = B 2 v d − B(v d · B).

(14.113)

v d is composed of the two components v⊥ and v :

vd = E ×

B B2

+ e0

E , m

(14.114)

(so-called E ×B drift) or cross-drift [814]. The center of gyration is accelerated in a direction which is normal to both fields, and its constant drift velocity vd amounts to E/B. Finally, vd depends neither on the charge nor on the mass because both forces E and B scale with the particle charge. The gyration radius of a heavier mass is larger, but its cyclotron frequency, however, is lower by the same factor (Fig. 14.20).

r [a. u.]

electrons ions

B, E

Fig. 14.20. For crossed electric and magnetic fields, electric carriers evade the electric field with the drift velocity vd = E/B on cycloidical trajectories. To observe this effect effect, |E| must be small compared to |B|. Dimensions are not to scale!

vz [a. u.]

In a quasi-neutral medium composed of an equal number of unlike charges, these charges move in response to the force, i. e. the field E, to which they are subjected, and two currents with opposed directions are generated. With respect to B, circular motions are caused with opposed helicity. The cross-drift, however, turns out to be equal for both types. In the direction of vd , E causes a mass separation without current, the trajectory is not a helix but a cycloid! The velocity in the direction of the magnetic field is exclusively determined by the electric field (conservation of energy). Furthermore, the amount of B must

606

14 Advanced Topics

exceed that of E, otherwise, the particle would be continuously accelerated in the direction of E!9 For constant magnetic field (B E), this results in a helix with constant radius, if the magnetic field varies with time (∂B/∂t Bωc ), and additional motion has to be taken into account according to the induction law. For a circular motion, application of Faraday’s law yields

∂B , (14.115) ∂t and the charged carriers are accelerated in response to a force that is directed perpendicular to the magnetic field; its magnitude can be calculated along with Eq. (14.110) to ∇ × E dA =

E · ds = 2πrc E = −πrc2

1 dωc −e0 dv⊥ = rc = E. dt 2 dt m Since v⊥ = ωc rc , the time derivation yields

(14.116.1)

drc dω c dv ⊥ = ωc + rc , dt dt dt and comparing Eqs. (14.116.1) and (14.116.2) shows that

(14.116.2)

dv⊥ 1 dωc drc rc = −ωc = : (16.116.3) 2 dt dt dt ωc (drc /dt) is the negative change of the normal component of the velocity with respect to time, and the entire change in angular momentum results in

dωc rc2 drc dv⊥ dL =m = mrc ωc + dt dt dt dt

= 0.

(14.117)

Increasing (decreasing) magnetic field causes a reduction (increase) of the precession radius. For a gradient of the magnetic field which is collinear to this field (∇B B z), we can state that ∇ · B = 0, which can be written in cylindrical coordinates, neglecting the azimuthal component (BΘ = 0 ∧ ∂B/∂Θ = 0), yielding 1 ∂(rBr ) ∂Bz + = 0. (14.118) r ∂r ∂z Assuming that Bz Br , the variation of the magnetic field with respect to the radial component does not change considerably, and we obtain the approximate result at r = 0: Br = −

rc ∂Bz , 2 ∂z

(14.119)

9 This principle is applied in velocity discriminators. In connection with deflecting magnets, E × B selectors can filter a monochromatic beam of defined mass out of a “colored” bunch of beams.


607

by which the occurrence of a radial force can be described whose magnitude is F = evBr .

(14.120.1)

Along with Eq. (14.119) and considering the Larmor relation, Eq. (14.120.1) can be written as 2 1 mv⊥ ∂Bz ∂Bz = −μ (14.120.2) 2 B ∂z ∂z with μ the magnetic moment μ = I A or Ekin,⊥ = μ B, and Ekin,⊥ the transverse kinetic energy rectangular to the magnetic field. The particle is accelerated along the gradient: for rising field, the parallel component will decrease, for weakening field, the parallel component will increase. For a stationary field, the kinetic energy is conserved for a thermal motion of the particle:

F =

dEkin,⊥ dEkin, + = 0, dt dt

(14.121)

which yields d dt

mv2 + μB = 0, 2

(14.122.1)

dv d(μB) + = 0. dt dt

(14.122.2)

or mv On the other hand, we have

v m

dv dB d dz dB dB = 1/2 mv2 = −v μ = −μ = −μ , dt dt dz dt dz dt

(14.122.3)

and by comparison of Eqs. (14.122.2) and (14.122.3), the variation of the magnetic moment in time will vanish. The magnetic moment is adiabatically invariant: dμ = 0. (14.123) dt Since the magnetic moment is defined by current times the enclosed area (μ = I A), we see that the constancy of μ yields (m0 : electron or ion mass)

2 e0 ω e0 v⊥ 2 m0 v⊥ = . (14.124.1) π 2π ωc 2 m0 ωc The magnetic moment equals the ratio of the transverse kinetic energy over the intensity of the magnetic field

μ=

14 Advanced Topics

rx [a.u.]

ry [a.u.]

608

Fig. 14.21. For sufficiently intense magnetic fields, an electron will be reflected in a longitudinal direction, but for decreasing intensity, the electron is subjected to an acceleration with respect to this direction. The pitch of the envelopping cone depends on the gradient of the magnetic field strength, being positive from left to right.

Bz [a.u.] Ekin,⊥ , (14.124.2) B0 and the transverse velocity scales with the square root of the magnetic field: μ=

v⊥ ∝

√

B,

(14.125)

the negative gradient of the magnetic field causes the carriers to move along a widening spiral along the magnetic field, electrons gyrate in a clockwise, ions in a counterclockwise motion. For a positive gradient, the rising magnetic field B causes an increase of the transverse velocity component v⊥ . Since the total kinetic energy must remain constant, the charged particle gyrates with an evertighter orbital motion along the line of increasing magnetic field, converting more and more of the parallel velocity component (translational) v into the transverse velocity component (rotational) v⊥ until v will eventually vanish (Fig. 14.21). For even higher B, the charged particle has to turn, retaining its helical direction. At the turning point, we find from Eqs. (14.124.2) and (14.125) for an arbitrary axial position 0 and the extraordinary position m (for max): 2 2 1 mv0,⊥ B0 1 mvm,⊥ 2 2 = ⇒ v0,⊥ = vm,⊥ . (14.126.1) 2 Bm 2 B0 Bm v0 has the two components v0,⊥ and v0, , vm consists but of the transverse component vm,⊥ :

vm = vm,⊥ .

(14.126.2)

Conservation of energy further requires m 2 m 2 m 2 2 ) = vm,⊥ , v0 = (v0,⊥ + v0, 2 2 2

(14.126.3)


609

or, with Eqs. (14.126.1) and (14.126.2),

2 2 2 2 2 2 v0, = vm − v0,⊥ ∧ v0, = vm − vm

B0 Bm − B0 2 2 ⇒ v0, = vm . Bm Bm

(14.126.4)

From the last equation it is evident that trapping of particles occurs under the restrictive condition 0 0 0v 0 Bm − B0 0 0, 0 0 0< . 0 v0,⊥ 0 Bm

(14.127.1)

The trap consists of a cone (Fig. 14.21), and writing the components v0,⊥ = v0 sin α and v0, = v0 cos α, i. e. v0 = v0 (sin α + cos α)

(14.127.2)

yields the pitch of the envelopping cone [with Eq. (14.126.3)] v2 v02 sin2 α v02 sin2 α B0 = 0,⊥ = = = sin2 α. 2 2 Bm vm vm v02

(14.128)

The adiabatic invariance opens the possibility to trap the plasma within a magnetic bottle by magnetic mirors. The confining surface is the loss cone. Particles which are generated or injected into the cone will spiral within the cone along the B-axis and are reflected by the magnetic mirrors at either end. If the ratio of the parallel velocity over the transverse velocity within the cone exceeds the limit given by the restrictive condition (14.127.4), then the particle evades the bottle (Figs. 14.22 and 14.23). The trapping condition is independent on charge and mass. In a collisionless plasma, electrons and ions can be captured by a magnetic bottle system with equal probability. However, when their temperature deviates from zero, the particles can alter their velocity by collisions. Since electrons exhibit a significantly higher frequency of collision, it is more difficult to confine them. That this derivation is only approximate becomes evident for the borderline case: For vanishing v⊥ , the magnetic moment would drop to zero as well. Consequently, it would not deviate due to the magnetic field at all. When approaching the cyclotron frequency limit, the magnetic moment cannot be considered adiabatically invariant any longer. The opposite of a magnetic mirror is realized in the divergent magnetic fields of ECR sources which serve as extraction system: The parallel velocity component v in the weakening magnetic field, and the ions are expelled from the plasma. The magnetic field of the earth, being weak at the equatorial region and strong at the poles, has created such a double mirror in an altitude far beyond the atmosphere. The van-Allen belts consist of two toroidal systems: an inner

14 Advanced Topics

ry [a.u.]

vy [a.u.]

610

Bz [ a.u. ]

.] a.u r x[

Bz [ a.u. ]

vx

u.] [a.

Fig. 14.22. A magnetic bottle, the magnetic field exhibits a minimum in the middle. LHS: radius of the gyrating electron as function of the magnetic field strength, RHS: transverse velocity of the electron, projected curves are r(B) and v⊥ (B). The transverse velocity rises with the square root of B, the cyclotron frequency scales linearly with B. A similar dependence is valid for the trajectory in a one-dimensional potential: The orbital velocity v⊥ is proportional to the square root of the radius r.

belt with a radius of about 6 000 miles contains fast protons, the outer belt contains fast electrons, and its radius is about 13 000 miles. 14.5.2 Modification of diffusion A magnetic field always causes an anisotropic behavior. If we focus on diffusion (B z), we consider the Lorentz force as an external force which counters the forces which are generated by pressure gradients along with the change of mean momentum with respect to time (caused by collisions, stationary state dv/dt = 0): ∇p = F = e0 v × B. n From kinetic gas theory, we know that mνm v +

(14.129)

1 < v2 > p = nm < v 2 > ∧D = , 3 3νm

(14.130)

and applying ωc = −e0 B/me , we find

∇n ωc = 1− × v, n νm which can be written in matrix form −D

(14.131)


611

v v : max v:0

v

a v : min v : max

vy v

vx v : max v:0

Fig. 14.23. An electron can be trapped in a magnetic bottle and will be mirrored by a high magnetic field, but accelerated in the axial direction for a weakening field. Electrons will move counterclockwise (CCW or R), ions clockwise (CW or L). The restriction for confinement is an upper limit for fast carriers.

⎛

⎞

⎛

1 ∂n/∂x D⎜ ⎟ ⎜ − ⎝ ∂n/∂y ⎠ = ⎝ − νωmc n ∂n/∂z 0

ωc νm

1 0

⎞⎛

⎞

0 vx ⎜ ⎟ 0 ⎟ ⎠ ⎝ vy ⎠ . vz 1

(14.132)

Since we are interested in the dependence of particle flux j with respect to the gradient in density [j = j(∇p)], the matrix will be inverted to yield ⎛

νm 2

2

ωc +νm ⎜ ⎜ − 2ωc 2 D= ⎝ ωc +νm 3 2

0

ωc 2 ωc2 +νm νm 2 ωc2 +νm

0

⎞

0 ⎟ 0 ⎟ ⎠, 1 ; νm

(14.133)

which is commonly written as [815] ⎛

⎞

D⊥ D× 0 < v2 > ⎜ ⎟ D= ⎝ −D× D⊥ 0 ⎠ 3 0 0 D

(14.134)

612

14 Advanced Topics

with • D⊥ the transverse, • D× the normal, and • D the parallel diffusion coefficient. The diffusion coefficient, which is orientated parallel to the magnetic field, D , remains unaffected. By means of the other two coefficients, however, the effects on the diffusion normal to the magnetic field is described. For the limiting cases of the strong and the weak magnetic field, respectively, we can note: • ωc νm : D⊥ scales with 1/ωc2 or 1/B 2 . The radial diffusion directed normal with respect to the applied field is suppressed. Vice versa, this means for fixed B: D⊥ ∝ < v 2 >/ωc2 ∝< v 2 > m2 , and the magnetic field causes a large acceleration of the ions, compared with that of the electrons. • For small magnetic fields, the antisymmetric terms in the matrix will vanish resulting in the symmetric diffusion tensor: ⎛

νm

2 ωc2 +νm ⎜ ⎜ 0 D= 3 ⎝

2

0

⎞

0

0 ⎟ 0 ⎟ ⎠,

νm 2 ωc2 +νm

(14.135)

1 νm

0

Inserting the symmetric diffusion tensor into the definition equation (5.30) yields

1 νm 1 1 1 1 = = 2 + 2 + 2. ∧ Λ2 D Λ2 Λx Λy Λz

(14.136)

Since B z, the diffusion length becomes νm 1 = νm 2 Λ2 νm + ωc2

1 1 + . Λ2x Λ2y

(14.137)

The magnetic field causes an enhancement of the diffusion length in all the directions which are orientated normal with respect to the magnetic field by the ratio 2 νm + ωc2 . 2 νm

(14.138)

14.6 Dispersion in a HF plasma

613

14.6 Dispersion in a HF plasma 14.6.1 Cutoff and skin depth The equation of motion of an electron in a transversal E-field that is supported by an external source E 0 is given by

me

dv e + νm v e = −e0 E 0 ei(ωt−kx) , dt

(14.139)

taking account of elastic collisions between electrons and neutrals; and the velocity of the electrons results in ve = −

1 e0 E 0 ei(ωt−kx) me iω + νm

(14.140)

which yields for the current density with j = ρv = σE j=−

ne e20 1 E 0 ei(ωt−kx) . me iω + νm

(14.141)

This equation comes to be better understood by realizing the loss of momentum me v e the electron experiences by every collision has to be weighted by νm collisions per second. Therefore, the change in momentum per second totals νm me v e [note that νm should not depend on the electron temperature which is only correct for the light gases (cf. Sects. 2.3 and 14.1)]. Mobility μ and conductivity σ are closely connected by σ = ρμ with ρ the charge density; and inserting the plasma frequency, we obtain for σ: σ = ρμ = ωp2

ε0 iω + νm

(14.142)

which yields for ∇ × H in an isotropic linear medium [only in these media, the simple relationship between D and E (D = ε0 εE) holds true (μ is assumed to equal unity)]: ∇×H =

σ + ε · ε0 iω · E, ε0 iω

(14.143)

or ∇ × H = (σ + ε0 εiω)E,

(14.144)

the first summand represents the conduction current whereas the second stands for the displacement current. Since we search solutions for plane electromagnetic waves, i. e. E = E 0 exp −i(k · x − ωt) which solve Faraday’s law ∇ × E = −iωμ0 H whence ∇ × ∇ × E = −∇2 E = −iωμ0 ∇ × H = k 2 E, it follows immediately

614

14 Advanced Topics ∇×H =−

∇×∇×E ⇒ (σ + ε0 εiω) iωμ0 E = −k 2 E; iωμ0

(14.145.1)

˙ = −iωE, E ¨ = −ω 2 E, and E = −k 2 E: or, with E ε ¨ E, (14.145.2) c2 the telegraph equation, from which the relation between E and H is given by ˙ + ∇2 E = σμ0 E

H=

k×E . ωμ0

(14.146)

From that equation, we can express the complex wave vector by

k2E = 1 −

iσ ω2 ε 2E ε0 εω c

(14.147)

which can be written, using Eq. (14.142)

ωP2 ω2 i k = 2ε 1− . c εω νm + iω 2

(14.148)

Separating into real and imaginary parts yields

ω2 ωP2 iνm k = 2ε· 1− 1+ 2) c ε(ω 2 + νm ω

2

.

(14.149)

Neglecting any inner polarizability of the ions, which is a rather good approximation below the plasma frequency of the electrons, ε saturates at unity and we obtain the complex refraction index squared

n2 =

k 2 c2 ωP2 iνm = 1 − 1+ , 2 ω2 ω 2 + νm ω

(14.150)

which turns into the dispersion relation ω2 k = 2 c

2

ω2 1 − p2 ω

⇒ ω 2 = ωp2 + c2 k 2

(14.151)

for νm = 0. In a plasma, the square of the refraction index below the plasma frequency remains smaller than unity.10 Considering these facts, the phase velocity exceeds c: 2 = vph

ω2 c2 = > c2 , k2 1 − ωp2 /ω 2

(14.152)

however, the group velocity vg = dω/dk remains below c: 10 This has to be taken into account when optical paths are considered (non-appearance of the phase jump for reflection and cutoff in the atmosphere).


615

dω c2 k dω c2 =c = ⇒ = dk ω dk vph

1−

ωp2 . ω2

(14.153)

For vph > c we find vg < c, and the refraction index becomes n = c/vph = ck/ω = 1 − ωp2 /ω 2 . For ω < ωp the refraction index becomes imaginary, and these waves cannot propagate in this tenuous plasma. In a collisionless plasma, the power cannot be transferred, and the wave is entirely reflected (case of the ionosphere). The wave vector becomes imaginary as well: k = i/c ωp2 − ω 2 : • The induced current density equals or exceeds the displacement current density. • Their phase angles are shifted by 180◦ . • The wavelength becomes infinite. • The wave is attenuated during its course (Figs. 14.24). It is customary to unite the reactive properties ε and μ as well as the resistive property σ. We start from Eq. (14.144): ∇ × H = (σ + iε0 εω)E

(14.144)

and define a complex dielectric constant σ + iε0 εω = iε0 ε˜

(14.154)

with ε˜, the complex number iσ iσ ⇒ ε = ε˜ + , ε0 ω ε0 ω

ε˜ = ε −

(14.155)

yielding

∇ × H = σ + iε0 ω ε˜ +

iσ ε0 ω

E,

(14.156)

and ∇ × H = iε0 ε˜ωE.

(14.157)

This equation differs from the simple equation for an insulator algebraically just by a term iσ , ε0 ω and remains in the same shape. We obtain11 ε˜ = ε −

(14.158)

11 With this approach, all the imaginary portions remain positive since in all optical materials the damping constant turns out to be positive which causes a reduction in intensity for transmission. The only exception is a laser which represents a source for radiation [816].

616

14 Advanced Topics

10

v/c

1 phase velocity group velocity

0.1

0.01

150

108

wp [sec-1]

109

1.0

1.2

1.4 1.6 w/wp

1.8

1010

2.0

wp [sec-1]

109

w p/2p = 2.45 GHz

1010

1011

100

d [cm]

l [cm]

100

50

10

operating frequency 13.56 MHz

l g = 12.25 cm

0 108

109

1010

1

1011

109

-3

n p [cm ]

1010 1011 np [cm-3]

1012

Fig. 14.24. Top: Below the plasma frequency ωp , the refractive index becomes imaginary: vph → ∞, vg → 0: Cutoff. Both velocities are rapidly changing functions of frequency in the vicinity of cutoff. Above, both velocities slowly approach unity. Bottom, LHS: This has the further consequence that λ → ∞, shown here for operation in the microwave band at 2.45 GHz with a vacuum wavelength λg = 12.25 cm. The cutoff density is 7.3 × 1010 cm−3 . Bottom, RHS: Within the skin depth δ, the penetrating wave is severely damped (to 1/e or ≈ 30 %), displayed for plasmas operated at 13.56 MHz.

• a complex dielectric constant: σ , ε0 ω

(14.159.1)

εR − iεI ,

(14.159.2)

ε˜ = εR − iεI ∧ ε = εR − i • a complex refraction index: n ˜ = n − iκ =

√

ε˜ =

√

• and a complex wave vector: iω k˜ = − n ˜. c

(14.159.3)


617

14.6.2 Complex properties It we write for n ˜ = n − iκ, i. e. • n ˜ 2 = n2 − κ2 − 2inκ, it is given by √ √ • |n ˜ |: n2 + κ2 = 4 ε2R + ε2I = εR 4 1 + (εI /εR )2 ), and • k 2 = ω 2 /c2

ε2I + ε2R ,

and the two parts of the refraction index are given by

n = κ =

εR +

√

−εR +

ε2R +ε2I 2

√

ε2R +ε2I

2

=

√

=

√

εR

1+

εR

√

−1+

1+(εI /εR )2 , 2

√

1+(εI /εR )2 2

⎫ ⎪ ⎪ ⎪ ⎬

(14.160)

⎪ ⎪ ⎪ ⎭

with εI = σ/(ωε0 ), and for the real and imaginary parts of the dielectric constant: 2 ωP 2 ω 2 +νm 2 ω νm P 2 ω ω 2 +νm

n2 ) = n2 − κ2 = 1 − εR = (˜

= 1−

εI = (˜ n2 ) =

=

2nκ

=

2 /ν 2 ωP m 2 , 1+ω 2 /νm

2 ωP 2 ), νm ω(1+ω 2 /νm

⎫ ⎪ ⎬ ⎪ ⎭

(14.161)

and we must investigate the following limiting cases (Figs. 14.25 and 14.26): • Case (1): Imaginary part exceeds the real part significantly, the real part is 2 negative and its amount is large against unity (εR εI , ω νm , νm ωp2 2 ⇒ ω ωp ), and the dielectric constant can be simplified to σ ε˜ = n ˜ ≈ −i ⇒n ˜= ε0 ω

2

σ (1 − i) , 2ε0 ω

(14.162.1)

from which we can infer n −→ κ −→

εI 2 εI 2

1+

εR 2εI

1−

εR 2εI

⎫

⎬ , ⎪ ⎭ . ⎪

(14.162.2)

The amounts of n and κ are almost equal, from which the phase shift between kI and kR , i. e. between κ and n yields tan α =

κ ≈ 1 ⇒ arctan 1 = 45◦ . n

(14.163)

The wave vector can be written as k = n ˜ i with i a unit vector, and we obtain for the ratio between E and H (the wave impedance) along with eq. (14.146):

618

14 Advanced Topics 103

102

102

10

10

1

1

n

k

103

10-1 10-2 10-3 10-4 1

10

100 w [MHz]

nm /w p = 0.0001 nm /w p = 0.001 nm /w p = 0.01 nm /w p = 0.1

10-1


10-2 10-3 1000

10-4 1

10000

10

100 w [MHz]

1000

10000

Fig. 14.25. Roots of the real part (LHS) and the imaginary part (RHS), resp., of the dielectric constant according to Eq. (14.160) in plasmas (driven by RF and MW, resp.) for several frequencies for momentum transfer νm . The plasma frequency (ωp = 5 640 kHz) separates the regime of evanescence (cutoff plasma) from the dielectric.

√ k H = = εR E μ0 ω

4

1+

εI εR

2

1 . μ0 ω

(14.164)

Since εI εR , the magnetic field turns out to be large against the electric 2 and νm ω these approximations are valid for n and field. For ωp2 νm κ:

n ≈ κ ≈

2 1 ωP 2 ωνm 2 1 ωP 2 ωνm

1−

ω 2νm

1+

ω 2νm

⎫

⎪ ⎬ , ⎪ ⎪ ⎪ . ⎭

(14.165)

√ κ scales with 1/ νm and will become more or less independent of ω. Since the penetration depth of electromagnetic waves δ depends on the operating frequency ω and κ, this yields c 1 c δ= ≈ ω κ ωp

For an inductively coupled plasma in argon with – ω = 2 π × 13.56 MHz = 85.2 MHz, – np = 1012 cm−3 , – νp = 9 GHz and ωp = 56.5 GHz, – p = 10 mTorr (1.3 Pa),

2νm ω 1− . ω 2νm

(14.166)


619 1.00

1.00 nm /wp = 0.0001 nm /wp = 0.001 nm /wp = 0.01 nm /wp = 0.1

n

0.75

0.75

0.50

k

0.50

0.25

0.25

0.00 0


1 (wp/w)2

2

0.00 0

1 (wP/w)2

2

Fig. 14.26. Roots of the real part (LHS), and the imaginary part (RHS), respectively, of the dielectric constant as a function of the normalized electron density n/nc = (ω/ωp )2 in high-frequency plasmas for some frequencies of momentum transfer νm . The plasma frequency (ωp = 5 640 kHz) separates the cutoff plasma (regime of evanescence) from the dielectric.

– Te = 4 eV, – νm = 95 × 106 sec−1 (1.3 Pa) δ equals 4.1 mm or δ/λ0 = 4.1 mm/22.1 m = 1.8×10−5 . In copper, the skin depth of electromagnetic waves would amount to (λ0 = 22.1 m) 160 μm for 13.56 MHz, but for 2.45 GHz only (λ0 = 12.25 cm) 12 μm. Lowering the operating frequency from 13.56 MHz to 2 MHz enlarges the skin depth to 1.74 cm, i. e. by more than a factor of 4. Neglecting all elastic collisions (νm = 0), the skin depth becomes: δ=

1 c c ≈ . = 2 2 k ω p ωp − ω

(14.167)

For a plasma with a density of 1012 cm−3 , the plasma frequency equals ω = 56.5 GHz. For an operating frequency of 13.56 MHz, the skin depth is approximately 5.3 mm. For some typical plasma densities, the skin depths are compiled in Table 14.1. • Case(2) ε2R > ε2I , εR < 0 : This is the relaxation range (so-called cutoff plasma or regime of evanescence), where ω/νm dominates in the denominators of εR and εI . The dropping absorption coefficient scales with ω −1 , and the imaginary part of n ˜ 2 becomes smaller than its real part which remains large and negative. We obtain:

620

14 Advanced Topics

Table 14.1. Skin depths δ according to Eq. (14.167) as function of plasma density for νm = 0, neglecting elastic collisions, referred to λ0 = c/13.56 MHz. plasma density [cm−3 ] 108 109 1010 1011 1012

plasma frequency [MHz] 565 1790 5652 17.9 × 103 56.5 × 103

skin depth [cm] 53 17 5.3 1.7 0.53

n −→ κ −→

√ √

−εR −εR

δ/λ0 0.024 0.0077

−εI , 2εR

1+

εI 8ε2R

⎫ ⎪ ⎬ ⎭ , ⎪

(14.168)

2 ω 2 ωp2 and for νm

n ≈

νm ωp2 2ω 2

κ ≈

ωp ω

δ ≈

c ωp

1−

2 νm 2ω 2

+

ω2 2ωp2

1−

2 νm 2ω 2

−

ω2 2ωp2

1+

2 νm 2ω 2

+

ω2 2ωp2

⎪ ⎪ ⎬

⎫

⎪ , ⎪ ⎪ ⎪

, .

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(14.169)

δ remains nearly constant and approximately equals one vacuum wavelength. That means for an inductively coupled plasma with – ω = 2 π × 13.56 MHz = 85.2 MHz, – np = 1012 cm−3 , – ωp = 56.5 GHz, – Te = 4 eV, – νm = 95 × 106 sec−1 (1.3 Pa) a skin depth of 8.6 mm (δ/λ0 = 0.00032), for an inductively coupled plasma with – ω = 2 π × 2 MHz = 12.57 MHz, – np = 1012 cm−3 , – ωp = 56.5 GHz, – Te = 4 eV, – νm = 95 × 106 sec−1 (1.3 Pa)


621

a skin depth of 15.16 cm (δ/λ0 = 0.0069), for a capacitively coupled plasma in argon with – ω = 2 π × 13.56 MHz = 85.2 MHz, – np = 1010 cm−3 , – ωp = 5.65 GHz, – Te = 4 eV, – νm = 95 × 106 sec−1 (1.3 Pa) a skin depth of 6.97 cm (δ/λ0 = 0.0032), and for an ECR driven plasma with – ω = 2 π × 2.45 GHz = 15.4 GHz, – np = 1012 cm−3 , – ωp = 56.5 GHz, – νm = 9.5 × 106 sec−1 (0.14 Pa) a skin depth of 5.3 mm (δ/λ0 = 0.043). 2 ω 2 ≈ ωp2 : In this range, none of the approximations can be • Case (3): νm √ applied. On the contrary: Within a very tiny interval, 2νm ωp , the skin depth changes by the factor of 2ωp /νm : Slight variations of the plasma frequency by only a few percent are sufficient to turn an opaque plasma into a transparent one and vive versa.

˜2 • Case (4) Dielectric matter with ε2R ε2I , εR > 0 : The real part of n becomes positive, and the absorption drops to very low values: √

n −→

√

κ −→

εR

1+

ε2I 8ε2R

εI . 2εR

εR

⎫

⎪ , ⎬ ⎪ ⎭

(14.170.1)

For the approximation ω 2 ωp2 :

n ≈ κ ≈

ωp2 , ω(ω−iνm )

1− νm ωp2 2ω 3

1−

ωp2 ω(ω−iνm )

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(14.170.2)

2 which becomes for ωp2 νm :

n ≈ κ ≈

1− νm ωp2 2ω 3

⎫ ⎪ ⎪ ⎬

ωp 2 , ω

1−

ωp 2 . ω

⎪ ⎪ ⎭

(14.170.3)

622

14 Advanced Topics The index of refraction does not depend on the frequency of momentum transfer between electrons and neutrals νm any longer. Low absorption means vanishing of the imaginary part of the refraction index, and applying Newton’s relation, we obtain the simple result [cf. Eq. (14.153)] ε=1−

ωp ω

2

.

(14.171)

In Figs. 14.24 and 14.25, the dispersion behavior according to Eqs. (14.160) is shown for some collision frequencies in the RF and MW range taking into account the reciprocal relation between the imaginary part of the refraction index and the skin depth δ [eq. (14.166)]. The skin depth δ for electromagnetic waves is shortest for a collisionless plasma and is lengthened with rising gas temperature. On the other hand, the large values for the skin depth are purely fictitious since the field strength has been weakened by collisions. These figures perfectly illustrate the softening of the hard cutoff condition for high discharge pressures which aggravates measurements of the plasma density by microwave interferometry. Compared with the penetration depth of electrostatic waves which is in the order of several Debye lengths (cf. Sect. 14.3), the skin depth of electromagnetic waves exceeds this value by a factor of several hundred. In the case of the longitudinal waves, this behavior is owed to the fact that the electrons are forced to oscillations by the electric field which is orientated along to the wavevector k. For frequencies ω ≤ ωp , the longitudinal field is damped by a sheath. For transverse waves, the oscillating electrons set up an additional magnetic field rather which further weakens the screening effect of the plasma against penetrating waves (Fig. 14.27). direction of motion of the electrons

E

k

+ _ + _ + _ + _ + _

+ _ + _ + _ + _ + _

E E

Fig. 14.27. Transverse (electromagnetic) waves can deeply penetrate the plasma (some hundred Debye lengths) [817] c Academic Press). (

plasma

From Eqs. (5.9), we infer the transferred power to scale with the plasma density (P ∝ np E 2 ), and the radial dependence of np becomes pronounced in capacitively coupled discharges with a centered maximum, whereas np (r) is less marked with an inverted behavior in inductively coupled discharges since diffusion processes which are directed downstream serve to even out these gradients,

14.7 Whistler waves

623

however, with respect to substrate level. As has been reasoned by Brown [818], high plasma densities at the confining parts of the barrel-shaped plasma source define the upper limit of power input. Starting with low plasma densities at the edge, the power is continuously absorbed on its way directed inward (Fig. 14.28 top, curves A + B). Having reached the critical density for which ω < ωp (curve C), we have reached the cutoff plasma, and the power input is effectively restricted to the dimensions of the skin depth [819]. As has been measured and modeled by Collison [820], the central plasma density can be significantly lower than in the confining areas (Figs. 14.28, bottom; cf. also Fig. 7.7). Fig. 14.28. Sketch of the radial dependence of the plasma density for three different (two undercritical and the critical) pressures in an inductively coupled plasma and measured overcritical dependencies (bottom, left) which have been modeled (bottom, right) [818, 820].

ne nP C B A

1.0

normalized ion saturation current

normalized ion saturation current

r=0

0.8 0.6


0.4 0.2 0.0 -300

-200

-100

0 100 r [mm]

200

300

1.0 0.8 0.6 5 mTorr 10 mTorr 40 mTorr

0.4 0.2 0.0 -300

-200

-100

0 100 r [mm]

200

300

14.7 Whistler waves 14.7.1 Plane waves 14.7.1.1 Formula of Appleton and Hartree. The Drude or Langevin equation consists of four parts: e0 E(t) = e0 v × B 0 − m

dv − νm v dt

(14.172)

624

14 Advanced Topics

• e0 v × B 0 : Hall term; • mdv/dt: term of electron inertia; • νm v: resistance term; and yields written in components with z B 0 : (iω + νm )vx +

e0 B0 vy m

⎫

= − em0 Ex , ⎪ ⎪ ⎪

− e0mB0 vx + (iω + νm )vy = − em0 Ey , (iω + νm )vz = and as matrix with ⎛

1 ⎜ ⎝ ε0 ωp2

ωp2

=

− em0 Ez

⎬

(14.173)

⎪ ⎪ ⎪ ⎭

ne20 /mε0 ⎞⎛

⎞

⎛

⎞

iω + νm ωc 0 jx Ex ⎟⎜ ⎟ ⎜ ⎟ −ωc iω + νm 0 ⎠ ⎝ jy ⎠ = ⎝ Ey ⎠ . jz Ez 0 0 iω + νm

(14.174)

This is the matrix equation σ −1 ij · j = E,

(14.175)

but we are interested in the solution j = σ ij · E :

(14.176)

i. e. we must invert Eq. (14.175) and obtain for the inverted matrix of σ −1 ij with A= ⎛

(iω + νm )2

A⎜ ⎝ ωc (iω + νm ) 0

1/ε0 ωw2 , 2 ] iω + νm [(iω + νm )2 + ωc,e

(14.177.1) ⎞

−ωc,e (iω + νm ) 0 ⎟ (iω + νm )2 0 ⎠, 2 0 (iω + νm )2 + ωc,e

(14.177.2)

and σ ij becomes ⎛

⎞

σ⊥ iσ× 0 ⎟ σ ij = ⎜ ⎝ −iσ× σ⊥ 0 ⎠ , 0 0 σ

(14.178)

following Heald and Wharton, we use theses abbreviations [821]: σ⊥ : σtransverse :

ne20 iω+νm 2 m (iω+νm )2 +ωc,e

= −iε0 ωp2

ω−iνm 2 (ω−iνm )2 −ωc,e

σ× :

σnormal :

ne20 −ωc,e 2 m (iω+νm )2 +ωc,e

= −iε0 ωp2

ωc,e 2 (ω−iνm )2 −ωc,e

σparallel :

ne20 1 m iω+νm

σ :

= −iε0 ωp2 ·

1 . ω−iνm

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

(14.179)

14.7 Whistler waves

625

If k B 0 , motions along (Θ = 0◦ ) are denoted longitudinal, and across to it (Θ = 90◦ ), transverse. For example, E and H are directed transverse to k. Rotations along the magnetic field are called L (CCW) or R (CW). The index “×” points to the components of a tensor which is generated by the cross product. D = eijE E

Et

El

Fig. 14.29. Directions of the field vectors B, E and D as well as the wave vector k and index vector n of a plane wave which moves through an anistropic plasma.

k

B

With the definition for the complex dielectric tensor with δ ij the unit matrix12 εij = δ ij −

iσ ij ε0 ω

(14.180)

which yields accordingly ⎛

⎞

ε⊥ −iε× 0 ε⊥ 0 ⎟ εij = ⎜ ⎝ iε× ⎠ 0 0 ε with ε⊥ : εtransverse : 1 − ε× : ε :

εnormal : εparallel :

(ωp /ω)2 (1−iνm /ω) ; (1−iνm /ω)2 −(ωc,e /ω)2

(ωp /ω)2 ωc,e /ω ; (1−iνm /ω)2 −(ωc,e /ω)2

1−

2

(ωp /ω) . 1−iνm /ω

(14.181) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(14.182)

After having evaluated the dielectric tensor, we seek the dispersion relation of a plane wave ψ = ψ0 exp i(k · r − ωt) in an anisotropic medium with μ = 1 ˜ the complex vector of but ε = εδ ij from which follows k n ⊥ D ⊥ B with n, the refraction index (Fig. 14.29). With Maxwell’s equations 12

For a cutoff plasma with εR ≤ εI , the following approach is recommended [822] iσij ε . εij = εR δij − ε0 ω

626

14 Advanced Topics k × E = iωμ0 H; k × H = −iωε0 εij E

(14.183)

we obtain for the wave equation k×k×E−

ω2 εij · E = 0 c2

(14.184)

and with n = −ic/ω k: n(n · E) − n2 E + εij · E = 0.

(14.185)

This is a set of three homogenous linear equations for the field components Ex , Ey und Ez : ⎛

⎞

⎛

⎞

nx Ex ⎟ ⎜ ⎟ (nx Ex + ny Ey + nz Ez ) ⎜ ⎝ ny ⎠ − (n2x + n2y + n2z ) ⎝ Ey ⎠ + n Ez ⎛ ⎞⎛ ⎞z = 0. εxx εxy εxz Ex ⎜ ⎟⎜ ⎟ + ⎝ εyx εyy εyz ⎠ ⎝ Ey ⎠ εzx εzy εzz Ez (14.186) Since n ⊥ D, but not necessarily n ⊥ E, we obtain non-trivial solutions only for the case of a vanishing determinant: 0 0 ε − n2 − n2 0 xx y z 0 0 εyx + ny n2z 0 0 εzx + nz nx

εxy + nx ny εxz + nx nz εyy − n2x − n2z εyz + ny nz εzy + nz xny εzz − n2x − n2y

0 0 0 0 0 = 0. 0 0

(14.187)

Let the wave propagate in the xz plane, and there might exist an angle Θ between the direction of propagation (wave vector k) and the static magnetic field B 0 (Fig. 14.30): nx = n sin Θ, ny = 0, nz = n cos Θ, εxx = ε⊥ , εxy = −iε× , εyx = iε× , εyy = ε⊥ , εzz = ε , which yields for the determinant 0 0 ε − n2 cos2 Θ −iε× 0 ⊥ 0 ε ⊥ − n2 iε× 0 0 2 0 n sin Θ cos Θ 0

n2 sin Θ cos Θ 0 ε − n2 sin Θ

0 0 0 0 0 = 0, 0 0

(14.188)

14.7 Whistler waves

627

z B0 q k

Fig. 14.30. The wave will not necessarily propagate parallel to the static magnetic field, and we split the wave vector into two components: k = k + k⊥ , k = k cos Θ, k⊥ = k sin Θ.

y g x

and we obtain the biquadratic equation of Appleton and Hartree for the refraction index: An4 − Bn2 + C = 0

(14.189)

with A = ε⊥ sin2 Θ + ε cos2 Θ, B = (ε2⊥ − ε2× ) sin2 Θ + ε⊥ ε (1 + cos2 Θ), C = ε (ε2⊥ − ε2× ).

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(14.190)

It is equivalent to solve for tan2 Θ; for this case, we obtain the implicit formula of Appleton and Hartree in the form of Astrøm [823]: tan2 Θ = −ε

[(n2 − ε⊥ ) − ε× ] [(n2 − ε⊥ ) + ε× ] . (n2 − ε )[n2 ε⊥ − (ε2⊥ + ε2× )]

(14.191)

For further discussion, it is first useful to diagonalize the dielectric tensor by introduction of the relations ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (ωp /ω)2 1 − 1−ωc,e /ω ; ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

εl = ε ⊥ + ε × = 1 − εr = ε ⊥ − ε× = ε⊥ = ε× =

εl +εr ; 2 εl −εr ; 2

and the unitary transformation

(ωp /ω)2 ; 1+ωc,e /ω

(14.192)

628

14 Advanced Topics ˜ij · U−1 ε˜ ij = U · ε

(14.193)

with ⎛

⎞

1 −i 0 1 ⎟ U= √ ⎜ ⎝ 1 i √0 ⎠ 2 0 0 2

(14.194)

and ⎛

U−1

⎞

1 1 0 1 ⎟ =√ ⎜ ⎝ i −i √0 ⎠ 2 0 0 2

(14.195)

yielding ⎛

⎞

εl 0 0 ⎜ ⎟ ε˜ ij = ⎝ 0 εr 0 ⎠ 0 0 ε

(14.196)

and the formula of Appleton and Hartree reads: tan2 Θ = −ε

(n2 − εl )(n2 − εr ) (n2 − ε )(ε⊥ n2 − εl εr )

(14.197)

or An4 − Bn2 + C = 0

(14.198)

with A = 12 (εl + εr ) sin2 Θ + ε cos2 Θ, B = εl εr sin2 Θ + 12 εl εr ε (1 + cos2 Θ), C = ε εl εr .

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(14.199)

From Eq. (14.198) we can derive first that two waves propagate through the plasma; one is directed CW, the other one CCW.13 Equation (14.197) allows a very simple method of determining the directions of propagation for the two special cases Θ = 0◦ and Θ = 90◦ , the so-called principal directions [824], and eventually the angular dependences of cutoff and resonance, respectively. 14.7.1.2 Cutoff and resonance. The frequencies of cutoff and resonance divide the dispersion diagram into bands of propagation and stop: Cutoff is defined for vanishing refraction index and infinite wavelength; if the wave is at resonance, the refraction index diverges, the phase velocity vph and the wavelength both vanish and the wave is strongly absorbed. But in both cases, the group velocity drops to zero. 13 We do not want to treat the case that the dielectric constant itself becomes a function of the refraction index; then, more than two waves are possible with spatial dispersion.

14.7 Whistler waves

629

• Cutoff: n2 → 0 ∧ vph → ∞. • Resonance: n2 → ∞ ∧ vph → 0. Varying the plasma parameters causes n2 to migrate through the complex plane. For n2 being real those points are sharp (cold collisionless plasma), for complex values of n2 , these points are diffuse (warm collisional plasma). For electromagnetic waves, the fields E and B are normal with respect to wavevector k. But this means that conduction current and displacement current will always cancel in longitudinal direction. At cutoff, this happens in the normal direction as well. This follows from the equation for a harmonic oscillation ∂B ⇒ ik × E = −iωB, (14.200) ∂t from which we can calculate a value for the phase velocity vph as ratio of the amounts of the two transverse components: ∇×E =−

0

0

ω 00 E 00 =0 0 (14.201) k B which causes | B | to diverge at cutoff and results in the extinction of the currents in transverse direction vph =

∂D . (14.202) ∂t To follow up the phenomenon of resonance, we note that vph → 0 ⇒ E → 014 from which we infer the electric field vector to be entirely longitudinal or zero. The current in transverse direction is calculated according to ε0 c 2 k × B = j +

j = −ik × H

(14.203)

and satisfies the two borderline cases: 1. j → ∞ for H = 0, or 2. j → finite for H = 0; both of them have been observed. 14.7.1.3 Dispersion relation. For the principal directions Θ = 0◦ and Θ = 90◦ , the dispersion relation is obtained by setting nominator or denominator of Eq. (14.197) to zero: Θ = 0◦ : ε = 0 ∨ εr = n2 ∨ εl = n2 , Θ = 90◦ :

ε = n2 ∨

εl εr ε⊥

⎫ ⎬

= n2 . ⎭

(14.204)

14 Approaching resonance, the phase velocity drops to very small values which explains the occurrence of Cerenkov radiation if an electron beam is directed into a plasma.

630

14 Advanced Topics

Table 14.2. Frequencies for cutoff and resonance of the principal waves wave ordinary extraordinary (X)

cutoff ω = ωp ω = 1/2 ±ωc + (ωc2 + 4ωp2 )

resonance — 2 ω 2 = ωp2 + ωc,e

R-wave

ωR = 1/2 −ωc + (ωc2 + 4ωp2 )

ω = ωc,e

L-wave

ωL = 1/2 +ωc + (ωc2 + 4ωp2 )

ω = ωc,i

From these equations, we can infer further for C = 0 and either A or B = 0, one of the roots must vanish: cutoff. C does not show any angular dependence, and the cutoff does not depend on the direction of propagation. C becomes zero for ε = 0 ∨ ε l = 0 ∨ ε r = 0 :

(14.205)

• ε = 0 ⇒ ωp2 /ω 2 = 1. • εl ∨ εr = 0 ⇒ ωp2 /ω 2 = 1 ± ωc,e /ω. • For Θ = 0◦ , the wave is resonant at tan2 Θ = 0◦ = −

ε , ε⊥

(14.206)

or ε⊥ = 1/2 (εl + εr ) → ∞ which happens for – εl → ∞: ion cyclotron resonance for positive ω, – εr → ∞: electron cyclotron resonance for positive ω. • For Θ = 90◦ , however, this happens for ε⊥ → 0, i. e. if 2 2 ωuh = ωc,e + ωp2 ,

(14.207)

where ωuh denotes the upper hybrid resonance.15 The cutoffs and resonances are compiled in Table 14.2. For ε , a cutoff and a resonance is demanded simultaneously for Θ = 0◦ . Since n (and therefore, also λ) can take any value at Θ = 0◦ , this is a longitudinal electron oscillation rather than a wave [Sect. 14.3, Eqs. (14.62)]. The extraordinary X-wave can propagate not only in the band between ωl and ωuh but also beyond of ωr which are separated by a stop band between ωuh and ωr . Below of ωl , no propagation is possible as well. 15

For Θ = 0◦ , resonance will be observed for ε⊥ → ∞ at the ion cyclotron resonance.

14.7 Whistler waves

631

For R-waves and L-waves, respectively, the frequencies for cutoff and for resonance are very close at low plasma densities [(ωp /ω)2 1], the stop band is very narrow and is located a little bit above of the resonance frequency. 14.7.1.4 R- and L-waves. The refraction index n = ck/ω will be modified by a static magnetic field:

n=

ωp 1− ω

2

−→ n = 1 −

ωp2 , ω(ω ± ωc )

(14.208)

or ωp2 c2 k 2 , (14.209) = 1 − ω2 ω(ω ± ωc ) in the case of propagation along the magnetic field (k B 0 ); i. e. for ωc,e > ω, the subtrahend becomes negative but the refraction index exceeds unity. ε = n2 =

• R-wave: The refraction index starts with values smaller than unity to vanish at the cutoff and to become singular at resonance; eventually it saturates at unity. For strong fields, n is larger than unity. k exhibits a singularity at ω = ωc,e : The R-wave is resonant at the frequency of the cyclotron motion of the electrons. Its energy is absorbed by the continuous acceleration of the electrons since its plane of polarization is coplanar with the plane of precession of the electrons [(Figs. 7.24 − 7.26). • L-wave: The L-wave does not show any resonance for positive values of ω since its direction of polarization is CCW. It could be resonant at ωc,i which would happen at very low frequencies since their ratio equals the mass ratio: ωc,e /ωc,i = me /mi . Its refraction index is always lower than unity. • Both the waves are reflected for low field strengths and high plasma densities, causing but a low correction of the dispersion function: The plasma becomes birefringent. In an anisotropic dispersive medium, the Poynting vector S is never found to coincide with the wavevector k! For a given electron density, R-waves are resonant at ωp ∝ n2e . However, their cutoff happens at lower strength of the magnetic field. It is therefore mandatory that the waves penetrate the plasma at a position where the intensity of the magnetic field exceeds the value required for resonance. Will they come from the low-field front, they will be reflected. Plotting the plasma density versus the magnetic field (the usual diagram shows ωc,e /ω → ωp2 /ω 2 ), we obtain the CMA diagram which is called after Clemmow, Mullaly and Allis. A simplified version for the whistler waves is pictured in Fig. 14.31.16 16 A complete CMA diagram indicates the course of the phase velocity as function of angle between static magnetic field and direction of propagation.

632

14 Advanced Topics 1.5 nL < 1 nR > 1

R-resonance

nR > 1

Lcu tof f

ff to cu R-

wc/w

1.0

0.5

Fig. 14.31. Simplified CMA diagram which describes the range for propagation of the R-waves and L-waves, respectively, along the lines of the magnetic field. In the hatched area, propagation is forbidden and the wavevector becomes imagiω2 ωp2 n ω nary ( ωp2 = np = 1 ± ωc,e with n the electron density).

nL < 1 nR < 1

0.0 0.0

0.5

1.0

1.5

2.0

2.5

2

(wp/w)

In Fig. 14.32, the dispersion diagrams of the R-waves and L-waves, respectively, are shown for ωp = ωc,e = 900 MHz. This case resembles that of the ionosphere with np ≈ 104 − 106 cm−3 which leads to an angular plasma frequency ωp between 6 and 60 MHz. For the magnetic field of the earth of 0.3 Gauss = 30 μT, ωc,e equals 6 MHz. For a more plausible representation, the quadratic refraction index, i. e. the dielectric constant in first order (ε = n2 ), is plotted versus frequency ω: The L-wave does show a range of imaginary k for ω < ωL , whereas this range is located between ωc, and ωR for the R-wave. In the high-frequency region, R-wave and L-wave coincide; but the low-frequency range is the region of the whistler waves: The plasma acts like a filter. The maximum in the phase velocity is obtained by deriving Eq. (14.209) at ω = 1/2 ωc,e . Inserting this value into Eq. (14.209), we find for c2 :

2 c2 = vph 1+4

2 ωp,e . ω2

(14.210)

It follows that vph is always smaller than c, even at its maximum. Neglecting 2 yields approximately electron inertia i. e. ω ωc,e and ωωc,e ωp,e k2 ≈

1 c2

2 ωp,e ω

ωc,e

−

ω3 ωc,e

⇒ k2 ≈

2 ωp,e ω, c2 ωc,e

(14.211)

and the condition for resonance will become fainter and fainter to eventually be lost. But both velocities display a strong dispersive conduct. The phase velocity vph = ω/k can be approximated to vph = and the group velocity is given by

c √ ωωc,e , ωp,e

(14.212)

14.7 Whistler waves

633 vg =

c √ dω =2 ωωc,e = 2vph . dk ωp,e

(14.213)

The velocity of the whistlers scales inversely with the square root of the plasma √ density: ωp,e ∝ √ np . Moreover, the propagation velocity falls with dropping frequency: vg ∝ ω. These whistlers are generated by atmospheric lightning and therefore exhibit a broad spectral width. Due to its intense dispersive character, a wave train which propagates along the lines of the magnetic field of the earth will melt away very quickly. The positive dispersion causes the whistling sound which shifts from treble to bass and can be listened to in the SW band. 50

100 R-waves (whistler)

10

R-waves (whistler)

n

2

25

n

2

0

1 L-waves

L-waves

-25

-50

0

R-waves

R-waves

0.1

500 1000 1500 2000 2500 3000 w [MHz]

0.01

0

500 1000 1500 2000 2500 3000 w [MHz]

Fig. 14.32. Dispersion diagram for R-waves and L-waves, shown here for the dependence of the dielectric constant for a model which resembles the ionosphere (ωc,e = ωp,e = 900 MHz). The branch top left is the range of the whistler waves; at the bottom, we see the branches of the L-wave (left) and of the R-wave (right), respectively which coincide in the high-frequency range.

For ωp,e ≈ ωc,e ≈ 6 MHz, the duration of a whistling sound turns out to be some seconds for distances of about 10 000 km (circumference of the earth: 40 000 km). Inserting the relations ωp2 = e20 ne /ε0 me and ωc,e = e0 B/me for ωp,e and ωc,e , we obtain eventually for k and λ:

k≈

2π μ0 e0 ne ω B ⇒λ= ≈ 5.6 × 105 B k ne ω

(14.214)

for B in T, ne in cm−3 and ω in Hz. For B = 10 mT, n = 1010 cm−3 and ω/2π = 13.56 MHz, λ ≈ 43 cm. 14.7.1.5 Dispersion relation for arbitrary directions. For arbitrary directions (the wavevector k is split into a component which is parallel with respect to the magnetic field k and one component which is orientated normal k⊥ and

634

14 Advanced Topics y x

Fig. 14.33. Resonance cone of a wave for which wavevector and static magnetic field form an acute angle.

q B0 z

2 k 2 = k2 + k⊥ ), we obtain the condition for resonance with Eq. (14.197) for n → ∞ or which is equivalent, for A = 0 according to

tan2 Θres = −

ε . ε⊥

(14.215)

This condition can only be met if ε and ε⊥ have opposite signs. For a magnetized plasma, thisis a range for which either 0 < ω < ωp,e , ωc,e (so-called 2 + ω2 > ω > ω ω lower branch) or ωuh = ωp,e p,e c,e (so-called upper branch). By c,e this equation, a resonance cone is spread out for the angle arctan Θ = ρ/z (with √ ρ = x2 + y 2 in Cartesian coordinates, cf. Fig. 14.33). Its axis is orientated parallel to the static magnetic field B 0 , its opening angle is determined by • ωp,e , • ωc,e and • ω, the angular operating frequency. The cone happens to be very narrow in the neighborhood of ωc,e and of 2 /ω 2 shows resonant cutoff: Θres → 0◦ . Therefore, the condition for cutoff ωp,e conduct, in particular for guided waves. For the extraordinary wave, the resonance cone becomes very flat at resonance: Θres → 90◦ . n2 yields for arbitrary directions to √ B 2 − 4AC B ± , (14.216) n2 = 2A 2A with the discriminant of the Appleton-Hartree equation and the identity εr εl = ε2⊥ − ε2× : D2 = B 2 − 4AC = sin4 Θ(εr εl − ε ε⊥ )2 + 4ε2 cos2 Θ(ε2⊥ − εr εl )

(14.217.1)

B 2 − 4AC = sin4 Θ(εr εl − ε ε⊥ )2 + 4ε2 ε2× cos2 Θ.

(14.217.2)

14.7 Whistler waves

635

ε2× remains always positive, and neglecting collisions, this holds also true for D2 . Since n2 is expected to fall short of unity, we start with the approach n2 = 1 − x which yields n2 = 1 −

1 √ , 2A(2A − B ± B 2 − 4AC

(14.218.1)

or (which is the same): 2(A − B + C) √ . (14.218.2) 2A − B ± B 2 − 4AC After cumbersome algebra, the explicit form of the Appleton-Hartree equation yields n2 = 1 −

n2 = 1 −

2ωp2 (1 − (ωp /ω)2 )

√ . 2 sin2 Θ ± ω 2 2ω 2 (1 − ωp /ω)2 ) − ωc,e c,e D

(14.219)

Depending on which of the two terms (either the sin term or the cos term) predominates, we can neglect the other one, the root in the discriminant vanishes and it is termed the quasi-transverse (QT) or the quasi-longitudinal approximation (QL) [825]:17 QL : sin4 Θ(εr εl − ε ε⊥ )2 4ε2 cos2 Θ(ε2⊥ − εr εl ),

(14.220.1)

QT : sin4 Θ(εr εl − ε ε⊥ )2 4ε2 cos2 Θ(ε2⊥ − εr εl ).

(14.220.2)

• Quasi-longitudinal (QL-l, QL-r): 1. Neglecting the sin4 Θ term in D leads to D ≈ 2ε ε× cos Θ. 2. Neglecting the sin2 Θ term in B leads to 2 sin2 Θ 2ω 2 (1 − ωpe /ω)2 ) B ≈ 2ε ε⊥ , also ωc,e ⇒ n2l,r ≈ 1 − ⇒

(ωp /ω)2 ω (ωp /ω)2 =1− ω ± ωc,e cos Θ 1 ± (ωc,e /ω) cos Θ ωp2 c2 k 2 = 1 − ω2 ω(ω ± ωc,e cos Θ)

(14.221)

with 17 For vanishing discriminant, both solutions turn into the same solution. The same result is obtained if the root is expanded first and is linearized afterwards.

636

14 Advanced Topics 2 k 2 = k2 + k⊥ ,

(14.222)

here, k and k⊥ are the components which are orientated in parallel and perpendicular fashion, respectively, with respect to B 0 . In this case the resonance is located at cos Θ = ω/ωc,e , and arccos ω/ωc,e is called the angle of the resonance cone which is spread out by the wavevector and the static magnetic field, neclecting electron inertia (neglect of the imaginary part which describes the damping behavior) c2 k 2 ≈

2 2 ωωp,e ωp,e c2 k 2 ∧ n2 = 2 ≈ . ωc,e cos Θ ω ωωc,e cos Θ

(14.223)

For plane waves, the single density independent criterion is neglect of electron inertia that ω ωc,e cos Θ. The difference of the two solutions (14.221) and (14.223) is a term of magnitude ω/ωc,e cos Θ. cos Θ = k /k becomes small for k k⊥ . The resonance cone approaches 90◦ , and the phase velocity becomes almost perpendicular to B 0 , the group velocity is almost parallel to B 0 . In the case of resonance, ω = ωc,e (condition for ECR), the whistler waves are strongly guided parallel to the lines of the static magnetic field B 0 ; the phase velocity and the opening angle of the resonance cone both vanish. The whistler band is symmetric with respect to ω = 1/2 ωc,e . • Quasi-transverse (QT−×, extraordinary, QT-o, ordinary): 1. neglecting the cos2 Θ-Terms in D yields: D ≈ (εr εl − ε ε⊥ ) sin2 Θ, 2. B = ε⊥ ε (1 + cos2 Θ) + εr εl sin2 Θ n2o ≈ 1 − n2× ≈

ωp ω

2

if

√

D2 > 0,

2 sin2 Θ √ 2 (1 − (ωp /ω)2 )2 ω 2 − ωc,e if D < 0. 2 sin2 Θ (1 − (ωp /ω)2 )ω 2 − ωc,e

(14.224.1)

(14.224.2)

• Quasi transverse (QT-o): n2o : no dependence on the static magnetic field (in particular, no angular dependence), since the wave is polarized in parallel fashion with respect to this field (Θ = 0◦ ∧ sin 0◦ = 0). There is no condition for resonance, but only a cutoff at ω = ωp,e . This mode is used to determine electron densities with the condition of cutoff. • Quasi transverse (QT-×): This wave exhibits singular properties. Whereas QL-r and QL-l are cir˜ B 0 ), this here is ˜ ⊥ B 0 ) and QT-o linearly polarized (E cularly (E

14.7 Whistler waves

637 B0

B0 B

B

E

E

kL B0

kR B0

E

B E

k0

kx

Fig. 14.34. Polarization of waves which propagate in the anisotropic plasma in directions perpendicular and parallel to the static magnetic field: t.l.: QT-l, t.r.: QL-r, b.l.: QT-o, b.r.: QT-x.

B

˜ ⊥ B 0 ), their velocity can be found between vpl elliptically polarized (E and vpr (Fig. 14.34): 2 2 2 2vp,× = vp,l + vp,r .

(14.225)

14.7.2 Bounded plasma 14.7.2.1 Introduction. Up to now, we have dealt with plane waves. In the case of guided waves we have to face an additional restriction: The freedom of action is confined, and the wave answers by changing its character. It does not remain a TEM wave any longer.18 Since currents are induced in a dielectric wall (displacement current) or in a conducting wall (conduction current), certain components of the electromagnetic field are generated which are orientated parallel to the direction of propagation even when magnetic fields are not present at all. Therefore, either TE-modes (all the components at least of the electric field are orientated perpendicularly to the wave vector) or TM-modes (all the components at least of the magnetic field are orientated perpendicularly to the wave vector) should be launched into the waveguide. Moreover, this modes should be “pure”, i. e. they should consist of only one mode. In contrast to the propagation of cylindrical modes in a hollow waveguide which is treated in textbooks, we must take into account the conductive medium and the anisotropy which is caused by the static magnetic field. Due to the confinement in radial and axial direction, certain boundary conditions have to be fulfilled for the electromagnetic fields prependicular to the 18 In a TEM wave, electrical and magnetic field are orientated mutually perpendicularly and are also perpendicular with respect to the wave vector.

638

14 Advanced Topics

direction of their propagation. In contrast to the plane waves, the wavevector k has to be split into two components k and k⊥ (as we have already seen in the case of propagation in arbitrary directions). As a first result, an electrostatic component is added to the electromagnetic character of the waves by which the dispersion behavior of the waves is significantly modified. In particular, we are now only interested in the behavior of the waves of the whistler band, i. e. the region between ωc,i and ωc,e . The interest in the whistler region has been mainly triggered by the experimental work of Rod Boswell, who found plasma densities of up to 100 % tubes of 10 cm in diameter with only 1 kW of RF power and 1 kGauss of magnetic flux density [273, 826, 827]. The high degree of ionization remained a mystery for more than a decade until Francis Chen suspected Landau damping as the main source of energy absorption and subsequent ionization [270, 828, 829]. Their collaboration culminated in a twin article in 1997 [830, 831]. 14.7.2.2 Extended Drude equation. If we neglect pressure gradients but consider elastic collisions which act as a viscous damping force, the equation of motion according to Drude of a cold plasma can be written down (for more detailed treatments, cf. [832]): F = me a = −e0 (E + v × B) − me νm v,

(14.226.1)

or, referred to the density (with n the plasma density): dv (14.226.2) = −ne0 (E + v × B) − me nνm v. dt Using the relations j = ρ v and j = −e0 nv, these equations can be written as me n

me dj j×B − ηj =E− ne20 dt ne0

(14.227)

with η = me νm /ne20 , the specific plasma resistance. This equation is denoted the generalized Ohm’s law. In the term on the left we recognize the term of inertia, the second term on the right is the Hall term, the third is the resistance term. For a stationary state, dj/dt vanishes, and if there is no magnetic field present, this equation simplifies to E = ηj. Likewise, we can write for η with the well known relations νm = nσv and < v >= kB Te /me √ η=

kB Te me σ. e20

(14.228.1)

Between collisions of two point charges, σ(= π r02 ) exhibits a 1/v 4 dependence (cf. Sect. 2.3.3). In a rough approximation, we equal potential and kinetic energy (e20 /r0 ≈ me v 2 ), yielding the formula

14.7 Whistler waves

639

η = π e20

me , (kB Te )3

(14.228.2)

which is sufficient to estimate the order of magnitude for η. 14.7.2.3 Wave equation. Considering only radial density gradients n = n(r) in a cylindrical plasma confined by conducting walls at r = a with an applied static magnetic field B 0 = B0 z where z denotes the unit vector which is ˜ field has to be modified orientated parallel to the static magnetic field, the E from Eq. (14.227) to ˜ E(r) =

˜ m e νm ˜ me dj˜ j˜ × B j. + + 2 n(r)e0 dt n(r)e0 n(r)e20

(14.229)

˜ where E, ˜ j˜ This equation can be linearized according to B = B 0 + B ˜ are assumed small quantities. From the generalized Ohmic law (14.229), and B ˜ = B(r) exp[i(ωt − k z − mθ)] with k the we seek for solutions of the type B axial component of the wavevector [Eq. (14.222)], and m specifies the azimuthal mode: ˜ ˜ = −iω B; ˜ ˜ = − ∂ B ⇒ −ik × E ∇×E ∂t

(14.230.1)

˜ ˜ = j × B0; E ρ

(14.230.2)

˜ ˜ = μ0 j; ∇×B

(14.230.3)

˜ ∂ j˜ ∂B ˜ = iω j˜ ∧ = iω B; (14.230.4) ∂t ∂t Equation (14.230.2) denotes the Lorentz force (j = ρv). The entire plasma current is determined by the E ×B drift of the electrons, i. e. no gyromotions of the electrons and ions (the frequency of the wave field should be low against the cyclotron frequency of the electrons, but high against the cyclotron frequency of the ions). In Eq. (14.230.3), we have neglected the displacement current, Eq. (14.230.4) expresses the harmonic perturbation, and by inserting Eq. (14.230.2) into Eq. (14.230.1)

∇×

j˜ j˜ 1 ∂ j˜ j˜ ˜ × B 0 = (B0 · ∇) = B0 = −(B0 ik ) = iω B, ρ ρ ρ ∂z ρ

(14.230.5)

we see that the current which is generated by the harmonic disturbance is orientated in parallel direction with respect to the magnetic field of the wave. All these things considered, we obtain for Eq. (14.229)

640

14 Advanced Topics

me · (iω + νm )j˜ = 0. e0 For disturbances of the plasma, we seek for the curl ˜ − j˜ × B 0 − ne0 E

˜ − ∇ × j˜ × B 0 − me (iω + νm )∇ × j˜ = 0. ne0 ∇ × E e0

(14.231)

(14.232.1)

With Eq. (14.230.5), this yields 2

˜ + (ik B0 ε0 c2 )∇ × B ˜ − me c ε0 (iω + νm )∇ × ∇ × B ˜ = 0. (14.232.2) −(ne0 iω)B e0 2 Since ωp,e = ne20 /ε0 me and ωc,e = e0 B/me , this gives 2 ˜ ˜ + (iω + νm )∇ × ∇ × B ˜ = 0, iω ωp,e B − ik ωc,e ∇ × B

(14.233)

a differential equation of second order which can be factorized according to [269] ˜ = 0. [∇ − k1 ] × [∇ − k2 ] × B

(14.234)

The general solution of this equation is ˜ = k1 B ˜ ∧∇×B ˜ = k2 B, ˜ ∇×B

(14.235)

˜ under conwhich can be eventually transformed by formation of the curl of B ˜ sideration of ∇ · B = 0 to ˜ ˜ = −k 2 B. ΔB 1,2

(14.236)

k1 und k2 are the roots of the equation k2 −

ω2 ω ωc,e k k + 2 p,e =0 iνm + ω c (iνm + ω)

(14.237)

and we obtain ⎡

k1,2

⎤

2 ωc,e k ⎣ ω + iνm ωωp,e ⎦. = 1 ∓ 1 − 4 2 2 2(iνm + ω) k ωc,e c2

(14.238.1)

Neglect of the resistance term ηj yields ⎛

k1,2

⎞

ω2ω2 ωc,e k ⎝ = 1 ∓ 1 − 4 2 2 p,e2 ⎠ : 2ω k ωc,e c

(14.238.2)

Every wave vector k which is orientated in parallel direction with respect to the static magnetic field B 0 , is connected with a pair of k, and, hence, a pair of 2 . k⊥,1 depends on the inertia of the electrons [+ in Eqs. k⊥ , since k 2 = k2 + k⊥ (14.238)], whereas k⊥,2 takes the Hall term into account [− in Eqs. (14.238)]. 2 2 /c2 ωc,e , there are imaginary k⊥ which When k2 becomes smaller than 4ωωp,e

14.7 Whistler waves

641

2 describe the damping of the wave. For k2 ωωp,e /c2 (ωc,e − ω), there exists only one real solution for k⊥ ! For lossless plasmas, however, k has to be real, which determines the lowermost value of k :

k ≥

2ωωp,e . cωc,e

(14.239.1)

From Eq. (14.237), we obtain for the lossless case (iνm + ω → ω)

k =

ω2 1 ω k 2 + p,e , k ωc,e c2

(14.239.2)

which sets the uppermost value for k for k⊥ = 0 to

ωp,e ω/ωc,e . k = c 1 − ω/ωc,e

(14.239.3)

From Fig. 14.35, we see that the minimum condition (14.239.1) generates an additional trigonal band in which propagation is forbidden in contrast to the whistler band of Fig. 14.32. With rising magnetic field (rising ωc,e ) the gap of evanescence widens. To the left of the minimum, we find the helicon wave, to the right, the ECR wave with k = k ωωc,e [cf. Eqs. (14.222) and (14.223)]. This curve considering electron inertia is called Trivelpiece-Gould wave (TG); in unbounded plasmas, these waves are whistler waves [833]. As we have seen in Sect. 14.7.1, these waves are strongly damped in the whistler wave regime. The limiting cases obtained from Eq. (14.238.2) k1 (+) →

2 1 ω ωp,e ωc,e c2 k

(14.240.1)

ω k ωc,e

(14.240.2)

k2 (−) →

are also shown. For intense static magnetic fields, the two regimes are very well apart, and k2 is much larger than k1 . For decreasing B0 , the values of approach each other, and the two regimes become mixed. For further decreasing B0 , the condition ω > 1/2 ωc,e will be reached: The helicon wave becomes evanescent, but the Trivelpiece-Gould wave can still propagate. Eventually, also the plasma becomes opaque for the TG-mode, and we have entered the ICP regime. Boswell first pointed out this connection between k and k and presented this diagram (cf. Sect. 14.7.2.6, Fig. 14.45) [826]. Especially in the vicinity of the roots of Eq. (14.237), this dispersion behavior has dramatic consequences for the phase velocity vph = ω/k , since Eq. (14.239) satisfy the condition vph ≤

cωc,e . 2ωp,e

(14.241)

For a plasma density of ne = 1012 cm−3 , we obtain a plasma frequency of 55 GHz. Applying a magnetic field of 100 Gauss (10 mT) yields a cyclotron frequency

642

14 Advanced Topics

9.7

1 k [cm -1]

20 100

0.1 500 875

0.01 0.01

0.1

1

10 k [cm -1]

100

Fig. 14.35. The dispersion curves for np = 1012 /cm3 and various values for the magnetic field strength (ω = 13.56×2π Hz) after [835]. For B = 100 Gauss, the limiting cases for the Hall curve and the electron-inertia curve are also shown. For B = 9.7 Gauss, ω = 1/2 ωc,e : lower limit of propagation of the Hall wave. Dashed: plane waves, vph = ω/k .

of 0.28 GHz and a phase velocity of 0.015 c, a field of 875 Gauss (87.5 mT) results in a higher phase velocity of 0.15 c. At any rate, this application of the static magnetic field results in the propagation of slow-travelling electromagnetic waves, and their phase velocity is comparable to the thermal velocity of the electrons (cf. [834]). It is this phase velocity kω that can be made low enough to match or even equal the thermal velocity of the electrons thereby allowing the energy transfer from the wave fields to the electrons [829]. 14.7.2.4 Cylindrical confinement. The cylindrical confinement limits the mobility of the electrons, and the solutions of the eigenvalue problem with boundary values (linearization and expansion of the Laplace operator in cylinder coordinates) results in Bessel functions of mth order. We seek a solution of the form B z (r) = A1 Jm (k⊥,1 r)e−i(k z+mθ−ωt) ,

(14.242.1)

B z (r) = A2 Jm (k⊥,2 r)e−i(k z+mθ−ωt) ,

(14.242.2)

applying the method of separation of variables, where A1 , A2 are amplitude constants, m denotes again the azimuthal mode (m < 0: CCW, m > 0: CW) and θ the azimuth, and k is the component of the wavevector k in parallel direction 2 2 = k12 − k2 , k⊥,2 = k22 − k2 ) in with respect to the static magnetic field B 0 (k⊥,1 the axial direction; in the following, we omit the ∼ denoting the time-varying field. The exponent consists of three terms, and we keep one dimension constant to investigate their helical conduct.

14.7 Whistler waves

643

• For fixed time, θ decreases for rising z to keep the argument constant: The wave pattern rotates counter-clockwise in the +z-direction for m > 0, and clockwise for m > 0. • For fixed z, θ must rise when t increases, hence the wave pattern rotates clockwise in the +θ-direction for m > 0. The wave equation (14.236) will be used to express the components Br and Bθ in terms of Bz [836, 837]:

1 ∂ ∂ r r ∂r ∂r

2 + k⊥ −

m2 Bz = 0. r2

(14.243)

Writing Eq. (14.235) in cylindrical coordinates gives the components for r and θ: im Bz − ik Bθ = kBr , r

(14.244.1)

ik Br − Bz = kBθ

(14.244.2)

from which we get the solutions for Br and Bθ : i Br = 2 k⊥

imk Bz + ik Bz r

1 Bθ = 2 k⊥

ik = k⊥

k m Bz + B , rk⊥ kk⊥ z

(14.244.3)

mk Bz + kBz . r

(14.244.4)

Considering the radial dependence of Bz Bz (r) = A1 Jm (k⊥,1 r) + A2 Jm (k⊥,2 r),

(14.245.1)

we obtain for its derivation with respect to r d d (k⊥,1 r) + A2 Jm (k⊥,2 r) Jm (k⊥,1 r) + A2 Jm (k⊥,2 r) = A1 Jm dr dr (14.245.2) which can be expressed by the derivative formula for the Bessel functions of order m: Bz (r) = A1

1 (Jm−1 − Jm+1 ) 2 and in particular for the Bessel function of order m = 0: (x) = Jm

J0 (x) = −J1 (x),

(14.246.1)

(14.246.2)

644

14 Advanced Topics

which are closely connected to the recursion formula for the Bessel function 2m (14.246.3) Jm (x) = Jm−1 (x) + Jm+1 (x); x [cf. Figs. (14.36) for details of the Bessel functions of orders m = 0 to m = 3 and their derivatives] and eventually for the three components Br , Bθ and Bz , and via Eq. (14.230.1) for the three components of the electric field Er , Eθ and Ez where Er =

ω Bθ , k

Eθ = −

(14.247.1)

ω Br , k

(14.247.2)

ω Bz . (14.247.3) k⊥ For zero current, Ez would vanish. To ensure ∇ · B = 0, there is always a z component for B. From Eq. (14.222), k has two components. From Eq. (14.230.1), it is evident that only the transverse component k⊥ is of importance. Ez = −

Br = Bθ =

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

mk1 Jm (k⊥,1 r) + k k⊥,1 Jm (k⊥,1 r) , r mk − kA2 1 Jm (k⊥,1 r) + k1 k⊥,1 Jm (k⊥,1 r) r ⊥,1 iA1 2 k⊥,1

,

Bz = A1 Jm (k⊥,1 r);

Br = Bθ =

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

mk2 Jm (k⊥,2 r) + k k⊥,2 Jm (k⊥,2 r) , r mk A2 − k2 Jm (k⊥,2 r) + k2 k⊥,2 Jm (k⊥,2 r) r ⊥,2 iA2 2 k⊥,2

,

ω k

iA1 2 k⊥,1

Er = − kω

mk1 Jm (k⊥,1 r) r

−A1 2 k⊥,1 ω

mk

r

+ k k⊥,1 Jm (k⊥,1 r)

,

Jm (k⊥,1 r) + k1 k⊥,1 Jm (k⊥,1 r)

ω k

Er = − kω

iA2 2 k⊥,2

mk2 Jm (k⊥,2 r) r

−A2 2 k⊥,2 ω

mk r

+ k k⊥,2 Jm (k⊥,2 r)

Ez = −A2 k⊥,2 Jm (k⊥,2 r).

(14.248.3)

⎪ ⎪ ⎪ ⎪ ⎭

Jm (k⊥,2 r) + k2 k⊥,2 Jm (k⊥,2 r)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

, ⎪ ⎪

Ez = −A1 k⊥,1 Jm (k⊥,1 r).

Eθ =

(14.248.2)

⎪ ⎪ ⎪ ⎪ ⎪ ⎭

Bz = A2 Jm (k⊥,2 r);

Eθ =

(14.248.1)

⎪ ⎪ ⎪ ⎪ ⎪ ⎭

,

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(14.248.4)

14.7 Whistler waves

645 1.00

1.00 J0(x) J1(x) J2(x) J3(x)

0.75

0.50

0.25

f(x)

f(x)

0.50

J0(x) J'0(x)

0.75

0.25

0.00

0.00

-0.25

-0.25

3.832 2.405

5.520

7.016 10.174 11.791 8.654

-0.50

-0.50 0

5

10

15

0

20

5

10 x

x

0.6 J1(x) J'1(x)

0.4

0.4

0.0

0.2 5.331 7.016 1.841

3.832

8.536

10.173 11.706

f(x)

f(x)

0.2

0.0

-0.2 -0.2

-0.4 0

5

10 x

0

5.136 6.706 3.054

9.970 11.62 8.417

J2(x) J'2(x)

3

6

9

12

x

Fig. 14.36. Bessel functions of orders 0 − 3 and their derivatives for the first three functions.

These triplets of coupled second-order differential equations describing the magnetic (electric) field inside a plasma-filled cylinder in terms of the three components Br , Bθ , and Bz (Er , Eθ , and Ez ). Each of these equations is a sum of two Bessel functions multiplied by an arbitrary constant to get a total of six arbitrary constants. The radial and the azimuthal components for both the fields exhibit a phase shift of 1/2 π, as the radial and the azimuthal components of either field do as well. In principle, six boundary conditions are required to find the dispersion relation ω(k). Before the dispersion relations are discussed, we investigate the boundary conditions. Boundary conditions. We can distinguish between two border cases: perfectly isolating wall and perfectly conducting wall.

646

14 Advanced Topics

1. Perfectly isolating wall: the current at r = a with a the diameter of the cylinder should vanish. From Eqs. (14.230.3) and (14.235), we find ˜ = k1,2 B ˜ = μ0 j ⇒ j(r = a) = 0 ⇒ B(r ˜ = a) = 0. ∇×B

(14.249.1)

2. Perfectly conducting wall: the tangential components Ez and Eθ should vanish at the surface (r = a). Hence, the r component of Eq. (14.230.1) along with the z components of Eqs. (14.230.3) and (14.232.1) gives im Ez − ik Eθ = Br . r

(14.249.2)

1 ∂ im (rBθ ) − Br = μ0 jz , r ∂r r

(14.249.3)

ne0 Ez = (iω + νm )jz ,

(14.249.4)

From these equations it can be seen that the equivalent boundary conditions for the magnetic field at r = a is Br = 0 ∧

1 ∂ (rBθ ) = 0. r ∂r

(14.250)

From Eq. (14.248.1), we eventually obtain for the nontrivial case mkJm (k⊥ a) + kaJ (k⊥ a) = 0 with

(k⊥ a) Jm

(14.251)

denotes the r-derivative of Jm (k⊥ r) at r = a.

14.7.2.5 Dispersion relation. In the nontrivial case, the dispersion relation is given by the vanishing determinant of the coupled equations Ez (a) = Eθ (a) = 0 [838], which, after some rearrangements where we have made use of Eq. (14.244.3), can be written down as 0 0 k⊥,1 Jm (k⊥,1 a) 0 0 0 0 0 m J (k a) + k1 J (k a) 0 ak⊥,1 m ⊥,1 k m ⊥,1

0 0 0 0 0 = 0, (14.252) 0 m k2 0 J (k a) + J (k a) 0 ak⊥,2 m ⊥,2 k m ⊥,2

k⊥,2 Jm (k⊥,2 a)

in which the wave vectors are to be inserted. k1 k means: k⊥,1 and k1 must dominate k⊥,2 and k2 . From the large dispersion relation (14.252) we obtain the high-density limit: k m Jm (k⊥,2 a) + Jm (k⊥,2 a) = 0. ak⊥,2 k2

(14.253)

14.7 Whistler waves

647

14.7.2.6 Azimuthal wave fields. Next, we will discuss the wave fields for the two lowermost azimuthal modes with radial mode number n = 1. The Eqs. (14.248) along with the mutual dependencies (14.247) yield the wave fields. Compared with the propagation in hollow cylindrical waveguides, the main difference consists in the existence of free charges which causes the electric field to possess a finite divergence. This can be evaluated as follows. From Eq. 14.230.2, we see that

˜ ˜ = ∇ · j × B0 ∇·E e0 np

(14.254)

which can be resolved to ˜ = ∇·E

1 ˜ B 0 · ∇ × j. e0 np

(14.255)

But according to Eqs. (14.230.3) and (14.235), k ˜ j˜ = B, μ0

(14.256)

and we arrive at ˜ = ∇·E

k ˜ (B 0 · ∇ × B). e0 np μ0

(14.257)

Together with Eq. (7.22), the remaining component which is orientated in parallel direction with respect to z yields ˜ = ω [∇ × B] ˜ z ∇·E k

(14.258)

˜ = ω k Bz : ∇·E k

(14.259)

or with Eq. (14.235)

The space charge is linearly proportional to the z-component of the magnetic wave field. m = 0. The wavefields result from Eqs. (14.247) and (14.248) with A=i

(A1 , A2 ) 2k⊥

(14.260)

to [829]: Br = −Ak J1 (k⊥ r) cos ϕ Er = Aω kk J1 (k⊥ r) sin ϕ Bθ = AkJ1 (k⊥ r) sin ϕ Eθ = AωJ1 (k⊥ r) cos ϕ Ez = 0. Bz = Ak⊥ J0 (k⊥ r) sin ϕ

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(14.261)

648

14 Advanced Topics

• The phase angle reduces to ϕ = k − ωt. • The boundary condition J1 (k⊥ a) = 0 yields k⊥ a = 3.83 for the lowest Er , no matter of the value of k/k . • Bz peaks on-axis (Figs. 14.37).

z

r Fig. 14.37. 3D-plot of the Bz mode which is peaked on axis.

• Br and Bθ both peak at r/a = 1.84(maximum)/3.83(border) = 0.48 (Fig. 14.38).

1.0 Bz

B [r. u.]

0.5

Fig. 14.38. Radial dependence of the wave fields Br , Bθ , and Bz for the m = 0 mode for the ratio k /k = 1/3 , i. e. an angle of the resonance cone of 70.5◦ (0.4 π), radial mode number: n = 1.

Bq

0.0 Br

-0.5 0.00

0.25

0.50 r/a

0.75

1.00

• The ratio Br /Bθ is invariant on the radius. • Fields

14.7 Whistler waves

649

– ϕ = 0: Er vanishes, E: only (azimuthal) component; Bθ vanishes, B: only (radial) component: TE01 mode, Fig. 14.39 (left). – ϕ = 1/2 π: Eθ vanishes, E: only (radial) component; Br vanishes, B: only (azimuthal) component: TM01 mode, Fig. 14.39 (right). – For all other phase angles: mutual dependence of E and B. That means that the wave fields, even for the helicon mode with m = 0, exhibit a hybridized E/B-behavior. This is mainly attributed to the mixing of an electrostatic component (TG) with an electromagnetic component (helicon). • For k /k 1 the radial, electrostatic component dominates the coupling behavior ⇒ the best performance is achieved by electrostatic coupling. Eventually, in the limit k /k = 0 or k⊥ /k = 1, the E-fields remain radial over the whole cycle [829].

E

H E

H

Fig. 14.39. At two discrete values of ϕ, the fields degenerate to the pure patterns of the TM01 mode (ϕ = 0) and to that of the TE01 mode (ϕ = 1/2 π). In between, they exhibit a helical pattern [829].

m = ±1. From the two modes with m = 1, only the CW wave with m = +1 will propagate and can accelerate the electrons, the m = −1 shows evanescent behavior. Er = Eθ =

−2Aω k [ J (k r) + kJ1 (k⊥ r)] sin ϕ k k⊥ r 1 ⊥ 2Aω k [ J (k r) + k J1 (k⊥ r)] cos ϕ k k⊥ r 1 ⊥

2A k [ J (k r) + k J1 (k⊥ r)] cos ϕ k⊥ r 1 ⊥ −2A k [ J (k r) + kJ1 (k⊥ r)] sin ϕ. k⊥ r 1 ⊥

Br = Bθ =

(14.262) Applying the recursion formulae (14.246) and considering J−n = (−1)n Jn , we can easily write down the components to m = +1 and m = −1, respectively

⎫ ⎬ ⎭

650

14 Advanced Topics

m = +1 ⎫

k Er = E0 (βJ0 − J2 ) sin ϕ Br = − ω E0 (βJ0 + J2 ) cos ϕ ⎬ k Bθ = ω E0 (βJ0 − J2 ) sin ϕ ⎭ Eθ = E0 (βJ0 + J2 ) cos ϕ

(14.263.1)

m = −1 ⎫

k Er = E0 (J0 − βJ−2 ) sin ϕ Br = − ω E0 (J0 + βJ−2 ) cos ϕ ⎬ k Eθ = E0 (J0 + βJ−2 ) cos ϕ Bθ = ω E0 (J0 − βJ−2 ) sin ϕ ⎭

(14.263.2)

with β=

k + k Aω ∧ E0 = − . k − k 2 k

(14.264)

• ϕ = θ + k z − ωt. • Er increases slightly from unity at r = 0 to about 0.6 at the edge (for β = 1). • Eθ falls from unity to zero at the edge. • Both the electric and magnetic components are mutually 90◦ out of phase, and • also the electric and the magnetic field are 90◦ out of phase. • The current jz is in phase with Bz . Furthermore, Chen showed that • the initial conditions require E to be perpendicular at the boundary, i. e. Br = Eθ = 0, and we can derive from Eqs. (14.263) the relation between k⊥ and β. For all k /k, the value of k⊥ is given by βJ0 (k⊥ a) + J2 (k⊥ a) = 0,

(14.265.1)

• which determines the radius r1 which separates those field lines which make it to the boundary from those who remain confined. This is condition Er = 0 or βJ0 (k⊥ r1 ) − J2 (k⊥ r1 ) = 0.

(14.265.2)

• The radial locus of most intense power transfer is given by J1 (k⊥r2 ) = 0. All these dependencies are shown in Figs. 14.40

14.7 Whistler waves

651 4

1.0 r1

Fig. 14.40. The dependencies of k⊥ , r1 and r2 on k /k.

k

0.8

3 r2

0.6

0.4 0.0

0.2

0.4 k /k

2 0.8

0.6

From these conditions, we get a final condition to construct the field pattern (Figs. 14.41 − 14.44): • The lines of the E-field are normally incident on the conducting boundary. • The lines of the B-field are perpendicular to the lines of the E-field, and they must be purely azimuthal at the edge ⇒ Conducting and isolating boundaries do have the same boundary condition. 14.7.2.7 Radial modes. In the last section, we found the wave fields for the first azimuthal mode numbers m = 0 and m = 1 by solving the dispersion relation for the lowermost radial mode n = 1. In fact, also the higher radial modes are excited. This topic has been extensively investigated mainly by Rod Boswell [273, 826, 827], Hans Oechsner [839, 840], and Peter Thonemann [269] (see Fig. 7.20). We want eventually to discuss the influence of the cylindrical confinement on the dispersion behavior of the higher radial modes in terms of electron inertia, plasma density and boundary conditions. For this end, we seek their asymptotic limits. Neglecting the inertia of the electrons (me → 0 in the term dj/dt, but not in the term ηj) yields from Eq. (14.238.2) with k 2ωωp,e /cωc,e , or vph cωc,e /2ωp,e ak1 ≈

aωc,e k aωc,e ≈ 1; ω vph

(14.266.1)

ω ≈1 k ω0 a

(14.266.2)

ak2 ≈

with 1/ω0 , a measure for the number of electrons per unit reactor length (electron density × radius2 ) which is defined according to ω0 =

ωc,e c2 B0 = . 2 a2 ωp,e μ0 ne0 a2

(14.267)

652

14 Advanced Topics

1.0

1.0 Br

0.5

Br

m=1 b= 1

B [r. u.]

B [r. u.]

0.5

0.0 Bz

-0.5

m=1 b=2

0.0 Bz

-0.5 Bq

-1.0 0.00

Bq

0.25

0.50 r/a

0.75

-1.0 0.00

1.00

0.25

0.50 r/a

0.75

1.00

0.50 r/a

0.75

1.00

1.0

1.0

Br Br

0.5

0.5

B [r. u.]

B [r. u.]

m=1 b=3

0.0 Bz

-0.5

m=1 b=6

0.0 Bz

-0.5 Bq

-1.0 0.00

0.25

0.50 r/a

0.75

1.00

-1.0 0.00

Bq

0.25

Fig. 14.41. Radial variation of the wave fields for the mode with m = +1 and radial mode number n = 1, which depends on J1 and J1 (via the recursion formulae, on J0 and J2 ) for three different values of k /k: Top, LHS: 0 (ϑ = 90◦ or 1/2 π), top, RHS: 1/3 (ϑ = 70.5◦ or 0.4π), bottom, LHS: 1/2 (ϑ = 60◦ or 0.3π), bottom, RHS: 5/7 (ϑ = 45◦ or 1/4 π).

• ω ωc,e : From Eq. (14.266.1), we see that k k1 , or k⊥,1 ≈ k1 . Electron inertia can be neglected for frequencies small against the cyclotron frequency, and the resonance cone opens towards π/2. Furthermore, this term can be neglected for vph aωc,e . From Eq. (14.266.2) we regain the case for plane waves [Eqs. (14.211)]: ω/k = vph ≈ aω0 , i. e. the phase velocity is inversely proportional to the plasma density, and inserting ωp,e and ωc,e into Eq. (14.266.2) yields the dispersion relation [Eq. (14.223)] k2 k ≈=

2 ωωp,e 2 2 c ωc,e

(14.268)

• ω → ωc : k becomes very large and the phase velocity will vanish (ECR 2 ). With the condition): ak2 → 0 and k⊥,2 a ≈ iak (k22 = k2 + k⊥,2 asymptotic formulae of the Bessel function for imaginary argument [841] (x 1)

14.7 Whistler waves

653

1.0

1.0

0.5

Br

0.5 Br

B [r. u.]

B [r. u.]

Bz

0.0 m = -1 b=1

-0.5

0.0 Bz

-0.5

m = -1 b=2

Bq

Bq

-1.0 0.00

0.25

0.50 r/a

0.75

-1.0 0.00

1.00

1.0

0.25

0.50 r/a

0.75

1.00

1.0 Br Br

0.5 B [r. u.]

B [r. u.]

0.5

0.0 Bz Bq

-0.5

-1.0 0.00

0.25

m = -1 b= 3

0.0 Bq

0.50 r/a

0.75

1.00

-1.0 0.00

Bz m = -1 b=6

-0.5

0.25

0.50 r/a

0.75

1.00

Fig. 14.42. Radial variation of the wave fields for the mode with m = −1 and radial mode number n = 1, which depends on J1 and J1 (via the recursion formulae, on J0 and J2 ) for three different values of k /k: Top, LHS: 0 (ϑ = 90◦ or 1/2 π), top, RHS: 1/3 (ϑ = 70.5◦ or 0.4π), bottom, LHS: 1/2 (ϑ = 60◦ or 0.3π), bottom, RHS: 5/7 (ϑ = 45◦ or 1/4 π).

E

H B0

Fig. 14.43. Development of the electric field of the m = 1 mode which rotates in ±θ direction. For a CW rotation of the purely transverse magnetic field (dashed), the electric field follows in a CW rotation which are purely radial at the boundary and perpendicular to the lines of the magnetic field.

654

14 Advanced Topics

H

E

Fig. 14.44. LHS: Approximate cross section of the mode for m = +1. The mode pattern unites elements from the normal circular TE11 mode and the circular TM11 mode through circular hollow waveguides, respectively (RHS: TM11 ).

Jm (x) −→

1 1 x , e 1+O 2πx x

(14.269)

we obtain for m = 1:

1 ak

1 1 − 2 2 ak⊥,2 ak⊥,1

+

1 1 J (k⊥,1 a) − · 1 =0 (ak2 )(ak⊥,2 ) (ak1 )(ak⊥,1 ) J1 (k⊥,1 a) (14.270)

The first two terms vanish for large ak , the third will become about iωω0 , hence, the forth terms remains finite as well. For large k becomes k⊥,1 large simultaneously, and the coefficient of the ratio of the Bessel functions behaves as 1/(ak )2 : iω0 1 J1 (k⊥,1 a) = , ω (ak )2 J1 (k⊥,1 a)

(14.271)

and we can insert the asymptotic terms of the Bessel functions with real argument [842] into the last term for x 1

Jm (x) = and obtain for the quotient

π π 2 cos x − m − πx 2 4

(14.272)

14.7 Whistler waves

655

tan k⊥,1 a −

π π . − 2 4

(14.273)

The argument must behave as (2n − 1) 1/2 π, which gives approximately nπ for large n. Inserting 2

(ak1 ) =

ωc,e ak ω

2

,

(14.274)

eventually yields (ak⊥,1 ) = (ak ) · 2

2

ωc,e ω

2

−1 .

(14.275)

For an assumed value of ωc,e = 20 ω this becomes approximately (n being the radial mode number) ak⊥,1 ≈

ak ωc,e ω nπ. ≈ nπ ⇒ ak ≈ ω ωc,e

(14.276)

Electron inertia. In Fig. 14.45, the influence of electron inertia on the dispersion is shown. The dimensionless property ak (with k the component of the wavevector in parallel fashion with respect to the static magnetic field and a the radius of the reactor, which consists of ideally conducting walls) is plotted versus ω/ω0 , which has been defined by Eq. (14.267). The shaded area denotes 2 2 /c2 ωc,e : no propagation is possible. The first three radial the area of k2 < 4ωωp,e modes are inverted. 3

1.5

2 k a

k a

1.0 1

0.5

7

5

4

3

1

2 3

1 2

4 5

0.0 0

1

2

w/w0

3

4

0

0

1

2

w/w0

3

4

Fig. 14.45. Dispersion curves for the azimuthal mode with m = 1 showing the radial mode dependence on electron inertia. LHS without and RHS with its consideration c C.S.I.R.O.). [826] (

656

14 Advanced Topics

In Fig. 14.46, this effect is focused by crossplotting the curves of Fig. 14.45 at ω/ω0 = 4. The waves are the solutions of the dispersion relation for radial modes up to n = 20 in different approximations. When electron inertia is considered but the Hall term is neglected, the refractive index is almost independent from ωP except for very small angles of the resonance cone. 0 2.0

k a

1.5

1.0

10

k a

20

30

with Hall term and electron inertia with electron inertia with Hall term

1 2

0.5 3

0.0 0

3

n

6

Fig. 14.46. Dispersion curve for ωc,e /ω0 = 80 and ω/ωc,e = 0.05 for three different approximations. Lower abscissa: number of radial eigenmodes, upper abscissa: argument of the Bessel functions with an estimated radial eigenmode spacing of c C.S.I.R.O.). k⊥ a = π [826] (

9

Even for a value of ω/ωc,e = 1/20 (ωc,e /ω0 is 80 in this case), the higher radial modes can be approximated with this approach [curve (2)], whereas the lower modes strongly deviate from this approximation. These can be approximated by the curve (3) which neglects electron inertia but considers the Hall term [Eq. (14.276)] yielding k ∝ ωp,e with ωp,e the plasma frequency of the electrons. This is the plane wave approximation. From curve (1) it becomes evident that the phase velocity decreases with rising radial mode number except for the very first modes. For high frequencies (ω → ωp,e ), the helicon approximation will fit the dispersion of the lower modes [curve (3)]. The dispersion of the higher modes, however, should be described by curve (2): electron inertia will begin dominating the dispersion, and the modes are forced to lie along the resonance cone, and their velocity decreases as the order of the radial mode increases. We have already identified the resemblance of the resonance cone with the radial modes of the TG waves higher in order. For lower frequencies, (ω → ωc,e ) electron inertia dominates the dispersion, which becomes almost independent from ωp,e . At ω ωc,e , the resonance cone reaches π/2, and electron inertia does not play any significant role for mode dispersion any more [Eqs. (14.221) and (14.222)]. By regarding electron inertia, we can explain the mixed character of the waves which exhibit both electromagnetic and electrostatic properties. • Low B0 and high ne : electromagnetic wave, classic helicon, smallest wavevector or longest wavelength, the phase velocity is almost perpendicular to the static magnetic field B0 .

14.7 Whistler waves

657

• high B0 and low ne (or ω → ωc,e ): electrostatic wave, TG-wave or ECRwave, phase velocity and group velocity are orientated perpendicularly, wave with shortest wavelength. Plasma density. Second, the dispersion behavior is strongly influenced by the plasma density. This is shown in Fig. 14.47 for fixed k. The density-related frequency is plotted versus the density-related gyrofrequency (np ∝ ωc,e /ω0 ). For high arguments of ωc,e /ω0 , the radial modes are ordered as in the high-density approximation [cf. Fig. 14.45 right, Fig. 14.46 curve (3)]; for low arguments of ωc,e /ω0 ), however, the mode structure is reversed [827]. The thick solid borderline denotes the highest velocity of the whistlers lying at k = 2ωωp,e /cωc,e . In between these two regions, mode mixing takes place because some modes will propagate with the same velocity.

k=2wwP/c wc n=1 n=2 n=3 n=4 n=5

4

w/w0

3

Fig. 14.47. Radial eigenmodes for ideally conducting walls for fixed k a = 0.68 but different densities (ω/ω0 ; c ωc,e /ω0 ∝ n/a2 ) [827] ( Cambridge University Press).

2 1 0

100 50 40 30 25 20 wc /w0

15

10

From Figs. 14.45−14.47 it is evident that different modes will propagate with the same phase velocity, which causes experimental difficulties for launching pure modes into plasmas of low density. For rising resistance, the TG-modes with higher mode number are attenuated more severely than the lower ones: After having propagated some wave trains into the plasma, the former modes are stronger damped. Boundary effects. From Figs. 14.48 and 14.49, we can judge the influence of geometry on the dispersion behavior. In principle, for rising radial mode number, the phase velocity decreases, and this conduct is more pronounced for smaller radii as can be seen from Fig. 14.47 where the inverted phase velocity is plotted versus the radial mode number with ωc,e /ω0 as parameter. Only for r → ∞, we regain the dispersion curve for plain waves as described by the Hall term. The influence of the reactor geometry on the dispersion behavior is shown most pronounced at the two boundary cases: perfectly conducting walls (Eθ =

658

14 Advanced Topics

4

Boswell Ferrari and Klozenberg 40

10 20

k a

3

2 80

1

0

oo 0

2

4

n

6

8

Fig. 14.48. Variation of ωc,e /ω0 for ω/ωp,e = 0.05. Comparison of the work of Boswell [826] (solid) and Ferrari and Klozenberg [843] (broken). For ωc,e /ω0 → ∞, we regain the dispersion curve for plane c waves from Fig. 14.46 ( C.S.I.R.O.).

10

0 at r = a) and rigid non-conducting walls [827] (Fig. 14.49). The cylindric geometry causes the group velocity of the modes to rise. The solution for nonconducting walls is expected to lie in between the solutions for plane waves and perfectly conducting walls. 4

Fig. 14.49. Dispersion for three different conditions: (1) perfectly conducting wall (dotted); (2) rigid non-conducting wall (solid); (3) plane wave (dashed) [827] c ( Cambridge University Press).

k a

3

2 perfectly conducting wall rigid non-conducting wall plane wave

1

0 0

4

8 wc/w0

12

16

14.7.2.8 Conclusion • For ω ωc,e or low radial mode numbers (generation by RF), the dispersion is controlled by the Hall term: the phase velocity rises with increasing mode number and falls with decreasing plasma density. • For ω ≤ ωc,e (ECR case) or high mode numbers, the dispersion is controlled by the electron inertia, and the modes are forced to lie along the resonance cone. The phase velocity of these modes, however, decreases

14.7 Whistler waves

659

with rising mode number and does not exhibit a strong dependence on the plasma density. The strong coupling leads to plasma densities between 10 and 100 %. The main mechanism is coupling of a helicon wave with a Trivelpice-Gould wave at the radial boundary which has been shown in a very lucid explanation by Chen [831, 835, 844]. At higher gyrofrequencies approaching ωc,e , another (collisionless) damping mechanism comes into play, and also the collision-free energy transfer mechanism of Landau damping is still discussed [272]. It took more than two decades from the very first suspicion to a perfectly worked-out theory.

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Register

Agrain, 234 Aisenberg, 432 Alkire, 482, 555 Allen, 316 Allis, 16, 631 α, 1st Townsend’s coefficient, 120 α-electron, 70, 83 α-ionization, 20, 30, 74, 75, 118, 120, 210, 212 α-mode, 155 α-reaction, 6 α-regime, 127, 210 amorphization, 292 analyzer, retarding field, 22, 194, 411 Anderson, 381 angle of main incidence, 501, 502 anisotropy, 264, 441, 445, 449, 463, 525, 527, 528, 530, 532, 537, 538, 550 – caused by a static magnetic field, 610 – definition of, 441 – degree of, 473 – dependence on E/p, 446 – enhancement of, 534 – etching, of, 402 – maximum of, 449 – of ECR etching, 499 – RIPE etching, 246 – static magnetic field, 235 anisotropy ratio, 441 anode fall, 95, 96, 419 – height of, 84, 97 – increase of, 96 anode sheath, 43, 44, 153 anode, disappearing, 410, 426 anodic oxidation, 412

Abdel-Fattah, 190 absorption coefficient, 619 absorption edge, 34 absorption spectroscopy, 369 AC mobility, 106, 107 acceleration grid, 280 – aperture of, 280, 282 – loss current at, 292 acceleration voltage, 282, 285, 287, 291, 292, 294 accelerator grid, 278, 284–286 – hole diameter of, 281 – orifice of, 284 – potential of, 291 accelerator hole, diameter of, 282 accelerator voltage, 293, 294 acoustic wave, phase velocity of, 588 actinometry, 346, 368, 526 – advanced, 346 activation energy, 292, 397 – transformation of, 519 activation type, 40 activation-type process, 389 adhesion – of ion-beam sputtered films, 436 – poor, 391 adiabatic approximation, 30, 31 adiabatic collision, 31 adiabatic parameter, 30 adsorption – non-dissociative, 519 – of radicals, 519 – radicals, 463 afterglow, 115 agglomeration, 387, 388

705

706 anodic zone, 95 antenna, 130 – RF driven, 246, 266 – slotted, 130 aperture, 279 – dimensions of, 279 – sheath thickness of, 285 aperture tunnel, 293 – length of, 282 Appleton + Hartree – equation of, 627, 628, 635 discriminant of, 634, 635 quasi-longit. approx., 635 quasi-transv. approx., 635, 636 applicator, 129, 130 AR coating, 436 arc, 45 arc discharge, 45, 410 arc, electric, 1 arcing, 142, 287, 289, 290, 377, 410, 425, 426 ARDE, 464, 479, 481–483, 487, 488, 513 area ratio, of electrodes, 160, 166, 451 Arnold, 482 arrangement – axial, 273 – planar, 273 – radial, 273 Arrhenius behavior, 525, 532 Arrhenius equation, 210, 402, 456 Arrhenius law, 541 ashing, cold, 444 Asmussen, 253 aspect ratio, 441, 447, 480–483, 499, 536, 544 – definition of, 441 Aston, 83 atmosphere, cutoff in, 614 atom, metastable, 99 attachment, electron, 116 Auciello, 396, 398, 399 Auger neutralization, 37 Auger process, 36, 37, 39, 40 Auger relaxation, 37 avalanche – charged carriers, of, 212 – electrons, of, 80

Register avalanche process, 89 avalanche, ionization, 560 azimuthal mode number, 238 Bachmann, 433 backdiffusion, 401 backscattering, coefficient for, 36 backside cooling, with helium, 292, 447 backsputtering, 391, 392, 412, 413 backstreaming, 294 – of electrons, 284, 290, 291, 294 Barber, 379 Barker, 522 Barkhausen, 235 barrel, 443 barrel reactor, 127, 440, 443–445, 533, 539 barrelling, 482 Barrett, 583 beach, magnetic, 253 bead, dielectric, 143 beam current, 293 beam diameter, 492 beam divergence, 286 – dependence on the perveance, 284, 286, 288 beam divider, 508 beam plasma – potential of, 290 beam potential, 280 beam, collimation of, 283 beamlet, 269, 276, 277, 279, 280, 294–296, 467 – collimation of, 284, 295 – deflection of, 284 – divergence of, 284, 294 – radius of, 276 – source of, 282 – space charge of, 284 Becker, 207 Beer, 7 Beer’s formula, 501, 506 Bell contact, 391 Benoit-Cattin, 199 Bernard, 199 Bernstein, 316

407,

529,

281,

283,

Register Berry, 19 Bestwick, 530 β-ionization, 28, 30 β-reaction, 6 Bethe, 23 Bethe formula, 345, 347, 349 bias potential, 246, 412, 428 bias sputtering, 376, 416, 425 bias voltage, 164, 172, 413, 439, 455, 485, 486 – zero, 499 Biehler’s model, 208 Bigio, 463 billiards, atomic, 380 birefringence, anisotropic plasma, 631 black snow, 550 blaze, 503 blistering, 393 blocking capacitor, 140, 163, 165, 166 Bohm, 77, 274, 576, 583, 589 Bohm criterion, 180, 193 Bohm edge, 59, 60, 62, 80, 93, 152, 154, 155, 160–162, 175, 176, 187, 216, 505, 579, 592, 596, 602 – initial velocity at, 193 – ion flux at, 478 Bohm potential, 54, 56, 155, 580 Bohm presheath, 60, 64, 74, 155, 193, 315, 319, 554, 596 Bohm velocity, 54, 56, 64, 115, 182, 193, 202, 216, 274, 277, 279, 280, 315, 430, 463, 554, 578, 580, 592 Bohm’s theory, 161 Bohr’s radius, 7 Bollinger, 439 Boltzmann, 350 Boltzmann equation, 192, 319, 574 Boltzmann factor, 54, 55 Boltzmann transport equation, 591 bombardment, electronic, 491, 519, 531 bombardment, ionic, 393–395, 400, 415, 416, 444, 457, 463, 490, 495, 518, 519, 522, 529, 531, 533, 555 bonding, 376 Born potential, 380–382, 385 Born-Oppenheimer approximation, 355 Born’s approximation, 23, 29

707 Boschi, 310 Boswell, 243–245, 592, 638, 641, 651 bottle, magnetic, 258, 259, 609 Boyd, 316 Boyle, 187 Braithwaite, 196, 198, 202 branching ratio, 345 Brandt, 195, 379 breakdown, 71, 118, 121 – condition for, 75, 118 – electric field for, 92, 118 – experiment for, 118 – field for, 120, 121, 123 – in a transmission line, 145 – optimum border of, 123, 124 breakdown field, 105, 282 breakdown voltage, 75, 98, 116, 152, 210, 225, 250, 252 bridge, plasma, 290 Brillouin diagram, 589 Brinkmann, 205 broad beam, 269, 276, 284, 294, 295 – flux of, 294 – radial uniformity of, 295 broad beam source, electrodeless, 264 broadband stub, 144 Brown, 623 Bruce, 478 bubble, formation of, 393 bulk plasma, isotropic, 575 bulk resistance, 174 bull’s eye, 462, 478 bull’s eye effect, 527–529 Bullard, 12 Burton, 542 Butler, 602 calotte, ground, 474, 475 capacitive coupling, 152 capacitive model, 172 capacitor, blocking, 152 carrier density, 299 carrier generation, by ionization, 121 carrier loss, by diffusion, 121 Carter’s theory, 400 cascade, collisional, 380 cascading, optical, 348

708 catcher anode, 451 cathode fall, 43–45, 73, 74, 83, 88, 97, 98, 210, 332, 424, 425 – abnormal, 45, 80, 81 – abnormal discharge, of, 82 – boundary of, 79 – field across, 85 – height of, 74, 75, 80–82 – increase of, 45 – normal, 45, 70, 72 thickness of, 74 – pressure dependence of, 156 cathode potential, normal, 80 cathode sheath, 56, 153 cathode, disintegration, 375 cavity applicator, 256 cavity resonator, 116, 256, 263 – cylindrical, 263 – design of, 262 cavity, cylindrical, 264 CCD matrix, 510 CCP, 441, 514 CCP discharge, 449 – asymmetric behavior, 451 CD, 471, 488 CD loss, 488 center-of-mass system, 9 Cerenkov radiation, 629 Chabot, 432 chamber conditioning, 373 chamber gap, 367 Chang, 474 channel formation, 387 Chapman, 455 characteristic wave impedance, 141 characteristic, isotropic, 457 charge density, 47 – highest, 66 charge transfer, 29, 30, 203, 284, 539 – asymmetric, 31 – cross section of, 31, 33 – current, 292 – double, 29 – measurement of, 32 – resonant, 31–33, 84, 182, 183, 193–195, 198, 201, 202, 205, 209, 386, 602 cross section of, 32

Register mean free path of, 196 – symmetric, 30, 31, 195, 196, 204, 212, 556 charge transfer current, 293 chemical etching, ion-induced, 548 chemisorption, 413, 534, 549 Chen, F.F., 56, 236, 242, 590, 638, 650, 659 Chen, M., 346, 368, 526 Cheung, 548 Child, 76 Child’s law, 161, 166, 198, 283 Child’s equation, 80, 82, 97, 154, 317, 592 – mobility limited, 78, 79, 526 – space charge limited, 77, 78, 83, 264, 280, 291, 324, 526, 599 chip separation, 484 Choi, 465 chopping, 535 circuit, sacrificial, 148 circulator, 129, 256, 265 Clemmow, 631 cluster, 397 – radius of, 387 clystron, 129 CMA diagram, 631 – simplified, 632 coalescence, 387, 388, 390 coating – conformal, 414 – optical surfaces, of, 411 coating rate – pressure dependence of, 417 coaxial cable, 143, 145 – characteristic impedance of, 143 – design frequency of, 143 – impedance of, 143 – magnetic energy in, 144 – solid, 143 Coburn, 200, 201, 346, 368, 419, 439, 448, 480, 522, 526, 530, 533, 537 cohesion – by interdiffusion, 391 – interfacial, 391 – oxidic, 391 coil configuration, 228

Register Ohmic loss, 225 collimation, 295 – ion beam of, 285 – requirement for, 285 collision – adiabatic, 31 – between ions, 271 – Coulombic, 125, 574 – elastic, 5, 201, 202, 212, 330 between electrons and neutrals, 613 – inelastic, 5, 201, 330, 574 – knock-on, 384 – ternary, 32 collision approximation, binary, 380 collision cascade, 378, 380, 381, 383 collision frequency, 13, 15, 16, 65, 66, 106, 118, 124, 125, 207, 252, 339 – effective, 335 volume-averaged, 336 – electrons, 155, 609 – of elastic scattering, 299 – transition of border of, 123, 124 collision number, 202, 203, 207, 575 – of electrons, 315 – of ions, 193, 315 collision rate, 46 collision, randomization of, 201 Collison, 623 column, positive, 42–45 compensating electrode, 301, 312 compressive stress, at high pressure, 425 Compton, 323 condensation coefficient, 416–418 condensor method, 32 conditioning, 373 conductance, 279 conducting wall, 637 conduction current, 173, 637 conductivity – complex, 19 – imaginary part of, 126 – real part of, 126 cone, 393, 394, 396, 398–400, 466 – angle, 441 – density of, 398, 399 – formation, 464

709 – formation of, 393, 394, 396, 398, 472 – growth of, 396 – structure of, 399 – truncated, 394 configuration, eclipsed, 275 Constantine, 499 contamination-induced mechanism, 400 convection, 92, 456, 460, 527 core model, 180 correlation function, 380 correspondence principle, 30, 31 corrosion, 444 cosine distribution, 375 cosputtering, 376, 414, 417 Coulomb collision, 237, 570 Coulomb potential, 19 coupling – capacitive, 127, 128, 153, 154, 156, 169–172, 274 – coaxial to waveguide, 145 – DC, 153, 154, 156 – inductive, 127, 128 – inductive/resistive, 169–171 – resistive, 171, 172 – resonant, 233 coupling factor, 166 Cox, 56 Cramer, 19 cross drift, 266 cross section – absorption, 10 – angular dependence, 12 – asymmetric, 195 – β-ionization, 29 – differential, 8, 9, 12, 19, 20, 27, 202, 207, 380, 381 – elastic, 11, 15, 106 measuring of, 19 – elastic collision, 30 – elastic scattering, 7, 16, 28, 33, 183, 203 – energy dependence of, 32, 85, 196 – for electron absorption, 329 – gas kinetic, 34 – inelastic, 23, 24, 62 – ion-neutral collisions, 181 – ionization, 23, 28, 30, 74, 84, 86, 87

710 electronic impact, 25, 26 energy dependence of, 73, 85 – momentum transfer, 9, 13, 416 – of ionization, 252 – photo ionization, for, 35 – thermal, 34 – total, 9, 13, 27, 28, 33, 193 cross-drift, 605 crosstalk, source of, 147 Cunge, 369 Cuomo, 415 current density – cathodic, 45 – ionic, 57, 75 – mobility limited, 449 – reduced, 82, 98 – space charge limited, 303, 449 current limit, 139, 142 current path, 80 current resonance, 139 cutoff, 234, 259, 628–630 – angular dependence of, 628 – definition of, 245 – L-wave, 631 – ordinary wave, 636 – R-wave, 631 – TEM mode, of, 143 cutoff condition, 622 cutoff frequency, 143, 146 – definition of, 628 – for Langmuir probes, 328 cutoff plasma, 618, 619, 623, 625 cycloid, 422, 424 cyclotron frequency, 420, 604, 641 – electronic, 235, 237 – ionic, 235 – limit, 609 – of electrons, 249 cyclotron resonance – electronic, 19, 246, 630 – ionic, 630 Czarnetzki, 302 damage, crystal, 444 Damk¨ ohler number, 460, 462 dark space, 48, 82, 93 – anodic, 41, 84, 95, 96

Register electron current density, 96 ion current density, 96 thickness of, 97 – Aston’s, 41, 69 – cathodic, 41, 43, 75, 85, 88 thickness of, 74, 76, 79, 97 – collisionless, 80 – Crooke’s, 1, 41, 43, 69, 85 – DC discharge, of, 93 – definition of, 52 – electric field across, 69, 210 – equilibrium, 95 – Faraday’s, 41, 42, 52, 85, 95 – ionization in, 81 – linear field decrease across, 195 – matching of, 451 – thickness of, 75, 76, 80–82, 85, 97, 376 reduced, 82, 83 – thickness, of, 81–83 dark space shield, 376 Davis, 195, 198, 202, 356 Davy, 1 DC bias, 140, 153, 160, 161, 169, 172, 201, 207, 232, 233, 245, 264, 376, 400, 430, 445, 449, 450, 452, 481, 499, 500, 513, 548–551 – drop of, 449 – power dependence of, 450 – pressure dependence of, 450 DC conductivity, 157, 176, 403 DC coupling, 152 DC discharge, 487 – fast changing polarity, 151 DC field – linear, 204 – uniform, 203 DC mobility, 107 DC offset, 152, 153 DC sheath, 154, 193, 583 – thickness of, 599 de-Broglie wavelength, 30 Debye, 48 Debye + H¨ uckel, theory of, 48 Debye length, 2, 49–52, 56, 57, 66, 76, 77, 80, 91, 153, 158, 193, 196, 209, 210, 264, 300–302, 313–315, 318, 326, 332, 588, 592, 593, 596, 622

Register Debye screening, 52 Debye shielding, 225 Debye wavevector, 66 decelerator grid, 286 decomposition, of InP, 551 DECR system, 256 Deichsel, 494 Demchisin, 389 density of states, 38 density, of radicals, 451 deposition – energy of, 427 – hard, 430 – soft, 430 deposition rate, 412, 413, 500, 512 – enhanced, 425 – reactive sputtering, 411 depth profile, of projectiles, 384 Derjaguin, 433 design frequency, 144 – standing waves at, 143 desorption rate, 519, 550 detector diode, 265 diamond – amorphous, 437 – phase purity of, 433 diamond crystal, 433 dielectric constant – complex, 615, 616 – imaginary part of, 404, 617 – negative, 66 – real part of, 403, 617 dielectric function, 587 dielectric loss – frequency dependence of, 145 dielectric wall, 637 dielectric, low-loss, 129, 156 diffusion, 2, 87, 88, 92, 111, 116, 326, 454, 460–462, 527, 610 – ambipolar, 84, 94, 114, 212, 228 – loss by, 121 – radial, 612 – reactants of, 460 diffusion coefficient, 89, 91, 96, 114, 396, 458, 460 – ambipolar, 91, 92, 94, 111, 114, 115, 124

711 – dependence on the electric field, 112 – electronic, 111 – enhancement by a magnetic field, 233 – free, 92 – ionic, 111 – normal, 612 – parallel, 612 – transverse, 612 diffusion control, 462, 476, 483, 532, 542 diffusion equation, 113 – spatial dependent, 118 diffusion length, 112, 114, 118, 121, 612 – definition of, 121 – enhancement of, 612 – enlargement of, 251 diffusion loss, 71, 86, 114, 115, 514 diffusion mode, 112, 113 diffusion potential, 228 diffusion profile, 460 diffusion rate, 396 diffusion tensor, symmetric, 612 diffusion velocity, 460 dilution – chemical, 448, 449 – physical, 448 Dimigen, 466 diode system, 376 discharge – abnormal, 81, 84, 99, 154 – asymmetric, 342 – capacitively coupled, 207–209 – coronal, 1 – dark, 83 – DC, self-sustained, 82 – dual-frequency, 111 – electrodeless, 127, 308, 313 – glow, 1 – hollow-cathode, 291 – low pressure, 1 – low temperature, 1 – microwave-driven, 110 – normal, 71, 84, 99 – self-sustaining, 80 – steady-state, 128 – sustaining of, 84 discharge current, 174 discharge voltage, 277, 292

712 dispersion relation, 238, 614 – cylindric whistler waves, 646 – high-density limit, 646 – imaginary part of, 590 – large, 646 – plane whistler waves, 625 – real part of, 590 dispersion, spatial, 628 displacement current, 637 – across the sheath, 169 dissociation – degree of, 368 – ionizing, 20 dissociation energy, of chlorine, 363 distribution – cosine, 414 – cosinoidal, 472 distribution function, 574 – velocity derivation of, 590 divergence, 285, 287, 294 – angle of, 275, 284 – ion beam, of, 282, 283 DLC, 432 Donnelly, 343, 346, 356, 369, 403, 448, 541 doping effect, 539 Doppler effect, 302 Doppler shift, 48, 353 dose, ionic, 393 double band, Bjerrum’s, 48, 354, 355 double electrode, 327 double probe, 313 double probe system, 308 dovetailing, 482, 484 downstream configuration, 429 downstream control, 452 downstream plasma source, 217 downstream process, 441 downstream zone, 228 drift, 119, 460 drift term, 91 drift velocity, 100, 106, 119 – electronic, 212 – ionic, 182 Drude, equation of, 248, 623, 638 Druyvesteyn, 84, 99, 350

Register Druyvesteynian distribution, 87, 188, 190, 305, 326, 329, 330, 352, 572, 574, 575 dual magnetron sputtering, 410 dual-frequency – discharge, 111 – system, 217 dummy load, 256, 265 duty cycle, 410 e-beam gun, 427 E-mode, 218, 227 E-type, 156 earth, magnetic field of, 235, 266, 632, 633 Eckstein, 379 Economou, 192, 482 ECR, 441, 514 ECR condition, 636, 652 ECR discharge, 449, 487 ECR heating, 233, 235, 247, 255, 591 ECR layer, 236, 253, 256, 258 ECR operation, 236 ECR-wave, 657 eddy current, 148 edge coating, 555 EEDF, 15, 64, 110, 125, 155, 188, 189, 236, 237, 304, 310, 326, 329, 350–352, 367, 574, 575, 591 – dependence on ω, 126 – Druyvesteynian, 110 – Franck-Hertz, 22 – frequency adjustment of, 110 – frequency dependence of, 118 – high-energy tail of, 107, 126, 236 – maximum of, 582 – Maxwellian, 110, 114, 125 – non-Maxwellian behavior of, 367 – normalized, 87 – operating frequency, 125 – plasma density influence of, 110 – shape of, 125 – tail of, 210 – time-dependent, 126 – time-independent, 576 effective field, 126, 176, 251

Register Efremov, 372, 373 Egerton, 404 eigenmode, radial, 657 Einstein-Smoluchowski, eq. of, 221 Einsteinian photo effect, 39 Einsteinian relation, 91, 114, 120, 124 elastic scattering, cross section of, 386 electric field – for breakdown, 111 – for maintenance, 111 electrode fall, 57 electrode gap, 155 electrode potential, 152, 163, 167, 168, 172, 308 – RF driven electrode, 166 – time-averaged, 172 electrode sheath, 56 electrode surface, 160 electromagnetic waves – penetration depth of, 618 electron – last, 85 – primary, 85 – secondary, 85 – thermalized, 85 electron attachment, 5, 20, 25, 26, 28, 114, 115, 352, 361, 362, 364, 538, 551 – dissociative, 25 – reaction of, 364 electron attachment, of chlorine, 363 electron clouds, 48 electron collection current, 304 electron collection region, 304 electron confinement, 271, 424 electron current, 316 – space-charge limitation of, 290 electron current density, 84, 96 electron density, 2, 50, 55–57, 92, 108, 112, 113, 116, 163, 305, 313, 449, 458, 632 – determination of, 362, 636 – exponential decay, 115 – finite, 176 – instantaneous, 181 – spatial dependence of, 115 – time-averaged, 598 – volume-averaged, 336, 337

713 electron energy, 93 – Maxwellian distribution of, 237, 350 electron inertia, 185, 593, 655, 656, 658 – neglect, 632, 636 electron oscillation, longitudinal, 630 electron repelling current, 315 electron saturation current, 304, 329, 330 electron sheath – instantaneous boundary, 594, 596, 598 – maximum amplitude of, 594 – pulsating, 594, 597 electron suppressor, 277 electron temperature, 2, 50, 55, 57, 105, 120, 125, 193, 210–212, 252, 253, 299, 301, 305, 306, 309, 313, 317, 319, 326, 329, 331, 344, 349, 359, 368, 369, 430, 452, 517, 572, 578, 582, 583 – finite, 316 – frequency effects, 186 – in ECR discharges, 264, 267 – inhomogenity of, 228 – radial homogenity of, 228 – RF discharge, in, 211 electron trap, 329, 419, 544, 551 electron velocity, 107 electroneutrality, 49, 65 electrons – inertia of, 651, 652 – non-Maxwellian behavior, 326, 329 – thermal velocity of, 642 electrostatic oscillation – dispersion of, 586 – group velocity of, 586 ellipsometry, 500, 502 Emeleus, 602 endpoint detection, 466 energy density – magnetic, 420 – material, 420 energy dissipation factor, 146 energy loss parameter – Langevin’s, 100 energy loss parameter, Langevin’s, 10, 116, 205, 567, 575, 591 energy transfer, 380, 381 – coefficient of, 381 – frequency of, 575

714 energy transfer function, 382, 384 Engemann, 130 ensemble, canonical, 345 ensemble, size of, 202 equilibrium, dynamic, 452 equipartition of energy, 59 equipotential surface, 143, 314 – Langmuir probe, of, 196 equipotential zone, 163 erosion, 466 – by sputtering, 400 Este, 427 etch characteristic, spatially dependent, 457 etch profile, 462 – rectangular, 489 – simulation of, 555 etching – fluorine-induced, 529 – ion-induced, 525 – neutral-induced, 525 – spontaneous, 532 etchrate, 290, 401, 440, 449, 480, 481, 500 – angle dependence of, 488 – angular dependence of, 467–469, 490 – anisotropy of, 446 – central, 527 – change by ion density, 478 – chemical, 463, 519, 526, 528 – chemical part of, 446 – dependence on etchrate and pressure, 450 – dependency of, 449 – diffusion control of, 481 – flow limited, 456 – flowrate limited, 456 – generation to reaction, 463 – horizontal, 246, 447 – horizontal component, of, 457 – increase of, 444, 519, 543 – initial effects of, 466 – ion-assisted, 463 – limitation of, 455 – loading dependent, 456 – maximum angle, 488 – maximum of, 449, 489, 551

Register – negative, 533, 551 – normalized, 480, 528 – physical, 526 – pressure dependence of, 401 – radial dependence of, 463, 528 – radial enhancement of, 527 – radial uniformity of, 228 – saddle-shaped design of, 450 – temperature dependence, 466 – time dependent, 447 – time-dependent, 480 – vertical, 246, 447, 532 – vertical component, of, 457 evanescence, regime of, 110, 618, 619 evaporation rate, 541 evaporation, by electron beam, 386 Ewald, 182, 183 excitation electrode, 163, 164 extraction system, 609 extraction, single-grid, 279 facet, 399 faceting, 464, 467–469, 472, 488, 489 – suppression of, 488 Faraday cage, 32, 443, 444 Faraday cup, 276 Faraday shield, 147, 148 Faraday’s dark space, 52, 94, 95 FCC, 126, 145, 233 Federal Communications Commission, 126, 127, 145, 233 Feltsan, 347 Fermi acceleration, 177 Fermi energy, 38, 47 Fermi level, 38, 539 Fermi velocity, 47 Ferreira, 574 field – effective, 117, 122, 251 – uniform border of, 121, 124 field drift, 105 field, effective, 108, 109, 126 field, RMS, 107, 109 filament, 80 filament current, space-charge limited, 290

Register film formation, theory of, 387 film, electrochromic, 411 filter function – after Blackman, 328 – after Gauss, 328 – square, 328 first wafer effect, 359, 373 floating potential, 52, 54, 55, 103, 153, 156, 169, 304, 306–310, 603 flow rate, 460 flow velocity, 456 fluorescence, 403 fluorescent lamp, 1, 3, 42 fluorine source, permanent, 534 Flynn formula, 416 fomblin oil, 552 foot, basic, 556 foot, of the etched structure, 477 force of friction, 192 forward voltage, 142 forward wave, 142 Fox, 22 Frank-Condon principle, 356 Franck-Hertz experiment, 21 Franklin, 1 Franz, 370, 494, 545, 549 free-electron model, 157 freon, 542 frequency – collisions, 9 – elastic collisions, 111, 118, 152, 251 – momentum transfer, 10, 338, 451 friction, 192, 194 Fridman, 591 Fujiwara, 481 G-star, 1 Gaebe, 371 Gaede, 27 Gaerttner, 415 γ-electron, 70, 80–83, 85, 88, 127, 151, 155, 172, 209–212, 417, 450 – yield of, 172 γ-ionization, 74, 75 γ-process, 69, 80, 88, 302, 600 γ-reaction, 6 γ-regime, 209

715 gap, 451, 452 – between the electrodes, 451 gap width, 452 Garscadden, 602 gas breakdown, 71 gas chopping, 535 gas feed, central, 464 gas flow, 452–454 gas shower head, 442 gauge component, 346, 351 gauge gas, 346 GDMS, 511, 512 GEC cell, 273 Geis, 493 Geissler, 1 Geltman, 24 generation rate, 86, 94 – ratio to etchrate, 463 geometric resonance, 140 global model, 58, 339, 351, 452 glow cathode, 427, 428 glow discharge, 45, 196, 427 – abnormal, 45 – normal, 43, 45 – obstructed, 97 – sustaining of, 97 glow discharge cleaning, 444 glow lamp, 151 glow negative, 86 glow, intensity of, 421 glow, negative, 6, 43, 44, 48, 52, 53, 59, 69, 74, 77, 82, 84–88, 95, 97, 195, 209, 212 – edge of, 87 – extension of, 86 – length of, 87 – potential of, 97 – radial shrinking of, 80 glowing cathode, 427 Godyak, 161, 177, 183, 188, 310, 312, 338, 591, 592, 600 Goedheer, 591 Gottscho, 27, 302, 356, 371, 403 Gould, 237 Gozadinos, 178, 179, 592 – pressure heating model of, 180 grain boundary, 400

716 gras, 394 Graves, 111, 185, 210 grid – aperture of, 284 – dished, 270 – micro-mesh, 285 – plane, 270 grid current, 291, 293 grid erosion, 285 grid optics, 269, 275, 284 – acceleration system of, 271 – conductance of, 282 – downstream side of, 292 – pattern of, 294, 296 – upstream side of, 292 Gross, 583, 589 group velocity, 578, 614 – at cutoff, 628 – at resonance, 628 – increase of, 658 – TG-waves of, 657 Grove, VII, 375 growth, columnar, 390 gun, e-beam, 386 gyration radius, 605 H-mode, 218, 227 H-wave, 256 Hacman, 403 Hagstrum, 38, 374 Hajicek, 396 Hall term, 638, 640, 656–658 halocarbon, 537, 542 halocarbon radical, 538 hard sphere model, 14, 201 hard sphere potential, 192 hard spheres, 204 hard wall model, 177 hare’s ear, 474 Harrison, 409 Hartree potential, 19 Heald, 624 heating – displacement current, 110, 174 – of electrons, 174 – of ions, 174 – Ohmic, 110, 158, 173, 176, 219, 220

Register plasma bulk,in, 176 – stochastic, 110, 155, 173, 174, 209, 211, 219, 313, 335, 339, 592 heating zone, 129 Heavyside layer, 235 Hebner, 371, 373 Heg gas, 100, 117 helical resonator, 368 helicon, 234, 238, 441, 656, 659 – phase velocity of, 236 helicon approximation, 656 helicon discharge, 326 helicon model, 243 helicon wave, 256, 258, 266 – standing, 244 helicon waves, 234, 243 – reactor for, 246 – reactor geometry, 266 – regime of, 233 – resonance condition, 239 helicons – dispersion of, 256 – phase velocity of, 658 helix, 247 Helmholtz equation, 89, 111 Herzberg, 505 hexode reactor, 156, 457 HF discharge – E-type, 127 – H-type, 128 – ionization mechanism, 210 high-density plasma, radial inhomogenity of, 266 high-pass filter, 152 Hoffman, 415, 425 hole, 393 hollow cathode, 170, 209, 269 – cylindrical, 98 hollow cathode discharge, 97, 98, 290, 427 – reduced diffusion to the walls, 97 hollow waveguide, plasma filled, 243 Hopwood, 221 horn, 130, 443 Horwitz, 171, 451 Howard, 547 Hu, 542, 547

Register Hudson, 391, 396 H¨ uckel, 48 Hussla, 403 hybrid resonance, upper, 630 hyperspace, parametric, 430 I(V) characteristic, 44, 45, 104, 300, 302– 310, 323, 325, 326, 331, 411, 421 IADF, 191, 202–204, 207, 208, 482, 527, 583 – energy dependence of, 555, 556 – fully developed, 207, 556, 557 – pressure dependence of, 203, 207 – upper limit of, 207 ICP, 441, 514 ICP reactor, 128 IE, 529 IEDF, 191, 196, 197, 199–204, 208, 439, 529, 583 – Ar+ -ions, of, 196 – bimodal, 199, 207 – broadening of, 198 – dependence on magnet field, 265 – frequency dependence of, 200 – maximum of, 529 – Maxwellian behavior, 207 – pressure dependence of, 196, 197, 202 – saddle-shaped profile of, 208 – shape of, 207 – sheath alteration, 305 ignition voltage, 104 impact – collisional, 400 – ionic, 478 impedance, 142 – characteristic, 141, 145 impedance matrix, 341 impedance scan, 341 implantation, 377, 382, 402 incandescent lamp, 1 incidence – angle of, 207, 437, 555 – normal, 489, 527 inclination angle, 467, 551 inclusion, 398, 400 inductive coupling, 127 inelastic scattering, electronic, 66

717 inertia – electronic, 483, 640 – ionic, 153, 419, 483 Ingold, 34, 79, 97 Ingram, 196, 198, 202, 482 ink-jet head, 563 interdiffusion, 392 interfacial layer, 391 interferometry, 500 invariance, adiabatic, 607, 609 ion acceleration, 264 ion beam – charge neutralization of, 289 – collimation of, 283, 284 – current density of, 287 – current neutralization of, 289 – neutralizion of, 290 – performance of, 295 – space-charge limited, 280 ion beam current density – constriction of, 283 – maximum constriction of, 283 ion beam deposition, 270 – secondary, 270, 285 ion beam divergence, 287 ion beam etching, 442, 474, 526 ion beam mode, 260, 263 ion beam reactor, processing chamber, 269 ion beam source – electron confinement in, 277 – inductively coupled, 272 – primary electrons in, 277 ion beam system, 442 ion beamlet, focusing of, 282 ion channeling, 498 ion cloud, 49, 50 ion current, 165, 316 – across the sheath, 169 – neutralization of, 283 – performance of, 284 ion current density, 84, 154 – mobility limited, 76 – potential dependence of, 526 – space charge limited, 76 ion density, 44, 48, 56, 57, 113, 158, 593, 596

718 – across the sheath, 332 – drop in, 478 – inhomogeneous, 181 – zone of equal, 260 ion dose, 397, 399, 400 ion energy, 483 – broadening of, 198 ion energy spectrum, bimodal, 201 ion etching, 128, 439, 441, 445, 446 ion flux, 186, 396, 399, 401, 478, 484, 492, 517, 526 – dependence on angle of incidence, 203 – energy dependence of, 203 – mobility-limited, 70 ion generation, 264 ion inertia, 191, 200 ion milling, 270 ion milling system, 269 ion optics, 287, 290, 294 – four-grid, 278 – sheath, 279 – single-grid, 278, 279 ion saturation current, 325 ion sheath, electric field of, 594 ion sound velocity, 578 ion source, broad beam, 280 ion temperature, 207, 301 ion thruster, 269 ion trajectory, 476 ion transit frequency, 446 ion velocity, 197 ion wave, 578 ion, radical, 444 ion, thermal velocity, 183 ion-beam current, 294 ion-beam current density, 291, 295 – maximum of, 291 – requirement for, 285 ion-induced etching, 555 ionic density, 64 ionization – avalanche of, 69 – cathodic, 69 – collision, 212 – cross section of, 87 – degree of, 2, 245, 345, 402

Register in a capacitively coupled plasma, 234 – efficiency of, 110 – electronic impact, 21, 25 – gaseous, 74 – impact, 6 – in the cathodic dark space, 69 – number of, 400 – surface, 74 – threshold of, 34 – yield of, 511 ionization degree, 427, 575 ionization frequency, 112, 113, 118 ionization potential, 16, 30, 31, 38, 43, 74, 82, 95, 105, 269, 330 – argon, 2nd, 271 – effective, 117 – first, of Ar, 86 – metals, of, 427 ionization rate, 186, 210, 211, 230 – electron density, dependence of, 210 – electron temperature, dependence of, 210 ionization threshold, by ionic impact, 29 ionization, degree of, 244 ionosphere, 615, 632, 633 ions, depletion of, 478 IR spectroscopy, 537 island growth, 387, 388 island of stability, HF discharge, 124 ITO layer, 410 Itoh, 512 Jensen, 210 John, 510, 515 Jones, 419 Jungblut, 195 Jurgensen, 555, 556 Kaufman, 292, 396 Kaufman source, 269, 271, 274, 285, 493 Kay, 200, 201, 408 Kelly, 398 Kelvin equation, 387 Kennedy, 591 Khan, 549

Register Kidd, 394 Kim, 549 Kino, 602 Kirkendall pore, 387, 391 Klick, 161, 333 Knoll, 415 Knudsen characteristic, 436 Knudsen regime, 480 Kobayashi, 433 K¨ ohler, W.E., 575 Koenig, 104, 165 Kollath, 12 Kushner, 191, 202, 370 L-matching network, 140 L-network, 140 L-wave, 235–237, 247, 248, 250, 561 – absorption of, 250 – cutoff, 250, 630, 631 – dispersion, 632, 633 – polarization, 637 – refraction index, 631 – resonance, 630, 631 – wavevector, imaginary, 632 L-waves, propagation of, 632 Laframboise, 316 Lam, 319 λ/4 window, 266 λ/4-layer, 411 Lambert, 7 Landau, 583, 589 Landau damping, 67, 217, 236, 240, 242, 246, 250, 266, 561, 575, 590, 591, 638, 659 Langevin, equation of, 248, 623 Langmuir, 41, 58, 76, 300, 317, 323, 375 Langmuir characteristic, 310 Langmuir curve, 245 Langmuir paradoxon, 87 Langmuir plasma, 52 Langmuir probe – double, 236 Langmuir probe, 45, 87, 252, 299, 307, 308, 310, 592 – characteristic of, 304 – double, 244 – effective area of, 305

719 – modified, 196 – potential of, 317, 322, 324 – probe current of, 308 – probe potential of, 304 – probe voltage of, 304 – retarding-field region of, 304, 305 – sheath potential of, 308 – transition region of, 304 Langmuir profile, 182, 458 Langmuir sheath, 583 Langmuir theory, 161 large-angle scattering, 192 Larmor equation, 420, 424 Larmor frequency, 237 Larmor radius, 247, 258, 265, 326 Larmor relation, 604, 607 laser interferometry, 403 lattice damage, 271, 285, 384, 440, 487 lattice defects, 377 lattice disordering, 498 Laubert, 379 layer growth, with ion bombardment, 430 layer, single-phase multi-element, 418 Lee, Y.I., 526 Lennard-Jones potential, 7, 201 LER, 216, 464, 471, 472, 488 leverage effect, 478 Lichtenberg, 183 Lieberman, 58, 161, 177, 183, 591, 592, 596, 598–600, 602 – collisionless sheath model of, 180 LIF, 330, 356 lifetime, flowrate limited, 456 lift-off technique, 386 LIGA, 563 Lincoln, 493 lithography, 442 Liu, 206, 529 load impedance, 139, 142 loading effect, 455–457, 525 – 1st order, 483, 505, 506, 513, 528 – 2nd oder, 457 – first order, 478 Loeb, 272 Logan, 171 longitudinal wave

720 – dispersion of, 587 – dispersion relation of, 586, 589 – group velocity of, 588 – penetration depth of, 622 – phase velocity of, 588 – skin depth, 587 longitudinal waves – dispersion relation of, 588 – skin depth of, 588 Lorentz force, 604 Lorentz plasma, 65 loss – by diffusion, 514 – conductive, 142, 143 – cone, 609 – dielectric, 142, 143 – mechanism, 2 – rate, 86, 94 Lotz, 24, 347 Lotz formula, 349 Loureiro, 574 low pressure plasma, 47 L¨ uthje, 466 Magistrelli, 310 magnetic bottle, 258, 259, 264, 266 – trapping condition, 609 magnetic field, divergent, 258, 259, 265 magnetic mirror, 266 magnetic moment, 609 magnetoplasma, 238 magnetron, 129, 250, 256, 271, 420, 421 – circular, 424 – cylindrical, 424 – planar, 424 Maier-Leibnitz, 22 Maissel, 104, 165, 451 Malyshev, 343, 368, 369, 372, 373 Mansour, 415, 416 Margenau distribution, 571, 572, 575 mask erosion, 464, 467 mass balance, equation for, 460 mass spectrometry, 500, 510 – conventional, 510 – energy resolved, 299 – energy-resolved, 198 – glow discharge (GDMS), 511, 512

Register – normal, 511 Massey, 12, 16, 30 matching network, 140, 243, 265, 340, 341, 376 – L-type, 342 – power loss in, 402 matching stub, 263 matrix sheath, thickness of, 76 Mattox, 391, 427 Mawella, 414 Maxwell, 13, 350 Maxwellian distribution, 47, 53, 86, 87, 99, 188, 190, 202, 207, 253, 352, 572, 574–576, 587, 589–591 – electronic, 18, 25, 56, 59, 86, 87, 305, 309, 310, 325, 326, 329, 330 with two temperatures, 305 – ions, 80 – maximum of, 66 Maxwellization, 591 Mayer, 522 McDonald, 391 McNevin, 448, 541 McVittie, 555 mean free path, 6, 7, 9, 15, 118, 124, 197, 203 – argon atoms, of, 7 – border of, 122, 124 – electronic, 27, 65, 72, 81–83, 85, 94, 98, 108, 121, 155, 172, 210, 247, 252, 271, 301, 315, 451 – ionic, 8, 56, 65, 70, 76, 79, 181, 183, 191, 193, 197, 202, 301, 315, 386, 446, 450, 555 – molecular, 530 – of molecular ions, 198 – optical excitation, of, 86 – sputtered atoms, of, 386 mechanism – power transfer, for, 266 – roughness-induced, 395, 396 Meeks, 370 Meijer, 591 memory effect, 534 MEMS, 563 meniscus, 280 MERIE, 217

Register metastable, 347 metastable species, 6 method of trajectories, of Harrison, 385 microdamage, 519 microdivergence, 276 microetch system, 269 microfeature, 440, 464, 478 microloading, 476, 483 micromasking, 555 microwave cavity, 260 microwave interferomery, 622 microwave interferometry, 299, 325 microwave window, 110, 130 migration, radiation-induced, 400 Minkiewicz, 455 mirror coating, 375 mirror, magnetic, 609 Misium, 181 mismatch, 143 Mitchell, 403 Mityureva, 348 mobility, 43, 613 – limit of, 82 mode, 237, 263 – cylindrical, 145 – dominant, 143, 263 – fundamental, 263 – ion beam, 260, 263 – plasma stream, 258, 263 – Q-factor of, 263 – stabilization of, 253 – TE-, 262–264, 637 – TEM, 143 – TM-, 261, 263, 637 resonance frequency of, 261 – Trivelpiece-Gould, 641 mode converter, 265 mode hopping, 250, 265, 266 mode jumping, 240, 241 mode mixing, 657 model potential, 201 Mogab, 455, 532 Moisan, 110, 111, 125, 574 molybdenum bronze, 392 momentum transfer, 442, 466, 536 – cross section for, 572

721 – frequency of, 571, 575, 618 Monte-Carlo calculation, 555 Monte-Carlo method, 186, 188, 189, 192, 201, 203, 204, 207, 383 Monte-Carlo simulation, 185, 201, 208, 399 Morgan, 86 morphology, columnar, 388 Morse, 16 Mott-Guerney equation, 78 Mott-Smith, 58, 300 Movchan, 389 M¨ uller, 233 Mullaly, 631 Mullius, 448 multi-wafer process, 461 Meyers, 27 Nakanishi, 18 native oxide, 392, 417, 444, 540 near field, 147 negative glow, 43, 53 – EEDF in, 87 negative ions – density of, 359 – mean free path of, 386 neutral density, 359 neutral flux, 478 neutralizer, 291 Niggebr¨ ugge, 550 Nojiri, 487 Norstrøm, 104 notching, 464, 482 nuclear stopping power, 381, 383 nucleation, 387, 392, 400, 491 nucleation theory, 396 number density, 13 – absolute, 368 O-star, 1 O-wave – cutoff, 630 – cutoff of, 636 – polarization, 637 – resonance of, 636 octopole, 246 Oechsner, 651

722 Oehrlein, 473, 530 OES, 299, 444, 500, 502, 503 Ohmic heating, 106, 109, 110, 158, 173, 176, 181, 211, 236, 334, 338 Ohmic loss, frequency dependence of, 145 Ohm’s law, generalized, 638, 639 OML, 325 O’Neill, 537 open-shell system, 444, 530 optical surface, coating of, 408 optical switch, 563 optics – three-grid, 285, 287 – two-grid, 285, 287 oscillation amplitude – border of, 123, 124 oscillation, ionic, 86 PACVT, 529 pair interaction, 380 pair potential, 29 Panagopoulos, 230 parallel resonance, 140 parallel-plate capacitor, 75 parallel-plate reactor, 92, 128, 156, 168, 441, 442, 445, 453, 457, 460, 461, 463, 512, 533 – configuration of, 266 Park, 385 partial wave method, 16 particle – primary, 393 – secondary, 393 particle-in-cell, 313, 592 Paschen, law of, 72, 98 Paschen curve, 73 Paschen minimum, 73, 82, 88, 120 Pashley, 387 path, mean free, 2 PE, 128, 529 Pearton, 499, 548 Peclet number, 460, 462, 463 PECVD, 128, 533 pendulum effect, 97 penetration depth, 292 – ions, of, 396

Register – of electromagnetic fields, 233 Penning, 84, 99 Penning effect, 73, 74 Penning ionization, 33, 511 Penning process, 419, 512 Person, 182 perveance, 281–284, 286, 288 – normalized, 282–284 Petrovic, 27 phase difference, 501 phase velocity, 66, 141, 578, 614, 629, 631, 636, 641, 642, 652 – at resonance, 629 – helicon waves of, 656 – TG-waves of, 657 phase width, 391 photodetachment, 371 photoeffect, 97 photoelectric effect, yield of, 84 photoelectrons, yield of, 39 photoionization, cross section of, 40 photoluminescence, 403 photoresist, degradation of, 292 physisorption, 408, 549 π-matching network, 140, 266 π-network, 140 PIC, 192, 592 pick-up, 327 Piejak, 188 pillar, 394 Fermi level, 498 pit, 393, 398–400 – facet of, 399 PL signal, 499 plasma – anisotropic resonance in, 634 – athermal, 5, 58, 576 – capacitance of, 600 – capacitively coupled, 168, 621 – collisionless, 120, 615, 622 – displacement current, 592 – ECR driven, 621 – electronic conduction current, 592 – filtering quality of, 632 – generation in the plasma bulk, 451 – half-length, 182

Register – inductively coupled, 618, 620 – Langmuir’s definition of, 51 – low-density, 2 – pulsed, 115 – spatially homogenous, 246 – tenuous, 2, 615 plasma ashing, 444, 445 plasma beam mode, 264 plasma boundary, 478 plasma bridge, 291 plasma bulk, 2, 48, 88, 109, 115, 153, 154, 157–159, 163–166, 172, 176, 178, 186, 195, 209, 212, 331, 335, 439, 444, 452, 495, 500, 511, 512, 517, 518, 525, 576 – as homogeneous zone, 303 – center of ionization, 210 – composition of, 512 – conduction current in, 209 – conduction current within, 161 – electric field across, 159, 165 – electric field in, 155, 181, 575 – electrical conductivity of, 336 – electron confinement within, 273 – electron density in, 154 – electron temperature of, 358 – electrons of, 91, 182, 424 – entering of γ-electrons, 450 – frequency effects, 186 – gas temperature of, 340 – glowing, 174, 340 – ion density in, 332 – ionization center, 211 – length of, 335 – plasma density in, 184 – positive ions out of, 412 – potential of, 155 – quasineutrality of, 576 – rapid electrons from, 392 – resistance of, 173 – surface of, 60 – trapped electrons in, 181 – trapping of electrons in, 420 plasma color, 245 plasma confinement, 217 plasma density, 2, 47, 51, 52, 57, 67, 77, 103, 112, 128, 129, 206, 225, 245, 246,

723 253, 266, 272, 299, 309, 316, 331, 335, 347, 359, 420, 427, 483, 487, 500, 514, 517, 548, 559, 578, 582, 583, 659 – at substrate level, 441 – at the Bohm edge, 602 – by resonant coupling, 267 – central, 623 – frequency effects, 186 – helicon reactor, 244 – high, 125 – high frequency generation, 186 – in a capacitively coupled plasma, 234 – in ECR systems, 266 – in electronegative plasmas, 361 – in helicon plasmas, 243 – in ICP discharges, 266 – in inductively coupled plasmas, 234 – in microwave discharges, 111 – in the ECR region, 326 – increase of, 111 – inhomogenity of, 228 – low, 125 – radial homogenity of, 228 plasma density, average, 367 plasma edge, 599 plasma etching, 128, 439–441, 445, 446, 528 – definition of, 445 plasma flickering, 265 plasma fluctuations, 310 plasma frequency, 66, 129, 234, 263, 274, 420, 613, 641 – definition of, 66 – electronic, 152, 155, 156, 158, 209, 235 – ionic, 152, 157, 163, 173, 209, 212, 431, 445 – parallel, 335 – serial, 335 plasma ignition, 115, 225 plasma impedance, 139, 331 – capacitive, 139 – inductive, 139 plasma oscillation, 87, 586 plasma oxidation, 412 plasma potential, 52, 54, 55, 152, 153, 155, 163–165, 167–172, 174, 187–189, 197, 213, 219, 220, 228, 232, 271, 277,

724 278, 291, 292, 302, 304, 306–310, 313, 324–327, 412, 420, 430, 443, 445, 451, 452, 485, 526 – constant, 592 – DC component of, 167 – definition of, 169 – determination of, 326, 330 – fluctuations of, 327 – for plasma etching, 445 – in helicon plasmas, 243 – instantaneous, 166, 191 – local, 155 – lowermost value of, 164 – maximum of, 167 – measuring of, 445 – minimum of, 167 – reduction of, 169 – spatial distribution of, 370 plasma resistance, 139, 638 plasma sheath, 110 plasma source – barrel-shaped, 623 – pressure in, 279 plasma speaker, 434 plasma stream mode, 258, 263 plasma tail, 575 plasma temperature, 85 plasma volume, 514 plasma wave, 66, 587 – generation of, 590 plasma, dusty, 550 plasma, high-density, 446 plasma-bridge neutralizer, 272 point defect, 498 poisoning, 426 Poisson-Boltzmann equation, 49, 55, 577, 580 polarization – degree of, 500 – extraordinary wave, 637 – i. anisotropic plasma, 637 – L-wave, 637 – of ordinary waves, 636 – ordinary wave, 637 – R-wave, 637 polymer, 471 – formation of, 450

Register polymer point, 533, 550 – bias threshold of, 550 polymerization, 534 – surface, 550 polywater, 433 Popov, 177, 236, 250 positive column, 42, 43, 48, 52, 67, 69, 95, 96 – field of, 86 – head of, 85, 86 – length of, 94 – potential of, 94 – striations in, 86 positive column, field, of, 94 potential – floating, 80, 310 – self-consistent sheath, 597 potential diagram, transition within, 356 potential function, 380 potential technique, retarded, 22 potential theorem – Koenig + Maissel, 451, 602 power factor, 339 power output, 139 power splitter, 412 Poynting vector, 631 pre-etching, 377 pre-sputtering, 377 precession plane, 631 precession radius, 606 presheath, 55, 56, 279, 319, 593 – Bohmic, 182, 313, 319, 592, 593 presputtering, 392, 512 pressing, isostatic, 407 pressure heating, 179 pressure sensor, 563 primary effect, 393 primary process, 6 principal direction, 628, 629 principal wave – cutoff, 630 – resonance, 630 principal waves, 630 probe potential, 197, 304, 315, 325, 330 propagation band, X-wave, 630 property, invariant, 81 proximity effect, 475

Register pseudopotential, 19 pump, differential, 510 pumping speed, 452 pyramid, 393–395, 399, 400 – formation of, 400 pyrometry, 403 Q-factor, 250, 252, 263, 264, 590 quadrupole, 246 quality factor, 132 quasineutrality, 51 – definition of, 50 quasineutraliy, 265 R-wave, 235, 237, 247, 248, 250 – absorption, 250 – band of propagation, 235 – cutoff, 630, 631 – dispersion, 632, 633 – phase velocity of, 236 – polarization, 637 – range of propagation, 237 – reflection, 631 – refraction index, 631 – resonance, 630, 631 – wavevector, imaginary, 632 R-waves, 236 – propagation, 632 Rabinowitz, 316 race track, 421, 424 radial configuration, 273 radial mode number, 238 radial uniformity, 273, 437 – beam, of, 437 radiation damage, 377 radiation damping, 105 radiation pattern – characteristics of, 130 – directivity of, 130 radical, 444 Ramsauer, 10, 12 Ramsauer effect, 16 Ramsauer minimum, 573 random phase motion, 106 random phase movement, 106 random walk, 89, 91, 208, 396 Rangelow, 488, 490, 535, 555

725 Ranjan, 274 rapid thermal annealing system, 404 rate coefficient, 345, 348, 349, 352 rate constant, 460, 480, 481 – 2nd order, 456 – area-dependent, 459 – etching reaction, 458 – for momentum transfer, 17 – recombination, 458 – second order, 458 rate-limiting step, 449, 525, 527 reaction energy, 466 reaction energy, temperature dependence of, 466 reaction rate, 534 reaction, heterogeneous, 89, 113 reactive ion etching – definition of, 445 reactive sputtering, 376 – deposition rate, 411 – pulsed, 410 reactor radius, 459 – normalized, 463 recombination, 2, 87, 88, 94, 113, 116, 326, 456, 458, 459 – rate of, 450 recrystallization, 378 rectification, across the RF sheath, 154 redeposition, 393, 398, 399, 401, 464, 472–475, 555 reference electrode, 301, 306, 308–310, 327 – potential of, 306 reference potential, 300, 310 – Langmuir probe, 313 reflected power, 142 reflected voltage, 142 reflected wave, 142 reflection, 234 – microwaves, of, 66 reflection coefficient – Beer’s, 502 – Fresnel’s, 501, 502 reflectometer, microwave, 340 refraction index, 244, 619 – complex, 616 – imaginary, 615

726 – imaginary part of, 617 – L-wave, 631 – modification in a static magnetic field, 631 – R-wave, 631 – real part of, 617 – squared, 632 Reinberg, 478 relaxation, non-radiative, 351 remote plasma, 444 remote plasma source, 217 residence time, 452, 453, 460 resistance term, 192, 638, 640 resistive model, 172 resonance, 629 – angular dependence of, 628 – coupled, 236, 246 – damping of, 132 – L-wave, 631 – of ordinary waves, 636 – Q-factor, 252 – R-wave, 631 resonance cone, 634, 636, 652, 656, 658 – at cutoff, 634 – at resonance, 634 – opening angle, 634, 636 resonance frequency – definition of, 628 – TE-mode, 262 resonant cavity, 129 resonant charge transfer, cross section, 32 retarding field analyzer, 194, 411 retarding potential difference, 22 retarding zone, 315 Reynolds, 316 Reynold’s number, 453 RF band, 111 RF compensation, 310 RF current density, 154 RF cycle – duration of, 191 – period of, 191 – phase of, 191 RF discharge – asymmetric case, 167, 168 – capacitively coupled, 196

Register – – – –

characteristic of, 209 geometrically symmetric, 600 high-frequency range of, 157–160 low-frequency range of, 157–159, 541, 543 – magnetron-supported, 425 – planar symmetric, 600 – symmetric case, 167 RF energy, coupling of, 104 RF modulation, 194, 200, 201, 212 – low-frequency regime, 199 RF noise, 310, 331 RF sheath, 209 – capacitively coupled, 583 – quasi-stationary, 193 – rectification across, 598, 601 – self-consistent, 597 RGA, 514 RIE, 128 RIE lag, 464, 479–481, 484, 488 Riemann, 81 RIPE, 244 RIPE etching, 245 RMS field, 107, 109, 251 Robinson, 396, 398, 492 roll-to-roll coater, 564 Rossnagel, 398, 399 rotational vibration spectrum, 48, 355 roughness-induced mechanism, 400 Rundle, 574 Rutherford scattering, 381 S-matrix, 341 Sabisch, 205 saddle profile, 201 safe-guard ring, 514 Saha equation, 61 Salpeter, 23 saturation current, 310 Savas, 602 Sawin, 356, 482, 555 scattering – by elastic collisions, 194 – by resonant charge transfer, 194 – elastic, 12, 202, 203, 208 scattering angle, 381 scattering mechanism, 190

Register scattering parameter, 9, 14 Scherzer, 84 Schottky, 76, 94 Schottky barrier, 497 Schottky diode, 499 Schottky profile, 182, 184, 228, 458 Schulz, 22 screen grid, 277, 280, 286 – aperture diameter of, 282 – aperture of, 284 – hole diameter of, 281 – losses at, 278 – sheath of, 280 – voltage of, 294 screen hole, diameter of, 282 screening length, 49 Seaward, 548 secondary effect, 393, 395 secondary electron, 69–71 secondary electrons – energy distribution of, 36 – flux of, 70 – generation of, 6 – production of, 71 – yield for, 83 – yield of, 35–38, 84, 379 secondary process, 6 Seeb¨ ock, 575 seed atom, 396 seed clustering, 396 Seeliger, 375 SEERS resonance, 336 selectivity, 439, 443, 444, 447, 448, 464, 467, 536, 537 – chemical, 440 – horizontal, 447 – vertical, 447 Self, 182, 183 self bias voltage, 370 self diffusion energy, 416 self-bias, 452 semiconductor, thermal conductivity of, 447 series resonance, 140 shadowing, 464, 475–477, 487, 555 Shaqfeh, 556

727 sheath, 41, 55, 56, 58, 93, 158, 331, 430, 478, 576 – acceleration process across, 591 – anisotropic, 575 – anodic, 43 – area of, 316 – at the grounded electrode, 332 – at the powered electrode, 332 – border of, 578 – breathing, 155, 162 – built-up, 302 – capacitance of, 165, 600 – capacitive, 170, 171, 600 rectifying behavior, 596 – capacitively coupled, 191 – carrier density in, 591 – cathodic, 43, 153 – characteristic of, 160 – clear definition of, 320 – collision free, 301 – collision number, 196, 204 – collisional, 201, 316, 591 – collisionless, 187, 198, 203, 246, 267, 314–316, 446, 577, 591, 592, 599 – composition of, 512 – conduction current, 154, 593 ionic, 157 – conduction current density, 157 – contraction of, 419 – DC, 41 – density of ions across, 155 – displacement current, 154, 157, 161, 174, 209, 592–594 – displacement current density, 157 – distribution of carriers across, 600 – divergence free, 192 – dynamics, 601 – electric field, 158, 159, 201 – electric field across, 591 – electrode, 209 – electron, 593 – electron boundary of, 155 – electronic conduction current, 592 – entering, 202 – field of, 159, 592 – harmonically excited, 597 – HF discharge, of, 93

728 – – – – – – – – – – – – – – – –

high-voltage, 166 ion boundary of, 155 ion distribution, 160 ionic conduction current, 592 ionic part, 592 ionization in, 208 isolated electrode, of, 56 low-voltage, 166 one-dimensional model, 161 oscillating, 201, 338 phenomenological introduction of, 57 physical function of, 52 planar, 317 positively charged, 153 potential of, 217 potential theorem Koenig + Maissel, 165, 451, 602 – pulsating, 176, 191, 209 – pulsating boundary of, 176, 594 – qualitative definition of, 52 – rectification across, 154 – resistive, 171 – RF field across, 592 – self-consistent thickness of, 599 – stability criterion, 599 – theorem Koenig + Maissel, 165, 451, 602 – thickness, 592, 596–599 – thickness of, 219, 317, 322 – thinning of, 283 – traversing, 190 – vicinity of, 577 – voltage across, 103 – voltage drop, 166 sheath boundary, 152, 155, 581, 595 – field at, 160 – ion density at, 582 – potential at, 581 sheath capacity, 160 Bohm’s, 578 sheath current, nonlinear, 342 sheath edge, 301 sheath equation, 82, 577, 578, 582, 583 sheath field, 159, 191, 202, 478, 495, 576 sheath heating, 182 sheath impedance, 154, 210

Register – capacitive, 139 – nonlinearities of, 209 sheath potential, 151–156, 164, 165, 168, 176, 198, 202, 209, 232, 306, 412, 431, 487, 526, 575, 599 – at the grounded electrode, 332 – at the powered electrode, 332 – definition of, 232 – for plasma etching, 445 – grounded electrode, 165, 167, 209, 232 – instantaneous, 207 – mean value of, 171, 556 – pressure dependence of, 155 – RF driven electrode, 165, 167, 168 – time dependent, 599 – time-averaged, 191 sheath resistance, 173, 174 sheath theorem – Koenig + Maissel, 165, 451, 602 sheath thickness, 44, 153, 158, 160, 166, 176, 179, 181, 187, 191, 196–199, 202, 203, 209, 279, 315, 317, 319, 424, 439, 446, 451, 491, 554 – at the RF electrode, 386 – calculated, 208, 386 – capacitive, 338 – cold electrode, 335 – DC component of, 191 – effective, 202 – electrically defined, 338 – helicon plasmas, in, 246 – hot electrode, 335 – measured, 386 – optically determined, 338 – pressure dependence of, 156 – reduction of, 421 – RF component of, 191 – shrinkage of, 446 – time-averaged, 199 sheath voltage, 163, 439, 578, 582, 583, 602 – DC component of, 161, 174, 445, 603 – maximum of, 198 – rectified, 173 – RF component of, 174, 603 – RF modulation of, 445 sheath voltage, mean, 191, 192

Register sheath width, 188, 189 – optical, 195 Sheward, 414 shield, 143 Shinoki, 512 shock wave, 396, 581 Shon, 370 short-range order, 391 shower head, 429, 458, 464, 514, 527, 528 Shul, 549 shutter, 377 sidewall bowing, 464, 482, 484, 555 sidewall charging, 482, 483 sidewall passivation, 440, 473, 479, 527, 528, 532, 533, 535–537, 542, 550, 551, 554–557 – perfect, 556 Sigmund, 379, 381, 395 Sigmund’s theory, 381, 382, 384 similarity rules, 97, 99, 100, 106, 196, 376, 449 simulation program, 555 single-wafer process, 461 sintering, 407 – theory of, 387 site, 550 – active, 463 – surface, 463 skin depth, 66, 110, 130, 143, 148, 221, 228, 234, 263, 274, 623 – electromagnetic waves, 588, 619–622 skin effect, 130, 142, 230, 233, 241, 273, 274, 335 SLAN, 120, 130, 272 sliding short, 130, 261, 263 slot, 478, 480 – system of, 483 small-angle scattering, 84 sodium vapor lamp, 1, 42 solenoid – plasma source, 244 – processing chamber, 244 Somekh, 391 Sommermeyer, 375 Sopori, 474 source impedance, 139 source resistance, 139

729 space charge, 75 – built up of, 43 spectral line – Doppler shift of, 354 – line width of, 48 – linewidth of, 353 – shape of, 353 spike, thermal, 398 Spitsyn, 433 spray discharge, 97 sputter cleaning, 392 sputter etching, 439, 442, 443, 446 sputter reactor, inverted, 442 sputtered layers, argon content of, 408, 409 sputtering, 128 – bipolar, 427 – cosine distribution, 375 – cross section energy dependence of, 379 – dual magnetron, 427 – magnetic materials, 424 – potential function for, 379 – reactive, 419 sputtering coefficient, 384 sputtering effect, angular dependent, 399 sputtering power, 277 sputtering rate, 104, 290, 376, 400, 401, 416, 417, 421, 478, 553 – angle dependence of, 488, 556 – at rising pressure, 401 – energy dependence of, 382 – enhancement of, 420 – model of, 382 – rise of, 420 sputtering system, 378 sputtering yield, 198, 288, 301, 379–382, 384, 394, 396, 398, 399, 402, 417, 499, 526 – angular dependence of, 399 – maximum of, 382 – pressure dependence of, 401 – sublimation energy, dependence on, 384 St. Elmo’s fire, 1 staircase process, 34 Stark, 375

730 Stark broadening, 302 Stark effect, 302 stationarity – condition for, 71, 72, 74, 75 – equation of, 84 Steenbeck, 98 step, rate-limiting, 478 Sternberg, 161 Stevens, 253 Stewart, 228 Stix, 236 stochastic heating, 176, 236, 313, 334, 339, 592 stop band – R-wave, 631 – X-wave, 630 stopping power, nuclear, 382 stray capacitance, 147 stress – compressive, 390, 431, 437 – tensile, 391, 431 stress control, 429 stress free deposit, 431 striation, 42, 43, 67, 86, 414 stripping, 29 structure – secondary, 396, 398, 400 – tertiary, 395 stub, 143, 256 – broadband, 145 substrate bias, 400, 412, 416, 418, 427, 499 – composition dependence on, 413 – dependence of the deposition rate on, 412 substrate biasing, 413–416, 429 – density increase by, 414 substrate damage, 291, 292 substrate plate, 457 substrate, unbiased, 264 Sugai, 190 Surendra, 111, 185 surface – coating of, 434 – effective, 115 – secondary, 400 – topography of, 394

Register surface contamination, 398 surface defect, 440 surface diffusion, 396, 398, 555 surface energy, 381 – minimum of, 390 surface film, 519 surface mobility, 389 surface morphology, 393, 547 surface oxide, 519 surface polymerization, 431 surface site, 550 surface temperature, 396 switch, micromirror-based, 563 synergy, betw. phys. and chem. processes, 555 Szmytkowski, 18, 348 tapering, 467, 468, 547 Tardy, 512 target, 146 – mosaic, 416 – multi-phase, 418 – sheath field of, 411 – sintered, 416, 417 target current – DC component of, 427 target voltage, 427 Tav, 27 TE-mode, 237, 637 telegraph equation, 614 TEM wave, 143, 637 temperature measurement, in-situ, 403 tensile stress, at low pressure, 425 term of inertia, 638 tertiary effect, 395 TG wave, 256 TG-mode, 657 thermal electron emission, 45 thermal velocity, electronic, 212 thermalization, 59, 326, 386 – deficient, 436 – ion beam, of, 197 thermometry, 403 thickness – geometrical, 253 – optical, 253 Thiele module, 460–462

Register Thomas-Fermi potential, 51, 380, 381 Thompson, B.E., 198, 202, 203 Thompson, M.W., 379, 384 Thomson, J.J., 29, 132, 375, 387 Thonemann, 651 Thornton, 387, 389, 390, 425 Thornton model, 389 three-stub tuner, 145, 256, 265 threshold of excitation, 5 tilting, 489 tilting angle, 488 time-of-flight method, 206, 385 TM-mode, 237, 261, 637 – resonance frequency of, 261 Tonks-Langmuir model, 58, 367, 370 Townsend discharge, 44 Townsend’s – approximation, 69, 72, 85 – discharge, 71 – equation, 83 – ionization coefficient, 70 – ionization theory, 70 – theory, 83 trajectory, ionic, 201, 555 transformer, 140 – coupling, 140 transhalogenization, 538 transmission line, 141, 253 – velocity of phase propagation, 144 transport, diffusion controlled, 386 trench, 393, 395, 400, 474 trenching, 464, 474, 476, 489, 555 trilevel technique, 468, 545, 547 triode system, 376, 427 triple electrode, 327 triple probe, 313 Trivelpiece, 237 Trivelpiece-Gould mode, 246 Trivelpiece-Gould wave, 256, 641, 656, 657, 659 Tsong, 379 Tsujimoto, 535 tungsten blue, 411 tungsten bronze, 411 tunnel, acceleration, 279 two-level process, 347

731 undercut, 541 uniformity – of etching, 485 – radial, 446, 513, 514 upper limit of, 292 upstream control, 452 utilization factor, 454 UV/VIS level, 86 van-Allen belts, 609 Vanderslice, 195, 198, 202 vapor deposition – chemical, 428 – physical, 428 velocity discriminator, 606 Vempaire, 369 Vender, 592 via hole, 483–485 vibration spectrum, rotational, 354, 355 Vlasov, 587 volatility, definition of, 444 voltage division, capacitive, 166, 173 voltage limit, 139, 142 voltage resonance, 132 volume polymerization, 430, 431, 550 wafer – backside of, 447 – topside of, 447 wall potential, 52, 94 Wannier, 24 wave – cylindrical, 257 – helicon, 258 – rectangular, 257 wave impedance, 617 wave riding effect, 177 wave surfing, 452 wave vector – complex, 616 – imaginary, 615, 632 waveguide, 145 – hollow, 256 – Ohmic loss in, 145 – radial, 256 – range of transmission, 145 – rectangular, 145, 256

732 waveguide applicator, 256 waveguide transformer, 257 waveriding resonance, 591 Wehner, 375, 394, 396 Wertheimer, 110, 111, 125 Westwood, 386, 427 Wharton, 624 whisker, 398, 399 whistler wave, 235, 238, 632, 633 – absorption of, 266 – absorption, resonant of, 266 – group velocity of, 632 – highest velocity of, 657 – phase velocity of, 234, 632 – velocity of, 633 whistler waves, 234 – atmospheric, 235 – low-frequency band of, 253 – propagation of, 257 – reactor, 245 – regime of, 237 – threshold of propagation, 235 Wiese, 345 Wiesemann, 310 Wild’s model, 208

Register Wilson, 394 Winkler, 310, 311, 567, 575 Winters, 87, 379, 408, 417, 439, 480, 522, 526, 529, 533, 537 Wood, 592 work function, 35, 38–40, 74, 301 Wright, 375 X-wave, 630 – cutoff, 630 – polarization, 637 – propagation band, 630 – resonance, 630 – stop band, 630 X-waves – polarization, 637 yield – electrons, of, 97 – ionization, of, 34 Yonts, 409 Zapesochnyi, 348 Zarowin, 154, 302, 529 zero beam, 12, 19

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