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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 61 CONTRIBUTORS TO THISVOLUME H. BREMMER D. S. BUGNOLO M. CAILL...

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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS

VOLUME 61

CONTRIBUTORS TO THISVOLUME

H. BREMMER D. S. BUGNOLO M. CAILLER GILBERTDE MEY J . P. GANACHAUD A. G . MILNES D. ROPTIN

Advances in

Electronics and Electron Physics EDITEDB Y PETER W. HAWKES Laboratoire d’Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France

VOLUME 61 1983

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers

New York London Paris San Diego San Francisco Sao Paulo Sydney Tokyo Toronto

COPYRIGHT @ 1983,BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED O R TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC O R MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS,INC.

111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. ( L O N D O N ) LTD. 24/28 Oval Road, London N W l

7DX

LIBRARY OF CONGRESS CATALOG CARDNUMBER:49-1504 ISBN 0-1 2-014661 -4 PRINTED IN THE UNITED STATES OF AMERICA

83 84 85 86

9 8 7 65 4 3 2 1

CONTENTS CONTRIBUTORS TO VOLUME 61 FOREWORD . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii ix

Potential Calculations in Hall Plates GILBERTDE MEY

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . I1. Fundamental Equations for a Hall-Plate Medium . . . . . . . . I11. The Van Der Pauw Method . . . . . . . . . . . . . . . . . . IV . Influence of the Geometry on Hall-Mobility Measurements . . . V . Conformal Mapping Techniques . . . . . . . . . . . . . . . . VI . Relaxation Methods . . . . . . . . . . . . . . . . . . . . . VII . The Boundary-Element Method for Potential Calculations in Hall Plates . . . . . . . . . . . . . . . . . . . . . . . . . VIII . Improvement o f t h e Boundary-Element Method . . . . . . . . IX . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1 . The Three-Dimensional Hall Effect . . . . . . . . Appendix 2 . On the Existence of Solutions of Integral Equations Appendix 3. Green’s Theorem . . . . . . . . . . . . . . . . Appendix 4 . The Hall-Effect Photovoltaic Cell . . . . . . . . . Appendix 5 . Contribution of the Hall-Plate Current to the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . Appendix 6 . Literature . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 7 9

10 15 18 38 48 49 52 54 57 58 59 59

Impurity and Defect Levels (Experimental) in Gallium Arsenide A . G . MILNES

I. I1. I11. IV . V. VI . VII . VIII . IX . X.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . Possible Native Defects and Complexes . . . . . . . . . . . Traps (and Nomenclature) from DLTS Studies . . . . . . . . Levels Produced by Irradiation . . . . . . . . . . . . . . . Semi-Insulating Gallium Arsenide with and without Chromium . Effects Produced by Transition Metals . . . . . . . . . . . . Group I Impurities: Li, Cu. Ag. Au . . . . . . . . . . . . . Shallow Acceptors: Be. Mg. Zn. Cd . . . . . . . . . . . . . Group IV Elements as Dopants: C. Si. Ge. Sn. Pb . . . . . . Oxygen in GaAs . . . . . . . . . . . . . . . . . . . . . . . v

. . . . . . .

.

64 65 76 81 91 100 108 116 118 123

vi

CONTENTS

XI . Group VI Shallow Donors: S. Se. Te . . . . . . . . . . . . XI1 . Other Impurities (Mo. Ru. Pd. W. Pt. Tm. Nd) . . . . . . . . XI11. Minority-Carrier Recombination. Generation. Lifetime. and Diffusion Length . . . . . . . . . . . . . . . . . . . . . . XIV . Concluding Discussion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

. .

i27

128 131 141 142

Quantitative Auger Electron Spectroscopy M . CAILLER.J . P . GANACHAUD. AND D . ROPTIN

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . 11. General Definitions . . . . . . . . . . . . . . . . . . . . . I11. Dielectric Theory of Inelastic Collisions of Electrons in a Solid . . IV . Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . V . Auger Transitions in a Solid . . . . . . . . . . . . . . . . . VI . Quantitative Description of Auger Emission . . . . . . . . . . VII . Auger Quantitative Analysis . . . . . . . . . . . . . . . . . VIII . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

162 167 173 185 187 213 244 289 289

The Wigner Distribution Matrix for the Electric Field in a Stochastic Dielectric with Computer Simulation D . S . BUGNOLO AND H . BREMMER

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . I1. The Differential Equation for the Electric Field Correlations . . . 111. Derivation of the Equations for the Wigner Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Related Equations for the Wigner Distribution Function . . . . . V . Asymptotic Equations for the Wigner Distribution Function . . . VI . Equations for Some Special Cases . . . . . . . . . . . . . . . VII . A Brief Review of Other Theoretical Methods . . . . . . . . . VIII . The Coherent Wigner Function . . . . . . . . . . . . . . . . IX . Computer Simulation of the Stochastic Transport Equation for the Wigner Function in a Time-Invariant Stochastic Dielectric X . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1. A Listing of Experimental Program Number Two for the Case of an Exponential Space Correlation Function . . . Appendix 2 . A Sample of a Computer Simulation . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

300 303

AUTHORINDEX . SUBJECT INDEX.

391 402

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

313 325 330 335 345 347 354 382 383 386 388

CONTRIBUTORS TO VOLUME 61 Numbers in parentheses indicate the pages on which the authors’ contributions begin.

H. BREMMER,31 Bosuillaan, Flatgebou’w Houdringe, Bilthoven, The Netherlands (299) D. S. BUGNOLO,*Department of Electrical and Computer Engineering, Florida Institute of Technology, Melbourne, Florida 32901 (299) M. CAILLER,Laboratoire d e Physique du MCtal, Ecole Nationale SupCrieure de MCcanique, 44072 Nantes Cedex, France (161) GILBERTDE MEY, Laboratory of Electronics, Ghent State University, B-9000 Ghent, Belgium (1) J. P. GANACHAUD, Laboratoire de Physique du Solide, Institut de Physique de I’UniversitC de Nantes, Nantes, France (161) A. G. MILNES,Department of Electrical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213 (63)

D. ROPTIN,Laboratoire de Physique du MCtal, Ecole Nationale SupCrieure de MCcanique, 44072 Nantes Cedex, France (161)

*Present address: Department of Electrical and Computer Engineering, University of Alabama in Huntsville, Huntsville, Alabama 35899. vii

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FOREWORD

The boundary between electron physics and solid state physics is by no means sharp, as several of the contributions to the present volume show: The articles on Auger electron spectroscopy and on gallium arsenide bring this out very clearly, and it is one of the functions of a series such as this that such topics, which span more than one discipline, can be explored at length. The other two contributions are more theoretical in nature; the article on Hall plates is of course still concerned with semiconductors, though here it is the calculation of the potential distribution that is of interest. The other theoretical article, dealing with the Wigner distribution matrix in a stochastic dielectic, covers one aspect of a topic that is currently attracting much attention; we plan to publish reviews of other aspects of the Wigner distribution in forthcoming volumes. It only remains for me to thank very warmly all the contributors to this volume. The customary list of forthcoming articles is given below, and as usual I encourage potential contributors to contact me, even if their plans are still in the preliminary stage. Critical Reviews: Electron Scattering and Nuclear Structure Large Molecules in Space Electron Storage Rings Radiation Damage in Semiconductors

Visualization of Single Heavy Atoms with the Electron Microscope Light Valve Technology Electrical Structure of the Middle Atmosphere Diagnosis and Therapy Using Microwaves Low-Energy Atomic Beam Spectroscopy History of Photoemission Power Switching Transistors Radiation Technology Infrared Detector Arrays The Technical Development of the Shortwave Radio CW Beam Annealing Process and Application for Superconducting Alloy Fabrication Polarized Ion Sources Ultrasensitive Detection The Interactions of Measurement Principles, Interfaces, and Microcomputers in Intelligent Instruments Fine-Line Pattern Definition and Etching for VLSI ix

G. A. Peterson M. and G. Winnewisser D. Trines N. D. Wilsey and J. W. Corbett

J. S. Wall J. Grinberg L . C. Hale M. Gautherie and A. Priou E. M. Horl and E. Semerad W. E . Spicer P. L. Hower L. S. Birks D. Long and W. Scott E . Sivowitch J. F. Gibbons H. F. Glavish K. H . Purser W. G. Wolber Roy A. Colclaser

X

FORE WORD

Waveguide and Coaxial Probes for Nondestructive Testing of Materials The Measurement of Core Electron Energy Levels Millimeter Radar Recent Advances in the Theory of Surface Electronic Structure Long-Life High-Current-Density Cathodes Microwaves in Semiconductor Electronics Applications of Quadrupole Mass Spectrometers Advances in Materials for Thick-Film Hybrid Microcircuits Guided-Wave Circuit Technology Fast-Wave Tube Devices Spin Effects in Electron-Atom Collision Processes Recent Advances in and Basic Studies of Photoemitters Solid State Imaging Devices Structure of Intermetallic and Interstitial Compounds Smart Sensors Structure Calculations in Electron Microscopy Voltage Measurements in the Scanning Electron Microscope Supplementary Volumes: Microwave Field-Effect Transistors Magnetic Reconnection Volume 62: Predictions of Deep Impurity-Level Energies in Semiconductors Spin-Polarized Electrons in Solid-state Physics

Recent Advances in the Electron Microscopy of Materials

F. E. Gardiol R. N. Lee and C. Anderson Robert D. Hayes Henry Krakauer Robert T. Longo J. L. Allen I. Berecz, S. Bohatka, and G. Langer J. Sergent M. K. Barnoski J. M. Baird H. Kleinpoppen H. Timan E. H. Snow A. C. Switendick W. G. Wolber D. van Dyck

A. Gopinath

J. Frey P. J. Baum and A. Bratenahl

P. Vogl H. C. Siegmann, F. Meier, M.Erbudak, and M. Landolt D. B. Williams and D.E. Newbury

PETERW. HAWKES

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS

VOLUME 61

This Page Intentionally Left Blank

.

ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS VOL . 61

Potential Calculations in Hall Plates GILBERT DE MEY Laboratory of Electronics Ghent State Wniversiry Ghenr. Belgium

I . Introduction ........................................ 11. Fundamental Equations for a Hall-Plate Medium

2 3 3 4 5 6

...........................

A . General Equations for a Semiconductor ............ B. Approximations for a Thin Semiconducting Layer ... C . Constitutive Relations with an Externally Applied Magnetic Field .......... D . Boundary Conditions with Externally Applied M q n e Ill . The Van Der Pauw Method ........................................... IV. Influence of the Geometry on Hall-Mobility Measurements ................... V . Conformal Mapping Techniques ..................... A . Basic Ideas ......................................................... B. Approximate Analysis of the Cross-Shaped Geometry .................... C . Exact Analysis of the Cross-Shaped Sample ............................. D . Properties of the Cross-Shaped Hall Plate . . .................... V1. Relaxation Methods .................................... VII . The Boundary-Element Method for Potential Calculations in Hall Plates ........ A . Introduction ...................... ............................... B. Integral Equation for the Potential Distribution in a Hall Plate . C . Numerical Solution of the Integral Equation .......................... D . Application to a Rectangular Hall Plate ..................... E . Zeroth-Order Approximation .............................. F. Numerical Calculation of the Current through a Contact . . . . . . . . . . . . . . . . . . G . Application to the Cross-Shaped Geometry .............................. H . Direct Calculation of the Geometry Correction .................. 1. Application to a Rectangular Hall Generator .................... J . Application to a Cross-Shaped Geometry ............... K . Application to Some Other Geometries ................. VIII . Improvement of the Bounda A . Introduction ................................................. B. Integral Equation ...... .................... C . Calculation of the Funct .......... D . Application to a Rectangular Hall Generator ................... E. Application to a Cross-Shaped Form ................................... F . Application to Some Other Geometries ................................. 1X. Conclusion ............................................................ Appendix 1. The Three-Dimensional Hall Effect .................... Appendix 2. On the Existence of Solutions of Int ations Appendix 3. Green's Theorem ............................................

7 9 10 10 11 13 14 15 18 18 18 20 21 22 24 25 21 29 30 33 38 38 39

40 43 45

46 48 49 52 54

1

.

Copyright P 1983 by Academic Press Inc. All rights oi reproduction in any form reserved . ISBN n-12-014661-4

2

GILBERT DE MEY

Appendix 4. The Hall-Effect PhotovoltaicCell.. ............................ Appendix 5. Contribution of the Hall-PiateCurrent to the Magnetic Field ...... Appendub. Literature .................................................. References .............................................................

57 58 59 59

I. INTRODUCTION

Hall plates are thin semiconducting layers placed in a magnetic field. Owing to the Lorentz force, the current density J and the electric field E are no longer parallel vectors. This means that a current in a given direction automatically generates a potential gradient in the perpendicular direction. With suitable contacts (the so-called Hall contacts), a Hall voltage can then be measured. In a first approximation one can state that the Hall voltage is proportional to the applied magnetic field, the externally supplied current, and the mobility of the charge carriers. Knowledge of the current and of the Hall voltage yields the product pHBof the mobility and the magnetic field B. This indicates two major applications of Hallcffect components. If the mobility is known, magnetic field strengths can be measured. On the other hand, if the magnetic field is known, the mobility can be calculated. The latter is mainly used for the investigation of semiconductors because mobility is a fundamental material parameter. A first series of applications is based on the measurement or detection of magnetic fields. Measurements of the magnetic fields in particle accelerators have been carried out with Hall probes provided with a special geometry in order to ensure a linear characteristic (Haeusler and Lippmann, 1968). Accuracies better than 0.1% have been realized. For alternating magnetic fields, Hall plates can be used at frequencies up to f l0,OOO Hz (Bonfig and Karamalikis, 1972a,b). For higher frequencies emf measurements are recommended to measure magnetic fields. The Hall probe can also be used to detect the presence of a magnetic field. This phenomenon is used in some types of push buttons. On each button a small permanent magnet is provided, and the pushing is sensed by a Hall plate. At this writing Hall plates combined with additional electronic circuitry are available in integrated-circuit form. A Japanese company has produced a cassette recorder in which a Hall probe reads the magnetic tape. The principal advantage here is that dc signals can be read directly from a tape, whereas classical reading heads generate signals proportional to the magnetic flux rate d4/dt. Magnetic bubble memories have also been fitted out with Hall-effect readers (Thompson et al., 1975). A survey of Hallcffect applications can be found in an article written by Bulman (1966) that mentions microwave-power measurements, the use of Hall probes as gyrators, insulators, function generators, ampere meters, etc. Even a brushless dc motor has been constructed using the Hall effect (Kobus

3

POTENTIAL CALCULATIONS IN HALL PLATES

and Quichaud, 1970). Finally, Hall plates can also be applied as transducers for mechanical displacements (Davidson and Gourlay, 1966; Nalecz and Warsza, 1966). A second series of applications, to which this article is mainly devoted, involves the measurement of mobilities. A Hall measurement carried out in a known magnetic field yields the value of p H . This constant is an important parameter for investigating the quality of semiconducting materials. Combined with the resistivity, it also enables us to calculate the carrier concentration. Knowledge of these data is necessary for the construction of components such as diodes, solar cells, and transistors starting from a semiconducting slice. The present article describes the Hall effect and its mathematical representation. The well-known Van Der Pauw method for Hall-mobility measurements is then discussed. The influence of the geometry on the Hall voltage is pointed out using physical considerations. This explains why the potential distribution in a Hall plate should be known in order to evaluate the so-called geometry correction. Then several techniques for potential calculations in Hall plates, such as conformal mapping, finite differences, and the boundary-element method, are outlined and compared. 11. FUNDAMENTAL EQUATIONS FOR

A

HALL-PLATE MEDIUM

A . General Equations f o r a Semiconductor

The fundamental equations for an n-type semiconductor (assuming low injection) are the following: ( a n / a t ) - 4 - 1 V . J , = (an/at)gen - [(P - ~ o ) / ~ p ]

+ q-'

(a~/at)

V *J p = (ap/af)gen- [(P - ~

(1) o ) / ~ p ]

(substitute [(n - no)/r,,] for a p-type layer); Jn

= nqpnE

V.E

=

+ qDn Vn,

-Vz+

Jp

= PqPpE

= (q/c,C)(p - n

- 4Dp V P

+ND -

(2) (3)

NA)

where n is the electron concentration, p is the hole concentration, J , is the electron current density, J,, is the hole current density, J = J , J p is the total current density, ND is the donor concentration, N A is the acceptor concentration, E is the electric field, is the electric poential, p, = qD,/kT is the electron mobility, p p = qD,,/kT is the hole mobility, T,, is the electron relaxation time, T~ is the hole relaxation time, no is the equilibrium electron concentration (in the p layer), po is the equilibrium hole concentration

+

+

GILBERT DE MEY

4

(in the n layer), L is the dielectric constant, and (a/dt),,, is the generation rate (e.g., due to incident light). Equations (1)-(3) are nonlinear for the unknowns n, p , and 4; however, for Hall generators several reasonable assumptions can be advanced so that the final problem becomes linear.

B. Approximations f o r a Thin Semiconducting Layer For thin-film semiconducting layers with contacts sufficiently distant from each other (order of magnitude in millimeters), one usually assumes that the layer is sufficiently doped to ensure that the contribution of the minority carriers becomes negligible. We shall work later with an n-type semiconductor; however, the same treatment can be carried out for a p-type semiconductor. One also assumes that no space charges are built up in the conductor. It can be shown that an occasional space charge only has an influence over a distance comparable to the Debye length. For an n-type layer, the Debye length is given by (Many et al., 1965) LD

=

[(totkT)/q2ND)]”Z

(4)

Normally, LD varies around 100-1000 A, so that a space charge can only be felt over a distance much smaller than the distance between the electrodes. Practically, space charges can only be realized at junctions or nonohmic contacts. Because Hall generators are provided with ohmic contacts, the space charge is zero everywhere. Hence the right-hand member of (3) should vanish; for an n-type material this gives rise to n = ND

and

p0.5), the error curve seems to diverge. This does not mean that the method should fail. The reason is that the relative error was calculated taking Eq. (33) as the exact value. However, there is a slight nonlinearity between the Hall voltage and the applied magnetic field B, a secondary effect not included in Eq. (33). Figure 25 shows the geometry correction as a function of h/l calculated for PHB = 0.1. The theoretical relationship (33) has also been drawn. For h/l < 1.5, numerical results are close to the theoretical ones, especially for high values of ZM,which is the number of unknowns for one side (the total number of unknowns is then 12ZM). For h/l 2 1.5, the deviations become remarkable. Nevertheless, the numerical results still converge to the exact values. In order to obtain insight into the practical consequences, let us take h/l = 2. The exact geometry correction is 0.2%, whereas the numerical value obtained with ZM = 9 turns out to be 0.5%. It makes no difference in practice if an experimental value is corrected by 0.2% or OS%, because this correction is much smaller than the accuracy of the Hall-mobility measurements. Only I 20

-

1

+

-ae 10

$

-

20 10

5-

5-

2-

2-

9' 0 z Q

0 Q b Q

: -

1-

I-

0.5-

0.2-

0.5

+

-

0.2

-

FIG.24. Relative error on the geometry correction versus IM for a cross-shaped Hall plate: (a) h/l = 0.5; p H B= 0.3 (+).

32

GILBERT DE MEY

!i

+8 z E

9

1-

0.5-

az

\

1

0.5

1

I

1

I h/l

I

1.5

I

I

2

*

FIG.25. Geometry correction as a function of h /lfor a cross-shaped Hall plate: p H B= 0.1 ; I M = 3 (A),5 (OL7 (+), 9 ( 0 ) .

=-

when the geometry correction is greater (e.g., 1%) can it be calculated with sufficient accuracy. The conclusion is that the method can be applied in actual cases. Only when the geometry correction is negligible can it not be calculated with high precision; but in these cases knowledge of it is of no importance. It should be noted that the cross-shaped geometry has four reentrant corners, which makes its use difficult for numerical treatments. Some equipotential lines have been drawn in Fig. 26. Owing to Eq. (16) these lines are also the current lines or streamlines. Only 50% of the Hall plate is shown because the remaining part can be found by symmetry. From Fig. 26 the hypothesis formulated in Section IV can be verified.

POTENTIAL CALCULATIONS I N HALL PLATES

33

FIG.26. Streamlines in a cross-shaped Hall plate.

K . Application to Some Other Geometries

We first consider a rectangular Hall generator provided with four finite contacts (Fig. 27). This type of Hall plate is often used experimentally because monocrystalline semiconductors are usually available in the form of a rectangular bar. It is not always possible to cut a monocrystalline semiconductor slice in the form of a cross; a rectangular Hall plate is then used to carry out Hall-mobility measurements. This geometry has been extensively studied theoretically by Haeusler, using conformal mapping techniques (Haeusler and Lippmann, 1968). Nevertheless, this shape is far from optimal. Even for small contacts, the geometry correction turns out to be 30-40%. A

34

GILBERT DE MEY

h

FIG.27. Rectangular Hall plate provided with four finite contacts.

small shift in the position of the contacts causes a nonnegligible change of the geometry correction and hence an error in the experiments. Figure 28 shows the numerically calculated value of A,uH/PHfor h/b = 2 as a function of a/b. The geometry correction increases rapidly with the contact length a and can attain very high values. For this geometry Haeusler found the following approximate formula valid for sufficiently small contacts (Haeusler and Lippmann, 1968):

Equation (67) is accurate up to 4% for h/b > 1.5 and a/b < 0.18. The first condition is met, but the second one is not fulfilled because the numerical BEM cannot be used if a contact is much smaller than the other sides of the Hall plate. Hence we expect that Eq. (67) will not coincide perfectly with the numerical results. For h/b = 2 and 8, small, Eq. (67) reduces to APH/PH

= 1 - (1 - e-")[l

- (2/a)(a/b)] = 0.0432 + 0.609(a/b) (68)

The linear relationship (68)has also been drawn in Fig. 28, and the agreement is fairly good. One observes a better fit when the number of unknowns per side IM increases. A second particular geometry is the octagon provided with four equal contacts (Fig. 29). Although this shape has not been used in experiments, it is the first geometry that has been studied theoretically by Wick using conformal mapping techniques (Wick, 1954). The reason is that the conformal mapping of the octagon into a circle leads to calculations which can be


35

< 0.L

0.5

0.6

0.7

0.8

O/b

FIG. 28. Geometry correction for the rectangular Hall plate: h/b = 2; pHB = 0.1 ; IM = 3 (+I, 5 (A), 7 ( x ), 9 (Oh I 1 13 (01, 15 ( 0 ) .

(m),

performed analytically. Figure 30 represents the numerically calculated geometry correction as a function of l / h for various values of the number of unknowns. One observes again that the geometry correction is high. Wick did not calculate any geometry correction, but from several graphs in his article it was possible to deduce approximately ApH/pH. These results are

t1

FIG.29. Octagonal Hall plate.

36

GILBERT DE MEY

+

I

I

I

0.2

0.3

i/h

0.4

I

*

0.5

FIG.30. Geometry correction for the octagonal Hall plate: p H B= 0.1 ; IM = 3 (+), 5 ( A ) , (m),1 3 ( 0 ) , 1 5 ( 0 ) .

7(x).9(0). I1

shown in Fig. 30 and referred to as “theoretical.” One sees good agreement between these theoretical points and the numerical results. Chwang ef al. (1974) analyzed the octagon using finite difference techniques. From the results published in their article, the geometry correction could be derived for small values of l/h. These results are not very accurate because they differ a lot from the results obtained by Wick (conformal mapping) and the BEM. It is also possible to obtain an approximate formula for ApH/pH for the octogon. Van Der Pauw published geometry corrections for a circular shape provided with finite contacts. An octagon can be approximated by a circle with a diameter h and a contact length 1. The geometry correction is then (Van Der Pauw, 1958) AA+/PH

=

4(2/nZHI/h)= (8/n)’(1/h)

(69)

This relation is also drawn in Fig. 30, and it turns out to be a good approximation for l / h % 0.5. For other l/h values the replacement of an octagon by a circle seems to be less adequate.

37


FIG.31. Unsymmetrical cross-shaped form.

A final interesting geometry is the unsymmetrical cross-shaped form. Thin-film semiconducting Hall plates are usually made by evaporating the materials through a metallic mask. For the contacts on the Hall plate another mask is required, and thus a shift between the mask positionings can never be avoided. This gives rise to an unsymmetrical cross-shaped form. A fortiori, when the contacts are made manually with a conducting ink, for example, a perfect symmetry cannot be guaranteed. We now investigate a cross-shaped form where one arm differs from the others (Fig. 31). The numerical results are shown in Fig. 32, representing the geometry correction ApH/pH as a function of u/l. The geometry correction increases when the arm length shortens, which can easily be understood. These results can also be found theoretically. In Section V,B, we made the assumption that each arm gives its own contribution to the geometry correction. This being the case, the following formula was found for four equal arm lengths:

Applying Eq. (70) to three arms with length h and one arm with length u, one obtains

*

PH = 4L nze x p ( i l n 2 =

For h / l

=

1.04,3[:exp(

- :)[:exp( -n)>

-n:)

+ aexp( -

31

+ aexp( - n 3 1

1, Eq. (71) reduces to ApH/pH = 0.03387

+ 0.2613e-"""

(72)

This relation has also been shown in Fig. 32, and the agreement with the numerical results is very good. This also proves the hypothesis that each

38

GILBERT DE MEY

13

+ 12

A

11

10-

-

s

9

L

z +

8

I

x

d

7

6

5

L

I

0.4

I

0.5

I

0.6

1

0.7

I

0.8

I

I

0.9

I

C

0 4

FIG.32. Geometry correction for the unsymmetrical cross-shaped Hall generator: p H B = IM = 3 (+), 5 (A), 7 ( U ) . 9 (0).

0.1; h / l = I ;

arm generates its own contribution to the geometry correction ApH/pH independent of the other ones.

VIII. IMPROVEMENT OF THE BOUNDARY-ELEMENT METHOD A . Introduction

In this section we present a modified BEM. By investigating the method it was found that the highest errors occurred at the corners of the geometry. Therefore, a method is given here which uses analytic approximations at the corners while the remaining part of the potential is calculated numerically. This technique can be seen as a combination of conformal mapping and the


39

BEM. The conformal mappings are used to calculate analytic approximations at the corners of the Hall plate. The remaining part of the potential distribution is found by numerical solution of an integral equation. The main advantage of this method is that singularities occurring at a corner can be represented exactly by the analytic approximation (De Mey, 1980).

B. Integral Equation We consider a Hall plate as having the shape of an arbitrary polygon (Fig. 33). Along each side zi the following boundary condition holds: a4

+ BV4.U" + yV4.q

= fo

(73)

Forametalliccontactatpotential Vo,onehasa = 1,p = y = O,andf, = Vo. For a free boundary a = 0, /?= 1, y = -pHB, and fo = 0. All types of boundary conditions along a Hall-plate medium can be represented by Eq. (73). It makes no difference in the subsequent analysis whether one is interested in the electrostatic potential or the stream potential II/. We assume now that in the neighborhood of each corner an analytic approximation for the potential is known. At the ith corner one has

4(r)

AiPi(r)

(74)

where cpi(r) is a known function satisfying the Laplace equation; Ai is a still unknown proportionality constant. The constants Ai will be determined together with the solution of the integral equation. The potential inside the Hall plate is now written as

4(r) =

Aicpi(r) i

+ f C p(r')G(rlr') dC'

(75)

The function cpi(r) includes all the singularities at the corners. Note that cpi(r) is not only defined at the ith corner, but in the entire region. This means that cpi(r) will not necessarely be zero at the other corners but that the functions cpi(r) will be chosen in such a way that cpi(r) is a smooth potential

\

z,-i

FIG. 33. Boundary condition at one side of an arbitrary polygon: aiQ yIvQ.u,=jO.

+ fii V+.u. +

40

GILBERT DE MEY

distribution at the other corners. The basic idea is that 1 Aiqi(r)is not a first-order approximation but a particular solution involving all singularities, especially those occurring in the gradient field. The remaining part will then be a perfectly smooth surface and is represented by the last term in Eq. (75). Applying the boundary condition (73) to the proposed solution (75) gives rise to C A i [ a q i + PVqi.un + ~ v q i . ~ ] i

- P+[p(r)l

+ (jcP(rWNG(rIr7

+ P V G ( r l r ’ ) - u , + yVG(rIr’)*u,]dC’ = fb(r)

(76)

Equation (76) constitutes an integral equation in the unknown function p. We remark that the constants Ai are also unknown and have to be determined simultaneously with the numerical calculation of p. C. Culculation of the Functions qi(r)

The purpose of this section is to determine the q i , potential functions being good approximations at corners, and to include all singularities: especially those occurring in the gradient fields Vq,.The method is outlined for the corner A of the Hall plate shown in Fig. 20. The corner can be replaced by two infinite sides. The function qi is then a potential satisfying the boundary conditions indicated in Fig. 20 along two infinite sides. This can be easily done, as it is always possible to map one corner onto another at which the potential is reduced to a constant and homogeneous field. This conformal

FIG.34. Approximation of a corner by two circular arcs.


41

mapping is done by the simple analytic function 2'. Some problems may arise, however. Consider again Fig. 20 for the particular case p H B = 0. In each corner one side has a constant potential and the other side requires that the normal gradient should be zero. The field is then homogeneous because the corners are rectangular. The potential pi then varies linearly with distance. The same situation is valid for all corners, however, which means that the four functions cpl, cpz, cp3, and cp4 will be linearly dependent. In this case it is not possible to determine the constants A , , A z , A 3 , and A, numerically. Linearly dependent functions should always be avoided. In order to eliminate this problem, a corner will not be approximated by two infinite sides but rather by two circular arcs, as shown in Fig. 34, representing a corner with angle cp. One must now solve the Laplace equation V Z q i= 0 with the condition cpi = 0 at one circular arc and Vcpi-u,+ p H BVqi u, = 0 at the other. In this case it is almost impossible to obtain linearly dependent solutions at two corners. Because the circles are tangent at the sides of the corners, qi is still a good approximation for the potential. The solution of this potential problem is carried out with conformal mapping techniques. The geometry bounded by two circular arcs can be mapped onto a semiinfinite plane using (Fig. 39,

-

2' =

A"z/(z -

Zo)]n'e

(77)

where A' is a complex constant. Let us consider now the particular case where c2 is a free boundary and c1 a metallic contact held at constant potential. We

FIG.

35. Conformal mapping on a corner with angle ( 4 2 ) - OH

GILBERT DE MEY

42

then have to transform into a corner with an angle 4 2 - OH ( w plane in Fig. 3 9 , because for this geometry the potential can be written immediately. The transformation from the z to the w plane is then w = A ’ ( 1 / 2 ) - ( @ H / 2[) z / ( z - zO)](n/2o)-(@H/9) = A [ z / ( z - ~ ~ ) ] ( n / 2 1 ) - ( @ H / O ) (78) For a point z located on c1 (Fig. 35), one has

- zo) = ( z / ( z - zo)l e-jqlz

z/(z

(79)

then, w = A 1 z/(z - z o )1 ( ~ / W - ( @ H / V ) exp( - j i n

+ jl-0 2 H)

(80)

A point z on c1 is mapped onto a point w situated on the real u axis. Hence Eq. (80) has to be real, from which A can be determined:

A = exp(jin - j + O H )

(81)

Because the function cpi is multiplied by a constant Ai in the proposed solution (75), the absolute value of A has no influence on our problem. From the transformation formula (78) one obtains dw/dz = w [ ( $ n - &)/~][-zO/z(z - ZO)]

(82)

Equation (82) is used later to calculate the transformation of the electric fields. In the w plane, the potential cp corresponds to a homogeneous field directed perpendicular to the u axis (Fig. 35) because the angle was taken to be $n - 0,. Hence pi can be written as Vi(U,O)

=

(83)

IJ

The potential function can be derived from the complex potential W : W = -jw

[cpi

= Re( W ) ]

(84)

with d W / d w = -j

The potential in the z plane can be easily determined because the complex potential W is invariant. The electric field components in the z plane are determined by dW - - - - E , + j E. dz

y

acpi ax

= - - j - = -acpi -= ay

dwdw dw dz

-j-dw

dz

(86)

Inserting Eq. (82) into Eq. (86) directly gives us the field components in the original z plane. It has been proved that the potential cpi in a corner can be determined by a simple conformal mapping.


43

It should be noted that cpi has only been determined up to a proportionality constant. Because the functions cpi are preceded by an unknown constant Ai in Eqs. (74) and (75), this causes no problems. For other situations at the corners, e.g., two adjacent free boundaries at a corner, the procedure to determine cpi remains identical; however, some formulas may be appropriately changed. D. Application to a Rectangular Hall Generator

Figure 36 shows a rectangular Hall plate with the circles for the approximation of the potential in the corners. First, the influence of the radius R was investigated. Figure 37 indicates the relative error of the Hall voltage and the current I as a function of the radius R for various values of I M , which is the number of unknowns per side. One observes that the error is almost independent of R. As for the current, it seems to be an optimum around R = 1; however, this is owing to the fact that the numerical value of the current minus the exact value changes its sign when R x 1, so one cannot consider it to be optimal. Figure 38 shows the relative error of the Hall voltage and the supply current as a function of IM. The numerical results obtained without the analytic approximation C Aicpi have also been drawn in Fig. 38. The accuracy of the Hall voltage remains practically unchanged, but the accuracy on the current calculation increases by almost two decades. Because both quantities are needed for practical applications, an overall improvement has been achieved. The geometry correction has also been investigated. Figure 39 shows the relative error on the geometry correction as a function of IM. The same

FIG.36. Square-shaped Hall plate with approximate arcs at the corners.

5-

-z

2-

6

E

1-

2 c

a5-

2 a20.1-

M5-l

-., t

a1

R

FIG. 37. Relative error as a function of radius R : p H B = 1 ; AVH/VH ( 0 .

(0.0. A).

A); A I / l

4

3

5

9

15

29

1M

FIG.38. Relative error on the Hall voltage and the supply current: R = 1 ; pHB = I , with ( + ) and without ( 0 )analytic approximation.


45

t

i

5

rb

IM

.?a

50

FIG.39. Relative error on the geometry correction as a function of IM:p,J = 0.I (+, x ); R = I ; with analytic approximation (+, A, 0 , 0); 0.3 (A, A); 0.5 ( 0 ,0 ) ; I (W, 0); without analytic approximation ( x , A , 0, m).

results without the analytic approximation are also drawn in Fig. 39. Generally, the introduction of analytic approximations yields better results, especially for low values of p H B ,when the Hall voltage is low and difficult to detect. E. Application to a Cross-Shaped Form

The cross-shaped form has also been analyzed using the same method. In this case, there are 12 corners and hence 12 qifunctions have to be evaluated. The geometry correction was calculated using the method outlined in Section VII,H. The results are shown in Fig. 40.The data obtained in Section VII,J are also drawn in Fig. 40. One now observes, contrary to the results found in the preceeding section for a square-shaped form, that the introduction of analytic approximations at the corners yields no improvement in the numerical results. Noting the fact that the calculation of the cpi functions involves computation time, one must conclude that the method of Section VII,H should now be preferred. These results indicate that the present method is only useful for a limited number of corners. The cross-shaped sample has also been analyzed using an analytic function at a reduced number ( c12) of corners, but this did not lead to any improvement.

GlLBERT DE MEY

46

A

1

a5

I

1

1

1

I

1.5

I

1)

2

h/ I

FIG.40. Geometry correction versus h / l for a cross-shaped sample; pHB = 0.1 ; with analytic approximation, IM = 3 (A), 9 ( 0 )without ; analytic approximation, IM = 3 (A),9 ).(

F . Application to Some Other Geometries

Because the number of corners seems to influence the numerical accuracy, two geometries with eight corners are now investigated. The rectangular and octagonal Hall plates treated in Section VII,K are now analyzed. It should be noted that the rectangular Hall plate provided with four finite contacts should be viewed as having eight corners. Because the electric field at both ends of each Hall contact shows a singular behavior, the introduction of analytic functions may be useful. Figure 41 represents the geometry correction as a function of a/b. Some results of Figure 28 have been redrawn. One sees that more accurate results are now found.

60

-

-A

M-

2

*5 z

Q

LO-

30

I

I

0.5

0.4

I

d.6

I

0.7

0.6

o/b

FIG.41. Geometry correction for a rectangular Hall plate: hlb = 2; pHB= 0.1 ; without analyticapproximation, IM = 3 ( A ) , 15 ( 0 ) withanalyticapproximation, ; IM = 3 ( A), 15 (0).

0.2

0.3

0.6

as

I/h

FIG. 42. Geometry correction for an octagonal Hall plate: pHB= 0.1, without analytic approximation, IM = 3 ( A ) , 15 ( 0 ) ;with analytic approximation, IM = 3 ( A ) , 15 ( 0 ) .

48

GILBERT DE MEY

Similar data for the octagonal shape are shown in Fig. 42. The same conclusion still holds because one observes that the numerical results with analytic approximations are closer to the theoretical curve. As a general conclusion of this section, one can state that the introduction of analytic approximations that include the singularities at the corners is only useful if the number of corners is limited to eight.

1X. CONCLUSION In this article two objectives concerning Hall-effect devices have been outlined. First, a review was given concerning several methods for calculating the potential distribution in a Hall plate. Second, an attempt was made to answer the question: What can be done with such a potential distribution? It turns out that the Hall voltage depends upon the geometry of the plate, an effect which can only be calculated if the potential problem is solved. Because many applications of Hall effects involve measurement of the Hall voltage, it is important to know all the influences on this voltage. When several methods for potential calculations were tested, not only did we draw streamlines or equipotential lines, but we emphasized the evaluation of the geometry correction to the Hall voltage. The first objective of this article was to check several methods for potential calculations in Hall plates. The potential satisfies the Laplace equation, but the boundary conditions are rather unusual, so that common techniques such as separation of variables or eigenfunction expansion cannot be applied. Three methods were found to give acceptable results: conformal mapping, finite difference approximation, and the BEM, the last two being purely numerical. Conformal mapping was the first method used to investigate the Hall effect. Some geometries can be analyzed with a completely analytical treatment; however, the calculations are very lengthly and complicated. Conformal mapping has also been applied successfully in a semianalytical approach : after the Schwarz-Christoffel transformation formula is written down further steps are performed numerically. In this way the high accuracy associated with conformal mapping techniques can be achieved with only moderate computational effort. The finite difference method is the most obvious numerical method, but complications arise owing to the special boundary conditions. This method can be used for a potential distribution, but if current and impedances are calculated the finite difference approach gives unacceptable results. In the BEM the special boundary condition presents no particular difficulties. Potential calculations can be done accurately. With some modifications, even small geometry corrections may be


49

calculated. Finally, we can say that either theconformal mapping or the BEM can be used. Conformal mapping yields high accuracies, but for every new geometry, calculations have to be done again. The BEM can be applied to all geometries but its accuracy is lower. One of the two methods will be appropriate for each individual case. In the last section a combination of these two methods was presented. The potential was still calculated by the BEM, but at the corners analytical approximations obtained with conformal mapping were introduced. I t turns out that this combined method only constitutes an improvement for geometries with a limited number of corners. The second objective of this article was to show the influence of the geometry on the Hall voltage, i.e., the geometry correction. Van Der Pauw’s theory gives a formula for the Hall voltage, but owing to geometic effects (i.e.,finitecontacts) a lower value will be measured in practice. Thiscorrection has been calculated for several geometries, and the conclusion is that the cross-shaped form requires the smallest correction. This may be useful because some geometric parameters are not always known with sufficient accuracy. The cross-shaped form is also fitted with contacts comparable to the dimensions of the Hall plate, so that the resistance between two contacts and hence the noise in the measuring circuitry will be minimal. APPENDIX 1. THETHREE-DIMENSIONAL HALLEFFECT The Hall generators studied in this article are essentially two dimensional. Van Der Pauw’s theory, as outlined in Section 111, cannot be extended to a three-dimensional semiconducting volume having four contacts on its surface and placed in a magnetic field. Even when all contacts are point shaped, a general formula for the Hall voltage cannot be given. For every geometry one has to solve the potential problem from which the Hall voltage is found. Neglecting terms of order &B2, it turns out that the potential still satisfies the Laplace equation (De Mey, 1974b). At a metallic contact the potential is given, but on a free surface one has where u, is the unit vector perpendicular to the semiconductor surface and u, is the unit vector in the direction of the magnetic field. Note that u, and u, need to be perpendicular, as in the two-dimensional case. Because u, and u, can include all possible angles, the right-hand side of Eq. (87) turns out to be a complicated boundary condition. A perturbation method is therefore used to solve the problem. Treating PHB as a small quantity, the potential 4 can be

GILBERT DE MEY

50

written as a zeroth-order approximation 4oand a first-order perturbation

4

=

40

+ P(HB4I

The equations and boundary conditions for do and

v24, = 0 v40 'U, = 0 4o = applied potential v24, = 0 V 4 , 'u, = (V40 x uz)*u, 41

=o

:

(88)

4, are then (89)

on a free surface

(90)

on a metallic contact

(91) (92)

on a free surface

(93)

on a metallic contact

(94)

These equations are to be solved for a cylindrical semiconducting bar, as shown in Fig. 43. If a potential difference V is applied, the zeroth-order potential is easily found: 40

=

(V/a)x

(95)

The zeroth-order current I, through the contacts is given by I0

= (V/p)(nR2/a)

(96)

For the first-order perturbation, an eigenfunction expansion is used. In a circular section, the Dirichlet eigenfunctions are

where N,, is the normalization constant, Jn is the Bessel function of the first kind of order n and xnpis the pth root of the transcendental equation The first-order potential

41 is then written

where

,

the other coefficients being zero. The normalization constant N, is given by

3

FIG.43. Cylindrical Hall medium placed in a magnetic field.

as

0

1

1

1

1

1

5

FIG.44. Hall voltage for a cylindrical semiconductor.

.-

52

GILBERT DE MEY

From Eqs. (96), (99), and (lOO), the Hall voltage V , between the points P and Q can be found for a given current I , : VH = -4/&BpI,(nR)-'c ( x t p - l)-'[l - cosh(x,,/R)(a/2)]-'

(102)

P

Figure 44 shows the Hall voltage normalized to pHBpIo/R as a function of u/R. If u/R + co, i.e., the semiconducting bar becomes infinitely long, the normalized Hall voltage equals 0.6324. This means that the cylinder generates the same Hall voltage as a two-dimensional Hall plate with the same width 2R and a thickness R/0.6324.

APPENDIX 2. ON THE EXISTENCE OF SOLUTIONS OF INTEGRAL EQUATIONS If the Laplace equation for a given geometry with suitable boundary conditions is replaced by an equivalent integral equation, generally a Fredholm equation of the first kind will be obtained. On the other hand, the existence theorems for integral equations deal only with equations of the second kind (Hochstadt, 1973).For these equations it is possible to construct an iteration procedure, which can be used to prove the existence theorems. With regard to Fredholm integral equations of the first kind, the literature is quite contradictory. Some authors claim that the existence has been proved (Symm, 1963), whereas other authors allege the reverse (Tottenham, 1978). A particular geometry is now treated in order to investigate the existence of a solution. The conclusion cannot be extended to arbitrary geometries, but it will provide us with a deeper insight into the problem. Also, the contradictions occurring in the literature will be better understood. As integral equations for potential problems are mainly used as numerical approximations, the treatment in this appendix is also done from a numerical point of view. According to Fredholm theory (Courant and Hilbert, 1968) an integral equation can be regarded as the limit of a set of algebraic equations whose number of unknowns goes to infinity. We shall proceed now in a similar way. In order to investigate the integral equation for the circular geometry shown in Fig. 45, the boundary is divided into n equal parts Acj. The integral equation fcp(r')G(rlrc)dC'= V(r),

is then replaced by the finite algebraic set

reC

(103)


53

I'

FIG.45. Circular geometry used to study the existence of a solution of the integral equation.

This set will have a unique solution if the determinant of the matrix [A,,] is not zero: (105) det[A,,] # 0 Similarly, the integral equation will have a unique solution if the Fredholm determinant does not equal zero. In our problem, the easiest way to check if Eq. (105) is fulfilled is to calculate all the eigenvalues of [A,,]. When all the eigenvalues are other than zero, the determinant will also be nonzero and vice versa. For the particular geometry of Fig. 45, one can easily verify that =

A(l-J)mOdn

(106)

if the boundary is divided into equal parts AC,. Equation (106) means that the matrix [A,] is circulant. It can be proved that the eigenvalues are then given by 11

i.,

=

C

(107)

A,,

k= 1 n- I

'm

=

1A l . k + l

k= 0

e - 2 n jmkln

,

m = l , 2 ,..., n - 1

(108)

If the matrix dimension increases, the matrix still remains circulant. In the limit for n + 00, the expressions (107)and (108)are then replaced by integrals.

54

GILBERT DE MEY

The eigenvalues of the integral equation (103) are then found to be r

i o= (2n)-'fClnlr - r,IdC

i,, = (2n)-

'

In I r - r, Ie-JmedC

where r, can be chosen arbitrarily on the boundary. FOPa circular geometry, the integrals (109) and (1 10) can be easily evaluated: j . , =

(R/n)

i,, = (R/n)

1; [i

ln(2R sin +6)d6 = R In R

(111)

ln(2R sin +6)cos m6 d6 = - R/2m

(1 12)

If R = 1, the eigenvalue I, turns out to be zero. The integral equation will then have no solution at all because the Fredholm determinant is zero. This fact has been investigated by Jaswon and Symm (1978) and Symm (1964). By using an appropriate scaling factor, this problem can be easily eliminated. It has been verified experimentally that eigenvalues of the matrix [ A i j ] (with dimension n) show behavior similar to the first n values l o ,R,, . . .,i n given by Eqs. (111) and (112). Especially, it was found that the smallest eigenvalue is inversely proportional to the matrix dimension n. This implies that I , , - , tends to zero when n + 00. One therefore concludes that the Fredholm determinant of the integral equation (103) will always be zero, meaning that Eq. (103) will have no solution at all. O n the other hand, as long as n remains finite, all the eigenvalues are other than zero and the algebraic set has a unique solution. It can thus be stated that the integral equation only has a solution if one considers it to be the limit solution of an algebraic set. If the solution of the set converges for n + co,one can define it as the solution of the integral equation. Many properties of Fredholm integral equations of the first kind have not been studied or explained because this matter turns out to be a difficult area of functional analysis. This explains why some numerical results, such as the l/n law for the relative error versus the number of unknowns, have not been declared yet. A theoretical background will indicate to us in which direction research should be continued in order to improve the BEM.

APPENDIX 3. GREEN'S THEOREM The method outlined in Section VIII,B is rather unusual for constructing an integral equation for the BEM. Normally, Green's theorem is used,

55


leading to an integral equation where the unknown source function p has a physical meaning. If the potential is given on a part of the boundary, the unknown function turns out to be the normal component of the gradient and vice versa. However, this can only be done if the so-called natural boundary conditions are given, i.e., the boundary conditions only involve 4 or V 4 sun. The boundary condition (18) is not a natural one. An attempt has been made to integrate the tangential component along the boundary in order to obtain the potential 4 rather than V 4 mu,. The procedure is outlined later. Consider the Hall plate shown in Fig. 11. The potential satisfies

v24(r)= 0

(113)

and the Green's function satisfies V2G(r(r')= 6(r - r ' )

(1 14)

Multiplying Eq. (1 13) by G and Eq. (1 14) by 4 and substracting the resulting equations from each other, one obtains after applying Green's theorem fc [4(r)VG(rlr')*un- G(rlr')V4-un]dC = 4(r')

(1 15)

Equation (1 15) holds for each point r' inside C and can be transformed into an integral equation if r' is placed on the boundary. When 4 is given (e.g., on a metallic contact), then V 4 * u, is the unknown function. However, along the side BB', Eq. (18) is valid and V4.u" is not given, so 4 cannot be treated directly as the unknown function. Some intermediate steps are required:

The last integral in Eq. (1 16) can be transformed by partial integration: pHB

jBB'

G V$

*

U, dC

jBB'4

= pHB[G(r1r')4(r)]:r:~,- pHB

VG -u, dC

(1 17)

Equation (1 16) now reads jBB'[4VG-un- GV4.un]dC

IBB'

= pHBG(rB,lr')vo-I-

[4VG*U, - p~BvG'll,]dC

(118)

The tangential component V 4 -u, has been eliminated, and only 4 appears as an unknown function in the intergrand. Hence it is possible now to consider 4

56

GILBERT DE MEY

the unknown function along BB’. The same procedure can be carried out along AA’, so that Eq. (1 15) can be rewritten as n

n

- pHBG(rB lr’)Vo +

[ V o V G . u , - G V d * u , ] dC s.,A.

d[VG.U, - p H B V G ’ U , ] dC

+ pHBG(rA,lr’)Vo,

CE

c

(119)

+ JAA,

I f r’ E C , Eq. (1 19) is an integral equation that can be solved numerically; however, in Eq. (119) terms of the form VoG(rB,lr’) occur. Since G is a logarithmic function, these terms will be singular at the corner points A ’ and B’. Generally, it can be stated that singularities will occur at the end points of every contact staying at a nonzero potential. These difficulties d o not happen with the technique presented in Section VII,B, because the Green’s function then always appears under the integral sign. The integral equation (119) with r’E C has been solved numerically. In the term G(rA,lr’), r‘ was put equal to ri, which is the center point of the ith boundary element. Singularities were hence eliminated. Figure 46 shows the relative error on the Hall voltage. These results are compared with those

t

\

5

APPROXIMATION

I

1

3

1

1

5

I

I

7

9

1

I1

I

I

13 15

I

I

n 19

+

n

FIG.46. Relative error on the Hall voltage calculated by using Green’s theorem.


57

obtained in Section V11. One observes that the error is very high, making the method unusable. For p H B = 0, accurate results were found because the singular terms in Eq. (1 19) vanish. These results explain why Green’s theorem is not used to construct an integral equation for the field problem in Hall plates. The methods outlined in Section VII should be preferred in this case. APPENDIX 4. THEHALL-EFFECT PHOTOVOLTAIC CELL The Hall effect can also be used to convert solar energy to electric power. The basic principle of all photovoltaic cells is the generation of electrons and holes in semiconductors owing to the absorbed light. If electrons and holes are accelerated in opposite directions a net current will be delivered to an attached load. In junction solar cells the separation of electrons and holes is done by the junction field. The same effect can be achieved with the Hall phenomenon because electrons and holes are deflected in different directions by the Lorentz force. A possible configuration of a Hall-effect photovoltaic cell is shown in Fig. 47. The incident light generates electron-hole pairs in the layer. Owing to the exponential decay of the light intensity, a concentration gradient for the charge carriers is built up. Hence electrons and holes diffuse in the y direction. Owing to the Lorentz force, charge carriers will be deviated along the x axis in such a way as to give a net current through the load resistor R,. A theoretical analysis has been published (De Mey, 1979).Assuming that one type of charge carrier has a much higher mobility than the other (which

gs

58

GILBERT DE MEY

is the case for InSb, InAs, etc.), the maximum attainable efficiency was found to be where p is the mobility, B the magnetic induction, E , the band gap of the semiconductor, L the diffusion length, and u the light absorption coefficient. For E , = 1 eV, and giving GILits optimum value, Eq. (120) reduces to = 0.00625(puB)2

(121)

For p B = 1, the efficiency turns out to be 0.6%, a low value compared with that of junction solar cells, which have efficiencies better than 10%. Note that p B = 1 is a rather high value because B = 1 Wb/m2 is difficult to attain, and there are only two semiconductors having p > 1 : InSb ( p = 7) and InAs ( p = 3). Because Eq. (120)was developed for monochromatic light, the efficiencywill be reduced at least by 0.44 in order to take the solar spectrum into account, and because both InSb and InAs have small band gaps, these conductors are not matched to the solar spectrum, which results in a much lower spectrum factor (< 0.44). The conclusion is that the Hall-effect solar cell is not suitable for energy production because of its low conversion efficiency.

APPENDIX 5. CONTRIBUTION OF THE HALL-PLATE CURRENT TO THE MAGNETIC FIELD A current must be supplied through two of the contacts of a Hall plate, which gives rise to a Hall voltage (in combination with an externally applied magnetic field). However, the current in the Hall place also generates its own magnetic field, which can also influence the Hall effect. The mathematical analysis of this secondary influence is rather complicated because the magnetic field caused by a current distribution in a plate turns out to be a three-dimensional problem. However, a crude approximate analysis indicates that the contribution of the currents in the Hall plate to the applied magnetic field is negligible. The situation changes if an alternating magnetic field is applied. Eddy currents are then generated in the Hall plate, creating an additional magnetic field and influencing the original Hall voltage. It is even possible to get a Hall voltage for a zero supply current, the eddy currents only being responsible for the Hall effect. If the frequency of the alternating magnetic field is high, the reaction of the eddy currents on the magnetic field can be important. A theoretical analysis carried out for a circular Hall plate in an ac magnetic field indicates that the parameter Q = 2n f o p o d / R should be considered (f,


59

frequency; 0 , conductivity; po , permeability; d, thickness; and R, radius) (De Mey, 1976d). If R < 0.1, the contribution of the eddy currents to the magnetic field is negligible. Eddy currents are calculated directly from the applied field, and the Hall voltage is found by integrating the electric field between the Hall contacts. For high values of R, the calculation of the eddy-current pattern constitutes a complicated field problem. For practical purposes (e.g., magnetic field measurements in electrical machines), one is interested in working under the condition R < 0.1. The quantity d/R can then be replaced by the quotient of the thickness and a typical dimension of the Hall plate if the shape is not circular.

APPENDIX 6. LITERATURE Most books on the Hall effect describe the physical aspects of Hall mobility. Putley’s well-known book (1960) describes the galvanomagnetic properties of a large number of semiconductors. The book also provides many references. An excellent work has also been published by Wieder (1979), which describes both the physical nature and the measuring techniques related to the study of galvanomagnetic properties.

ACKNOWLEDGMENTS I wish to thank Professors M. Vanwormhoudt and H. Pauwels for their continuous interest in this work. I am also grateful to my collegues B. Jacobs, K.Stevens, and S. De Wolf, who collaborated on several topics treated in this article. I want also to thank Ms.H. Baele-Riems for careful typing of the manuscript and Mr. J. Bekaert for drawing the figures.

REFERENCES Abramowitz, M., and Stegun, 1. (1965). “Handbook of Mathematical Functions,” pp. 888-890. Dover, New York. Bonfig, K. W., and Karamalikis, A. (1972a). Grundlagen des Halleffektes. Teil 1. Arch. Tech. Mess. 2, 115-118. Bonfig, K. W., and Karamalikis, A. (1972b). Grundlagen des Hall-effektes. Teil 11. Arch. Tech. 3, 137-140. Brebbia, C. (1978a). “The Boundary Element Method for Engineers.” Pentech Press, London. Brebbia, C., ed. (1978b). “Recent Advances in Boundary Element Methods.” Pentech Press, London. Brown, I. C., and Jaswon, E. (1971). “The Clamped Elliptic Plate under a Concentrated Transverse Load,” Res. Memo. City University, London. Bulman, W. E. (1966). Applications of the Hall effect. Solid-State Electron. 9,361-372. Chwang, R., Smith, B., and Crowell, C. (1974). Contact size effects on the Van Der Pauw method for resistivity and Hall coefficient measurements. Solid-Stare Electron. 17, 12171227.

60

GILBERT D E MEY

Courant, R.,and Hilbert, D. (1968). “Methoden der Mathematischen Physik I,” pp. 121-124. Springer-Verlag, Berlin and New York. Davidson, R . S., and Gourlay, R. D. (1966). Applying the Hall effect to angular transducers. Solid-State Electron. 9,47 1-484. De Mey, G. (1973a). Field calculations in Hall samples. Solid-Slate Electron. 16,955-957. De Mey, G. (1973b). Influence of sample geometry on Hall mobility measurements. Arch. Elektron. Uebertragungsrech. 27, 309-3 13. De Mey, G .(1973~).Integral equation for the potential distribution in a Hall generator. Electron. Lelt. 9, 264-266. De Mey, G . (1974a). Determination of the electric field in a Hall generator under influence of an alternating magnetic field. Solid-State Electron. 17,977-979. De Mey, G. (1974b). An expansion method for calculation of low frequency Hall etTect and magnetoresistance. Radio Electron. Eng. 44,321-325. De Mey, G. (1975). Carrier concentration in a Hall generator under influence of a varying magnetic field. Phys. Status Solidi A 29, 175- 180. De Mey, G. (1976a). An integral equation approach to A.C. diffusion. In:. J. Heat Mass Transjer 19,702-704. De Mey, G . (1976b). An integral equation method for the numerical calculation of ion drift and diffusion in evaporated dielectrics. Computing 17, 169- 176. De Mey, G. (1976~).Calculation of eigenvalues of the Helmholtz equation by an integral equation. In:. J. Numer. Methods Eng. 10, 59-66. De Mey, G. (1976d). Eddy currents and Hall effect in a circular disc. Arch. Elektron. Uebertragungstech. 30, 312-315. De Mey, G . (1977a). A comment on an integral equation method for diffusion. In!. J. Heat Mass Transjer 20, I8 1 - 182. De Mey, G . (1977b). Numerical solution of a drift-diffusion problem with special boundary conditions by integral equations. Comput. Phys. Commun. 13,81-88. De Mey, G. (1977~).A simplified integral equation method for the calculation of the eigenvalues of Helmholtz equation. In:. J. Numer. Methods Eng. 11, 1340-1342. De Mey, G. (1977d). Hall effect in a nonhomogeneous magnetic field. Solid-State Electron. 20, 139-142. De Mey, G . (1979). Theoretical analysis of the Hall effect photovoltaic cell. IEE Trans. Solid Slate Electron Decices 3,69-71. De Mey, G . (1980). Improved boundary element method for solving the Laplace’ equation in two dimensions. Proc. In/. Semin. Recent Ado. Bound. Elem. Methods, I980 pp. 90- 100. De Mey, G., Jacobs, B., and Fransen, F. (1977). Influence of junction roughness on solar cell characteristics. Electron. Lert. 13, 657-658. De Visschere. P., and De Mey, G. (1977). Integral equation approach to the abrupt depletion approximation in semiconductor components. Eleclron. Lelt. 13, 104- 106. Edwards, T. W., and Van Bladel, J. (1961). Electrostatic dipole moment of a dielectric cube. Appl. Sci. Res. 9, 151-155. Ghosh. S. (1961). Variation of field effect mobility and Hall effect mobility with the thickness of deposited films of tellurium. J. Phys. Chem. Solids 19,61-65. Gray, R. M. (1971). “Toeplitz and Circulant Matrices: A Review,” Tech. Rep. No. 6502-1, pp. 16- 19. Information Systems Laboratory, Stanford University, Stanford, California. Grutzmann, S. (1966). The application of the relaxation method to the calculation of the potential distribution of Hall plates. Solid-State Electron. 9,401-416. Haeusler, J. (1966). Exakte Losungen von Potentialproblemen beim Halleffekt durch konforme Abbildung. Solid-Stare Eleclron. 9,417-441.

POTENTlAL CALCULATlONS 1N HALL PLATES

61

Haeusler, J. (1968). Zum Halleffekt Reaktanzkonverter mit vier Elektroden. Arch. Elektr. Uebertragung 22,258-259. Haeusler, J. ( I 971). Randpotentiale von Hallgeneratoren. Arch. Elekrrotech. (Berlin) 54, 77-81, Haeusler, J., and Lippmann, H. (1968). Hall-generatoren mit kleinem Linearisierungsfehler. Solid-Stare Electron. 11, 173- 182. Hochstadt, H. (1973). “Integral Equations,” Chapters 2 and 6. Wiley, New York. Jaswon, M. A., and Symm, G. T. (1978). “Integral Equation Methods in Potential Theory and Electrostatics.” Academic Press, New York. Kobus, A., and Quichaud, G. (1970). Etude d’un moteur a courant continu sans collecteur a commutation par un generateur a effet Hall en anneau. RGE, Rev. Gen. Electr. 79,235-242. Lippmann, H.,and Kuhrt, F.( l958a). Der Geometrieeinflussaufden transversalen magnetischen Widerstandseffekt bei rechteckformigen Halbleiterplatten. 2.Naturforsch. 13,462-474. Lippmann, H., and Kuhrt, F. (1958b). Der Geometrieeinfluss auf den Hall-effekt bei rechteckformigen Halbleiterplatten. Z. Naturjorsch. 13,474-483. Madelung, 0. (1970). “Grundlagen der Halbleiterphysik,” Chapters 37-39. Springer-Verlag, Berlin and New York. Many, A., Goldstein, Y., and Grover, N. B. (1965). “Semiconductor Surfaces,” p. 138. NorthHolland Publ., Amsterdam. Mei, K.,and Van Bladel, J. (1963a). Low frequency scattering by rectangular cylinders. IEEE Trans. Antennas Propagation AP-I 1,52-56. Mei, K., and Van Bladel, J. (1963b). Scattering by perfectly conducting rectangular cylinders. IEEE Trans. Antennas Propag. AP-11, 185-192. Mimizuka, T. (1971). Improvement of relaxation method for Hall plates. Solid-Stare Electron. 14, 107-110. Mimizuka, T. (1978). The accuracy of the relaxation solution for the potential problem of a Hall plate with finite Hall electrodes. Solid-State Electron. 21, 1195-1 197. Mimizuka, T. (1979). Temperature and potential distribution determination method for Hall plates considering the effect of temperature dependent conductivity and Hall coefficient. Solid-State Electron. 22, 157- 161. Mimizuka, T., and Ito, S. (1972). Determination of the temperature distribution of Hall plates by a relaxation method. Solid-State Electron. IS, 1197-1208. Nalecz, W., and Warsza, Z. L. (1966). Hall effect transducers for measurement of mechanical displacements. Solid-State Electron. 9,485-495. Newsome, J. P. (1963). Determination of the electrical characteristics of Hall plates. Proc. Inst. Electr. Eng. 110, 653-659. Putley, E. H. (1960). “The Hall Effect and Related Phenomena.” Butterworth, London. Shaw, R. (1974). An integral equation approach to diffusion. In:. J . Hear Mass Transjer 17, 693-699. Smith, A,, Janak, J., and Adler, R. (1967). “Electronic Conduction in Solids.” Chapters 7-9. McGraw-Hill, New York. Stevens, K., and De Mey, G . (1978). Higher order approximations for integral equations in potential theory. In:. J. Elecrron. 45,443-446. Symm, G. (1963). Integral equation methods in potential theory. Proc. R. Soc. London 275, 33-46. Symm, G. (1964). “Integral Equations Methods in Elasticity and Potential Theory,” Res. Rep. Natl. Phys. Lab., Mathematics Division, Teddington, U.K. Symm, G., and Pitfield, R. A. (1974). “Solution of Laplace Equation in Two Dimensions,” NPL Rep. NAC44. Natl. Phys. Lab., Teddington, U.K.

GILBERT DE MEY

62

Thompson, D. A., Romankiw, L. T., and Mayadas, A. F. (1975). Thin film magnetoresistors in memory, storage and related applications. IEEE Trans. Magn. MAG-11,1039-1050. Tottenham, H. (1978). "Finite Element Type Solutions of Boundary Integral Equations." Winter School on Integral Equation Methods, City University, London. Van Der Pauw, L. J. (1958). A method of measuring specific resistivity and Hall effect of discs of arbitrary shape. Philips Res. Rep. 13, 1-9. Wick, R. F. (1954). Solution of the field problem of the Germanium gyrator. J . Appl. Phys. 25,741-756.

Wieder, H. H. (1979). "Laboratory Notes on Electrical and Galvanomagnetic Measurements." Elsevier. Amsterdam.

ADVANCES I N ELECTRONICS A N D ELECTRON PHYSICS, VOL. 61

Impurity and Defect Levels (Experimental) in Gallium Arsenide A. G. MILNES Department of Electrical Engineering Carnegie-Mellon University Pittsburgh, Pennsylvania

..................................

1. Introduction 11. Possible Nati

mplexes . . . . . . Ill. Traps (and Nomenclature) from DLTS Studies IV. Levels Produced by Irradiation ............. V. Semi-Insulating Gallium Arsenide with and without Chromium ............... VI. Effects Produced by Transition Metals .........................

....................................... IX. Group IV Elements as Dopants: C, Si, Ge, Sn, Pb ........................... A. Carbon .... ...... B. Silicon. ............................................................

91

116 I18 120 121

.............................................

............................................. ...........................

X. Oxygen in GaAs ...... XI. Group VI Shallow Donors:

.... ......................................

123

e X11. Other Impurities (Mo, Ru, Pd, W,Pt, Tm, Nd). ............................. A. Molybdenum ............................................ B. Ruthenium ......................................................... C. Palladium . .

127 129

E. Platinum . ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .........................

130

.....

129

...........................

133

.............................................................

142

A. In LPE-Grown GaAs Layers

XIV. Concluding Discussion References

63 Copyright (* IPR3 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-1m 4 ~ 1 - 4

64

A. G .

MILNES

1. INTRODUCTION

The present understanding of defects and deep impurity levels in GaAs is represented by a substantial literature but still leaves much to be desired after two decades of study. Speculation on the physical nature of complexes and defect levels is still very tentative. It is not even possible to say with any assurance that simple gallium or arsenic vacancies (VGaand V,,, respectively) have been related to particular energy levels. Gallium arsenide specimens grown by various techniques' such as LEC, Bridgman, VPE (chlorine process), MOVPE, LPE, and MBE are now recognized as likely to have different properties in terms of energy levels within the band gap due to differences in defect levels and trace impurities or impurity complexes. Gallium arsenide grown by LPE from a gallium melt, for instance, may be expected to be low in gallium vacancy defects and possibly high in arsenic vacancies. Liquid-phase epitaxy GaAs usually contains energy levels at E, + 0.41 and 0.71 eV (where E, is the energy of the valence-band edge). Vapor-phase epitaxial layers are grown under arsenicrich conditions and almost always contain an important level at about mid-band gap. This is still not understood. Also, depending on the construction of the system and the care with which it has been prepared and cleaned, the VPE layer may contain another handful of energy levels. So a crystal grower can no longer feel his obligation fulfilled by producing material characterized only in terms of mobility and dislocation count. He is now interested in deep levels because of the effect they have on ( 1 ) the carrier concentration from shallow doping; (2) the effect on mobility because of ionized impurity scattering; (3) trapping effects that can affect transport and frequency and switching performances of devices; and (4) the recombination diffusion lengths and generation in bipolar structures such as solar cells and other optical sensing devices. Section I1 begins by listing some of the many defects and complexes that may be expected to exist in GaAs. Section 111 presents many of the energy levels obtained by deep-level transient spectroscopy (DLTS) and related methods for determining trap and recombination levels in GaAs. Energy levels produced by high-energy irradiation are next considered. Section V is a discussion of the effects of Cr in GaAs, since this may be used to produce semi-insulating GaAs. The possible role of oxygen or other levels in this respect is also considered. The effects of Group I-VI impurities are then considered. Section XI11 discusses recombination and generation and minority-carrier lifetime and diffusion-length effects.

'

LEC, liquid encapsulation Czochralski ; VPE, vapor-phase epitaxy ; LPE, liquid-phase epitaxy ; MBE, molecular beam epitaxy ; MOVPE, metalorganic vapor-phase epitaxy.

65

IMPURITY LEVELS IN GALLIUM ARSENIDE

11. POSSIBLE NATIVEDEFECTS AND COMPLEXES

The number of possible native defects in a binary semiconductor such as gallium arsenide is large, as may be seen from Table I. None of these has been identified with any confidence. Experiments for studying such defects tend to be too uncontrolled. There may be changes of electrical properties after heating under various conditions of Ga and As overpressure (and other conditions are mentioned later). This is particularly true since it is difficult to obtain gallium arsenide with trace-impurity levels below the mid- 10’ atoms cmand difficult to perform heating operations on specimens under conditions of comparable cleanliness. A hundredth of a monolayer of metallic impurities (say, I O I 3 atoms cm-’) entering a GaAs layer to a depth of, say, 10 pm has the potential for producing contamination levels with concentrations as high as 10I6 cm-j. Such effects probably invalidate many experimental studies of the past, and the literature must be considered in this light. Complexing of impurity atoms with native defects may also be expected. Table I1 sets out some possibilities, more to stimulate thinking rather than to be exhaustive or to represent the conditions that have been suspected. Several nomenclatures exist for describing traps and their actual and effective charge states. In this review, various forms are used to match the TABLE I POSSIBLE DEFECTS I N GALLIUM ARSENID~ Gallium vacancies involved

Arsenic vacancies involved

Antisite defects

Interstitials involved

Threecomponent defects”

Asi GaiAsi

a

Subscript indicates lattice sites and i stands for interstitial. Many others are possible besides the three listed as examples.

66

A. G . MILNES

TABLE I1 SOME HYPOTHETICAL COMPLEXES OF NATIVE DEFECTS ELEMENTAL IMPURITIES FOR GALLIUM ARSENIDE'

AND

Group I1 impuritiesb

Group IV impurities'

Group VI impuritiesd

Transition impurities'

various styles used in the literature. The nomenclature of Kroger (1977) is used in some papers (although it is not in widespread use in the GaAs literature) and so is explained as follows: Defects can be represented by symbols, atoms being represented by their normal chemical symbol, vacancies by V. Subscripts indicate the lattice site, characterized by the atom normally occupying that site. Thus V, is a vacancy at an A site. Interstitial sites are indicated by a subscript i, Ai being A at an interstitial site. Electrons are represented by e, holes by h. Defects, like normal atoms, can have charges. It is useful to distinguish actual charges and effective charges. The former are the charges present inside the bounds of the defect. Effective charges are the difference between the actual charge at a site and the charge normally present at that site if no defect is present. Coulombic interaction between charged defects is determined mainly by the effective charges. For interstitials, actual and effective charges are the same; for substitutional atoms or vacancies, the two types of charges are different. Actual charges will be represented by superscripts + and -, effective charges by dots (.) and primes ('). An effective charge zero is indicated by a superscript x. Thus A; is an interstitial n atom with a single positive charge, V; is an A vacancy with a double negative effective charge. Similar symbols can be used for normal crystal constituents, A: representing a normal A at an A site with effective charge zero.

Use of these symbols occurs in fallowing Logan and Hurle's discussion (1971) of the behavior of GaAs in contact with As vapor. Consider the


~~

67

~~~

"Reproducedby permission of Her Britannic Majesty's Stationery Office.

significant species to be Vga and (VGa);, which, if they accept an electron, assume the forms Vba and (VGa);;V;, is considered to be a donor that is capable of shedding an electron to become V;,. If we limit ourselves to these defects, then the reactions of GaAs in contact with As vapor are those shown in Table 111. The K values are estimated by Logan and Hurle, who conclude that defect concentrations in the range 10'8-10'9 cm-3 might be expected at a temperature of 1100°C (40°C below the melting point of GaAs) and a pressure pAs2of about 0.3 atm. The equation 0 + V&, + V;, corresponds to Schottky disorder, for which the heat of formation is believed to be less than 5 eV (Van Vechten, 1980).The threshold energy for Ga Frenkel-pair formation is believed to be 17 eV and that for As Frenkel-pair formation to be even higher. However, Hurle (1979 ) has developed revised equilibria equations in which Frenkel disorder on the arsenic sublattice is considered dominant. The important reaction is then AS,, + Vi = Asi + V,,, and it is supposed that V,, can become ionized to form donors V;,. An activation energy for this of 0.25 eV has been suggested by Il'in and Masterov (1976). The arsenic interstitial is considered to be either neutral or a singly ionized donor. Hurle considers that there are several substantial experimental studies, notably the lattice measurements of Bublik et al. (1973), that suggest that Frenkel disorder is present to a very important extent. Analyses based on simple vacancy models at high temperatures, however, may not predict quenched-in effects. Grown-in defects as a sample cools down might be expected to cluster to form neutral complexes such as VGaAsGaVGa. As Van Vechten (1976) points out: these should be effective nonradiative recombination centers because they quickly bind both electrons and holes in deep states tightly localized in the same region of the crystal, so that electronic matrix elements will be large, and because large lattice distortions occur about these defects so that the configuration coordinate mechanism may proceed. Such defects would appear as a cylinder one atom wide, 12 A long and oriented in the ( I 10) direction. The recently developed direct lattice imaging, DLI, technique of electron

68

A. G.

MILNES

microscopy allows the observation of such defects and they have in fact been found in the size orientation and concentration (10” cm-3) predicted in GaAs (LEC melt-grown material).

Other workers have looked for such structures with no success. Antisite defects must also be considered, rather than limiting the modeling to vacancies. Van Vechten remarks ( 1 976): Consider the antisite defects, e.g., As: and Ga;:, and the bound pairs of these, As,,Ga,,, which constitute a neutral defect. Obviously it takes no energy at all to interchange the two atoms in the unit cell of Ge. It can not take a large energy at all to interchange the two atoms in GaAs, but it will take more energy to interchange the two atoms in ZnSe, and still more for CuBr. It is interesting to compare the QDT (quantum dielectric theory) estimates to those of Pauling’s theory of electronegativities. The two estimates are quite close in many cases, e.g., 0.70 and 0.72 eV for GaAs. No one has given an argument that the energy of formation of the pair defect should be much larger. These energies for 111-V’s are much less than the energies of formation of vacancies or interstitials and imply that antisite defects are present in compound semiconductors in substantial concentrations. In addition to the DLI observation of antisite-vacancy complexes noted earlier, high concentrations of antisite defects have been identified by infrared absorption and are inferred from EPR and other experiments. No description of a compound semiconductor can be complete without them.

However, it would be good as a start to find a simple and certain way of producing a vacancy such as V,, so that its properties may be explored in a convenient temperature range. Electron or proton irradiation may be thought of as such a way; but in studies of Si, the time required for a single vacancy created by electron irradiation to migrate to and complex with an impurity in the purest available material is of order lo3 sec at T = 100 K, so the time required for a single vacancy to complex with an impurity at room temperature is of order l o v 3sec or less. Therefore, in silicon and presumably also in GaAs, after a growth process, no feasible quenching procedure can prevent the single vacancies which were present in their equilibrium concentration at Tho, from forming complexes during the quench and before Hall-effect measurements can be performed (Van Vechten, 1976). In our pursuit of the effects of vacancies, we are tempted to turn to experiments in which As overpressures are involved. In one such experiment in which GaAs was annealed under low pressures of As, donors were formed, and under greater As pressure, acceptors were formed (Chiang and Pearson, 1975a,b). So it may be surmised that V,, (or V,, complexes) are donors and that V, (or V,, complexes) are acceptors. For this to be accepted as true, many repetitions of this experiment under conditions of highest purity in many laboratories must consistently yield similar results (a state we have not yet arrived at). The modeling of Fig. 4 (to be discussed later) assumes V,, to be acceptors and V, donors.

69

IMPURITY LEVELS I N GALLIUM ARSENIDE TABLE IV LUMINESCENCE SPECTRAL PEAKSAND VACANCY ASSOCIATIONS" Luminescence peak (eV)

Vacancy association

Luminescence peak (eV)

Vacancy association

a Proposed by Chang er a / . (1971). Reproduced with permission of the American Physical Society. Created after Cu diffusions. Enhanced after 0 implantation.

Chang et al. (1971), in studies of luminescence spectra for GaAs subjected to various treatments with controlled As pressures, arrive at the tentative assignments shown in Table IV. (The possibilities of interstitials of As being involved were neglected.) Most of these assignments are now no longer accepted. In other work from studies of density, lattice constant, and internal friction, the energy of formation of defects has been ranked as follows: UGa,

>

UvAs> UvGa> U A ~ ,

(1)

so arsenic institials may have to be considered as possibly occurring in the experiment [see also Van Vechten and Thurmond (1976)l. Consider now some evidence for the formation of complexes. Hurle (1977) considers the incorporation of donors, particularly Sn and Te, in the lattice, and postulates that donors combine with gallium vacancies to produce complexes such as Sn,,V&, and TeAsV&,. By rejecting the common idea that Sn, is the dominant acceptor, it is possible to explain why in Sn-doped meltgrown, VPE, or LPE GaAs, the ratio of N D / N A is usually of the order of 4, (a result that would otherwise certainly not be expected for LPE material). Hurle's model also explains why the free carrier concentration ceases to be proportional to the Sn or Te doping at high concentrations. Two such curves are shown in Figs. 1 and 2. Some of the features of Hurle's model (without going into all the details) are as follows. The gallium vacancy is assumed to have a double negative is formed charge in n-type material so that the complex acceptor Sn,,V,, by the reaction (2) snd, + vE; = Sn,,V,,

70

A. G . MILNES

’

- lou8

i ,/

m

c

’1

,

I Ol6

-’?8

-

1

I

MELT CONCENTRATION (Tell

FIG. 1. Carrier concentration versus Te concentration in an LPE melt; T = I123 K. Data from Goodwin er 01. (1969). (See Hurle, 1977; reproduced by permission of her Britannic Majesty’s Stationery Office.)

Hurle then shows that V& and ND/NA should be independent of PAs2and that this explains the similar compensation results for VPE and LPE layers. It is not argued that Sn, does not exist, but only that it is not the dominant acceptor. Furthermore, for Si doping, it is suggested that SLaV& occurs but that Si, can be dominant. During the annealing of n-type GaAs(Te), the formation of compensating acceptors is a complicated function of the initial doping, arsenic pressure, and 1

.

-

10”

(100) (211)Go

- Colculoted

n

z“

-

I

0

z

1015

I

1014

, , ,.,,..I

. . ,..,..I

10‘6

I

*

1-

10’8

N,+NA ( ~ r n - ~ ) FIG.2. Carrier concentration versus total ionized impurity concentration for Sn-doped

VPE layers. Dashed curve is expected behavior if electron-hole generation is dominant at the growth temperature. Full curve is expected dependence. Data from Wolfe and Stillman (1973). (See Hurle, 1977; reproduced by permission of her Britannic Majesty’s Stationery Office.)


71

ARSENIC PRESSURE ( T o r r )

FIG.3. Acceptor density in the heat-treated GaAs crystals at IO00"C and 900°C as a function of arsenic vapor pressure. Heat treatments were performed for 67 hr. The initial electroncarrier concentrations were [published with permission from Nishizawa ef 01. (1974)j: Symbol

Sample

0

A C F I

0 A

0

9 1.8 5.4 9.5 2.8

K

0

Conc. (cm-') x

1016

x 1017 x 10i7 x 101' x 10"

TH("C) 1000

900 I000 I000 I000

temperature as shown in Fig. 3. Hurle explains the cusped minima as follows. He supposes that upon annealing, As atoms exchange with the ambient arsenic pressure but that Ga atoms diffuse very slowly and no exchange is possible. Following Hurle's (1977) explanation, Te:,

+ Vk;

= Te,,V&

(3)

and upon raising the crystal to the annealing temperature arsenic Frenkel defects and some gallium Frenkel defects form AS,,

+ Vi = VA, + AS^

(4)

Ga,,

+ Vi = V,, + Gai

(5)

where denotes a vacant interstitial site. Whilst from mass action, the Frenkel product [V,,][Asi] is constant, the ratio of [V,.] to [Asi] will depend on the ambient arsenic pressure since exchange with the ambient phase is deemed to occur. With the gallium Frenkel defects, however, we have: [VG.] = [Gail since no exchange is allowed

(6)

72

A. G. MlLNES

We suppose that the gallium vacancies are strongly “gettered” by the donors via reaction (3) to form complexes. For an arsenic pressure for which [As,] [V,,], the following may be expected to occur as the crystal is cooled. Firstly the arsenic Frenkel defects can recombine leaving the net excess of As,. The latter can combine with the Ga, via the reaction Ga, + Asi = Ga,As, (7) and the interstitial pairs can aggregate at a somewhat lower temperature to form the observed interstitial loops. Since the gallium Frenkel reaction is supposed to occur to a significant extent only when the V& are “gettered,” the absence of interstitial loops in heat-treated undoped GaAs is explained. For low annealing pressures, corresponding to [As,] < [ViJ, we can hypothesize that the unstable Ga, is removed from solution by the formation of the antistructure defect: Ga,

+ V,,

= Ga,,

as proposed by Van Vechten (1975). If now the arsenic pressure corresponds exactly to stoichiometry, namely [Asi] = [V,.], then there are now no sinks for the Gai as the crystal cools save for the complex formed by reaction (3). We may therefore suppose that under these conditions, the interstitial annihilates the complex to restore the donor state: Te,,V&

+ Ga, = Teis + 2e.

(9)

In consequence there will be a sharp (cusped) minimum in the number of acceptors in the vicinity of the pressure corresponding to the stoichiometric composition p:,, as observed experimentally.

In a discussion of lattice dilation associated with VGaGaiTei(Dobson et al., 1978),a cusped minimum is also found, and Hurle speculates that this is

caused by the Te,,V,, complex. Following his development of a point-defect equilibria model based on Frenkel disorder, As, V,,, Hurle has further considered Te, Sn, and Ge doping effects (Hurle, 1979b,c,d; see also Logan, 1971). One problem in the study of native defects is that of distinguishing between vacancies and interstitials. For instance, treatment under Ga-rich conditions may be expected to result in material with either As vacancies or Ga interstitials, and there is need for an experiment that establishes which is dominant. One method that has been attempted is a study of positron lifetimes in GaAs. Because of the repulsive force between ion core and positron, the positron, after thermalization in the crystal, can be trapped at vacancy-type defects, but not at interstitials. The trapping may result in an increase in positron lifetime because the electron density at vacancies is lower than that in the bulk. The results obtained by Cheng et al. (1979)for such an experiment did not show evidence for the existence of As-vacancy defects in heat-treated ( 1000°C, plus quench) samples. This suggested three possibilities:

+

-

(1) The positron cannot be effectively trapped at As vacancies (2) The arsenic deficiency exists only in a narrow region (x < 200 pm) below the surface

73


(3) The arsenic deficiency is not mainly in vacancy form (e.g., Ga interstitials are dominant). It is possible to muster some arguments in favor of all of these possibilities, and in the present state of knowledge it is difficult to choose between them. However, the experiment did generate some evidence for the presence of Ga vacancies or multiple vacancies in LEC-grown materials. This discussion has touched on only a few aspects of model-building problems in understanding defect interactions in GaAs, and it should serve to indicate some idea of the complexities of defect studies. Electron paramagnetic resonance (EPR) studies of GaAs do give evidence of site locations and symmetries for some impurities. Unfortunately, the EPR studies do not work as well for GaAs as for Si because of the hyperfine linebroadening effects and high nuclear spins of Group 111 and V elements (Eisen, 1971). Rutherford backscattering can also sometimes distinguish between substitutional and interstitial location of impurities in GaAs; however, it is generally difficult to infer anything about trap structure except by very indirect methods (Kudo, 1979). An example of such a method comes from the study of Schottky barrier formation on n- and p-type GaAs as monolayers of metal (or oxygen) are added to “clean” surfaces. The Fermi-level pinning is found to be at about E, - 0.75 eV for n-type GaAs and at about E, + 0.5 eV for p-type GaAs and is nearly independent of the metal used, as shown in Fig. 4a. eV

1.2

/ / / / / / / / / / / C8M

-

GoAs (110)

0.00.4

‘.4c73

-

A

A

0.75 eV ACCEPTOR DUE TO MISSING A s

GoAs

A

t

0

M I S S I N G As 1.4

Eg

0.5 eV DONOR DUE

0

TO M I S S I N G G o

0 DONOR

(b)

(C

1

FIG.4. Schottky barrier pinning and the unified defect model of Spicer el a/. (1980): (a) final pinning position of the Fermi level for n ( 0 )andp ( A ) GaAs; (b) postulated defect levels; (c) interface states.

74

A. G. MILNES

Spicer et al. (1980) explain their unified defect model in terms of states produced in the GaAs near the interface, as suggested in Fig. 4b and c. They comment that these probably represent the simplest stable defects since the adatoms were applied very “gently” and the samples were not heated. Thus, the energy levels in Fig. 4(b) which are considered accurate to 0.1 eV, represent the simplest defect levels stable at room temperature. The energy levels are most accurately known; the acceptor or donor nature of the defect is next-best known; whereas, there is the least certainty about the identity of the missing atom. Based on the low coverage needed for pinning and independence of pinning energy on extreme changes in the chemistry of the adatom, it is concluded that the pinning mechanism must be indirect, i.e., that the pinning can not be due to levels directly introduced by the adatom, i.e., states that depend on the electronic orbitals of the adatoms. Rather the new states must be generated indirectly, i.e., the perturbations of adatoms must create new energy states which do not depend on the electronic structure or other characteristics

of the adatom. Most importantly, there is a close correspondence in the pinning positions found in these experiments with submonolayer coverages of adatoms and those found in practical MOS or Schottky barrier devices with many monolayers of adatoms. Once the indirect nature of the formation of the new states becomes apparent, it becomes very attractive to assume that these new states are due to lattice defects induced by the adatom since this provides the simplest mechanism consistent with the large range available. For oxygen, both valence-band spectroscopy and LEED show that the surface becomes disordered at low coverage. lt is easy to conceive how, in such a situation, vacancies or more complex lattice defects may be created. The fact that the clean surface is heavily strained by the large surface rearrangement makes it easier to see how the surface may be disordered by a relatively small oxygen adsorption. For thick oxides, the 111-V atoms move through the oxide to the outer surface of the oxide before they react with the oxygen. This is in contrast to Si-Si02 where the oxygen moves through the oxide to react with the Si at the Si-oxide interface. The outward movement of the 111-V atoms can aid in the formation of additional defect states as the Ga and As atoms near the interface are consumed in the growth process. This is complicated by the nonuniform chemical spatial distribution in thick oxides. The situation with regard to Schottky barrier formation is also interesting. One of the most surprising results in the work leading to the unified defect model was the discovery that, even for metals where no appreciable chemical reaction was expected, the semiconductor material moves out into the deposited metal in surprisingly large quantities. This is now well documented in the literature, and a mechanism has been suggested for it based on the heat of condensation of the metal on the semiconductor. There has been a mystery concerning Schottky barriers on Ill-IV’s which the unified defect model clears up. On Si the Schottky barrier height varies strongly, depending on the cleanliness of the surface on which it is deposited. In contrast, 111-V Schottky barriers have been found to be surprisingly insensitive to the surface oxygen or oxidation. This is just what would be expected on the basis of the [Spicer]defect model since the same defect levels and thus the same pinning position would be produced by oxygen on the metal; thus, the Schottky barrier height on Ill-V’s would not be affected (to first order) by addition of oxygen to the surface before the metal.

The defect model of Spicer et al. (1 980) therefore suggests consideration of E, - 0.75 eV and E, + 0.5 eV ( f0.1 eV) as native defect levels in bulk GaAs. They are less certain, however, about the identity of the defects, although

75


speculating that the midgap state is related to missing As. A review by Brillson ( 1982) discusses the matter. From photo-EPR measurements, Weber and Schneider (1982) find double-donor action in a center they identify as AsGa,and the energy levels involved (10 K) are Do/D+at E, - 0.77 eV and D+/D2+at E, + 0.50 eV, which are close to the Fermi-level pinning energies of Spicer et al. (1980) (see also Ikoma and Takikawa, 1981). A case can be made that the E, - 0.75 eV level is associated with missing Ga since a level between 0.75 and 0.83 eV, presumed to be the Spicer level, is commonly seen when growths are made under As-rich conditions (as in VPE). Growth under various ratios of AsH,/GaCI in the range l/3-3/1, by Miller et al. (1977), shows that the mid-gap-level (E, - 0.82 eV) concentration increasesas the ASH, is increased. Several other studies identify the E, - 0.75 to 0.82 eV level as Ga vacancy related (E, is the energy of the conduction-band edge). An example is the study of Bhattacharya et al. (1980) with variations of the As/Ga ratio in organometallic growth of GaAs. Another interpretation of As-rich growth results would be that the level or AsGaVAs. Bhattacharya is AsGarelated or even a complex such as AsGaVCa et al. also find an electron trap at E, - 0.36 & 0.02 eV. If we exclude impurities such as 0 as the cause of this level, the candidates for this level might by V, or AsGa(or, again, some complex of these). Arsenic-rich growth conditions might also be expected to produce arsenic interstitials. Little is known about the energy levels expected for Asi. Arsenic interstitials have been suspected by Driscoll et al. (1974; also Driscoll and Willoughby, 1973) as possibly responsible for a shallow donor level at E, - 0.035 eV. Pons and Bourgoin (1981a,b) conclude that levels at E, - 0.45 eV, -140 meV, and -0.31 eV involve arsenic sublattice Frenkel pairs. Yet another interpretation of the Miller results has been proposed by Zou (I98la). He assumes the following relationship. PAsH3IPGaCI OC [VGa]/[VAs]

OC [VGa]2/[VGal[VAs]

OC [VGa12

(lo)

So, for the mid-band-gap trap (0.75-0.83 eV), Zou assumes that its configuration must be related to (VGa)2 and/or AsGa. He further assumes that these are related by the reaction AS,,

+ (V,,);

= As,

+ V,,V,,

+ e-

(1 1)

Thus the concentrations of (VGa); and As Ga might be proportional to each other. Ozeki et al. (1979) stated that the mid-band-gap level may, in fact, be two levels quite close to each other in energy, and Zou tentatively suggests these may be (VGa); and AsGa. Let us leave for the moment consideration of the mid-band-gap level(s) and consider possible As-vacancy-related states. In studies of resistivity of material grown by the VPE-Ga/AsCI,/H, process, Saito and Hasegawa (1971)identify a deep acceptor at approximately 0.5 eV from the valence-band

A.

76

G. MILNES

edge, which they infer to be related to an arsenic vacancy. Spicer’s model requires a Ga-vacancy-related donor at E, + 0.5 eV, and so acquires no support from the Saito and Hasegawa observation. Growth under Ga-rich conditions, as in LPE growth from a Ga melt, normally results in two minority carrier (hole) traps in n-type GaAs layers. These hole traps termed A and B are at E, + 0.41 eV and E, + 0.71 eV, respectively. The electron trap at E, - 0.75 to 0.83 eV is not seen in the material. The physical natures of the A and B levels have still to be determined, but one surmise is that they have to involve either an As vacancy or a Ga atom on an As site. Perhaps the A level at E, + 0.41 eV may be a manifestation of Spicer’s interface defect level at E, + 0.5 f 0.1 eV (assuming their identification to be changed to missing As), but this assignment is very hypothetical. One attempt to make an assignment of the A and B level, is that of Zou (1981a). He concludes that the hole trap A (E, + 0.41 ev) might be GaAsVGa, and hole trap B (E, + 0.71 eV) might be AsGaVGa.The argument is involved and is not presented here. To accept V, complexes in LPE material grown under such Ga-rich conditions, it is necessary to assume that the V, diffuses from the substrate into the LPE layer during growth. It should be noted that AsGaVGais a possible candidate for the mid-band-gap electron trap EL2 (E, - 0.75 to 0.83 eV) seen in material grown under As-rich conditions. Another candidate that has been suggested for EL2 from VPE growth is ASG,VC~VA~ (Zou, 1981b; Zou et al., 1982). Attempts to show that in LPE material hole trap B is identical with electron trap EL2 of VPE material fail, however, because the electron-trapping action is not observed in DLTS tests of material containing trap B. 111. TRAPS(AND NOMENCLATURE) FROM DLTS STUDIES

In 1970, Sah and co-workers showed the considerable merits of studying traps by emission in junction depletion regions from the associated capacitance changes. The circuit-processing technique known as deep-level transient spectroscopy (DLTS) added to the convenience of this type of measurement (Lang, 1974; Lang and Logan, 1975; Wada et a!., 1977). The group at the Laboratoires d’Electronique et de Physique Appliqee (LEP) has led this technology in France and has published useful catalogs of electron and hole traps in bulk, VPE, LPE, and MBE GaAs. Deep-level transient spectroscopy studies yield the emission coefficients en,pwhich are functions of temperature. For electron emission, the theoretical expression from the detailed balance of emission and capture events is el = ( 0 1 (v1

)NrJ,go /Sl) exp( - A E / W

(12)


77

where a1is the minority-carrier capture cross section, (v l ) the mean thermal velocity of minority carriers, N,, the effective density of states in the minoritycarrier band, g1 the degeneracy of the trap level, go the degeneracy of the level not occupied by the electron (often equal to unity), and AE the energy separation between the trap level and the minority-carrier band. However, the cross section ol may be activated thermally with an energy Eb, and the trap depth AE may vary as a function of temperature (often as a nearly linear function of temperature with c1 the variation in eV K - I . Furthermore, N,,, depends on TZ. Application of these corrections yields e,,(T) = ~,,T2e,g,(expa/k)exp[-(El~+ Eb)/kT]

( 12a)

where a,,, is the extrapolated value of a,,for T = 00, E I o the extrapolated value of E l for T = 0, g,, the degeneracy factor, and y,, a constant equal to 2.28 x 10’’ cm-’ sec-’ K-’ in GaAs. The plot of T’/e,, as a function of as well as E I o + Eb (denoted by 1/T yields a,gn exp(a/k) (denoted by ofla), E m ) ;these two parameters, emand Ens, are actually the “signature” of a trap, even if they do not have a direct physical meaning (Martin et al., 1977b). The lines presented in Figs. 5 and 6 are plots of T’/e,, and T2/e, versus 1000/T for electron and hole traps in GaAs (Mircea et al., 1977; Pons, 1980). The corrections for Eb and AE have not been applied in Figs. 5 and 6 or in Tables V and VI that provide the keys to the data. Table V shows the capture cross sections of the electron traps and the class of material in which they were observed. Martin et al. (1980a,b) have revised certain of these values. In particular, the lines for T2/e, of the EL1 and EB1 have been raised by over an order of magnitude. This also matches better the ni (intrinsic carrier concentration) studies of Blakemore (1982). The letters T, I, B, and L in the labels represent the names of various laboratories. It seems probable that EL2 = ETl = ESl = EB2 = ECl, and this is a commonly found electron trap at about E, - 0.83 eV. (In the past, this was considered to be oxygen related, but is now thought to be a complex that involves V, or AS,, or AsGaVGa). The dotted lines are some trap levels seen in VPE GaAs (-mid-10” cm-3 n type) at Carnegie-Mellon University, Pittsburgh; EC3 is probably the same as EI1; EL4 is similar to EB5; and EL5, EB6, and EC5 are probably the same trap. Other likely candidates for equivalence may be seen by inspection of the energy levels and the locations on the plot (the location being dependent on both the energy and the capture cross section). Discussion of this catalog is not appropriate at this time, except to say that the nomenclature is now in general use. It may be remarked that the levels seen in the vicinity of E, - 0.17 eV are thought to be Cu related. Figure 6 shows a collection of hole-trap data similarly plotted and Table VI provides the key to the data (Mitonneau, 1976; Mitonneau e f al., 1977).

78

A . G . MILNES

FIG. 5 . Electron-emission coefficients plotted as T 2 / e , versus IOOOIT for electron traps communicating with the conduction band in GaAs. (After Martin er a/.. 1977b; reproduced with permission of IEE, London.)

2

4

6

8

1000/T (K-')

FIG.6. Plots of T 2 / e ,as function of IOOO/T for all hole traps: (T) University of Tokyo, (S)University of Sheffield, (B) Bell Telephone Laboratories, (L) LEP work. The curves B giving Lang's results are extrapolated from his DLTS spectra. (After Mitonneau et a / . , 1977; reproduced with permission of IEE, London.)


79

TABLE V ELECTRON TRAPS IN GALLIUM ARSENIDE"

Label in Fig. 5

Activation energy E,, (eV)

Cross section b p . (cm)

ET 1 ET2 ES I EFI El 1 El2 El3 EB I EB2 EB3 EB4 EB5 EB6 EB7 EBS EB9 EBlO ELI EL2 EL3 EL4 EL5 EL6 EL7 EL8 EL9 ELI0 ELI I EL12 EL14 EL15 EL16

0.85 0.3 0.83 0.72 0.43 0.19 0.18 0.86 0.83 0.90 0.71 0.48 0.41 0.30 0.19 0.18 0.12 0.78 0.825 0.575 0.51 0.42 0.35 0.30 0.275 0.225 0.17 0.17 0.78 0.215 0.15 0.37

6.5 x 1 0 - 1 3 2.5 x 1 0 - l 5 1.0 x 10-13 7.7 x 1 0 - 1 5 7.3 x 10-16 1 . 1 x 10-14 2.2 x 10-14 3.5 x 2.2 x 3.0 x lo-" 8.3 x 2.6 x 2.6 x 1 0 - 1 3 1.7 x 10-14 1.5 x 10-14 Imprecise Imprecise 1.0 x 1 0 - 1 4 (0.8-1.7) x (0.8-1.7) x 10-13 1.0 x l o - " (0.5-2.0) x 1.5 x 1 0 - l ~ 7.2 x 7.7 x l o - " 6.8 x 1.8 x 1 0 - 1 5 3.0 x 4.9 x I o - I z 5.2 x 5.7 x 10-13 4.0 x l o - ' *

Observation* BM BM BM Cr-doped BM VPEM VPEM VPEM Cr-doped LPEM As-grown VPEM El M EIM As-grown MBEM EIM As-grown MBEM As-grown MBEM EIM EIM Cr-doped BM VPEM VPEM As-grown MBEM VPEM BM As-grown MBEM VPEM VPEM As-grown MBEM VPEM VPEM BM EIM VPEM

a After Martin er a/. (1977b). Reproduced with permission of the IEE, London. * BM, bulk material; VPEM, VPE material; LPEM, LPE material; MBEM, MBE material; EIM, electron-irradiated material.

Full confidence cannot be placed in the values given for several reasons that reflect limitations in the measurement techniques. The activation energy (T2corrected) measured from a T2/e, vs. 1000/T chart gives an energy E , , Eb.To convert this to the true defect energy level E I o ,it is necessary to know Eb.If this correction is desired, one must determine Eb from the

+

80

A . G.

MILNES

TABLE VI HOLEPARAMETERS F,,* A N D up, OF TRAPSAS CALCULATED FOR FIG.6"

Label in Fig. 6

Activation energy E,, (eV)

HTI HS I HS2 HS3 HBI H B2 H8 3 H B4 HB5 H B6 HLI H L2 H L3 H L4 HL5 H L6 H L7 H L8 H L9 HLlO HLI I HLl2

0.44 0.58 0.64 0.44 0.78 0.71 0.52 0.44 0.40 0.29 0.94 0.73 0.59 0.42 0.41 0.32 0.35 0.52 0.69 0.83 0.35 0.27

Emission section up, (m-')

1.2 2.0 4.1 4.8 5.2 1.2 3.4 3.4 2.2 2.0 3.7 1.9 3.0 3.0 9.0 5.6 6.4 3.5

Type of sample

VPE LPE LPE 1 0 - 1 ~ LPE lo-'(' Cr-doped LPE As-grown LPE Fe-doped LPE lo-'* Cu-doped LPE lo-" As-grown LPE lo-'' Electron-irradiated LPE lo-'' Cr-doped VPE As-grown LPE lo-'' Fe-diffused VPE lo-'' Cu-diffused VPE As-grown LPE 10VPE with p ' layer lo-'' As-grown MBE lo-'' As-grown MBE lo-" As-grown VPE lo-'' As-grown VPE Melt grown lo-'' Zn-contaminated LPE

Excitation modeb

Chemical origin

x lo-"

x lo-''' x x x x x x

x x x x x

x x x x

x

1.1 x

1.7 x 1.4 x 1.3 x

Cr Fe cu

EO EO E EO EO E E E 0 EO 0 E

Cr Fe

cu

" Reproduced with permission of the IEE, London (after Mitonneau ef al., 1977). EO, electrical and optical; E, electrical; 0, optical.

variation of capture cross section with temperature. For electrons this is approximated over some temperature range as a,,= a, exp(-Eb/kT)

where a, is the limiting value of a,,at T + 00. This can lead to quite a significant correction. For example, one electron trap in VPE GaAs was measured as having a T2 corrected activation energy of E, - 0.48 eV, but Eb was determined to be 0.09 eV; hence the corrected defect level was inferred to be 0.39 eV. Sometimes, Eb is found to be negative in value and one then is concerned that the model (or the experiment) is not adequate [see also Majerfeld and Bhattacharya (1978) and Sakai and Ikoma (1974)l. Other sources of uncertainty in DLTS measurements are the effects of high electric fields in depletion regions (Vincent et al., 1979), the effects of trap gradients, and the possibility of traps closely spaced in energy-level


81

interacting in the output and producing a nonexponential capacitance transient that is misinterpreted as a true exponential by the processing electronics, which then yield an incorrect answer. For the EL2 level, Mircea and Mitonneau (1979) have found that if the electronic field is high ( 1.9 x lo5 V/cm), the electron emission is large when the electronic field is (111)oriented, pointing from As to Ga, and is significantly lower for the opposite field orientation. For the (100) and (110) directions, the emission rates take intermediate values. This suggests that the defect structure must initially have nearly tetrahedral symmetry and that it tends to favor a simple structure for EL2 such as an isolated vacancy rather than a complex such as an impurity coupled to a vacancy. In a further study of the EL2, however, Makram-Ebeid (1980a) has shown that at low temperature and moderate fields, the EL2 kinetics are dominated by a combination of hot-carrier capture and impact ionization. At high electronic fields the kinetics are dominated by phonon-assisted tunnel emission of electrons. Thus the anisotropy observed by Mircea and Mitonneau (1979) may be explained by these effects. Noise studies have been made by Roussel and Mircea (1973) and photocapacitance studies by Vasudev et al. (1977). Another important measurement technique that has been developed is deep-level optical spectroscopy (DLOS). In this technique, photostimulated capacitance transients are studied after electrical, thermal, or optical excitation of a junction or Schottky barrier. This technique provides the spectral distribution of both o;(hv) and o,O(hv), the optical cross sections for the transitions between a deep level and the conduction and valence bands. Besides its sensitivity, DLOS is selective in the double sense that o:(hv) and o;(hv) are unambigously separated, and that the signals due to different traps can be resolved from one another. As a result, the aO(hv)spectra are measured from their threshold up to the energy gap of.the semiconductor, over a generally large temperature range (Chantre and Bois, 1980; Chantre et al., 1981). In GaAs, it has been applied to a study of the EL2 level, the “Cu” level at E, + 0.40 eV, and the ELI, EL3, and EL6 levels. A comparison between thermal and optical ionization energies allows the Franck-Condon energy caused by lattice relaxation to be determined. For the EL6 electron trap the thermal activation energy is 0.31 eV, but the optical ionization : is much larger (0.85 eV), corresponding to a Franck-Condon energy for o energy of about 0.6 eV.

-

IV. LEVELS PRODUCED BY IRRADIATION Irradiation of GaAs and GaAs devices has been carried out primarily to determine the effects on performance of FETs, Impatt and Gunn diodes, LEDS, heterojunction lasers, and solar cells; however, in a few instances, it

A. G. MILNES

82

provides interesting clues to the physical nature of defects. The radiation may be in the form of high-energy electrons ( > O S MeV), protons, neutrons, or high-energy gamma or X rays. Such radiation may produce displacement defect traps that alter device performance unless annealed out again. Ion-implantation damage and subsequent annealing are not discussed here. The effects of radiation on FET-device performance have been reviewed by Zuleeg and Lehovec (1980).They report that GaAs and Si FETs operating with the channel electrons in thermal equilibrium (the normal mode of FET operation) degrade almost to the same extent under fast-neutron action. In short-channel devices with hot-electron action, the GaAs FET has some advantages over Si. The present technology for small-scale integrated GaAs circuits appears to allow a radiation tolerance for neutrons of lo5 neutrons/ cmz and for ionizing radiation dose lo7 rad with a dose rate for logic upset of 10'' rad/sec. This may be compared with a total dose hardness of lo6 rad in Si MOS integrated circuits (Berg and Lieberman, 1975; Kladis and Euthymiou, 1972).The effect of radiation on GaAs diodes has been examined by Wirth and Rogers (1964) and by Taylor and Morgan (1976). In GaAs solar cells, the main effects causing performance deterioration are the creation of deep centers that cause recombination-generation in the depletion region and a decrease of the minority-carrier diffusion length. Since GaAs is a direct gap material, photon absorption takes place in only a few micrometers of depth. This makes the junction depth an important parameter in the radiation-resistance behavior. In general, GaAs solar cells [including (A1Ga)As-GaAs structures] are more resistant than Si cells to radiation damage by both electron and proton irradiation. The efficiencies of electron-damaged GaAs cells tend to recover, in part, with anneals at temperatures as low as 200-300°C (Li et al., 1980a). We consider now some of the indications of the nature of defects obtained from irradiation studies. One of the principal effects of radiation is that acceptor-type defects are produced that remove electrons from n-type GaAs. Conversion to p-type has been studied by Farmer and Look (1979) and is attributed to the development of a relatively shallow acceptor at -0.1 eV. The carrier removal rates observed by many investigators for 1 MeV electrons are shown in Fig. 7. There is a range of a factor of 10 in An/4 values but no systematic dependence on donor type or concentrations. Large carrier removal rates seen by some investigators seem to be related to larger electron fluxes (so the rate of damage creation can be a factor), but Farmer and Look (1980)find the removal rate to be dependent on the Fermi level but independent of the irradiation flux. No great differences are seen for LPE, VPE, or bulk material, from which it may be inferred that the created defects tend to be intrinsic in nature and not affected directly by defects or impurities in the starting material. Table VII shows a collection of data assembled by Lang (1977) for energy levels seen by a number of investigators.

-


83

The introduction rate for several defect levels by 1 MeV electrons shows a crystal-orientation dependence and suggests that these defects (notably El, E2, and E3, at E, - 0.1 3, E, - 0.20, and E, - 0.3 1 to 0.38 eV) can according to Lang et al. (1977) be related to Ga-site displacements, or according to Pons and Bourgoin (1981a,b) to As-site displacements. Understanding of the knock-on process dependence with energy now favors the As-site displacement interpretation.

-----O lO *'

+ (1 MeV electrons/cm2 1 FIG.7. Carrier removal An against 1 MeV electron fluence Q for n-GaAs. Note that An/Q ranges from 0.5 to 5.0 cm-', but with no systematic dependence on donor concentration or species (after Lang, 1977; reproduced with permission of the Institute of Physics, UK): n0

Symbol 0

(cm 4

Aukerman and Graft (1962) Thommen (1970) (Stage 3) lo'* (Si, Te) Kahan er a/. (1971) I015(Sn) Kalma and Berger (1972) loi7(Si) Kalma and Berger (1972) 1OI6 Pegler et al. (1972) I O ~ ~ ( V P E ) Dresner ( 1974) I0I6(LPE) Lang and Kimerling (1975)

1016

7 x

0

6 9 2 2.5

X

1.3 x

0 A A V

x

x x

x

1 x

Reference

tos

1015

1017

84

A. G . MILNES

TABLE VII ENERGY LEVELS AND INTRODUCTION RATESOF DEEPLEVELSINTRODUCED INTO GaAS BY 1 MeV ELECTRON OR Cob' IRRADIATION AT ROOMTEMPERATUR~ Energy levelsh(eV) Electron traps Measurement technique

El

E2

€3

E4

Hole traps

E5

HO

H1

References

Thermal emission activation energy: As measured -0.08 0.19 With T' correction 0.18 With a ( T )correction Hall coefficient 0.12 0.20 activation energy 0.13 0.17

0.45 0.76 0.96 -0.09 0.32 Lang (1974). Lang 0.29 and Kimmer0.41 0.71 0.90 0.31 ling (1975) 0.38 0.10 - Vitovskii el a / . ( 1964) 0.31 0.10 - Brehm and Pearson (1972) Photoconductivity 0.38 0.52 0.72 Vitovskii er d. threshold ( 1964) 0.54 0.10 - O'Brien and Corelli (1973) 0.10 0.29 Best value 0.13 0.20 0.31 - 0.7 Lang and Introduction rate N/qi 1.8 2.8 0.7 0.08 0.1 Kimerling (cm-'), I MeV electrons, 300 K (1975) From Lang (1977). Reproduced with permission of the Institute of Physics, UK. Energies of electron(ho1e) traps are measured with respect to the conduction(va1ence) bands. a

The threshold energy for atomic displacement in GaAs appears to be of the order 10 eV (which corresponds to an electron energy of about 0.25 MeV). Hence, 1 MeV electrons have sufficient energy to displace more than one may be atom, and so the possibility exists that divancancies such as V,,V, produced in addition to (VGa+ Gai) and (VAs + Asi). Attempts to observe divacancy production by varying the electron energy over the range 0.25- 1.7 MeV were, however, inconclusive (Pons et al., 1980). This was explained by suggesting that if such centers are created, either the energy levels are too close to the band edges to be readily observed or that the centers anneal at below room temperature. Annealing stages after electron irradiation have been found at 235, 280, and 500 K. Pons et a / . (1980) propose that there is a recombination (due to Coulomb attraction) of a divacancy with a neighboring interstitial (Asi or Ga,), resulting in single vacancies (V,, or V,,) and in associations of antisite defects (one Ga atom on an As site or vice versa) with single vacancies: (Ga,, VGa) and (As,, + V,J. These reactions could occur in two separate stages: one stage due to the Asi jump and the other one due to the Ga, jump.

+

85


10 10-6

I (015

1

I

4 0’6

101’

10’6

no (cm-3)

FIG.8. Anneal rate (200°C) for the E2 [ A , Lang er a/. (1976)l and A2 [ O , Ackerman and Graft (1962)] levels seen in GaAs after irradiation. The proportionality to n2’3 suggests that the annealing is by cornplexing with donors. (After Lang, 1977; reproduced with permission of the Institute of Physics, UK.)

The suggestion is, therefore, that the two low-temperature anneal stages are in some way associated with such reactions. Pons and Bourgoin (1981b) at present favor the view that the El, E2, and E3 levels at E, - 0.045, -0.140, and 0.330 eV are manifestations of As vacancies or interstitial Frenkel pairs. The annealing around 500 K involves the El and E2 levels and tends to be proportional to the 2/3 power of the preirradiation donor concentration (see Fig. 8). Lang remarks that this functional dependence is exactly what one would expect if the donor atoms were acting as sinks for the defect (or defects) responsible for the (El, E2) group. This can be seen from the following simple argument. The average distance between donors is proportional to the inverse cube root of the donor concentration. The average number of random jumps needed to travel a given distance is proportional to the square of that distance. Hence the average number of jumps necessary for a defect to travel to a donor should be proportional to the 2/3 power of the donor concentration.

Since the (El, E2) production rate is independent of doped species and material type (LPE, VPE, LEC), it suggests that these defects at E, - 0.13 eV and E, - 0.20 eV are predominantly intrinsic in nature and annihilate by complexing with donors (or acceptors related to donor concentration) during the 500 K annealing stage. Another interpretation is proposed by Pons and Bourgoin (1981b) since their experiments show that the charge state of the trap influences the

86

A . G. MILNES

annealing rate and that the dopant concentration is not significant in the way proposed by Lang. The model proposes that El and E2 are different charge states of the same defect site and that when El is lying above the Fermi level at low donor doping concentrations (i.e., is empty of electrons), its annealing rate is less than when it is below the Fermi level (Pons, 1981). Annealing of the E3, E5, and H 1 levels ( E , - 0.3 1 to 0.38 eV, E, - 0.9 eV, and E, + 0.29 eV) seems independent of the donor concentration. There are some indications that the E5 (E, - 0.9 eV) and HI ( E , + 0.29 eV) introduction rates vary from specimen to specimen, presumably depending on the impurity content of the preirradiated material. Annealing has also been studied by Walker and Conway (1979). Electron paramagnetic resonance studies of semi-insulating GaAs:Cr after 2 MeV electron irradiation show signals that are assignable to the antisite defect AsGa(Strauss et a/., 1979; Kaufmann and Schneider, 1982). The possibility that the signal comes from a complex such as AsGa-X, however, cannot be entirely ruled out. Kennedy et al. (1981; Wilsey and Kennedy, 1981) believe that the AsGa antisite defects may be formed by room-temperature diffusion of As interstitials to Ga vacancies. Fast-neutron-irradiated GaAs also develops the Ash: signal and this anneals out between 450 and 500°C. No direct ESR evidence yet can be offered for the formation of the complementary GaAsantisite defect. Elliott et a/. (1982) have attributed an acceptor level at 78 MeV to the GaAsantisite defect, since the defect is tetrahedrally coordinated and appears in LEC material grown from Ga-rich melts. Ta et a/. (1982 ) agree that this level is associated with excess Ga, but find some evidence that the level increases with concentration if there is increased B in the crystal. From simple consideration of a group V element on a group 111 site, AsGamay be expected to be a double donor. However, the energy level(s) associated with AsGahave not yet been established. The complex (As,,VA,) is considered by Lagowski et al. (1982) to be a candidate for the EL2 trap, and some evidence for double-donor action exists. Fast-neutron-irradiated GaAs may be expected to have incurred considerable damage. The primary-knock-on effect is likely to transfer as much as 50 KeV to the atom that is displaced and this, because of secondary collisions, may produce many further displacements (Worner et al., I98 I). This seems to be borne out by the 400-600°C high annealing temperatures required. However, Coates and Mitchell (1975) suggest that close interstitial vacancy pairs may also be involved. Turning again to consideration of electron-irradiated GaAs, Farmer and Look (1980) reach the following conclusions: (i) Room-temperature defect production is nearly always sublineal with h e n c e . This can be explained by a simple model of stable (or nearly stable) Ga vacancies competing with traps and sinks for the mobile Cia interstitials.


87

(ii) The wide variation of reported free-carrier removal rates cannot be accounted for by a flux dependence. Some of the variation is probably due to the formation of defectimpurity complexes, which may include E5 and H I . Another factor, quite important in high purity samples, is the position of the Fermi level, which can change the proportionality between the defect-production rate and the carrier-removal rate. (iii) The 200°C annealing stage includes two first-order substages with annealing rates close to those reported in the literature. Besides being first order, the first substage (1,) is also nearly independent of sample growth conditions and doping levels and has a relatively low prefactor. These attributes can be explained by a model in which the i , substage involves Ga-vacancy related defects annealing by interactions with Cia interstitials which are themselves emitted by interstitial traps. The I, substage is also best described by a dissociation process with decreasing n o , although the exact relationship is not clear. (iv) The defect model most consistent with all of the data includes E3 as a donor, HI as an acceptor, and El and E2 as the two charge states of a double acceptor. The exact identifications of these defects are, however, somewhat in doubt and must await further experimentation.

Interesting observations have been made on the E3 level introduced into LPE (loo), ( 1 lo), ( 1 1 1 ) and (111)material by 1 MeV electron irradiation. The dependence of production rate of the E3 level on orientation is shown in Fig. 9. The production rates are large for orientations that favor easy knockon of Ga atoms. This anisotropy seems also to be true for the El and E2 defects, which also anneal at 500 K and are therefore also believed to be related to Ga-site displacements. The annealing actions seen at 235 and 280 K most probably are Asi and V, related. The high mobility of Asi

C

-

C

Go + As

easy 0

8 0 >

t110> Go

As

t110>

hard As easy Go

hard

Ga+As

Go

1 MeV electron beam direction

Ga+As

(110)plane: Go As unit cell I:> interstitial o As Go site

FIG.9. Orientation dependence of the introduction rate of the E3 leve Four samples with (100). ( 1 lo), ( I I1)Ga. and ( I I1)As surfaces exposed to the I MeV electron beam were irradiated simultaneously to a total fluence of 4 = I x 10” cm-’. The right-hand side of the figure is the (1 10) plane of the GaAs unit cell and illustrates three of the four sample orientations used. The “easy” and “hard” notations are explained in the text. (After Lang er ol. 1977; reproduced with permission of the American Physical Society.)

88

A. G.

MILNES

at room temperature then suggests the possibility that the formation of VG, is followed by creation of an antisite defect AsG,. Lang and his co-workers (1977), as a speculation, comment that the AsGaantisite defect should be a double donor and might be expected to have two levels near the conduction band, such as El and E2. If El and E2 are donors and are present in quantity, then some other level that is an acceptor must be produced in quantity to provide the electron-removal action, such as the 0.1 eV, relatively shallow acceptor seen by Farmer and Look (1979). Frenkel-pair defects are unlikely to be involved for El and E2, according to Lang, since the need for 500 K for annealing suggests long-range motion of the defects and not close-pair recombination. However, this view does not concur with that of Coates and Mitchell (1975) and has been challenged by Pons and Bourgoin (1981), who regard the El, E2, and E3 levels as related to As-site Frenkel pairs. Weber and Schneider (1982) also have discussed the AsGaantisite defect. In the work of Lang et al. (1977) the El, E2, E3, and other levels have been studied as a function of crystal composition in the range GaAsA1,,,Ga0~,As with the results shown in Fig. 10. The E3 level is seen to be the

p

-

-

1.2

0

-

(,

-

n

L

zl

-

a t.0 W

z

-

W

0.6

- --

FB

-

-

-

0

0.4,;

-

-

C u a

IP

-

0.2 0

1

1

I

I

I

I

I

89


only one that tracks the valence-band edge. Since vacancies, according to Lang, tend to be created from valence-band-type wave functions, this suggests that E3 is a vacancy, and the assignment V, is made because the E3 production is greatest in situations that Lang considers to favor Ga knock-on. Both views have been questioned (Wallis et al., 1981a). Displacement effects produced by electrons have also been considered by Watkins (1976). Consider now the traps produced by proton and neutron irradiations. Protons of energy 50, 100, and 290 keV applied to n-type GaAs (LPE) layers with fluences of lo'', lo", and lo'* protons/cm2, resulted in the trap levels shown in Table VIII. The proton penetration depths are 0.5 pm for 50 keV, 1 pm for 100 keV, and 2.5 pm for 290 keV. The trap species produced depend not only on energy but also on dose. Comparison of these results with those of Table VII for the traps produced by 1 MeV electron irradiation and Tables V and VI for electron and hole traps in as-grown material show a substantial commonality. The following traps appear frequently in electronand proton-irradiated and as-grown material (with no specific deep impurity dopant added) :

E, - 0.1 1 to 0.130 eV

E, - 0.71 to 0.83 eV

E, - 0.18 to 0.20 eV

E,

E, - 0.31 eV

E,

+ 0.71 to 0.73 eV

+ 0.40 to 0.44 eV E, + 0.31 eV

E, - 0.52 eV

The uncertainty (range) values are as indicated or may be taken as about k0.02 eV. Loualiche et al. (1982)find dominant levels acting as acceptor-like TABLE VIll

DEFECT LEVELS OBSERVED IN LOW-ENERGY PROTON-IRRADIATED GaAs" Defect level (eV)

Trap level Electron traps

Hole traps

(I

E, E, E, E, E, E, E, E, E, E, E,

- 0.I 1 - 0.14 - 0.20 - 0.31 - 0.52 - 0.71 + 0.059 + 0.17 + 0.44 + 0.57 + 0.71

100 keV 10"

290 keV

100 keV 10"

290 keV

10"

protons/cm2

protons/cm2

protons/cm2

protons/cm2

10"

X X

X

X X

X

X X

X

X X

X

X

X

X X

X

X

X

X

From Li el al. (1980b). Reproduced with permission of the Metallurgical Society of AIME.

90

A . G . MILNES

traps at E, - 0.22 eV and E, - 0.33 eV after proton irradiation of VPE n-type GaAs. Shallower hole traps around 0.17,0.l0, and 0.06 eV are also observed in irradiated material but not commonly seen in as-grown GaAs. As-grown LPE material tends to contain only hole traps at E, + 0.41 eV and E, 0.71 eV. The Ga melt presumably suppresses traps related to V, and also acts as a gettering sink for impurities that might exist on the specimen surface or the inside of the specimen growth system. (Direct studies aimed at showing the residual contamination of the Ga melt in LPE appear not to be reported, but variations of growth purity have been reported for a sequence of growths from the same melt. One LPE growth technology (Ewan et al., 1975) uses a deep Ga melt system, and one of the advantages of this is that frequent system “cleaning” and melt loading is not needed.) Where clustering of defects occurs, as in high-energy neutron, proton, or heavy-ion bombardment, one may expect many interacting energy levels to result (Ludman and Nowak, 1976; McNichols and Berg, 1971; Stein, 1969). Thus, a broadening of DLTS peaks occurs as may be seen from Fig. 11 for O + and He+ ion bombardment. The effects of ion implantation for doping of GaAs and the annealing of the associated damage by bulk thermal or laser heating are not considered

+

I MeV (e‘) X0.25

E2

€3

€4

E5

TEMPERATURE ( K )

FIG. 1 1 . DLTS spectra of four n-GaAs samples irradiated at room temperature with 1 MeV electrons, 600 keV protons, 1-8 MeV He’ ions, and 185 keV 0’ions. Note the general trend toward a broader and deeper spectrum as the mass of the high-energy particle increases (Lang, 1977; reproduced with permission of the Institute of Physics, UK).


91

here. Papers available include those of Davies (1976), Degen (1973), Eisen (1971), Hemment (1976), Donnelly (1979, Donnelly E t a/. (1975), Elliott er al. (1978), Favennec et al. (1978), Hunsperger and Marsh (1970), Hunsperger et al. (1972), Moore et al. (1974), Ilic et al. (1973), Littlejohn et al. (1971), Sansbury and Gibbons (1970), Sealy et al. (1976), Surridge and Sealy (1977), Takai et al. (1975),Woodcock and Clark (1979, Yu (1977),and Zelevinskaya (1973). WITH V. SEMI-INSULATING GALLIUM ARSENIDE

AND WITHOUT

CHROMIUM

Chromium in gallium arsenide produces material of high resistivity, typically in the range 0.3-2 x lo9 0-cm at 300 K. Calculations from known effective masses suggest that perfectly pure (intrinsic) GaAs of band gap 1.42 eV should have n, = pi = 1.5 x lo6 cm-3 at 290 K. One may also calculate n , ( T )from (n*p*)'/', where n* = (e,/c,) and p* = (eP/cp),for two successive charge conditions of a deep level. For the Cr2+ P Cr3+ transition, Blakemore (1982) obtains ni = 1.8 x lo6 cm-3 for the 300 K value in very satisfactory agreement. The experimental evidence supports a value 1.7 f 0.4 x lo6 cm-3 (Look, 1977). The Fermi level should then be at E, - 0.637 eV. Semi-insulating GaAs may be obtained with substantial addition of Cr (the simple model suggests that Cr is a deep acceptor at about E, f 0.73 eV). For high-resistance material, sufficient Cr must be added to compensate for the residual (ND - NA) shallow doping. The electrically active solubility of Cr in GaAs has been estimated to be in excess of 5 x IOI7 cm-3 and therefore can, in principle, compensate net shallow donors up to the low 10I7cm-3. If an attempt is made to reduce the net shallow doping to a very low level, for instance, by LEC growth in B 2 0 3 capped conditions in BN crucibles, the material obtained is frequently semi-insulating even though no Cr has been added (Thomas et al., 1981). The reason for this is not certain but some investigators believe that it is related to EL2, a deep impurity level (Fig. 5 ) at about midgap that is almost always present in a ~ discussed . in Section 11, it seems probable concentration about 1 O I 6 ~ m - As that this level is gallium vacancy or AsGa related. If oxygen is not a component of the level, possible candidates might be AsGaVGa,AsGaVAsrand VGaAsGaVGa. Any study of semi-insulating GaAs with Cr present must recognize the existence of the EL2 donor level and other possible levels (such as H10). From ESR studies, Cr tends to be present on the Ga site. The outer electronic shells of Ga are 3d1°4s24p' where the 4 . ~ ~ electrons 4~' participate in the lattice bonding. The corresponding structure for Cr is 3d34s24p1and therefore neutral Cr in a Ga site may be characterized by Cr3+(3d3).This neutral center condition in another nomenclature system might be termed Cro or NOT. If one electron is trapped from the conduction band, or picked up from the valence band by optical pumping, the electron goes into the 3d shell

92

A. G. MILNES

and the level may henceforth be described as Cr2+(3d4)or N ; . The first excited state of this level is believed to be a few tens of meV into the r conduction band (Eaves et al., 1981 ). On the possibility that chromium might accept a second electron to become Cr' +(3d5), there is convincing evidence that this level is 1 15 meV into the r conduction band at 300 K and therefore can be seen only if the conduction-band edge is raised by the application of pressure (Hennel et al., 1980, 1981a,b; Hennel and Martinez, 1982). Conceivably, Cr3+(3d)could exhibit electron-donor action in GaAs that would result in it becoming Cr4+(3d2),but there is no convincing evidence that this happens. Chromium in GaAs, when excited optically (say with an argon laser), exhibits an luminescence band at about 0.8 eV with a zero phonon line at about 0.839 eV. This is luminescence from inside the Cr atom involving relaxation in the 5E-5T2 crystal field scheme. The 5T2 is the ground state of the Cr that has trapped an electron and become Cr2+(3d4).The crystal-fieldlevel structure has the form shown in Fig. 12, according to Vallin et al. (1970), Williams et al. (1981), and Clerjaud et al. (1980). See also Bishop (1981),

FIG. 12. Energy levels of Cr2+(3d4)as predicted by crystal-field theory for an undistorted, substitutional cation site of Td symmetry. The ground state is the P, of 6T, (k is small, about 0.060 meV for GaAs). (After Vallin er al., 1970; reproduced with permission of the American Physical Society.)


93

Grippius and Ushakov (1981),Glinchuk et al. (1977),Klein and Weiser (1981), Krebs et al. (1981), and Krebs and Strauss (1977). The luminescence at about 0.8 eV was thought at one time to involve a Cr-conduction- or valence-band transition. Now that it has been shown to be a Cr internal transition, it is of less importance with respect to carrier trapping activities. The structure of the luminescence is now almost elucidated (White, 1979; Picoli et al., 1980,1981; Eaves, 1980; Eaves et al., 1981b; Williams et al., 1981; Deveaud and Martinez, 1981. The 0.838 eV Line is strong in emission but weak on absorption and exhibits (111) axial symmetry. Eaves considers it to be a complex (Crii -X) where X is an unidentified impurity (oxygen is not excluded). There is also a strong 0.82 eV absorption line that is considered to represent the 5T2+5E absorption of simple substitutional chromium, CrGa(Clerjaud et al., 1980). For practical purposes, as opposed to detailed physical study, the amount of Cr in GaAs can be estimated by gross comparison of the optical absorption in the region between about 0.8 and 1.2 eV for various slices as shown in Fig. 13. The absorption is associated with electron transfer from the valence band to the Cr3+(3d3)state, causing it to become Cr2(3d4).Absorption with a threshold around 0.67 eV may involve further pumping of the CrZ+state, and the electron is handed on to the conduction band. The shape of the absorption band (either at 300 or 4.2 K) does not allow accurate determination of the energy transition (at 4.2 K it is roughly 0.85 eV for the transition Cr3+ evB+ Cr2+)since the Lucovsky model does not fit, and the temperature effects suggest that lattice relaxation or electron-phonon interactions may be taking place. The segregation coefficient for Cr inferred from these absorption measurements is 8.9 x

+

Energy (eV)

FIG. 13. Absorption coefficient at room temperature as a function of the energy of the photons for four different samples: undoped (sample A) or doped with Cr (samples C, E, and 1). (After Martin er al., 1979; reproduced with permission of the American Physical Society.)

94

A. G. MILNES

More detailed absorption curves for 300, 77, and 4.2 K are found in the studies of Martinez et al. (1981), Hennel et al. (1981b), and Martin et al. (1981b). The latter group also examine the variation of EL2 throughout ingots and conclude that it must be a complex defect involving lattice defects growing in the presence of stress. Others take the view that EL2 is the simple antisite defect AsGa.Yet others consider it to be due to complexes involving AsGa(Li et al., 1982) or AsGaVAs (Lagowski et al., 1982) formed by As movement into Ga vacancies or divacancies. Martin and Makram-Ebeid ( 1 982) consider the outward diffusion of EL2 level at temperatures above 600°C. They consider that EL3 or HL9 may be the AsGadefect. Photoconductivity in epitaxial GaAs: Cr shows evidence of optical threshholds between 0.75 and 0.79 eV for low and high illumination levels. A broad level of trap distribution (225 meV) may be involved, and an electroncapture cross section of 2 x ~ m - which ~ , is unexpectedly large, results from the modeling. The Lucovsky model is not a good fit. Better success is had with a photoionization theory that assumes a Is)-ls) transition involving a tight orbit with thermal broadening described by a single phonon energy of 0.03 eV and a Huang-Rhys factor of 3, and taking into account nonparabolicity in the band by k.p. (Kronig-Penney) theory. The enthalpy of the 0.75 eV level is therefore 0.66 eV (Amato et al., 1980). Some departure from the Lucovsky model is also found by Vasudev and Bube (1978). See also Monch et al. (1981) and Vaitkus et al. (1981). In other studies of photoconductivity and photo-Hall spectra, Look (1977)gives the energy-band diagram of Fig. 14. In more recent studies, Look shows that mixed conductivity effects that complicate the interpretation of Hall and resistance measurements in very high resistance material are not Cr-SE(rxcit)

t

CB

FIG. 14. A proposed room-temperature energy diagram for GaAs: Cr (CB, conduction band; VB, valence band). (After Look, 1977a; reproduced with permission of the American Physical Society.)

95


important if the resistivity is below 5 x lo8 R-cm. From analyses of many Cr-doped and undoped semi-insulating specimens, Look ( 1980) concludes that E, - E,, is 0.64 eV (if go/gl = 4/5) and that another dominant level is at E, - 0.59 eV for go/gl = 1/2 (which he terms E, without necessarily implying that it is oxygen related). Look finds that for Cr-doped crystals, the Fermi levels at room temperature are slightly lower than for undoped (or 0 doped) semi-insulating crystals. Other important studies have been made of the properties of semiinsulating GaAs. We have, for instance, the models of Zucca (1977),Lindquist (1977),and Mullin et a/. (1977). The Mullin group comments that transport studies have until recently led to a consensus view of Cr-doped GaAs that interprets the activation energy for conduction as due to a Cr level at -0.70 eV from the conduction band. They declare that this is a misleading interpretation and go on to show that their material exhibits two dominant levels, termed Em and E,, at 0.40 and 0.98 eV from the conduction band, respectively. The Cr concentrations (10" cm- 3 , exceed the measured values of E, ( 1014-1015cm-3), although Cr was considered clearly responsible for forming the species E,. They suggest that the E, level may be related to a Cr-0 complex. Martin et al. (1980b) have presented one of the most recent studies of compensation mechanisms in GaAs. Hall measurements made for their ~ ) undoped semi-insulating maCr-doped (6 x 10" - 4 x 10'' ~ m - and terials are given in Tables IX and X. The variations between different specimens are quite large, the Hall-effect mobilities need interpretation, and the

TABLE IX RESULTSFROM HALL-EFFECT MEASUREMENTS A N D OPTICAL ABSORPTION MEASUREMENT ON SEMI-INSULATING MATERIALS DOPEDWITH C r CONCENTRATION RANGING BETWEEN l o i 6 A N D l o i 7 c m - j

Material DI Bridgman D2 LEC D3 Bridgman D4Bridgman D5 LEC D6 LEC

Optical absorption data, concn. of Cr (cm-j) 3.5 3.5 5 3 9 9

x

lo'*

x IOl* x 10l6 x 10l6

x lo'* x

Hall-effect data

(cm' V-l sec-I)

Slope of In(R,T3'*)-' (eV)

I600 I800 2800 3200 760 I200

0.762 0.762 0.700 0.757 0.716 0.776

(yRH)-'

at 400 K (cm-j) 1.5 7.5 1.2 7.0 1.6 8

x x x x x x

lo1' 10' 10"

10' 10" 10'

p,, at 400

K

Concn. of Si (cm0 0

A few x 10" (1-2) x lo1* 0 0

" After Martin el (11. (1980a). Reproduced with permission of the American Physical Society.

96

A. G . MILNES

TABLE X

RESULTSON HALL-EFFECT MEASUREMENTS CARRIED OUTON SEMI-INSULATING GaAs NOTDOPED WITH C r

A1 Bridgman A2 Bridgman A3 Bridgman A4 Bridgman A5 Bridgman A6 Bridgman A7 LEC A8 Bridgman A9 LEC A1 LEC

1.1 x 10'1 1.7 x 10" 1.3 x 10"

2.5 x 1.5 x 2.8 x 1.6 x 8 x 4 x 1.1 x

1013

1013

1013

10"

IO" 109 1010

4500 4500 5500 4300 4300 2500 2800 I700 I850 2600

0.757 0.729 0.750 0.350' 0.422' 0.392' 0.60W 0.164' 0.728' 0.762d

3 x loi6 3 x 10l6 4 x 10l6 5 x 10l6 5 x 10l6 5 x 1017 4 x 101~

0.539 0.604 0.533 0.352 0.370 0.348 0.526 0.312 0.653 0.619

After Martin er 01. (1980b). Reproduced with permission of the American Physical Society.

' Deduced from p,,.

Slopeof In(R,T3/4)-1 = f ( l / T ) . Slope of ~ I ( R , T ~ ' ~=) -l(l/n. '

Fermi levels are quite variable. From an analysis of the Cr-doped specimens, two dominant deep levels are found, namely, the Cr acceptor at E, 0.73 eV and a deep donor (termed EL2) at E, - 0.67 eV (400 K). As seen in Fig. 15, their deep donor lies below the deep acceptor, in reasonable agreement with Fig. 14. The EL2 donor is considered to be dominant in creating high resistance undoped (or very lightly Cr-doped) material. The shallow doping concentration has some effect on the Fermi-level position as shown in Fig. 16. If the electron and hole concentrations in GaAs at 400 K are to be equal, then the Fermi level will be at E, - 0.637 eV. For very high resistivity GaAs with both electrons and holes contributing to mixed conduction, the apparent Hall mobility is strongly dependent on the precise doping conditions. Figure 17 shows calculated values expected if the level EL2 is supposed not to be present. On the other hand, Fig. 18 shows the curves if EL2 is present in a concentration 10l6cm-3. The lines are seen to be in general agreement with experimental results for some materials from Tables IX and X. In optical DLTS studies of Cr-doped GaAs, Jesper el a/. (1980)find a level with an electron activation energy of E, - ET = 0.72 eV. The photon cross section of this increases with temperature, but not in a way that conforms to

+

97


1.375 a\

0.73130 rv

FIG. 15. Scheme of the band gap with different important deep levels, at 400 K: A, conduction band; B, shallow donor ND(Si); C, deep acceptor NAA(Cr);D, deep donor N,,(EL); E, valence band. (After Martin er a/., 1980; reproduced with permission of the American Physical Society.)

=02

T= 400 K

bond

band FERMI-LEVEL

POSITION ( e V )

FIG. 16. Shockley diagram corresponding to a semi-insulating GaAs with a given concentration of the deep Cr acceptor and the deep donor EL2, with or without shallow donors. (After Martin e r a / . , 1980; reproduced by permission of the American Physical Society.)

98

A. G . MILNES

5000

J J

a

I

1000 -

CONCENTRATION OF CHROMIUM FIG. 17. Expected variation of the observed Hall mobility in semi-insulating GaAs, as a function of the Cr concentration, for different values of N, - N, (the concentrations of shallow donors and acceptors) assuming that the concentration of the deep donor EL2 is zero; T = 400 K. (After Martin el al., 1980;reproduced with permission of the American Physical Society.)

multiphonon effects. The level may be detected as either an electron trap or a hole trap, depending on its charge state and the nature of the experiment. The electron-capture cross section is about 2 x cm2 and is not temperature sensitive. The authors conclude that the DLTS signal in n-type semiinsulating material is dominated by hole emission [i.e., by electron pickup fromthevalence bandasCr3+(3d3)+Cr2+(3d4)]andopis -3 x lo-’’ cm2. The values found in the work of the last decade for the acceptor level of Cr tend to range from 0.7 to 0.9 eV. This scatter is a measure of the difficulty of interpretation of results as a function of temperature when additional deep levels such as EL2 are present. In theory, the Fermi function from Martin et a / . (1980a) is

where

99


Y

I

\\

-NA levels)

1016

1017 CONCENTRATION OF CHROMIUM

1018

-

FIG. 18. Expected variation (full lines) of observed Hall mobility in semi-insulating GaAs, as a function of the concentration of Cr, for different values of N, - N, (the concentration of shallow donors and acceptors), and for a given concentration of the deep donor EL2 equal to 10l6 cm-’; T = 400 K. Points correspond to experimental results on different materials: ( 0 )Bridgman grown under 0, overpressure; ( 0 )LEC; ( A ) Bridgman, not intentionally Bridgman, doped with Si. (After Martin et d.,1980; reproduced with doped with 0,; (0) permission of the American Physical Society.)

is the free energy of ionization of the level, AH its enthalpy, AS its entropy; go is the degeneracy of the level not occupied by an electron and g , the degeneracy of the level occupied by an electron. The detailed electrical characterization of a level allows the determination, as a function of temperature, of the following terms:

ET - kT ln(g,/g,) E’ = AH - T AS - kT ln(g,/g,)

E’

=

E,

-

(16) (17)

The band gap for GaAs as a function of temperature is EG

=

1.519 - r(5.4 x 10-4)TZ/(T+ 204)] eV

(18)

and from hole-emission and capture-rate data, Martin et al. (1980a) conclude that for T > 50 K, the Cr level above E, is possibly represented by E’(Cr) = 0.81 - [ ( 3 x 10-4)T2/(T+ 204) + kT In 0.93 eV

100

A . G. MILNES

At 300 K, this yields

E'(Cr) = ET - E, = 0.81 - 0.05357

+ 0.0018 = 0.805eV

(19)

and at 400 K, Eq. (19) yields 0.73 eV. The variation with temperature is seen to be quite significant and the level cannot be regarded as pinned to either band edge. Detailed DLTS and ODLTS (optical deep level transient spectroscopy) studies of the Cr3+ and Cr2+ levels in GaAs have been made by Martin et al. 1980a. The electron-capture cross section of the chromium site for the cm2. The hole-capture cross section process Cr3+ + e + Cr2+ is 4 x for the process Cr2+ + h + Cr3+ is much larger, perhaps 2.5 x 10- cm2. From comparisons of trap densities and chemical analyses for Cr, it is concluded that most of the Cr atoms (within a factor of 2) are electrically active. Recombination and trapping has also been considered by Li and Huang (1972). Field-effect transistors may be made by the creation of n channels by ion implantation into Cr-doped semi-insulating expitaxial layers and subsequent annealing. The redistribution of Cr into the implanted region is significant. In VPE and MOCVD (metalloorganic chemical vapor deposition) layers, the out-diffusion of Cr into the implanted region has been represented by D = 4.3 x lo3 exp( -3.4/kT). At high Cr concentrations, the behavior is more complicated. Semi-insulating GaAs can also be produced by MBE (Morkoc and Cho, 1979). METALS VI. EFFECTS PRODUCED BY TRANSITION Transition metals produce many interesting effects in GaAs. We have just reviewed the role of Cr in producing semi-insulating material; Fe also gives a fairly deep acceptor (E, + 0.5 eV) and so can contribute to producing fairly high resistance material ( - lo5 Q-cm, 300 K). TABLE XI

MULTIPLE CHARGE STATFSOF TRANSITION-METAL ACCEPTORS I N 111-V SEMICONDUCTORS' State

3d2

3d3

3d4

3dS

3d6

3d7

3d8

A'

V3+

Cr3+

Mn3+ Cr2+

Fe3+ Mn2+ Cr

co3

Ni3+

Fez+

Co2+

Cu3+ Ni2+

A-

V2

+

A?S a

+

Fe+

+

I

312

2

Kaufmann and Schneider (1980).

512

0, 1/2, 1

312

3d9

Cu2+

Ni+ 0

112

101


The core charge states of transition metals on a gallium site are shown in Table XI. Although A’- states are shown, it does not follow that they occur or have been identified in GaAs. Following Kaufmann and Schneider (1 980), we can say that: Paramagnetic resonance studies of the Fe acceptor in GaAs have shown that it resides on the Ga sublattice and has a deformed state of a 3d wave function for a free atom. Similar observations have been made for V, Ni, Cr and Co. This is also the case for Mn in ingot material, though in Mn doped LPE GaAs the ESR spectrum shows reduced symmetry due to the formation of (what is probably) a Mnca-VAs complex. The E, + 0.15 eV Cu level is believed to correspond to Cu on the Ga lattice, with another level at E, 0.45 eV corresponding to Cuz-. However, recent examination of the 0.15 eV defect suggests that its symmetry may be lower than a simple Cu,, assignment, possibly due to complexing with another defect. Thus, it appears that the Fe group transition elements (Sc-Cu) in GaAs generally act as deep acceptors, and occupy sites on the Ga sublattice.

+

A. Energy Levels Considered in Relation to Atomic Size and Strain

The nonionized core state of Fe is Fe3+ (3d5), and on accepting an electron from the valence band (equivalent to emitting a hole to the valence band), it becomes Fe2+(3d6).The situation is thus similar to that of an isoelectronic impurity, except that the inner 3d shell is incomplete and a hole may be thought of as localized in the atomic core. For isoelectronic impurities it is known that the energy of binding a hole is determined by the deformation of the lattice about this impurity (J. W. Allen, 1968). Thus an impurity atom of radius R,, which replaces a lattice atom whose radius is 6 R larger, forms a field of elastic stresses around the impurity, and the activation energy of the hole is given by

6E = f i d R & 6 R / a 3 = KR& 6 R Here, a is the distance between nearest neighbors in an undeformed crystal, d is the deformation potential, and K is a proportionality constant given by K =fid/a3

The anisotropy of the activation energy can be neglected to a good approximation. It has been shown for substitutional isoelectronic impurities that displacements of the nearest neighbors result in a redistribution of the valence charge such that the activation energy of a hole is reduced to near zero. The residual activation energy, of the order of 0.01 eV, is determined by the Born-Mayer forces between the impurity atoms and the nearest neighbors. The covalent, tetrahedral radii of various elements is a function of their position in the periodic table. The radii depend on the number of electrons in the 3d shell, and data can be taken from compounds in which the transition metal ion has a formal charge of +2. If these metal ions had filled atomic

102

A. G . MILNES

core shells like Ca2+ and Zn2+,they would presumably act as isoelectronic traps with near-zero activation energy. It is interesting (although rather crude in a fundamental sense) to follow up this line of thought by postulating that the binding energy of iron-group transition elements in GaAs may be given by Eq. (20) with 6R equal to the difference between the metal atomic radius R, and the extrapolated radius of the atom with a filled (3d'O) shell, R,. This was first done by Bazhenov and Solov'ev (1972) and more recently by Partin et al. (1979e), with the results shown in Fig. 19. Several factors complicate the task of identification of the Ga-site singleacceptor level. In most cases EPR data are not available to establish the symmetry and absence of complexing, so the modeling must be taken as very tentative. The level predicted for Mn (0.36 eV) deserves special consideration since it differs significantly from the experimental value of 0.09 eV. This difference can perhaps be understood from the fact that Mn has a d shell which is exactly half full (3d5)and thus, like Zn (3d"), is stable with respect to transfer of a 3d electron to a higher energy crystal bonding orbital. Thus, Mn acts almost like a normal acceptor, except that its radius differs significantly more from that of a Ga atom than does the Zn radius. There has been speculation that a hole-trap level at E, 0.33 or 0.37 eV observed in VPE material may be related to Mn (Mitonneau et al., 1980); however, the weight of the experimental evidence supports E, + 0.09 eV as the first Mn

+

Ec ELEMENT

FIG.19. Calculated (0) versus experimental (+) energy levels in GaAs. (The Co experiment value should be 0.16 eV.) (After Partin er al., 1979; reproduced with permission of Pergamon Press, Ltd., UK.)


103

acceptor (Mn”). It may also be remarked that since the study was made, it has become almost certain that the CoGalevel is at 0.16 eV and not the value of 0.56 eV also shown. Whether or not the undulations in the experimental values of Fig. 19 are adequately explained by the 6R model, one thing that is apparent from the diagram is the definite downward trend of ET - E, as the transition elements are changed from Ti to Cu. The trends of energy levels associated with transition elements in GaAs have been considered by Il’in and Masterov (1977) using a Green’s function method within the framework of a semiempirical description of the impurity Hamiltonian. The downward trend of energy as one proceeds from Ti to Cu results from the study. The problem has also been considered by Jaros (1971, 1982) in a wave-function study in which screened long-range Coulomb potentials and short-range core potentials are taken into account. Understanding of chemical trends of deep energy levels in semiconductors has been greatly increased in a semiquantitative sense by the Hjalmarson theory (Hjalmarson et al., 1980), involving notions of hyperdeep and deep levels and related to neglecting the long-range potential of the impurity and simply inputting the band structure of the host, the positions of the atoms, and a table of atomic energies (Sankey et al., 1981). B. More on the Transition Elements

A few further comments on the behavior of transition elements from Ti to Ni in GaAs follow. Titanium in GaAs produces photoluminescence peaks at 0.525 and 1.3 eV (Kornilov et al., 1974). From Hall measurements Bekmuratov and Murygin (1973)assign a value E , - 0.45 eV to Ti as an acceptor. This is in general agreement with other work, such as Gutkin et al. (1972, 1974), from which the level is E, + 0.98 eV. Photoluminescence and photoconducting studies of V in GaAs suggest an energy of 0.8 eV for transfer of electrons from the valence band to the V3+ neutral state. There is also luminescence at 0.17 eV and absorption at 1.01.2 eV (probably involving intracenter transitions) (Vasil’ev et al. (1976). Martin (1980) identifies the EL19 level as the electron trap associated with the presence of vanadium. The ESR spectrum of the neutral acceptor state V3+(3d2)has been discussed by Kaufmann and Schneider (1980). Manganese in GaAs exhibits a characteristic photoluminescence peak at 1.405 eV, arising from the process of electron capture from the conduction band to a neutral Mn level, A’ + e - + A - + hv [or, in another nomenclature, Mn3+(3d4)+ eCB+ Mn2+(3d5)+ hv]. This photoluminescence is not an intracenter transition within the 3d shell as exists for most other transition elements in GaAs (and as exists for Mn in Gap). There is also an

104

A. G . MILNES

infrared absorption band starting at about 0.11 eV and peaking at about twice the energy, that is the photoionization complement, namely, Mn3+(3d4)+ eVB+ Mn2+(3d5),of the 1.405 eV luminescence. The curve which can be fitted by Lucovsky’s model is shown in Fig. 20. From such behavior, Mn has been found to be a common impurity in VPE-grown GaAs. Possibly it comes from the stainless-steel components of ), growth systems. The solubility of Mn in GaAs is large ( > 10’’ ~ m - ~and the diffusion coefficient is hn= 6.5 x lo-’ exp(-2.49/kT) cm2/sec according to Seltzer (1965).Manganese is sometimes implicated in the thermal conversion to p type of GaAs surfaces (Klein et al., 1980). Consider now Fe in GaAs. The solubility has a maximum active concentration of 1.3 x 1OI8cm-3 at 1100°Cand is strongly retrograde, as shown in Fig. 21a. The result of a typical diffusion at this temperature is shown in Fig. 21b. Boltaks et al. (1975) conclude that the Fe atoms diffuse through both interstices and the vacant sites; the interstitial atoms diffuse far more rapidly than the substitutional atoms. The interstitial atoms are transferred to the vacant sites, and vice versa; transition from interstices to lattice sites remains predominant until the Fe-atom concentration in the sites and interstices attains its equilibrium value at the given temperature. The diffusion coefficients of the bulk sector were determined by comparing the experimental curves with the theoretical curves of In N vs. x. The temperature dependence of the diffusion coefficient of Fe in GaAs is represented by the exponential function D = 4.2 x l o p 2 exp{(- 1.8 f

t

-I

-5

I

120

-

l

l

I

I

I

-

N

- 3 9

5

‘0 I

I-

z

w

2 I I . W

0

u

z

0 I-

n. E

sm a

FIG. 20. Infrared absorption associated with the ionization of the neutral manganese acceptor in GaAs. (After Chapman and Hutchinson, 1967; reproduced with permission of the American Institute of Physics.)

105


O.l)/(kT)} as shown in Fig. 21c. Other studies show the diffusion coefficient to be very dependent on the conditions of As overpressure (Uskov and Sorvina, 1974). The activation-energy level for the transition Fe3+(3d5) Fe2+(3d6) that takes an electron from the valence band (i.e., emits a hole) is about E, + 0.50 eV. The maximum resistivity that can be created by Fe in GaAs tends to be about lo5 Q-cm at room temperature, so Fe cannot be considered a useful dopant as an alternative to Cr in creating truly semi-insulating GaAs (Ganapol’skii et al., 1974). However, Fe has been found by one laboratory to be the predominant deep acceptor in LEC GaAs grown in either SiOz or PBN crucibles, the concentration being 6 x 1015 cm-3 (Wilson, 1981). Intracenter electron transitions in Fe give photoluminescence in the range 0.34-0.37 eV (Bykovskii et al., 1975, Fistal et al., 1974). Excited states of Fe in GaAs have been studied by Demidov (1977). --+

-

(00

-?1ot

200

300

X (pm)

I03/T (K-’)

(b)

(a)

1 (“C)

1160 1100 1000 900 830 I

1

-” 10-9

10-10

0.7

0.8

0.9

10 ’/ T ( K - ’ ) (C

1

FIG. 21. Behavior of Fe in GaAs: (a) solubility versus temperature; (b) diffusion profile of Fe in GaAs: T = 1 loPC, z = 24 x lo4 sec; (c) diffusion coefficient versus temperature for the bulk region. (After Boltaks et al., 1975.)

106

A. G . MILNES

In LPE-grown layers of GaAs doped with -10l6 Fe cm-3 (2.7 x lo3 R-cm), photoionization cross-section studies at 98 K (Kitahara et al., 1976) show a photoionization threshold Eiof 0.5 eV and a reasonable fit to the Lucovsky theory which gives the photoionization model as o(hv)

=

C(Ei)”2(hV - Ei)3’2/(hV)3

(22)

A lower threshold at 0.37 eV and a higher one around 0.80 eV also were present in this material. Lebedev et al. (1976) and other workers also have seen levels at around E, + 0.38 for Fe-doped GaAs. Whether this level is directly Fe related is uncertain. Other elements such as Cu and probably Ni cause a level to appear at or near this position in the band gap, and there is also, of course, the E, + 0.41 eV defect level seen in LPE material that is still due to unknown causes. The main energy level of Fe has been found to vary with temperature as EFe - E, = 0.52 - (4.5T/104).

(23)

This temperature dependence is somewhat similar to that of the band gap of GaAs (Haisty, 1965; Omel’yanovskii et al., 1970). There has been a report (Schlachetzki and Solov, 1975) that in LPE, Fe may give rise to n-doped conductivity; however, another study (Hasegawa et al., 1977)finds only the usual acceptor at E, 0.52 eV with a hole-capture cross-section of about cm2. Ikoma et al. (1981) find that the E, 0.51 eV hole trap appears in LPE GaAs only after electron-beam irradiation and annealing above 200°C in Fe-doped wafers. This suggests that Ga vacancies are necessary for Fe to form a deep state. For VPE-grown FETs it has been reported that the use of an Fe-doped lo5 R-cm buffer layer on the Cr-doped substrate has a beneficial effect on the active-channel-region mobility and therefore on the transconductance gain gm (Nakai et al., 1977). Iron doping of bipolar npn GaAs transistors has some beneficial effect apparently on the trapping actions in the base region that improves the frequency response (Strack, 1966). Cobalt in GaAs, according to Haisty and Cronin (1964), has an energy level at E, + 0.155 eV. Photoionization studies by Brown and Blakemore (1972)and Andrianov et al. (1977a) showed a steep rise in response at 0.16 eV and so confirmed the previous work. On the other hand, tunnel-junction spectroscopy studies of Fistul’ and Agaev (1966) showed some action at 0.54 eV. The E, 0.16 eV level is supported by photoluminescence seen at 1.30 eV (Ennen et al., 1980), which is considered to result from electron capture from the conduction band according to A0(3d6) ecB+ A-(3d7). Ennen et al. also observed optical absorption at 0.501 eV (the zero-phonon

+

+

+

+

107


line) and assign this to the intra-d-shell transition 4A2 + 4T2(F) of the A negatively charged acceptor state Co2+(3d7).Kornilov et al. (1974) observed photoluminescence at 1.29 and 0.49 eV. For diffusion of Co at 1000°C at atmospheric pressure of As, the surface concentration is about 10" cm-3 and the penetration depth is about 20 pm after 24 hr. The effective diffusion coefficient for No = N, erfc[~/2(DT)"~] is D 2 x lo-" cm2/sec for the above conditions. The effects of 6oCo irradiation have been studied by Share (1975). Consider now the energy level(s) created by nickel in gallium arsenide. Nickel is likely to occur in VPE-grown GaAs as a result of contamination from the growth system when HCl is used. Nickel is also a component of a standard ohmic-contact recipe (Au-Ge, Ni) for n-type GaAs. Nickel is a medium-fast diffusing species in materials with Ga vacancies, and so it may enter the crystal during contact formation. This is of significance because Ni can have an adverse effect on minority-carrier hole lifetime in II GaAs. At various times, energy levels at 0.08, 0.15, 0.21, 0.35, 0.42, and 0.53 eV have been attributed to Ni in GaAs (Damestani and Forbes, 1981). Neutral nickel on a gallium site can be represented by Ni3+(3d7)and after accepting an electron by Ni2+(3d8)(Andrianov et al., 1977a; Murygin and Rubin, 1970; Fistul' and Agaev 1966). Certain workers favor E, + 0.20 eV as the energy level for the Ni(3d7) neutral acceptor (Bimberg et al., 1981). Other groups believe that nickel substituting on a gallium site has an acceptor energy E, + 0.42 eV. For instance, optical absorption in GaAs:Ni has been studied by Bazhenev et al. (1974). The absorption spectra exhibited maxima at 0.84 and 1.14 eV and the absorption coefficient c( in the 0.8 eV region was proportional to the amount of Ni in the crystal (determined chemically). This led Bazhenev et al. (1974) to attribute this to the acceptor photoionizing from the valence band and, following the Lucovsky model, the value of 0.42 eV (0.84/2) is assigned for the threshold process, Ni3+ eVB+ Ni2+, with an activation energy E, + 0.42 eV. This is in agreement with the Hall measurements of Matveenko et al. (1969). On the other hand, from attempts to dope LPE GaAs with Ni in the Ga melt, Kumar and Ledebo (1981) conclude that the E, + 0.40 eV level observed is caused by trace Cu contamination and that no level specific to Ni is seen. Optical absorption attributed to the process A' + A- plus hole photoionization, with a Hall activation energy of 0.35 eV has been observed by Suchkova et al. (1975, 1977). Suchkova and co-workers find an absorption band at 1.15 eV which they attribute to electron transition within the d8 shell of the compensated Ni2+ ions. Ennen et al. (1980) find a strong zero-phonon line at 0.572 eV in GaAs: Te: Ni, which they tentatively assign to an intra-d-shell transition of the two-electron-trap state A2- of Ni+(3d9).Kaufmann and Ennen (1981) N

+

108

A. G . MILNES

also report sharply structural bands of absorption and emission in the 2-pm region in nickel-diffused n-GaAs that they assign to intra-d-shell transitions of Ni acceptor-shallow donor near-neighbor associates of Ni-S, Ni-Se, Ni-Te, Ni-Si, Ni-Ge, and Ni-Sn. Deep-level transient-spectroscopy studies of VPE GaAs :Ni (Partin et al., 1979a) show an energy level acting as a hole trap at E, + 0.39 eV produced by Ni diffusion. The uncorrected energy value was 0.48 eV, but the temperature dependence of cross section leads to the E(T = 0) value of 0.39 eV. An electron trap was also found at E, + 0.39 eV after Ni diffusion, and the concentration was about that of the E, + 0.39 eV level. The physiochemical origin is unknown. No evidence was found for a level at E, + 0.2 eV either before or after the Ni diffusion. For the E, + 0.39 eV level, the holecapture cross section was about cmz at 207 K. The electron-capture cross section, however, was small, about 5 x lo-" cm2, and so the center at E, + 0.39 eV was not an important hole-recombination site. (These capture cross sections agree with those that Ledebo attributes to Cu.) On the other hand, the E, - 0.39 eV level appears to act as a strong minority- (hole) carrier recombination center, as discussed in Section XIII. VII. GROUPI IMPURITIES: Li, Cu, Ag, Au A. Lithium

Although studies of Li in GaAs have been informative in the past, there appears to have been little work with Li during the last decade (Grimm, 1972; Norris and Narayanan, 1977). The following summarizes briefly the understanding of Li in GaAs reached in earlier work. The solubility of lithium at 800°C is 1.6 x l O I 9 cm-3 in undoped GaAs and about 2.3 x 1019 cm-3 in GaAs doped heavily with Te or Zn. Lithium diffuses rapidly into GaAs with an interstitial diffusion coefficient D = 0.53 exp( - l.O/kT), where kT is in electron volts (Fuller and Wolfstirn, 1962). When nominally pure crystals of GaAs are saturated with lithium by diffusion at temperatures greater than 500°C and cooled to room temperature, they are iompensated to a high resistivity. The compensation phenomenon is caused by the action of Li' which permeates the crystal and tends to form donor acceptor complexes. The presence of the Li' interstitials may be expected to increase the gallium vacancy concentration by orders of magnitude. The possible reactions that follow result in the species Li'V-, Li-, Liz-, Li'Li2-, and (Li')zLi2-, where Li' is interstitial and Li- and Liz- are substituted ions. On cooling to room temperature after a high-temperature lithium diffusion,

IMPURITY LEVELS I N GALLIUM ARSENIDE

109

excess Li' readily precipitates because of its high mobility. This leaves relatively immobile excess acceptor complexes which give rise to the observed p-type compensation. The activation energy observed is 0.023 eV from the valence-band edge. Lithium is found to compensate n-type GaAs that has tellurium concentrations as high as 5 x lo'* ~ m - ~ . Chemical interaction among defects can result in ion-pair formation in semiconductors. The evidence for pairing is mostly indirect; however, when ion pairs have an IR-active local vibrational mode, IR spectroscopy provides a sensitive tool for studying the pairing reactions and pair structure. Pairing is recognized by the reduction of site symmetry and the consequent lifting of degeneracies of vibrational modes. Local mode frequencies of unpaired impurities must be known in order to recognize site symmetry reduction. If more than one stable isotope is available for one of the impurity species, the change in the absorption-band frequency due to changing the isotope can be used to study impurities which are heavier than host lattice atoms, provided they are paired with a light atom. When the behavior of electrically active impurities is studied in GaAs by local-mode IR spectroscopy, the free-carrier concentration introduced by the impurities must be reduced by compensation with another electrically active defect. This reduces the nonlocalized modes of vibration contributions to the absorption. The behavior of Li has made it a good choice for the compensating impurity: it is a rapid diffuser, and two stable isotopes are readily available. Usually, it is the local mode of the Li paired with the original dopant which is experimentally observed. The bandwidth range of interest is typically 300-460 cm- In GaAs, the pair systems studied include Te-Li, Mg-Li, Cd-Li, Zn-Li, Mn-Li, and Si-Li (Lorimor and Spitzer, 1966; Allred et al., 1968; Spitzer and Allred, 1968; Leung et al., 1972). In addition to the local modes observed in GaAs doped during growth, absorption bands have also been reported in pure GaAs into which Li has been diffused; the corresponding absorption centers are believed to be Li complexed with native defects. Other impurities in the 1A column such as Na and K have apparently received no further study since the tentative report of Na as an acceptor (Hilsum and Rose-Innes, 1961). Potassium however has been found to be very adverse as a surface-eroding contaminant in early GaAs devices (Leedy et al., 1972).

'.

B. Copper

Copper is a fast interstitial diffuser in GaAs and converts to a substitutional site CuGa that is capable of acting as a double acceptor. The cm2 sec-' at 500°C and the interstitial diffusion coefficient D' is 1.0 x

110

A. G . MILNES

diffusion activation energy is 0.53 eV. The solubility of copper in GaAs depends on the other impurities present, but may be in excess of lo'* cm-3. Surface and bulk diffusion has been studied by Boltaks et a!. (1971) who find surface diffusion coefficients of the A and B faces to be unequal and therefore the bulk impurity profiles near these faces to be different. See also Hasegawa (1974). Deep-level determinations up to 1970 suggest that the main energy levels associated with copper in GaAs are 0.14-0.15 eV above E,, presumably the first acceptor level (Cu-), and 0.44 eV, presumably the second acceptor level Cuz-. A level seen at E, + 0.24 eV was thought to be related to a pair of copper atoms. A level at E, + 0.19 eV was attributed to a Cu-Te complex, and miscellaneous levels at other depths were thought to be related in some way to Cu. The photoluminescent studies showed a broad band at 1.35 eV, which was tentatively ascribed to VAsCuGa pairs or DAsCuGa pairs where VA, and DA, denote, respectively, an arsenic vacancy and a donor impurity occupying an arsenic site. Since 1970, there have been numerous investigations of the behavior of Cu in GaAs that have added to previous knowledge. Norris (1979) finds that in melt-grown GaAs: Cu, a broad cathodoluminescent band centered at 1.36 eV shows injection-level and temperature characteristics that are consistent with conduction-band-acceptor transitions. Norris (1979) comments that if we follow the suggestion that the peak at 1.36 eV represents a nophonon transition, then thermal quenching of the 1.36 eV band should proceed with an activation energy of approximately 0.14 eV since the band gap of GaAs is close to 1 S O eV at 80 K. Indeed, transport measurements have shown that Cu in GaAs introduces an acceptor level with a depth of 0.140.15 eV. More importantly, a previous study of the temperature dependence of the 1.36 eV luminescence in GaAs:Cu showed quenching with an activation energy of 0.147 eV, thus making it plausible that the 1.36 eV transition terminates on the same acceptor levels identified in the transport measurements.

Norris states, however, that frequency-response measurements conflict with this attribution and indicate that the kinetics of the 1.36eV luminescence are distinctly nonexponential. In heavily Cu-contaminated material the kinetics of the edge emission and the 1.36 eV band are remarkably similar. Nevertheless, variation in injection level causes a strong variation in relative prominence between the 1.36 eV band and the edge emission; this injectionlevel dependence is not related to the kinetics of the two bands. Norris therefore suggests that the 1.36 eV transition terminates on the lower configuration-coordinate branch of a compact Cu-related complex with strong coupling to the lattice and that the upper branch of the same complex is degenerate with the conduction-band continuum and in slow exchange with the same.


111

Copper in LPE GaAs has been studied by Chiao et al. (1978). From photoconductivity and other measurements they state that if the Cu impurity is diffused into an n-type LPE GaAs layer after growth, it behaves as a conventional acceptor and converts the layer to p-type conductivity; if the Cu impurity is present in the melt during the growth of an LPE GaAs layer in a C-H,-SiO, system, an increase in electron density by a factor of 10 is observed compared to an undoped layer grown under the same conditions.

Hypothetical mechanisms have been suggested by which the Cu impurity might increase the final electron density in n-type layers, but an unambiguous description is not presently available. The Cu impurity may form a complex with a lower oxidation state in such a way as to make more donor impurities available: Cu might react with 0 to produce Cu,O,, thus removing 0 from SiO, in the melt and freeing Si to act as a donor in the GaAs; Cu might enter interstitial sites, acting as a shallow donor; or Cu and 0 may form a neutral complex that serves, in turn, as a reaction center for the remaining excess 0 to form donors. Copper might also react with lattice defects, the Cu serving as a reaction center for the formation of electrically active defects. Chiao et al. (1978) continue as follows: One particular impurity level is found in both Cu-diffused p-type GaAs layers and in n-type GaAs layers grown with Cu present in the melt. This level is associated with an impurity center that acts as a sensitizing center for n-type photoconductivity in the Cu-diffused layers. Its energy level lies 0.43 eV above the valence band, its hole-capture cross section is about cm’, and its electron-capture cross section is about lo-’’ cm’. Its presence in the Cu-diffused layers is associated with the acceptor action of the Cu impurity that dominates under these conditions; its presence in the layers grown with Cu in the melt indicates that some of the Cu still occupies acceptor-like sites, but that the major effect of the presence of Cu overwhelms this behavior. This 0.43 eV level associated with Cu appears identical in every way to the corresponding level reported for Cu in bulk-grown GaAs crystals.

from The distribution coefficient of Cu in these LPE studies was emission spectroscopy but 10 - 6 from photocapacitance effects. These numbers can be compared with the previously reported for meltgrown GaAs where the temperature is much higher than the 700-750°C for LPE growth. The considerably smaller values of distribution coefficient given by capacitance technique indicate that the active Cu for these measurements is only about 1% of the total. Possible reasons for this difference are (1) photocapacitance gives only a lower limit of the Cu concentration, since it assumes that all Cu centers have been emptied under the steady-state photoexcitation; (2) Cu may diffuse away from the junction region in the electric field after junction formation; (3) a major proportion of the Cu present may be involved in nonelectrically active complexes.

112

A. G. MILNES

In these studies, levels were seen also at E, + 0.20,0.65, and 0.70 eV, but these were not clearly copper related, and there was no sign of the E, + 0.14 eV level observed in other work (for instance, Ashirov et al., 1978). Boborykina et al. (1978) in photocapacitance studies at 77 K use a configuration-coordinate diagram approach and find the impurity energy level to be E, + 0.37 f 0.02 eV with Stokes losses E, = 0.10 k 0.01 eV, and a phonon energy of 2.5 k 3 meV. The luminescence band is a maximum at 1.035 eV. Kolesov et al. (1975) believe the 1.24-1.26 eV luminescence in GaAs :Cu to be related to donor-acceptor pairs which they suggest are Cui-Oi complexes. Dzhafarov (1971) suggests that since the vacancies in gallium arsenide are charged (V& and VLJ it follows that positive copper ions migrating along the interstices should precipitate at negatively charged gallium vacancies V,, . This is also favored by the fact that the radius of Cu (1.35 A) is closer to the covalent radius of G a (1.58 A) than to the radius of As (0.96 A). Copper which is precipitated at gallium vacancies exhibits acceptor properties. If a copper ion becomes lodged alongside a positively charged impurity (or vacancy), it may form a complex with this impurity or vacancy because of the Coulomb attraction. In the surface region of gallium arsenide, where the nonequilibrium concentration of the arsenic vacancies is high, copper may form complexes mainly with one or several arsenic vacancies. In the bulk of gallium arsenide, where the equilibrium concentration of the vacancies is low, the formation of complexes between the copper and the original impurity is more likely. However, coppervacancy complexes may also form in the bulk region. Since a 0.18 eV level is observed only in the case of the simultaneous presence of Cu and Te, we can attribute this level to Cu-Te complexes.

Dzhafarov (1971) continues by stating that investigations of the spectra of bound excitons in samples of GaAs doped with copper by diffusion have established the symmetry of impurity centers in the surface region. These centers have been identified as complexes of copper and two arsenic vacancies {VAeCuGaVAs}-. The acceptor level at 0.5 eV found in the surface region in our crystals, may be attributed to the same {VAsCuGaVAS}centers. A copper ion bound to two arsenic vacancies in the surface region should have a low mobility and this is confirmed by our study of electrotransport. The jump of a copper ion, bound into a complex, from one equilibrium position to another requires the breakup of the bond between the copper ion and the arsenic vacancies. This means that the activation energy should increase by an amount equal to the binding energy of such complexes. In the bulk of a crystal, where the equilibrium con) centration of vacancies is low and the concentration of copper (8 x 10” ~ m - is~ higher ) , copper atoms are not bound than the concentration of tellurium (- 1 x 10’’ ~ m - ~many and they may diffuse freely in gallium arsenide. Thus, the experimentally observed rapid fall of the copper concentration near the surface of gallium arsenide is the result of the formation of complexes in the vacancy-rich region and the presence of these complexes reduces strongly the effective diffusion coefficient of copper.

Zakharova et al. (1 972) have examined the effects of Cu diffused into n-type GaAs: Si at 900°C. Acceptors are formed having ionization energies of about


113

0.1 or 0.14 eV. Slow cooling and diffusion at an elevated arsenic pressure favored the formation of acceptors having an energy 0.1 eV, whereas quenching and diffusion under an equilibrium arsenic pressure at the same temperatures led to the appearance of the level 0.14 eV. The 0.1 eV level is attributed to Cui and the 0.14 eV level to Cu;,. However, Willmann et al. (1971) conclude from the symmetry of the IR excitation spectrum of the E, + 0.15 eV level that this acceptor level cannot be a simple substitutional Cu ion on a Ga site. The inference is that Cu induces the centers in conjunction with lattice defects, but the actual physical nature is unknown although the symmetry has been established (Willmann et al., 1973). Kamar and Ledebo (198 1) have shown that Cu in LPE GaAs produces an acceptor at E, + 0.4 eV. The behavior of the “Cu” level (HL4) at E, + 0.40 eV has been examined with deep-level optical spectroscopy (DLOS) by Chantre et al. (1981). This level has optical and thermal activation energies that are identical, whereas many other traps typically show shifts of 0.1-0.2 eV because of lattice relaxations (Bltte and Willmann, 1971). The discussion of Cu in GaAs is concluded with a few further short observations. One is that the Ga-As-Cu phase diagram has been constructed by Panish (1967).The second is that Cu contamination enhances the degradation of GaAs electroluminescence diodes (Bahraman and Oldham, 1972). Indeed, Cu can move 30 pm or more in a few hours in illuminated pn junctions (Boltaks et al., 1972). Copper in high concentrations precipitates in GaAs (Morgulis et al., 1973) and may also be involved in the generation of interstitial dislocation loops in GaAs:Te annealed above 380°C (Hutchinson and Dobson, 1975).

C . Silver Solubility and bulk and surface diffusion studies of silver in GaAs have been made. The diffusion studies suggest that the bulk solubility of Ag in the temperature range 500-1 160°Cis 2-8 x 1017cm-3 (Boltaks and Shishiyanu, 1964). In other work, 3 x 10l6 cmP3 has been reported for the solubility at 800°C. The distribution coefficient is low, less than 4 x Distribution coefficient measurements for impurities in GaAs have been summarized by Willardson and Allred (1966). The surface solubility of the Ag is in the range ~ . diffusion is interstitial and the line marked 7 x 10’’ to 4 x 10’’ ~ m - The Ag in Fig. 22a roughly represents the behavior in bulk melt-grown (Bridgman) GaAs. This illustration, taken from Kendall’s (1968) review of diffusion in 111-V compounds, is helpful for visualizing the overall diffusion pattern for GaAs. See also Fig. 22b, from a review by Crawford and Slifkin (1975) that contains some newer information. The original literature should be consulted where possible because diffusion profiles tend to be complex in shape

114

A. G. MILNES

and dependent on the As overpressure. Self diffusion in GaAs has been studied by Palfrey et al. (1982). Phosphorus diffusion has been studied by Jain, Sadana, and Das (1976). From photoluminescence studies, Blatte et al. (1970) report the acceptor level of Ag to be E, 0.238 eV (4 K) for diffusion at 950°C into bulk n-type GaAs to a concentration of 1017 cm-3 in a sealed ampule with As added. This Ag acceptor at E, 0.235 & 004 eV has been confirmed by Hall measurements. In addition, a shallower acceptor level at E, 0.107 eV is seen with a concentration about two orders of magnitude less than the Ag level at 0.235 eV. The concentration NA of the 0.107 eV energy level is dependent on the diffusion time; a longer diffusion time results in an increase of the relative concentration of this level in the range 1-3 x l O I 5 ~ m - This ~ . level has previously been reported after Cu treatments of GaAs so it is not certain to be Ag related. From DLTS studies the E, 0.23 eV level for Ag diffused into LECcm2 (Yan and grown GaAs has a hole-capture cross section of about Milnes, 1982). The Ga-As-Ag and Ga-As-Au ternary-phase diagrams have been obtained by Panish (1967). The equilibrium phase diagram for the binary gold-gallium system has been established by Cooke and Hume-Rothery (1966). In general, if Ag is a component of a GaAs ohmic contact technology

+

+

-

+

+

I

IOOO/T ( K ) 0.60

0.70

'

I

I

I

I

IOOO/T(K) 0.60

0.70

0.90

I

I

I

I

0.90

1 2 c i -

0

loe

AQ -10

10

Au Be Tm Mn

z -12 0 10 v) IA. 3

Sn S

k I614 a

-161 I

10

1200

I

I

1100

1000

900

-16

10

I

1200

As Se

I

1100

1000

900

FIG.22. Diffusion coefficients in GaAs at the low concentration limit: (a) after Kendall (1968); (b) after Crawford and Slifkin (1975).


115

such as (Ag-Ge:Ni), the alloying temperature needed is more than 100°C higher than for the corresponding (Au-Ge :Ni) technology. The Ag-Ge eutectic temperature is 650"C, and the Au-Ge eutectic temperature is about 350°C.

D. Gold The solubility of Au in GaAs varies from 2.5 x 10l6 to 1.6 x 10" cm-3 for the temperature range 900-1140°C. A diffusion at 1000°C for 4 hr produced an acceptor concentration of 4.5 x lOI7 cm-3 in bulk-grown ~ . measurements over a GaAs that was initially n-type 1.1 x loi5 ~ m - Hall range 150-300 K can be fitted by an energy level of E, + 0.405 f 0.002 eV. A shallow acceptor state at E, + 0.05 eV also appeared with a concentration of loi5 ~ m - Hiesinger ~ . (1976) comments that this is consistent with photoluminescence measurements of Au-diffused bulk GaAs, but the photoluminescence of Au-doped layers grown from a saturated Ga solution did not reveal the shallow level. Therefore, the shallow acceptor may be a center formed by Au and Ga vacancies introduced during diffusion. In other studies an acceptor at 0.04 eV detected by photoconductivity measurements in n-type GaAs may be formed as a result of an interaction between gold and defects associated with the presence of copper. Hiesinger found no evidence of the earlier published level of 0.09 eV for Au-doped GaAs. From DLTS studies the E, + 0.40 eV trap in Au-diffused LEC-grown GaAs has a hole-capture cross section of about cm2 (Yan and Milnes, 1982). This is significantly smaller than the cm2 cross section for a Cu-related hole trap seen at about E, + 0.40 eV when Cu is diffused into GaAs (Kumar and Ledebo, 1981). A complex of Au-Ge is found in degenerate germanium-doped n-type layers of GaAs grown from a gold-rich melt and produces an energy level at E, + 0.16 eV (Andrews and Holonyak, 1972). The diffusion of Au in GaAs shows a higher concentration surface branch and a lower concentration deep branch (Sokolov and Shilshiyanu, 1964).The deep branch can be represented approximately by an erfc function. The deep diffusion coefficient varies from 6 x lo-' to 6 x cm2 sec-' in the temperature range 740-1025°C. The alloying of Au on the surface of GaAs at 550°C causes a heavily disordered near-surface region. Outward diffusion of As may occur at temperatures as low as 350°C and at higher temperatures, there is evidence of considerable inward diffusion of Au and outward diffusion of Ga and As. Transmission electron microscope studies have shown the presence of hexagonal AuGa (Magee and Peng, 1975). Other studies have shown the formation of Au7Ga2. Schottky junctions of Au on n-GaAs deteriorate in barrier height after heating to 350°C for 30 min (Kim et al., 1974). Gettering of Au from GaAs

116

A. G . MILNES

can be achieved to some extent by back-surface damage followed by an anneal at 800°C (Magee et al., 1979). VIII. SHALLOW ACCEPTORS: Be, Mg, Zn, Cd The commonly used acceptors in GaAs are Be, Mg, Zn, and Cd: Be and Mg are used for implant purposes; Zn and Cd tend to diffuse readily (Kadhim and Tuck, 1972); and in LPE, Be or even Ge (an amphoteric dopant) may be preferred for the fabrication of multilayer structures. Beryllium is the acceptor of choice in MBE because its sticking coefficient is much higher than that of Zn or Cd. Implantation of Be at 100 keV in GaAs followed by activation at 550°C produces good p-type doping results (Anderson and Dunlap 1979). Other studies of Be implants in GaAs have been made by Bishop et al. (1977), Kwun et al. (1979), Lee (1980), McLevige et al. (1978), and Nojima and Kawasaki (1978). Magnesium and cadmium have been studied by Aoki et al. (1976), Yu and Park (1977), and Williamson et al. (1979), who observed surface pileup and outward-diffusion effects. Laser annealing of implanted acceptor (Zn, Cd) and donor (Se, Sn) ions in GaAs has been studied by Badawi et al. (1979) and other workers. For Cd solubility see Fujimoto (1970). A systematic study has been carried out by Ashen et al. (1975) of the incorporation of shallow-acceptor dopants in highly refined VPE and LPE gallium arsenide. A detailed study of the bound exciton, free-to-bound, and pair luminescence of doped layers resulted in an identification catalog for the various elements. Many specimens were specially prepared for the study and several growth methods were used. The layers intentionally doped with Zn and Cd were grown by VPE; Be- and Mg-doped layers were obtained by LPE. Magnesium has a fairly large segregation coefficient (-0.2) and is relatively easy to use for lo'* cm-, doping, but in more lightly doped melts the loss by oxidation presents a control problem. The Be distribution coefficient is greater than 1.O, but small additions of Be are affected by oxidation or reaction with the C of the LPE growth boat. Attempts to add Ca and Sr in LPE growths showed no new photoluminescence lines. Carbon doping in VPE growths (AsCl, :H, :Ga) was achieved with CC1, addition. Ashen and his co-workers (1975) comment that Silicon-doped VPE layers were grown with the Si dopant situated in the reactor immediately downstream from the source. None of the layers which were deposited under the standard growth conditions, showed any evidence of Si acceptors, only additional donors. Silicon is a ubiquitous contaminant in both VPE and LPE layers. An appreciable portion invariably occurs as an acceptor in LPE growth as it does with all deliberately Si-doped LPE layers. In the exceptional VPE layer grown in the presence of considerable excess of CCI,, Si acceptors were identified, together with C acceptors and residual Zn acceptors.

117

IMPURlTY LEVELS IN GALLIUM ARSENIDE

It is believed that in this case the excessive C deposit which built up on the walls of the reactor during the run was responsible for the reduction of significant amounts of silica and that these non-standard growth conditions were favourable for the introduction of the acceptors.

The C acceptor was almost always found in LPE layers and Ge contamination was frequently found in undoped LPE layers. High-purity VPE layers show a line at 1.4889 eV that Ashen et al. (1975) tentatively assign to O, . The results for specimens obtained from many sources are well illustrated in Fig. 23. The values assigned to the binding energies of C , Si, Ge, Be, Mg, Zn, and Cd are 26.0, 34.5, 40.4, 28.0, 28.4, 30.7, and 34.7 meV, respectively.

L L L L V V V V L L L 1.495

L V V L L L V

--

C-

V V V V V V V L V V \

--

-

--

-

-

1.490

Zn-

eV sil*

1.480

Ge

-

1.475

FIG.23. Energies of photoluminescence bands in GaAs below 15 K produced from a variety of sources. The spectra have been shifted by an energy which brings the shallow exciton luminescence into coincidence. The arrows marking the positions of the elements on the energy axis represent the free-to-bound transition for a conductive electron captured at a neutral acceptor site. The dotted levels represent emission bands observed between 10 and 15 K-some tend to become obscured at very low temperatures due to the growth of the pair luminescence. (After Ashen et al., 1975; reproduced by permission of Her Britannic Majesty's Stationery Office.)

118

A . G. MILNES

In other work, there have been reports of deep levels apparently associated with Zn and Cd doping. A deep level, E, 0.65 eV, possibly associated with Zn doping, has been observed by Su et al. (1971) in LPE-grown GaAs. A deep acceptor level at about 0.36 eV above the valence band has been observed with Cd doping and is attributed to lattice defects or to Cd lattice-defect complexes (Huth, 1970). However, systematic confirmation of such levels is needed before general acceptance since trace contamination (for instance, of Cu) must always be considered as being possibly present. Kanz (according to Kendall, 1968) in a diffusion of Hg203 at 1000°C found low surface concentration (5 x 10” cm-3) and low diffusion coefficient (5 x cm2 sec-I). Mercury, therefore, does not appear to be an interesting dopant in GaAs.

+

IX. GROUPIV ELEMENTS AS DOPANTS: C, Si, Ge, Sn, Pb For future device applications, large-area GaAs slices are needed, and as discussed in Section V these must be semi-insulating yet relatively free of Cr. Since the Cr is added to compensate Si donors, there is need to reduce the Si content of the crystals. Acceptor impurities such as C are also undesirable because of possible effects of ionized impurity scattering on electron mobility. There therefore is a trend toward LEC growth (that is, GaAs pulled from a melt encapsulated with boric oxide) from crucibles of BN rather than S i 0 2 or C. The result is that the GaAs contains boron. Laithwaite et al. (1977) comment as follows: After samples are rendered transparent, either by electron irradiation or by copper diffusion, the optical absorption due to localized modes of vibration (LVM)indicates that a large fraction of the boron atoms occupy gallium lattice sites; this is not surprising as boron and gallium are both group 111 elements. However, other boron atoms are located in another site which we have labelled B(2). Because of the existence of the two naturally occurring isotopes loB and “B, it was possible to show that a B(2) centre involves only one boron atom. These centres may be remote from other defects, but considerable pairing with Si,,Se,,, or Te,, donors also occurs; similar complexes were observed in gallium phosphide (Morrison et a/., 1974). It was deduced that an isolated B(2) centre had Td symmetry and that the paired boron in the B(2) complexes was surrounded by four nearestneighbor gallium lattice atoms, apart maybe from one such atom in the [B(Z)-Si,,] pairs. There would appear to be only two possibilities: (i) the boron atom occupies a substitutional arsenic site; or (ii) the boron is in the tetrahedral interstitial site with gallium neighbows. Because of the strong pairing with the known donor species it was argued that a B(2) centre should be an acceptor.

Laithwaite et al. (1977) conclude that the B2 level is an acceptor but the structure of the defect remains uncertain: they incline to the view that it is possibly an interstitial boron atom rather than the antisite defect BAS.

119


Undoped GaAs prepared under the purest possible conditions tends to have many residual impurities in concentrations of the order of 5 x l O I 3 cm-j. Wolfe et al. (1977) and Stillman et al. (1976) have shown that it is possible, in favorable circumstances, to determine the chemical nature of shallow donors in concentrations somewhat higher than this from highresolution far-IR photoconductivity measurements of transitions between the ground state and the first excited state. The shallow donor levels are very nearly at the energy level 5.737 meV predicted from the hydrogenic model, but central cell effects can cause small chemical shifts (increases) of the ground-state energies. The shifts and observed line shapes are magnetic field dependent and this helps in the resolution. Table XI1 shows the central cell shifts (above 5.737 meV) observed when known dopants are added and some observations of levels that can then be interpreted for residual donor levels seen in VPE and LPE specimens given similar examination. The ground-state values arrived at by Wolfe and co-workers for Se, Si, S, and Ge are 5.854, 5.808, 5.890, and 5.908 meV, respectively. Some idea of the precision, and difficulty, of the method may be obtained from the similar experiments of Ozeki et al. (1979) that yield values 5.812, 5.799, 5.845, and 5.949 meV, respectively. Further work will no doubt give closer agreement, but the use of the technique is limited to dopings less than 5 x 1014 cm-3 because of spectral broadening. The work of Wolfe et al. (1977) in preparing the lightly doped specimens led to estimates of segregation coefficients for their growth processes. Their numbers and some other recent determinations are added to Table XIII, which is otherwise the collection compiled by Willardson and Allred (1966). TABLE XI1 CENTRAL CELLCORRECTIONS OF GROUND-STATE ENERGY ABOVE HYDROGENIC LEVELOF 5.737 meV FOR KNOWNDONORLEVELS AND TENTATIVE IDENTIFICATIONS OF RESIDUAL DONORS"

THE

Residuals observed (meV) Known donors (mev) VPE

LPE

Pb

0.036 -

0.064

0.041

-

Se Sn Si

0.071

-

-

0.080

-

0.079 0.117

0.117 0.117

-

Residuals observed (meV) Known donors (meV) VPE S Te(?) Ge C

0.153 0.155 0.171 0.200 0.200

Reproduced with permission of the Institute of Physics, UK.

LPE -

120

A. G. MILNES

TABLE XI11 DISTRIBUTION COEFFICIENTS OF IMPURITIES IN GALLIUM ARSENIDE~

Doping element” Al Ag Bi Be Ca C Cr co cu Ge In Fe Pb Mg Mn Ni P Pb Sb Se Si S

Te Sn Zn

Haisty and Cronin (1964)

Edmond (1959)

Weissberg (1961a,b)

Whelan et al. (1960)

3 0.1

Willardson and Allred (1959)

Wolfe et al. (1977)

0.2 4 x 10-3 5 x 10-3 3 2 x 10-3

< 0.02 0.8

6.4 x 10-4 8.0 x 10-5 0.03 0.1

2 x 10-3 0.018 3 x 10-3

2.0 x 10-3 0.047 0.021 6.0 x 10-4

< 0.02 0.03

5.7 x 10-4 4 x 10-4 < 2 x 10-3 0.01 7 x 10-3 1 x 10-3 < I x 10-5

1-2 x

0.1

> W , the two-hole density of states departs considerably from a one-hole density self-convolution. The band is split into two parts separated by a gap of width U . The low-energy part contains N,,(N,, - 1) states (Nalis the number of sites). Its width is approximately 2 W . It essentially corresponds to states having two holes at different sites. The high-energy band is very narrow, its width being b W 2 / U ,and resembles an atomic band. It contains N,, states, most of them corresponding to cases where the two holes are at the same site (although they are moving in the solid). However, as indicated above, Auger emission is a local mechanism. Thus, the two-hole densities of states that we have,to consider to describe the transition are principally local densities where the two holes are at the same site. Under these conditions, the Auger CVV spectrum presents a narrow quasi-atomic peak and a large weak band on its low-energy side. The ratio R of the integrated intensities for the peak and for the other part is approximately

-

R

=

( U / W )- 1

(97)

It is thus possible to obtain an estimation of the correlation energy from the Auger spectrum: U

rr

b ( R + 1)

(98)

209

QUANTITATIVE AUGER ELECTRON SPECTROSCOPY

Sc Ti

V

Cr Mn Fe Co Ni Cu Zn

Sc

(a1

Ti

V

Cr Mn Fe Co Ni

Cu Zn

(b)

FIG.11. (a) Two-hole effective interaction energy in the final state of the Auger transitions L3-M4,5M4,5(U), M2,3-M4,5M4,5( O ) ,M1-M4,5M4,5(*) of the first series of the transition metals. (b) Comparison between the two-hole effective interaction energy in the final state of the Auger transition L3-M4,5M4,5,the width of this Auger line, and the width of the 3d band for the first series of the transition metals: ( 0 )2r (M4,5);(*) r (L3-M4,5M4,5);(U) Ci(M4,5M4,5). From Jardin (1981). courtesy of the author.

For copper (Madden et al., 1978), the L,,,VV lines and their satellite structures bear a rather good resemblance to those calculated with a purely atomic approximation (McGuire, 1978) but with the contribution of CosterKronig L, L3V and L,L,V transitions, which reinforce the L,-hole population (in a Coster-Kronig transition, one of the final-state holes is in the same shell as the initial-state hole). M,VV and M,VV lines have a narrow structure, in good agreement with the line-shape calculations made with an atomic model by McGuire (1 977). The theoretical approaches of Cini and of Sawatsky have been generalized by Treglia et al. (1981) to the case of unfilled d-band metals. Starting from the two-hole Green's function, correlation effects between the hole in the final state of the Auger transition and the holes of the ground state are accounted for by replacing the usual one-electron density of states n(E)by a one-particle spectrum C(E) defined in terms of a local self-energy Z(E), evaluated for an arbitrary number of holes in the band. The two-particle spectra are, in fact, strongly sensitive to the hole population of the d band. The values of .the effective interaction energies between the two holes have been measured by Jardin (1981) for the first series of transition metals and for L3cM2,?.M4,5 L3cM4,5M4,5 M1-M4,5M4,5 , and M2,3cM4,5M4,5 Auger transitions. The results for the three transitions involving the M4,5M4,5 final configuration are shown in Fig. 1 la. The effective interaction energy for the L3-M2,3M4,5Auger transition has a slightly smaller value but presents a quite similar variation with 2. The effective interaction energy U of the L3-M4,5M4,5 Auger line is compared with 2r (where r is the M4,5d-band width) in Fig. 1 Ib. As can be 3

9

210

M . CAILLER, J . P. GANACHAUD, AND D . ROPTIN

seen from this figure, U exceeds 2r as soon as Z > 28. Consequently, the Auger line must be of the bandlike type and have a width 2r for Z < 28. For higher Z values, it must be of the quasi-atomic type. This conclusion is confirmed by analysis of the shapes and widths of the LVV and MVV Auger lines (Jardin, 1981). The M,VV spectrum has, on its high-energy side, a broad shoulder which does not appear in L,VV and M,,VV lines. This shoulder has been initially explained by Madden et al. (1978) as being the broad structure predicted by Sawatzky (1977). Particularly because theoretical predictions lead to an intensity too weak for this structure and because it does not appear in L,,VV and M,,VV lines, which show the same final states as the M,VV line, this interpretation has been questioned by Jennison (1978~)and described by a convolution of the partial s and d densities of states in the valence band. The ratio of the intensities coming from the atomic d-d part (sharp structure in the M,VV spectrum) and from the s-d part (large shoulder on the highenergy side) does not match MacGuire's theoretical predictions within the assumption of a 3s- '(d'Os) initial configuration where the initial hole is not screened. On the contrary, Jennison has shown that the agreement could be largely improved by considering an initial 3s-l(d1 Os) configuration where the initial hole is screened by an s electron. Thus, we are led to consider now the case of a weak interaction between the two final-state holes ( U i ] 4 , J E P ) N f J = YA,~NA(~)

(117')

where Np(i) is the number of primary electrons crossing the ith atom layer. As can be seen from Eq. (117'), Nfsx depends on the number of atoms A per unit volume in the ith layer, on Np(i),and on the average value of the cosine of the propagation angle 6 of the primary electrons. Neglecting the dependence on i in this last factor, one can write N f , x = UA,~NA(~)NP(~)

(191)

Among all these Auger electrons, only a fraction will reach the surface and be collected in the spectrometer. Their number will be Nf$

= aL,,NA(i)Np(i)kfPX

( 192)

In fact, because of the contribution of the backscattered electrons, Eq. (192) has to be replaced by

N;;; = GY~,,NA(i)NP(i)[l + rA,x(i)]kf'X

(192')

The product N,(i)[l + TA,&)] can be approximated by an exponential N exp(id/u) term, where N is the number of efficient electrons (primary or backscattered) in the first atom layer, d is the thickness of an atom layer, and a is a length to be evaluated, which depends on the Auger line considered. For a CMA, can be approximated by a second exponential term: k f S x

so that N f $ can be expressed by the relation N;,:'

= UL,,NA(i)ki,,

(194)

where the factor kA,x (to be raised to the ith power) is the product of two exponential terms : and The total contribution of all the layers is

k2 = exp

262

M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN

For example, for a pure sample, NA(l’)is independent of i and equals NA. Thus N;.” = UA,~N(A)NA(~ - ICA,~)-’

(1 96’)

where N(A) is the number of efficient primary electrons in the first atom layer. It must not be confused with NA,which is the numerical density of the pure element A. The ratio of the Auger intensities due to the pure elements A and B is

In the absence of experimental observations, the ratio N(A)/N(B) is taken ~ ) is evaluated from Eqs. (189) equal to the ratio (1 + rAJ/(l + T ~ , and and (190). By using Eq. (197), Le Hericy and Langeron (1981) have determined the ratio crb/a;, from measurements on TiO, Ti,03, and TiOz samples. For that they evaluated kA,xand k,,y from Eq. (195) and the Seah and Dench’s universal curves for the MFP. They obtained ab/a& N 1,2. They have also discussed the cases of the heterogeneous samples and reported that in these cases only approximations can be obtained.

Quantitative Analysis from the Auger Line Shapes

Some papers have lately been devoted to the development of a quantitative analysis method using the Auger line shapes. This new field of study is very promising. Turner et al. (1980) performed a quantitative analysis of the surface composition of sulfur-bearing anion mixtures. The anions have a characteristic LVV line shape which depends only slightly on the nature of the cation. To study a binary mixture for instance, the AES spectra of the individual components have to be made comparable. For that, the AES spectrum of one component is normalized to that of the other with respect to the XPS area of the S2p peak and to the beam currents. Then, the component concentrations are determined by adjusting a linear combination of the component spectra to the mixture spectrum. Gaarenstroom (1 98 1) made a very interesting application of the principalcomponent analysis method to the study of the Auger line shapes either in the case of a component mixture or for a depth profiling. The technique is especially devoted to the analysis of the continuous evolution of the line shape from one mixture to another (or from one depth to another) when the Auger lines of the components are in the same energy range. The principles of the method are the following:


263

(1) The line shapes of the n different mixtures are digitized for p energy channels giving a data matrix D with n rows and p columns. (2) The matrix W which is the product of D by its transpose (W = DD‘) is of order n and rank m where m is the number of components in the mixtures (m < n and m < p). (3) The number rn of components is equal to the number of nonzero eigenvalues of the matrix W. This result can be proved mathematically. Owing to the uncertainties inherent in the experimental data, the numerical results have to be interpreted with some care. For instance, a statistical test must be used to decide when a given eigenvalue is “physically” different from zero. (4) Moreover, it is possible to obtain quantitative information about the mixture compositions if the Auger spectra of the pure components are known and included in the data matrix D. In other words, these pure component spectra are digitized and constitute two rows of the data matrix D. In this case, and for binary mixtures A-B, the n eigenvectors of the matrix W have two principal components x and y. Especially normalized eigenvectors associated with the pure elements A and B have as components (xA, yA), (xB,ye). They determine in the plane (x, y ) two vectors OA and OB which represent the pure components. So they determine a system of oblique axes on which each mixture eigenvector OM can be projected. The lengths of these projections determine the concentrations CA and CB in this particular mixture (see Fig. 24).

Although, to the authors’ knowledge no mathematical evidence was given to this result, the method seems to work well. However, this quantitative analysis rests on the assumption that the Auger lines are constitutive. In other words, it is assumed that the Auger line of the mixture can be determined as a linear combination of the pure component Auger lines. Such an assumption was also the basis of the analysis by Turner et al. (1980), but presumably still deserves further confirmation for other systems.

FIG.24. Determination of concentrations: Gaarenstroom’smethod.

264


D. Sample Preparation A first method consists in elaborating the sample in a chamber by evaporation-deposition on a substrate. Such a procedure was used by Goto et al. (1978) to elaborate a series of Cu-Ni alloys by coevaporation. In the case of an evaporation onto a liquid nitrogen-cooled substrate, the surface concentrations given by AES are linearly related with the bulk concentrations measured by atomic absorption spectroscopy. In the case of a coevaporation at room temperature, there is an enrichment of the surface in copper, typical of a thermal segregation. This is only one example of the difficulties encountered in the preparation of samples. When the evaporation speed is well controlled and is slow, it is possible to observe in the time variance of the Auger line intensity of the deposit some discontinuities typical of a layer-by-layer growth. This was observed by Guglielmacci and Gillet (1980a) in the case of Ag deposits on a (11l)Pd surface for a deposition speed of 1 A min-l and a substrate temperature of 20°C. This represents a means of calibrating the thickness of a film, particularly for submonolayer deposits. More commonly, especially for technological-like samples, the preparation of the target consists in suppressing the contamination of the surface induced by the gases or the vapors of the atmosphere or by the pretreatments suffered by the sample. Bouwman et al. (1978) quote four decontamination procedures: (1) chemical etching in which a chemical reaction is induced at the surface of the target as, for example, in the case of the removal of carbon by exposing the surface to oxygen at 500°C; (2) plasma etching involving a chemical reaction by intervention of a reactive plasma so that the etched material is changed into a volatile compound. (3) thermal desorption by heating or flashing the sample in uacuo; and (4) ion etching, which is the procedure most currently used and consists in sputtering the target with ions, most frequently low-energy inert-gas ions. Ideally, any method could allow elimination of the contaminating species without modifying the masked surface which represents the central object of study. One often wants to make a surface initially submerged in the bulk of the sample free. The reason could be that the surface presents some interesting characteristics, for instance, grain boundaries in the case of the grainboundary embrittlement of steels or superalloys. For that, a fracture is performed in situ in the UHV chamber (McMahon et al., 1977; Briant and Banerji, 1978, 1979a,b,c). The reason could also be that once the surface is free of contamination, it can be taken as a standard reference for studying the physicochemical surface mechanisms (catalysis, gas adsorption, oxidation, gaseous corrosion, epitaxial growth, surface segregation, etc.). For this, one


265

usually proceeds to a fracture in uucuo or to removing some matter by either scribing or lapping. Finally, one also very often performs a profile analysis in depth. This technique has as its essential aim the study of the variation of the elemental concentrations in the bulk of the target. Some parts of the material are gradually removed so that surfaces which were initially merged at more and more important depths in the bulk can be analyzed by this method. In the case considered here, the material to be eliminated is nearly always sputtered by bombarding the target by ions more energetic than for a simple decontamination. The analysis is made by AES. Another mode of etching by a high-power laser technique has been considered by Papagno et al. (1 980). All these methods for preparing the sample raise some questions with respect to the quantification of the results. Some of these problems are now reviewed in the following sections. 1. Preparation by Fracturing in Vacuo

As mentioned earlier, this technique of preparation can be used in two very different cases. In the first case, the fracture aims to prepare a virgin reference surface. To achieve this, the rupture must occur either by cleavage, for brittle materials, or by transgranular fracture, for ductile samples (Van Oostrom, 1976, Rehn and Wiedersich, 1980). The assumption that the cleavage surface or the transgranular fracture domains are representative of the bulk solid is probably satisfactory, except if voids or particles of a second phase are present. Even in this case, however, partial analyses leading to the determination of the concentrations of some elements are still possible (Mulford et al., 1980). Nevertheless, characterization of the fracture surfaces by other techniques must accompany AES. In the second case, the object of the studies is the embrittlement of materials by grain-boundary segregation. Therefore, the characterization of the embrittling solutes which segregate to the grain boundaries requires that the fracture be intergranular. There are few studies to determine the dependence of the results of Auger analysis on the characteristics of the fracture surface. To our knowledge, the only studies on this topic were carried out by Rowe et al. (1978) and by Mulford et al. (1980). By using a highresolution scanning Auger microscope they proceeded to a grain-by-grain analysis of the fracture surface. They made the following observations :

(1) The diffusion along the surface of contaminants from the outside edges of the fracture surface does not appear to be a problem in the alloys studied (stainless steels, Fe-Si alloys, low-alloy steels). (2) At a grain boundary, the impurities are more or less randomly separated by fracture between the two sides of the fracture surface. On an average, the quantities of the impurities on each side are equal (50-50%),

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M. CAILLER, J . P. GANACHAUD, AND D. ROPTIN

but they can vary from one grain to another between the two limits 40 and 60%. Thus, no side is systematically favored. (3) On a side of the fracture surface, the impurity concentrations vary smoothly from place to place on an individual facet. Thus it is shown from observations (2) and (3) that the fracture surface follows the central region of the boundary. However, Mulford et al. (1980) caution against generalizing this result to systems other than those they studied. (4) The heights of the Auger peaks of the segregated elements vary from grain to grain because of the variation of the incidence angle; however, this angular effect can be minimized by means of a normalization technique. For instance, in the case of steel this technique consists in dividing the height of the Auger peak of the segregated element by the height of the iron Auger peak. Under these conditions, the angular effect is no longer dominant, and the results so obtained give a correct measure of the impurity concentrations on each grain. It is then possible to study the variations of the elemental concentrations at the grain boundaries, for instance, during a heat treatment of the sample. 2. Preparation by Scribing

This mode of preparing a virgin surface was used by Betz (1980)and Betz et al. (1980a,b) to compete with the method of fracturing in uacuo in order to characterize the elemental concentrations at the surfaces of binary and ternary alloys. It is assumed that the results of the Auger analysis are characteristic of the bulk composition of the target if they are the same in both modes of preparation of the virgin surface. The identification of the surface concentrations as measured by AES with the bulk concentrations as measured by an electron microprobe, for instance, requires good calibration of the Auger technique (Holloway, 1977). The reference state having been defined in this way, it is then possible to study the subsequent variations of the concentrations at the surface by measuring the variations of the intensities of the Auger peaks. Various tools have been used to scribe or scrape the sample, such as a diamond or a stainless-steel point or a tungsten carbide blade on a linear rotary-motion feedthrough (Sliisser and Winograd, 1979) or a tungsten brush (Yu and Reuter, 1981a,b). 3. Preparation by Lapping

This mode of preparing the sample is intended for depth profiling where it can compete with sputtering. This method was known for a long time but


267

was only recently used in relation with AES (Tarng and Fisher, 1978). It consists in removing part of the sample in order to cause either a spherical surface or an oblique flat surface to appear. Lapping has been particularly used to prepare an overlayered sample in view of the analysis of the interface between the substrate and the deposit. Then, in the case of spherical lapping, the diameter of the ball used to lap the sample must be sufficiently large so that the interface area which is available for the analysis is much larger than the electron beam. Then, depth profiling is realized by laterally moving the electron beam on the sample with the help of an electrostatic deflection. However, if the area which has to be analyzed is so large that the electrostatic deflection will lead to a wrong use of the analyzer, it will then be necessary to mechanically translate the sample in front of the electron beam. In practice, spherical lapping has been realized with the help of a steel ball freely rotating in contact with the sample while it is fed with diamond paste (Thompson et al., 1979; Walls et al., 1979; Lea and Seah, 1981). In some cases (brittle or soft materials), however, the method may not work. In any case, lapping, as described above, must be followed by cleaning the sample before introduction into the UHV vessel and by decontamination treatment after introduction into it. Removal of the contaminants is accomplished by a low-energy ion sputtering. The advantage of the method is that it is possible to obtain very smooth spherical surfaces (roughness of 100-500 A by using 0.1-pm diamond paste). Therefore, the lapping-depth resolution Azp, which is limited by the roughness of the lapped surface, can be compared with the sputtering-depth resolution. Such a comparison was made by Lea and Seah (1981) in the case of an overlayered sample. Five situations corresponding to different roughness conditions for the free surface and the interface were distinguished. From their study, it appears that for deposit thicknesses of the order of 100 nm, sputtering will lead to a better depth resolution than lapping; however, for thicknesses higher than 1 pm, the interface broadening obtained by very smooth lapping is smaller than that obtained by ion sputtering. It should be noted that the theoretical comparison by Lea and Seah of the lapping and ion-sputtering depth resolutions was made by using the results of the sequential-layer sputtering (or SLS) model of Benninghoven (1971) and Hofmann (1976). In this SLS model the contribution A+ to the depth resolution, inherent in the mechanism of sputtering itself, is considered to be proportional to the square root of the thickness of the removed layer. Seah et al. (1981) now doubt the validity of this square-root dependence. Taking into account the site dependence of the sputtering probability of an atom, they modified the SLS model. They showed (Section VII,E) that as soon as the thickness t of the sputtered layer exceeds about ten atom layers, the

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M. CAILLER, J. P. GANACHAUD, AND D . ROPTIN

depth resolution Azs becomes independent of t and can be neglected, compared with the other interface broadening contributions (atomic mixing, diffusion effects, etc.). The theoretical comparison of the lapping and ion-sputtering depth resolutions must therefore be reformulated.

4. Preparation by Sputtering Sputter etching a target by ion bombardment is the most widespread method for achieving depth-profile measurements by AES. Most frequently one uses noble gas ions, especially argon; however, other types of ions have been used, for instance, 0; and N i , to obtain a better sputtering yield or to avoid the chemical reduction of the target or to obtain a better resolution in depth as well. The cleared crater is always of large dimension compared with the diameter of the Auger probe. In view of its quantitative elemental analysis, etching a surface by ion bombardment raises some problems caused by the following: (1) There are uncertainties about the depth at which the analyzed surface was located before sputter etching. These uncertainties come partly from the roughness of the stripped surface, either its initial roughness (before sputtering) or that induced by the ion bombardment. In the latter case, it may be due to irregularities in the ionic current density, to inhomogeneities or anisotropies in the target, or simply to the random character inherent in the sputtering effect. We may consider that these uncertainties are purely geometrical in character. They exist prior to any Auger analysis, and their degree of intervention is partly determined by the transverse dimensions of the Auger microprobe. Smith (1976) has studied the sputtering effects in aluminum oxides and has established contour maps of the film thickness by ellipsometry. The sputtering provoked by the ions is not uniform and there are more or less hollow domains. The electron beam, which should be incident on an oblique microsurface, could be deviated to strike the surface in a hollow part. (2) There are some uncertainties connected with the finite value of the zone thickness explored by the probe, that is, by Auger electrons. This thickness runs from about one to a few Auger electron MFPs. It increases with increasing electron energies, except at low values (5100 eV). Strictly speaking, these uncertainties depend on the surface roughness but are generally considered to be independent of the sputtering result. (3) There are surface-composition modifications from (a) preferential sputtering of one or several elements, leading to a surface depletion in these high-sputtering-yield elements; (b) implantation of primary ions and introduction of impurities from the surface;


269

(c) surface diffusion of some elements; (d) ion-induced atomic mixing, particularly near the interfaces ; (e) creation of point defects and the resulting enhancement of the diffusion ; (f) chemical reactions induced by the ion beam. These compositional changes could, in turn, produce some uncertainties about the value of the initial depth of an interface when used as a means to observe this interface. In this case, the uncertainties generated in this way will combine with the uncertainties associated with the roughness, the probe, and the initial lack of parallelism between the interface and the free surface of the sample. The consequence of all these effects will be that the variation of the signal used to characterize the target will not present the expected steplike variation at the crossing of the interface (supposed to be ideal), but it will exhibit a continuous variation. A measure of the interface broadening (Fig. 25) is then given by the difference Az between the two “depth” values corresponding to signal amplitudes of 84.15 and 15.85% of the maximum amplitude of the analyzed signal. The depth values are just mean values, estimated from sputtering times. It seems that no overall theoretical study of the total interface broadening has been made to date. Any of the contributions to the broadening can be related to a separate term Azi so that

AZ = (1A z ? ) ~ ” i

if one assumes that they represent independent contributions. This is not necessarily true, and Eq. (198) gives an upper limit for the total interface broadening. The interface broadening AzR due to the initial roughness of the target in the case of a homogeneous and isotropic overlayer having a thickness zo along the vaporization and condensation direction has been the object of

’oo%3L

, 84.15%

1565% 0%

FIG.25. Interface broadening.

270


studies by Seah and Lea, 1981. If evaporation is done normally to the mean surface of the substrate and if the ionic bombardment is done under an incidence angle B with respect to this mean surface, the interface broadening is given by where z is the sputtered thickness and M , is the standard deviation of the angular distribution supposed to be Gaussian B

=

(f- 1)tanB

when

fl > m 0

(200)

with

f

3: HSub/(49.5Ali2)

(201)

In Eq. (201), A is the atomic weight of the target and Hsubis its molar sublimation enthalpy (in kilojoules per mole). In this case the interface broadening would be proportional to the thickness of the material to be sputtered and the weaker the angle p, the weaker the broadening. It would then be desirable to operate at normal incidence to reduce AzR ; however, under these conditions Eq. (199) is no longer valid and has to be replaced by

AzR = 1.66~;ji

(204

We note that according to Eq. (198), the total interface broadening Az differs from AzR due to the other causes of broadening. Minimizing AzR does not necessarily minimize Az. For instance, Seah and Lea have considered what would happen if the interface broadening due to atomic mixing would compete with that due to initial roughness. Assuming that the former effect does not depend on fl, they deduced that for M, > lo, it is better to operate with a normal incidence, whereas for a, < 0.35", an oblique incidence could lead to a better depth resolution. One may note from Eqs. (199) and (202) that AzR is determined by the distribution of the microfacet orientations, which are characterized by a, and not by the height of the rugosities. This conclusion agrees well with the experimental results of Mathieu et al. (1976). For elements such as Au, Ag, Pd, and Cu, f is less than unity. With this condition, application of Eq. (199) raises some problems because its RHS member is then negative. One would be led to admit that for fl > M , for these elements, the ion bombardment tends to cause the initial roughness to disappear. At least it is clear that for Ni-Cr alternate sandwich structures, the interface broadening is much stronger for an initially rough sample than for a smooth one (Hofmann et al., 1977). Roughness can be induced by sputtering. Thus, for example, it has been observed that a ridge and valley structure (with heights of about 50 A )


27 1

appeared by sputtering on amorphous SiO, targets (Cook et al., 1980). For a crystalline material, roughness can be induced by preferential sputtering along some special crystallographic directions or by channeling effects. These effects can be particularly severe for polycrystalline materials. Roughness can also be induced by inhomogeneities, especially by foreign materials, at low sputtering rates. According to Hofmann (1977),the interface broadening due to these defects or to the different orientations in the crystal lattice would be proportional to the sputtered thickness. According to Bindell et al. (1976) and Laty and Degreve (1979),very often it is the surface roughness which represents, in practice, the limitation for the resolution in depth. The interface broadening AzA due to Auger analysis is of the order of the Auger electron inelastic MFP (or a few times this MFP). Thus it is of the order of a few angstroms, and the weaker the energy of the Auger electrons used for the analysis, the smaller the broadening. It is usually admitted that AzA is independent of the sputtered thickness. The influence of surface roughness upon this interface broadening is not well known, however. In the literature one usually uses the term “preferential sputtering” in the case of multicomponent targets in which at least one element is removed more quickly than the others. Nevertheless, in the case of pure targets one can also consider that preferential sputtering occurs for some atoms but not for others, even though they are similar in nature. This greater or lesser aptitude of the atoms for being ejected is related to position in the surface layer. The fewer the atoms in a given layer, due to sputtering, the lower the coordination numbers of the remaining atoms. Consequently, the instantaneous binding energy of these atoms decreases as bonds are broken, and their sputtering is made easier than that of atoms in an intact layer. This effect has been introduced by Seah et al. (1981) in the model for sequentiallayer sputtering of Benninghoven (1971) and Hofmann (1976). This model is based upon simple statistical arguments in which the solid has a layer structure, and sputtering takes place only in that part of the atom layer exposed to the ion beam, with a rate i. In the basic model, the rate is constant, but Seah et al. assume that this rate is a linear function of the fractional coverage 8, between the two limits 6 = 0, where only one atom is still present, and 0 = 1, where the layer is unaltered. The ratio k of the rates i(0 = 0) and i(O = 1) is given by the ratio of the coordination numbers of the atom in these two extreme situations. In theory, this ratio varies from 1 to co with respect to the orientation of the atom layers, but in practice it could not exceed a value of 3. For k > 2.5, the resolution in depth Azs due to the sequential sputtering mechanism becomes nearly constant beyond 10 evaporated layers. On the other hand, as soon as k exceeds the value 1.5, which seems to be the usual case [for

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M . CAILLER, J. P. GANACHAUD, AND D. ROPTIN

instance, k = 3 for the ( 1 11) plane of an fcc system], the influence of the sequential sputtering quickly decreases so that the resolution in depth is governed by other causes. By computer simulation Harrison et al. (1978) have determined the differences between the sputtering yields of the (loo), (1 lo), and (1 1 I ) faces of copper for 600-eV argon ions, the sputtering yield being defined as a mean value in the impact zone of the ion beam of the number of atoms ejected per incident ion. Their statistical technique, compared to the statistical model of Benninghoven, plays the same role in the description of the intrinsic sputtering mechanism as that played by a computer simulation of Auger emission compared to the analytical resolution of the Boltzmann transport equation. The enormous advantage of computer simulation is that it makes it possible to take into account more realistic physical assumptions than those obtained by analytical description. The values obtained are, respectively, 3.93, 3.54, and 6.48 for the (loo), (1lo), and (1 11) faces. The tendencies one notes from these results agree well with experimental observations. Following Hofmann (1977), the interface broadening Az, due to inhomogeneities in the intensity distribution of the primary ions would be proportional to the thickness of the sputtered material. On the other hand, the interface broadening Az, of the knock-on and ion-induced atomicmixing effects would be independent of the sputtered thickness and would be functions of the energy and the mass ratio between the target atoms and the primary ions (Hofmann, 1977).It would also depend on the angle of incidence of the ions. Liau et al. (1978, 1979) have established that the thickness of the mixed layer was approximately proportional to the energy of the ions for Ar+ in the energy domain from 10 to 160 keV. By extrapolating at 2 keV, one obtains a mixed-layer thickness of 30 A which is higher than the escape depth of the Auger electrons (at least those of the low energy). Broadening by knock-on effect and by atomic mixing can be reduced by lowering the energy of the ions. Thus for argon ions of less than 1 keV, there would be no appreciable broadening (Mathieu et al., 1976),and the important factors would be the surface roughness, the escape depth of the Auger electrons, and the fact that the interface is itself ill-defined. The broadening effect by atomic mixing can also be lowered by using heavy ions or grazing incidences. To sum up (Hofmann, 1979), the escape depth of the Auger electrons, the knock-on effect, the preferential sputtering, and the diffusion are determining for small values of the etched thickness ( c10 nm). Beyond these values, the irregularities of the beam, the initial roughness of the surface, the sequential sputtering, the surface diffusion, and the crystalline character of the sample are the most important factors.


273

Two examples are now given to illustrate this analysis : (1) Hofmann and Zalar (1979) have studied the depth resolution in Ni-Cr sandwich structures, sputtered by argon ions of 1 keV. They approximated their experimental results by the relation

Az

=

2(az

+ A’)’’’

(203)

with a = 0.3 nm and 1 = 1 n m The values deduced from Eq. (203)and the experimental results have good agreement when the etched thickness exceeds 50 nm. As an illustration, the experimental depth resolution given by Eq. (203)would be 11 nm for an etched thickness of 100 nm. The depth resolution : ions, but the difference is not significant is slightly smaller if one uses N (10 nm for an etched thickness of 100 nm). (2) Roll and Hammer (1979) have studied the Ni-Mo and Co-Mo sandwich structures and approximated their results for the depth resolution by the relation

AZ = C I Z ’ ’ ~ + /I

(204)

For z = 100 nm, the depth resolution is of the order of 7-12 nm, depending on the couples of materials and according as crossing the interface leads from material A to material B or the reverse. E. Efects

of Sputtering on the Surface Composition of Multicomponent Materials

Multiphase systems should be studied apart from alloys or homogeneous compounds, which will be considered here. Numerous studies have been devoted to the analysis of homogeneous binary and ternary targets. The results obtained show that, generally, surface composition after sputtering differs from that obtained immediately after fracturing or scribing. Ionic bombardment in fact creates an altered layer at the surface which spreads over several atom layers (Ho, 1978) and has a composition which is different from the mean composition of the bulk. The thickness of this Iayer increases first with the dose of ions (this is the transient state). It tends to a constant value when the sputtering times increase. In a rather general way the values proposed for the altered layer (for the steady state), at ambient temperature and for argon ions with an energy of about 1 keV, range from a few angstroms to 30 A (Ho et al., 1976, 1977; Watanabe et al., 1977; Betz, 1980), which is of the order of the penetration depth of the ions (Mathieu and Landolt, 1979;Kim et al., 1974);however, the thickness of the altered layer is much smaller than the projected range of the ions (Kim et al., 1974). In the case of Al-Cu alloys sputtered by 1-keV Xe

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M. CAILLER, J. P. GANACHAUD, A N D D. ROPTIN

ions, however, Chu et al. (1976) have reported a value of 300 A for the thickness of the altered layer. In any case, the altered-layer thickness values cannot be neglected by comparison with the MFP and the escape depth of the Auger electrons. Therefore, the measured Auger signals are highly dependent on the altered layer. The effect is normally more important for low-energy Auger electrons than for high-energy Auger electrons. Thus one has a simple means of collecting information on the altered-layer composition. Several elementary mechanisms contribute to the variation of the alteredlayer composition: preferential sputtering enhanced diffusion thermal segregation atomic mixing precipitation

No global theoretical description has apparently been given for the altered layer. The simplest model takes into account the preferential sputtering of some elements compared to others. When steady-state conditions have been reached for sputtering, the target is sputtered in a stoichiometric way with respect to its bulk composition (1) if S(A) and S(B) are the sputtering yields of elements A and B which constitute the alloy or the homogeneous compound; (2) if &,(EA, x) and I,,(&, y ) are the intensities of the Auger lines x and y of elements A and B before sputtering and Is(EA,x) and Zs(EB,y ) the Auger intensities of the same lines in the sputtering steady state, one can write (Shimizu and Saeki, 1977)

By measuring the ratios of the intensities of the high- and low-energy peaks of elements A and B, one can approximate the ratio S(A)/S(B) so long as the intensities of the high-energy Auger peaks can be considered, in an approximate way, representative of the bulk composition of the sample. In the case of solid solutions or compounds, the steady state is quickly reached, as soon as the thickness of the eliminated layer becomes comparable with the penetration depth of the primary ions. For multiphase systems, preferential sputtering can also take place if the phases have different sputtering rates. The steady state will be reached only after the elimination of a layer having a thickness representative of a few


275

grain diameters (Henrich and Fan, 1974). In this case, the formation of cones and the accumulation of species difficult to sputter leads to very complicated sputtering behavior. Two parameters determine preferential sputtering. The first is related to the way in which kinetic energy is transferred from the energetic ion to the atoms. For most compounds, the energy of the ion is transmitted principally to light atoms which would be preferentially sputtered if this parameter were preponderant. This kinematic model leads to surface enrichment of the heavy material (Haff and Switkowski, 1976; Haff, 1977; Kelly, 1978). The second parameter is the binding energy of the atoms, and its principal effect is to reduce their recoil energy. For the extreme case where only the bindingenergy effect plays a role in preferential sputtering, the result is enrichment in the material having the highest binding energy. The role played by these two parameters appears in the theory of Sigmund (1968, 1969a,b) who expressed the sputtering yield of pure amorphous or polycrystalline substances in the form where f ( M , , M i ) is a function of the atomic weights, M , for the target and Mi for the incident ions; Eiis the energy of the primary ions; U , is the binding energy of the target atoms, a measure of which is given by the enthalpy of sublimation. The product S V, is called the recoil energy density. If the function f varies slowly with the atomic weight of the target, as soon as the atomic number of this target exceeds 30, the sputtering yield of a sample will be lower the stronger its binding energy is. If, on the other hand, two elements have the same binding energy, the heavier one will have the higher sputtering yield, the function f increasing with M , . An empirical relation for the sputtering yield has been proposed by Okajima (1980): S = K’(M,/E,)k

(207)

in which E, is the cohesion energy, M , is the atomic weight of the target, and k and K’ are constants. According to this relation, the sputtering yield depends only on the ratio M , / E , . The value obtained for k is the same for 10-keV argon ions and for 45-keV krypton ions ( k N 413). For 45-keV krypton ions, K‘ N 40 when E , / M , is expressed in J kg-l. The yields for the sputtering of pure elements by argon ions of 500-1000 eV have also been analyzed by Seah (1981), who showed that there was good overall agreement among the experimental results once they were corrected for the effects of contamination by oxidation and the theoretical results of Sigmund. Thus the well-known quasi-parabolic variation of the binding energy in each of the three transition metal series, related to the filling of the

276

M . CAILLER, J. P . GANACHAUD, AND D . ROPTIN

electronic levels, leads to a systematic but inverse variation for the sputtering yields. An examination of Fig. 3 of Seah’s paper, which presents the variations of the ratio of the measured and predicted yields for 500-eV argon ions as a function of the enthalpy of reaction with oxygen per gram atom of oxygen, seems to indicate that there is not just a single relation but several between these two quantities, a systematic variation appearing, for instance, for each transition metal series. Taking into account this multiplicity allows a slight amelioration of the correlation established by Seah between the corrected experimental yield and the predicted yield. It is not obvious that the sputtering yields of an element in an alloy or in a binary or ternary compound are the same as in the pure material. One might even expect that this will not be the case. It seems, however, that according to the studies by Betz (1980) for alloys and compounds having recoil energy densities SU,, nearly equal, the surface is enriched in the element having the smallest sputtering yields in its pure state. When the components have quite different recoil energy densities, one can see from Eq. (206) that the effect of the difference of weights can play an important role in the evaluation of the sputtering yields, and that enrichment in the heavier component can take place. Table IV shows the results obtained for preferential sputtering surface enrichment. Here, the elements have been classified for both entries along increasing enthalpies of sublimation. A cross (for an alloy) or a circle (for a compound) has been placed in the column of an element for which surface enrichment occurs. Table IV shows that, except in relatively exceptional cases, there is enrichment by the component which has the strongest enthalpy of sublimation, and the weight factor plays a secondary role. The exceptions in Table IV are related to compounds involving silicon, which is a rather light element presenting covalent bonds in the pure state and for which the amorphous state is easily reached. This makes uncertain the validity of the use of a crystalline heat of sublimation to characterize the binding energy at the surface after sputtering (Sigmund, 1969a). However, it seems that the effect which is concerned here is principally a difference of weight effect (Ho et al., 1979). It it possible to express the sputtering yields of multicomponent targets in terms of the sputtering yields of pure elements? According to Betz, the sputtering yield of solid solutions seems to increase linearly with the concentration from the value for the low-yield component to that of the high-yield component. This seems to indicate that in such a case, each element keeps the sputtering yield it had in the pure state. For nonmixable systems and for the major part of the concentration domain, however, the sputtering yield of

277


the target would first stay near that of the element having the lowest yield and would thereafter increase rapidly to the yield value of the other element. Except in the domain of strong concentrations in the high-yield element, one can assume that the surface bombarded by the ions is covered with atoms of the component having a low sputtering speed. The observation shows that after sputtering has ceased, the surface gets covered through a bulk diffusion process by an overlayer of the high-yield component. In the case of Ag-Pd alloys, enrichment in Pd would occur by formation of a segregated alloy layer, rich in Pd (Slusser and Winograd, 1979). The thickness of this layer would decrease with the bulk concentration in Pd. Rivaud et a!. (1981) have observed the precipitation of a phase rich in In at the vicinity of the surface of oversaturated Cu-In alloys. Preferential sputtering is not the only contribution to the formation of the altered layer. Liau et al. (1979) have imagined a model in which the altered layer had a given thickness t, for a given target and a given ion beam (t, N 700 8, for the Pt-Si system and for 250-keV X: ions). Due to atomic mixing, the element concentrations are at every moment uniform in the TABLE IV

PREFERENTIAL SPUTTERING SURFACE ENRICHMENT' As Pb In Ga Ag Sn Al Be Cu Au Pd Cr Fe Ni

As Pb In Ga Ag

Si

U

Pt Mo Nb

X

X X

x

x

x

Sn

0

A1 Be cu All

.

o

x x

Pd Cr Fe Ni Si U Pt Mo Nb

X

x x

X

X

x X X

X X

X X

0

0 X

x , alloy; 0 ,compound

278


altered layer. Owing to preferential sputtering, however, continuous depletion in the high-sputtering-yield element takes place (Si in the quoted example). With the aid of this model and by continously recording, it is possible to relate the measured profile to the real one. Ho (1978)has proposed a model of the altered layer based on preferential sputtering in the very first atom layer at the surface (emission layer) and on diffusion in the altered layer. This diffusion becomes strongly enhanced by virtue of the creation of a large number of point defects (vacancies) in the vicinity of the surface. The model is based on two mass balance equations. One is related to the outermost surface and takes into account the diffusion and preferential sputtering fluxes. The second is related to the rest of the altered layer and takes into account only the diffusion flux. This model leads to the introduction of an effective thickness parameter:

6 = Du-’ (208) where D is the diffusivity in cm2 sec-l and u is the sputtering speed. It also introduces a dependence in z for the composition of the altered layer. All along the sputtering transient state, 6 varies. At equilibrium, which is reached after sputtering material about 56 thick, the altered layer extends over an area about 46 thick. So, for high-diffusivity alloys, the altered layer can extend up to large depths owing to the enhanced diffusion. This would explain the high value of the altered-layer thickness in Al-Cu alloys found by Chu et al. (1976). Conversely, it is possible, with the aid of Eq. (208) to estimate the diffusivity during ionic bombardment. The values obtained for D in the case of Cu-Ni systems are much higher than those obtained without sputtering. Thus, sputter damage seems to be a very efficient process for reinforcing diffusivity in the altered layer (Watanabe et al., 1977; Ho, 1978). In Ho’s model the composition of the altered layer varies monotonically from the surface composition. In the steady state, for a binary alloy, the concentration of atoms A at a depth z is given by the relation: CA(z) = CA(0) + CA(O)(1 - C A ) ( b A - dg)[1-

~XP(-Z)]

(209)

where Z = zv/D

(210)

and

is the concentration of atoms A in the emission layer; CAis the concentration of atoms A before the sputtering of the sample, supposed to be homogeneous;


279

cAand oB are given by the relations OA

= SA/Nal(A),

gB

= SB/Nat(B)

(212)

where N,, represents the numerical atomic density for the pure element. Equation (209) shows that in the general case CA(Z) exhibits an exponential dependence starting from the surface z = 0; however, this dependence disappears as one would expect for a pure target (CA= 1) or when the reduced sputtering yields 0, and 0, are equal. One can also see from Eq. (209) that CA(2) > CA(0) when CTA > 0,. Chou and Shafer (1980) have extended Ho’s model to the case when primary-ion implantation in the target, initially homogeneous, can no longer be neglected. They obtained for the concentration at the surface

G(0) = PI/(aWat) (214) where Na, is the numerical atomic density of the sample before sputtering and =1

+ JPI/(VhT,,)

(21 5 )

where J is the primary-ion flow and PI is the fraction of these ions which impregnates the target. In the absence of ion implantation (PI = 0), CI(0) becomes null, o( becomes equal to 1, and Eq. (213) reverts to (211). Hofmann and Zalar (1979) have observed that in the Ni-Cr system sputtered by l-keV N: ions, the relative amount of implanted ions was inversely proportional to the sputtering speed. The intervention of the enhanced diffusion raises the problem of the role played by temperature in the variation of the composition of the altered layer. Sputtering yields of pure metals are nearly independent of target temperature up to the melting points. In the case of bombardment with noble-gas ions (Kaminsky, 1965) and for coevaporated Cu-Ni systems, there exist no significant differences in the sputtering mechanism at liquid nitrogen and room temperature (Goto et al., 1978). This is not the case for higher temperatures. Various studies by AES and SIMS have been devoted to the influence of temperature on the sputtering of Cu-Ni systems (Nakayama et al., 1972; H. Shimizu et al., 1975; Rehn and Wiedersich, 1980; Yabumoto et al., 1979; Shikita and Shimizu, 1980; Okutani et al., 1980). The results obtained converge and provide evidence, as early as 200°C, of a surface (or Gibbsian) segregation in addition to preferential sputtering, enhanced diffusion, and ion implantation. In the absence of sputtering, this surface segregation leads to enrichment of the surface in Cu. The effectsof surface segregation and of preferential sputtering are antagonistic. Therefore,

280

M. CAILLER, J . P. GANACHAUD, AKD D. ROPTIN

overall, the altered layer is enriched in Ni owing to preferential sputtering, but enrichment of the outermost surface is less important than in the subsurface region owing to surface segregation. In other words, the concentration in Ni does not vary in a monotonic way in the vicinity of the surface but reaches a maximum in the subsurface region. There would even be enrichment in Cu in the outermost atom layer, indicating that preferential sputtering is weak for this element. For very high energy argon ions, of the order of 1 MeV, the profile in depth of the Cu-Ni system exhibits depletion in Cu (or enrichment in Ni) at the surface, then enrichment in Cu in the subsurface region (Rehn et af., 1981),interpreted in terms of a radiation-induced segregation (RIS). The RIS produces surface enrichment of the undersize solute element (Ni) and depletion of the oversize solute elements (Cu), the migration of the former taking place via interstitial fluxes or vacancies. The RIS departs from enhanced diffusion in that it no longer exists in the absence of radiation. Kelly (1979) has reviewed thermal effects in sputtering. Two types of effects are possible: (1) prompt thermal sputtering, where vaporization of the target results from a local increase of the temperature, and (2) slow thermal sputtering, which implies no temperature increase. For elemental targets and metal alloys, prompt thermal sputtering cannot occur under the usual conditions of temperature, except, perhaps, in some limited cases (Na, Ga-As, Al-Mg). On the other hand, important stoichiometric changes could be observed due to prompt thermal sputtering in some oxides (oxygen loss) for most of the halides and for some organic molecules. Slow thermal sputtering can take place whether the target (oxides or halides) is bombarded by ions, photons, or electrons. In this case, the temperature of the target is well defined. In any case, loss of halogen or oxygen occurs by electron sputtering (or beam-induced dissociation). Halogen or oxygen ions are neutralized by electronic interactions (formation of a relaxed hole, Auger decay, direct ionization), and the neutralized atom is then either vaporized or ejected (electron sputtering) so that the surface becomes enriched with the metal. Vaporization of the metal takes place if it is sufficiently volatile. This happens for compounds containing Cs, Rb, K, Na, and Cd when they are bombarded at temperatures greater than 100°C.

Bombardment b y Ions Other Than Inert Zons: Chemical Reactions

The secondary-ion yield of metallic targets can be enhanced by several orders of magnitude with respect to that of inert-gas ions when the ions used for sputtering lead to ionic bonds with the atoms of the target. The same occurs when the target is sputtered by noble-gas ions in an oxygen atmo-


28 1

sphere (Benninghoven, 1975). This enhancement effect is related to the formation of oxide. Yu and Reuter (1981a,b) have studied the emission of singly ionized positive metal ions from binary alloys A-B under the action of an 0;ion bombardment or an argon-ion beam, but within an oxygen atmosphere. In both cases, when A has a stronger oxide bond than B, the presence of B in the alloy reduces the emission of the A + ions, whereas the presence of A reinforces the emission of the Bf ions. This “reactive preferential sputtering,” to be distinguished from the “nonreactive preferential sputtering” with inert-gas ions (Yu and Reuter, 1981c) leads, in the case of reactive binary alloys, to enrichment of the surface with the element having the weakest metal-oxygen binding energy. The Cu-Ni and Ag-Pd alloys, for which the oxidation is weak, do not present marked differences with respect to nonreactive preferential sputtering by argon. It appears that in reactive preferential sputtering, the surface composition reajusts itself to minimize the energy required for preferential sputtering of the oxygen, whereas in a simple oxidation there is surface enrichment with the metal which makes the strongest bond with the oxygen. Bouwman et al. (1978)used hydrogen particles at 800 eV to clean copper and steel samples. In the case of copper, the major part of the contaminating species (C, 0, S) was eliminated within a 6-A depth after 30 min of bombardment. In the case of steel, bombardment by Ar’ eliminates 0 and s, but is inefficient for C. Bombardment by H eliminates S and part of C, but 0 is unaffected, which indicates that H attacks only the outermost layer. This surface cleaning could be a chemical process comparable with plasma etching, eliminating the contamination species without any appreciable destruction of the sample under the contaminating layer. Taylor et a/.(1978) have bombarded Si, SiO, and Si02 targets by 500-eV N: ions. Before they reach the surface of the sample, these N i ions are neutralized by charge exchange and dissociated so that bombardment is in fact achieved by N atoms having virtually half the kinetic energy of the N : ions. Besides elimination of the contaminating species C and 0 at the surface, bombardment leads to reactive ion implantation which affects the chemical nature of the surface by silicide formation. In the case of silicon, the thickness of the altered layer is of the order of 20 A. Among the chemical reactions generated by ionic bombardment, the most frequently quoted is certainly oxide reduction (Kim et al., l974,1976a,b; Thomas, 1976; Buczek and Sastri 1980). According to Kim et al. (1974), the fact that an oxide is reduced or not can be linked to its free energy of formation. The oxides which are reduced are those which have a low free energy of formation.

282

M. CAILLER, J . P. GANACHAUD, AND D . ROPTIN

The nature of the ions used for sputtering also has some importance. Thus the reduction of PbO (Kim et al., 1976b) is much stronger with inert-gas light ions than with heavy ions. So, Kr+ and Xe+ do not reduce the oxide. Another way to avoid reduction, in some cases, is to use 0; ions.

F . Electron-Beam-Induced Effects in AES

Some disturbances can be induced by an electron beam at the surface of the sample. Therefore, the results obtained by AES for these damaged targets have to be interpreted with circumspection. This topic has been reviewed in papers by Fontaine et al. (1979), Lang (1979), and Fontaine and Le Gressus (1981). Readers interested in these topics will find much of value in these papers. In some cases, the effects induced by an electron beam result only from the interaction of the sample and the electron beam. However, quite frequently a third system (residual gases in the UHV chamber) can intervene. The importance of the disturbances depends on the characteristics of each interesting system. The sample is characterized by its physicochemical parameters such as: (1) chemical nature for a pure compound, (2) impurity distribution, (3) thermal conductivity, and (4) electrical conductivity. The role played by residual gases depends on their composition and on the partial pressures of their constituents. Finally, the current density, the acceleration voltage, and the dose used for Auger analysis are the important parameters for the electron beam. The main disturbances which can be observed are the following: (1) physical modifications in the sample (temperature increase and electrical charging at the impact point of the electron beam), and (2) chemical changes. However, this classification is rather arbitrary because the physical modifications induce chemical changes and vice versa. 1. Temperature Increase

According to Roll and Losch (1980)and Fontaine and Le Gressus (1981), when the acceleration voltage is lower than 10 keV and the diameter of the electron beam wider than 1 pm, the temperature increase AT of the sample can be evaluated by the surface-energy dissipation model of Vine and Einstein (1964). Therefore,

AT = aw/Kd (216) where W is the electron-beam power, d is the half-height width (fora Gaussian profile of the electron density), and K is the linear thermal conductivity


283

coefficient of the sample material. For bulk aluminum, Eq. (216) becomes AT

N

103Wfd

(217)

and W is expressed in watts and d in micrometers. For identical values of the ratio W/d,the temperature increase AT varies as the inverse power of K . Thus for Si, SiO,, and KCI, AT would be 1.5, 17.5, and 2.3 times higher than AT in aluminum. For a bulk metallic target AT is small, but for the case of metal films deposited on a glass substrate, the temperature increase can become important. For instance, for W = 30 mW, d = 80 p m , AT is of the order of 240°C (Roll and Losch, 1980).

2. Charge Efects

The occurence of electric charge effects under the primary-electron-beam spot has long been recognized in studies of secondary-electron emission from insulating targets. For an electrically isolated target, the net variation of the electric charge per second is

AQ

=

-eN,(l - 0)

(218)

where Npis the primary-electron-beam flow, e is the electron charge, B is the total secondary yield of the target, and 0 varies as a function of the acceleration voltage V,. Usually, for insulators, Q is higher than unity in some energy range (V,, , Vp2)for the acceleration voltage. In this range, a positive charge appears on the target, whereas outside this range, 0 is lower than unity and the target gains a negative charge. In the case of a conductor, the potential of the sample is kept constant by connection to an electric source which supplies the charge -AQ, thus insuring the neutrality of the target. In studies of secondary-ion emission of insulators, various methods have been used to reduce charge effects. The first method consists in working with conditions such that 0 = 1. This is the method most frequently used in AES. However, the samples are never perfect insulators so that part of the surfacecharge increase AQ is eliminated by electrical conduction if the insulator is connected to an electric source. For dc, the compensation does not have enough time to take place so pulse techniques have been used. These techniques have the same result as reducing the number of primary electrons that knock the target every second. Another method consists in neutralizing the charge increase by either depositing positive charges (by ion bombardment) when 0 c 1, or depositing low-energy electrons when c > 1. In the usual case of a positively charged surface, one observes a general shift of the Auger spectrum towards low

284

M. CAILLER, J . P. GANACHAUD, AND D . ROPTIN

energies. When the acceleration voltage of the primary electrons is sufficiently important, CJ can become lower than unity and the sample becomes negatively charged. The formation of metallic islands (alkaline or alkaline-earth islands on MgO or Pd on mica), at low secondary yields, on an insulating sample can partially prevent the secondary emission of the sample by an antagonistic potential effect. The secondary-electron yield is then lower than unity. In that case, the Auger peaks are shifted towards high energies.

3. Physicochemical Effects The main physicochemical modifications induced in the sample by ion bombardment are (1) desorption of the adsorbed species, (2) dissociation of the adsorbed molecules and of the sample, and (3) stimulated adsorption. The electron-beam stimulated desorption (ESD) is a mechanism frequently encountered in surface studies. It provokes the elimination of species initially adsorbed at the surface of the sample. When combined with ion mass analysis, it represents a method for studying chemisorption. Joshi and Davis (1977) proposed this method to obtain images by scanning-electron stimulated desorption. Most frequently, in quantitative AES, the production of the adsorbed layer is achieved in a more or less uncontrolled way. It occurs as a contamination of the sample to be analyzed in given UHV conditions. Its desorption does not represent by itself a central aspect of the study but just a stage of the analysis, which can lead to some artifacts. Ion-beam stimulated desorption has been interpreted with the MGR model proposed by Menzel and Gomer (1964) and by Redhead (1964); however, some aspects in this model have been questioned (Feibelman and Knotek, 1978; Antoniewicz, 1980). The MGR model works in two successive steps. In the first step, it is assumed that an electron of the incident beam collides with a valence electron of the adsorbed species. A bond in this species is suddenly broken due to the jump of the valence electron from a bonding to an antibonding level, and thus submitted to the action of a repulsive potential, the species will desorb. Gersten and Tzoar (1977) have analyzed the maximum kinetic energy of the dissociated species and have compared it with the substrate-species binding energy when these species are either in a molecular state (undissociated)or in an atomic state (dissociated).They have shown that desorption most frequently takes place in the undissociated state. Two exceptions have been noted : hydrogen on tungsten, which is dissociated during the desorption, and oxygen on molybdenum, which can be either dissociated or not. Feibelman and Knotek (1978)pointed out that the MGR model could not


285

explain the desorption of 0' ions in the case of transition-metal oxides in their maximum valence states: TiO,, W 0 3 , V,O, (ionic surfaces). As a matter of fact, before electron bombardment, oxygen is quite presumably in a negatively charged state and would have to suffer double ionization to become positively charged. This is not very consistent with the energy of the primary beam when it is very low (0-100 eV). Moreover, the stimulated desorption threshold energy of oxygen is not determined by the atomic levels of this element but by those of the metal. To account for these two aspects, these authors built a new model involving an interatomic Auger transition. In this model, the initial ionization takes place in the core of the metal ion. The reorganization leads to the creation of two holes in the valence band; more precisely in oxygen quasi-atomic states. The concerned oxygen atom is thus forced to desorb because it is surrounded by a repulsive Madelung potential. Woodruff et al. (1980) have studied the desorption of F f , Cl', and Of from W( 100)and found that for the first two ions, part of their results were in agreement with the model of Feibelman and Knotek, but not all. Antoniewicz (1980) has considered what occurs following the first step of ESD in order to answer the question of whether neutral or negative ions are emitted. He noted that the lifetimes of excited atomic states and of electron relaxation process are much shorter than the ejection time of the species after it has been ionized. This poses the problem of the existence of an antibonding state having a lifetime sufficiently long to cause the ion to gain enough kinetic energy to desorb. For oxides this is plausible; however it would not be the case for metals. In any case, the Madelung potential scheme can hardly account for the desorption of neutral and negative ions. Antoniewicz has therefore proposed the following explanation for the latter case. Due to ionization, the atom adsorbed on the substrate becomes reduced in size so that the new equilibrium position is closer to the substrate after ionization than before. It starts moving to reach this new equilibrium position, gaining kinetic energy in this way. During this displacement it becomes neutralized by electron tunneling from the substrate or by Auger neutralization. Recovering its initial size, it rebounds on the substrate, flies in the opposite direction with the kinetic energy it has acquired, and is ejected into the vacuum. The emission of negative ions implies two tunneling processes. Feibelman (1981) has shown that the reneutralization rate of a doubly ionized surface species was 10-100 times weaker than that of a singly ionized state. Thus reneutralization is much less likely in Auger-induced desorption than in MGR desorption. The kinetics of ESD is described by N ( t ) = N ( 0 )exp( - N,,ot)

(219)

286

M . CAILLER, J . P . GANACHAUD, A N D D . ROPTIN

where N ( t ) is the coverage (in atoms cm-') at time t, N ( 0 ) is the initial coverage, Np is the flow of incident electrons, and (r is the desorption cross section. The stimulated desorption cross sections range from lop2, to cm2 (Fontaine and Le Gressus, 1981) and depend essentially on the adsorbate-surface binding energy. For the species usually studied in AES, (r N 10-22-10-23 cm2. This allows the use of current densities of about 10-'-10-2 A ern-,, with no appreciable desorption for the normal duration of an Auger analysis (1000 sec). To avoid the desorption of weakly bound species, one has to use current densities much lower than the above values.

4. Electron-Impact-StimulatedAdsorption The existing studies concern a limited number of systems. One of the most frequently encountered problems is that of the adsorption enhancement of CO and 0, on semiconductors. For silicon (Coad et ai., 1970; Joyce and Neave, 1973; Joyce, 1973; Kirby and Dieball, 1974; Kirby and Lichtman, 1974) generally, the presence of carbon is reinforced all over the surface of the sample, whereas for oxygen this effect occurs only in the impact zone of the electron beam. The mechanism invoked to explain these results is based upon a three-step model: (1) CO and 0, adsorption; (2) dissociation; and (3) surface diffusion of carbon, electron-induced desorption, and slow diffusion of oxygen into the bulk. The surface concentration in oxygen still corresponds to one monolayer, but the Auger signal of oxygen keeps increasing owing to the diffusion of oxygen in Si. The amplitude of the oxygen signal increases with the primary energy of the electrons and the pressure of the gases, but decreases with increasing temperatures of the target. Fontaine et ai.(1979)have expressed some doubt about the validity of this model. In fact, they have remarked that no carbon enrichment occured in silicon whatever the residual pressure was (lo-' or to lo-'' Torr) when the sample itself contained no carbon. On the contrary, if the target contains some carbon, this carbon will appear by diffusion under the action of the beam. It will diffuse towards the surface, up to the boundaries of the bombarded zone, where it remains frozen. Under lo-' Torr, the oxygen concentration increases very rapidly at the surface of the sample in the impact area of the electron beam, whereas at 2 x lo-'' Torr, it remains undetected except when the target has been Torr, the oxygen previously ion etched in an oxygen atmosphere. At results from residual gases, whereas at 2 x lo-'' Torr, it results from the diffusion from the bulk of the sample. Unlike silicon, the oxidation of a certain number of other semiconductors, reinforced by the electron beam, leads to the formation of oxides in the impact area. This is the case for


287

germanium (Margoninski et al., 1975), for AsGa (Ranke and Jacobi, 1975), and for InP (Olivier et al., 1980). For AsGa and InP, a volatile oxide is formed (As,O, and phosphorous oxide). Here carbon monoxide plays no role, and the oxidation is interpreted as resulting from adsorption of molecular oxygen followed by dissociation at the impact of the beam and by a reaction between the substrate and the atomic oxygen. Enhancement due to the excitation of oxygen in the gaseous phase plays a negligible role (Ranke, 1978). Oxidation has been observed in InP, even for oxygen partial pressure as low as 5 x lo-'' Torr, the enhanced adsorption remaining localized in the irradiated area. For oxidation of Si, one can remark that the silicon oxides become dissociated by electron bombardment, leaving an excess of elemental Si on the surface (Delord et al., 1980). Nickel has also been the object of several studies. Verhoeven and Los (1976) observed that in an oxygen atmosphere (10Torr), an ordered arrangement of chemisorbed oxygen can be formed, up to 0.5 monolayer, followed by a nucleation of NiO, giving a passive film of about 2 monolayers. Oxidation speed could be increased by a factor of 5 by the action of the electron beam. Frederick and Hruska (1977) note that in a C O or CO, atmosphere, the height of the carbon peak varies with time according to the law

+

H = A[1 - exp( - t/~)] H,

(220)

where the value of z is in both cases about 30 or 40 min and H , is the value of H in the absence of the electron impact ( H , N 4'5). The scheme considered here is one previously encountered : adsorption, CO and CO, dissociation, and oxygen-stimulated desorption. Tompkins (1977) noted that H,O interacted with the surface of nickel in the presence of the electron beam to form, in the bombarded area, a stable oxide film several tens of angstroms thick; H,O could be physisorbed in small quantities, but for sufficiently long times, to allow interaction with the incident electrons. Lichtensteiger et al. (1980) have observed that a clean, strongly nonreactive surface of CdS in an H,O atmosphere (even at lo-" Torr), submitted to electron impact, could adsorb oxygen. This stimulated adsorption could come from local activation of the surface rather than from dissociation of previously adsorbed species. In the first step, oxygen becomes bound to the sulfur atoms. In the second step, oxidation of CdS takes place with increasing quantities of oxygen atoms bound to the Cd atoms. The electron beam can also stimulate the adsorption of C, even for pressures lower than 10- l o Torr of CO, CO,, and CH, (Joyner and Rickman, 1977). At the end of the experiment, the deposit is graphitic, and the saturation coverages are independent of the carbon source and close to the density of the

288

M. CAILLER, J . P. GANACHAUD, AND D. ROPTIN

atoms ern-,). The scheme proposed is (0001) plane of graphite (3.8 x the same as for Ni, plus the last phase that adds graphite. 5. Beam-Induced Dissociation This dissociation mechanism plays an important role as a step for the desorption and adsorption of species present on a substrate, these processes being stimulated by the electron beam. The dissociation of these species can induce an artifactual contamination, principally in carbon, due to the presence of CO (Hooker and Grant, 1976) and of hydrocarbons in the residual atmosphere of the chamber. The higher the resonance energy per n electron, the more delocalized the 71 electrons, and the less the species will be damaged (J. T. Hall et al., 1977). Moreover, the dissociation cross section varies with respect to the primary energy E , of the electrons according to the relation (J. T. Hall et al.) d E P )21 (WEp)WpP)

(221)

where this dependence comes mainly from the factor l/Ep. This shows that the dissociation cross section is essentially inversely proportional to the primary energy. The target itself can also become dissociated. Most studies have been devoted to SiO,. For such a target, the problem of charging is also set. Carriere and Lang (1977) suggested that secondary electrons were responsible for the creation of surface charges and for bond breaking, producing elemental silicon and oxygen. Ichimura and Shimizu (1979) studied the topography modifications and dissociation provoked in thin SiOz films by a scanning Auger electron microscope. They showed that the damages were strongly connected with the amount of energy dissipated in the sample. Bermudez and Ritz (1979) indicated that small Si clusters of a few atoms appeared in the irradiated area. In multiplexing, the simultaneous bombardment of a given zone of the sample by ions and electrons provokes enhancement of the sputtering. In the case of SiO, ,the electron bombardment alone reduces the concentration in oxygen, whereas the simultaneous impact of electrons and ions leads to the sputtering of the Si atoms by the ions and to the elimination of the oxygen atoms at the surface by the electrons. Consequently, sputtering speed is enhanced (Ahn et al., 1975). 6 . Migration and Difusion

An electron beam can also provoke migration of some atoms. This is the case for sodium and potassium, for instance, during analysis of glasses (Pantano et al., 1975, 1976; Dawson et al., 1978; Malm et al., 1978). This


289

migration can be substantially slowed by decreasing the temperature. Diffusion induced by increasing the temperature in the case of targets deposited on glass substrates have been observed for multilayer thin films of Cu-Ni (Roll et al., 1979).

VIII. CONCLUSION In this review article we have described some of the major questions related to the development of quantitative Auger electron spectroscopy. We outlined all the steps going from theoretical first principles to a real sample analysis. As already mentioned, the path is far from complete, but with the help of very diverse approaches such as the simulation method, phenomenological description, calibration and sensitivity factors, matrix effects, and ion- and electron-beam effects, progress has been made. All these approaches are essential for improving the quantification of this method of analysis; however, there is now a need for more rigorous standard procedures, and this will no doubt result in further progress. Particular attention was paid here to line-shape analysis. The authors’ interest in this topic was initiated by important work done by Gaarenstroom (1981)which could lead to the development of a new quantitative procedure. As mentioned in Section I some topics have been entirely omitted, such as evaluation of Auger line intensities or the angular aspects of Auger emission. It was merely a matter of choice and of time. ACKNOWLEDGMENTS The authors are indebted to the following copyright owners for permission to reproduce tables and formulas: McGraw-Hill Book Co., New York (formulas from Slater, 1960), John Wiley and Sons, Inc., New York (formulas from Kittel, 1967), Cambridge University Press, London (table from Condon and Shortley, 1970).

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Kolmogoroff spectrum in the inertial region: ~ - 1 1 / 6

-Y

p.

Dissipation region

* FIG. I . A typical example of an isotropic Kolmogoroff spacewise spectrumfor the dielectric fluctuations: (a) I,, mean scale size of the dielectric fluctuations; I,, characteristic scale size of the dissipation region; (b) K = k - k’.

308

D. S. BUGNOLO AND H. BREMMER

inertial and dissipation region. Using the root-mean-square (RMS) scattering angle previously defined by Bugnolo (1960a),

ORMS E [ ( E ~ ) R / ~ , ] ~ / ~ where R is the range or distance traveled by the wave in the stochastic dielectric. It follows that the wave number K will lie within the inertial region of the spacewise spectrum of the dielectric fluctuation if and only if the wavelength of the electromagnetic (EM) wave A, is such that

It follows that Eq. (13) will be most relevent when the range R is less than

(W3.

In any case, a complete solution of the stochastic propagation problem will require a detailed knowledge of the dielectric fluctuation spectrum. Although Bugnolo (197213)has already addressed the problem of turbulence, it is our opinion that more work is required in the theory of turbulence if progress in the corresponding propagation problem is to be forthcoming. For this and other reasons, such as the answer to the pointed question, What is actually really known on a sound statistical basis about a complete propagation path?, we shall restrict ourselves for the time being to wavelengths which satisfy Eq. (13). Some numerical examples are in order. For a typical propagation path in the earth's troposphere, we might use consequently A, >> lo-' m, a conthe values 1, E 10 m and (c2) z dition that is most certainly satisfied at frequencies below the far infrared. As our second example we shall use the turbulent interstellar plasma which is of interest in the study of pulsar pulse shapes (Bugnolo, 1978). Here we take I , = 3 x 10" m and d , = 1014 m at 300 MHz. Similar considerations indicate that we may use our theory in the frequency range below about 10 GHz. Prior to concluding this section we should like to note that whereas the region bounded by the condition dB >> 1, is well posed and easily calculated when I,, may be determined experimentally, a large body of literature exists for wave propagation at optical frequency that clearly violates this simple condition unless 1, is assumed to be very small.

B. Evaluation of the Functional Derivative in Eq. ( 9 ) According to our second physical assumption, SE should propagate as if the E disturbance were absent almost everywhere; therefore we may proceed with the solution of Eq. (11). We shall solve this equation for the vector SE

STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION

309

with the aid of operational calculus. We introduce the four-dimensional “image functions” ej(P, 4) = PlP2P34

s

dr dt exp[ -(P

+ qt)] GEj(r,t,

*

f

(16)

aj(P, 4) = PlP2P34 dr dl exp[ -(p’r

f

4t)]Xj(r,

where p is a vector with components (pl,p 2 , p3). The vector equation corresponding to Eq. (11) is now “imaged” into [ p 2 - (q2/c2)]e - p(p-e) = q2a,

xj E

1

c - ~ ~E~,E,

(17)

r

The solution for the three components ej of e may be found with the aid of a determinant to yield

In view of the “operational images”

and the transformation rule for the convolution products, we obtain the following “operational original” of Eq. (18):

This involves the explicit expression 1 J {a2xj(rff,t”) GEj(r, t ) = - - dr” dt” atif2 4n -

6(t - t” - Ir - r”l/c) Ir - r”l

F m a2

c2

After applying partial integrations, we obtain

-

c2

1Xr(r”, t”)

6(t - tf’ - Ir

-

~

r

Ir

- rnI

r”I/c)


310

Substituting the definition of the components of x , we find 6Er,(r“, t“)Es(r”,t“)

~

4.n -

1 c2

a2

r

I

SEjr(r”, t”)Er(rrf, t”) -

~ (-t

atn2

a2

ax: ax:,

t” - Ir - r”l/c) (22) (r - r n )

Remembering that the disturbances BE,, are confined to an infinitesimal domain around r” = r’, t“ = t’, we obtain the functional derivatives

The dependence of the functional derivative on the second-order partial derivative may be explained physically by noting that the dielectric constant normally enters this analysis by way of a second time derivative in the wave equation [see Eq. (5)]. C . The Ensemble Average of &;jCjk,a Most Relevant Parameter

The third-order correlation function involving the’dielectric fluctuations and the electric field intensity is required if we are to reduce Eq. (8). A similar problem must be addressed in the parabolic equation formulation of the mutual coherence function (for example, see Ishimaru, 1978, Vol. 2, pp. 412-414). It is best to begin by writing the ensemble average under consideration in its complete form: (Eij(r

+ r‘/2, t + t’/2)Cjk(r,t ; r’, t ‘ ) )

This may be transformed by subtracting r‘/2 from r and t‘/2 from t to yield

(Eij(r,t)Er(r, t)Ek(f - r‘, t

-

t’))

(24)

We now proceed with the aid of the Novikov relation [Eq. (9)] and the property that the functional derivative of a product can be determined in the same way as the derivative of an ordinary product: (Eij(T,

t)Ej*(r,t)Ek(r - r’, t - t’))


31 1

Next, applying Eq. (23) and its complex conjugate and using the definition (Eik(f,

t)&ik(r’,t‘)) = hik(r - r’, t - t’)

we obtain

s

(cij(r, t)Ej*(r,t)Ek(r- r’, t - t’)) 1

=-

4n

dr” dt“hij(r - r”, t - t“) (E,(r - r‘, t - t‘)ET(r“,t“))

x(+---)

axi!

ax; ax;

a2

c2 a t r r 2

h(t - t” - [r - r”[/c) (EF(r, t)Ej(r”,t”)) Ir - r“I

c2 atfr2

b(t - t’ - t” - Ir - r’ - r”I/c) Ir - r’ - rrrl

d{

+

By a shift of the arguments, while making use of the definition for the Ciis [Eq. (5b)], we further obtain

=

4n

j

drf’dt”h,, (r r‘

x {(Cjk(i -

+ r‘ - r”, t + -t’2 - t” rft t - t‘

+T,

t” r‘ t‘ + ?; -r + - + r’f, -t + + t” 2 2 -

The other quantity needed for the evaluation of Eq. (8)’ (&kj(r- r’/2, t - t’/2)Cij(r,t ; r’, t’))

is obtained from the preceding one by, in succession, (1) interchanging i and k, (2) replacing r’ and t’ by -r’ and - t’ and (3)taking the complex conjugate. We have yet to apply the identity

Cg(r, t ; r’, t’) = cki(r, t ; -r’, -t’)

(28)

312


After some algebra, the resulting equation reads r'

1 47t

=-

dr" dt"hkj(r -

r'

-

t' r", t - - - t" 2

-r + -r'+ - r" -t + - +t' - - ;t"r + - - rr'" , t + - - t " 2

r

4

2'2

4

2

r' + r" -t - t' t" + -; -r 2'2 4 2

2

t'

2

+ r'2 + r", -t + -t'2 + t" ~

D. The Equation for the Ensemble Average of the Electric Field Correlations

If we substitute Eq. (29) in Eq. (8) and replace r" and t" by new variables of integration defined by s = r f r'/2 - r'' and z = t t'/2 - t", we obtain after some algebra


313

where CS(;,~) marks the Kronecker 6 function 6!, henceforth represented by 6(a,b) should this representation become simpler. In the present case, a = i or k and fl = j . This is our equation for the ensemble average of the electric field correlation (C i k )under the condition of the two physical assumptions.

111. DERIVATION OF THE EQUATIONS FOR THE WIGNER

DISTRI~UTION FUNCTIONS A. Zntroduction of the Wigner Functions and Derivation of Their Equations

The matrix for the Wigner function is related to the matrix for the electric field correlations by Wik(r, t ; k, W )

-

- dr‘ dt‘ exp[i(k r‘

-

ot‘)]Cik(r,t ; r’, t’)

(31)

The inverse of the Fourier transform reads n

Cik(r, t ; r’, t’) =

J

dk’ do‘ exp[ - i(k’* r’ - dt’)]Wik(r,t ; k , o r ) (32)

In terms of the Wigner function, Eq. (30) may be written

x

1

dk d o exp[ - i(k * r‘ - at’)]( Wik(rrt ; k, 0))

314


dk do exp[ - i(k. r’ - cot’)]( Wjk(r, t ; k, o))

x

-

+

4y

(? 2 4ac2 at - at

{J

x

+

s

ij

ds drh,(s, z) k

dk do exp{ -i[k-(r‘ T S) - d t ’ T

.)I>

k

((I;;

t’ z dk do exp[Ti(k.s - oz)] W j j r 2 - - -, t k - - -; k, co 2 2

b(~ f t’

- (ST r‘l/c)

(33) Evaluating the effect of the operators

x

x

a/&’

and d/dt’, we obtain

1

dk do exp[ - i(k * r’ - at’)]( Wik(r,t ; k, co))

J m( do exp[ - i(k.r’ - cot’)]( Wjk(r, t ; k, 4) ij

1


315

In order to recognize in all terms a Fourier transform with respect to the variables r’ and t’, we yet introduce a corresponding Fourier integral for the operant at the end of the second term on the RHS, as well as for the last term on the RHS. Thus the new RHS reads

ds dz h,(s, z) k

4

s

-

dk do exp[ - ik (r‘ f s) + iw(t’ T T)] z

Wj, r - -, t - -; k, w)) x ( ij( 2

s

+-6 4 1n 5 c 2 F sdsdzh..(s,z) X

x

{

dk’ dw‘ exp( - ik‘ r’ a

[dk’

dw‘ exp( - ik’ r’

6(~ I+)

(- a2asj 2 -)d2 as,

6’

- c2

IS[

az2

k

dkdoexp(Tik.sf i o z )

s

dh d p exp(ik‘

s

dh d p exp(ik’. h - iw’p)

+ iw‘t’)

+ iw’t’)

-

- iw’p)

We next replace the operator a/at k 2a/at’ by a/at & 2io‘ and omit the operator sdk do exp(-k.r’ + iwt’) in order to get rid of the variables r’ and t’; however, we first have to interchange k and k’, as well as o and ID’,

D. S. BUGNOLO AND H.BREMMER

316

in the second term on the RHS and then replace k by k‘ and o by o’in the last term. This leads to

{(i

T 2iky

-

-

(& T

$(:

2ikJ

2 i o y ] ( Wik(r,t ; k, w ) )

(& T

2ikj) ( Ydr, ij t ; k, a))

k

- 1 - 471c2

ds dthy(s, T ) exp(fik s T iwz)

-

+ 2iw

(it-

>’

S k

+-

1

dh d p exp(ik-h -

We must further apply the substitutions s = - 2r’ and z = - 2t’ in the first term on the RHS; h = + s & 2r’ and p = + T f 2t‘ in the second term on the RHS; and d(2r) = $6(r). We then obtain, after some reordering,

+ r ’ ~ + t’;

(Wjkcr

a2

d2

ky

ij

(d.:a.;

-

7s

1 +-64n5c2

s

k

dk’ do‘ dr exp[fi(o’

6(t’

7 p)

-

o)z]

s

dr’

+ Ir’l/c) lril


exp( f 2ik r’) ds hij(s,z) exp[ k(k - k’) s] S

k

( Wjj(r+ r’, t

exp[i(k.h - op)] ($(r

T

5,

S

317

dt‘

+ t’; k’, 0‘))

t T !)$i(r

$ -

?, h t

&

i))

(37)

B. Preparation for Further Analysis

According to the Wiener-Khintchine theory it is customary to pass, in the treatment of stochastic media, from the correlation function for the permittivity to a power function, connected to the former by a Fourier transform. In our vectorial treatment we have to introduce a new matrix F, the elements of which are given by

‘S

Pik(k, 0) - da dz exp[i(k a 16z4

+ wz)]hik(a,z)

(38)

We may consider Pi, to be a component of the spatial frequency spectrum of the dielectric fluctuations. The inverse relation reads

h,(a, z)

=

s

- + 0‘7)]Pik(k’,

dk’do‘exp[ - i(k‘ a

0’)

(39)

The space-time correlation function h, is a real even function in a and z; namely, h,(a, z) = & ( - a , -7). As a consequence, Pik is also real and even, i.e., Pik(k, a)= Pik( -k, - 0)

(40)

An inspection of Eq. (37) indicates that the correlation matrix and integrals of the form of Eq. (38) occur in both the first and second terms on the RHS. If the h, terms are replaced by integrals of the form of Eq. (39), this in turn will lead to integrals of the form

Integrals of this type occur frequently in stochastic propagation theory. They may be reduced in the following novel manner. Let us introduce a set of coordinates in the r‘ space such that the polar axis is taken in the direction

318

D . S. BUGNOLO AND H. BREMMER

dr‘exp(Tiar‘)[exp(TilVlr‘) - exp(+i(~lr‘)]

Hence dr‘ exp[ f i ( a

:j

-

In view of the condition Im a yielding

+ IVJ)r’]

dr‘ exp[ f i(a - (Vl)r’]

5 0,

both integrals converge at r’-+

P(a; V) = &(2.ni/lV1){T[i/(a+ Iv~)] & [i/(a - Iv~)]} or P(a; V) = 4z/((Vl2 - a’), Im a must apply the formula

J:

5 0. On the

00,

(44)

other hand, if a is real we

dr’ exp( T ipr’) = n6(p) f i/b

(45)

where the 6 function gives use an additional contribution. Recalling Eq. (43), we may write for the &function contribution to P(a; V),

If we use the identity

we obtain the contribution in question:

+4n2i sgn a6(1VI2- a’) Consequently, for a real we find,


319

Since we must in our analysis deal with the real quantities k, w, etc., it would appear that the more complex result of Eq. (49) must be used rather than the more simple result given by Eq. (44).We shall prove that

In fact, lim

&-.+I)

4a

IV(’ - (a T lim 4a

=

E’+O

i&)2

((V(’ - a’ (lV12 - az

+ 2 )T 2 i a ~ + &2)2 + 4a2E2

Taking t = IVl’ - a’, we must evaluate T

=

lirn [ ~ / ( (+t E’)’

e++O

+~ C X ~ E ~ ) ]

the limit of which may be obtained with the aid of the following reduction, assuming provisionally that a > 0: 1

&

(t

+ &’)’ + 4a2E2

2aE - i(t

+m:J

+ &’) + 2aE -k i(t + E 2 )

do exp[(2as W

do exp[i(t

+ it + is’)o]

I

+ &’)a - 2acloll

(53)

Hence, taking E = 0, we have, 71

. = ‘ , f a

4a

-w

do exp(ito) = 201 d(t)

(54)

Finally, substituting for t, we have derived from Eqs. (50)-(52), for the case of a positive a,

320


Since the second term in Eq. (55) is odd in a,a factor sgn a may be added to it such that the result may be valid for all real a.This concludes our proof that Eq. (50) is the equivalent of Eq. (49). We shall use as the abbreviation for the positive value of E + +0, simply +O. We thus have proved the following:

Before proceeding further, we note that the following requires the introduction of the Fourier integrals for the Wigner functions, and these are defined as follows with the aid of a transform F: Wik(r,t ; k, co) =

s

dh dp Fik(h,p ; k, co) exp[i(h * r

+ pt)]

(57)

All of the above considerations are used to further the reduction of the RHS of Eq. (37). The first term on the RHS may be termed the “radiation” contribution since the argument of the W, terms is independent of k , and consequently is not associated with the mutual interference between different k waves. This type of term is indicative of a general emanation or radiation that results in an attenuation of the wave. It may also be attributed to some absorption effect. On the other hand, the integrations over k’, w’ in the second term on the RHS of Eq. (37) reveal an “interference” between the waves of differing wave numbers. Finally, the last term on the RHS of Eq. (37) may be associated with a source effect.

C. Reduction of the Radiation Contribution R,

As explained above, the first term on the RHS of Eq. (37) shall be termed the “radiation” contribution. Returning to Eq. (37), we proceed by substituting Eq. (39) for the correlation function hij and the Fourier integral Eq. (57)] for the Wigner function. This leads to the following expression


32 1

for R,;

x exp[ f 2i(k * r‘ - wt’)]

-

y

+ 2iw

(it -

d l dp Fjk(I.,p ; k, w ) ij

x exp[iI.*(r k

or

R, =

-

X

X

exp[i(i * r

+ pt)]

s

mC. do”Pij(k,of’) dr’ dt’ s

k

exp[i(l T 2k - 2k”)’r’ + i ( p & 2w

+ 2w’‘)t’I (59)

X

The last fourfold integral, a, say, over r‘, t’ may be reduced by partial integrations over the coordinates x’, y’, z’,and t‘. Since each of these integrations extends from - co to + 00, the integrals tending to zero at these boundaries, we may write the following after applying partial integrations :

-(5 T 2k - 2 k ) i - ( 5 f 2k - 2k”)j + a ( p k

20

C’

x exp[i(l f 2k - 2k”)- r‘

+ 2~”)’

+ i ( p k 20 + 2w”)t’l

(60)

Next, we integrate over t‘ to obtain =

{

exp[i(I. T 2k - 2k”)*r’- i ( p f 2 0

+ 20”)lr’l/c]

Ir‘l

To this result we next apply Eq. (56). Hence, ~~

[-(I. T 2k - 2k”)i, (I. T 2k - 2k”)j + d c , *

@=4n‘

j)

(pk 2 0

+ 2w”)2

0’ c

((I. T 2k - 2k”)(’ - ~ - ~ &( 2w p + 20’’ - iO)2

(61)

322


Substituting this result into Eq. (59) for R , , we find

-(A T 2k - 2 k ) i k ’ ( 1T 2k - 2k”)j X

x

11 f Fjk(k, ij

+

(p f 20 S(i,j)

+ 2o”)2

c2 2k - 2k12 - ~ - ’ ( pf 20 + 20” - i0)’

p ; k , o)exp[i(l*r

+ pt)]

This may be put into a more convienient form by again using operato to remove the term in brackets from the integrations on h’ and p. Since then follows that the remaining integration over these variables equals tl Wigner function, we may write, taking k“ =. k ,

x W$r, t; k , a> V

This concludes our present reduction of the radiation term.

D. Reduction of the InterJerence Contribution R, The second term on the RHS of Eq. (37) was identified as the interferen term. It shall be evident later that this term, in the limit of single scatterir yields a first Born approximation for the Wigner function. Again, using Ec (38),(39),and (57), we obtain 1 R, = 647csc2 x

1

fdk’do’ f

dz exp[ fi(o’- o ) z ]

dr’ exp[ f2ik.r’l

- + o”z)]Pij(k”,w”)

d k do” exp[ - i(k” s

x exp[ k i(k - k ‘ ) s]

f

s

dt‘

k

exp(T 2iot‘)

-

>’

+ 2io

(it-

I

ds

323


x [dh dp Fjj(I,p ; k’, w ’ )

The t’ integration yields 2nS[o” T (w‘ - w)], enabling the This yields

7J

1 R2

=

dk’ do‘

J

J dr‘ exp(k2ik-r’)

ds

x jdk”P$k; +(a’- w)] exp[i(-k” f k T k)-s] x exp( f 2iwt’) -

(:t-

>’J

+ 2io

0’’ integration.

s

dt‘

d I dp Fjj (I, p; k’, w‘)

By following a procedure similar to that of the previous section, we may eliminate the integrations over r’, t’, s, and k” to yield 1 R2 = 2 x

1

dk‘ d o ’ P i j [f ( k - k’); f(of- o)] k

I

-

+ 2io>’j d I d p Fjj(h, p ; k’, 0’)

(it-

x exp[i(h-r

f 2k)k + (Sf/c2)(p f + pt)] -(I f(h2k)i*(I f 2k)’ - ( p T 2 0 i0)’/c2 -

20)’

(67)

Again, I may be replaced by the operator -a/& and p by - ia/at, so that in view of Eqs. (57) and (40),we obtain the desired result:

“J

dk’ j

J

dw’Pij[(k - k’); (o- o’)]Wjj(r, t ; k‘, m’) k

We note that the “interference” term samples the spectrum of the dielectric fluctuations at the wave number (k - k ) and the frequency (w - 0’).

324


E. Final Operational Equation f o r the Wigner Function This may be obtained by substituting Eqs. (64) and (68) into Eq. (37). After a minor transformation of variables by way of k” = k - k‘ and w” = w - w‘ in R , , we obtain the desired result:

{:(

f 2ik)i -

-

(&

-

7(& 4 2)i 7j

T 2ik;)

k

-

$ (i+ 2iw)2}( Wik(rrt ; k, a))

A(w C2

T

2 at

x {a;(k‘, CO’;

f

( W,k(r, ij t ; k, w ) )

dk‘ dw’Pij(k - k’, w - a’) k

a/&, a/&)( Wjk(r,t ; k, w ) + ij

k

x ( k , CO;

2ikj)

a/&, slat) ( W j j ( r ,t ; k’, a))} + S:

(69)

where

is a useful operator, and where the source term is given by

x exp[i(k.h - up)] (E;(r f

h

2, t f

;)(r

h

f 2,t k

;))

(71)

In Eq. (69), (i, k ) = 1, 2, 3. As a consequence we have a total of nine equations for the ensemble average of the Wigner function. These individual equations are elements of the matrix equation for ( W ) . Equation (69)may be put into a more compact form as follows: Let Vc

and let the matrix

= (a/&)

T 2ik,

V;

= (a/&) rfl

2iw

(72)

A’ be defined by

Finally, let the elements of the R H S of Eq. (69) be used to define the matrix

325


R i . It then follows that Eq. (69) with the upper sign may be represented by {(Vk+)’ - C-’(V:)’}( Wik) - C (A’)ij( Wjk)

=

Ri

(74)

=

RLk

(75)

j

whereas that with the lower sign may be represented by Wik) - C ( x - ) k j ( Wij)

{(VL;)’ - C-’(V;)’}(

j

From the properties of the V and A functions defined above, it is selfevident that the two equations are not independent, one being the complex conjugate transpose of the other, since the Wigner function is itself Hermitian.

Iv. RELATED EQUATIONS FOR THE WIGNER DISTRIBUTION FUNCTION A. An Integral Equation of the Second Kind for the Wigner Function

Equation (69)is a complete operational equation for the ensemble average of the Wigner function. In its present form it is striking that even the terms remaining in the case of a homogeneous medium, i.e., for P = 0, are rather complicated. One possible way around this difficulty is the following. We begin by writing Eq. (74) in matrix form: [((VZ)2

-

c-2(V,+)2}1- A + ] ( W ) = R +

(76)

The matrix solution of Eq. (76) reads

( W )= [{(VZ)2

-

c-2(V:)2}1

-

,+I-%+

(77)

In the above, I is the unit matrix. In order to further reduce Eq. (77), we must use the general formula which may easily be proved:

Applying Eq. (78) to Eq. (77), we obtain

Once again using the definition for V: and ments of the ( W )terms are Ri

(w)ik

=

C2 -

[ ( a p t ) + 2ioI2

(L

-

A+, it follows that the ele-

2iki

1

(”-)+ axi

2 i a 2ik) - ;;Z(’

~

(a:j 2 2io)

- 2ikj

)

Ri (80)

326


This, in turn, yields the following integrodifferential equation for the Wigner function:

{(z

- 2ik)i- ?(% i a

+ 2iw)i}

(Wik)

dr

=Ri

-

- 2iki) 5: (& - Zik,) + 2ioI2 (2 axi

C2

[(a/&)

R$

(81)

Although this equation is somewhat simpler in form than Eq. (69), it is still quite complicated in a homogeneous medium. This difficulty may be removed as follows. Let the RHS of Eq. (81) be denoted by the function G such that

{(:

- 2ik)i

-

f (E + 2io)i} (Wik(r,t; k,

0)) =

G(r, t ; k, w ) (82)

We can transform this to a more tractable form by defining a function such that

-

( Wik(r, t; k, w ) ) = exp[2i(k r - wt)]$J(r,t ; k, o)

4

(83)

Substituting Eq. (83) in Eq. (82) we find that [(a/&) - 2ikI2(Wik) = exp[2i(k*r - at)] A similar relation may be obtained for the operator

[(apt) + 2io]’(Wik)

(84)

+ 2io),

(a/&

- wt)]d24/dt2

= exp[2i(k.r

(85)

We thus arrive at the following inhomogeneous four-dimensional wave equation for 4:

(..- f $)4(r,

-

t; k, a)= exp[ - 2i(k r - wt)]G(r, t ; k, w )

(86)

In the absence of an inhomogeneous scattering medium, G = 0 and Eq. (86) becomes homogeneous with the solution for the homogeneous background 4pr.By treating the RHS as a source function, we may arrive at a formal ‘‘solution’’:

M,t; k, o)= &(r,

‘s

t ; k, w ) - 471

dr’ dt’

d(t - t’

- Ir - r’l/c)

Ir

-

x exp[ -2i(k*r‘ - ot‘)]G(r’, t‘; k, a)

r‘( (87)

This is a formal integral equation of the second kind for 4. It is of some importance to develop such an integral equation for the ensemble average of the Wigner function itself. There are many reasons for


327

such an approach. The first is that a first Born approximation to the integral equation is very easily obtained. Another is that the first-order smoothing approximation by way of the diagram method may be used to obtain a solution valid in the multiple-scattering region. For examples of the smoothing approximation, see Frisch (1968) and Bugnolo (1972a). Integrating Eq. (86) over t’ we obtain

4@,t ; k, 4 = 4pr-

exp(2iot)

4n

s

dr‘

+

expi-2iCk.r‘ (o/c)lr - r’l]) Ir - r‘l

Now according to Eq. (81), the G function can be written as G(r, t ; k, o)=

1

- zitti)(&

-

2ikj)]Rf

(89)

j

In order to remove the integration over r’, we introduce the Fourier integral Rj+k(r,t ; k, w ) =

s

ds d t b j k ( s ,

1

t;k, 0) exp[i(r

*s

+ tt)]

(90)

Substituting and applying the operators yields

4b.7 t ; k, 4 = 4prx j

exp(2iot)

s

471

dr’

+

exp{ -2i[k-r’ (o/c)(r - r’l]} Ir - r‘l

ds dz bj&, 7; k, 0)

c2 (isi - 2iki)(isj- 2ikj) (it 2i0)~

+

In order to apply Eq. (56),we must change the order of integration as follows:

x

{

exp(i7t) 8; -

c2

(t

+ 242

- 2 k J ~ j- 2kj)

( ~ i

+

exp(i[(s - 2k)ar’ - (t 2co)lr - r‘l/c]> )r - r’)

(92)

328

D. S. BUGNOM AND H. BREMMER

From this point we follow the usual procedure. Using Eq. (56), we then translate the { } outside of the integral over S and z. We then use the defining equation for Q [Eq. (70)]. This yields

-

M . 3

c2 exp[ - 2i(k r - wt)] t ; k, 0) = 4pr- 7 [w - +i(d/dt)I2 x

1Q$(k, co; d/dr, d/dt)R$(r, t ; k, w )

(93)

j

Returning to the Wigner function by way of Eq. (83),

( W i k ( r , t ; k, @I) = wikpr(r, t ; k, 0) x

C

O;

C2

410 - 3i(d/dt)12

a/ar, d/dt)Rj+k(r,t ; k, 0)

(94)

.i

This may be put into final form by using the RHS of Eq. (69) for R i k with i replaced by j :

( W i k ( r , t ; k, O)> = Wik,,(C, t ; k, 0 ) -k

x

[F /

1a&&,CD; a/&, d/dt) i

dk’ do’Pi,(k - k’; w - w’){Q;(k’,

0’; d/dr,

d/dt)

It is self-evident that we may obtain a first Born approximation for the ensemble-averaged Wigner function by replacing (Wlk) and (W,,) on the RHS by Wpr. This result may prove to be of use in certain special circumstances.

B. The Transport Equations for the Wigner Function

In our preceding analysis we considered two simultaneous equations for the ensemble average of the Wigner function which, however, are not independent. This dependence is connected by the requirement that ( W i k ) be Hermitian. That this is the case is self-evident from the definitions given by Eqs. (31) and (7).


329

We may construct a stochastic transport equation for the ensemble average of the Wigner function by taking the difference of the equations with the upper and lower signs. The same result may be obtained by using the imaginary part of the other equation. For example, from Eq. (81) we obtain

This result is complicated by the fact that the RHS depends on the other components of the Wigner function. We shall return to this later. A number of comments are in order. Stochastic transport equations of the type described above constitute a generalization of the transport concept to the case of stochastic media in the large. Whereas their most simplified forms appear to be similar in a superficial manner to the classical transport equations of astrophysics (see, for example, Chandrasekhar, 1950), they are quite dissimilar in detail as a result of the fact that we are dealing here with stochastic media in the large. In fact, any similarity may be superficial and some caution is advised. On the RHS of Eq. (96), the operator L where

indicates that the rate of change of ( Wii) in the fi direction is to be measured while moving in this direction with the velocity

Vobs = (C2/4k In fact, the rate of change per unit time is given by

(98)

(99) On the other hand, the phase velocity of the special plane waves associated with ( Wik),i.e., exp(k r - wt), must satisfy the relationship D/Dt

=

(c2/w2)L

V,.k = w or k.Vph= w/k (100) It is evident that since lVp,,l may exceed c, we must satisfy the relativistic constraint We shall return to the stochastic transport equation later.

330


v. ASYMPTOTIC EQUATIONS FOR THE WIGNER DISTRIBUTION FUNCTION A. The Forward-Scattering Approximation

Forward scattering of electromagnetic waves by stochastic dielectrics is a special case of interest with many applications. By this we mean such applications where the forward-scattered component of the Wigner tensor is of the utmost importance. Before proceeding, it is best to review some fundamentals. Given an isolated volume of space characterized by the stochastic dielectric E, and given that the volume is of linear extent L such that L is small compared to the MFP for scattering by the stochastic dielectric [our dB of Eq. (12)] and yet large compared to the mean scale size of the dielectric fluctuations, we may proceed as follows. Let the dielectric fluctuations be anisotropic so that the mean scale size is a tensor related to the correlation function hij by I-

loij

E

J hij(s)ds

By invoking conventional approximations, we may define a scattering cross section per unit volume, per unit solid angle by

a(K) = 2nk: sin’ xP(K)

(103)

In this result a(K) is obtained by way of a first Born approximation for the scattered field, and P(K) is defined by Eq. (38), z = 0. In effect, in this particular formulation the fluctuations are characterized by a time-invariant space correlation function. The geometry for this approximation is illustrated in Fig. 2. The details of the above may be found in Ishimaru (1978, Vol. 2,

Direction of scattering

FIG.2. Geometry for scattering for the case of a plane wave incident on a stochastic dielectric of Volume V : K r 2k, sin(O/2).


33 1

Section 16-2). In order to proceed we require a model for the dielectric fluctuation spectrum P(K) defined by Eq. (38). Before continuing it may be judicious to indicate the differences between our formulation and that commonly used by authors in the U.S.S.R. Some time ago they noted that a stochastic dielectric such as the earth’s troposphere might be characterized in terms of a “structure function’’ defined by

Df(4 = ( I f ( r 1 + r) - f ( r , ) I 2 )

( 104)

(See Ishimaru, 1978, Vol. 2, Appendix B or Tatarskii, 1961.) As an example, consider the case of an isotropic turbulence producing, in turn, an isotropic P(K) and Df(r). Existing theories indicate that the spectrum for such a case will be Kolmogoroff in form. This yields for the structure function the form

where 1, is the mean or outer scale size and Id is the dissipation scale size. For this case, Tatarskii (1961, 1971) obtains a spectrum of the form

P ( K ) = 0.033C,2K-”’3 exp( - K2/KH)

(106)

for K > 271/1, where K , = 5.91/ld. We note that although this form may only be applicable for K > 27&, it does not depend on the outer scale size 1, explicitly. This is usually justified by noting that the outer scale size I,, is difficult to measure and is dependent on the geometry, whereas 1, is somewhat independent of the geometry. Instead of the above we have used a formulation originating with Norton (1960). In our notation, Norton defined a space correlation function for the dielectric fluctuations of the form

where r ( p ) is the gamma function and K , the modified Hankel function. If this formulation is used, then

We note that this formulation depends on the outer scale size 1, explicitly. It does, however, have the disadvantage of being restricted to wave numbers less than 27&. This latter problem has been addressed by Bugnolo (1972b) for the case of a weakly turbulent gas or plasma. Now the upper bound on the frequency which resulted from the second physical assumption of Section II,A,2, i.e., Eq. (13), is based on the North

332

D. S. BUGNOLO AND

H. BREMMER

model for the case when the parameter p = 113, the Kolmogoroff case [Eq. (108)], rather than Eq. (105). We suggest that since the outer scale size 1, can indeed be measured in the troposphere and in the ionosphere, this seems to be a better approach.

1. Complete Equations for the Wigner Function Returning to Eq. (103) and using Eq. (107), we note that the scattering will be well directed in the forward direction, i.e., small 8, and in fact will be concentrated within a cone of half-angle

ezq24,

(109)

It follows that the scattering will be primarily in the forward direction when the wave number of the electromagnetic field is such that

(kl0)-' ki+2cor2

+-

x Im( Wlk(r,t ; k, w ) )

- -Re

X

(

- 4 k.= - 4 1 j

+ --

r:

(YZ26.i -

Re( Wik(r,t ; k, 0))

it) -

Si

)[[F

kikj

dk’ dw’Pjl(k - k’, o - 0’)

Im( Wlk(r,t ; k, 0))_ -.n 6(k‘ - (o’(/c) k’2 - u 1 2 / c 2 2 w’/c


x ( WJr, t ; k', 0'))

335

I1

Equation (120) is a stochastic transport equation for the real part of the Wigner function, whereas Eq. (119) is such an equation for the imaginary part of (Wik).At present we are of the opinion that Im( Wik) is of less importance then Re( Wik).However, this observation may require further study. Equations ( 1 19) and (120) are some of the most important results of this work. We note that our stochastic stochastic transport equations are valid in the absence of stochastic fluctuations in the medium as well as in the most general case, which includes dielectric fluctuations moving with the mean wind and subject to internal decay as predicated by the theory of turbulence. 3. Integral Equation of the Second Kind for the Wigner Function

Finally, we write the forward-scattering approximation for the integral equation of the second kind for the Wigner function [Eq. (95)]. By following a procedure similar to that above, we obtain

(W i k ( r , t ; k, o))

VI. EQUATIONS FOR SOME SPECIAL CASES In order to reduce our general results [Eqs. (1 19) and (120)] in the forwardscattering region to some special cases of interest, we must first address the stochastic media itself and extend our characterization of Section V,A. We be& by notiqg that our general result €or forward scattering contains a matrix expression for the space-time spectrum of the dielectric fluctuations.

336


In view of this it may be applied to the most general cases of an anisotropic stochastic medium such as those found in many places as a result of natural phenomena. The effects of propagation through such a medium may be illustrated in a physically quantitative manner by the following description. Let us begin with a plane-wave source of monochromatic spectrum at some distance from a semi-infinite, time-variable stochastic medium. As this wave begins to penetrate the medium it will first suffer from a slight loss of phase coherence, an effect that may easily be described by a first Born approximation or by the WKBJ approximation of our Eq. (121). Such phase perturbations of course require that the original spectrum of the wave, say 6(0 - w,), be spread somewhat. As our wave moves further into the medium, the phase fluctuations of the wave will increase and its amplitude will begin to fluctuate. In addition to this, we will begin to find an appreciable amount of energy in wave numbers concentrated in a cone around the original direction of propagation k, . The angular width of the initial cone will be on the order of 342.111,. Following our wave further, we soon encounter the distance marker d,, the MFP for stochastic scattering as given by Eq. (12). At this point the phase spread will be exactly equal to 2.11. This point marks the beginning of strong multiple-scattering effects. In fact, we may actually estimate the number of such events (see, for example, Bugnolo, 1960~).As a result of this multiple scattering, our wave will also be spread in wavenumber space itself, a fact that will become evident later in this work. Before proceeding, let us briefly illustrate the event probability for multiple scattering. Let us assume an isotropic process for the dielectric fluctuations because this does not affect the fundamental characteristic of the discussion. Let us further assume that all frequency shifts due to eddy Dopplers are negligible. Under these conditions, it follows that the spectral function P, defined by Eq. (38), may be written as P,(k - k , w - 0’) = 6(0 - d)P(lk - k’l) 6:

(122)

For this case we may easily define a total cross section for scattering by the stochastic dielectric (Bugnolo, 1960a,b, 1972a) such that Q,

E

s

dkP(lk1) = dB1

(123)

Substituting for P(k) from Eq. (108) and proceeding in a manner similar to that used in reducing Eqs. (41), we obtain in the high-frequency limit defined by 2.11lo/,l > 10:

337


t

1.01

-0

3

2

1

4

5

*

( x ) Range in mean-free paths (dB) ( x = Range/d8)

FIG.3. Probability of multiple scattering by an isotropic scattering dielectric. Curves: A, at least once; B, at least twice; C, at least three times.

In the forward scattering of interest we may also proceed here, as in Bugnolo (1960c), to show that the probability of any ray of the incident plane wave being scattered at least N times in the distance R is given by P,(r I R) = [Q:/r(n)]

r"-l

exp(-Q,r) dr

(125)

jOR

Integrating, we obtain the following: PI = 1 - exp(-Q,R), f ' , = f'z

P , = 1 - (1 - t(Q,R)'

+ Q,R)exp(-Q,R),

~ X P-(Q,R)

(126)

Equations (126) have been plotted in Fig. 3. We note that P , crosses the equiprobability line (1/2) at a distance of Q,R = 1.7, or at 1.7 MFPs. At greater distances it is only reasonable to expect extensive multiple-scattering effects. We shall next address this interesting special case in greater detail by way of Eqs. (119) and (120). A. The Case of a Plane Wave Incident on an Isotropic Stochastic Dielectric Half Space We shall consider the case of an isotropic stochastic half space free of sources such that the effects of the source are contained in the boundary conditions fork, positive (i.e., into the half space). The geometry is illustrated by Fig. 4. In accordance with the discussions of Section V, we begin with Eq. (120) for the ensemble average of the Wigner function. In view of the

338


Hermitian property of the Wigner function, it follows that Im( Wii) = 0 and Eq. (120) thus reduces to the following for the diagonal terms of the Wigner matrix, i.e., for i = k ,

71 6(k’ - IO’I/C) -Re( Wji(r,t ; k, 0)) 2 O‘/C X

The source term has been retained for possible later reference. We have also used the rather obvious condition for the isotropic dielectric: P,(k - k‘; o - 0’) = 6:P(lk - kl, o - w ‘ )

(128) It is clear that the left-hand side (LHS) of Eq. (127) follows directly without approximation from Eq. (120) under the condition that i = k. However, we are still faced with components of the imaginary part of the Wigner tensor on the RHS of Eq. (127). We must still consider the effects of the Im( Wji)’swithin the integral and compare this contribution to that of the Re( Wji)’s.In order to do so, it is convenient to introduce the concept of “blurred” and “sharp” resonances. Y

i 4

Stochastic dielectric

* X

FIG.4. Geometry for a plane wave incident on a stochastic half space.


339

Blurred resonances are here defined as those obtained from the integral dk’ dw’

P(lk

(5

- k’l, cu - 0’) - 0’2/cz Im(

Wji(r, t ; k, 0))

k’2

- k;’)

(129)

whereas sharp resonances are here defined as those obtained from the integral

s

dr’ dw‘P((k - k‘l, o - a’) X

6(k’ - Iw’I/c)

Re( WJr, t; k‘, w‘))

W’/C

Since 6(k’ - w’/c) = c6(o‘ - k‘c), the latter yields after integration over w’, c Re( Wji(r, t; k, w))

s

dk‘

P((k - k l , o - 0’) (k’ - k:’) k

(131)

whereas the blurred resonances depend on the integral,

Integrals such as these must always be considered as “principal values” if they are not well defined otherwise (Whittaker and Watson, 1952). This follows from the method that was initially used to construct the transport equations, namely, Eq. (50). In view of this, Eq. (133) wil! 5e real if P is real, which is always the case in this work. In addition, it will vanish as a “principal value.” Having so disposed of the contributions of the blurred resonances we may rewrite Eq. (127) as follows:

c2

+ I1

x (Wjj(r,t ; k’,0’)) - Q Irn sji

(134)

340


Equation (134) is our final, complete stochastic transport equation for the case of a plane wave incident on an isotropic stochastic dielectric half space, when observed within the latter. In order to compare some of the results of our theory to those obtained by other theoretical methods it is convenient, when possible, to make another assumption. Let us specialize our result to the case of a normally incident, monochromatic plane wave of the form

E

= TE0 exp[i(k,x

- coot)],

Ex = E,

=

0

(135)

propagating in the 2 direction. It is evident that the only component of the Wigner tensor at the boundary that does not vanish is given by W,,(O, t; k, 4 = 6(w - o 0 ) W o - k , ) W q 6 ( k z ) ( G )

(136)

If the stochastic dielectric is sufficiently weak, then the Wigner tensor within the stochastic dielectric will also approximately be given by the single element W,, . This suggests that a self-consistent approximation could well be obtained by neglecting all terms in Eq. (134) for whichj # i. It follows that the Wigner tensor for x > 0 will also be approximated by

(k - 2 + ar

4) ( Wii(r, t ; k, w ) )

c2 at

-(-

= - 71 o2-

k i ) ( Wii(r,t ; k, 0))

2 c2 6(k‘ - Io‘I/c)

x SdL’ do’($ - ki”)P(lk - k‘l, o - 0‘)

o’lc

o2 + -271 6(k - Iwl/c) (7

k f ) ’ S dk’ dw’P(1k - k’l, o - o‘)

x ( Wii(r, t ; k’, w ’ ) )

(c2/4w2) Im S z

-

@/C

-

(137)

If we finally take ki = 0 and k! = 0, which implies that the wave-number vectors in the direction of the primary field y have a negligible effect on (Wii), we are left with the following equations when we take, possibly, o’ o = ck, if o occurs in a special factor without other terms:

-

71k4

= - -(Wii(r, t ; k, o>>

2

34 1


x x

s

6(k’ - Io’l/c) + $zk36(k - ~o(/c) WlfC

dk’ dW’P(1k - k’l, w - 0’)

dk’dw‘P(1k - k’l, o - w’)( Wii(r,t ; k’, w’))

1 4k

- ?Im

St

(138)

Our final result for the approximate stochastic transport equation for the ensemble average of the Wigner function is similar in form to the stochastic transport equation previously proposed by Bugnolo (1960a,b). This previous result was defined for the related function, the power spectral density S i i , which in our present notation may be written as Sii(r, t ; k7 W) 1

dr‘ dt’ exp[i(k.r’ - wt’)](ET(r,t)Ei(r + r’, t

+ t’))

(139)

In order to illustrate this similarity we must first define a new function F such that F and W are related by Wii(r,t ; k, w) = 6(k - lwl/c)Fii(r,t ; k, w )

( 140)

Substituting Eq. (140) into Eq. (138), we obtain the following stochastic transport equation for the related function F :

s1

nk4_ - -_ (Fii(r, t ; k, 0)) 2

X

6(k’ - IW’ ~ / C )

+ z k2 3 ~

of/c

x (Fii(r, t ; k‘, 0’))

dk’dw’P(lK(,o - w’)

dk‘ d~’P(lK1,w - w‘)S

- (1/4k2)Im SiT

(141)

where SiT = 6(k - ~w~/c)S; This approximate form for the stochastic transport equation may be compared to that of Bugnolo [(1960a,b), Eq. (36)]. Let us first consider the differences. Our present result includes a time dependence on the LHS and a source term on the RHS. These results are new. The two integral terms may be reduced to those proposed previously by noting that the presence of the 6 function within the integrals reduces the integration over k’ to an integration over dn’k. Although a similarity in mathematical form does indeed exist, we propose that our present result is an extension of the previous one

342


in that the integrations over k' are now well defined and indicate that the wave number k of the incident wave will be affected in both direction and magnitude. This will be particularly the case when we must deal with strong fluctuations of the dielectric.

B. Monochromatic Waves in an Isotropic Time-Invariant Stochastic Dielectric

This is another special case with a possible hint of an number of interesting applications. We may consider the stochastic dielectric to be time invariant if the time constant associated with its change z is very long compared to the time constants associated with the wave itself. For this special case we may substitute in all of our equations Ei(r, t) = exp(-jq,t)Ei(r)

(142)

Since the Wigner function is defined by (Wik(r, t ; k, 0)) dr' dt' exp[i(k * r' - cot')] x (ET (r

+ f.t +

i)

E, (r -

1

f.t ;) -

(143)

it follows that we may reduce this by way of the following steps: (Wik) =

dr' dt' exp[i(k- r' - at')]

exp[io,(t

+ t'/2)1

exp[ - io,(t - t'/2)]E,

s

s g)

1 ( Wik)= 16.n4 dt' exp[i(o, - o)t'] dr' x exp(ik * r') (ET (r

+

E, (r -

g)

(Wik) = 6(o - ad( Wik(r, k)) We also require a reduced form for the P function of Eq. (38), since for this case &(a, T) + hik(a).A procedure similar to the above yields Pik(k,

O) =

b(w)Pik(lk1)6F

(145)


343

and likewise for the source term we have

s

PO

S; = i - o o 6 ( o - w 0 )

2A3

dlexp(t-h)

where we have also taken the time dependence of the source current distribution to be Ji(r, t) = e x d - icoot)I;(r)

(147) The equivalent of Eq. (127) for this case may also be obtained from Eq. (120). Taking i = k, we have

a

k * - ( Wii(r, k)) dr

d5 exp(ik * 5)(E J 7 )

I1

Using our usual arguments to minimize the effects of the blurred resonances we may reduce Eq. (148) to a more useful result by removing the term in Im( W), hence,

x sdk’P(1k - k’I)S(k’- k,)(kt -

(6jk;

x

-

j

_ _ _ _1 8n3~,oO

kikj)’

i

n

ki2) + -S(k - k,) 2kO

dk’P(1k - k‘l)( Wjj(r, k’))

s

(bjkg - kikj)Re dh exp(ik * 5)( E J f )

(149)

Again we argue that a self-consistant equation may be obtained for the case when the stochastic medium is sufficiently weak by neglecting all terms in

344


Eq. (149) for whichj # i. This yields

a

k * - ( Wii(r, k)) ar 71 - - _

2kO

+

71 ~

2kO

( k i - k:)( Wii(r, k)) 6(k - k,)(ki

-

J

dk‘P(1k - k’1)6(ko - k’)(ki - k12)

s(

kZ)2 dk‘P(1k - k’l)( Wii(r,k’))

dh exp(ik.1) Ei (r - ’;>IT (r If we once again take ki

-

=

+

i))

(150)

0 and kf = 0, we are left with Eq. (151):

s s

nki 2 ( Wii(r, k))

-~

rcki

+ 26(k - k,)

dk’P(1k - k‘1)6(ko - k‘)

dk’P(1k - k’l)( Wii(r, k’))

With the exception of the source term, Eq. (151) has previously been obtained by Howe (1973) by a completely different method, starting with a very general Lagrangian density. In mmparing his result with ours, we must take 6(k2 - k;) = 6(k - k,)/2ko (152) On the other hand, a derivation based on the wave equation has previously been presented by Barabanekov et al. (1971) for a transport equation similar to Eq. (151). We again note that our result as given by Eq. (151) may be transformed by defining a function F such that Wii(r, k)

6(k - ko)Fii(r,k)

=

(153)

Substituting this into Eq. (151), we obtain the result, omitting the source term:

a

k . - (Fii(r, k)) ar

= -

71k3 9 (Fii(r, k))

s

+2 2 71k3

s

dk’P(IK1)6(ko- k‘)

dk‘P(IKI)G(K - ko)(Fii(r, k’))

(154)

STOCHASTIC DIELECTRlC WITH COMPUTER SIMULATION

345

We may compare Eq. (154) to Howe’s (1973) result particularly his Eq. (5.25). Our result is similar to his if we set k = k , in his Eq. (5.25) and use the definition for the 6(k2 - k;). Setting k = k , is, in our opinion, proper in the case of weak dielectric fluctuations. However, this is not the case for strong fluctuations of the dielectric in that the wave number of the incident wave will be affected in both magnitude and direction by the fluctuations. Again, we also note that setting k = ko in Eq. (154) results in a stochastic transport equation which is similar to that first proposed previously by Bugnolo (1960a,b). By this we seek to indicate that Eq. (154) in its present form does include the effects of the medium on both the magnitude and direction of the vector k, whereas the previous result obtained some time ago was limited to the effects of the medium on the direction of the vector k alone and thus consequently limited to the case of weak fluctuations in the medium.

C. Monochromatic Waves in an Anisotropic Stochastic Dielectric By following a procedure similar to that of Section VI,B above, we may obtain a self-consistent transport equation for the ensemble average of the Wigner function. We find that this may be written as

-

xki -2 <Wii(r,k))

+ nk3 6(k - k,) 2

J

f

dk’dw’P(k - k’, o - co’)6(k0 - k‘)

dk’ do’P(k - k‘, o - o’)(Wii(r, k’, co’))

(155)

This concludes our discussion of special cases. We shall return to these later in this work.

VII. A BRIEFREVIEW OF OTHERTHEORETICAL METHODS A number of other theoretical methods have been proposed for dealing with the problem of strong fluctuations of the field in the multiple-scattering region. In brief these are the diagram method, the integral equation method, and the parabolic equation method. Of these three, the parabolic equation method has shown the most progress and popularity to date. The interested reader is referred to Ishimaru (1978) for an introduction to the latter method as well as to the extensive list of references. It is not our intention here to duplicate this effort.

346

D. S . BUGNOLO AND H. BREMMER

More recently, the method of Feynman’s path integrals has been applied to the solution of the parabolic equation for scattering by a stochastic dielectric (Dashen, 1979).This is probably one of the more advanced works to be published as of this date. The parabolic method contains four physical assumptions (Ishimaru, 1978). The first of these, which we shall label assumption 1, is contained in the approximation used to formulate the parabolic equation. For a wave propagating in the x direction, we must take Assumption I

Ik(dv/&c)l >> ld2u/8x21

as long as 1, >> il

(156)

(see Ishimaru, 1978, Eq. 20-6b, p. 408). In effect this is a high-frequency condition similar to condition (i) of Dashen (1979). The second physical assumption is similar to our own, namely, that the stochastic dielectric must be characterized by a Gaussian random process at any space point. We shall label this assumption 2, which is given in Ishimaru [1978, Eq. (20.8)] and is used later in a conditional manner by Dashen (1979, Section 2): Assumption 2 {p} is Gaussian or

kL(p’)“’ 0. Any further work on this subject must address the problems inherent in the solution of the more complete Eq. (162). Suggestionsfor further work in this field are contained in Section IX,F.

Ix. COMPUTER SIMULATION OF THE STOCHASTIC TRANSPORT EQUATION FOR THE WIGNER FUNCTION IN A TIME-INVARIANT STOCHASTIC DIELECTRIC In order to better understand the results of Section VI, we propose to simulate the stochastic transport equation for the Wigner function for a monochromatic wave in a time-invariant stochastic dielectric. Such a simulation is made possible by the large-scale mainframe computers of today and by the availability of fast integration subroutines. We hope that this will prove to be a much better method of solution than that proposed some years ago by Bugnolo (1960a,b). We begin by recalling Eqs. (153) and (154) for the special case under consideration:

a

k . - (Fii(r,k)) dr

= -

nk3

(Fii(r,k))

s

dk’P((KI)G(k,- k’)

dk‘P(IK1)G(k’- ko)(Fii(r,k ) )

(154)

with (Wii(r9k)) = d(k - b)(Fii(r, k))

(153)

We propose to simulate Eq. (154) under the following conditions: (1) The source is monochromatic and consists of an antenna array such that the initial wave is launched within a narrow beam of width BETA, normally incident on the stochastic half space. (2) The time-invariant stochastic dielectric is everywhere homogeneous and isotropic within the half space and characterized by way of the spacewise spectrum P(IK1) and weak fluctuations. (3) The solution of Eq. (154)is observed by an ideal device that is designed to collect all components of (Fii) or ( Wii) that are contained within the beam of angle BETA. All other components are rejected.

Of these three conditions, the third is probably the most restrictive. However, conversion of our results to a more practical receiver is not a simple matter, because this would depend on the details of the receiving array.


355

A. An Integral Equation for a Very Narrow Beam We next address the problem of finding an integral equation equivalent to Eq. (154)for the geometry of Fig. 8. We begin by noting that the field launched by the source is initially contained within the cone of angle BETA and is characterized by an electric field vector E , in the R direction, propagating with the wave number k , in the 9 direction such that k , = q,/c. We observe that the first term on the RHS of Eq. (154) is just that considered previously in connection with our discussion of the coherent wave. As we again address the case of weak fluctuations of the dielectric, we expect that our integral equation will contain terms of the type given previously by Eq. (173), as applied to the geometry of Fig. 5. The launched wave will again have the approximate form F X X ( 0 , k,) =

- ~ o ) W x ) W , ) ~-( k,)E% ~,

( 178)

We note that this wave is deterministic and is a reasonable approximation for the case of a very narrow beam. A wide beam would require all three initial tensor components. With these conditions we write Eq. (154)as follows:

We next address the problem of the integration of Eq. (179) for the geometry of Fig. 8. We again invoke our very narrow beam condition so as to permit the approximation (R- r ( r R - r, thus placing all distance measurements on the z axis. We must also consider the boundary condition at z = 0 in the (x, y ) plane. Here our source is an “antenna” of effective aperture size A, radiating a uniform beam of width BETA into the stochastic medium X

A

Y

FIG.8. Narrow-beam geometry: [ R - r’l z R

-

r‘.

356


z > 0. In order to simplify our notation, we have placed the extensive mathematical details into our model for the function (F,,(O, k , ) ) such that the integral over the first term in our solution is well behaved as R + 0. For the details in a somewhat similar case, see Bugnolo (1960b). We therefore obtain as our approximate result for a very narrow beam

+

x

$ ~XPC J dk'P(IKl)6(k' JoR

dr'r'2

- Qs(ko)(R - 1'11 16z2(R - r')'

- ko) 0. At the boundary r = 0 or z = 0, F,, must take its proper value at the boundary, that form given by Eq. (178). We must therefore begin our simulation at the boundary z = 0 where F,, is known and proceed into the stochastic half space step by step, evaluating the RHS of Eq. (183) at each step. Any proposed method of numerical integration requires that the volume V illustrated by Fig. 10 be subdivided into a number of sections. This requires that we subdivide the cone of angle BETA, into, say, M sections, as illustrated by Fig. 9. However, we note that Eq. (183) is no ordinary integral equation, but a stochastic integral equation in the large. By this we mean to emphasize the conditions used to obtain a model for the differential cross section per unit volume per unit solid angle o ( K ) .The extent of the volume covered by the integrals used to define this function must be of extent L such that L >> lo, the mean scale size of the dielectric fluctuations. We note in passing that this problem does not exist in the problem of multiple scattering of a wave by very small particles. From the previous discussion, it follows that we may write 1, are “weak.” We therefore expect that our results for the attenuation of the Wigner function relative to its free-space value (computer by way of 1/167c2R2)will be affected by the number of sections M or by RIM, the actual physical length of each section. A study of our examples indicates that this is indeed the case here. Consider for example our Fig. 10a for the case where the wavelength is 3 mm, the mean scale size I , is 10 m,and ( E ’ ) ranges from lo-’’ to lo-”. Our “weak” bounds for these cases become 10 m

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