ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS
VOLUME 31
CONTRIBUTORS TO THISVOLUME P. L. Bargellini P. H. Dawson W. J...
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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS
VOLUME 31
CONTRIBUTORS TO THISVOLUME P. L. Bargellini P. H. Dawson W. J. Fleming G. H. Kimbell Sherman K. Poultney E. S. Rittner J. E. Rowe Richard 0. Rowlands Edward S. Yang
Advances in
Electronics and Electron Physics EDITED BY L. MARTON Sniithsoniun Inst it u tion, Washington, D. C. Assistant Editor CLAIRE MARTON
BOARD EDITORIAL E. R. Piore T. E. Allibone M. Ponte H . B. G. Casimir W. G . Dow A. Rose L. P. Smith A. 0. C . Nier F. K. Willenbrock
VOLUME 31
1972
ACADEMIC PRESS
New York and London
COPYRIGHT 0 1972, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. N O PART OF THlS PUBLlCATlON MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION I N WRITING FROM T H E PUBLISHER.
ACADEMIC PRESS, INC.
1 1 1 Fifth Avenue, New York. New York 10003
United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/24 Oval Road, L o n d o n N W 1
LIBRARY OF
CONGRESS CATALOG CARD
NUMBER: 49-7504
PRINTED IN T H E UNITED STATES OF AMERICA
CONTENTS CONTRIBUTORS TO
FOREWORD . .
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Vii
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Chemical Lasers P . H . DAWSONA N D G . H . KIMBELL
I . Introduction .
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11. General Conditions for Lasing Action . 111. The Vibrational Transition . . . . 1V . The Pumping Reaction . . . . . . V . Relaxation of the Excited State . . . V1. Characteristics Favorable to Chemical Laser
VII . Some Laser Systems . VIII . Conclusion . . . . . . References
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3 5 9 16 26 29 35 36
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Single Photon Detection and Timing: Experimentsand Techniques SHERMAN K . POULTNEY I . Introduction . . . . I1 . Single Photon Detection
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39 42 69
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Advances in Satellite Communications P. L . BARGELLINI AND E . S . RITTNER
I . Introduction .
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11. The INTELSAT System 111. The Orbita System . .
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1V . Systems Considerations .
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. . . . . . Spectrum and Orbit Utilization . . . . Modulation, Multiplexing. and Multiple Access Electron Devices . . . . . . . . . Materials Technology . . . . . . . . . . . . . . . . Future Trends References . . . . . . . . . . . . V
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119 123 . . . . . . . . . 127 . . . . . . . . . 128 . . . . . . . . . 130 . . . . . . . . . 134 . . . . . . . . . 137 . . . . . . . . . 155 . . . . . . . . . 157 . . . . . . . . 158 .
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CONTENTS
AcoustoelectricInteractions in In-V Compound Semiconductors AND J . E . ROWE W J FLEMING
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I . Introduction . . . . . . . . . . . . . . . . . . . I1 General Theory of Off-Axis Acoustoelectric Interactions . . . . . . 111. Exact Solution of the Acoustoelectric Interaction for Collinear Static Fields IV . Solution of the Acoustoelectric Interaction for Arbitrarily Oriented Static Fields and On-Axis Acoustic-Wave Propagation . . . . . . . . . V . Solution of the Acoustoelectric Interaction for Arbitrarily Oriented Static Fields and Off-Axis Acoustic-Wave Propagation . . . . . . . . . VI . Solution of the Acoustoelectric Interaction for Electron-Hole Carrier Transport and Off-Axis Acoustic-Wave Propagation . . . . . . . . . VII . Summary . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .
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185 206 234 242 244
Current Saturation Mechanisms in Junction Field-Effect Transistors EDWARD S . YANG
1. I1. 111. IV . V. VI . VII . VIII .
Introduction . . . . . . Review of the Literature . . . Governing Equations . . . Gradual Channel Approximation Saturation Models . . . . Numerical Calculation . . . Current Saturation in MOSFETs Conclusion . . . . . . References . . . . . .
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Electronic Engineering in River and Ocean Technology RICHARD 0. ROWLANDS I . Introduction . . . . . I1. Water Quality Measurement 111. Surface Waves . . . . 1V. Tides . . . . . . . V . Ocean Currents . . . . VI . Navigation . . . . . VII . Sonar Application . . . VIII . Fishing . . . . . . . IX . Helium Speech Processing . X . Data Transmission . . . References . . . . . AUTHORINDEX . . . . SUBJECTINDEX . . . .
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261 268 214 216 217 280 283 293 295 295 299
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CONTRIBUTORS TO VOLUME 31 P. L. BARGELLINI, COMSAT Laboratories, Clarksburg, Maryland P. H. DAWSON,Centre de Recherches sur les Atomes et les Molecules, Universite Laval, Quebec, Canada W. J. FLEMING,* Electron Physics Laboratory, Department of Electrical and Computer Engineering, The University of Michigan, Ann Arbor, Michigan
G . H. KIMBELL, Defence Research Establishment Valcartier, Courcelette, QuCbec, Canada SHERMAN K. POULTNEY, Department of Physics and Astronomy, University of Maryland, College Park, Maryland
E. S. RITTNER, COMSAT Laboratories, Clarksburg, Maryland J. E. ROWE,Electron Physics Laboratory, Department of Electrical and Computer Engineering, The University of Michigan, Ann Arbor, Michigan RICHARD 0. ROWLANDS, Ordnance Research Laboratory, The Pennsylvania State University, University Park, Pennsylvania
EDWARDS. YANG,Department of Electrical Engineering and Computer Science, Columbia University. New York, New York
* Present address: Research Laboratories, General Motors Corporation, General Motors Technical Center, Warren, Michigan 48090. vii
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FOREWORD The present volume encompasses a greater variety of subjects than some of its predecessors. The phenomenal growth of laser research and development is reflected in Dawson and Kimbell’s review of chemical lasers. Parallel, however, to the development of more sophisticated light sources, work proceeds toward the detection of fainter and fainter sources of light. Thus it is very timely to review single photon detection experiments ; Poultney’s review covers techniques as well as the experimental results. The review of Bargellini and Rittner tackles an important aspect of the problem of communication. The use of satellites for communication is a relatively new field, but sufficiently advanced to warrant a review of its present stat us. The next two contributions take us into semiconductor technology. First, there is a discussion, by Fleming and Rowe, of acoustoelectric interactions as they appear in 111-V compound semiconductors. A different aspect is treated in Yang’s review of current saturation mechanisms in junction fieldeffect transistors. The final contribution, by Rowlands, covers a field which we have not treated for several years: oceanography. The title of the review, Electronic Engineering in River and Ocean Technology, is indicative of the scope of the discussion. In our next few volumes we expect to publish reviews on the following subjects : Galactic and Extragalactic Radio Astronomy Image Formation in the Electron Microscope Recent Advances in the Field Emission Microscopy of Metals Multiple Scattering and Transport of Microwaves in Turbulent Plasmas Microfabrication Using Electron Beams The Effects of Radiation in MIS Structures Small Angle Deflection Fields for Cathode Ray Tubes Sputtering Interpretation of Electron Microscope Images of Defects in Crystals Optical Communication through Scattering Channels Wave Interactions in Solids ix
Wrn. C. Erickson and F. J.
Kerr D. L. Misell L. W. Swanson and A. E. Bell V. L. Granatstein and D. L. Feinstein A. N . Broers Karl Zaininger R. G. E. Hutter and H. Dressel M. W. Thompson
M. J. Whelan Robert S . Kennedy Morris Ettenberg and Vural
B.
X
FOREWORD
Hollow Cathode Arcs Channelling in Solids Physics and Applications of MIS-Varactors Ion lmplantation in Semiconductors Self-Scanned Solid State Image Sensors Quantum Magneto-optical Studies of Semiconductors in the Infrared Gas Discharge Displays Photodetectors for the i p to 0.11” Spectral Region High Resolution Nuclear Magnetic Resonance in High Superconducting Fields The Photovoltaic Effect Application of Single Photon Techniques
The Future Possibilities for Neural Control Electron Bombardment Ion Sources for Space Propulsion Recent Advances in Hall-Effect Research and Development Semiconductor Microwave Power Devices The Gyrator Electrophotography Microwave Device Technology Assessment The Excitation and Ionization of Ions by Electron Impact Whistlers and Echos Experimental Studies of Acoustic Waves in Plasmas
J. L. Delcroix R. Sizmann and Constantin Varelas W. Harth and H. G. Unger S. Namba and Kohzoh Masuda Paul K. Weimer Bruce D . McCombe and Robert J. Wagner R. N. Jackson and K. E. Johnson David H. Seib and L. W. Aukerman
H. Sauzade Joseph J . Loferski Sergio Cova, Mario Bertolaccini, and Camillo Bussolati Karl Frank and Frederick T. Hambrecht Harold R. Kaufman D. Midgley S. Teszner K. M. Adams, E. Deprettere, and J. 0. Voorman M. D. Tabak and T. L. Thourson Jeffrey Frey and Raymond Bowers John W. Hooper and R. K. Feeney Robert A. Helliwell J. L. Hirshfield
We would like to express our pleasure that the request for suggestions on subjects and authors, voiced at the end of the forewords of previous volumes, has borne fruit. A number of valuable suggestions and proposals were received; this volume, as well as earlier volumes, reflects those results. We would like to repeat our request and amplify it by saying that inquiries from potential authors are most welcome. L. MARTON CLAIRE MARTON
Chemical Lasers P. H . DAWSON Centre de Recherches sur les Atonies et les Molecules, Universite Laval, QuPbec, Canada
AND
G. H. KIMBELL Defence Research Establishment Valcartier, Courcelette, Quebec, Canada
Ill. The Vibrational Transition. .....................
...........
V. Relaxation of the Excited State . . . . . . . . . . 16 V1. Characteristics Favorable to Ch VII. Some Laser Systems. . . . . . . . . . . . . . . . . . . . 29 A . A True Chemical Laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. A Flame Laser.. . . .
I.
INTRODUCTION
Gas lasers (I) can be characterized according to the means by which the necessary population inversion is achieved. This mechanism may be, for example, excitation by energetic collisions-such as electron bombardment in a gas discharge, by photoh impact, or by the direct production of excited species in chemical reactions (chemical lasers). Chemical lasers were the last type to be realized but represent a rapidly evolving technology. 1
2
P. H. DAWSON AND G . H. KIMBELL
In 1958, Schawlow and Townes (2) made the first proposal for an “optical maser ” using population inversion in a gas. This was to utilize optical pumping. However, the first practical device was the result of work by Javan et al. (3) in 1961 and used a gas discharge. The following ten years have resulted in the development of a large number of these “physical” lasers. The carbon dioxide system, in particular, is capable of high power (up to 60 kW cw has been realized using gas dynamic principles) and, i n certain cases, operation at atmospheric pressure (4). The electrical efficiency (power out/power in) is generally of the order of 5 The possibility of a chemical laser was recognized in the early days of the gas laser ( 5 ) and in 1964 a symposium was organized on this subject (6); at about the same time the first chemical laser was devised by Kasper and Pimentel (7) using the reaction between atomic chlorine and molecular hydrogen to form vibrationally excited HCI. The reaction was initiated by the production of chlorine atoms in flash photolysis. The laser action, of course, only occurred as a short duration pulse. The use of pulsed electrical discharges to produce reactive atoms was a subsequent development (8) and the first continuously operating lasers (cw) were only reported late in 1969 (9,ZO). Since this breakthrough, there has been a rapid advance in the development of chemical laser systems although the chemical reactions utilized for cw lasers have been largely limited to the formation of hydrogen halides and to the formation of CO in a reaction between oxygen atoms and carbon disulphide (11). One system has been found (J2) which is a true chemical laser and requires no external energy input to create the reacting atoms, i.e. to initiate reactions. Scientifically,chemical lasers are of importance as tools which give deeper knowledge of chemical reactions. They are of potential technological importance because of the possibilities of high efficiency and freedom from the dependence on an external power source. Much development work is still required to increase power, and, more especially, efficiency. Ultimately it is hoped to achieve efficient operation at atmospheric pressure or in flames without external power input. The art of chemical lasers is, nevertheless, still in its infancy. In this article we discuss first the necessary conditions for the achievement of chemical lasing action and then describe, in turn, the important processes involved in lasing systems such as (a) the vibrational transition which gives the emitted radiation, (b) the production of vibrationally excited species by reaction, and (c) the vibrational-vibrational and vibrational-translational energy exchanges which can cause depopulation of the excited levels. The aim is to elucidate the kind of characteristics one must look for in the search for new and improved chemical lasers. Finally we describe some typical examples of chemical lasers to illustrate the progress achieved. Since the
x.
3
CHEMICAL LASERS
field of chemical lasers is evolving so rapidly, there is no attempt at comprehensive coverage of the achievements. Such a review would be outdated before its publication. Rather, the intent is provide a general view of the field and to illustrate present experimental trends. 11. GENERAL COND~TIONS FOR LASING ACTION
The laser is based (13) on the fact that a molecule i n an excited state can be stimulated to emit some or all of its excitation energy by an incoming photon that has exactly the right energy. The incoming photon may therefore be amplified as it passes through the lasing medium. If the incoming photon meets a molecule in the lower (less excited or ground) state, it may be absorbed. Therefore a necessary condition for overall amplification is that there is a large population of molecules in the excited state. This provides the lasing medium (analagous to a negative resistance). Since, in many systems, the amplification or gain per meter of path length is not very large, the cascade process is increased by having the lasing medium within an optical resonance cavity. The cavity consists of a pair of aligned mirrors which reflect the laser radiation. Figure 1 illustrates a simple geometrical arrangement which is frequently used. More details of experimental systems are given in Section VI. Let us look in more detail at the conditions which are favorable for lasing action. Consider a simple two-level system. The absorption coefficient cq, in a medium, for radiation at the center of the transition (when natural line broadening is small in comparison with Doppler broadening), is given by (14)
'I
TO -2450Mc
~
EX HA U ST
MICROWAVE
FLAT FRONT-SURFACED MIRROR Se ON Ca FE 2%TRANSMISSION ~~~
E"
120CM
~
4 M RADIUS ALUMINIZED MIRROR ~
-
~
~-~-
FIG.I . The geometrical arrangenientsof a longitudinal flow, continuous wave, chemical laser.
4
P. H. DAWSON AND G. H. KIMBELL
where e is the electron charge, rn the electron mass, f21 the oscillator strength of the transition from state 2 to state 1, N, the population density of molecules in the lower level (state l), N, the population density in the upper laser level, g1 and g2 are the statistical weights of the two states, and A v D is the Doppler width of the transition and can be expressed as AvD =
(2vo/c)[(2KT/M) In 2I1l2,
(2)
where vo is the center frequency of the transition, K the Boltzmann constant, T the absolute temperature, and M the atomic mass. For optical gain, a0 must be negative. That is, g1lg2 . N , > N , .
(3)
To satisfy this condition, one must have an appreciable excitation rate to the upper level and it is desirable that the lifetime of the upper level (due to radiation and other deexcitation) be much greater than that of the lower level. For laser oscillation there must be a net gain in the system. For a medium of length I, the gain is exp( -uo I ) - 1 where a0 is now negative. For small gain, this can be approximated by - uo 1. For a net gain in the system -u,I > L, where L represents the total of diffraction and reflection losses involved in the single pass through the optical cavity.
Laser action is therefore favored for transitions with small line widths and large oscillator strengths. Obviously optically allowed transitions are desirable. By chemical reaction, it is possible to form species in rotationally, vibrationally, or electronically excited states. Rotational energy is rapidly thermalized since only a few molecular collisions are required to sec at 1 Torr of a molecular gas). Vibrational level lifetimes are of the order of 10-2-10-3 sec. Most electronically excited levels have shorter lifetimes and it is difficult to form these excited states, by chemical means, at a high enough rate to obtain a population inversion. As expected, laser action on vibrationally excited levels has proved to be the easiest to achieve. Use of purely rotational transitions has been reported (15) but, so far, chemical lasers do not employ electronically excited levels. An intriguing idea for the latter involves molecules which are only stable in the excited state, the lower (or ground) level then always having a very low population.
5
CHEMICAL LASERS
The characteristics of the vibrationally excited levels and the associated transitions are now examined in more detail.
111. THEVIBRATIONAL TRANSITION We consider the transitions of diatomic molecules for simplicity and because present chemical lasers involve mainly such molecules. The simplest model for a vibrating molecule is a harmonic oscillator (16) for which the energy levels are equidistant. The levels are separated by hvo .
E(v) = hvo(u
+ 9,
where v is the vibrational quantum number and can have integral values 0, 1 , 2 etc. and vo
= (1/2Zl(WP)>
where p is the reduced mass and k is the force constant. A much better approximation is the anharmonic oscillator of which the energy values are given by E(u) = hv,(v
+ +) - hv,x,(u + *)’ + hv,y,(v +
4)3
...,
(5)
where ye 4 x, 6 1. Figure 2 gives an example (16) of an anharmonic potential curve and the associated vibrational levels. Each vibrational level is actually made up of a number of closely spaced rotational levels. A suitable model is the nonrigid rotor (two mass points connected by a massless spring) of which the rotational energy levels are given by
E,.(J) = hc[BJ(J
+ 1)
-
DJ’(J
+ l)’],
(6)
where J is the rotational quantum number and B is a rotational constant (in cm-’) which is inversely proportional to the moment of inertia. D is much smaller than B but is larger the smaller the vibrational frequency (i.e. the less rigid the rotor). Energy levels of a vibrating rotator are illustrated (16) in Fig. 2b. The constants B and D vary a little according to the vibrational level. The selection rule for transitions between the rotational levels of different vibrational levels is AJ= + I (provided that A = 0; the Q branch, AJ = 0, can be seen for molecules such as NO and OH). Furthermore, although 2’ can change by any integral amount since the oscillator is anharmonic, Ao = 1 (the fundamental) still gives far
6
P. H. DAWSON AND G. H. KIMBELL
la)
V"
(L
W W
z
-0
INTERNUCLEAR DISTANCE (b)
4 "'I
I
3 2 '0
ENERGY LEVELS 4 v
=o
3 2 '0
IIII.illlI P
R
SPECTRUM Y
FIG.2. (a) An anharmonic potential energy curve for a diatomic molecule, showing schematically the positions of the vibrational levels. (b) Rotation-vibration energy levels showing the allowed transitions, AJ = & 1, and the resulting P and R bands of the spectrum.
more intense transitions than the overtones (I Avl > 1). As indicated in Fig. 2b, those transitions for which the rotational level is highest in the lower vibrational level are known as the P branch and vice versa as the R branch. Now consider the population of the various levels. The rotational and vibrational temperatures will only be identical under equilibriuni conditions. For a temperature T,,, the distribution between the vibrational levels is given by the Maxwell-Boltzman law Nu e - E v l K T v . (7) However, for the rotational levels each level has a (25 + ])-fold degeneracy (the levels coinciding in the absence of a magnetic field) and therefore
N J K (25 + I ) e - E J / K T r as in Fig. 3.
7
CHEMICAL LASERS b
02-
J
FIG.3. Illustration of the population distribution among rotational levels, expressed as N J / x N J as a function of J , where N J = N J U l ) - ‘ h c ’ K T ” B ~ J ( J’ +) . The rotational temperature T, is 300°K. Curve A represents HCI with B equal to 10.59 and curve B represents N 2 0 for which B = 0.41.
+
Let us compare two adjacent vibrational levels and the conditions for which a population inversion between appropriate rotational levels will occur. If the vibrational temperature is negative (i.e. > N , ) and the rotational temperature is positive (the rotational energy will in fact be thermal in most cases), then there will be a population inversion for all P branch transitions and for some R branch transitions. For the P branch transitions the population distribution within the rotational levels provides a “natural assistance towards the inversion. For the R branch transitions the population distribution works against the inversion. Figure 4 illustrates some calculated gains for various conditions. The condition > N u has become known as a “ total population inversion.” When T, is positive, but T, > T , , a population inversion can exist between certain rotational levels of adjacent vibrational levels. This is called a “partiaI population inversion.” Owing to the rapid redistribution of energy between vibrational levels, many diatomic gas lasers operate under conditions of partial inversion. The inversion is again the result of the shape of the distribution curve for the rotational levels. ”
8
P. H. DAWSON AND G . H. KIMBELL
0
I
1
I
I
I
I
4
8
12
16
20
24
Upper level
*
J
FIG.4. Normalized gain curves calculated by Patel (33) for the P and R branches of the 7-6 transitions of CO, plotted as a function of the upper level J for T = 300°K and population ratios between the 7 and 6 levels given by N,,/Nu,= 0.8,0.9, 1.0, 1.1, and 1.2.
Consider the necessary conditions for the population of the level ( u + I , J - I) to be greater than that of ( v , J ) (i.e. P branch transition), with the simple approximation that the rotational constant B is the same for both levels and that the vibrational populations are represented by a vibrational temperature T, and the rotational level populations by a temperature T , . For inversion Nv+l,J-l
or
'Nu,
J ,
CHEMICAL LASERS
9
Tv/Tr> vo/2JBc.
(1 1)
If J is large, this simplifies to
This condition can obviously be satisfied if T, > T , and if J is sufficiently large [a situation exploited in gas dynamic lasers (17)]. The inversion is favored for molecules with a large B ; that is, a small moment of inertia. In such a case it tends to occur at smaller J values where the absolute population densities are higher and laser gain will also be higher. The condition (11) can be more accurately expressed
TJT, > vo/c[2JB,+ aJ(J - I)],
(12)
where a = B, is the constant of interaction between rotation and vibration. The derivation of a condition analagous to (11) but for an R branch transition N u + , , , + , > shows that an inversion can only occur if the vibrational temperature is negative. As shown in Fig. 4, even for total inversion the P branch transitions tend to dominate. However, R branch transitions have been observed in many cases and can be isolated by the use of selective tuning of the appropriate wavelength using a grating within the optical cavity (18).
IV. THEPUMPING REACTION The pumping reaction must, of course, be exothermic, must lead to product molecules in an inverted population, and must be sufficiently fast to maintain an appreciable inversion in the system. Measurement of product energy distributions have generally been made by studying infrared chemiluminescence, as will be described later, or by molecular beam techniques. Theoretical approaches to predicting the factors favorable to vibrational excitation will be considered first. A . Theoretical Approaches
The main effort has been in the use of classical trajectory calculations of reaction dynamics, especially by Polanyi and co-workers (19-21). We will give a qualitative account of such calculations. Consider the three-body exchange reaction A-tBC
-
ABIC,
Assumption of a suitable potential energy hypersurface is the starting point of the calculation, since ab iniiio calculations of these surfaces are only just beginning to be carried out. Figure 5 shows some simplified schematic
10
P. H. DAWSON AND G. H. KIMBELL
‘E
rAB
‘AS
FIG.5 . A schematic representation of three types of potential energy surfaces for the collinear reaction A BC + AB C. (a) A repulsive surface, the energy barrier is in the retreat coordinate; (b) an intermediate case; (c) an attractive surface, the energy barrier is in the approach coordinate. The broken lines represent possible trajectories giving rise to varying degrees of vibrational excitation of the products.
+
+
examples for collinear reaction. Three-dimensional computations have also been carried out (22,23). The classical Hamiltonian equations of motion are integrated numerically for motion on the potential energy hypersurface, involving 12 simultaneous differential equations. Different choices of initial conditions can be made and generally “ batches” of trajectories (perhaps 300) are calculated and the results averaged to simulate the normal Boltzmann distributions of translational and vibrational energy in the reactants. The examples of Fig. 5 show surfaces which differ in the location of the saddle point ; the energy barrier being located in the retreat coordinate (a) (called a repulsive surface), in an intermediate position (b), or in the entry valley or approach coordinate (c) (called an attractive surface). Although the families of trajectories are necessary to obtain quantitative results, the qualitative trends are readily discernable in examining individual trajectories. For example, in the figure the dotted lines show trajectories for cases where reactants have a translational energy only slightly in excess of that necessary to pass over the saddle point and a small initial vibrational energy. Figure 5a shows little of the released energy appearing as vibration; Fig. 5c shows much of the energy appearing as vibration and Fig. 5b is intermediate. That is, the location of the energy barrier in the approach coordinate favors vibrational excitation of the reaction product. One can perhaps better understand
CHEMICAL LASERS
11
this by considering a rolling ball on an analogous potential surface. Other interesting results are that initial translational energy is effective in increasing the reaction cross section on the Fig. 5c attractive ” potential surface but the fraction of the energy appearing as vibration tends to decrease. Initial vibrational energy does not help in this type of reaction. For the “repulsive” case Fig. 5a, the opposite situation applies. Initial vibrational energy enhances the cross section and increases the small fraction of energy appearing as vibrational excitation of the products. To increase the value of these results, Mok and Polanyi (21) have attempted (for some related families of reactions) to correlate the barrier location with other better known properties of reaction. They conclude that for substantially exothermic reactions the barrier is in the entry valley. For decreasing barrier height the barrier moves to successively earlier positions along this valley. It has also be found (19) that the most probable trajectory depends on the relative masses of the particles involved and this is important particularly for the intermediate case (Fig. 5b). The optimum situation for the release of energy as vibrational excitation of AB occurs when M A N M , 4 M c . The case of M A 6 M , N M , is unfavorable. Polanyi has recently produced movies derived from the computed trajectories, vividly illustrating the collision dynamics. Quantum mechanical solutions for collinear reactive systems have also been obtained (24,2.5) and these include the possibility of “ tunneling ” through the potential barrier. The degree of vibrational excitation predicted was quite low but the quallitative features outlined above were confirmed. At low energies (just sufficient to overcome the barrier) quantum tunneling increases the reaction cross section if the barrier is narrow and more of the available energy becomes vibrational excitation. This is more likely to be important for the H atom; for example, with H + C1, tunneling would be effective below about 500°K. “
8. Experimental Studies
Of concern in the experimental investigations is the measurement of the initial vibrational energy distributions; that is, the rate constants for reactions into levels L’ = I , v = 2, . . . , etc., before any relaxation has occurred. The relaxation processes can then be studied as a separate factor (Section IV). Two methods of approach have been developed-the method of arrested relaxation and the method of measured relaxation (26). As the name suggests the principle of the first method is to prevent relaxation from influencing the observations while directly observing the vibrational energy distributions by means of the emitted infrared radiation. It may also permit the observation of initial, rotational energy distributions.
12
P. H. DAWSON AND G. H. KIMBELL
The second method consists of making observations of vibrational distributions at known times after mixing and deducing the initial distribution, often by a simple extrapolation. Figure 6a shows a reaction vessel used by Polanyi and co-workers in the method of arrested relaxation. It is a large chamber with internal mirrors and walls that can be liquid nitrogen cooled. Experiments are carried out at pressures in the l o w 4Torr range so that the mean free paths are comparable to the cell diameter. Product molecules should be removed from the vessel by capture at the walls before significant gas phase relaxation can occur. The cold walls appear to work quite efficiently for the hydrogen halides (26) but in removing vibrationally excited OH, room temperature walls coated with silica gel (27) were found to be more effective. The population of the vibrationally excited products can be determined by recording the fundamental or first overtone regions of the infrared luminescence. The first overtone is frequently used, because of the sensitivity of lead sulphide detectors in that region. Conventional infrared grating spectrometers have been used but Fourier transform spectroscopy (27) has been recently applied to the H + 0, reaction as illustrated in Fig. 6a. The latter gives advantages in efficiency which can be important in low pressure studies. Figure 7 shows the results obtained by Polanyi et al. for OH formed in the reaction H O 3 OH* 0 2 ,the populations being given relative to u = 9. At 3 x Torr the distribution of products is quite different from that at 5 x low4Torr suggesting that gas phase relaxation became important. That the spectra of Fig. 8 at 5 x Torr represent initial distributions is supported by observation of the rotational energy distributions (Fig. 8) for which the populations of the higher levels are very much in excess of room temperature values and where the distributions are clearly nonBoltzmann. The method of arrested relaxation has also been applied to the reactions H CI, HCI* C1(26), H + Br, -+ HBr* + Br (26), CI + HI --t HCI* + I (28), and H SCI, HCI* SCI (29). A mean fraction of the available energy in the vibrational mode as great as 6 5 % has been observed. In favorable cases contours of equal detailed rate constant as a function of vibrational and rotational energy of the products [and therefore also of the translational plus internal energy ( E ’ ) ] can be determined. Figure 9 shows an example for H SCI, -+ HCI + SCI. Figure 6b shows a reaction vessel for the method of measured relaxation, as described by Polanyi et al. An analagous system has also been used by Jonathan, Melliar-Smith, and Slater (30). Measurements are made of infrared emission at successive points along the line of flow; that is, at different times after mixing and reaction. Extrapolation to zero time may give indications of the initial distribution. Polanyi’s vessel has three circular bands of highly reflecting gold (shaded in the figure) which sharply delineate the
+
+
-+
+
+
-+
+
-+
+
+
FIG.6a. Apparatus of Polanyi et al. (27) for the study of the initial distribution of vibrational states of OH formed by H O3 + O H O2 by the method of arrested relaxation. A Fourier transform spectrometer was used to give greater sensitivity and the ability to work at very low pressures.
+
”,
+
-+J I cm
TOP
MIDDLE
BOTTOM
FIG.6b. Flow tube (31) used in the method of measured relaxation. a, Wood’s discharge tube; b, CI2 inlet; c, wire screen; d, pressure probe; e , lateral wall of vessel; f, slide valve; g, liquid nitrogen trap.
14
P. H. DAWSON AND G . H. KIMBELL
o,2
0
1
5x
Torr
3
4
5
6
7
8
9
U
FIG.7. Distribution of vibrational levels of OH (formed by H i '03) relative to it = 9, observed by Polanyi ef a / . (27) by the method of arrested relaxation at 3 x lo-' Torr and 5 x Torr. The latter represents the initial vibrational energy distribution.
volumes viewed via the corresponding windows. The distance between adjacent windows corresponds to a time difference of about I . 1 msec. Pacey and Polanyi (31) have made a detailed computer analysis of the diffusion, flow, radiation and deactivation occurring in their flow tube, and find that the K, obtained by detailed calculation are fairly close to those obtained by simple extrapolation, but the latter does tend to give rate constants for the lower vibrational levels which are too large. Obviously, the details of gas mixing and flow will be different in different systems and simple extrapolation
15
CHEMICAL LASERS
v = ?
-.._ ---__
0
I
0
10
1
I
I
I
1
1
1
1
1
1
20 3.0 40 50 60 ROTATIONAL ENERGY (kcol m o l d )
1
1
7.0
I
.
8.0
FIG.8. Relative rotational populations within vibrational levels u = 7, 8, and 9 for the experiment at 5 >: Ton shown in Fig. 7. The A curves refer to t h e n J i t substate; the B curves to the HI,* substate. The dashed curves are Boltzniann distributions drawn on the assumption that the lowest rotational levels have been thernialized to correspond to the wall temperature.
to zero time, unsupported by detailed calculation, may be misleading. According to Pacey and Polanyi, their corrections for relaxation are not very sensitive to the relaxation model, suggesting that their flow tube is not suited to the determination of relative relaxation rates. For H + CI, -+ HCI C1 vibrational distributions are in quite good agreement with those measured by the method of arrested relaxation as shown in Fig. 10.
+
16
P. H. DAWSON AND G . H. KIMBELL
,
60
~
0,4
,
0,8
,
1,2 H+SC12
,
1,6
-
,
2,o
,
HCI(V'J')+SCI
2,4 e l
eV 24
EXOHERMIC
50
,
v ' = 6 lkf=OOO)
20
R' (kcal mole-')-
+
FIG.9. Contours of equal detailed rate constant (29) for the reaction H SCI2+ HCI SC1 plotted versus product (HCI) vibrational energy V', rotational energy R', and translational energy plus 'internal energy of SCI, symbolized E', the latter being given by the diagonal dashed lines. The total energy available is 48 kcal mole-'. The rate constants are given relative to a total rate of 1.OO into the u' = 3 level (the most populated level).
+
V. RELAXATION OF THE EXCITED STATE We have considered the formation of vibrationally and rotationally excited states by chemical means. In the laser, we are concerned with the subsequent evolution of the excited population after the initial excitation. This requires, firstly, an evaluation of the individual processes of relaxation of the excited state and, secondly, studies of the combined effect of such processes. As stated previously, rotational relaxation, at least for heavy molecules, is completed within a few collisions because the energy changes involved are small compared with the average translational energy. Since a partial population inversion occurs when the rotational temperature is less than the vibrational temperature, chemical lasers generally have provision to encourage rotational relaxation, such as by cooling the walls and adding sufficient helium to the gas mixture to ensure efficient heat transfer. For molecules with small
17
CHEMICAL LASERS
0
02
06
04
08
10
f",
FIG.10. Semilogarithmic plots of the relative rate constants K ( d ) for the reaction H CI2 --f HCL,. GI versus f"', the fraction of the available energy which goes into product vibration. The curves were obtained ( 3 1 ) by the methods of measured relaxation (MR) and of arrested! relaxation (AR).
+
+
moments of inertia, relaxation may require numerous collisions; for example, 300 for H, (32) unless the dipole moment is large (HCI requires an average of about seven collisions). Relaxation of vibrationally excited levels can occur through spontaneous emission of radiation, through vibrational-translational energy transfer in molecular collisions, or through vibrational-vibrational relaxation, that is to say, by transfer of vibrational quanta between molecules of the excited species and other acceptor molecules which may be present. We have already described the nature of the emission processes for vibrationally excited states. It is perhaps sufficient to give some examples of the speed of the processes for some molecules of interest i n chemical lasers. Table I (38) shows lifetimes calculated by Pate1 for CO using the relation
18
P. H. DAWSON AND G . H. KIMBELL TABLE I CALCULATED EMISSION COEFFICIENTS AND LIFETIMES (33) OF THE v = 1 TO v = 10 VIBRATIONAL LEVELSOF CO ( x ' C + ) V
1 2 3 4 5 6 7 8 9
A"-"-, (sec-')
(sec-')
30.30 59.33 89.66 111.2 134.8 157.7 178.5 197.8 215.2
0.55 1.62 3.21 5.20 7.66 10.50 13.72 17.25
r, = l/CA,
A,+,-2
(
4
33 x 16.7 x 11.0 x 8.7 x 7.1 x 6.0 x 5.3 x 4.7 x 4.3 x 4.0 x
10
10-3 10-3
10-3 10-3 10-3
10-3 10-3 10-3
10-3 10-3
and values of the ratios of the matrix elements for vibrational transitions (R) as obtained by Cashion (34). The probability of decay by emission of a single is always much greater than that for two quanta quantum although less so at higher u. Also A,-,,.-l is approximately proportional to u, so that the lifetimes of highly excited states become quite short. The shorter lifetime of the more excited level tends to be unfavorable to laser action (see Section 11) except that the stimulated emission coefficients are also larger. TABLE 11 RADIATIVELIFETIMES OF THE v = 1 VIBRATIONAL LEVELSOF SOMESIMPLE MOLECULES
(msec)
co NO HCI HF H Br
33 145 29 5 I39
vt-lo (cm-')
2143 1876 2886 3958 2559
Table I1 gives a comparison of the lifetimes of the u = 1 level for several molecules. The rate of stimulated emission is given by ZB, where Z is the appropriate radiation density and B, @hestimulated emission coefficient. The ratio A J B , is proportional to l / v 3 so that laser action is harder to achieve the lower the frequency of the transition involved.
CHEMICAL LASERS
19
Vibrational relaxation by collision has been determined by a variety of methods; ultrasonic absorption (35,36), flash spectroscopy (37), and the quenching of fluorescence (38). However most of the data only apply to the lowest vibrational level. Recently, detailed measurements applying to several levels have begun lo be obtained by chemiluminescence studies (39,40).Rapid progress may therefore be expected i n both experiment and theory. Some theories were developed many years ago because of the importance of thermal relaxation in sound absorption. From the viewpoint of classical mechanics, considering the collinear approach of a " billiard ball " atom A to a spring BC, energy transfer is most likely when the interaction time is small compared with the vibrational period ( 4 / ) ;that is, when Iv/s is small, where v is the frequency of vibration, s is the velocity of A, and I is a measure of the range of the interaction. It seems, then, that short range forces between the molecules would be the most important. The quantum mechanical approach derived by Schwartz, Slawsky, and Herzfeld (34,4/,42)-widely known as the SSH t heory-considered as an approximation only the short range forces which are, in fact, repulsive forces. For example, in a V-T (vibration to translation) exchange between A and BC, if x is the separation of A from the center of gravity of BC and X is the separation between B and C (during vibration), then, for a collinear collision, the distance between A and B is
The repulsive interaction energy can therefore be expressed as ~ = ~ ~ e x p [ (ll7g x -f
1 7 7 ~
where V, and 1 are the interaction parameters. The SSH theory assumes that the probability of inelastic scattering is small compared with the probability of elastic scattering, thus providing a basis for the distorted wave approxiniation in which the interaction causing the energy transfer is treated as a perturbation. It may only be valid, therefore, when the probability of energy transfer per collision, is not too large. This probability is found to be a product of three factors. Pi+f(P)= P o x Pf!+,(s)x P;yf,
(15)
where Po is called the steric factor. P;LFis the translational part, and P;Zf is the oscillator part.
20
P. H. DAWSON AND G. H. KIMBELL
The Oscillator Part P?fc is s independent and is given by
This is normally integrated by an approximation of the exponential expression in terms of a Taylor expansion. That is, it is assumed that the amplitude of oscillation is small compared with 1. For a single quantum transition in the collision A + BC, the integral gives
where p B c is the reduced mass of BC and ui is the initial vibrational quantum number of BC. For two quantum transitions PoSc(Au= 2 ) ‘V [ P o s c ( A=~l)j2/2,
(18)
so that the probability is much reduced. Since A is just as likely to collide with A close to C instead of B, Eq. (17) should be modified to
The Translational Part The integral for the translational part can be evaluated to give
where q = 4np1/1z and p is the momentum. Note that this expression is.independent of V,, so that I remains the only parameter required. The remaining problem is to average PI; over the range of velocities wherein collisions occur in the gas. For a Maxwellian velocity distribution, an analytical expression can be derived, provided that the energy exchange is fairly large (1.e. the process is not close to “resonance,” A&= 0). The resulting expression is
( )’
p!r= ( 2~)”~ AE [(Ac)’ If nKTo K
where
-1
TTo
[
exp - -3 [(A&)’ 1 2 K T T ,
1”’
--A & ]
2KT ’
21
CHEMICAL LASERS
The Steric Factor The steric factor takes into account the three-dimensional nature of real encounters (not one-dimensional as in the above theory) and the fact that the probability of interaction of the incoming A with the molecular vibration will be reduced for other than collinear approaches. For a diatomic molecule BC, Po has been calculated to be about 1/3. The above discussion has concerned a V-T energy exchange and the general usefulness of the theory in calculating orders of magnitude is shown by the data assembled by Herzfeld and Litovitz (35), with the exception of dipolar molecules or molecules likely to form chemical complexes during collisions. For V-V exchange between two molecules AD and BC, the same approach can be employed. The oscillator part becomes
*
v1(vz
+ 11,
(22)
where the v1 level of AD and the v 2 level of BC become (ul - 1) and (u2 + l), respectively. The oscillator part is less than for a V-T exchange but the translational part is much larger because AE,the energy which must be converted to the translational form, is greatly reduced. Equation (21) applies with
AE = h( v’& - VZ;
I)
- h( ~2~ - v::
’).
The sign of the second term of Eq. (21) then depends on whether the process is endothermic or exothermic. For two diatomic molecules Po N 1/9. Although only the short range repulsive potential is considered in the theory, some account of other forces can be made through the choice of the parameter I to best fit a known potential (such as the Lennard-Jones 6-12 intermolecular function). Although the choice of I is important, (dp/p N -EM//), a value of 0.2 8, is frequently used for order of magnitude calculations. Figure 11 shows some calculated and experimental values for the process (43) CO(u = 0 )
+ CO(u
=
u’)
-
CO(u = 1)
+ CO(u =
1” -
1).
Curve D is derived from the above SSH theory using I = 0.2 A. Curve E is a modification of the SSH theory, removing the approximation in integrating Eq. (16) by performing a direct integration and numerically averaging the translational part over all velocities to avoid the divergence close to resonance (v = 1). The agreement with the experimental results (curves A and B, to be discussed later) is much improved. Figure 12 compares interaction of several different vibrational levels calculated in the same manner.
22
P. H. DAWSON AND G . H. KIMBELL
10-2
P
lo-:
lo-’
lo-!
5
10 V’
FIG.1 1 . Calculated and experimental probabilities for vibration exchange per collision for the processes CO(u = 0)
+ CO(u
=
u’)
-
CO(u = 1 )
+ CO(u = u‘
-
1).
Curves A and B are experimental values by different workers (39,40).Curve D is given by SSH theory, which considers short-range forces, curve E by a modified SSH theory (43). and curve C by a modified Sharnia theory (39) which considers long-range forces.
The weaknesses of the SSH theory lie in the neglect of the rotational energy mode and the neglect of long range forces. Attempts have been made to consider these factors by other methods. Cottrell and Matheson (44) suggested that rotation to vibration transfer can be significant in the excitation of hydrogen containing molecules which tend to have very fast excitation and relaxation (45) (“the light atom anomaly”). Mahan (46) postulated that for
23
CHEMICAL LASERS
P
c
I
,
I
0
5
10 "I
FIG.12. Probabilities of some vibration-vibration energy exchanges of the type
CO(0 = n,) i-CO(0 = n 2 )
_3
CO(v = 12, - 1)
+ CO(0 = nz i -1)
calculated by a modified SSH theory (43).
infrared active vibrations and near-resonant vibrational exchange the interaction could occur by means of long range dipole-dipole coupling. This was developed further by Sharma and Brau (47,48) in calculations for N, and the asymmetric stretching mode of CO,. At temperatures below 1000°K the long range interactions dominated. It was necessary to also include the rotational transitions because of the strong angular dependance of the dipolequadrupole interaction. The agreement with experiment was good below about 1000°K but above this probabilities increased with temperature, as predicted by SSH theory, and it seems that the short range forces again become important. A modified version of Sharma's theory has been applied to the CO V-V exchange problem by Hancock and Smith (39) again taking into account simultaneous rotational transitions (AJ = ? 1 for dipole-dipole coupling). As shown by curve C in Fig. 12, the agreement with experiment is excellent close to resonance but rapidly deteriorates at high u numbers. Yardley (49) has extended the long range interaction theory to nonresonant exchanges. Further attempts to better approximate the full interaction between colliding vibrationally excited species, perhaps including both long
24
P. H. DAWSON AND G. H. KIMBELL
and short range forces are to be expected, especially now that experimental data are becoming available. Hancock and Smith (39) have recently determined rates of energy transfer between CO in u = 4 to 13 with ground state CO, OCS, NO, N,,N 2 0 , and CO, . Suart, Arnold, and Kimbell (40) have made similar measurements for CO, OCS, and N,O. There is a disagreement over the absolute values, Suart et al.’s values being five to ten times lower (for example, curves A and B, Fig. 11) and showing less dependence on the u level of the CO. However the general trends are clear. Table 111 shows some of the results of Hancock and Smith. The experimental method is to observe the infrared chemiluminescence of CO formed in the reaction between 0 and CS with and without the presence of the added gases. The experiments are best carried out at low reactant pressures or extrapolated to low reactant pressures to allow for self-quenching effects. Wall effects are a problem in interpreting the measurements but can be minimized by the presence of a large excess of buffer gas such as argon. However, the interpretation in terms of rate constants is based on an assumption of homogeneity within the observed reaction volume, and it may be the evident violation of this assumption which gives rise to the differences between different experimenters. The simplest way to examine the kind of effects that are caused by relaxation in a laser flow is to carry out computer simulations of the evolution of the populations of the various vibrationally excited states. The CO formed in the CS,-0, laser is a good example because levels up to u = 14 may be initially populated by the reaction. Figure 13 shows (43) how an initial population (arbitrarily assumed) evolves with time when (a) CO-CO vibrational exchange is considered and (b) when 1 Torr of CO in the ground vibrational state (i.e. “cold” CO) is added to the mixture. The theoretical values such as those in Fig. 13 were used in these computations. In these simulations, stimulated emission, which would considerably modify the distributions, has not been included. However, one can clearly see the importance of the relaxation processes in three factors: ( I ) strong interactions between neighboring levels which quickly smooth out any fine structure in the initial population distribution, (2) the time for which the population inversion can be maintained and therefore, perhaps, the upper pressure limit for lasing action, and (3) the modification of the distribution and the enhancement of population inversion by the selective relaxation of the lower levels such as by the addition of “cold” CO. In fact, in the CS,-0, laser, an excess of oxygen is usually also present. The oxygen tends to de-excite levels above u = 12 and, with the added CO de-exciting those below u = 5, this gives an enhancement of lasing action in the intermediate region. This double use of relaxation effects may be of more general value in lasing systems.
TABLE I11 VALUESOF P ","- l a
v1 , d c n i - ')
V
4 5 6 7 8 9 10 11
12 13
FOR
DE-EXCITATION OF CO, BY He, CO, NO, 0, , N z , OCS, NzO, AND C02'
2143.3
1876
1556.2
6.7 ( - 3 y 4.3 (-3) 2.1 (-3) 1.1 (-3) 5.9 (-4) 3.8 (-4) 2.0 (-4) I .53( - 4) 1.1 3( -4) -
1.9 (-3) 2.9 ( - 3) 4.0 ( - 3) 7.2 (-3) 1.47(- 2) 2.1 (-2) 2.7 (-2) 3.0 (-2) 3.2 (-2) 2.8 (-2)
-
2330.7
v,(cm-')
2062
2224
2349
v,.,,,-,(cni-') 2064.0 2037.7 201 1.5 1985.4 1959.3 1933.6 1907.2 1882.7 1856.1 1829.9
c, -
-
4.2 (-7) 6.2 (-7) 9.1 (-7) 1.4 (-6) 1.67(-6)
f'E,v-l is the probability of de-exciting CO, According to Hancock and Smith (39). '6.7(-3)~6.7 X
--f
-
7.1 (-5) 1.5 (-4)
CO,-l per collision.
7.9 (-6) 5.6 (-6) 3.7 (-6) 2.9 (-6) 2.4 (-6) 1.7 (-6) 1.3 (-6) 7. (-7)
1.1 (-1) ~ 1 . 5(-1) 1.4 (-1) 9.0 (-2) 3.6 (-2) 1.79(-2) 9.9 (-3) 7.8 (-3) 4.8 (- 3) 2.5 (-3)
1.6 7.7 4.3 2.9 2.6 2.6 2.9 3.2 3.0 2.7
(-3) (-4) (-4) (-4) (-4) (-4) (-4) (-4) (-4) (-4)
1.0 (-4) 5.6 ( - 5 ) 5.4 (-5) 5.1 (-5) 5.4 (-5) 7.0 (-5) 1.12(-4) 1.75(-4) 2.9 (-4) 4.0 (-4)
8
i>3 r
r
ii
z
P. H. DAWSON AND G . H. KIMBELL
A = 0 I msec
8 = 0 5 m sec
C = I 0 msec \
0 : 2 5 msec E
:4 0
msec
I
V
(a)
FIG. 13(a) FIG. 13. (a) Computer simulation (43) of the evolution of the vibrational energy distribution of CO molecules. Allowance has been made for spontaneous mission and for the presence of 1.25 Torr of oxygen (as is likely in a CS2-02 laser system). The main process of interest is the V-V energy exchange and values calculated as in Fig. 12 were used. An initial distribution was arbitrarily assumed. (b) The same conditions as in (a) but with the addition of 1 Torr of "cold" ( u = 0) CO which undergoes V-V exchange more rapidly with the lower vibrational levels than the upper.
Detailed computer simulations of chemical laser systems including rates of reaction and V-T and V-V processes have been reported for the H,-CI, (50) and the CS,-02 (51) systems. VI. CHARACTERISTICS FAVORABLE TO CHEMICAL LASERACTION Before considering some of the particular achievements in obtaining laser action, it is valuable to summarize the sometimes conflicting requirements for systems favorable to chemical laser action as they have emerged from the earlier discussion. 1. The reaction must be exothermic.
27
CHEMICAL LASERS
A
= 0 I rnsec
8 = 0 5 rnsec C = IOrnsec
D = 25rnsec E = 4 0 rn sec F = 7 0 rnsec I n i t i a l distilbullon
,4--A --,
\
-
I I
\
I
5
"
I0
(bi
FIG.l3(b). See Fig. 13(a) for legend.
2. The reaction must be fast enough to maintain an inversion. It must have a low energy barrier and low steric factors. Atom abstraction reactions A - I BC
-
ABt C
are particularly likely to be useful. 3 . (a) Vibrational excitation of the products is favored if the energy barrier ir i n the approach coordinate (the most exothermic reaction in a related family of reactions), i.e. A is strongly attracted to BC. (b) Energy is more likely to be in the vibrational form if M*
=M,
% M,.
28
P. H. DAWSON AND G . H. KIMBELL
4. The excited levels must not be too rapidly relaxed, e.g. the vibrational levels of diatomic and triatomic molecules are good. 5 . The oscillator strength of the transition should be high. 6. Lasing is easier in conditions of partial inversion for molecules with small moments of inertia. This is important in diatomics where the energy tends to equilibrate over the available vibrational levels (rapid V-V transfer). The rotational temperature must also be kept low. 7. Concentration of the vibrational excitation into a few Ievels, e.g. the use of energy transfer to, or chemical production of, linear triatomic molecules may improve the chemical efficiency. 8. The presence of a chain reaction is useful. Even with a small external energy, output may then be large. 9. Suitable selective vibrational exchange reactions with added species can enhance the population inversion. 10. The high vibrational levels have larger stimulated emission coefficients and will give a greater gain for a given population inversion. The aim, of course, is to produce lasing systems of high chemical and electrical efficiency, operating at useful power levels at pressures near atmospheric to obviate the need for bulky pumping systems. Many of the characteristics listed above (fast exothermic reactions, chain reactions) seem most likely to be found in gas mixtures which are self-igniting under certain pressures and temperatures. Indeed lasing action has been recently observed in a freeburning flame (see below). One conflicting requirement is then the desirability of maintaining a low rotational temperature. Some of the reactions so far employed in chemical lasers are listed in SOMEOF
THE
Reaction mixture
csz + 0
2
csoz + 0
+
2
C1tO Ht Br2 RH Clz RH
+ +
TABLE IV REACTION MIXTURES USEDIN CHEMICAL LASERS Output
Pulsed Pulsed cw cw Pulsed Pulsed cw cw Pulsed Pulsed cw Pulsed Pulsed Pulsed Pulsed
Initiation
Power
Transverse discharge ca. 1 kW hv Low 3w Discharge Flame ca.1 mW hv Low Discharge High (>1 MW) Discharge Low > I kW hv Low hv Low Discharge Low Discharge Low hv Low Discharge Low Discharge Low
29
CHEMICAL LASERS
Table I V together with the nature of the output (pulsed or cw), the means of reaction initiation (photolysis, discharge), and in significant cases the power output. Photodissmociation and photoelimination lasers are not included in the table.
V11. SOMELASERSYSTEMS The following examples have been chosen to cover a broad spectrum of the experiments to date, emphasizing particular acievements in power output, various methods of initiation, and novel features such as spontaneity of reaction and use of energy transfer. A . A True Chernical Laser
Cool and Stephens have achieved cw chemical laser operation merely by fluid mixing (52). Figure 1 shows the type of apparatus. A source of fluorine atoms was provided by mixing NO with a flow of F, and helium by using the reaction Fz -t NO
-
NOF t- F.
This reaction is fairly rapid at ambient temperature. The combined flow is rapidly mixed with deuterium and carbon dioxide to produce vibrationally excited DF by the chain reaction F f Dz D+Fz
-
___*
DF*$ D, DF*+F.
These reactions aire capable of populating vibrational levels of D F up to u = 4. There is a subsequent intermolecular transfer of energy to the CO,. That is, DF(u
= n)
+ CO,(OO"O)
-
DF(u = n - 1 )
+ C02(Oo"1),
and this is followed by the "classical" CO, laser operation at 10.6 p between the (00O1) and (lOO0) levels. A power of 8 W (53)has been achieved with flow rates of He = 3800 pm/sec, CO, = 1550 pM/sec, D, = 370 pM/sec, F, = 365 pM/sec, and N O = 24 ,uM/sec and a chemical efficiency (coefficient of conversion of chemical energy to laser radiation) of 4 % .The reaction tube of the type shown in Fig. 1 was a 21 cm long, 9 mm bore Teflon tube. All interior wall surfaces of the laser were coated with H,BO, (54) to minimize atom recombination. Figure 14 shows how the laser output was related to the flow rates of the various components, the flow rates having been normalized by the optimum value for each gas. Detailed kinetic analysis of these results is not presently available.
30
P. H. DAWSON AND G . H. KIMBELL
FLOW RATE IN RELATION TO ITS OPTIMUM
+
FIG. 14. Power output (52) of the pure chemical laser: NO F 2 + N O F + F, F + D2+ DF* t D, D -tFZ+ DF* -1 F, DF(u = n) COz(OOoO)-+ DF(u = n - 1) COz(OOol),as a function of partial flow rates of each gas.
+
+
B. A Flame Laser It was recently reported (55) that, in a free-burning flame of CS2 and 0 , at low pressures (of the order of 1 Torr), population inversions occurred on vibrational-rotational bands of CO. This was shown by measuring the distribution of energy in the first overtone region of CO. Thus these workers predicted the use of this system in the establishment of a true free-burning flame laser.
31
CHEMICAL LASERS
In a recent communication workers at N R L (56) have succeeded in achieving laser action from this system. The burner consisted of a horizontal array of 24 parallel tubes. Each tube was 60 cm long, 6 nim o.d., and had 50 evenly spaced 1 mm holes along the top. A low loss cavity was formed by two dielectrically coated mirrors mounted internally within the vacuum system. Continuous wave laser action was observed at 5.216 11, u = 8 + u = 7 P(11); 5.297 p , u = 9 -+r = 8 P(12); and 5.421 p , u = 1 1 -+zi = 10 P(10). Total output power was on the order of ImW.
C. A
CIV
Carbon Monoxide Laser
The cw CO chemical laser depends on initiation of reaction by production of oxygen atoms in a microwave discharge and a subsequent series of reactions with CS, . A longitudinal gas flow apparatus, similar to that of Fig. 1, has given a power output of 2.3 W (57) with a chemical efficiency of at least 1 and an electrical efficiency of about 0.65 %. The gas mixing, accomplished here by admitting the CS, through a radial array of 12 holes each 0.5 mm diameter downstream of the O,/He inlet, has to be rapid to obtain high power. The 2.3 W was only obtained with the addition of vibrationally “cold” CO to enhance the population inversion as discussed in Section VI. The analysis of the spectral output showed the complete inversion of several vibrational levels. The observed transitions were the P(8) to P(11) bands of the 7 + 6 transition, P(13) to P(7) of the 8 - + 7 transition, P(13) to P(8) of the 9 -+8 transition, P(13) to P(7) of the 10 - + 9transition and P(11) to P(7) of the 11 -+ 10 transition. The strongest line in the presence of the added CO was the P(8) branch of the 8 + 7 transition. At maximum power output the gas flow rates were 67 liters/min for He, 6.8 liters/niin for O,, 1.1 liters/min for CS,, and 3.4 liter,s/min for added CO giving a total pressure of 9 Torr. The helium helps in the oxygen dissociation by the microwave discharge and serves to maintain a thermalized rotational distribution (Section 111). The mechanism of reaction has been analyzed in some detail (57). The essential steps are as follows: Rate constants
o+csz -+ cs -1- so 0-I-cs -+ CO*-tS S t - 0 , - so + o O,+CS -+ co +so so I-so szo2
2.1 x 2.1 x
S 2 0 - t 0,
--b
-
SO+SOZ
1013
1.2 x 10’2
-b
--b
(cm3 mole-’ sec-’)
109
s2o+soz
32
P. H. DAWSON AND G. H. KIMBELL
The reaction between 0, and CS has been suggested as an alternative pumping step but the evidence from analysis of flash photolysis experiments makes this seem unlikely. The reaction
so+o2
soz+o
__*
gives the possibility of a chain reaction but it seems that this only becomes important at temperatures above 500°K. The oxygen atoms are competed for by the CS radicals formed in the first step and by CS,. If CS, is in excess, some of the oxygen atoms are used to create CS which can no longer be converted to CO except by the slow reaction with 0,. The conversion efficiency is reduced by the excess CS,. Figure 15 shows relative power as a function
I N I T I A L 0 ATOM CONCENTRATION
OL 0
t 1
1
I
I
I
2
[CSz]
I
4 X I 0 - O ADDED
I
6
FIG.15. [CO] formed or relative power as function of [CS,] added. Curves (a) and (b) show the result of varying the rate constant for the pumping reaction 0 $- CS + CO I- S; curve (c) the relative power (58); curve (d) the relative power (59); curve (e) neglects the SO. above reaction and assumes a rate constant for CS O2 +CS
+
+
of the CS, present as observed by Jeffers and Wiswall (58) and by Suart, Kimbell, and Arnold (59). Also shown is a prediction of the extent of formation of CO in a lasing condition given by a computer simulation of the chemically reacting system (57). D . A Transversely Sparked HF Laser
The use of transverse multiple-spark discharges to initiate lasing action was developed for the CO, “physical” laser and later applied to CS,/O,/He
33
CHEMICAL LASERS
mixtures (60,6/). A similar electrically pulsed laser operating with niixtures of CF,, C2F6, C4Fe, or SF, and H,, CH,. C2H6, C,H,, or C4H,, will function even at atmospheric pressure (62). Figure 16 shows the apparatus. The discharge cavity was a 5 cm diam pyrex tube. Premixed gases enter at one end of the tube and are exhausted at the other. The electrodes were parallel monel rods (1.3 cm diam and 1.2 m long) and the high tension electrode had 200 monel pins oriented to form a transverse electrode array with an interlectrode gap of 1.3 cm. The spark discharges were obtained from a 0.01 pF capacitor charged to 30 kV by a dc power supply, the initiation being triggered by a spark gap.
RESISTANCE CHAIN INITIATING SPARK GAP RGlNG VOLTAGE CAPACITANCE CROSS -SECTION OF T U B E
FIG.16. A transverse discharge laser apparatus.
At pressures less than 50 Torr, the addition of helium did not greatly affect performance but it was necessary at higher pressures to obtain lasing action. The maximum observed powers were greater than 0.5 MW. The maximum pulse energy was 40 mJ for a SF,/C,H,/He mixture in the ratio 6.4/1/78 at a pressure of 115 Torr and using a cavity formed by a concave mirror and a 35 7,;reffecting flat. This gas combination was superradiant (no mirrors on the cavity) at pressures between 20 and 760 Torr. At atmospheric pressure the optimum peak power was about 30 kW. The greatest number of the observed transitions were in the L' = 2 + r = 1 series although the I -0 and 3 + 2 transitions were also present. The pumping reaction is F
+ RH+
HF* + R
1 HF 2.8 pm laser radiation
The maximum gain was for J = 4 indicating that the gain (and population inversion) was very high.
34
P. H. DAWSON AND
G . H. KIMBELL
E. A Flash-Induced OH Laser The formation of OH(u s 2) in the flash photolysis of O,, H,, and N, mixtures was reported by Basco and Norrish (63). Photolysis of ozone in the ultraviolet produces O(’D) atoms which react as follows : O(’D)+H,
-
-
OH*+H.
There is then a possible further reaction H+01
OH*+02.
The latter reaction has been studied at low pressure Torr) by infrared chemiluminescence by Polanyi et al. (27), who find that the OH is formed predominantly in the u = 8 and u = 9 levels (Fig. 7) but that even at Torr there is a rapid relaxation by vibrational exchange or chemical reaction (see Section 111). However, levels up to u = 9 have recently been observed by EPR at higher pressures (64), so that the possibility of using this reaction remains open. Recently Callear and Van Den Bergh (65) have observed stimulated emission from OJH, mixtures in the ratio of 1 :10 and at pressures of 1 to 10 Torr. Flash energies were in the range of 400-2000 J. The peak power output was quite low. Emission was observed from the 3 +2, 2 + 1, and even the 1 4 0 fundamentals. The observed frequencies are about 3200 cm-’. Flash photolysis laser studies, such as this, are particularly important in studying new potential lasing systems.
F. Transverse Flow Lasers The emphasis in cw lasers has now switched to transverse flow systems instead of the longitudinal flow described in Section VII, A and C above. That is, the gas flow is made transverse to the optical axis. Such systems have been described by Jeffers and Wiswall(58) and by Cool, Shirley, and Stephens (66). A recent design by Foster and Kimbell (67) is shown in Fig. 17. In this device oxygen atoms from a microwave discharge are mixed rapidly with carbon disulphide which is injected into the flow through a series of tiny teflon tubes. Combustion then occurs, rapidly producing CO, the laser medium. The two rows of tubes are inserted to allow for the possibility of the addition of a vibrationally “cool” gas to enhance the inversion as described earlier. The advantage of the transverse flow is that the “spent” gases are rapidly removed from the optical field of view, avoiding undesirable absorption effects, and measured output powers can be several times greater than equivalent longitudinal flow systems. In suitable cases, the normal cavity volume can be extended by the use of a multiple reflecting system of mirrors (66). A fivefold
35
CHEMICAL LASERS
PURGE
I
A
BREWSTER WINDOWS
OPTICAL AXIS
.................... .................... csz 7
"TOTAL" REFLECTOR (1.3m r o d of curvature)
ADDED GAS --I'
iDISCHARGED
Ge FLAT
(98% reflecting 05 . 2 ~ )
He, 0, MIXTURE
FIG.17. A transverse flow laser system.
configuration has been used with two plane mirrors of 99.4% reflectivity, one 10 m radius of curvature niirror with 99.4'j, reflectivity, and, for decoupling, a partially transmitting mirror of 10 ni radius of curvature. This device using the DF-CO, chemical system has produced 167 W with a chemical efficiency of 4.6 %. The transverse flow system should also provide a useful diagnostic technique since the cavity can be moved to different positions along the flow (different times after mixing), and used to examine the relaxation processes that are occurring.
VIII. CONCLUSION The chemical laser represents a method of obtaining new laser frequencies; in conventional molecular gas lasers one is restricted to molecular gas mixes wherein population inversions of the parent molecules can be sustained during electrical discharge. In a very short time, rapid progress has been made, as is demonstrated by the development of the purely chemical laser (no external power source), the free-burning flame laser, and several cw lasers. All this despite an evident lack of fundamental knowledge of many of the processes involved. These lasers exploit the chemical nature of the excitation process and hold the promise (or dream?) of portable, efficient, laser sources. Further progress, i n the direction of higher efficiencies and operation at higher pressures, will undoubtedly be slower and may involve the exploration and utilization of more complex chemical systems.
36
P. H. DAWSON AND G. H. KIMBELL
There is, however, considerable promise in the exploitation of what might be termed “hybrid” systems such as the transversely sparked H F laser described above which already gives high peak power and atmospheric pressure operation. Another possibility is integration of the flame chemical laser in a gas dynamic type of laser system. In any case, the technology is unlikely to progress without further advances in our knowledge of the chemical creation of excited species and the subsequent loss and exchange of the excitation energy.
REFERENCES 1. C. K. W. Patel, “Lasers” (A. K. Levine, ed.), Vol. 2,p. 1. Dekker, New York, 1968. 2. A. L.Schawlow and C . H. Townes, Phys. Rev. 112,1940(1958). 3. A. Javan, W. R. Bennett, and D. R. Hehiot, Phys. Rev. Lett. 6 , 106 (1961). 4. A. J. Beaulieu, Appl. Phys. Lett. 16,504 (1970). 5. J. C.Polanyi, J . Chem. Phys. 34, 347 (1961). 6. Appl. Opt. Chem. Laser Suppl. 2 (1965). 7. J. V. V. Kasper and G . C . Pimentel, Phys. Rev. Lett. 14, 352 (1965). 8. T.F.Deutsch, AppL Phys, L e f t . 11, 18(1967). 9. D. J. Spencer, T. A. Jacobs, H. Mirels, and R. F. Gross, Int. J . Chem. Kinetics 1, 493
(1969). 10. T. A. Cool, R. R. Stephens, and T. J. Falk, Int. J . Chem. Kinetics 1,495(1969). 11. R. D.Suart, G . H. Kinibell, and S. J. Arnold, Chem. Phys. Lett. 5, 519 (1970). 12. T.A. Cool and R. R . Stephens, J . Chem. Phys. 51, 5175 (1969). 13. G. C.Pimentel, Sci. Amer. 214, 32 (1966). 14. A. C.G . Mitchell and M. W. Zernansky, “ Resonance Radiation and Excited Atoms.” Macmillan, New York, 1962. 15. T. F. Deutsch, Appl.Phys. Lett. 11, 18 (1967). 16. G. Herzberg, “Spectra of Diatomic Molecules.” Van Nostrand, Princeton, New Jersey, 1950. 17. E. A. Gerry, BuIl. Amer. Phys. SOC. 15, 563 (1970). 18. D . W. Gregg and S . J. Thomas, J . Appl. Phys. 39, 4399 (1968). 19. P. J. Kuntz, E. M. Nemeth, J. C. Polanyi, S. D. Rosner, and C . E. Young, J . Chem. Phjx 44, 1168 (1966). 20. J. C . Polanyi and W. H. Wong, J..Chem. Phys. 51, 1439 (1969). 21. M. H. Mok and J. C . Polanyi, J . Chem. P ~ J J51, S . 1451 (1969). 22. D.L. Bunker and N. C. Blais, J . Chem. Phy~..41,2367 (1964). 23. M. Karplus and E. M. Raff, J . Chem. Phys. 41, 1267 (1964). 24. C. C.Rankin and J. C . Light, J . Chert. Phjlv. 51, 1701 (1969). 25. D. Russell and J. C . Light, J . Chem. Phys. 51, 1720 (1969). 26. K. G. Anlauf, P. J. Kuntz, D. H. Maylotte, P. D. Pacey, and J. C . Polanyi., Disc. FaradaySoc.44,183(1967). 27. P. E. Charters, R. G . Macdonald, and J. C . Polanyi, Appl. Opt. 10,1747(1971). 28. K.G.Anlauf, J. C. Polanyi, W. H. Wong, and K. B. Woodall., J . Chem.Phys. 49, 5189 (1968). 29. H. Heydtmann and J. C . Polanyi, Appl. Opt. 10, 1738 (1971). 30. N.Jonathan, C. M. Melliar-Smith, and D . H . Slater, Mol. Phys. 20, 93 (1971). 31. P. D. Pacey and J. C . Polanyi. To be published.
CHEMICAL LASERS
37
32. M. S . Dzhidzhoev, V. T. Platonenko, and R. V. Khokhlov, Sou. Phys.-Usp. 13, 247 (1970). 33. C. K. W. Patel, Phys. Rev. 141, 71 (1966). 34. K. Cashion, J . Mol. Spectrosc. 10, 182 (1963). 35. K. F. Herzfield and T. A. Litovitz, “Absorption and Dispersion of Ultrasonic Waves,” Academic Press, New York, 1959. 36. J. D. Lanibert, Quart. Rev. Chern. Soc. 21, 67 (1967). 37. A. B. Callear, ‘’ Photochemistry and Reaction Kinetics,” Chapter 7. Cambridge Univ. Press, London and New York, 1967. 38. R. C. Millikan, J . Chem. Phys. 38, 2855 (1963); 43. 1439 (1965). 39. G. Hancock and I . W. M. Smith, Chem. Phys. Lett. 3, 573 (1969); Appl. Opt. 10, 1827 (1971). 40. S . J. Arnold, R . D. Suart, and G. H. Kimbell, private communication. 41. A. B. Bhatia, ‘‘ Ultrasonic Absorption.” Oxford Univ. Press (Clarendon), London and New York, 1967. 42. R. N. Schwartz.. Z. I . Slawsky, and K . F. Herzfield, J . Chem. Phys. 20, 1591 (1952). 43. P. H. Dawson and W. G. Tam, Can. J . Phys. 50, 889 (1972). 44. T. L. Cottrell and A. J. Matheson, Trans. Faraday Soc. 59, 824 (1963). 45. M. G. Ferguson and H . W. Read, Trans. Faruday Soc. 63, 61 (1967). 46. B. H. Mahan, J . Chem. Phys. 46, 98 (1967). 47. R. D . Sharma and C . H. Brau, Phys. Reo. Lett. 19, 1273 (1967). 48. R. D. Sharnia and C. H. Brau, J . Chem. Phys. 50, 924 (1969). 49. J. T. Yardley, J . Chem. Phys. 50, 2464 (1969). 50. N. Cohen, T. A . Jacobs, G. Emanuel, and R. L. Wilkins, Znr. J . Chem. Kinetics 1, 555 (1 969). 51. B. R . Bronfin, Conf. Mol. Energy Transfer, Cambridge, England, 1971. 52. T. A. Cool and R. R. Stephens,J. Chem. Phys. 51, 5175 (1969). 53. T. A. Cool and R. R. Stephens, Appl. Phys. Lett. 16, 55 (1970). 54. E. A. Ogryzlo, C a n . J . Chem. 39,2556 (1961). 55. K . D. Foster and G . H. Kimbell. J . Chem. Phys. 53, 2539 (1970). 56. H . S. Pilloff, S. K. Searles, and N. Djeu, Appl. Phys. Lett. 19, 9 (1971). 57. R. D. Suart, P. H. Dawson. and G. H . Kimbell, J . Appl. Phys. 43. 1022 (1972). 58. N. Q. Jeffers and C. E. Wiswall, Appl. Phys. Lett. 17, 67 (1970). 59. R. D. Suart, G. H. Kimbell, and S . J. Arnold, Chem. Phys. Lett. 7, 337 (1970). 60. T. V. Jacobson and G . H . Kimbell,J. Appl. Ph.vs. 41, 5210 (1970). 61. M. C. Lin and W. H . Green, J . Chem. Phys. 53. 3383 (19701. 62. T. V. Jacobson and G. H. Kimbell. J . Appl. P h j 42, ~ 3402 (1971). 63. N. Basco and R . G. W. Norrish, Proc. Roy. Soc. Ser. A , 254, 317 (1960). 64. K. P. Lee, W. G. Tam, R. Larouche, and G . A. Woonton, Can. J . Phys. 49, 2207 (1971). 65. A. B. Callear and H. E. Van Den Bergh, Chern. Phys. Lett. 8, 17 (1971). 66. T. A . Cool, J. A . Shirley, and R. R. Stephens, Appl. Phys. Lett. 17, 278 (1970). 67. K. D. Foster and G. H . Kimbell, to be published.
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Single Photon Detection and Timing: Experiments and Techniques SHERMAN K. POULTNEY Departnient of Physics and Astronomy, University of Maryland, College Park, Maryland
I. Introduction
...................
. . . . . . . . . . . . . . . . . . . 39
11. Single Photon
C . Photodevice and Background Noise and Their Reduction D. Practical Photodevice Techniques and Photon Detection
C. Single Photon Timing Methods.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Single Photon Precise Timing Experiments. . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
I. INTRODUCTION Single photon detection and timing means essentially the detection and timing of a single photoelectron released by light from a photosensitve surface of a photomultiplier, channel multiplier, avalanche multiplier photodiode, or other photodevice. An electron released by a particle or photon bombardment of a windowless multiplier is also thus included. Single photoelectron detection requires sufficient low noise amplification to overcome the Johnson noise in the first resistor of the eventual electronic circuit plus additional amplification to operate standard discriminator detection circuits. The time response is usually quite fast due to the desire to count at both low and high rates, to time closely spaced events, or to time intervals to highest precision, 39
40
SHERMAN K. POULTNEY
and so most of the gain usually comes before the electronic circuits. This gain is obtained by secondary electron emission processes or their equivalent down a chain of electrodes or a continuous channel and is subject to statistical fluctuations. The resultant fluctuations in the output charge can affect timing with single photoelectron pulses, the length of time needed to achieve a light flux measurement to a given precision, the efficiency of single photoelectron detection, and the discrimination against some types of device noise. The major type of device noise has a similar output charge distribution, however, and this dark noise must be reduced by cooling the photodevice or by other techniques in most weak light applications. Background light noise also has a similar distribution and must be reduced when present by narrow spectral and spatial filters. Correct technique can help minimize device noise, but is often neglected. It is important to have standard experimental tests to evaluate the single photon counting performance and the noise performance of photodevices. Use of certain photoemissive materials and reduced photosurface size can also reduce device noise. However, for single photon counting, the choice of material is usually dictated by quantum efficiency considerations at the wavelength of interest. A number of opaque, reflection-mode photosurfaces are now available which extend the spectral range of high sensitivity. The older transmission-type photosurfaces show worthwhile increases in sensitivity by means of external and internal quantum efficiency enhancement techniques. The total quantum counting efficiency of a photodevice can often be lower than the photosurface quantum efficiency. The discrepancy may be caused by those photoelectrons that fail to get collected by the multiplier or by those anode pulses that fail to exceed the discriminator threshold due to the multiplication statistics. A low counting efficiency is just as serious as a low photosurface quantum efficiency and should be evaluated for any photodevice under consideration. Experiments involving single photon detection are usually photometric or spectrophotometric measurements of a weak light beam. The weak intensity may be due to scattering with small cross sections as in laboratory studies of Raman and Brillouin lineshifts or due to weak astronomical sources being viewed directly or through a many-channel spectrophotometer. The length of time needed to obtain a given precision is limited by background noise, device noise, or signal shot noise depending on which cannot be reduced. Digital storage of standardized discriminator pulses has both theoretical and practical advantages over other storage and detection methods. If the light source can be effectively modulated, digital synchronous detection can be used at least for convenience and stability in much the same manner as analog synchronous detection. If photometry over a wide range of intensities is required, the single photon counting method will be limited by photodevice or circuit time responses or dead times.
SINGLE PHOTON DETECTION AND TIMING
41
A second class of experiments requires both single photon detection and precise (nanosecond or better) timing. Timing of this precision depends on both the photodevice and the arrival time detection circuit. The photodevice must be designed to minimize electron transit time spreads; especially between the photosurface a.nd multiplier and in the first stages of the multiplier. The timing capabilities and limitations of some representative photomultipliers are examined closely in order to understand the timing capabilities and limitations of improved photomultipliers and other new photodevices. The arrival time detection circuit should be of the constant-fraction-of-pulse-height type or its equivalent to minimize any timing spread due to the multiplication fluctuations of tht: photodevice output signal. A photodevice which itself reduces these fluctuations in some manner also helps to minimize this time walk. A short time interval between two events of interest is best measured by sending the now standard timing pulses to a time-to-pulse-height converter which can be interrogated by a suitable analog-to-digital converter. The time distribution of intervals between repeatable physical events can be stored in a multichannel pulse height analyzer for on-line or later use. If the time interval becomes too long 10 obtain the desired resolution with a time-to-pulse-height converter, either the time-to-pulse-height converter has to be used in conjunction with a suitable digital time-interval meter or another timing method must be used. Methods for measuring the timing precision and accuracy of photodevices with single photons are outlined with special emphasis on the necessary fast light pulses. Methods of calibrating and monitoring the stability of the ti.ming circuits are also outlined. Typical photon timing experiments are the measurement of atomic and molecular fast fluorescence decay times and the lunar ranging experiment. The former measurement often has plenty of light signal, but this signal is attenuated to single photon levels in order to measure the fast decay time statistically so that it is limited only by the photodevice transit time spread. The width of any short light pulse can similarly be examined. Reference to the large body of work at higher light levels with scintillators is made since this work sets the basis for the terminology, theory, and past performance of photodevices for fast timing. In the lunar ranging experiment, the very low signal cannot be significantly increased. In spite of the high signal attenuation, high background noise, and difficult pointing problem, nanosecond timing of a 2.5 sec interval is now being done at the single photon level with current nanosecond laser transmitters. Ultra:short pulse lasers will leave the photodevice as the limiting factor in achieving the goals of that experiment. A third class ol' experiments requires single photon detection in addition to moderate timing capabilities. The time information may be added to the light signal by effectively modulating it in order to employ synchronous detection to aid in the separation of signal from noise. Optical radar is a
42
SHERMAN K. POULTNEY
pulsed version of this modulation and can be used for probing of the structure of the atmosphere, for example. The measurement of the arrival time distribution of photons in a light beam can characterize the statistical properties of that light beam. If the beam properties are known, this measurement on scattered light can yield information about the internal behavior of a scattering medium undergoing statistical fluctuations in some property. These fluctuations may be due to particles undergoing Brownian movement in a liquid or to driven or spontaneous excitations of the medium. The spontaneous thermal excitations can be either propagating as in the case of optical and acoustic phonons which yield Raman and Brillouin scattering, respectively, or nonpropagating as in the case of density fluctuations at constant pressure which yield Rayleigh scattering. The measurement of photon arrival time is thus a valuable complement to spectroscopic studies of the above phenomena in that it is able to resolve the narrow linewidths. Typical experiments considered are the study of particles in Brownian motion and the measurement of the statistical properties of a narrowband thermal light source. The question of time-correlated photodevice noise can be studied with these same techniques. All of the above topics about single photon detection and timing cannot be treated in detail here. Wherever possible, reference is made to current reviews in the literature about individual topics. It is hoped that the following introduction will allow new workers to use these techniques in their fields of interest and will provide old hands with a current survey of the field which integrates many loose ends. 11. SINGLE PHOTONDETECTION
A . Necessary Photodevice Gain and Detection Circuits Consider an electron produced by photoemission from the photosurface of a photodevice (e.g. Fig. 1). This electron must be detected by electronic circuits in order to be of further use. It constitutes a charge pulse of 1.6 x C . Such a charge pulse is often collected on an RC network between photodevice anode and detection circuit. To be detected, the corresponding voltage pulse must exceed some preset level in the integral pulse height discriminator or single channel pulse height analyzer which then generates a standard counting pulse (I). If the time constant of the RC network can be much greater than the photodevice impulse response time, then the peak of the corresponding voltage pulse Vp = elC
43
SINGLE PHOTON DETECTION AND TIMING
t PHOTCl
TIMING CIRCUIT
.ECTRON
COUNTER
STOP
START
L
I MULTIPLIER GAIN G,
SLOW
FAST
SYNC 'IGNAL
DISCRIMINATOR
NOD*
t AMPLIFIER
REGULATED HIGH VOLTAGE
FIG.I . Schematic of single photon detection and timing with a photomultiplier. Diagram is not to scale. Co is bypass capacitor used when anode is at high potential.
which is a measure of the total charge of the charge pulse. In this case, V, wouldequal 25 nV if C is taken be the stray capacitance 40/2n p F and certainly needs amplification1 to be detected. The amplification must be of a special kind because charg,e fluctuations in the output resistance, R, of the photodevice due to Johnson noise will be the basic limitation to detection. These spontaneous charge fluctuations in R at a temperature T yield a voltage fluctuation for the RC network given by
wrm5(kT/e)(r/C)
(2) where k = 1.38 x J/"K. In order for the emitted photoelectron to be detected above Johnson noise during the observation time, it must be amplified without the use of it resistance by a charge gain G1 such that it is larger than the fluctuation charge. The necessary G, is thus given by (
=
'
GI 2 ( C / e M Tie),
(3)
where kT/e equals 0.025 V at 300°K. In the present case, G, must be greater than lo4 which yields a voltage pulse with a peak of 0.25 mV. An additional gain G, of at least 200 is needed to activate the discriminator. This gain can be provided by an amplifier for slow counting applications. Kowalski (I) discusses the many aspects of amplification, pulse-shaping, and detection. Both the RC integration method and slow electronics were developed for precise measurements of the total charge contained in the anode pulse and for operation with counting dead times of about 1 psec.
44
SHERMAN K. POULTNEY
The need to count at higher rates or to time short intervals makes it necessary to treat the photodevice anode pulse in a different manner. Neither single photon detection nor timing requires a precise measurement of charge. The RC time constant of the output circuit can then be made less than or equal to the pulse width or circuit resolution time whichever is longer. If the device is already being operated to minimize C, the time constant can only be lowered by decreasing R. The peak of the voltage pulse across RC now becomes lower than that given by (1) because it corresponds to less than the total charge. The gain of the device G1 must be raised to overcome the Johnson noise. For example, a good approximation for the anode current pulse of a photomultiplier is given by i(t) = i, exp [- ( t - h)2/21’] = (Gle/@A)
exp[-(t - h)2/21Z1, (4)
where the full width of the current pulse at half-maximum is given by 2.36 A and where h is the average transit time in the device. The voltage pulse across R of the network of Fig. 1 can be obtained with some reinterpretation from the work of Lewis and Wells (2). Ge 2c
[
V ( t ) = - exp - ( t - h )
If
RC
+
-1
(erf
2RC
[% J-
- ___ A ,RC]
v(t)is expressed as v(t>= (G,e/C)Wt, h, 1,RC),
(6)
one can display the solutions as in Fig. 2 for several 1/RC ratios. Note well the units along the abscissa. This unorthodox scale allows either A or RC to 1.21
,
I
I
,
,
I
,
,
,
-1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 t-h RC
-
FIG.2. Voltage waveform from RC network of Fig. 1 corresponding to a Gaussian current pulse from photomultiplier anode. Parameter is d2 / \ / R C . Figure after Fig. 7.3 of Lewis and Wells (2). (Courtesy of Pergamon Press.)
SINGLE PHOTON DETECTION AND TIMING
45
be varied. For RC = ,/i A, the peak voltage is 0.4 of that in (1) and GI must be 2.5 x lo4 for detection above Johnson noise. A typical 1for a fast photomultiplier is about 1 nsec (i.e. full width of 2.4nsec) so that if RC = ,/2 A the R equals 220 R for the above stray capacitance C. Standard fast detection circuitry is usually lbased on 50R so that G,must be lo5 or greater for fast detection of single photoelectrons above the Johnson noise. The standard fast integral discriminators or timing discriminators typically have thresholds of 50 mV so that the additional G, of at least 200 is still needed. For fast counting .(e.g. lOOMHz), this G, may be provided by a fast amplifier. For precise timing, it is essential that the whole gain be provided by G, of the photodevice, which here must be 2 x lo7. Discussion of the necessary fast circuits can be found in Kowalslti (I), Meiling and Stary (3), or the product literature. Precise timing with photodevices is considered in Section 111. Other constraints on single photodetection and photometric measurements are discussed here in Section 11. The above gain considerations are important when examining the suitability of new photodevices for single photon detection. They are also important as a guide to the redesign of present devices. In trying to improve count rates and timing capabilities of photomultipliers, designers now try to reduce A and, concomitantly, the stray capacitance C since R is probably fixed. The former is done by electrode and field design. The latter is accomplished by the use of a coaxial structure for the anode output which also reduces pulse distortion by eliminating electrical discontinuities (4). These reductions lead to a corresponding decrease in the necessary gain by (4) as long as the detection circuits are fast enough. Some photodevices may be subject to cooling along with the resistance R and so would also have a lower gain requirement since the Johnson noise is lowered by ( 2 ) . Operation of many photomultipliers at high gain can lead to noise problems as discussed in Section II,C. Some photodevices are subject to gain nonlinearities under certain conditions. In a photomultiplier, the gain is subject to saturation caused by the basic limitation of spacechargeeffectsin the final stages (5,6).This gain saturation limits the peak anode current. Am increase in the applied high voltage then increases the total charge gain and so broadens the pulse width. If gain saturation is reached before the level of single photon detection, one can sacrifice speed for gain by using an amplifier G, in conjunction with a lower G , or by using a larger RC. If sufficient gain G, is available and maximum count rates are not required, saturation may have a beneficial effect on timing precision as mentioned in Section II1,B. At high gains thcre is also the danger that the high peak or average currents can damage the electrodes of the photomultiplier. The above and other photomultiplier nonlinearities as well as corrective procedures are discussed by Kowalski (I), Pietri and Nussli (5), and RCA Staff ( 4 ) .
46
SHERMAN K. POULTNEY
B. Mechanisms and Statistics of Photodevice Gain 1. Introduction
Photodevice gains up to lo9 can be attained without the use of a resistance by means of secondary electron emission. If a photoelectron from the photosurface in Fig. 3 is given sufficient kinetic energy before striking an electrode, INCIDENT RADIATION SEMITRANSPARENT PHOTOCATHODE
PHOTOCATHODE TO DYNODE No 1 ELECTRON OPTICS
TYPICAL PHOTOELECTRON TRAJECTORIES
VACUUM ENCLOSURE ELECTRON MULTIPLIER
FIG.3. Schematic of 4 typical photomultiplier employing a focused multiplier. Electrodes 1-1 2 are dynodes, 13 is anode, 14 are focusing electrodes, and 15 is the photocathode. Some electron trajectories are shown. RCA Staff (4). (Courtesy of RCA.)
a number g of electrons will be emitted from that electrode. A chain of these multiplications can be arranged at separate electrodes as in Fig. 3 for a photomultiplier or along the continuous surface of a curved channel as for a channel multiplier if the cascading charge packet is suitably directed. The time response of the photodevice depends on the secondary electron trajectories, their energy distribution, and their spatial distribution ; the multiplication at each electrode; and the number of multiplication steps. These limitations to time responses in the nanosecond region are discussed in Section II1,A. A solid-state analog of the secondary gain of a multiplier is being developed in
SINGLE PHOTON DETECTION AND TIMING
47
the form of a carrier avalanche process in semiconductor diodes (7). Gains of 10’ to lo4 have been reached in small area diodes (8). If the electron multiplier were an ideal device, the anode output pulses resulting from single photoelectrons would have exactly the same charge. However, in a real device, not only might some of the photoelectrons go uncollected by the multiplier, but the secondary emission process is a statistical process. The inultiplier gain G , is an average gain and the multiplication factors g are average values. Output charge pulses therefore have a spectrum of sizes. These fluctuations in the output charge need not affect single photon detection as outlined in Fig. 1 as long as the spectrum is well-behaved at the lower end where the level of the integral discriminator would be set. Otherwise, a number of 1 he smaller single photon pulses would be lost and a number of small noise pulses may be added. The pulse height spectrum can also affect system time resolution as discussed in Section III,B. 2. Secondary Electron Emission
Within the last few years, important improvements have been made in secondary emission (and photoemission) from materials. In order to understand these improvements and their relation to multiplication fluctuations and fast timing, a description of the emission process is given in Section II,E,2 for photoemission and the differences for secondary emission are given here. Secondary emitters are either semiconductors or insulators with a band structure similar to that in Fig. 6 (9). An energetic electron incident upon a secondary emitter excites a number of electrons to the conduction band. Some of these electrons mlove towards the surface and escape if their energies are greater than the electron affinity energy of the surface. During the transport to the surface, the electrons lose energy as a result of phonon or electron collisions. In conventional secondary emitters, one thus expects and observes the multiplication factor g to increase with the energy of the primary. However, a more energetic primary excites electrons at greater depths in the material from which escape is much less probable. Consequently, g reaches a peak at some primary energy and then decreases. This peak value is typically from 6 to 10, but at primary electron energies somewhat higher than typical for photodevice electrode potentials. The distribution in energy and direction of the secondary electrons has an important bearing on the timeresolutionof the photodevice as discussed in Section III,A,l. Conventional emitters have quite large spreads in energy (i.e. 1 to 1OeV) (10). Improvement in secondary emission has been obtained by modifying the band structure with asurface layer of electropositive Cs on GaP as shown in Fig. 7 and as discussed in Section II,E,2. Under these conditions even the thermalized secondary electrons in the conduction band can escape the surface.
48
SHERMAN K. POULTNEY
Secondary electrons thus have a much greater escape depth and the multiplication at an electrode can be expected to be much larger than for the same primary energy incident on a conventional material. Multiplications of up to 50 have been observed in production photomultipliers for a single GaP dynode (IZ). These authors also point out the expectation that the energy distribution of secondary electrons will be quite narrow with the highest energy about l e v . Such an energy distribution would lead to an improvement in multiplier time resolution. However, this improvement is limited by the long diffusion times (e.g. 100psec) of excited electrons which are related to the greater escape depths.
3. Gain and Gain Statistics for Discrete Dynode Multipliers
A number of multipliers with high gain, well-focused structures (e.g. Fig. 3) have an output pulse height distribution for single photoelectrons that is peaked with a fairly narrow width. A typical single photoelectron distribution is shown in Fig. 4. These distributions can be obtained with the circuit
2i
20
CHANNEL NO. 5
leF
2e6
3e8
4e6
PULSE HEIGHT
FIG. 4. Single photoelectron integral and differential pulse height distributions. RCA type 4501 (similar to 8575). Photocathode K,CsSb. Counting time 10 min. Morton (7).
of Fig. 1 as discussed in Section II,D. By varying the level of the integral discriminator, one obtains the integral curve which intersects the ordinate at the total number of electrons entering the multiplier during the observation period. This integral curve allows one to calibrate the abscissa of the pulse height distribution curve in equivalent photoelectrons. Note that the spread of the distribution due to multiplication statistics causes some of the single photoelectrons to appear as doubles and others to be lost below a discriminator threshold. If the threshold could be set at the lower end of a very narrow
SINGLE PHOTON DETECTION AND TIMING
49
distribution without admitting any low amplitude device noise, one could expect stable single photon detection in spite of gain changes due to causes other than multiplication statistics. Theoretical derivations of single electron response (i.e. SER) amplitude distribution by a number of authors are summarized by Donati, Gatti, and Svelto (12) with particular emphasis on scintillation spectrometry. Various stochastic processes are used as models for secondary electron emission from an individual multiplier electrode. The SER distribution is obtained from an appropriate probability generating function which cascades the distribution successively down the electrode (or dynode) chain. The mean gain of the multiplier GI is then the product of the mean multiplication factor g of each dynode and results in an output pulse with mean amplitude A . If a Poisson process is assumed to describe the secondary emission at a dynode and if each dynode has equal mean gain, one can computean SERamplitude distribution that resembles Fig. 4 for a g of about 4. One can also calculate the percentage of single photoelectrons lost in the multiplication process for various g as in Table I (13). For g = 4, this loss is less than a few percent. It is often conTABLE I PERCENTAGE OF SINGLE PHOTOPULSESLOST I N A MULTIPLIERFOR DIFFERENT VALUES OF DYNODE GAIN9''
ELECTRON
9
Percent lost
1.5 2.0
42 20
2.5 3.0 5.0
11
Lombard
6 0.70
and
Martin
(13). (Courtesy of American
Institute of Physics.)
venient to use the ratio A of the full width of the SER amplitude distribution at half-maximum to the peak amplitude as a measure of the observed (or expected) width for a particular photomultiplier with dynode gain g. Here A is related to the relative variance by 2.35 E ~ The . relative variance is given by Morton (14) as 2 &A
= gEg2/(g-
=
l/(g - 1)
(7)
50
SHERMAN K. POULTNEY
for a Poisson process (cg2 = I/g) at each dynode and for a sufficiently long chain of dynodes, Much work has gone into trying to resolve the discrepancy of observations with theory for older photomultipliers. Relative variances larger than predicted by (7) are ascribed to non-Poissonian statistics in the secondary emission process itself or to a nonuniformity of gain on a dynode surface. In either case, the gain variance of a dynode is better expressed by t g 2= (bg + 1)/g where b expresses the deviation from Poisson behavior. The closer b is to 1, the larger the dynode variance, the amplitude variance and the A, Stable and efficient single photon detection requires a photomultiplier with a fairly narrow SER amplitude distribution (i.e. g 2 4). The width of the SER amplitude distribution can be decreased in several ways. With conventional dynode materials, one can increase dynode gain to its maximum value of 6 to 10 by raising electrode potentials. Fortunately, the first dynode has a dominant influence on the tA2as might be expected and only the first dynode or two need the high potentials. By regarding the first dynode as the source of the group of g1 electrons, one can compute the for a photomultiplier with remaining dynodes of gain g and for Poisson statistics using the results of Morton (14). &A2
=g/(g - l)gI.
(8)
The A for a photomultiplier with g1 = 10 and g = 5 would be 0.83 which is an improvement of 40% over a uniform gain photomultiplier. One can narrow the A much more dramatically by using the high gain dynode material on the first dynode. The A for a photomultiplier with g1 = 43 and g = 5 would be 0.40 which is close to that measured by Morton, Smith, and Krall(1.5). Such a photomultiplier is an excellent device for single photon detection. I t allows stable, efficient counting in addition to the capability of studying its noise behavior, as disccused in Section lI,C,l. Figure 5 shows the remarkably narrow peak of the distribution for single photoelectron noise (as well as other noise). Photomultipliers employing discrete dynodes and static, crossed electric and magnetic fields have been built to obtain fast time response as discussed in Section III,A,2. Their usefulness for single photon detection can be judged on the basis of their total gains which are 2 x lo4 to lo5 and of their dynode gains of 3 to 4. 4 . Gain and Gain Siatistics f o r Continuorrs Midtipliers Several varieties of continuous electron multipliers exist. Heroux (16) reviews the detection performance of a crossed-field continuous multiplier with respect to a discrete dynode multiplier. The crossed-field continuous multiplier uses a straight strip of high resistance dynode material. The electrons
SINGLE PHOTON DETECTION A N D TIMING
51
lo4 Small pulses
l-
a
I I : c I-
Single electron pulsrs
~
10
'
I
Large pulses I
I
PULSE
HEIGHT
FIG.5 . Diagrammatic noise pulse height distribution. RCA type C31000D (similar to 8575 but with GaP first dynode). Single electron noise represented by dotted line. Coates ( 2 1 ) . (Courtesy of The Institute of Physics.)
are guided between successive impacts on the dynode strip by means of an electric field between the dynode strip and a parallel (field) strip and by means of a crossed magnetic field. Gains high enough to detect single photoelectrons (e.g. G, = lo') can be obtained, but the SER amplitude distribution is very broad and lacks a well-defined peak. This distribution is attributed to a low value of dynode multiplication (e.g. g = 1.5) at the first impact and to an effective nonuniformity of the niultiplication factor due to photoelectrons striking the dynode strip at different positions. Moreover, the percentage of single photoelectron pulses lost in the multiplier would be expected tobehigh (e.g. 42). from Table I. Other impediments to the use of the crossed-field magnetic multiplier for single photon detection would be additional loss in efliciency due to photoelectron collection problems between photosurface and multiplier and a regeneration noise problem. The channel multiplier overcomes most of the above problems as described in a review by Wolber (17) and makes an attractive photodevice for single photon detection and timing. The channel multiplier is a hollow glass tube with a secondary-emitting coating deposited on its inner surface. Its length is relatively long compared to its inside diameter which is typically 1 mm. A voltage is applied to the ends of the channel to establish a uniform field along its axial length. An electron injected into the channel will be
52
SHERMAN K. POULTNEY
accelerated down the channel until it hits the wall to begin the multiplication process. The secondaries again are accelerated and collide with the wall due to their transverse energy components. Gains of lo7 are easily obtained. The SER amplitude distribution might be expected to be very broad from the discussion above. Such a behavior is found for moderate gains. However, at higher gains, SER amplitude distributions are obtained that are as narrow as the one in Fig. 5 for the high gain dynode photomultiplier. The cause of this narrowing is output pulse saturation. The mechanism for this saturation has been most recently studied by Harris (18). Stable single photon detection can thus be expected. Injection of the photoelectron into the multiplier is usually done by an accelerating potential of about 200V so that the first multiplication factor is about 3 and only a small loss is present inside the multiplier. The channel is usually curved to minimize regeneration noise. Due to its small size, the channel multiplier has other advantages with respect to device noise and to fast timing as discussed below in Sections II,C,2 and 111,A,3, respectively. The saturated output pulse is typically 10 to 20 nsec long. For single photon detection, the considerations of Section II,A indicate that load resistances of 2000 R or greater are optimum. The channel multiplier is limited in maximum counting rate to about lo5 counts/sec by current limitations and not by the pulse width. However, maximum count rates up to lo7 counts/sec have been obtained by Zatzick (19) using an auxiliary amplifier. An elaboration of the channel multiplier is a bundle of channels called a wafer. One such michrochannel wafer photomultiplier is described by Boutot and Pietri (20). It consists of about 5 x lo4 microchannels with individual diameters of 40pm.The useful photocathode diameter was about I cm and the multiplier gain was about lo5. The mircochannel wafer thus retains many of the channel multiplier advantages with the addition of a larger photosurface which is a necessity for many workers. However, the SER amplitude distribution has not yet been reported.
C. Photodevice and Background Noise and Their Reduction It does no good to be able to detect single photons if they are obscured by background and/or photodevice noise. An ideal photodevice would probably be one in which the only noise source was thermionic emission of electrons from the photocathode. In such a case, noise photoelectrons would exhibit the same SER amplitude distribution as signal photoelectrons. The same would be true for background noise photoelectrons. The only way to separate signal from noise is then to reduce the noise to tolerable levels by an appropriate technique or by choice of an appropriate photodevice. Figure 5 shows the noise pulse height distribution of a photomultiplier. Its behavior is not ideal in that there are additional pulses both smaller and larger than the well-
SINGLE PHOTON DETECTION AND TIMING
53
defined peak of single thermal electron pulses. The narrow SER amplitude distribution here allows one to set the threshold of the integral discriminator of Fig. 1 so as to accept most of the single electron pulse and reject most of the small pulses. The causes and reduction of photodevice and background noise is discussed below. The limits that the various noise sources set to single photon detection are discussed in Sections II,G; ll,C,l; and IV, both ingeneral and for particular photodevices.
I . Device Noise in Photomultipliers and Its Reduction Noise in photomultipliers depends greatly on the type of photomultiplier, the type of photocathode, the operating gain, technique, and the history of the particular photomultiplier selected. Attention is directed here to a recent study by Coates (21) of the dark noise performance of the photomultiplier with the high gain dynode discussed in Section II,B,3. The narrow SER amplitude distribution allows one to separate, identify, and study its various noise sources. Figure 5 shows a diagrammatic pulse height distribution for this photomultiplier at gains of about lo7 when the photocathode is shielded from external light sources. The contribution from single noise electrons is indicated by the broken line. Subtraction leaves two other classes of noise; small pulses below 0.2 electrons and large pulses above 2 electrons which account for 5 to 10% of the total counts. The small pulses were attributed to three sources; thermal emission from the dynodes, internal ohmic leakage between anode and the dynodes, and electron emission from the dynodes as a result of bombardment by ions. The small pulses are of minimal importance here because they can be discriminated against with little loss in single photon detection. The large pulses were also divided into three groups; ion pulses, afterpulses, and pulses due to cosmic rays. Ion pulses are single large pulses occurring at random and are due to ion bombardment of the photocathode. They are the major source of large pulses in this photomultiplier and their count rate increases with both gain and temperature. Afterpulsesare due to ions formed i n the region between photocathode and first dynode by the collision of an electron with gaseous impurities in the photomultiplier. The event is characterized by correlations in time between single electron pulses and large pulses. A characteristic time for this photomultiplier was 0.4psec. In photonlultipliers of different design, afterpulses coming 40nsec later are seen and are due to feedback of light produced at the anode by charge pulses ( 4 ) . Cosmic ray pulses are due to the production of Cherenkov radiation in the photomultiplier window by the fast particles. These ultrashort pulses occur at the rate of about 0.3 counts/sec for this 5 cm diameter photomultiplier and appear to saturate the output at about the 15 photoelectron level for typical gains. The cosmic ray particles also excite fluorescence in the window with a decay
54
SHERMAN K. POULTNEY
time of the order of 20 to 50psec and so generate more correlated noise pulses. The large noise pulses do not have a particularly serious effect on single photon detection since they are each standardized to one count in the discriminator. At low signal rates with an otherwise quiet photomultiplier, the cosmic ray pulses or pulses due to radioactivity in the tube envelope can be serious. At low rates, however, one could use a single channel analyzer to eliminate both small and large noise pulses. The predominant photomultiplier noise is single electron noise. Its magnitude depends on both the particular photosurface and particular photomultiplier in use. Coates (21) studied photomultipliers with bialkali photosurfaces. A typical room temperature noise rate was 200 counts/sec. The single electron noise consisted of two parts: thermal emission from the photocathode and an excess noise apparently generated by field emission. The thermal emission could be eliminated by rooling the photomultiplier to 0°C or below where the excess noise became dominant at about 50 counts/sec. The excess noise shows a nonrandom behavior (approximately Ilf in character) and will affect single photon detection (Section II,G,I). It also usually increases greatly with photomultiplier gain and so may become serious if high gains are required. If the application allows, one might use an amplifier to lower the gain operating point of the photomultiplier. Red sensitive photosurfaces show much higher thermal dark noise so that selection and noise reduction become very important for single photon detection. Table 111 lists typical room temperature noise rates for various photocathodes. Cooling of the photodevice is a popular method of noise reduction. A rough rule of thumb is that the noise rate drops by a factor of ten for each 20°C drop until the excess single electron noise dominates. Such a rapid dependence of noise on temperature requires good temperature control for single photon detection of weak sources. Foord, Jones, Oliver, and Pike (22) give a recent review of noise reduction by cooling in a paper concerned with many aspects of photon counting. Cooling does have the disadvantages of condensations and of Cherenkov radiation produced in windows added to prevent condensation. The other noise reduction techniques require that the incoming light signal be collimated to a small spot or narrow beam. Oliver and Pike (23) studied a photomultiplier with an effective small photosurface which was also cooled. They obtained a dark noise rate of 0.459 counts/sec at -20°C which was excess noise, random, and ascribed to radioactivity in the tube window. Another method is to eliminate from the multiplier input optics the electrons from all but a small portion of a large photosurface by means of a magnetic or electrostatic lens. Manufacturers are now making these available. A clever use of a magnetic lens was made by Topp et al. (24) who combined it with a device to enhance the quantum efficiency of their photomultiplier. Even more serious than dark noise for many applications is noise genera-
SINGLE PHOTON DETECTION AND TIMING
55
ted by the light signal. If noise electrons can produce afterpulses, photoelectrons can do likewise. Coates (21) found about one afterpulse for every 100 photoelectron pulses. Foord et a/. (22) investigated signal afterpulsing for a number of photomultipliers and pointed out their effects on both photon counting and correlation experiments (see Section IV,B) and photometry of weak sources. In some photomultipliers, prepulsing has been observed as mentioned by Meiling and Stary (3, p. 25). The signal-correlated noise can be studied by either viewing a light source known to cause random emission of photoelectrons in ii series of photon counting intervals or viewing it with a multichannel time analyzer as outlined i n Section 1V,B,3.
2. Deuice Noise in Channel Photomultipliers The dark noise pulse height distribution of the channel photomultiplier is effectively the same as the single photoelectron distribution when the channel multiplier is operaied in the pulse saturation mode. However. the amount of noise is substantially lower compared lo a photomultiplier. Wolber (17) reports noise counts of from 1 to 10counts/sec for an S-20 photosurface at room temperature and attributes them to thermionic emission from the photosurface. The low noise rate is due primarily to the small size of the photosurface ( 1 tnm'). Again collimated light signals are required if a single channel is used. A reduction of 20 in the noise was observed upon cooling to -2O"C, but no further reduction was observed at lower temperatures. Replacement of the photocathode with a blank disk reduced the room temperature noise rate to 0.01 counts/sec and indicated the cosmic ray contribution. The discrepancy between the coolecl rate and the blank rate is probably due to excess noise although the smoothness and continuity of the channels should minimize that source. The conditions for ion noise and time-correlated afterpulses are much more restrictive in the small channels, but they may be present. Afterpulses would cause the same problems in applications as with photomultipliers and should be studied. 3. Background Noise
In some applications, single photoelectrons from a light background could obscure the signal photoelectron. Reduction is again the only alternative in the form of spatial filtering, spectral filtering, or time gating. An extreme example of the background limitation is the detection of a single photon against the combined background of bright moon and bright sky as in the Lunar Laser Experiment (see Section III,D,2). A 6 arcsec field of view and a 1 A wide filter still allowed about 300 kcounts/sec of noise using the 2.7 m diameter McDonald observatory telescope. The final reduction
56
SHERMAN K. POULTNEY
technique required the setting of a microsecond time gate about the expected time of arrival of the lunar return by electronic means. A number of repeated rangings were, of course, necessary to be certain of a signal return. It is also of interest to note here that the photomultiplier used had to be operated at very high gains and so was subject to excess noise. The ERMA photosurface which was used exhibited a noise rate of 30 kcounts/sec at room temperature and several kcounts per second when cooled to about 0°C.
D. Practical Photodevice Techniques and Photon Detection Performance Tests It is important to be able to measure the SER amplitude distributions of both single electron signals and noise in order to evaluate the single photon detection capabilities of a particular photodevice. These tests also enable one to be sure that correct photodevice techniques are being employed. Poor practices can greatly increase photodevice noise and may lead to destruction of either the photosurface or the device. Table I 1 gives a checklist of good TABLE II
CHECKLIST OF Goon PHOTONCOUNTING PRACTICE WITH PHOTOMULTIPLIERS Housing design a. Optical isolation b. Electrostatic, magnetostatic, and rfi shields c. Insulating material choice and plncement d. Stable, high frequency electrical components e. Dissipation of heat from divider resistors f. Stable cooling method free of electrical noise g. Cooling scheme free of condensation h. Tapered divider and charge-storage capacitors
Practice a. Test housing before inserting tube b. Ground photocathode (if at all possible) C. Never expose tube to strong light d. Keep clean of cloth fibers, moisture, etc. e. Use as low a gain as possible f. Allow device noise to stabilize g. Use stable, low noise high voltage supply h. Good high frequency wiring technique
practices; a number of which seem always to be ignored i n homemade and commercial photodevice units. Recent discussions of performance tests and good photodevice techniques have been given by Morton (7), Zatzick (19), and RCA staff ( 4 ) . If afterpulsing can seriously affect a particular single photon detection experiment, additional tests for time correlations should be made as discussed in Section 1V,B,3.
SINGLE PHOTON DETECTION AND TIMING
57
Morton (7) gives a concise summary of tests for single photon counting performance of photodevices based on Fig. I . First in importance is the determination of the location and shape of the single photoelectron peak. An approximate location of this peak can be found by setting the threshold of the integral discriminator at its lowest value and by examining the dark noise as a function of tube voltage. The presence of a peak in the single electron pulse height distribution will be recognized by the occurrence of aplateau in this integral voltage curve. If the noise rate on this plateau is sufficiently low, one can study the single photoelectron peak by illuminating the photosurface with a steady, weak light source which yields a count rate about ten times the noise rate and examining the count rate as a function of discriminator level. The result should be an integral curve similar to that in Fig. 4. To express the abscissa in terms of equivalent electrons, one sketches in the rectangle shown with height equal to the total number of electrons observed and whose area is the area under the integral curve. The intercept of this rectangle on the abscissa gives the height of one photoelectron. The slope of the integral curve gives the SER amplitude distribution. The use of a multichannel pulse height analyzer obtains the amplitude distribution directly and greatly reduces the time necessary for these tests. However, the tests should be done as close as possible to the operating conditions of a particular experiment which may preclude the use of an analyzer. I t is sometimes necessary to subtract the integral curve of the device noise in order to interpret the single photoelectron distribution. In any case, a careful study of the device noise amplitude spectrum (e.g. Fig. 5) is necessary in order to set a lower (and upper if used) discriminator level to optimize noise discrimination with a known loss of single photoelectromn pulses. If cooling is necessary, the tests should be done under these conditiions. SER amplitude distributions can be examined in the presence of high device noise by using the time-gating method discussed in Section lll,C, I . Fiinally, one should investigate the statistics of the counting process for device noise by repeated counting since this is one of the factors limiting detection measurements of weak sources. Deviations of the signal counting statistics from random behavior is discussed in Section IV,B.
E. Quantutn Eficienry of'Photosurfaces I . Introduction The photosurface of the photodevice converts incident signal photons into photoelectrons which enter the multiplier to be detected. The efficiency of this conversion, I], at the wavelength of interest, I , , is therefore very important for single photon detection. It is called the photosurface quantum efficiency and is usually quite a lot smaller than 100%. If one were simply
58
SHERMAN K. POULTNEY
trying to detect a signal photon in a gated interval in the absence of noise, it is obvious that the probability of detection increases directly with the quantum efficiency at the wavelength of interest. If a background photon noise rate, N B ,and a device noise rate, N , , are present, the probability of detection depends on the accumulated photoelectron signal, qn, , becoming larger than noise fluctuations in the expected interval after repeated trials. Assuming that both noise and signal fluctuations can be described by Poisson statistics, one can obtain the signal to noise ratio, SIN, as
SIN
=
qn,/(qn,
+ q N , T + Nd T ) ' / 2 ,
(9)
where it has been assumed that the noise has been measured in an equal time interval where the signal was hot present and T is the total time of observing both intervals. A similar result is obtained for photometry of weak sources in Section II,G,l. If device noise dominates and the photoelectron signal is written in the form of a rate, (9) reduces to the expression for the figure of merit of a photodevice for photon counting given by Morton (7). In this case, it is possible that one would select a photodevice with a lower noise to maximize this figure of merit rather than a higher q. In all other cases, the choice is the highest quantum efficiency. A history and description of most available photosurfaces is given by Sommer (25). A review to mid-I970 is given by Bell and Spicer (26).Table 111 is compiled from these sources plus RCA Staff ( 4 ) and includes the properties of some of the older photocathodes as well as those of the new photosurfaces TABLE I l l
TYPES A N D CHARACTERISTICS OF VARIOUS PHOTOSURFACES~
Cathode type
S-lb S-I1 S-20 Bialkali ERMA GaAs GaAsP GaInAs
Stoichiometry
Ag-0-Cs Cs3Sb-O Na,KSb(Cs) K,CsSb Na-K-Sb-Cs GaAs(Cs) GaAs, -xP, InCaAs-CsO
Wavelength of peak response
Peak quantum efficiency
Wavelength at I % peak quantum efficiency
(A)
( %)
(A)
8000 4000 4100 3800 5600 3400 3300 4000
0.5 15 19 30 10 12
12,000 6300 8300 6600 9300 9000 7400 10,500
17
22
Dark noise rate at 20°C (e/cm2/sec) 5 x lo6 100
400 30 I 03 104
I 04
Dark noise rates from Zatzick (19). Other data from Sonirner (25) and RCA staff (4. Above 4000 A.
SINGLE PHOTON DETECTION AND TIMING
59
that have been developed for use in the near infrared. The ultraviolet response of conventional photosurfaces is generally limited by the device window. New materials are sought for this region primarily for lack of sensitivity at longer wavelengths. Table I11 lists peak quantum efficiency, a measure of the span of the spectral response, and dark noise rates. Spectral response curves can be found in the references. The quantum efficiency at a given wavelength is related to the commonly used radiant spectral sensitivity, c, in mA/W by
where Al is in Angstrom units.
2. The Pliotoemiss,;onProcess A concise description of the photoemission process will be useful at this point for a number of reasons. It aids in the understanding of the improved yields of photoemission and secondary emission of the new materials, their effects on device time resolution (see Section IIl,A), and quantum efficiency enhancement in thle older photosurfaces. Even prior to the development of the new materials. a basic description of the efficient conventional photosurface was achievled on the basis that they were semiconductors (25). Figure 6 shows an energy model for a semiconductor photoemitter which includes the space dimension normal to the surface. The valence band is the highest filled energy band for electrons. Immediately above the valence band is an energy gap of width EG for which no energy states exist for electrons. Above this forbidden band, there is a band of permitted energy states (i.e. the conduction band) which at ordinary temperatures contains very few electrons. VACUUM LEVEL
CONDUCTION BAND
SOLID
7’, EA
-
I
---L
VACUUM
FIG.6. Energy band model for a semiconductor photoemitter. Distance from surface also shown. Threshold for photoemission is EPH= E, E G . EA is electron affinity
+
energy.
60
SHERMAN K. POULTNEY
An electron at the bottom of the conduction sees a potential barrier for emission to the vacuum represented by EA which is called the electron affinity energy. Photoemission from the semiconductor for a photon of energy E,, may then be viewed as a three-step process; absorption of the photon by the material and excitation of an electron from the valence band to the conduction band, movement of the excited electron to the vacuum interface, and escape of the electron over the potential barrier into the vacuum. Therefore, the minimum photon energy necessary to produce a photoelectron is the sum of EG and E A . This minimum sets the limit of red response. For example, the S-20 photosurface has an EG of I .OO eV and an EA of 0.55 eV. The absence of thermal electrons in the conduction band and the typical E A which are much smaller than impact ionization thresholds mean that the electron excited to the conduction band can escape from moderately large depths compared to a metal ; being limited only by energy loss through electron-phonon interactions. This large escape depth leads to high quantum efficiencies as long as the wavelength-dependent absorption coefficient is high enough. Typical escape depths for conventional photosurfaces are about 200A compared to ~ o Ain metals. The addition of the concept of negative electron affinity to the above description started a search for suitable photoemitters among the more familiar semiconductors. Negative electron affinity means that the energy bands near the surface of the semiconductor are bent in such a way that the bottom of the conduction band deeper inside the surface lies above the vacuum energy as shown in Fig. 7. This bending can be accomplished by the adsorption of electropositive metal atoms on the surface of a sufficiently p-type bulk semiconductor (such that its Fermi energy is at or near the edge of the valence band). The depth of the bent band region should be as small as possible. The great advantage of materials with negative electron affinity is
CONDUCTION BAND VACUUM L E V E L FORBIDDEN BAND
FERMl
SOLID +--
I
-VACUUM
Fro. 7. Energy band model of a semiconductor photoemitter with negative electron affinity. Distance from surface also shown.
SINGLE PHOTON DETECTION AND TIMING
61
that the majority of electrons excited to the conduction band and traveling slowly toward the surface can be emitted even though they lose energy in phonon collisions. Even those electrons excited deep within the material (up to 10,OOOA) have a reasonable probability of being emitted as photoelectrons into the vacuum. Consequently, the photosurface quantum efficiency is greatly enhanced, especially near the threshold. The limit of red response is approximately equal to the semiconductor band gap (e.g. 1.4 eV in GaAs-Cs). Scheer and van Laar (27) reported the first photosurface (GaAs-Cs) built on this concept in 1965. The practical utilization of these concepts had to await the development of high quality semiconductors with appropriate band gaps and doping levels and with good minority carrier transport properties. Most promising for practical and commercial applications are the 111-V compounds. The incorporation of the new photoemissive materials into photomultipliers is proceeding rapidly. Reports on the performance of these new photomultipliers have not yet reached the general literature although some numbers are available in Table 111. The energy distribution of photoelectrons may be narrowed as for secondary electron emission, at least for shorter wavelengths. A greater effect on time resolution could come from the long electron diffusion times (e.g. 100 psec) associated with the greater escape depths. This effect does not at present limit the time response of photodevices, but may if materials with greater escape depths are used to obtain higher quantum efficiencies. Cooling of a photosurface to reduce dark noise can affect the quantum efficiency of the phiotosurface. Boileau and Miller (28) report that a lowering of the temperature usually increases the short wavelength efficiency, but decreases the long wavelength sensitivity. The increase is minor while the decrease can be as much as 2% per "C near the photosurface threshold. If one is working in the long wavelength region and is cooling the photosurface, one should measure the quantum efficiency at the operating temperature. In any case, the temperature of the photosurface should be kept constant in in order to have a stable efficiency photomultiplier. It is best to assume that a given photosurfact: has a large temperature coefficient in the far red until proven otherwise. 3. Measurement of' Photocathode Quantum Eficiency It is often necessary to measure the quantum efficiency of a photosurface in order to check its uniformity over the surface, to check its relative spectral behavior, to examine its temperature dependence at a given wavelength, or to measure an absolute value at a certain wavelength. These and other basic tests of a photodevice are' described in the IRE Standards (29). The absolute measurement is difficult to carry out accurately. Relative measurements can
62
SHERMAN K. POULTNEY
be made more easily (30). The photodevice is connected as a diode if possible with all other electrodes at a potential of several hundred volts. The cathode current is then measured for a constant amount of light at the wavelength of interest as the conditions arevaried. If the wavelength is varied, a thermopile or some other secondarystandard is needed to monitor the light intensity. If a rough measure of the quantum efficiency is sufficient (i.e. to about 10 %) and the secondary standard is stable and previously calibrated, the quantum efficiency in electrons per photon can be calculated.
4. Quantum Eficiency Enhancement Quantum efficiency enhancement really consists of the reduction or elimination of losses which occur at the photocathode (25). Before the new materials, all photocathodes were deposited as thin layers on opaque or transparent substrates. Aside from the obvious reflection losses at the photocathode interfaces, there are two other types of losses: transmitted light and light absorbed beyond the escape depth. These losses are wavelength dependent because the absorption coefficient of the photosurface material usually varies with wavelength. One method of enhancement increases the light path (and absorption) without going beyond the escape depth by depositing the photosurface on a reflecting substrate. The enhancement is usually greatest in the red where the absorption coefficient is lowest. A further improvement is to make the photosurface part of a reflection interference filter which, however, limits the range of useful wavelengths. These techniques are not particularly useful with the new materials which are already near optimum absorption and escape depth and which are small bulk semiconductor crystals mounted as an integral part of the multiplier. In addition, most older photomultipliers were designed for use with diffuse light sources and so required large, transmission-type photosurfaces. These transmission-type photocathodes have thicknesses chosen to maximize their response toward the blue. Much of any incident red light is transmitted through the photosurface resulting in low quantum efficiencies. Thicker photosurfaces would help, but would soon be limited by the escape depth problem in addition to lowering the blue response. These photocathodes can have their red response enhanced by external optical techniques that prolong the light path inside the photocathode without causing it to move too far from the surface. Gunter, Grant, and Shaw (3f)give a good review of these techniques and practical results with several tube types. Enhancements of about two were obtained in the blue and up to six in the red and near-infrared. Optical enhancement requires narrow, collimated beams as do the photomultipliers with the new photosurfaces. Finally, Crowe and Gumnick (32) report enhancements with conventional photosurfaces as a result of the application of external electric
SINGLE PHOTON DETECTION AND TIMING
63
fields. This enhancement may be due to a lowering of the vacuum potential barrier shown in Fig. 6. Enhancements of 3 to 6 were observed near 9000A. The same enhancement technique may be applicable to the new photoemitting materials.
F. Total Quantum Counting Eficiency The photoelectrons released from the photosurface of a photodevice may not all be detected as counts at the discriminator for a variety of reasons. The total quantum counting efficiency o f the device will therefore show a discrepancy compared to the photosurface quantum efficiency. The discrepancy may occur due to inefficient collection of photoelectrons at the first multiplier surface, losses in the multiplication process, or rejection o f the smaller pulses o f the SER amplitude distribution by the discriminator. Pietri (33)measured the collection efficiency of an early high gain, focused photomultiplier to be 807; at 4200A for a 2 cm2 spot at the center of its photosurface. He was able to redesign the input optics in a later tube to raise the collection efficiency to near 100%. The collection efficiency was found to decrease with an increase in the area o f the photosurface used and to increase as the wavelength of the incident light was increased. The collection efficiency of most photodevices and its behavior with wavelength and position can be predicted by computer simulation o f electron trajectories in the photocathode-first dynode region ( 4 ) . Focused photomultipliers are designed to give collection efficiencies from 85 7”to 98 ”/, depeoding on the size of the photosurface used. Other geometry photodevices may have lower collection efficiencies. Wolber ( 1 7 ) quotes a lower bound of from 70 to 90 for a channel multiplier. Loss of counts in the multiplier would riot be expected to affect discrete dynode multipliers, as discussed in Section II,B,3. However, this loss might be expected to be more serious in devices which have low initial multiplication factors. The possible source of discrepancy most easily studied is the rejection of small pulses by the discriminator. Although it can be taken into account i n experiments by knowing the SER amplitude distribution and the setting of the integral discriminator, observing times still depend on the detected signal rate. The high gain, first dynode photomultiplier discussed in Section II,B,3 should enable one .to work on a counting plateau where a very large percentage of the single photoelectrons are detected and thereby to achieve a counting efficiency very close to the photocathode quantum efficiency. Lakes and Poultney (30) have made a direct measurement o f these two quantities for this photomultiplier arid find about a 25 %, discrepancy even though an excellent counting plateau was observed. Birenbaum and Scar1 (34) later found a 407; discrepancy, but with a plateau of significant slope at their operating gain. They attributed the discrepancy to an unexpected number of photoelectron
64
SHERMAN K. POULTNEY
pulses much smaller than the well-behaved single photoelectron peak. Figure 5 gives an indication of the deviation at small pulse heights. Coates (35) has studied such a deviation of the SER amplitude distribution from the predictions (Section II,B,3) for this particular photomultiplier and was forced to propose an edge effect after considering many other causes. The edge effect consists of a loss of electrons from a charge packet due to striking either the edge of a dynode or a macroscopic inhomogeneity on a dynode. Again this photomultiplier allows a very detailed investigation of its own behavior. It is not now clear whether this otherwise excellent photomultiplier has a unique problem or whether the poorer SER ampitude distributions of other photomultipliers manage to mask the effect. In any case, workers should be aware of the possibility that the total quantum counting efficiency of a photodevice may be lower than the photosurface quantum efficiency. The discrepancy can cause longer observing times, in the least, and unexpected systematic errors in photometric applications.
G . Typical Photon Detection Experiments 1. Introduction
There are a wide range of photometry and spectrophotometry experiments that require the measurement of a weak light intensity. The weak intensity may be due to a weak source, scattering with small cross sections, or many spectral channels viewing a stronger source. With the assumption that the photodevice has been selected for optimum quantum counting efficiency, qo , one can define weak light intensity as about 100 counts/sec or less. The photodevice is operated as shown in Fig. 1 for single photon detection which is also called photon counting. The anode charge pulse is detected at the discriminator, standardized, and recorded by a digital counter. If there are no correlated afterpulses and if the device and background noise rates are much smaller than the signal rate, the fractional precision of a measurement (i.e. the inverse of the signal to noise ratio) is given by (S{N)-’ = I/(qoNpT/2)1’2,
(1 1)
where N P is the signal photon rate and T/2 is the measurement period. The signal counts are Poisson distributed in time or shot noise limited, at least for coherent and broadband light. Alternate detection and recording methods do exist in which analog recording (i.e. a rate meter or current meter) can be used on either standardized or nonstandardized photodevice output pulses. Jones, Oliver, and Pike (36) show theoretically and experimentally that photon counting yields the best precision as given by (11) in a given observation
SINGLE PHOTON DETECTION AND TIMING
65
period. A rate meter (i.e. analog recording) viewing standardized counts is shown to yield a fractional precision lower by 4’2 and therefore requires twice the measurement period to obtain the same precision. This result can easily be obtained from Campbell’s theorem for the situation in which one observes for a period four times the integration time of the rate meter. An additional lowering of the fractional precision of a measurement in a given period enters if nonstandardized counts are recorded. For an SER amplitude distribution whic.h fits the theory outlined in Section II,B, the fractional ) 1 + ( A / 2 . 3 5 J 2where precision is lowered by the square root of ( I E ~ * = and A are defined in Section II,B. This factor becomes \/2 for a photodevice without a peak. The particular choice of detection method depends on the signal rate. Photon counting is superiof at very low signals rates, but does have a high signal limit. Near or above this limit one must switch to analog recording of nonstandardized pulses. At signal rates in between these two limits, the method is probably determined by past practice and simplicity of record keeping. One advantage of photon counting is that it allows one to make a detailed study of the photodevice being used. Further advantages of photon counting are discussed below for a number of typical experiments, The presence of ‘lo in (1 1) should also be noted. Assume now that the photodevice in use is being operated under optimum conditions as discussed previously so that a counting plateau exists and the dark noise and background rates have been reduced to about 1 count/sec. As the signal rate approaches 1 count/sec, the observation period needed to obtain a certain precision will increase over that predicted by ( I I ) . This increase can be obtained from
+
(SIN)-’
= (‘loN
,
+ 2q0 N , + 2Nd)”‘/q0 N P ( T / 2 ) ’ ” ,
where N , , N , , and Nd are the signal photon, background noise photon, and dark noise rates, respectively. The signal has been separated from the noise by effectively modula.ting the source with a square wave of period T and the noise is also assumed to be Poisson distributed in time. Again, any of the alternate detection methods can be used with the fractional precision ( 1 2 ) lowered as discussed above for the shot noise limited expression (1 I ) . It should be pointed out that synchronous analog detection has additional noise discrimination advantages over a straight current measurement (37), due to low frequency circuit noise not considered above. Synchronous digital detection has no such advantage over photon counting. Its advantage over all other methods beyond those already discussed is its low drift operation with time which allows very long observation periods (of the order of hours). Synchronous photon counting was first reported by Arecchi, Gatti, and Sona (38) and somewh,at later by Oliver and Pike (23). A more sophisticated
66
SHERMAN K . POULTWY
version with provisions for readouts of signal and noise as well as the difference and sum for assignment of statistical uncertainty was recently reported by Zatzick (19).Their results are consistent with the determination of a signal of 0.1 count/sec with a precision of 10% ( S I N = 10) in the presence of 1 count/sec Poisson-distributed noise using a counting period of one hour. Predictions for other signal and noise rates can be obtained from (12). Again note the advantage of the highest possible total counting efficiency. The high signal rate limit of photon counting is determined by the response times of photodevice, amplifier, discriminator, or digital counter. Usually, the dead time of either the discriminator or counter sets the limit. The recent version of the synchronous digital photon counter has a limit of 85MHz yielding a practical intensity range of about lo*. However, this upper limit may have to be lowered considerably if it corresponds to an anode current limitation of a photodevice or if afterpulses are present. One way to eliminate the effect of afterpulses is to gate the detector off until after the characteristic time of afterpulsing (e.g. 10psec). Rather than switch to other detection methods from photon counting, one has the final alternative of attenuating the incoming signal with calibrated filters. The deviation of the signal and noise rates from a Poisson behavior due to the photodevice will lower the precision of a measurement below (12) in a fixed observation time. The same is true for deviations due to fluctuations in the light intensity. However, here a study of the deviations in the counting statistics would reveal information as to the nature of the light source, or the scattering medium as discussed in Section IV,B. The deviations due to the photodevice which are discussed in Section II,C, 1 can also dictate the choice of detection method. For example, the residual noise component due to cosmic rays occurs at a very low rate, but yields very large pulses. If measurements are made on weak sources using nonstandardized pulses, the large pulses can cause serious systematic errors (39). In addition, afterpulsing can only be eliminated in the photon counting mode. The most serious deviation due to photodevice from a diagnostic viewpoint would be the presence of single electron afterpulses several microseconds after every dark or signal count. Neither pulse height spectra nor a cursory counting statistics study would uncover it and a systematic error in source intensity would result. Probably only a study of the arrival times of each count by the methods of Section II1,C would catch it. 2. Laboratory Photometry and Spectrophotometry
Weak source photometry in the laboratory includes measurements of the angular and polarization dependence of light scattered from either single particles (40) or from the cooperative effects of many particles (41). These
SINGLE PHOTON DETECTION AND TIMING
67
measurements can be used to study the sizes and other properties of individual particles and to study the correlation lengths for cooperative particle or density fluctuation effects. Low light level spectrophotometry includes the measurement to medium or high resolution of the spectra of any weak source or of any scattered light. The threefold advance of recent years in laser light sources, high resolution (and high background rejection) spectrometers, and photon counting has had a profound effect on the study of matter by light scattering. Smith (42) gives a brief review of the many different collective motions in matter now being studied by combinations of these techniques. Accessible with taindem monochromators are weak Raman-shifted frequencies (10” to 10l3 Hz) of light scattered from molecules and from various optical modes of excitation in matter as well as smaller frequency shifts previously obscured by conventional source linewidths. Accessible with high resolution, scanning Fabry-Perot interferometers are weak Brillouin-shifted frequencies ( lo9 to 10’ Hz). Chu (43) briefly reviews recent Raman and Brillouin scattering studies, discusses Rayleigh scattering studies using optical mixing spectroscopy, and recommends the photon counting statistics discussed in Section IV,B as an alternate to that new spectroscopy. Single photon detection is only used in these types of experiments for the weakest intensities. Reynolds (44) reviews an interesting series of photon detection experiments with interferometers and slit systems which demonstrate that the interference patterns are obtained as expected with very weak sources. Both photomultipliers and image tubes were used. Arecchi et al. (38) used their synchronous photon counter to measure the angular distribution of Raman scattered light from the 992 cni-’ vibrational frequency of benzene in a direction parallel to the polarization plane. A He-Ne laser at 6328A was used a.s the excitation source. Signals as low as IOcounts/sec were measured to 10% precision with observation times of 1 min in the presence of noise of 500counts/sec from the uncooled RCA 7265. An example of photon detection as applied to Brillouin scattering in gases is given by Greytak and Benedek ( 4 5 ) . They use an ITT FW 130 with 3 dark counts/sec and state that signal shot noise limited the precision. Their peak signal rate was 300 counts/sec. Lineshifts of about 10‘ Hz and linewidths of about 3 x lo7 Hz (nearly equal to instrumental width) were measured. Background scattered light had to be subtracted. Barrett and Adams ( 4 6 ) give an excellent description of the many related techniques used to improve signal to noise ratios in a study of Raman scattering from rotation-vibration lines in ultrasmall gas samples. These techniques include sample illumination, polarization discrimination against Rayleigh-scattered light, optimum collection of scattered light, and single photon detection. An EM1 62568 photomultiplierwas used at a peak signal rate of about 10 counts/sec resulting in a total of a few hundred pulses for a typical rotational line in the vibration band. The dark count was
68
SHERMAN K. POULTNEY
46 counts/sec at room temperature and one count every 5 sec when cooled to about -40°C. A rate meter-type recorder was also provided for viewing the standardized pulses. The monochromator was scanned at as low a rate as 1.8 cm-'/min for a total scan time of three hours in the case of the rotationvibration (1-0) band in nitrogen. An integration time of 10 sec was used for counting. The magnitude of the background was not mentioned. Mooradian (47) considers the elimination of unwanted excitation radiation as one of the most difficult experimental problems of laser light scattering spectroscopy and refers to several electronic and optical subtraction techniques that have been successfully employed. Most of these techniques are coupled with synchronous detection. The unwanted radiation becomes most troublesome in the spectral region near the laser line itself and even several monochromators in tandem cannot reject it sufficiently. The resolution, scan rates, and integration times of weak source spectroscopy will depend on the aims of the particular experiment as well as the signal to noise ratio. The decision to move to single photon detection and most likely to synchronous digital detection probably requires the recording and processing of the data in digital form. Commercial manufacturers of spectrometer systems now offer the whole range of detection and recording methods mentioned in Section II,G,I.
3. Photometry and Spectrophotometry in Astronomy Interest in the detection and measurement of faint astronomical sources has always constituted an impetus to single photon detection. An early review of this work can be found in the articles by Baum, Johnson, and Lallemand in the volume edited by Hiltner (48). Photometry and spectrophotometry in astronomy differ from those in the laboratory in that other natural phenomena affect the measurements and other workers are waiting to use the costly facilities. The natural phenomena include atmospheric " seeing," scintillation, and extinction. In addition, the night sky background may fluctuate due to physical processes. Background can be subtracted by means of a two-channel photometer with one detector viewing a nearby patch of dark sky or by means of a one-channel photometer synchronously switched from star to dark sky in conjunction with the synchronous digital detection mentioned in Section II,G,l. The switching can be done either in the telescope optics or in a photomultiplier designed for magnetic deflection of a viewing spot on the photosurface. Tull (49) describes a photon counting system for high resolution spectrophotometry and includes a discussion of the atmospheric and other noise limitations. Zatzick (19) briefly describes a 32channel photon counting spectrometer attached to the Mt. Palomar 5 m telescope. Each photomultiplier views a different region of the spectrum and is interfaced with a computer for data recording, background subtraction, and
SINGLE PHOTON DETECTION AND TIMING
69
limited data analysis. The typical light flux may be about 1 or 2 photons/sec so that spectral information from stars of 22nd magnitude can be obtained. The need to perform calibration measurements on bright stellar objects with known properties; requires the additional capability of fast counting devices and circuitry. Dennison (50) has recently described the philosophy and practice of electronic optical astronomy and includes therein new developments with single photon detection in astronomy. Rather than use 32 or 80 photomultipliers one would like to use a single image tube with an electronic readout of the image of a weak source or spectrum. At present, the image intensifier is used as the front end of such an image tube and its single electron detection properties determine those of the whole device. Methods of evaluating the single photon detection capabilitiesof imageintensifiertubes have been pioneered by Reynolds (51) and have many similarities to the evaluation of photomultipliers. 111. FASTTIMING WITH SINGLE PHOTONS
Fast timing with single photons means that the timing system consisting of a photodevice, timing circuits, and timing methods is capableof measuring the interval between single photon events to nanosecond or better accuracy. The capability of counting photons at high rates is closely related; especially if the interval between events being timed is very small. In addition to this timing capability, the fast system must possess all the single photon properties outlined in Section 11. In special cases such as background noise limited detection or a surplus of signal photons, this last requirement can be relaxed by allowing a high device noise or by allowing a poor counting efficiency respectively as noted in Section II1,D. A . Photodevice Timing Capabilities and Limitations
Photomultipliers with discrete linear dynodes and electrostatic focusing (see Fig. 3) have long held the lead in precision timing of intervals between single photon events or between a time marker and a photon event. Many of the properties which placed them foremost i n efficient single photon detection also aid in the attainment of this fast timing and counting. Efficient photon detection is not, however, a sufficient condition for fast timing. A review of the timing capabilities and limitations of this particular type of photomultiplier will establish a framework for the comparison of present devices and for the contemplation of new devices. Several of these new devices can be expected to yield an order of magnitude improvement over present photomultipliers.
70
SHERMAN K. POULTNEY
Most devices operating on the principle of external photoemision possess three basic elements; a photosurface with electron optics which directs the photoelectron into the electron multiplier, the electron multiplier which amplifies the single electron by secondary emission, and the anode which collects the charge packet to provide an external signal (see Fig. I). The time of travel from the photosurface to the external output lead is called the electron transit time, 11, of the device. It can be measured using a short light signal of single or multiple photon intensity which has a synchronous electrical pulse. The arrival time of the resultant charge pulse at the external output must be defined with respect to a characteristic of the pulse such as its centroid or a point part way up its leading edge. This arrival time will fluctuate from one measurement to another, especially for single photon signals. This fluctuation in device transit time will limit the timing precision for single photon events and will be assumed to consist of transit time fluctuations or spreads between photosurface and multiplier, e K M ,in the multiplier, eMS, and between multiplier and anode, These spreads originate both in the geometry of the device and in the distribution of initial velocities of photo and secondary electron emission. For example, the geometry may be such that all possible electron paths between two device elements are not isochronous for even the electrons emitted at rest. The individual spreads will be expressed as standard deviations and added as such with suitable weighting factors. Values for the individual transit time spreads and their dependence on photodevice design are discussed below. It might be expected that initial stages of a photodevice contribute with the greatest weight to the total transit time fluctuation. In the later stages, the growing number of electrons in a charge pulse provides many samples of the transit time of a stage and should reduce the transit time fluctuation of that stage in the manner of the standard error of a mean (i.e. the transit time variance of that stage divided by the number of electrons sampling the paths). With the assumptions that all transit time spreads after E K M are equal cSS, that all stage multiplications are equal (i.e. g), and that multiplication fluctuations do not have a significant effect on the spread weighting factors, one can express the total transit time fluctuation for a photomultiplier as & H ;
= & k M f &&/(g -
l),
(13)
Gatti and Svelto (52).The first stage or dynode often has a larger gain g, as well as a larger eSS as discussed below. In this case, the single photon transit fluctuation of the photomultiplier would be approximated by H ;‘
= &kM
+ &;1S2/g1 + &;S/(g
- lbl.
(14)
Thus the high multiplication factors that yield good SER amplitude distributions also yield better time resolution. Time resolution, R, is related to the
SINGLE PHOTON DETECTION AND TIMING
71
transit time fluctuation by 2.35 cpHand is the experimentally observed quantity (see Section 111,C). Single photon time resolutions of as low as 320psec have been measured for a particularly fast photomultiplier by Birk, Kerns, and Tusting (53). If the charge pulse collected at the anode is neither saturated nor ringing, the anode pulse width can also be estimated from the transit time spreads, but with different weighting factors. The anode pulse for a single photon is called the single electron time response and can often be approximated by (4). Its full width at half-maximum, P, is given by
where n is the number of identical multiplications and II is the width of the single electron response in terms of a standard deviation. The front stages no longer dominate so that a device with a very narrow anode pulse need not have exceptional time resolution. Gatti and Svelto (52) also show that the variance of II is much smaller than the total transit time fluctuation epHfor the usual case of a number of multiplications in a multiplier at moderate gains. This conclusion further supports the use of a standard shape for the single electron time response of a photomultiplier. Birk and co-workers (53) find a single electron response of roughly Gaussian shape with a full width P of 0.8 nsec for the fast photomultiplier. This response is best measured in real time with single (or multiple) photon light pulses of sufficiently short duration. This time response sets the limit to the counting rate at high levels and, through a Fourier transform, to the frequency response of the photodevice. Great care is necessary in the design of the anode to minimize ringing and saturation which will cause an additional decrease in counting rate and frequency response. Pietri (54) outlines a number of these precautions as well as raising the specter of a transit time fluctuation related to saturation of larger pulses. RCA staff ( 4 , pp. 24, 107) give some advice about preserving the device time resporise at the anode and at the external connection. Saturation and ringing tend to affect the falling portion of the time response and usually do not limit timing resolution. When these are present, connection with (15) can still be made flor a study of cSS by using the rise time of the pulse. A general theory of the statistical time behavior of a photodevice could be constructed as summarized by Donati et al. (12). The probability density functions of transit times in each element of the photodevice would have to be calculated or measured and then convoluted together. To calculate the probability density functions, one would need to know photodevice geometry, element potentials, element multiplication factors g, and the distribution of photoelectrons and secondary electrons in energy and angle of emission. This general theory would also necessarily include the amplitude fluctuations
72
SHERMAN K. POULTNEY
previously treated. However, there may be a correlation between the probability density functions of successive stages which would render even a numerical convolution inaccurate. The analysis of the electron trajectories and transit times in a device with given geometries and potentials is in practice done numerically by computer simulation [Krall and Persyk (55a)l. The assumption made above of the Gaussian behavior of individual spreads and the use of (13), (14), and (15) is an attempt to estimate and compare the timing capabilities of various photodevices without doing the detailed calculations.
1. Photomultipliers a . Conventional fast photomultipliers. Conventional fast photomultiplier here means the type in Fig. 3 with transmission-type photocathode, electrostatic focusing, and linear, discrete dynode multiplier. Typical single photoelectron transit time fluctuations cpHare about 0.5 nsec. Pietri (54) and Pietri and Nussli ( 5 ) give a particularly clear account of the design, characteristics, and improvement of a photomultiplier of this type. The large contributor to the cpHof this and many other types of photodevices is the transit time spread cKSl in the photocathode-first dynode transit time. This spread shows up of course only after repeated measurements with single photons and after allowance for later dynode compensation. In a photomultiplier, the cKSl depends on the geometry, electric fields, and the process of photoemission in the KSl region. The electrode geometry is usually adjusted so as to equalize the flight times to the first dynode of photoelectrons emitted at rest anywhere on the photocathode. In addition, the photocathode is curved, fields are made uniform in its vicinity, and some compensation is introduced in later dynodes. Any remaining difference for the larger photocathodes is quoted as a transit time difference across the photocathode (e.g. 0.8 nsec). Here it is assumed that only the central portion of the photocathode is used so that the transit time difference is zero. The central photoelectronscan, however, leave the photocathode with different velocities which depend on the process of photoemission as well as the wavelength of the incident light. The transverse velocity component (perpendicular to the mean path of the photoelectron) on average spreads out the target spot on the first dynode. The focusing fields must be configured so as to keep this spot smaller in size than the active part of the dynode for every point on the photocathode in order to achieve good collection efficiency. The transverse velocity spread need not cause a transit time spread except for the need to leave the axial symmetry of the KSl region and enter the planar symmetry of the multiplier. This change in symmetry means that the first dynode intercepts the electron paths at about an angle of 45"(Fig. 3). The deviation of the next two dynodes from
SINGLE PHOTON DETECTION AND TIMING
73
the linear multiplier geometry is introduced to compensate transit time differences due to transverse velocity spread. This compensation will be assumed sufficient to leave the spread in normal emission velocities as the prime contributor to &KS1. Assuming the electric field between K and S1 uniform and due to a potential cp over the distances sI and with the energy W, corresponding to the normal component of initial velocity small compared to ecp, one can esimate the transit time spread by
d t , / t , = - ( W,/ecp)"2. The KSI transit time t , is given by
t, =s,(3 x
lO-*)/q1'2,
where sI is in cm and cp in V. Large photocathode photomultipliers usually have s1equal to or larger than their diameters to aid in flight time equalization and collection efficiency maximization. 'Typical values of s,, cp, t , , W, might be 7cm, 300V, 12nsec, and 0.3eV. The eKSl is then about 3.5 o/, of the transit time or 0.42nsec. An increase in cp would decrease the eKSl in direct proportion, but the increase is finally limited by either photomultiplier construction or the need to be near the peak of the secondary emission curve. The transit time fluctuation in the multiplier also depends on the geometry, electric fields, and the process of electron emission. The linear cascade of opaque dynodes in Fig. 3 is used for good time response as well as minimizing ion and light feedback noise and providing a good SER amplitude distribution. It quickly becomes a regular array after the coupling region where the dynodes are shaped and positioned to compensate transit time differences. Additional focusing fields are provided near each dynode so that the cross section of the electron cloud becomes narrower and narrower as it travels down the multiplier. If it is assumed as above that the geometrical problem has been solved, the secondary emission velocities again determine the transit time fluctuations of each stage of the multiplier. These velocities correspond to energies W , about ten times greater than those of photoelectrons for conventional materials. An estimate for e,, can be obtained from ( I 6) and ( 17) slightly modified assuming that only the velocity spread along the path is important. Typical parameters are s, = 2cm, cp = 135V, and W,, = 3eV. The modifications consist of a substitution of 2cp for cp in (16) and a substitution of s,/2 for s, in ( 17). These modifications enter because of the effect of the n 2 dynode on the electrons traveling between n and n + I dynodes. For uniform potentials between dynodes, the field at the n dynode is doubled. Thus the transit time between dynodes is about 2.6nsec and the transit time fluctuation E,, is 10% or 0.28nsec. The transit time fluctuation in the first stage of the multiplier eSlS2 is much larger at about 0.90nsec, due no doubt to the extreme difficulty of making geometric compensation for this stage.
+
74
SHERMAN K. POULTNEY
The total transit time of a twelve-dynode conventional fast photomultiplier is 43 nsec and the total transit time fluctuation E , , ~is, by (14), equal to 0.60 nsec for a g1 = g = 4. The single electron time response by (1 5) is 3 nsec if the anode does not broaden it and if the center of the photosurface is used. Some manufacturers quote a pulse response for full illumination of the photosurface and so include the photosurface transit time difference. Table IV TABLE IV SINGLE PHOTON TIMING(AND DETECTION) CHARACTERISTICS OF SELECTED PHOTODEVICESa
Identification
Gain
A
Transit time
Time resolution R
Rise Pulse time width T, P
Amperex 56AVP Amperex XP1020 Amperex XPI 21 0 RCA 8575 RCA C31000E/F RCA 8850 RCA C3 I024 RCA C3 1OOOK C3 1034 RCA 31024B RCA C70045
10'
(1.5)
1.2 0.66 0.38 1.1 0.6/0.45
2 1.8
-
lo6
0.9 0.8 1 to(1.2) (0.6) (0.4) 0.5 1.5
41 28 20 31 34 31 -
0.47 0.60
lo6 5 x lo6
0.6 0.8
34
0.32
0.8 0.45
Static crossed field
Sylvania 502
2 x 104
1.5
0.1
-
0.1
Channel photomultiplier
Bendix RX 754 Amperex HR300
107
(0.4)
105
-
2 I
Photodevice tY Pe
____
Photomultiplier with linear focused multiplier of electrostatic type
lo8 107
10' 10'
lo9 lo6
-
-
0
10
8
z
w 0 G w
6
4
0
p/n O 3 8 m m
A
n/p O 2 8 r n m
0
n/p 0 15 mm
2
0 I MeV
ELECTRON
FLUENCE ie/cm2)
FIG. 1 1 . Efficiency degradation in 10 R-crn silicon solar cells produced by 1 MeV electron irradiation.
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P. L. BARGELLINI AND E. S. RITTNER
the same amount of damage as a space exposure at synchronous altitude of seven years. The 0.28 mm thick nip cell, which is the same one for which I-V characteristics are shown in Fig. 10, suffers a decrease in efficiency of about 14% under a fluence of this magnitude. The thinner nip cell exhibits a lower initial efficiency but declines to about the same value at 3 x 1014 electrons/cm*, corresponding to a decrease of about 7%. The explanation is that the thicker cell absorbs relatively penetrating photons of energy in the neighborhood of the cell threshold to which the thinner cell is transparent. However, with sufficient radiation damage, the additional photocarriers generated in the thicker cell die out before reaching the junction, thus wiping out the advantage of the higher initial efficiency. The somewhat thicker p / n cell exhibits the highest initial efficiency of the three and a more rapid fall-off with electron fluence corresponding to about a 35% decrease after exposure to 3 x l O I 4 electrons/cm2. In addition to the thickness effect, the 2.8-fold lower diffusion constant for holes in n-material over that for electrons in p material is the cause of the poorer radiation performance in this instance. It is also clear from Eq. (8) that any improvement in initial efficiency resulting solely from increasing the minority carrier lifetime in the base region will be lost under sufficient radiation exposure. Use of float-zone silicon, which contains a smaller oxygen content than conventionally pulled crystals, leads to a small but significant improvement in performance under electron irradiation (24). A reduction in base resistivity produces a higher voltage output and hence a higher initial efficiency. However, because of the accompanying higher damage constant, the fall-off in efficiency with fluence is more rapid. Thus, the optimum choice of base resistivity lies in the general range of 1-10 R-cm, the precise value depending upon transfer orbit and mission life. Since the minority carrier lifetime saturates with doping density for a Shockley-Read type of recombination center, whereas the open circuit voltage should increase monotonically with doping density up to the solubility limit, the possibility exists of a rise to a second (hopefully higher) maximum in the end of life efficiency at a very much lower base resistivity. For nonpenetrating radiation, e.g. protons of energy sufficient to penetrate the cover slide but insufficient to pass completely through the cell, the situation is more complex than that described above. Near the end of its range, a proton loses nearly all of its remaining kinetic energy in elastic collisions that produce displaced lattice atoms. Thus, for mono-energetic protons stopping in the cell, an appropriate theoretical model describing the nonuniform damage has been advanced (25),consisting of two damaged regions and one undamaged region. Equation (8) is assumed to apply in each of the damaged regions but with a higher damage constant near the end of range. The resulting efficiency-fluence curve falls off somewhat more rapidly than for the uniform damage case. Also, charge carrier removal may
ADVANCES IN SATELLITE COMMUNICATIONS
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become more important near the end of the proton range as a displacement density comparable to the doping density is possible. This could lead to the creation of an undesirable junction which would degrade cell performance still more. A special case of great importance arises from the simultaneous presence of nonpenetrating radiation and a small region on the cell surface which is unprotected by the coverglass. The junction underneath the unprotected area is damaged, leading to a highly significant internal conduction path shunting the external load and to a performance degradation out of all proportion to the fraction of exposed area (26). A promising approach (27) towards improving radiation resistance to proton damage is fabrication of a pin cell where the predominant dopant in the n-material is lithium. Lithium is a highly mobile solute, and especially so at the elevated temperatures expected in sun-oriented arrays. Annealing of the radiation damage, or at least a major portion thereof, proceeds spontaneously at the cell operating temperature. The recovery mechanism is not yet fully understood; however, it is hypothesized that the protons produce large clusters of damaged regions which are negatively charged, thus constituting giant recombination centers for holes. The lithium ions are presumably attracted to, and electrostatically shield, these centers. 2. Heterojunction Solar Cells Shortly after the discovery (20) of the silicon solar cell, it was concluded (28) from a general theory of the p/n homojunction cell that the optimum bandgap for solar energy conversion is in the neighborhood of 1.5-1.6 eV, an energy substantially larger than the bandgap of silicon (1.1 eV). This has stimulated fabrication (29) of a p / n solar cell of gallium arsenide, early cells exhibiting an efficiency of four percent as compared with the five percent efficiency o f t he earliest silicon cells. Extensive subsequent development work has led to higher values of efficiency, the highest reported value to date being thirteen percent with typical values of about eleven percent (30). This is disappointing i n view of the theoretical upper limit of about twenty-five percent. The relatively poor performance has been attributed (31) to a high surface recombination velocity, together with a high optical absorption coefficient over most of the spectral sensitivity range of the GaAs cell. A promising approach to overcome this difficulty is to prepare a p/n heterojunction between GaAs (or any other material with a bandgap appropriate for solar energy conversion) and another material of much larger bandgap upon which the light is incident. Carriers generated at a high rate near the interface of the two materials are readily collected by the junction. An efficiency of eleven percent has been claimed (32) for a heterojunction between GaAs and an amorphous nonstoichiometric form of silica (SiO,)
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P. L. BARGELLINI AND E. S. RITTNER
of low resistivity and large bandgap. Another example is a heterojunction between n-CdS and p-Si for which a conversion efficiency of 5.5% has been reported (33). Still another heterojunction type of solar cell is the thin filmp-Cu,S/n-CdS cell (34) which was not recognized to be of this type until relatively recently (35). Although exhibiting an efficiency of only several percent (reaching 6% as a result of recent improvements), it is notable for a relatively high output per unit weight and for high flexibility. Both attributes are highly advantageous in deployed arrays, particularly of the roll-out variety. Both are consequences of the use of a thin flexible plastic substrate and of a thin vacuum evaporated CdS layer, only a minute portion of which is converted to Cu,S. A further advantage of the thinness of the cell is improved radiation resistance. The cell differs from the heterojunction cells discussed above in that the light is incident upon the lower bandgap material (the Cu,S), in which substantially all ofthe effective optical excitation occurs. Since Cu,S is a polar material, the thermal bandgap (36) (0.9 eV) is smaller than the optical bandgap (1.2 eV), in accordance with the Franck-Condon principle, and further removed from the optimum value than that of silicon. An unusual feature of the Cu,S-CdS cell is that much of the photoeffect appears to arise from the direct photoemission of carriers from the Cu,S over the barrier into the CdS (35,36). Instabilities in the cell performance, however, have heretofore frustrated actual usage in satellite power supplies. There are two distinctly different sources of the instabilities, one ionic and the other electronic. The first of these arises from electrochemical reduction of the Cu,S with the formation of shorting copper nodules whenever the photovoltage exceeds 380 mV, corresponding to the free energy at 25°C of the reaction (37):
cu,s
-
cus + c u .
(9)
Fortunately, the voltage across the cell corresponding to the power maximum is below this critical electrochemical potential ; however, even in this case, electronic instabilities occur (36). These are associated with electron trapping at interface states between the two semiconductors and at states in the CdS near the heterojunction. The trapping occurs as a result of photoexcitation of electrons by long wavelength radiation and produces a time-dependent fatigue effect. This is compensated by hole injection into the CdS produced by short wavelength excitation and subsequent untrapping, producing a time-dependent sensitivity increase. Thus, the electronic instability is highly sensitive to the spectral distribution of the exciting radiation. For a wellsimulated solar spectrum, the competing trapping-untrapping processes are in relatively good dynamic balance and the instability manifests itself as a minor, long term net fatigue effect. A highly similar thin film heterojunction type of solar cell employing a
ADVANCES IN SATELLITE COMMUNlCATIONS
149
heterojunction between p-Cu,Te and n-CdTe is also under development (38). The performance and advantages are fairly similar to those of the p-Cu,S/ n-CdS cell. A major difference physically from the latter cell is that the effective optical excitation takes place in the CdTe layer, the Cu,Te serving as a higher bandgap window and p-type contacting material.
C. Energy Storage Spacecraft power during launch and during eclipse periods associated with vernal and autumnal equinoxes is generally supplied by rechargeable nickel-cadmium batteries. Since the weight of the batteries is comparable to that of the solar array, whereas power is drawn from them for only about one percent of the total mission time, improvements in energy storage are badly needed. The most promising approach to this problem appears to be use of a fuel cell i n a rechargeable mode. Development work on hydrogenoxygen cells for this purpose is being vigorously pursued.
D. Transponder Electronic Derices With reference to Fig. 5, it may be noted that the amplifying devices typically employed in satellite transponders are tunnel diodes and traveling wave tubes. Conventional crystal oscillators, frequency multiplier circuits, and mixer diodes are also employed to effect the frequency translation from the up-frequency (6 GHz) to the down-frequency (4 GHz).
I . Turinel Diodes The choice of tunnel diodes for the front end of the satellite receiver has been dictated by the need for linear amplification at high carrier frequency (6 GHz), large bandwidth (500 MHz), and relatively low noise. A larger area, higher current version is also suitable as a linear amplifier at the downconverted frequency (4 GHz) but is limited in power output; hence, further gain i n a low level TWT is required prior to subdivision of the full bandwidth into smaller channels in the transmitter. The tunnel diode consists of aplnjunction. both sides of which are doped so highly that the Fernii level resides within the conduction band in the n-side and within the valence band in thep-side. Under these conditions the junction width is so small that electrons readily penetrate the barrier via tunneling. Thus, biasing the n-region negatively with respect to the p-region causes a net electron current flow from n to p . As the forward bias is progressively increased. at first the electron reservoir " sees" an increasing density of empty states i n the p-material, and later a decreasing density as the reservoir rises
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P. L. BARGELLINI AND E. S. RITTNER
above the top of the valence band. At still higher biases, both electrons and holes have sufficient energy to surmount the barrier. The result is a currentvoltage characteristic of the type shown by curve A in Fig. 12, the negative resistance portion of which is employed i n the amplifier. 24 2.2
2 0 1.8
-
16
E
14
1
a
i 00
4
06
1i
04
02 0
1 100
200
300
400
500
600
VOLTAGE ( m v )
FIG.12. I-V characteristics for typical Ge tunnel diode at 300 K . Curve A, Initial; curve B, after degrading; curve C, after extensive degrading.
While tunnel diodes have been fabricated with a large variety of seniiconductor materials (e.g., Ge, Si, GaSb, GaAs, InAs, InSb, PbTe, Sic), germanium (the material with which the original discovery was made) (39) is most commonly employed in commercial units. Since germanium is an indirect bandgap semiconductor, one would expect on theoretical grounds (40)phonon assisted tunneling and a tunneling probability some three orders of magnitude lower than for the direct bandgap case. In the case where the n-type side of the Ge is doped with antimony, fine structure is observed (41) i n the 1- Vcharacteristic at very low temperatures at biases corresponding to the energy of acoustic phonons. Moreover, the observed tunneling current is only in the microampere range. When the n-type dopant is changed to arsenic (the choice in commercial diodes) or phosphorus, the fine structure disappears and the tunneling current increases to the millianipere range. While the higher solubility (42) in Ge of As and P relative to Sb is certainly an important factor, it does not raise the Fermi level sufficiently to populate
ADVANCES I N SATELLITE COMMUNICATIONS
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the higher lying (0, 0, 0) valley with electrons so as to permit direct tunneling between bands. To avoid this difficulty (and a similar one in silicon where the energy separation between corresponding bands is even greater), it is postulated ( 4 3 ) that As and P i n Ge (and most donor impurities in Si) produce strongly localized central cell potentials which introduce components into the electron wave function from many points in momentum space, including all of the valleys. This presumably would allow tunneling into any region of momentum space consistent with energy conservation, a phenomenon labeled “ impurity assisted tunneling.” A quantitative theory and supporting experimental proof of this postulate are still lacking. I t has recently been discovered (44) and verified (45) that there is a negative capacitance component, C,, associated with the band to band tunneling regime. The region of existence of the negative capacitance coincides with the region of negative resistance and both quantities peak together. On theoretical grounds it has been shown (44) that
c, = gr, where g is the differential negative conductance and Y is a characteristic time. Interpretation of measured values ( 4 4 , 4 5 ) of C, and g leads to values of the time constant of the order of 100 psec and 10 psec, respectively, much longer than expected tunneling transit times. Moreover, the value of Y varies with bias, both facts suggesting that the measured Ys are probably circuitdiode time constants. At higher forward biases, i n the neighborhood of the valley of the I-V curve, the observed current is higher than the expected sum of the band to band tunneling and the minority carrier injection currents. This “excess” current is of considerable concern, as the growth of this quantity during diode service is the major cause of diode degradation. The origin of the excess current is tunneling at constant energy between conduction electrons and localized energy states associated with defects within the junction, followed by a recombination-like transition to the valence band (or a transition of a conduction electron to a deep lying localized defect state followed by tunneling into the valence band). Since the barrier height is reduced as the forward bias is increased with an attendant large increase in tunneling probability, a relatively modest defect state density can contribute an excess current component comparable to the band tunneling component. This may increase the magnitude of the negative resistance at the diode operating point, thus causing loss in gain (see curve B of Fig. 12). In the limit of extensive defect introduction into the junction, the negative resistance may even disappear (curve C, Fig. 12). Diode behavior with respect to valley current increase is widely varied. I n extreme cases one diode may exhibit an unacceptable increase even under
152
P. L. BARGELLINI AND E. S. RITTNER
shelf storage while another withstands the highest temperature allowed by the diode construction (230°C) without significant change. A recent extensive experimental study (46) of this question has produced considerable physical insight and has provided preselection guidelines for weeding out potential failures. The method of fabrication of the diode and the resulting physical structure have been of considerable importance in developing this insight. The diode is fabricated by ball alloying of an arsenic tin alloy into a gallium doped p-type Ge substrate at relatively low temperature to minimize diffusion and to produce a sharp junction. The small junction area required for high frequency performance (- lo-’ cm’) is then obtained by etching the germanium under the alloy ball, producing the mushroom-like structure shown in the scanning electron micrograph of Fig. 13. The “cap” diameter in this instance is about 50 pm, the neck diameter at the junction is about 5 pm, and the length of the “stem” is about 25 pm. Contact to the ball is made by means of a metal mesh supported by plastic rods (not shown in the micrograph). Thus, the junction is subjected to rather substantial stress which can vary considerably among diodes. When the stress exceeds the elastic limit, and after an incubation period, plastic deformation occurs. The initial phase of this deformation, called
FIG.13. Scanning electron micrograph of typical 6 GHz tunnel diode. Magnification 1520 x , (Courtesy of T. D. Kirkendall.)
ADVANCES IN SATELLITE COMMUNICATIONS
153
creep, results in generation and motion of dislocations and point defects. Arrival of these defects in the junction region produces the allowed energy states leading to the excess current. The incubation time, t i , before which the introduced strain is very small and after which the strain increases linearly with time, is given by the following equation: ti = cd exp[(Q - aa)/kT],
(1 1)
where c and a are constants, d is the junction diameter, Q is an activation energy, and c is the stress. At a critical stress in the neighborhood of lo9 dyn-cm-2, the exponent changes sign and correspondingly the incubation time flips from extremely large to extremely small values. This explains the wide variation in excess current growth from diode to diode. Moreover, if diodes are heated to an elevated temperature (in the range of 100-140°C) for appropriate times (of the order of 1-100 hr), and the room temperature valley current monitored, it should be possible to select out the diodes with higher built-in stress that constitute potential failures. The noise of the 6 GHz tunnel diode amplifier is about four times higher than that corresponding to the ultimate noise limitation imposed by fluctuations in the radiation field incident upon the satellite antenna. Improvement in this regard is possible with the use of an uncooled parametric amplifier with some penalty in weight and power. Pump reliability is also of concern here. Another possibility is the use of the Schottky barrier gate gallium arsenide field effect transistor (47) which is undergoing rapid development and which exhibits advantages of simplicity (three terminal devices), small size, and low power consumption. At higher frequencies, the increased fragility of the tunnel diode may dictate other choices for the receiver front end. Again, an uncooled parametric amplifier or an integrated circuit version of a down-converter are prominent candidates.
2. Traveling Wave Tubes The choice of a traveling wave tube for the output stage of the transmitter section of the repeater has been dictated by the needs for broad bandwidth, high gain, light weight, long life, high reliability, and high efficiency. This latter requirement is of the utmost importance in the output stage which consumes more power than any other constituent of the satellite. A low level traveling wave tube employed in the receiving section of the repeater, although still relatively efficient, is designed for moderately low noise. Figure 14 displays photographs of the completely encapsulated output stage TWT employed in INTELSAT I11 transponders, and of a cross-sectioned tube identical to those employed in the output stage of JNTELSAT IV
154
P. L. BARGELLINI AND E. S. RITTNER
FIG.14. INTELSAT Ill encapsulated output TWT and cross-sectioned INTELSAT IV output TWT.
transponders. With reference to the photograph of the cross-sectioned tube, the electron gun is visible on the left, the helix surrounded by the periodic permanent magnetic (Pt-Co) beam focusing structure is visible in the center, straddled by the input and output connectors, and the collector portion is seen on the right. The performance characteristics of major interest are listed in Table I. The relatively high efficiency is a consequence of low beam interception by the helix, a high beam-helix interaction efficiency, and the use of a single stage of collector depression. The latter refers to the use of a retarding field between helix and collector to reduce the kinetic energy of the landing electrons and the corresponding thermal dissipation in the collector. The slightly lower efficiency in the INTELSAT IV tube is a deliberate trade-off for a lower phase retardation produced by application of less overvoltage. (Overvoltage is the TABLE I PERFORMANCE CHARACTERISTICS OF TRANSMITTER TWT’s Characteristic Overall efficiency (%) Phase retardation (deg) Overvoltage (%) Small signal gain (dB) Saturation output power (W) Bandwidth used (MHz) Weight (kg) Design life (yrs)
INTELSAT 111 (Tube #H235)
INTELSAT IV (Tube #H261)
34-36 40 4-5 56 12-23 250 0.45
30-32 25 0-1 65 5-6 40 0.68 10
I
ADVANCES IN SATELLITE COMMUNICATIONS
155
voltage applied to the helix in excess of that which corresponds to exact synchronism between the electron beam and the RF wave.) The lower phase shift reduces nonlinearity in the phase shift versus input power characteristic which, in turn, reduces the conversion from amplitude to phase modulation and the accompanying intermodulation interference. The higher gain in the INTELSAT 1V tube results from an increase in the length of the slow wave structure. The lower power output and much smaller bandwidth used in this tube are consequences of the system design, which calls for twelve output transmitters in INTELSAT IV relative to the two employed in INTELSAT 111. This yields advantages of flexibility, less attrition percentagewise in the event of tube failure (also mitigated by the use of 100% redundancy), and amelioration of the problem of cross-talk. The design life is mainly associated with the life ofthe cathode. The cathode employed is an oxide type with a double carbonate coating upon a highly purified nickel base containing a few percent of tungsten and about 0.1% of zirconium activator. The intricate chemistry resulting from the simultaneous presence of the tungsten and the activator, and the reasons why this results in a very well activated and a long-lived cathode, have been discussed long ago (48). Operation of the cathode at a temperature just sufficient to supply the emission needs is also essential to long life, but cannot be achieved unless painstaking efforts are made to avoid a poisoning ambient inside the tube. These efforts include the use of all metal and ceramic parts to permit vigorous outgassing and exhausting of the tube during processing, a getter, an ion barrier between the accelerating anode and the helix provided by appropriate potentials on these electrodes, and extremely careful focusing of the electron beam as it traverses the helix to minimize interception and resulting outgassing. The depressed collector also serves as a sink for ions. The most troublesome aspect of the TWTs, the nonlinearity in the amplitude input-output relationship and in the phase-input power relationship, arises from inherent nonlinearities in the beam-wave interaction. The former arises near the end of the helix from the space charge field associated with the tightly bunched electrons, whereas the latter occurs over the entire length of the helix. Hence, the effects with respect to intermodulation generation tend to be cumulative (49).
VIII. MATERIALS TECHNOLOGY A very wide variety of materials is employed i n satellites, ranging from metals to plastics. All of them have to be space qualified (as do indeed all of the electron devices in the spacecraft),which means passing tests for volatility in high vacuum, vibration, mechanical shock, thermal cycling simulating eclipse, and radiation damage. A few interesting examples of materials employed in INTELSAT IV will be presented here.
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P. L. BARGELLINI AND E. S. RITTNER
The surface structural members of the satellite body are constructed from aluminum honeycomb with a very high rigidity to weight ratio. Solar cells are attached to the cylindrical solar panel surfaces with a flexible epoxy adhesive. Use is also made of the honeycomb construction for the platform for the antenna farm which is faced with an 0.25 mm thick aluminum sheet attached via an epoxy polyamid film. The platform is covered with a conical sun shield constructed from a 3 mm thick aluminum core honeycomb faced with an 0.008 mm thick aluminum sheet to which metallized quartz mirrors are bonded. The metallization is vapor-deposited silver protected by a nickel alloy coating. The rotor shaft for the despun platform is made of titanium. The global antennas are fabricated from aluminum honeycomb with an 0.05 mm thick aluminum bonded inner face sheet and a fiberglass outer face sheet. The spot beam antenna parabolic reflectors are also made of aluminum honeycomb sandwiched between epoxy fiberglass surfaces with a gold coated polyester mesh embedded in the concave face sheet. The reflector feed horn is attached to an elliptical waveguide of boron fiber-reinforced epoxy which also serves as a structural support for the horn. The propulsion system makes use of a number of exotic alloys. The fuel tanks are made of a titanium alloy containing 6 wt.% aluminum and 4 wt.% vanadium because of its chemical inertness in the presence of hydrazine and its relatively light weight and high mechanical strength. The connection between the tanks and various stainless steel components in the propulsion system is made with special co-extruded tubing of graded composition with length, starting with the Ti alloy and ending with the steel. Valve seats and poppets are constructed from tungsten carbide in a cobalt matrix. The thrusters are constructed of Inconel alloy 600 (72% Ni, 14-17% Cr, 6 - 8 x Fe, 1.75-2.75”/, Nb, 0.1% C max) because of the need for hot strength. Lubrication in the presence of the ultrahigh vacuum space environment constitutes an especially difficult problem. The bearings for the despun antenna platform are lubricated with Vac Kote, an apiezon C oil with a proprietary additive. The nutation damper is lubricated with MoS, . I n the electronic units, the circuit boards are constructed from copperclad glass epoxy laminates with electro-plated solder protection. The multiplexer waveguide filters in the transmitter portion of the repeaters are constructed of silver plated Invar because of its low thermal expansion coefficient which facilitates meeting the stringent requirements on frequency bandpass stability. Since the filters are major contributors to the weight of the communications package, extensive efforts are underway to develop new materials or new approaches to reduce filter weight. Prominent possibilities are fiberreinforced composites of low thermal expansion or surface wave acoustic filters, respectively.
ADVANCES IN SATELLITE COMMUNICATIONS
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IX. FUTURE TRENDS Communications via satellite will expand greatly in the future. Forecasts based on prediction of traffic growth and extrapolations of present technology are useful to define a lower bound of this expansion. The definition of a higher bound is more difficult, but can be attempted in terms of possible advances of new technologies resulting from a vigorous research and development program. As new services such as domestic and international videophone, high speed data transmission, audio and visual conference calls, mobile aeronautical and maritime services, telemail, library and educational services, etc. will be made possible, even the most optimistic prediction of the growth of present services and traffic patterns will be exceeded. Future communications satellites will provide communications capacity much greater than those available today. Progress will be made possible by advancements over a broad technological front. The spacecraft stabilization will no longer be obtained by spinning the spacecraft. Inertia wheels will provide stability along the three axes of the spacecraft and the stability will be augmented by microthrusters controlled by orientation sensors. Thus, extremely stable space platforms will be available capable of pointing the high gain antenna beams in the desired directions, with tolerances of at least 0.1" or better. Another advantage of this approach is the theoretical n-fold increase i n electrical power over that obtainable from solar cell panels of equal area on a spinning spacecraft. With multiple independent beams, the same frequency band can be reused many times with a resultant increase of communications capacity. Furthermore, a twofold increase of communications capacity can be obtained by using orthogonal polarization in each beam. A large fraction of the increase in communications capacity will be made possible by adding to the presently available functions of reception, amplification, and transmission on the spacecraft, the switching of signals from and to different earth stations on board, and by using all-digital techniques with time division multiplexing combined with time and/or space division multiple access techniques. Substantial contributions to all these advances in terms of more efficient, more reliable, and novel electron devices are expected.
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P. L. BARGELLINI AND E. S. RITTNER
ACKNOWLEDGMENTS Thanks are due to the Members of the Technical Staff at COMSAT Laboratories, on whose work this contribution is largely based. One of the authors (E. S . Rittner) is particularly grateful to A. Meulenberg, R. Strauss, and P. Varadi for supplying information, and to them and to R. Arndt, P. Fleming, J. Lindmayer, A. Revesz, R . Rostron, and A. Verbin for critical review of parts of the manuscript.
REFERENCES 1. J. R. Pierce, “The Beginning of Satellite Communications.” San Francisco Press, San Francisco, 1968. 2 . A. C. Clarke, Wireless World, 51, 303 (1945). 3 . J. B. Wiesner, in “ Lectures on Communication System Theory” (E. J. Baghdadi, ed.), Chapter 22. McGraw-Hill, New York, 1961. 4 . H . Rosen, in “Space Communications” (A. V. Balakrishnan, ed.), Chapter 17. McGrawHill, New York, 1963. 5 . S . Metzger, Astronaur. Aeronaut. 6 , 42 (1968). 6. W. L. Pritchard, IEEE Int. Conv. Digest, New York, March 22-25, 1971, pp. 24-25, (Paper I A.2.) 7 . J. G . Puente, W. G . Schmidt, and A. M. Werth, Proc. IEEE 59, 21 8 ( I 971). 8 . K. L. Plummer, Spaceflight 12, 322 (1 970). 9. G. E. Mueller, Proc. IRE 48, 557 (1960). 10. R. W. Sanders, Proc. IRE 48, 575 (1960). 11. A. G . Smith, Proc. l R E 4 8 , 593 (1960). 12. Radio Spectrum Utilization in Space, Joint Technical Advisory Council of the IEEE ai7d rhe Electronic Industries Association, IEEE, New York, 33 (1970). 13. W. E. Bradley, Astronaut. Aeronaut. 6, 34 (1968). 14. P. L. Bargellini, IEEE Inr. Conf. Commun., Boulder, Colorado 5, 37-25 (Paper 37-4). (1969). 15. W. G. Schmidt, et al., IEEE Int. Conf. Commun., Boulder, Colorado 5, 15-13 (Paper 15-3).(1969). 16. R . W. Rostron, AIAA 3rd Commun. at ell ire Systenis Conf., Los Anyeles, California. Paper 7 0 4 8 I ( 1970). 17. F. S . Johnson, J . Meteorol. 11,431 (1954). 18. M. P. Thekaekara and A. J. Drummond, Nature(London)Phys. Sci. 229, 6 (1971). 19. D. P. LeGalley and A. Rosen, “Space Physics,” p. 673. Wiley, New York, 1964. 20. D. M. Chapin, C. S. Fuller, and G. L. Pearson, J . Appl. Phys. 25, 676 (1954). 21. W. Shockley and W. T. Read, Jr., Phys. Rev. 87, 835 (1952). 22. E. S . Rittner, in “Photoconductivity Conference” (R. S . Breckenridge, B. R. Russell, and E. E. Hahn, eds.), pp. 250-256. Wiley, New York, 1956. 23. J. Lindmayer, COMSATTech. Rev., 2, 105 (1972). 24. A. Meulenberg and D. Curtin, private communication. 25. R . Arndt and L. Westerlund, COMSATTerh. Rev. 1, 117 (1971). 26. R. G. Downing, 5th Photovolraic Specialists, 2, D-7, Conf., Greenbelt, Md., October 1965; R. L. Statler and D. J. Curtin, IEEE Trans. Electron Devices 18, 412 (1971); R. W. Rostron, Energy ConverJion, in press. 27. J. Wysocki, Conf. Record 6th Phorovolraic Specialists Conf. 3, 96 (1967).
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E. S. Rittner, Phys. Reo. 96, 1708 (1954). R. Gremmelmaier, Z . Naturforsch. A 10, 501 (1955). A. R . Gobat, M. F. Lamorte, and G. W. McIver, IRE Trans. Mil. Electron. 6,20(1962). B. Ellis and T. S . Moss, Solid Sfate Electron. 13, 1 (1970). T. L. Tansley, Opto-Electron. 1, 143 (1969). H. Okimura and R. Kondo, Jap. J . Appl. Phys. 9, 274 (1970). D. C . Reynolds, G. Leies, L. L. Antes, and R. E. Marburger, Phys. Rev. 96, 533 (1954). A. E. Potter, Jr. and R. L. Schalla, 6th Photovoltaic Specialists Conf. Record, 1,2434, March 1967. 36. J . Lindmayer and A. Revesz, Solid Sfate Electron. 14, 647 (1971). 37. L. Clark, R. Gale, K. Moore, R. J. Mytton, and R. S. Pinder, Proc. Colloq, Int. Cellules Solaires, Toulouse, July 1970, pp. 605-621, Gordon and Breach, London (197 1). 38. J. Lebrun, Colloq. Int. Cellules Solaires, Toulouse, July 1970, Conf.Record 8th Photovoltaic Specialists Conf., Seattle, August 1970, pp. 33-39. 39. L. Esaki, Phys. Reo. 109, 603 (1958). 40. E. 0. Kane, J. Appl. Phys. 32, 83 (1961). 41. J. J. Tiemann and H. Fritzche, Phys. Rev. 132, 2506 (1963). 42. F. H . Trurnbore, BellSystern Tech. J. 39, 205 (1960). 43. H . Fritzsche, “Tunneling Phenomena in Solids” (E. Burstein and S. Lundquist, eds.), pp. 167-180. Plenum, New York, 1969. 44. B. Pellegrini, Aha Freq. 39,429 (1 57E) ( I 970). 45. P. L. Fleming and L. E. Foltzer, Int. Microwave Syrnp. (G-MTT), Washington, D.C., May 16-20, 1971, Paper XI-7 (unpublished). 46. A. G . Revesz, J. Reynolds, and J. Lindmayer, Solid State Electron. 14, 1137 (1971). 47. W. W. Hooper, R. D . Fairrnan, and N . G . Bechtel, Int. Electron Devices Meeting, Washington, D.C., October 11, 1971, Abstract 3.5, p. 32 (1971). 48. E. S . Rittner, Philips Res. Rep. 8, 184 (1953). 49. A. L. Berman and C . E. Mahle, IEEE Trans. Comm. Technol. 18 (I), 37 (1970).
28. 29. 30. 31. 32. 33. 34. 35.
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Acoustoelectric Interactions in 111-V Compound Semiconductors W. J. FLEMING*
AND
J. E. ROWE
Electron Physics Laboratory, Department of Electrical and Computer Engineering, The University of Michigan, Ann Arbor, Michigan
I. Introduction. ..................................... A. Historical Background and Device Applications. ........................ B. Acoustic Noise Generation Manifested by Microwav 11. General Theory of Off-Axis Acoustoelectric Interactions A. Introduction.. . . . . . . . . . . . . . . . . . ......................... ......................... B. Derivation of the General Dispersi C. Reduction to an Equivalent One-Dimensional Dispersion Equation. . . . . . . . . D. Derivation of the rf Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Exact Solution of the Acoustoelectric Interaction for Collinear Static Fields. . . . . A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Derivation of the Acousto-Helicon Dispersion Equation . . . . . . . . . . . C. Solution of the Acousto-Helicon Dispersion Equation. IV. Solution of the Acoustoelectric Interaction for Arbitrarily ........... Fields and On-Axis Acoustic-Wave Propagation. . .
.......................... B. Solution for the rf Conductivity.. . . . . . . . . . .
162 166 166 170 174 176 177 185
. . . . . . . . . . 186
C. Solution for the Growth Rate of the Acousto D. Incorporation of Empirical Field Factors into the Theory of Acoustoelectric Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Solution of the Acoustoelectric Interaction for Arbitrarily Oriented Static Fields and Off-Axis Acoustic-Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Introduction.. . . . . . . . . . ......................... B. Solution for the Effective oupling Parameters. . . . C. Solution for the Growth Rate of the Off-Axis Acoustoelectric Interaction. . . . VI. Solution of the Acoustoelectric Interaction for Electron-Hole Carrier Transport and Off-Axis Acoustic-Wave Propagation. . . . . . . . . . . . . . . . . . . . . . . . A. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Solution of the Static Carrier Transport System. . . . . C. Solution for the Growth Rate of the Acoustoelectric VII. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..........................................
199 206 206 207 215 234
244
* Present address: Research Laboratories, General Motors Corporation, General Motors Technical Center, Warren, Michigan 48090. 161
162
W. J. FLEMING AND J. E. ROWE
I.
INTRODUCTION
A . Historical Background and Device Applications The collections of charge carriers responsible for the transport of electricity in a semiconductor interact with the crystal lattice and exchange energy. In many cases the collections of conduction charge carriers can be almost regarded as an isolated subsystem of the semiconductor which interacts weakly with the lattice and the properties of the subsystem can be studied separately. In this case, the interactions of the drifting conduction carriers with the lattice are treated as collisions where the carriers gain or lose units of quantized lattice-vibrational energy called phonons. However, if the subsystem of conduction carriers is strongly coupled to the lattice by means of acoustoelectric effects, interactions between the drifting carriers and the acoustic lattice-vibrational modes may give rise to an acoustoelectric gain mechanism. In acoustoelectrically active semiconductors, piezoelectric and deformation potential effects generate internal electric fields which accompany the acoustic modes of lattice vibration. Whenever the conduction carriers are drifting faster than the velocity of sound, the space-charge electric fields of the drifting carriers couple to the acoustic waves of the lattice and acousticwave amplification may occur. Analogous to the electron-field interaction in a traveling-wave tube, the acoustic waves grow in amplitude, therein making possible the realization of a solid-state traveling-wave amplifier. Acoustic-wave amplifiers have recently emerged as an important component in signal processing systems which require long signal delay. The essential elements of the acoustic-wave amplifier consist of a bar-shaped specimen of an acoustoelectrically active semiconductor with ohmic contacts at each end across which the drift field is applied. Acoustic transducers are fabricated directly on top of these contacts, and provide the input and output ports of the amplifier. Since the signals propagate at the sound velocity which is some lo5 times lower than the velocity of electromagnetic waves, a 2-cm long specimen will introduce delay times on the order of 10 psec. Moreover, the existence of the acoustic amplification mechanism compensates for transducer and delay media loss, thereby permitting the development of practical devices. Acoustic-wave amplification was first observed in 1961 by Hutson et al. (Z), using CdS as the acoustoelectrically active semiconductor. For several years thereafter experiments were largely confined to this material because of its superior acoustic and semiconducting properties. The results of these early investigations are discussed in detail by McFee (2). In the course of
ACOUSTOELECTRIC INTERACTIONS
163
these investigations it was found that after sufficient amplification of the acoustic waves, acoustoelectric forces bunch the drifting conduction carrier stream into narrow high-electric field domains which manifest themselves as rf current oscillations. These oscillations have in turn been the subject of investigation, since their presence limits the utility of the acoustic-wave amplifier . Acoustoelectric current oscillations i n 111-V compound semiconductors were first investigated by Bray et al. (3). They found that the presence of a transverse magnetic field dramatically enhances the acoustoelectric instability in high-mobility semiconductors such as n-type lnSb and GaAs. As a result of this observation, Bray et a / . (3) suggesfed that the transverse magnetic field gives greater synchronism between the conduction carrier drift velocity and the acoustic-wave propagation velocity. Following this suggestion, both Steele (4) and Turner et a ] . (5) reported identical theories which show that this is indeed the case. The transverse magnetic field reduces the effective rf mobility of the conduction carriers, thereby decreasing the diffusion relaxation time constant and increasing the dielectric relaxation time constant. Under these conditions maximum acoustoelectric gain can occur at significantly lower drift velocities with correspondingly low dc power dissipation levels. This discovery has stimulated interest i n the acoustoelectric properties of high-mobility 111-V compound semiconductors. Particular attention has been given to n-type InSb which, when cooled to a temperature of 77"K, exhibits electronic mobility values on the order of 1 x lo6 cm2/V-sec. High-gain acoustic-wave amplification has been realized by several investigators (6-8). Typically, in the presence of a transverse magnetic field greater than 3 kG, there exists 60 to 70 dB/cm of electronic acoustic gain at frequencies of I to 2 GHz. However, the electronic gain is offset by an acoustic attenuation loss of 10 dB/cm inherent to the semiconductor, and by transducer loss of approximately 40 dB. Nevertheless, net terminal gain in excess of 10 dB and signal delay times up to 10 psec are obtainable. The aforementioned device depends on the bulk amplification of acoustic waves. a process which is inherently inflexible. Since the acoustic signal is inaccessible during its transit time i n the semiconductor bulk, these devices are confined largely to fixed delay components, whereupon additional circuits are required to further process the signal. On the other hand, the recent development of acoustic surface-wave technology permits multiple direct access to the acoustic waves for signal-processing functions. A variety of microwave system components such as amplifiers, isolators, and phase shifters can be constructed on a single substrate package, therein forming circuits capable of autocorrelation, Fourier transformation, and correlation functions (9).
164
W. J. FLEMING AND J. E. ROWE
B. Acoustic Noise Generation Manifested by Microwave Radiation A degrading phenomenon, common to both bulk-type and surface-type acoustic-wave devices is the generation of spurious off-axis acoustic waves. These off-axis waves manifest themselves as acoustic noise, therein masking the properties of the active on-axis acoustic wave. Active on-axis acoustic waves traverse well-defined paths along principal crystal axes. The characteristics of these waves are the subject of the majority of investigations on the acoustoelectric interaction. In the present work, however, the characteristics of off-axis acoustic waves are studied, and considerable physical insight is made possible by investigating one specific example in detail. Particular attention is given to the acoustoelectric interactions which occur in n-type InSb. This 111-V compound semiconductor is especially interesting due to the myriad of phenomena which manifest the existence of the acoustoelectric interactions. Electrical instabilities such as current oscillation (10,11), microwave noise radiation (12,13) and acoustic-wave noise (14) all are manifestations of the acoustoelectric interactions in InSb. A strong on-axis acoustoelectric interaction dominates the off-axis interactions in specimens of n-type InSb which are longer than approximately 1 cm. In shorter length specimens, off-axis acoustoelectric interactions are nucleated and are manifested by the generation of microwave noise radiation (1.?,24). No acoustic transducers are required to generate this radiation. Electrical contacts are made on the ends of a bar-shaped InSb specimen and a static electric field is applied. If the specimen is cooled to 77°K and a static magnetic field is present, microwave noise radiation is spontaneously generated when the applied fields exceed certain threshold values. The radiation has been conclusively related to acoustoelectric effects by several investigators (10,12-14); it occurs whenever the electric field exceeds a. few volts per centimeter and a magnetic field greater than a kilogauss is oriented transverse to the direction of current flow. A physical model of the radiation process can be developed on the basis of the following observations. Due to the high electron mobility in n-type InSb, the presence of a large transverse magnetic field component gives rise to localized high-electric-field regions at diagonally opposite corners of the crystal (15). As a result, localized impact ionization (16) or hole injection (17) may exist at the localized highfield regions and plasma-type instabilities (18) may occur, but rapid recombination of the nonequilibrium holes would abruptly quench the interaction. Furthermore, it has been found that the maximum intensity of the radiation does not occur directly at the contacts, but rather is observed near the center of the specimen (13). In an investigation of the acoustoelectric current oscillations in long InSb specimens, Seifert (19) found that, under conditions for which the radiation will exist, there is a statistical buildup of acoustic
165
ACOUSTOELECTRIC INTERACTIONS
domains in the bulk of the InSb specimens. These observations indicate that the microwave radiation is dependent on the existence of acoustic noise disturbances. Thus, the presence of both a localized plasma interaction region and a bulk acoustoelectric interaction has been observed experimentally in conjunction with the radiation. The radiation process is depicted in the following way. As shown in Fig. 1, the application of a transverse magnetic field B, creates a region of localized plasma at the edge of the contact. I t is assumed that either plasmatype interactions or simply shot noise acts as a nucleation source which RAUATION EMANATES FROM TME REGION OF BULK ACOUSTOELECTRIC CATMOOE
/
/
FIG.1. Physical model of the generation of radiation by off-axis acoustoelectric amplification.
excites, via piezoelectric forces, acoustic disturbances. These disturbances, represented by the wavevector q, are continuously nucleated and emanate from the localized plasma region into the bulk of the specimen. After sufficient amplification of the acoustic disturbances, acoustoelectric forces on the drifting electrons create transient oscillating space-charge dipoles which act as point radiators and generate the microwave emission. Hence, the acoustoelectric instability by itself is not sufficiently strong to excite the microwave radiation, and large-amplitude acoustic disturbances must first be generated by localized plasma-type instabilities. In the localized plasma regions, the streaming carriers flow in all possible off-axis directions due to Hall-field shortingeffects at the contacts (15); the on-axis direction corresponds to travel along the long dimension of the specimen. Representation of the acoustic disturbances by the set of all possible off-axis acoustic waves takes the contact shorting effects into account. As seen in Fig. 1, the off-axis acoustic wave q propagates in the direction (defined by the inclination angle [) for which the acoustoelectric gain is maximum (20).
166
W. J. FLEMING AND J. E. ROWE
The physical mechanism of the radiation is the electromechanical conversion of acoustic noise energy into microwave energy by means of the acoustoelectric interaction. Thus, it is assumed that the intensity variations of the radiation are proportional to the corresponding variations of the acoustoelectric gain.* In the present work, attention is given to the process of off-axis acoustoelectric interaction. 11. GENERAL THEORY OF OFF-AXIS ACOUSTOELECTRIC INTERACTIONS
A . Introduction
In this section the general equations which describe the acoustoelectric interaction between the streaming carriers and the propagating acoustic wave in a cubic 111-V compound semiconductor will be derived (mks units are used throughout). A small-signal theory which is valid in the bulk of the semiconductor is adequate because of the weak piezoelectric coupling in 111-V compound semiconductors which limits the growth rate of acoustoelectric instabilities. Acoustoelectric coupling can also occur via the deformation potential. However, at the frequencies of interest, f 5 10 GHz, the piezoelectric coupling is more than an order of magnitude stronger than the deformation potential coupling; therefore, these effects are neglected in the present study. Details of the piezoelectric and elastic tensor notation used here are discussed by Mason (21). B. Derivation of the General Dispersion Equation
It is assumed that acoustic disturbances exist in the bulk of the semiconductor and can be depicted as plane waves. As shown in Fig. 2, the acoustic wave propagates in the direction 4 with the displacement
6 = 6'exp(jor - j q * x), where 6' is the initial nucleation value of the acoustic-wave displacement, q = q ( M , -trn& + n.2,) is the wavevector, q = w/u, is the wavenumber, v, is the sound velocity and 1, m, and n are the respective direction cosines of the propagation direction 4 = q/q which is measured relative to the crystal coordinate system x = (xl, x 2 , x3).
* Indeed, recent work by T. Ishii [Radiation of electromagnetic waves in piezoelectric semiconductors with amplified sounds. J . Phys. SOC.Jup. 32, 574 (1972)l verifies that the conversion of acoustic energy to microwave radiation is sufficiently intense to account for the observed radiation levels. The radiation is generated by space-charge dipoles, occurring in the current flow, which are created by acoustoelectric forces which stimulate the afore-mentioned interaction.
167
ACOUSTOELECTRIC INTERACTIONS
FIG.2. Acoustic-wave orientation in the crystal coordinate system (xl, x 2 , xj).
The semiconductor provides an elastic medium wherein the acoustic displacement 5 is related to the stress T by Newton’s law of motion,
a2ri
p
-=In
dt2
C~T;~ ax, ’
where i and k index from 1 to 3 and pm is the lattice mass density. An electric field perturbation E, supported by the drifting carriers, and the acoustic disturbance 5, supported by the lattice, are coupled together by the piezoelectric equations of state. It is convenient to write the equations of state in the contracted subscript notation (21) as follows:
T, = c,, S, - e,, Ei
(3)
s, + E i k Ei ,
(4)
and Dk = e k ,
where r and s index from 1 to 6, i and k index from 1 to 3, c,, is the elastic tensor, T, and S,yare, respectively, the stress and strain written in contracted notation, e,, and & i k are, respectively, the piezoelectric tensor and the lattice dielectric tensor, and Dk and E, are the electric flux and field vectors. I n the contracted notation, T, and S, are related, respectively, to Tikand Sik in the following manner (21):
Tr=Tii,
S,=Sii
forrs3,
i=r
T, = T i k ,
s, = 2 s j k
for r 2 4,
i#k
(5)
168
W. J. FLEMING AND J. E. ROWE
and
For cubic I TI-V compound semiconductors (zinc-blende structure, crystal class 4 3m), the physical parameters c r S , eir and & i k exhibit a high degree of symmetry, namely (21) c11 c12 c12
0 0
c12
0 0
O
C
0
4
4
0
0 0 0 0
O e 1 4
=
ri
0 ;&
0
O C 4 ,
0 0 0 el4 and &ik
(7)
c44
0 0 0
0 0
0
0 0
0" El I
1.
(9)
Equations (1)-(9) are combined t o give the following set of six equations:
and
[%,1
0 =-jqel4[;
n m
7 a][:]
r,
where the components of the effective elastic tensor a are given by
+ (m2 + n2)cd4, a Z 2= m2c1 + ( I z + n ' ) ~ ~ ~ , a33= nZcl + (12 + r n ' ) ~ ~ ~ , a, 1
=
Pc, 1
a12 = W c 1 2 a13
and
=
+ C44L -k
c44>3
(1 1)
ACOUSTOELECTRIC INTERACTIONS
169
For convenience, Eqs. (10) and (1 1) are rewritten as c‘g = a .
-
6 - j - e14 ( e E) 4
(13)
where c’ = p , ( o / q ) 2 = pmvs2is the effective elastic constant, = e l l is the lattice dielectric constant, and the matrices a and e are as given below:
[;;; ‘12
a=
[. j. 0
‘13
%;
eA
n m 0 1 m 1 0
(15)
Hence the propagation of the acoustic waves defined by Eq. ( I ) is governed by the piezoelectric wave equations, Eqs. (1 3) and (14). A general acoustoelectric dispersion equation is obtained by determining the effective permittivity E‘ of the semiconductor, where E’ is defined by the relation D = E / E’E. The effective permittivity is established solely by the electric field fluctuations supported on the conduction carrier subsystem. If it is assumed that all field fluctuations can be represented by plane waves as in Eq. (l), the wave equation for electric field fluctuations is obtained from Maxwell’s equations, and can be written as
where q is the wavenumber, c1 = ( p l E , ) - ’ ” is the velocity of light in the semiconductor, p l is the lattice permeability, and E , is the lattice dielectric constant. Solution of the transport equations yields the rf conductivity u which is defined by J = u -E. (The details of this solution are discussed i n Section I1.D.) Substituting this result into Eq. (16) gives the effective permittivity E’ which is defined by
where I is the identity matrix and 44. E is a dyadic product. Note that the rf conductivity u must be derived in the crystal coordinate system x = (xl, x 2 , x3). It is most convenient to derive u in a different coordinate system; for example, the system r = (x, y , z ) where the wave propagation direction 4 is parallel to the 2-direction, the transverse magnetic field component B, is parallel to the ?-direction and the carrier drift velocity is in the x-y Lorentz force plane. Then a rotation matrix M which rotates the (x, y , z ) system into the (x,, x2, x3) system must be determined such that r = Max.
170
W. J. FLEMING AND . I . E. ROWE
If u is the rf conductivity in the (x, y, z ) system, it followsthat u’ = M-’ * u *M is the rf conductivity in the (xl, x 2 , x3) system. When Pq. (17) is used, the piezoelectric wave equations, Eqs. (13) and (14), become (a
- c’l) - 6 - j e14 - (e - E) = 0 4
(18)
and ( E ~ E-‘ c l I). E +,jqe,,(e.
5) = 0.
(19)
The general acoustoelectric dispersion equation follows directly from the simultaneous solution of Eqs. ( 1 8) and (19) and can be written as
-
det[(a/c’ - I). C 1 (E’ - I) - ~ ’ e ] = 0,
(20)
where K’ = e:4 /c’E~is the electromechanical coupling constant and e - l is given by
(21) If no piezoelectric coupling exists, x2 = 0 and the piezoelectric equations are no longer coupled, yielding the dispersion equations det(a -dI) = 0 and det(d - I) = 0. Inspection of Eq. (21) shows that, if the acoustic wavevector 4 lies in any plane formed by the unit crystal axes, any combination of I , rn, and n will contain at least one zero and e - ’ is undefined. I n practice, therefore, whenever e - ‘ is undefined it is more convenient to write Eq. (20) as
-
det[(E’ - I) - t i 2 e * ( a i d - I)-’ e ] = 0.
(22)
C. Reduction to an Equirialent One- Dimensional Dispersion Equation The complete solution of the general acoustoelectric dispersion equation is extremely complex and simplifications must be made in order to obtain mathematically tractable results. When the direction of wave propagation coincides with a principal crystal axis such as a (100) or ( I 10) direction, the acoustoelectric dispersion equation can be easily solved. However, if the direction of wave propagation is in an arbitrary off-axis direction, it i s expedient to use an equivalent one-dimensional acoustoelectric dispersion equation. The derivation of the desired result follows. When the relation J = E is used, the wave equation, Eq. (16), and the second piezoelectric equation of state, Eq. (l4), can be combined, thereby yielding the following result: us
171
ACOUSTOELECTRIC INTERACTIONS
It is convenient to define a relaxed permittivity tensor cR and a piezoelectric polarization field vector 8, as A
c R = &[(I
+ J ~ ~ / w E ~and)
With these definitions and the relation 4
G ,6 jye,,(e = W / Z I ~Eq. ,
*
&)/el.
(24)
(23) becomes
The longitudinal electric field EllA E . 4 and the transverse electric field E, 4 E x 4 can be obtained directly from Eq. (25). Taking the scalar product of Eq. (25) with 4 makes the left-hand side zero and yields the longitudinal electric field Ell
+ E,(ER
*
t',). 4 .
Taking the cross product of Eq. (25) with the transverse electric field
4
and noting that
(26)
4
x
4
=0
gives
Although Eq. (27) is not an exact solution for E,, since the electric field also appears on the right-hand side, it is taken as a reasonable estimate for the magnitude of E,. It is clear that El is reduced by a factor of ( L ) , / C ~ ) *z l o p 9 ; this occurs because of the electromagnetic nature of the transverse electric field components. On the other hand, the longitudinal electric field Ell given in Eq. (26) is comparable to the magnitude of the piezoelectric polarization field and will be dominant in the acoustoelectric interaction. I n fact, if the electric field is purely longitudinally polarized, the wave equation reduces to Gauss' law; this is called the quasistatic approximation. If the quasistatic approximation is used. the equivalent one-dimensional acoustoelectric dispersion equation for off-axis waves can be derived from the piezoelectric wave equations, Eqs. (13) and (14). Since only the longitudinal field components are significant, the electric field flux density DII is related to the electric field E l l by the simple expression Dll = - ( ~ l l / j ( o ) E ~ l ~
(28)
where o , ~is the longitudinal component of the rf conductivity tensor u written in the carrier transport coordinate system (x, y , z ) with 4112. Hence, the longitudinal electric field components i n the crystal ( x l ,x 2 , x 3 )coordinate system are Ell= El14= EII(12,
+ n?t2 + d 3 ) .
(29)
172
W. J. FLEMING AND J. E ROWE
It can be shown (22) that the piezoelectric force term on the right-hand side of Eq. (13) has only a perturbation effect on the normal acoustic modes of vibration. Thus, the normal modes of acoustic vibration can be assumed to be only slightly affected by the presence of Elland the zeroth-order normal modes are determined by the uncoupled eigenvalue solutions for (a - c’l).
6 = 0.
(30)
Here, a is defined by Eq. (15) and cn = pmu$ = c’ is any one of the three possible eigenvalue solutions (effective elastic lattice constants in the absence of piezoelectric coupling) which have the corresponding eigenvectors g A which in turn represent any one of three possible acoustic waves. The set of three possible acoustic waves consists of one which is called the longi(3 z 1 and two other waves which tudinal (compressional) wave because are called the fast- and slow-transverse (shear) waves because g,. 4 NN 0. Note that the approximate eigenvalue solutions obtained from Eq. (30) are correct to the order of ef4/(EIaik) z When the piezoelectric force term is included as a perturbation and the quasistatic approximations of Eqs. (28)-(30) are used, the piezoelectric wave equations, Eqs. (13) and (14), become
perturbation term
and
Here, kA = (11,t 2 ,13)Ais any one of the three acoustic-wave eigenvectors and cAis the corresponding eigenvalue. Squaring both sides of Eq. (31) yields
.
first-order perturbation term
negligible term
If the first-order perturbation term is retained and the negligible term is dropped, a perturbation expansion of the square root of Eq. (33) can be made which yields
ACOUSTOELECTRIC INTERACTIONS
where the acoustic displacement scalar 51
A (51 *
5A)”2
173
has been defined as = 151
1.
The right-hand side of Eq. (32) represents the piezoelectric contribution to the electric field but, since the quasistatic approximation is valid, only the longitudinal component of polarization field enters into the acoustoelectric interaction. Thus, taking the scalar product of Eq. (32) with 4 yields the longitudinal field equation
If Eqs. (34) and (35) are combined by elimination of the scalar variables and E,,, the one-dimensional acoustoelectric dispersion equation is obtained, namely, P m o2- I)(]% + 1) - 2 = 0,
(2
where by
K~
is the effective electromechanical coupling constant which is given K’
= ep2/&,ci.
(37)
I n Eq. (37), c1 is the effective lattice elastic constant determined from Eq. (30) and e p is the effective piezoelectric coefficient given by
Since the piezoelectric matrix e is symmetric [see Eq. (15)], the dyadic identity, (e . 4). = (e * &A). 4, holds and is used to obtain Eq. (38). When the values of e, 4 and t1 are substituted into Eq. (38). i t can be written explicitly as 2(mn5, e p = e14
+ / t i t 2 + lmt,), 15iI
(39)
r3
Here, the subscript A denotes that c1, i2, and are the components of the acoustic-wave eigenvector corresponding to the eigenvalue ci [as determined from Eq. (30)] and /, m, and n are the direction cosines which define the acoustic-wave propagation direction 4. The dispersion equation determines the propagation characteristics of‘ the acoustic wave. Equation (36) is the dispersion equation for the onedimensional acoustoelectric interaction and is solved by means of a perturbation procedure. A convective-type instability with phase velocity i n the neighborhood of the sound velocity is considered. Deviations from the sound velocity are small and the wavenumber can be expanded as q
% f3/2Js
+j a ,
I x I < w/u,,
(40)
174
W. J. FLEMING AND J. E. ROWE
where u = aRe+ jut,,, is complex. The acoustic-wave displacement, given by Eq. (I), will vary as exp[ccR,(q* x)]. If uRe> 0, the acoustic wave exhibits spatial growth. Substituting Eq. (40) into Eq. (36) and solving for Re(a) = aRe yield the acoustic-wave growth rate, (41) Examination of Eq. (41) shows that the growth rate of the acoustoelectric interaction depends primarily on the longitudinal rf conductivity (T,, . I n the next section, the transport and field equations which determine (T,, will be given. D. Derivation of the rf Conductivity
The degree of generality of the conduction carrier transport equations which yield the rf conductivity determines the complexity of the theoretical analysis. There is a hierarchy of special conditions which must be considered either for applicability or simply for utility. I n the present work, the transport equations are developed under special conditions of restricted validity and not only is the mathematical analysis simplified, but the simplifications give valuable insight into the physical basis of the acoustoelectric interaction. The following assumptions are utilized in the present analysis (these assumptions are applicable for the conditions of practicable acoustoelectric device operation) : 1. Quantum effects are negligible and a classical description of the carrier transport, which ignores nonisothermal rf effects, is adequate. 2. Nonlocal effects, such as cyclotron resonance, Landau damping, and intercarrier Coulomb interactions, are negligible and a hydrodynamical model of the conduction carrier transport is utilized. 3. The transport equations are linearized by considering only small-signal perturbations X from the static solution X , of all dynamic variables X u . Perturbations are taken i n the form
Xu(r,0
=
Xdr)
+ .Ur, 0
(42)
and are substituted into the field and transport equations. 4. First-order perturbations are equated and solutions for the rf conductivity are found by assuming that the perturbation can be represented by a plane wave (finite boundary effects are ignored) of the form X(r, r)
= X’(r)exp(jwt -
,jq*r),
(43)
where q is the wavevector of the perturbation which is measured in the carrier transport coordinate system r = (x, y , z).
ACOUSTOELECTRIC INTERACTIONS
175
If the above assumptions are used, a hydrodynamical description of the rf conductivity is obtained. The drifting carrier stream and attendant spacecharge waves are described by the equation of momentum conservation, Faraday's law, and the equations of continuity of current which are, respectively [taking small-signal perturbations as indicated in Eqs. (42) and (43113 CV=T
Q, (E + U O x B + v
m
x B,)
qxE=wB
+ j -V T 2 Nq, NO
(44) (45)
and
where E, , J, and V are defined by
E o = p ~ l . ~ g . J o = Q e N o ~ o , V = V + ~ ( W - U , . ~ ) . (47) Here u, (the carrier drift), No (the carrier concentration), E, (the electric field), B, (the magnetic field), and J, (the current density) are the static equilibrium transport and field variables; v, N, E, B, and J are the corresponding first-order perturbed values; the assumed-constant phenomenological constants v, (33, m*,z ' ~ ,and f ; are, respectively, the momentumrelaxation collision frequency, the electron charge, the effective carrier mass, the average thermal velocity of the carrier, and the trapping factor (which is the fraction of space charge Q, N which arises from the drifting conduction carriers); and po is the static drift mobility tensor which must be calculated from a knowledge of the applied field conditions. Since all other parameters are known, Eqs. (44)-(46) are combined to yield one equation which is only a function of v and E, namely,
Here the following definitions have been made :
and
a, = w - j ;u()' q.
176
W. J. FLEMING AND J. E. ROWE
Equation (48) can be rearranged into the form v = p - E and the rf mobility p is obtained. Combining Eqs. (46) yields the result
Equation (49) is only a function of v (all other parameters are specified) and it can be written as J = p * v, where p is the rf charge density. Combining Eqs. (48) and (49), which are written respectively as v = pa E and J = p - v , yields J = p - ( p * E) = E, where u = p - p, the rf conductivity, is the desired result. The analysis can be extended to include the streaming of different carrier species i by assuming that each species separately satisfies its own continuity equation, q . Ji - w Q i N i= 0. In this case, an aggregate rf conductivity can be defined simply as u = utot= u i ,where Ji = u i E and J,,, = Ji . The rf conductivity ui for each charge species i is still obtained from Eqs. (48) and (49); however care must be taken to assure that the proper field and transport variables are assigned to each species. (Note that the static drift velocities of the carriers in a many carrier system must be obtained from a self-consistent analysis of the dc transport equations.) Some general comments are in order here. First, the trapping factorf; represents the fraction of space charge which is mobile and takes into account the effect of the acoustic wave on the electrons residing in bound states (either donors or acceptors) which exist in the energy gap. The trapped space charge is equal to (1 - f ; ) Q , N and it is assumed that charge bunching in the bound states is in phase with the charge bunching in the conduction band. In this case,f, is a real number between 0 and 1. For in-phase charge bunching to occur, the bound states must reach equilibrium with the conduction band in time intervals which are short compared to the period of the acoustic wave. Finally, the extension of the analysis to include different charge species should be qualified. Different charge species i will separately obey their own continuity equations only when their recombination times are long compared to the period of the acoustic wave. For typical acoustoelectric device operation, the first comment regarding trapping effects is restrictive, especially at microwave frequencies; however, the latter comment is always well satisfied. u
s
xi
xi
INTERACTION FOR 111. EXACTSOLUTION OF THE ACOUSTOELECTRIC COLLINEAR STATICFIELDS A . Introduction Although a simplified one-dimensional dispersion equation was derived in Section II,C, it is instructive to solve the general dispersion equation of Section II,B. The special case of acoustic-wave propagation coincident with
I77
ACOUSTOELECTRIC INTERACTIONS
the ( I lo) crystal direction will be solved exactly. This analysis has been motivated by reports of a distinct mode of acoustoelectric interaction in n-type InSb which depends on the presence of a magnetic field oriented parallel to the direction of current flow (13,23). It is first shown that acoustic waves propagating along a (1 lo) crystal axis may possess both longitudinal- and transverse-field components of piezoelectric coupling. The parallel magnetic field does not affect the longitudinal wave interaction. However, a bulk-type acoustoelectric interaction may arise from the coupling of the transverse electric field components of the interaction between a helicon wave supported by the streaming electrons and an acoustic wave supported by the lattice. This interaction will be called the acousto-helicon interaction. There has been some preliminary analysis of this type of interaction ( 2 4 ) ; however only the simplest case is considered, namely, the interaction between the helicon wave and an acoustic wave propagating along a (100) crystal direction. For the (100) direction of wave propagation, two of the acoustic waves are degenerate and cannot be distinguished. Moreover, no longitudinal piezoelectric coupling exists and the acousto-helicon dispersion equation is greatly simplified (24). On the other hand, for the situation of experimental interest when wave propagation coincides with a ( 1 10) crystal direction, the acousto-helicon interaction has not yet been investigated. When the general theory of the acoustoelectric interaction described in Section 1I.B is used, the derivation of the acousto-helicon dispersion equation in (1 10)oriented lnSb can be described. B. Derivation of the Acousto- Helicon Dispersion Equation
Assume that the long dimension 2 of the lnSb specimen coincides with the [ I 101 direction, then 2 = ( I , 1, O)/J2. By symmetry, identical results will be obtained for any arbitrary (110) axis. Hence, for on-axis wave propagat ion, 4 = 2 = (1, 1, oyJ2, (50) whereupon the components of the effective elastic tensor a of Eq. (15) become a,, = a 2 2
= (c11
and 0 1 2 = (c12
+ c44)/2.
+ c44)/2,
a13
a33 =
c44
= a23 = 0.
(51)
In order to determine the effective eigenvalues cA, the eigenvalue equation (a - ~ ‘ 1 )6- = 0 must be solved, where a is defined by Eq. (15). Equations (51) are substituted into the eigenvalue equation, thereby yielding a11
- c’
a12
0
0
a33- c‘
(52)
178
W. J. FLEMING AND J. E. ROWE
The characteristic equation of Eq. (52) is (a33- c’)[(a11
- c’)2 - a f 2 ]= 0,
(53)
which has the eigenvalue solutions c,1 = a33 = c44,
and The corresponding eigenvectors kn are found by back substitution of each eigenvalue cA= c’ into Eq. (52) and these are represented as
kL1 =(O,
0, 11,
ka2 = (1,
1, O h
and
kA3 = (1,
- 1,O).
(55)
Since 4 . kd1 = 4 - kL3 = 0, waves 1 and 3 are transverse acoustic waves; since cd1 > c d 3 ,wave 1 is the fast-transverse wave and wave 3 is the slowtransverse wave; and since 4 * {, = I , wave 2 is the longitudinal wave. A normalized piezoelectric polarization field 8, is defined as
where e is given by Eq. (15). Thus, the polarization fields associated with each of the acoustic waves are, respectively,
d,,
= (1,
1,O)/J2,
b,,
=
(O,O, I),
and
b,,= (O,O,
0).
(57)
Wave 1, the fast-transverse wave, possesses a purely longitudinal polarization field (since b,, * 4 = 1); wave 2, the longitudinal wave, possesses a purely transverse polarization field (since b,,. 4 = 0); and wave 3, the slowtransverse wave, does not possess a polarization field. Hence, there is a possible acousto-helicon interaction with the longitudinal acoustic wave. The rf conductivity is derived in the conduction carrier coordinate system (x, y , z ) which is shown in Fig. 3. For simplicity all field and transport variables are assumed to coincide with the A-direction. It is assumed that electrons drift with velocity uo = uoA, parallel to the magnetic field B, = Bll = Bll2. Substitution of these quantities into Eq. (48) yields
where G D= o,f/v = w2/q2f,Dis the Doppler-shifted diffusion frequency; yf = G f / o = 1 - f t u o / u , and y = W/w = 1 - uo/u, are, respectively, the
ACOUSTOELECTRIC INTERACTIONS
179
A
x3
A XI
J
FIG. 3. Field configuration for the acousto-helicon interaction.
conduction and the total space-charge drift ratios: ti,, =jioB,,, and jio = p o v / i j = p = Q,/m*ij. [Note that the approximation q FZ w/u, IS used in Eq. ( 5 8 ) . ] Equation (58) is rearranged into the form v = p - E and the rf mobility tensor p is found. Similarly, Eq. (49) can be written out, combined with Eq. (58), and arranged into the form, J = E; whereupon the rf conductivity tensor u is given by us
0
0 (59)
where
C0 = a0v/ij = Q, No ,Go,
v = v + j G = v +joy, lib/ - jdO,), byy= bzz= y/(l + hi),
bxx
=
and
When the wave equation [Eq. (17)] is used, the rf conductivity must be expressed in terms of the crystal coordinates x = ( x l ,x 2 , x3). Inspection of Fig. 3 shows that the r = (x, y , z ) system is simply rotated 45" about the
180
W. J. FLEMING AND J. E. ROWE
1S3-axis [since L = (1, 1, O ) / J 2 is the long dimension of interest here] and the rotation matrix M, defined by r = M * x, is given by
(60) Thus the rf conductivity in the ( x l ,x 2 , x 3 ) system becomes u' = M-' us M, where u is defined in Eq. (59). After performing the transformation, it is found that u' can be written in the form 0'
where
5
[
a'12
4
2
-a;3
4 1 = (oxx + a,,)/2, a;
=
0;3]
4,
- 4 3
6 3
4
4
2
(61)
9
3
= (gxx.-
0,,)/2,
- a,,/J2
and a;3 = O Y Y .
In the acousto-helicon interaction it is assumed that 9112, that is, = (1, 1,O)/J2; thus when u' is taken from Eq. (61), the wave equation [Eq. (17)] becomes
(3
where
and Since the direction cosine n of ij equals zero, the general acousto-helicon dispersion equation is obtained from Eq. (22). The solution of Eq. (22) requires knowledge of the elastic tensor a and the effective piezoelectric tensor e. For wave propagation 4 = (1, 1, O)/J2, these tensors reduce to the following:
ACOUSTOELECTRIC INTERACTIONS
181
,,
where a , a 1 2 ,and a33 are defined by Eq. (51). If Eqs. (63) are used, the expression which appears on the right-hand side of Eq. (22) is evaluated and it becomes, after algebraic simplification,
However, as seen in Eqs. (54), the eigenvalues cA1and cA2are equal to, respectively, a33and (a, + a12).Thus, the terms a; and a;, can be written as
,
,
a; 1 = 1/(2C,,/C’ - 2) and Ui3
= l/(c,*/c’
- 1).
(65)
where cA1and cA2are the eigenvalues which correspond to, respectively, the fast-transverse and the longitudinal acoustic waves. When Eqs. (62), (64), and (65) are used, the general dispersion equation [Eq. (22)] becomes
where - 1 - t i 2 / ( 2 C A l / C ’ - 2),
=
El;
&y3
EY2 = C i 2
- K2/(2CIl/C’- 2),
ag3 =
- 1 - K2/(CA2/C’ - I),
ti2
=&i3,
= e:L/c’&,,
and c‘ = p m d / q 2 .
If Eq. (66) is written out, the general dispersion equation is obtained and, after simplification, it can be written as (&;I
+
E ; ~ ) [ E ; ~ ( E; I E;Z)
+ 2&;:]
= 0.
(67)
182
W. J. FLEMING AND J. E. ROWE
There exist two uncoupled solutions to Eq. (67). The first solution can be written [using Eqs. (61) and (62)] as EY1
+ &Y2
=
[ ( j Z- 1) -
lc2
(CdllC’ -
1)
]=o,
and the second solution can be written as
where A1 = j a y y / m + l q 2 c I 2 / o 2- 1
and A2 = Oyz/WEl.
It can easily be shown that Eq. (68) is the dispersion equation for the quasistatic interaction between the space-charge wave and the fast-transverse acoustic wave, and Eq. (69) is the desired acousto-helicon dispersion equation. Substitution of the expressions for crxx defined by Eq. (59) and c‘ = pmw2/q2 into Eq. (68) yields the usual quasistatic acoustoelectric dispersion equation [e.g., Eq. (8-21) of Steele and Vural (24)],
where v, = (cd1/pm)1/2= ( ~ ~ ~ /is the p ~sound ) ~ velocity / ~ of the fast-transverse = 0-frquo, wave, u p= (Q,2No/e,m*)1/2is the plasma frequency, 0 = 0 - q u o , f, is the trapping factor and ti2 = e:4/~44el I S the electromechanical coupling constant. It is convenient to rearrange the acousto-helicon dispersion equation [Eq. (69)] into the form
”/
(A1
- Az)(A1
+ AJ
(71)
Substitution of the expressions for o y yand o y z defined by Eq. (50) and c’ = p m d / q 2into Eq. (71) yields the acousto-helicon dispersion equation, namely,
183
ACOUSTOELECTRIC INTERACTIONS
+
+
where us = ( ~ ~ ~ / p , , = , ) ~[(c,, ” cI2 2~~~)/2p,,,]”’ is the sound velocity of the longitudinal wave, cI is the velocity of light in the semiconductor medium, w, = vbll = (Q,/m*)BII is the cyclotron frequency, W = w - quo ] electromechanical and ti2 = e:,/ca2 E~ = e:s/[&l(cl + c I 2 + 2 ~ ~ ~ ) is/ 2the coupling constant. Equation (72) is the exact acousto-helicon dispersion equation for ( 1 10)-oriented wave propagation. It should be noted that Steele and Vural (24) have analyzed the acoustohelicon interaction for wave propagation along a (100) crystal direction. For this direction of wave propagation, the analysis is greatly simplified and the acousto-helicon dispersion equation [Eq. (8-7 I ) of Ref. (24)] becomes
(2, )
(A1 f /I2) 7- 1
= ti2,
(73)
where A , , A 2 , and c‘ are defined in Eq. (69); cA1= c 4 4 , K’ = e:,/c,,&, , and us = (C44/p,,,)1’2is the sound velocity of the shear acoustic wave. C . Solution of the Acousto-Helicon Dispersion Equation
In this section, the acousto-helicon dispersion equation, Eq. (72), is solved by means of a perturbation procedure. Equation (72) is rewritten as
where A4 A, Li A3 - jv v k jw,’
Ax 2 A, - j ’
CI2
A3 = 1 - q 2 w2
and
A4 V v2 + w,2 ’
V = v+jS,
A4 = wP2W/w2.
A convective-type instability with phase velocity in the neighborhood of the sound velocity is considered. Deviations from the sound velocity are assumed small and the wavenumber is expanded as
q
= W/D, + j a ,
l a ( 4 w/u,,
(75)
where z = uRe+ joc,, is complex. If a convective instability exists, ocRe must be positive. Substituting Eq. (75) into Eq. (74) and solving for Re(cc) = aRe yield the following:
184
W. J. FLEMING AND .I. E. ROWE
where all terms in the quotient A X / L A, are evaluated for q = o / u , , i.e., the perturbation terms ct in this quotient are negligible. Thus, the factors A3 and A4 become [for ( c J / v J 2 % 11 A3 =
and
-(C,/U,~)~
(77)
A4 = w p 2 y / o .
Equation (76) is valid as long as the following inequality is satisfied: 11\31
If Eq. (77) and I can be written as
%
lh/Vl*
(78)
w 1 0 1 (the most restrictive case) are used, Eq. (78)
9
(;)2
($
(79)
where uo= ( o ~ o ~ )uT” is~ the , electron thermal velocity, and c J is the velocity of light in the lattice. At microwave frequencies, Eqs. (78) and (79) are well satisfied. When the definitions of Eq. (74) are used, the quotient in Eq. (76) is found and, after some algebra, is written as
where 6,
and
&/A3 V
a,
V2/( Vz
+ oC2)
Since the inequality of Eq. (78) holds, the parameter 6, is much smaller than unity and Eq. (80) can be rearranged by retaining first-order perturbations of 6, and it becomes Ax
A-A+
1 z-(1 A,
+j6,av).
Equations (76), (77), and (81) are combined and the growth rate of the acousto-helicon interaction is given by
where y = 1 - uo/us,
Vie = 1 +bf,
vim = w,(l - b i t )
and b;l = bll/(l
+w,~)”~,
bll = W , / V = Q,Bll/m*,
O,
=w ~ / v .
Although the growth rate is positive, indicating a convective instability, for uo > u s , the growth rate is diminished by the factor ( v , / c , )z~ Thus, at the frequencies of interest given by Eq. (79), only vanishingly small
ACOUSTOELECTRIC INTERACTIONS
185
growth rates are possible for the acousto-helicon interaction. This interaction cannot, therefore, account for the existence o f the acoustoelectric interactions enhanced by the presence of a parallel magnetic field. Nevertheless, the derivation and solution o f the acousto-helicon dispersion equation serve as an illustrative example for the application of the general analysis of Section
11,B.
Iv. SOLUTION
OF THE ACOUSTOELECTRIC INTERACTION FOR ARBITRARILY AND ON-AXIS ACOUSTIC-WAVE PROPAGATION ORIENTED STATIC FIELDS
A . Introduction
A general theory of acoustoelectric interaction was developed in Section 11. An analysis as well as a solution of this theory, specialized for the case of
on-axis wave propagation along the (100) crystal axis, is the subject of this section. Whenever the transverse component of the applied magnetic field and the sample current in InSb exceed certain threshold levels, acoustic domains form in sufficiently long-length specimens o f ( 1 10)-oriented n-type InSb. Although the presence of these acoustic domains limits the utility of the acoustoelectric amplification process (6,7) the domains are interesting in themselves. The acoustic domains are manifested by the appearance of rf current oscillations and microwave noise radiation which are the subject of several related investigations (ZO-ZJ). As discussed in Section 1,B, it has been found experimentally that both phenomena possess nearly identical applied field threshold dependences and exhibit two distinct modes of operation. Mode I occurs for high current values and low magnetic field strengths, whereas Mode 11 occurs in the opposite limits. Recently it has been concluded by several investigators that, although a magnetic field enhancement of the hydrodynamical theory of Steele ( 4 ) and Turner et d.(5) can account for Mode I1 operation, more detailed microscopic theories (6,25,26) are required to adequately account for the existence of both Modes I and 11. The results presented here will show that a hydrodynamical theory of the acoustoelectric interaction does not only account for both Mode I and Mode I1 operation, but also yields valuable insight into the physical basis of the acoustoelectric effect in InSb. In order to obtain a tractable solution, the quasistatic approximation is made wherein the rf electric field components, transverse to the direction of wave propagation, are neglected. As shown in Section II,C, the analysis reduces to a one-dimensional problem and the growth rate of the acousto-
186
W. J. FLEMING AND J. E. ROWE
electric interaction is given by Eq. (41). The growth rate is primarily dependent on the determination of the longitudinal rf conductivity o,,which in turn is found from a solution of the field and transport equations, namely, Eqs. (48) and (49).
B. Solution for the vf Conductivity The configuration of the applied fields and transport variables is defined in the conduction carrier coordinate system (x, y , z ) which is shown in Fig. 4. It is assumed that only electrons are present. The electron drift velocity uo is directed along an arbitrary direction [' in the 2-9 Lorentz force plane. For on-axis acoustic-wave propagation, the acoustic wavevector q is taken parallel to the 2-direction which in turn is parallel to the long dimension 2 of the crystal. Since a hydrodynamic theory is utilized and the quasistatic approximation is made, the parallel component of applied magnetic field does not enter into the longitudinal rf conductivity and is ignored. For convenience, since InSb has an isotropic band structure, the transverse component of applied magnetic field B, is chosen in the %direction. Inspection of Fig. 4 shows that
B,=B,i, and
+
q=qA
uo = uOx2 uo,j = (uo cos [')a - (uo sin
[')9.
(83)
h
Y
Z
FIG.4. Field configuration for the acoustoelectric interaction with arbitrarily oriented static fields [shown in the conduction carrier coordinate system (x, y , z ) ] .
ACOUSTOELECTRIC INTERACTIONS
187
Substitution of Eqs. (83) into Eq. (48) yields
0 )is the Doppler-shifted diffusion frequency; where OD = wD(?/ti) = o*/(qY; yr = W,/w = 1 - f t u o J u , and y = W/o= I - uox/u, are, respectively, the conduction and the total space-charge drift ratios; b = ,Go B, and iio = p o v / ? = Q,/(m*i). Note that the approximation q z w/u, is valid in Eq. (84) since all perturbation terms c( [from Eq. (40)] are reduced by the in the solution of the dispersion equation [Eq. (41)]. factor 12% Equation (84) is rearranged into the form v = p - E and the mobility tensor p is given by
where
and
Substitution of Eqs. (83) into Eq. (49) yields the rf charge density p which is given by
where po = Qe N o , p X z = I/y,, p y x = f, 0 % )and pyV= p Z z = 1. The rf conductivity is determined directly from Eqs. (85) and (86) and is given by
188
W. J. FLEMING AND J. E. ROWE
where
and = y(det,).
Q,,
Although the complete solution for the off-axis acoustoelectric interaction could now be obtained from Eqs. (17), (20), and (87), the quasistatic approximation is utilized here. The value of el, is found directly from Eq. (87) and is given by ell = 0,
eJdet,
=
ao/[yf(l
+ 6’) - jw/OD].
(88)
It is convenient to rewrite Eq. (88) into the form wR &I
ell =
Y~ v‘ - j o b D’
(89)
where oR= C J , / E ~ is the dielectric relaxation frequency, w D = V U , ’ / ( U , ~ ~ , ) is the diffusion frequency at synchronism, v’p ( V / v ) [ l + ( b v / i ~ )is~ a] factor which takes into account rf magnetoresistance and electron inertial terms, and V / v = 1 + j w , . When the magnetic field is zero, b -+ 0 and v’ -+ S / v , rf magnetoresistance is no longer present. When electron inertial terms are neglected, o,-+ 0, the factor v‘ becomes equal to 1 + 6’ in accordance with the theory of Steele ( 4 ) and Turner et al. ( 5 ) .
C . Solution f o r the Gro\d-th Rate of the Acoustoelectric Interaction Defining v’ = vie =jv;,,, and substituting Eq. (89) into Eq. (41) gives the desired expression for the acoustoelectric growth rate :
where vLe = 1
+ bf2, b’2
W,
Y=1
v;, = w,(I- b”),
b’/( I
=W/V,
- uox/us,
+ w,’), b
= po
Yf = 1
B, ,
- ftUOX/DS,
ACOUSTOELECTRIC INTERACTIONS
and
w
wR/o
189
+ w/oD.
Equation (90) can be rearranged into a more meaningful form as
where wR/(l
wk
+ bI2),
W’
w;l/w
+ w/wb
and A
wb = wD( 1
+ b ’ 2 ) /1[ - y
f
w,( 1 - b”)wD/w].
In Eq. (91), wk and wb are identified respectively as the effective dielectric relaxation frequency and the effective diffusion frequency. For simplicity, it is assumed that trapping effects are negligible and that space charge arises solely from the conduction electrons which drift along the direction of acoustic-wave propagation 4 (the solution for off-axis wave propagation is described in Section V). Moreover, for on-axis wave propagation in (1 10)-oriented InSb, Eq. (57) shows that the fast-transverse acoustic wave is piezoelectrically active in the quasistatic interaction. When these conditions are utilized, a number of the defined terms in Eqs. (90) and (91) simplify and become
f,= 1,
uox =
ug
O D = Vlls2/VT2,
,
yf
=
y
=
I - U0/I,, ,
2lF = ( C 4 4 / p m ) l ’ 2 ,
and K~
= e?,/E,
c
~
~
.
(92)
For these specialized conditions, Abe and Mikoshiba (27) have previously published a result similar to Eq. (91); however, their work is in error. I t can be shown (28) that in order to correct this error the term q ( y - 1 ) must be dropped from the definition of 4 in Eq. (9) of Ref. 27, where y~ equals o, of the present analysis and Eq. (8) of Ref. 27 remains as written. The derivation and notation of the present theory follow that of Steele ( 4 ) and Turner et al. ( 5 ) and other related work (29, 30); the theory of Steele ( 4 ) and Turner et al. (5) will hereafter be referred to as the STVW theory. When inertial terms are neglected, OJ, -+ 0, Eq. (91) reduces to the corresponding hydrodynamical result of the STVW theory. When the magnetic field is zero, b 40, Eq. (91) reduces to the corresponding hydrodynamical result of Kino and Route (29); and when both inertial terms and magnetic field effects are neglected, w, + 0 and b 0, Eq. (91) reduces to the corresponding hydrodynamical result of White (30). --f
190
W. J. FLEMING AND J. E. ROWE
TABLE I PHYSICAL PARAMETERS FOR N-TYPE lnSb Parameter Electron effective mass Carrier concentrationa Lattice dielectric constan Thermal velocity” Low-field dc mobility4 Elastic constant Elastic constant Elastic constant Lattice mass density Piezoelectric constant Specimen length’ Specimen width‘
Symbol
AT
77°K
Value
Reference
0.013 mo I x 1014cm-3
(31)
17.8 E O
(32)
3 x 107cm/sec 6 x lo5 cm2/V-sec 6.87 x 10” N/m2 3.75 x 10”N/m2 3.12 x lo1’ N/m2 5.77 x lo3 kg/ni3 0.07 C/m2 10 mm 1 mm
(33) (33) (33) (31) (34)
The values of N o and po are chosen arbitrarily and are suitable for high-purity n-type InSb. The average electron thermal velocity is determined by uT = (k,T,/m*)1’2,where k , is Boltzmann’s constant and T, is the average carrier temperature which is taken here as 77°K. The values of L and w are chosen arbitrarily and are typical for the specimens under investigation.
I. Field Dependence The physical parameters (31-34) in Table I are used and the growth rate of the acoustoelectric interaction, given by Eq. (91), is normalized to K’. The quantity u R e / ~ is ’ labeled “gain,” and is plotted as a function of a nornialized drift ratio I- (u0 - uS)/cT;the results of this calculation are shown in Fig. 5. In Fig. 5a the results for the STVW theory are shown and in Fig. 5b the results for the present theory, using Eqs. (91) and (92), are shown. A fixed maximum value of the acoustoelectric interaction growth rate, equal to r ~ ~ ( t o , ~ ~ ) ” ~is/ given 8 1 ~ , by , the STVW theory and it occurs at - y( I bz> = Wnli,, where W,,lin = 2(0,/W,)’” occurs for frequency ,fo = ( w ~ w ~ ) ~ /The ~ / ~parameters ~ T . of Table I give f o = 1.45 GHz, us = ( ~ ~ ~ / p=~2.33 , ) ’ x/ lo5 ~ cm/sec and the STVW theoretical maximum gain ( c ( ~ ~ / =K 4.89 ~ ) x ~ lo3 ~ ~ Np/cni. I n Fig. 5 and in subsequent figures, the Np/cm units are deleted and the acoustic gain is treated as a dimensionless quantity. It is seen from Fig. 5b that when inertial terms are included, the maximum value of the acoiistic gain becomes strongly dependent on the strength of the applied magnetic field. Comparison of Fig. 5b with the
+
191
ACOUSTOELECTRIC INTERACTIONS
20
0
40 (U,-V,)/~~
(a)
80
I0
NEGLECTING INERTIAL TERMS
.M
0
60
:r
.YO
1.0
(uo-~s)/vT=r (b)
Fici. 5 . Acoustic gain
INCLUOING INERTIAL TERMS CI~JK’
as a function of drift velocity
at 1.45
GHz.
numerical results o f corresponding microscopic theories (6,26) shows that no essential difference exists between the results o f the hydrodynamical theory shown here and the more complete microscopic theories. Indeed, another set o f acoustic gain characteristics has been calculated and is compared in Fig. 6 to the acoustic-wave amplification measurements of Route and Kino (6). I n order to obtain best agreement, parameters appropriate to the Route-Kino experiment have been substituted into the theory; namely. = 1.3 x / l o = 5.1 x lo5 cm’/V-sec and N o = 1.4 x I O l 4 cni-3. Acoustic gain in units of dB/cni is calculated from Eq. (91) using the re la t ion (dB/cm) (93) Acoustic gain = 10 10g,~[exp(2x,,)], which, upon evaluation, becomes Acoustic gain
=
8.68xR,,
(dB/cni).
The correlation of theory to experiment is excellent except for the zero magnetic field curve. Although the microscopic theory (f6)gives better agreement
192
W. J. FLEMING AND J. E. ROWE
0
5
10
15
20
Uo/v,-I=-Y
FIG.6. Comparison between the theoretical on-axis acoustic gain and the Route-Kino experiment (6) (solid curves denote theoretical results).
for the zero magnetic field curve, the hydrodynamical theory shown here gives as good, if not better, agreement for the remaining ( B , # 0) acoustic gain curves. Due to the collision-dominated nature of the acoustoelectric interaction in InSb, resonance effects are not pronounced and do not play a major role in the basic mechanisms of the microwave acoustic gain. Figure 6 shows that the essential features of corresponding microscopic theories (6,26)are retained in the present hydrodynamical theory. The relative simplicity of Eq. (9 1) has permitted a detailed investigation of its behavior (35). For example, level curves can be computer-generated using the numerical methods of interval halving and false position. One set of level curves, corres.ponding to the gain characteristics of Fig. 5b, is shown in Fig. 7. The emergence of Mode I operation (10,11) in the region of low magnetic field strengths and high drift velocities is apparent. The general structure of the level curve contour in Fig. 7 is very similar to the corresponding results from microscopic theory shown in Fig. 2 of Harth and Jaenicke (25). Further discussion of the field dependence of the acoustic gain is given elsewhere (35).The effects of carrier trapping on the field dependence of the acoustic gain have also been investigated and are discussed by Fleming (36).
193
ACOUSTOELECTRIC INTERACTIONS
f,
I
f=f,=I4 5 GHz
0
2.00
1.00
u.w
3.00
5.00
TRRNSVERSE MRGNETlC FIELD. k G
FIG.7. Level curve contour of acoustic gain at 1.45 GHz.
2. Physical Mechanism of Mode I Operation
In order to determine the ultimate limit of Mode I gain, as depicted by the hydrodynamical theory, the acoustoelectric growth rate of Eq. (91) is calculated for values of electron drift velocity which are far in excess of the thermal velocity. Acoustic gain is calculated from Eqs. (91)-(93) using the parameters of Table I. A value of electromechanical coupling K~ equal to 1 x is utilized in this calculation and is obtained using Eq. (92) and the appropriate parameters of Table I. The results of the calculation are shown in Fig. 8. Here the present theory given by Eqs. (91)-(93) is also compared to the corresponding theory of Abe and Mikoshiba (27). As discussed elsewhere (28),the theory of Abe and Mikoshiba is incorrect. It is seen that no significant error is present in their theory for electron drift velocities less than approximately 3 x 106cm/sec; however, at large drift velocities, their theory noticeably diverges from the present theory and does not yield high gain peaks. The locations of the sharply peaked maximum gain values which exist for uo > uT can be determined directly from Eq. (91) subject to the simplifications of Eq. (92). The peaks occur when the factor W’ in Eq. (91) equals zero, and this condition is given, in good approximation, by 2
Y 2 = %u s
[I
+
ey(I
+h,’)],
(94)
194
W. J. FLEMING AND J. E. ROWE
lo5
I
10’
I06
VS
I
Io8
VT ELECTRON DRIFT VELOCITY, cm/sec
FIG.8. Comparison of the present theory to the theory of Abe and Mikoshiba (27) at 1.45 GHz.
where wo = ( O R O D ) ” ~ = 2nf0 and b’, 4 b2v/wR. From Eqs. (91)-(94), maximum gain is given, in good approximation, by
[I (Acoustic gain),,,
WR
= 8 . 6 8 ~ ’-
+ (2)2 (1 -k
2v7’ [ I -k
(z)2+ (1
(95 )
At the frequencyf=f,, substitution of the parameters of Table I into Eqs. (94) and (95) yields (Acoustic gain),,, = 623 dB/cm which occurs at uo = 4.26 x lo7 cm/sec for b = 0. This value for maximum gain is 14.7 times greater than the corresponding maximum gain given by the STVW theory; however, it can only occur for uo > vT and b x 0. In the limit, w B wo and b= 0, Eq. (95) becomes (Acoustic gain),,,
= 8 . 6 8 ~ ’ w , / 2 v ~ , (dB/cm).
(96)
For the parameters of Table I, Eq. (96) gives a maximum achievable acoustic gain of 883 dB/cm. Although this maximum value can only be obtained for a drift velocity approximately equal to the thermal velocity, it is. 4 ( 0 R / v ) ” 2 times greater than the maximum gain of the STVW theory and this ratio ranges from 10 to 20, depending on the value of transport parameters wR and v. It is significant to note that the maximum gain value of Eq. (96) corresponds to the maximum attainable gain of White’s theory (30)in the limit wD-+ a,y = -v&, . This observation provides an important clue in the
195
ACOUSTOELECTRIC INTERACTIONS
investigation of the physical mechanism of Mode I operation. In the hydrodynamical theory of the acoustoelectric interaction, it can be shown (24) that diffusion effects are characterized by a diffusion time z D which is approximately equal to w,/w2. When inertial terms are included, as done in the present theory, and no magnetic field is present, the effective diffusion frequency becomes, from Eq. (91), 0;= wD/(l - rZ).As uo + vT and -+ 1, the effective diffusion frequency 0;approaches infinity. At a given frequency, this gives an infinitely long diffusion time z D which allows for more effective space-charge bunching of the streaming electrons. In fact, some preliminary algebra using the equations for conservation of momentum and continuity of current, Eqs. (44) and (46), shows that the condition r2= 1 [which for clarity can be rewritten equivalently as jOv, =j(vT2q2/O)v,] corresponds to exact cancellation of the diffusion term by the inertial term in the equation of momentum conservation. The essential point here is that the fundamental physical mechanism of Mode I operation, as depicted by the hydrodynamical model, is the quenching of diffusion effects by the driftenhanced inertial effects of the streaming electrons, thereby allowing for more effective bunching of the electrons, which results in the emergence of the Mode I operation. Although the hydrodynamical theory is analyzed here for conditions which exceed its estimated range of validity, the refinements of the more complete microscopic theory do not yield results which are significantly different from those of the hydrodynamical theory (35).It appears therefore that the inclusion of electron inertial terms in a hydrodynamical theory suffices as a first-order approximation of the acoustoelectric interaction, even at microwave frequencies. From Eq. (94) it is seen that for w 4 and b’2 9 1, Eq. (97) shows that diffusion quenching effects are eliminated and the Mode I gain mechanism is diminished.
197
ACOUSTOELECTRIC INTERACTIONS
+
However, as shown in Fig. 9, for r2< the effective diffusion frequency given by Eq. (97) is enhanced by the presence of a magnetic field and the STVW Mode I1 gain mechanism emerges.
3. Frequency Dependence Further analysis reveals that unlike the STVW theory the gain expression of the present theory, Eq. (91), is not log-symmetric about the frequencyf,. The frequency dependence of the acoustic gain for the STVW theory is shown in Fig. 10a and the corresponding results for the present theory are shown in Fig. lob. It is seen in Fig. 10b that for low drift velocities with r 5 0.75 the frequency of maximum gain is less thanf, , whereas for larger 2 0.75 the frequeacy of maximum gain occurs at drift velocities with frequencies greater than,f, .
I x 106
1x10'
I x 109
I x 10'0
IXlOll
FREOUENCY, Hz (a) NEGLECTING INERTIAL TERMS
~f=fo:145GHz B,=I
I I
kG
I 015,
I
z
a? L3N
? I
IOl'
FREQUENCY. Hz ( b ) INCLUDING INERTIAL TERMS
FIG.10. Frequency dependence of the acoustic gain for B,
=
I kG.
198
W. J. FLEMING AND J . E. ROWE
Although in Eq. (91) the gain expression of the present theory appears relatively simple, substitution of all the defined quantities shows that, as an explicit function of frequency, it is given by a ratio of a sixth-degree polynomial to an eighth-degree polynomial in terms of frequency. Because of this complexity it has been analyzed numerically, using the method of interval halving, in order to solve for the frequency of maximum gain. When fixed values of magnetic field and drift velocity are taken, Eq. (91) is numerically searched for the frequency at which maximum gain is achieved, and the resulting solutions have been plotted in Fig. I I . As before, the parameters of Table I and the simplifications of Eq. (92) are utilized. Figure 1 Ib shows that the maximum gain of the present theory is generally smaller than the maximum gain of the STVW theory; however, at sufficiently large drift ratios the maximum gain of the present theory greatly exceeds that of the STVW theory. A more detailed discussion of these results is given by Fleming and Rowe (35).
J 0
I a0
60
40
( uO- v S i / v T =
.80
I0
r
( a ) FREQUENCY AT WHICH MAXIMUM GAIN IS A T T A I N E D
0
.V@
( uO- v S ) / v T
.60
r
:
( b ) M A X I M U M ATTAINABLE GAIN
FIG.I I . hlaxitnum acoustic gain
1.m
199
ACOUSTOELECTRIC INTERACTIONS
D. Incorporation of Empirical Field Factors into the Theory of Acoustoelectric Interaction In Section 1V.C it was assumed that carrier heating effects are negligible and the dc carrier mobility remains constant, independent of field strength. The constant mobility assumption was made in order to facilitate the comparison of the present theory to other related theories and, more significantly, in order to obtain physical insight. However, in order to obtain agreement with experimental measurements, the theory must be modified to include empirical field factors (28). Field-dependent effects such as mobility saturation and geometrical magnetoresistance are especially important. Although it is possible to obtain analytical expressions which describe these effects (15,37), there is considerable mathematical complexity. Furthermore, in a real semiconductor, imperfections such as bulk inhomogeneities, nonohmic contacts, and surface trapping effects all affect the resultant transport of conduction current. Since it is impractical to take all these factors into account, only the effects of mobility saturation and geometrical magnetoresistance are considered. Moreover, since the other factors are ignored, the effects of mobility saturation and geometrical magnetoresistance are adequately described by simplified empirical field factors. An empirical relation for the mobility saturation is obtained by a least-squares fit to measured I-V characteristic data for three specimens of high-purity n-type InSb (36). For convenience a parameter Y is defined as
where R, = Vo/Iois the dc resistance of the specimen and Rb is the low-field ( E , 5 20 Vjcm) resistance of the specimen. Since no evidence of bulk impact ionization is observed in the measured I- V characteristics, all variations in R , are attributed to variations of the carrier mobility and the parameters Y can also be written as Y = &/po - I , where p,!, is the low-field electron mobility. Hence, Y is also a measure of the carrier mobility variation. Because mobility saturation is an energy-dependent effect, Y is expanded in even powers of J , and the empirical function Y
+~
= a2JO2
~
5
,
~
(99)
is chosen; ci2 and u4 are constants to be determined. Corresponding to each data point i the parameter Roi/RAi = Y i + I is calculated and i t is plotted vs. the applied current density in Fig. 12. Here, 120 data points measured in the absence of an applied magnetic field for IJ,I ,-. L * '
n cl
5 < 5 2
zn
none
5
2 ' .
i 2
2
i>L2
5iL2 5 .> L1
none none
;1 s >
i,
(124)
where b i A / p o , B, 1 . In Eq. (124), the carrier drift velocity is given explicitly in terms of the applied fields. If Eq. (124) is written out separately for each carrier species e and h, and the field configuration of Fig. 35 is used, a set of four equations is obtained. Upon the elimination of the static electric field components Eo, and Eo, , the four equations reduce to two equations which give the components of hole drift velocity in terms of electron drift velocity. The resulting equations can be written as follows :
where
and aXy= - a y x = be
+ b,, .
The above expression is particularly useful since the specification of the drift velocity of majority carrier electrons and the value of transverse magnetic field directly determine the drift velocity of minority carrier holes. It is desirable to note certain general properties of the electron-hole transport system. For example, when the sum of the squares of carrier drift components is taken using Eq. (125), it can easily be shown that the magnitude of the hole drift velocity is related to the magnitude of the electron drift velocity by the expression /UO,hl = Iu0,el
‘ (kLO,h/LLO,e)[(l
f
+ bh2)1”z.
( 1 26)
ACOUSTOELECTRIC INTERACTIONS
237
Similarly, Eq. (125) also shows that the angle O', measured between the electron and hole drift velocities (as shown in Fig. 35), is fixed by the value of magnetic field and is given by 0'
= 7c -
(0,+ ah),
(127)
where 0' L cos-'(G O , h . i i O , e )
and
OiLttan-'IbiJ.
In the absence of carrier recombination and generation, the static conduction current density Jois given by JO
=
QeJ'Ou0.h
- QeNouo,e,
( 128)
where Q, is the magnitude of the electronic charge, and Po and N o are the equilibrium hole and electron carrier densities. As before, the field and transport variables are expressed in terms of the drift velocity u ~ of, the ~ majority carrier electrons. If the field configuration of Fig. 35 is used, Eqs. (125) and (128) can be combined to yield the following relation:
where No' A N o + Pou'axx and Po' P o u ' u X y Moreover, . the magnitude of the static electric field can be determined from Eq. (124) and written as follows:
Similarly, the ratio of the Hall electric field component Eoy to the drift electric field component Eo, can also be found from Eq. ( I 24), and is equal to EoyIEox = -tan(@,
- i O , e),
(131)
where 0, A tan-'(po,, B,) is the electron Hall angle and l o , ,A tan-' ,) is the angle of electron flow shown in Fig. 35. x (uoy, A general solution for the static electron-hole transport system of Fig. 35 can be obtained by utilizing the following procedure. Values for the electron and hole densities, N O and P O , are specified (note that the value of N o should be chosen greater than Po because electrons are assumed to be the majority carriers). Appropriate values for the electron and hole mobilities are used, a magnetic field B , is specified, and a value of electron drift velocity luO,,l is chosen. First, the magnitude of the hole velocity and the direction of hole drift relative to electron drift 0' are found, respectively, from Eqs. (126) and (127). Then, the direction of electron drift ( o , c relative to the crystal long dimension L is chosen and the components of conduction current density J, and static electric field E, are found using Eqs. (l29)-( I 3 I ) .
238
W. J. FLEMING AND J. E. ROWE
It is significant to note that once the magnetic field B, is specified and the electron-hole drift parameters are chosen, the angular relationship of the vectors u,, ,, u,, ,, and E, in Fig. 35 is fixed. Thus, as the angle of electron drift l o ,,is increased, these vectors maintain their relationship to one another and simply rotate as a unit about the a-9 origin. The angle ('o,e is usually chosen to correspond to certain boundary conditions. For example, if the net transverse current J o y is set equal to zero, Eq. (129) indicates that the ratio of transverse to longitudinal electron drift velocity must be chosen as follows : Lloy, e l u o x , e
(1 32)
= Po'lNo'.
By definition, it is true that l o , 4k tan-'(u,,, Juox, ,); hence, for J , , = 0, the angle to, is specified by Eq. (I 32). When this value of (',, is substituted into Eq. (131), the ratio of Hall electric field to drift electric field, Eoy/Eox, can be determined and it agrees with the corresponding result given by Eq. (5) of King (37). Other boundary conditions of interest are (,, = 0 and = 0, which correspond, respectively, to the Suhl effect hole injection mechanism described by Shockley (47) and to the short-circuiting of the Hall field ( E o y= 0) at the contact-semiconductor interface. When high values of electric field are applied to a semiconductor, nearly equal densities of electrons and holes are generated by bulk impact ionization. Under these conditions, a diffusion gradient is established across the lateral dimension of the specimen by the presence of a transverse magnetic field. For this case, Swartz and Robinson (18) have shown that electrons and holes flow collinearly in a thin layer located adjacent to the lateral specimen surface toward which the Lorentz force is directed. Hence, the present analysis only applies for moderately large values of electric field, whereupon bulk impact ionization does not prevail. Nonetheless, the present analysis adequately depicts the static electron-hole flow when the minority carrier holes are generated by localized generation and injection mechanisms. Note that the effects of spatially dependent recombination processes are ignored here ; this implies that the above analysis is necessarily limited to local regions of the semiconductor wherein carrier recombination is slow and the density of minority carrier holes is relatively uniform.
co,
C. Solution .#or the Growth Rate of the Acoustoelertric Intevuction As before, the off-axis acoustoelectric interaction is described by the equivalent one-dimensional dispersion equation (derived i n Section I1.C) and the rf conductivity is determined from the small-signal field and transport equations (derived in Section lI.D). The growth rate of the interaction is
239
ACOUSTOELECTRIC INTERACTIONS
given by Eq. (41), wherein the values of the electromechanical coupling constants l i 2 and u, are found using the solution method of Section V.B, and the longitudinal rf conductivity o,,must yet be determined. If the static carrier transport system is analyzed using the solution method of Section VI.B,the electron and hole drift velocities u ~ and , u ~ ~ can . be ~ determined. Specification of the azimuthal orientation 4" of the transverse magnetic field and the assignment of a long dimension direction L locate the directions of the drift velocities u~,, and uo, ,, in the crystal coordinate system, as shown in Fig. 36. It is assumed that an acoustic wave propagates
LORENTZ FORCE , PLANE
A +
x2
L
FIG.36. Relative orientations of the transverse magnetic field B,, the acoustic wave 4 and the electron and hole drift velocities uo. and uo,,, [shown in the crystal coordinate system (x,, s2, xd].