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
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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
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.
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
p<< n
(5)
The Poisson equation (3) is then reduced to the simpler Laplace equation -V.E = V24= 0
(6)
From Eq. (5), it also follows that V n should be zero. This means that the current density J , only consists of the drift component qp,,nE. The hole current J , can be put at zero because both p and V p are negligible. One obtains for the current density J = J , = NDqp,E = CJE
(7)
A thin semiconducting layer can be seen simply as a sheet with a constant conductivity CJ. For sufficiently high doping concentrations ND, the minority-carrier concentration p can be put equal to its equilibrium value p o , hence p - po = 0. In the time-independent case, Eq. (1) reduces to
POTENTIAL CALCULATIONS IN HALL PLATES
5
if we suppose that the generation term (an/dt),,, vanishes, which will be the case if the layer is not illuminated or irradiated. Equations (6) and (8) constitute the fundamental equations for a semiconducting layer, J and E being related by Eq. (7). From these equations, the boundary conditions can easily be deduced. At a metallic contact the potential 4 should be equal to the applied voltage. At a free boundary the current density must be tangential:
J-U,= 0 (9) where u,, is the normal unit vector. Owing to Eq. (7), the boundary condition (9) is equivalent to (10) v(p*u, = 0 The potential problem in a thin semiconducting laver is reduced to the solution of the Laplace equation in a given geometry, 4 or Vq5.u, being known on each point along the boundary. This is a classical potential problem with mixed boundary conditions.
C. Constitutive Relations with an Externally Applied Magnetic Field The fundamental equations (1) and (2) are still valid in the presence of an externally applied magnetic field. Only the constitutive relations (2) have to be extended in the following way (Smith et al., 1967; Madelung, 1970): j
where the index i denotes the ith component in a rectangular coordinate system. A similar expression can be written for the hole current density Jp,i.The tensorial mobility pij in the presence of a magnetic field B is found by the well-known Jones-Zener expansion, and it turns out that pij has to be replaced by
Applying Eqs. (1 1) and (12) on a flat n-type semiconductor layer, one obtains (De Mey, 1975)
J
=
oE
+ qDVn - opH(E X B) - qC(HD(Vn X B)
(13)
Using the approximation (9,as has also been done in the foregoing section, the diffusion components in Eq. (13) can be dropped, which yields
J = CE - opH(E x B)
(14)
This relation is used later for potential calculations in Hall plates. It has also been assumed that the semiconductor is isotropic, which explains why only one pH coefficient remains in Eq. (14).
6
GILBERT DE MEY
Equation (14) is derived from the Jones-Zener expansion (12) neglecting terms of order pAB2 that describe the physical magnetoresistivity. Hence Eq. (14) can be inverted to E
=
pJ
+ ppH(J x B)
(15)
where p = l/o denotes the resistivity. Equation (15) is also correct to terms of order pHB. The fundamental equations are still V J = 0 and V x E = 0, but for a flat semiconducting Hall plate in a uniform magnetic field B, one can show the following: V * J = C T V . E - I T ~ H V . ( EBX) = a V . E - a p H ( V x E ) . B = o V * E = O V x J = O V x E - opc,V x (E x B) = -opH(B.V)E
+ opH(V.E)B = 0
-
in which (B V)E vanishes since B is directed perpendicular to the Hall plate, whereas E is parallel to it. To solve the problem one can use either V x E = V - E = 0 or V x J = V - J = 0; E can be derived from a potential 9. For reasons which are explained in Section II,D, the current density J can also be derived from a potential function i,b in the following way: J = U, x Vi,b
(16)
where u, is the unity vector directed perpendicular to the Hall plate. From V x J = 0 it can be easily proved that i,b also satisfies the Laplace equation.
D. Boundary Conditions with an Externally Applied Magnetic Field Let us start with the electrostatic potential 4. At a metallic contact 4 is equal to the applied contact voltage. At a free boundary, the current density should be tangential: J - U , = oE'u, - ~/AHBE*u, =0
(17)
or v4.11, = ~ H B V $ . U ,
(18)
where u, = u, x u, is the unit tangential vector along the boundary (Fig. 1). It should be noted that at a free boundary in a Hall plate the electric field can show a nonvanishing normal component. For a p-type semiconductor, the same calculations can be carried out. A minus sign will then appear in the right-hand member of Eq. (18). For Hall plates the current but not the potential through a contact is usually given. It is then easier to introduce the current potential defined by Eq. (16). At a metallic contact E should be perpendicular, or the tangential
POTENTIAL CALCULATIONS IN HALL PLATES
7
FIG. 1. Hall-plate configuration for outlining the stream potential.
component E*u, must vanish. With Eq. (14) this gives rise to ~ E . u=, J . 4
+ pH(J x B).u,= 0
(19)
+ ~ H B V $ * U=, 0
(20)
or V$.U,
At a free boundary, J must be tangential, and due to Eq. (16) $ must be a constant (Fig. 1). At BB' the boundary value can be taken $ = 0. At the opposite side AA', the value $o can be found from the known current 1 injected through the con tact s :
I
=
- JAB' J . u , d l =
JAB
V$*u,dl = $(B') - $ ( A ' ) = -$o
(21)
A similar treatment can be performed if more than two contacts are involved. 111. THEVAN DERPAUW METHOD
Van Der Pauw (1958)has presented an ingenious method for carrying out resistivity and Hall-mobility measurements on thin layers with arbitrary shape. In this article we shall restrict ourselves to Hall-mobility measurements. It should be noted that Van Der Pauw's theory is only valid if the following four conditions are fulfilled: The layer must be perfectly flat (2) The four contacts must be point shaped and placed along the boundary (1)
8
GILBERT DE MEY
(3) The layer must be homogeneous (4) The geometry must be that of a singly connected domain Only the second condition (point-shaped contacts) is difficult to meet. This article is therefore mainly devoted to the influence of finite contacts on Hall-mobility measurements. In order to determine the mobility pH,a current I is fed through two opposite contacts A and C (Fig. 2). The voltage across the two other contacts B and D is then measured with and without the magnetic field B. The difference gives us the so-called Hall voltage. If the magnetic field B = 0, the voltage drop V , between the contacts B and D turns out to be
When the magnetic field B is applied, one finds a voltage V2 between B and D given by V2 =
=
pl:J.dr
+ p p H I I ( Jx B).dr
(23)
Because the contacts are assumed to be point shaped, the boundary condition J-u, = 0 holds along the entire boundary. Since the basic equations and boundary conditions for J are unaltered by the magnetic field, one concludes that the current density field J remains unchanged. The J vector in Eq. (22) is thus the same as in Eq. (23). The Hall voltage is then found to be f-B
VH
=
V2
- V , = P,UH
J (J x
B).dr
D
with B = Bu,(u, directed perpendicular to the Hall plate):
FIG.2. Hall generator of arbitrary shape placed in a magnetic field B.
POTENTIAL CALCULATIONS IN HALL PLATES
9
where d represents the thickness of the layer. Equation (25) enables us to determine the mobility pH by measuring the Hall voltage provided that p, B, I, and d are known.
OF THE GEOMETRY ON 1V. INFLUENCE HALL-MOBILITY MEASUREMENTS
Equation (25) of Van Der Pauw is only valid if the contacts are point shaped. Actually, Hall generators always show finite contacts, and this will alter the Hall voltage in a still unknown way. It is shown further on in this article that the so-called geometry correction can be calculated provided that the potential problem in a Hall plate is solved. At this stage it is necessary to emphasize that there are two different kinds of size effects in semiconductor components. The first (and best known) effect is of a purely physical nature. Consider a semiconductor slice that is very thin, in which the mean free path of the charge carriers can become comparable to the thickness. One can easily understand that the mobility will then depend upon the size of the sample (Ghosh, 1961). But in our case the geometry (i.e., finite contacts) has no physical influence on the mobility but will affect the measured Hall voltage. This second kind of size effect is of a purely metrological nature. We shall now review Section 111 for the case of finite contacts on a Hall plate. Equations (22) and (23) are still valid; however, as J-u, = 0 no longer holds along the entire boundary, the current field J will change in the presence of a magnetic field. It is then necessary to replace J by J AJ in Eq. (23), where AJ is the change in the current field caused by the magnetic field B. The Hall voltage VH is then found to be
+
rB
VH
= V , - V, = p ]
AJ-dr - ppHB(l/d)
D
The absolute error ApH introduced by neglecting the influence of the finite contacts is then rs
ApH= (d/Bl)J AJ-dr D
The problem is now to develop a geometry for which the correction (27) is as small as possible in spite of the finiteness of the contacts. One has to find an integration path from D to B in an area where AJ zz 0. This can be done if contacts are placed at the ends of rectangular strips (Fig. 3). From field
10
GILBERT DE MEY
FIG.3. Hall generator with contacts placed at the ends of strips.
calculations presented in the next sections, it is shown that disturbances due to the magnetic field (i.e., AJ # 0) only occur near the contacts. These areas are shaded in Fig. 3, and an integration path is drawn which only goes through the shaded areas at D and B. In Hall measurements, however, these contacts are used to measure potentials, and hence the total current through them is zero. Therefore J and, afortiori, AJ are very small in the neighborhoods of D and B. The conclusion is that the correction (27) will be small for the geometry of Fig. 3 where the contacts cannot be considered “point shaped.” Finite contacts with dimensions comparable to those of the Hall plate offer several advantages. The resistance between two contacts is low. Hence the noise induced in the measuring circuitry will be reduced. Small contacts require precise mask positioning and hence cause technological problems which can be avoided by using bigger contacts. The only disadvantage is the introduction of an error ApH; however, in the following section we prove that ApH can be calculated from the potential distribution in a Hall plate. It is then rather convenient to call ApH a geometry correction instead of an error.
V. CONFORMAL MAPPING TECHNIQUES A. Basic Ideas
Conformal mapping is a very useful method for solving the Laplace equation, provided that a geometry can be found for which the problem is solved by inspection. For a Hall-plate problem such a geometry can be found (Fig. 4). Indeed, for the parallelogram of Fig. 4, field lines are perpendicular to the contacts AB and A’B‘. The current lines are parallel to the free sides AA’ and BE’. The homogeneous E field obviously satisfies the Laplace equation (6). One can easily verify that the boundary condition (18) is
POTENTIAL CALCULATIONS IN HALL PLATES
11
A
FIG.4. Homogeneous field and current lines in a Hall plate having thz shape of a parallelogram.
exactly fulfilled if the angles at A, B, A', and B' have the values (n/2) - OH or ( 4 2 ) + as shown in Fig. 4. By mapping a given Hall-plate geometry on the parallelogram of Fig. 4, the potential problem can be solved for a given value of PHB = tan 8,. Wick (1954) was the first to use conformal mapping techniques for the study of the Hall effect. By using the Schwarz-Christoffel transformation formula, the mapping function could be determined. For quite regular Hall plates, however, calculations turned out to be very complicated, even for a simple rectangular shape with two contacts (Lippman and Kuhrt, 1958a,b). A considerable amount of work has been done by Haeusler (1966,1968,1971) and Haeusler and Lippmann (1968), who calculated the Hall voltage for a rectangular Hall plate provided with four finite contacts. The calculations are so complicated that this method cannot be considered a general rule, i.e., applicable to all shapes of Hall plates. We present a semianalytical version of the technique in which the Schwarz-Christoffel transformation formula is still used, but the evaluation of various integrals and constants is done numerically (De Mey, 1973a,b); we shall give an example of this later.
B. Approximate Analysis of the Cross-Shaped Geometry A very useful geometry for Hall-mobility measurements is the crossshaped sample (De Mey, 1973b). It combines broad contacts with a low geometry correction (27), which is now calculated. The cross-shaped form (Fig. 5b) can be mapped onto a circle by the following equations (Haeusler and Lippmann, 1968): y = tan2 26
h/l
=
{2[(k/h) - l]}-'
(28)
12
GILBERT DE MEY
b c d e f g H I J K L M N 0 A B C D D , E F G
FIG.5. Conformal mapping of a cross-shaped geometry (b) onto a parallelogram (a).
where F is the hypergeometric function. In the case that c sz 0, i.e., the mapped contacts on the circle are small, one can introduce several approximations in Eqs. (28)-(30) so that one obtains finally (De Mey, 1973b)
2c = 2.58e-"hi'
(31)
In his famous article, Van Der Pauw (1958) has given the following formula for the geometry correction in the case of a Hall sample with circular geometry and one finite contact: ACIHICIH
=
2ch2
(32)
13
POTENTIAL CALCULATIONS IN HALL PLATES
A
2010-
P
\
-
5-
\s-
2 2I=
*0.50
0.5
I.5
I h/l
2
FIG.6. Geometry correction as a function of h / / for a cross-shaped sample.
Note that Eq. (32) is only valid if 6 is sufficiently small. For four contacts, one can simply multiply Eq. (32) by 4. For the cross-shaped sample, this yields Ap(H/p(H = 4(2€/a2) = 1.0453e-"h/'
(33)
This relationship is shown in Fig. 6. Note that for h = 1, the correction ApH/pH becomes 4.5%, whereas for h = 21 a negligible value of 0.19% is found. A correction of 5% means that experimental pH values have to be increased by 5% in order to obtain the exact value. C . Exact Analysis of the Cross-Shaped Sample
In order to calculate the potential distribution and hence the Hall voltage in a cross-shaped geometry, the Hall plate has to be mapped onto the parallelogram of Fig. 5a (De Mey, 1973b). For this simple geometry, the field distribution is homogeneous. The current density is found to be IJI = I/(IANIcos8(H)
(34)
I
where I is the current supplied through the contacts AN and HG. I AN denotes the length of the contact A N . For unit values of I and p, the Hall voltage is nothing other than the vertical distance between the Hall contacts CDE and JKL (Fig. 5a), or VH = IEI(IACI - ILNI)COS8H
X
(IACI - I L N I ) / ( A N )
(35)
14
GILBERT DE MEY
According to Van Der Pauw's theory, V , should be equal to pHB= tan OH, and by comparing it with the value given by Eq. (35), the geometry correction &/pH is found. The conformal mapping of a circle into a semiinfinite plane is well known and is described by the relation x = tan(O/2)
(36)
This half-plane is mapped onto the cross-shaped geometry according t o the Schwarz-Christoffel transformation formula:
From Eq. (37), the values of h and I can be found:
I
=
h=
I-+; (u2
-dxx 2 ) 1 ' 2
job
((c2
dx [(b - x)(x - a)]'"
>"'
(b2 - x2)(f2 - x 2 ) - x2)(e2 - x2)(g2 - x2)
x)(b2 - x 2 ) ( f 2 - x') (x + u)(c2 - x2)(e2- x2)(g2- x 2 )
(
(b
-
These are Gaussian-type integrals and can be easily evaluated numerically (Abramowitz and Stegun, 1965). The conformal mapping of the half-plane onto the parallelogram is performed by dwldz = A ( z - u ) - ~ ~ / ~ (C )z- ( P ~ / ~ ( Z - d ' ) ( z - e)-v2/n ( z - g)-(Pl/n x (z
+
+ C ) - ~ ~ / ~+( Zd " ) ( z + e)-'+'lln(z + g)-'+'2/n
U)-+'~/~(Z
(39)
The unknown constants d' and d" in Eq. (39) are determined by saying that the point E should coincide with C and J with L. From Eq. (39) one can determine the distances I ACI - I L N I and I A N I needed to evaluate the Hall voltage V,. These calculations also lead to Gaussian-type integrals, but further details about the numerical procedure are omitted here. Figure 6 shows several numerically calculated values of ApH/pH. We observe good agreement with the approximate equation (31). These results also prove that conformal mapping can be adequately applied to rather complicated geometries if one starts immediately with numerical integrations once the Schwarz-Christoffel transformation formula has been written. D . Properties oJ the Cross-Shaped Hall Plute
The cross-shaped sample has contacts whose lengths are comparable to the dimensions of the semiconductor layer. This means that the resistance
POTENTIAL CALCULATIONS IN HALL PLATES
15
between two contacts is low, which offers many advantages when low Hall voltages are to be measured. Technological problems caused by small contacts no longer occur. Moreover, the geometry correction ApH/pH is moderate. Even for h = 1, less than 5% is found. This also means that the position of the contacts is not very critical. High-precision mask positioning before evaporation of the contacts is no longer required.
VI. RELAXATION METHODS The relaxation method is the only purely numerical method that has been used to calculate the potential distribution in a Hall-plate medium (Newsome, 1963; Grutzmann, 1966; Chwang et al., 1974; Mimizuka, 1971, 1978, 1979; Mimizuka and Ito, 1972). If one is only concerned with the potential distribution for given voltages at the contacts, the relaxation method gives good results; however, if the current through a contact has to be calculated, the method is far from accurate. Impedance calculations carried out by Mimizuka (1978) show relative errors of 10%. On the other hand, the relaxation method allows us to include nonlinear effects such as the temperature effects inside the Hall plate, a problem which cannot be treated with other techniques (Mimizuka and Ito, 1972; Mimizuka, 1979). We now use the relaxation method to calculate the geometry correction ApH/pH for a cross-shaped plate. Because this involves knowledge of both the Hall voltage and the supply current, we expect moderate accuracy. This is mainly owing to the tangential derivative V 4 u, in the boundary condition ( 1 8), which is difficult to represent numerically. In order to represent the Laplace equation numerically, the cross-shaped form has to be divided into a mesh (Fig. 7). The Laplace equation is then approximated by the well-known five-points formula : 44i.j - 4i+1 . j - A-
1.j
-
4i.j+1
-
4i,j- 1 = 0
(40)
At a contact, the known potential values have to be inserted into Eq. (40). The boundary condition (18) can be written as (Fig. 8) 1 4i,j - 4i- 1 . j = pHBT(4i,j+ 1 - $i,j-
(41)
1
-
At a free boundary, the Neumann boundary conditionV4 u, = 0 is written as (42) 4i.j = 6(4i,j+1 + 4i.j-1 + W i - 1 . j ) which corresponds to putting 4’ = 4i-l . j (Fig. 8). This condition (42) gives
good results in numerical calculations. Note that Eq. (41) cannot be reduced to Eq. (42) when p H B tends to zero. If pHB = 0, Eq. (41) reduces to 4i.j=
16
GILBERT DE MEY
0
0
0
0
0
0
0
0
0
0
0
FIG.7. Grid pattern for the finite-difference approximation in a cross-shaped Hall plate.
FIG. 8. Boundary conditions for the finitedifference approximation.
THEORETICALVALUE= L 51 90
: s : : : : : : : : : : : 10
15
+
M f=NI
FIG.9. Geometry correction as a function of M :p H B= 0.1 ;h/l = 1.
POTENTIAL CALCULATIONS IN HALL PLATES
17
h/I
N
FIG. 10. Geometry correction versus h/l calculated with finite difference approximation: and 13 ( A ) .
= 7 ( 0 ) .10 ( W),
4i- a boundary condition leading to a reduced accuracy and hence never used. For Hall plates, only Eq. (41) can be applied. This fact can explain the poor results. Figure 9 represents the numerically calculated geometry correction ApH/pH for h = 1 as a function of M (= N).Note that the results show errors of more than SO%, and the convergence is extremely poor when the mesh number M increases. Figure 10 represents the geometry correction A/.iH/pH as a function of h / l . The analytic approximation (33) has also been drawn. Only for low h / l values were acceptable results obtained. For higher h / l values the calculations are meaningless. We conclude that the relaxation method is not suitable for calculating the geometry correction ApH/pH of a Hall plate. Several reasons can be advanced. It turns out that contact current cannot be calculated with sufficient accuracy because it involves numerical differentiations. Also, the numerical representation (41) of the boundary condition (18) is not optimal from a numerical point of view. It should be noted, however, that the high relative errors found here are made on a geometry correction, i.e., the relative difference between the numerical and theoretical Hall voltage, two numbers that are normally close together. The accuracy on the geometry correction will hence be much lower than the accuracy obtained on the Hall voltage.
18
GILBERT DE MEY
VII. THEBOUNDARY-ELEMENT METHODFOR POTENTIAL CALCULATIONS IN HALLPLATES A . Introduction
Several years ago, the boundary-element method (BEM) was a rather unknown technique. Its popularity has since grown considerably, and the method now competes with the finite-element method (FEM). In recent years, some textbooks on the BEM have been published (Jaswon and Symm, 1978; Brebbia, 1978a,b). The BEM replaces the given Laplace equation and the boundary conditions by a single integral equation. This equation involves only the boundary of the given geometry. For a numerical solution the boundary has to be divided into a number of elements. This explains the term boundaryelement method. Because only the boundary is involved, the complexity of the problem is reduced. The method requires less storage allocation and computation time if it is solved numerically. The BEM can also be programmed on small-size computers (e.g., 60K memory) in contrast to the FEM, which requires a large memory. The integral equation technique was very well known to people working in the area of electromagnetic fields. Presumably, the reason is that many problems in electromagnetic theory can be directly formulated in terms of integral equations (Edwards and Van Bladel, 1961; Mei and Van Bladel, 1963a,b). Integral equations have also been used in other fields such as thermal diffusion problems (Shaw, 1974; De Mey, 1976a, 1977a), driftdiffusion problems (De Mey, 1976b, 1977b), elastic problems (Brown and Jaswon, 1971; Symm and Pitfield, 1974),calculation of eigenvalues (De Mey, 1976c, 1977c), semiconductor-component analysis using abrupt depletionlayer approximation (De Visschere and De Mey, 1977; De Mey et al., 1977), and potential distribution in Hall plates (De Mey, 1973c, 1974a, 1976d, 1977d). B. Integral Equation f o r the Potential Distribution in a Hall Plate
We consider the rectangular Hall generator shown in Fig. 11. The potential satisfies the Laplace equation (6) and the boundary conditions
4=
V,
at
A'B',
4 = 0 at A B
V4.u" = p H BV ~ . U , at AA' and BB'
(43)
In order to construct an integral equation, the Green's function G(r I r') of the Laplace equation is used: G(r1r') = (27c)-' lnlr - r'I
(44)
POTENTIAL CALCULATIONS IN HALL PLATES
19
FIG. I I . Rectangular Hall plate.
One can easily verify that G(r I r’) satisfies the Laplace equation provided that a delta function is put in the right-hand member: V:G(rlr’) = d(r - r’)
(45)
The solution of our problem 4(r) is now written as
where p(r) is called the source function and is only defined along the boundary C of the Hall-plate medium. Owing to Eq. (49, the proposed solution (46) automatically satisfies the Laplace equation for every point r inside C. In order to determine the unknown source function p(r), one has to impose the boundary conditions (43) on the proposed solution (46). This gives rise to &fcp(r’)lnlr
- r’(dC’=
V,,
0,
rEA’B’ rEAB
(47)
where r E AA’ and r E BB’. Equations (47) and (48) constitute an integral equation in the unknown source function p. Once this integral equation has been solved, the potential 4 (and hence all quantities which can be deduced from 4) can be found at an arbitrary point r by evaluating the integral (46). The first term - p ( r ) / 2 ~occurring in Eq. (48) is a so-called fundamental discontinuity. It is caused by the discontinuity of the normal component V$*un at the boundary. The gradient V+ can be calculated at an arbitrary point r by V4(r) = (271.)-’ $C
p(r’)[(r - r’)/(lr - r’l’)]
dc‘
(49)
20
GILBERT DE MEY
C FIG. 12. Coordinate system (t, q ) to explain the fundamental discontinuities around ro.
If r lies on the boundary, the integrand of Eq. (49) becomes infinite because r can coincide with r'. For the normal component the integrand should be treated carefully by taking r close to the boundary at the interior side (Fig. 12). The boundary C can then be approximated by a straight line and p(r) can be considered a constant p. Introducing a (5, q ) Cartesian coordinate system, one has
5 2
By taking the limit 5 + 0 (i.e., r tends to the boundary C), the term -p(r)/2 remains. This problem does not occur for the tangential component (r - r')-u, because this integrand is an odd function of q. There are no fundamental discontinuities for tangential components.
C. Numerical Solution of the Integral Equations
In order to solve the integral equations (47) and (48) numerically, the boundary C has to be divided into n elements ACi. In each of them, the unknown source function p is replaced by an unknown constant p i . Denoting ri as the center point of the element ACi, the expression (46)for the potential can be rewritten as n
(#l(r)=
1 p.G(r(ri)IACjl J
(51)
j= 1
where IACjI denotes the length of the element ACj. In a similar fashion, the integral equations (47) and (48) can be discretized. However, if r coincides with one of the ris, one obtains G(ri 1 ri) In 0 = co ! In order to avoid this infinity, the self-potential, i.e., the potential at ri due to the source pi itself on
-
POTENTIAL CALCULATIONS IN HALL PLATES
21
Aci, has to be calculated otherwise. Replacing pi by a uniform constant function along Aci, the divergence no longer occurs and the self-potential turns out to be
The integral equations are then replaced by PiIAciIClnOlAciI) 1 + C" -pjIn(ri
j=l27L
11 -
rj(IAcjl =
V,,,
0,
riE A'B' ricAB
(53)
j#i
= 0,
riE AA', BB'
Equations (53) and (54)constitute a linear algebraic set with n equations and n unknowns pi, which can be easily solved numerically by the Gauss pivotal elimination method, for example. The potential can then be calculated at an arbitrary point r by using Eq. (51).
D. Application t o a Rectangular Hall Plate
The rectangular Hall plate shown in Fig. 11 has also been treated by conformal mapping techniques (Lippmann and Kuhrt, 1958a,b; De Mey, 1973a). The Hall voltage V , induced between the opposite point contacts P and Q has been calculated accurately. For a square-shaped Hall plate, one has V , = 0.522654 if V , = 1 and 0, = 45" or c(HB = 1. By comparing this value with the numerical results, the accuracy of the BEM can be checked. Figure 13 represents the relative error on the Hall voltage V , as a function of n, where n is the number of unknowns on each side (hence 4n unknowns have to be determined for this problem). One observes that good accuracies are obtained. Precision of better than 1% can be attained with a moderate number of unknowns. The linear behavior of the results shown in Fig. 13 is remarkable. When solving a problem with the BEM the relative accuracy always varies according to a l/n law. In contrast to most other numerical methods, this l / n law does not change when other algorithms for representing the source function are used. It should also be noted that a theoretical explanation for this phenomenon has not yet been published.
22
GILBERT DE MEY
I
I
I
3
1
1
5
1
7
1
1
1
1
1
1
9 11 13 151719
*
n
FIG.13. Relative error on the Hall voltage versus n for a rectangular plate: p H B= I ; V, = 1.
E. Zeroth-Order Approximation
It is a general experience in numerical analysis that higher order approximations yield more accurate results for the same number of unknowns. For example, we can cite that the fact that Simpson's rule for the numerical evaluation of an integral will be more accurate than the trapezoidal rule. In the preceding section a rather crude approximation for the source function p was introduced. Instead of replacing p by a series of constants pi concentrated at the points ri, one can see p as a piecewise constant function. In each element Aci, p equals a constant pi.The expression (51) for the potential now has to be written as
The integral equations (53) and (54) now read
1" p j g1I A c J l n l r -i r ' I d c ' =
j= 1
V,,
0
riEA'B' riEAB
where ri E AA', BB'. This method has also been applied to the square-shaped Hall generator. Figure 14 represents the relative error of the numerically calculated Hall voltage VH. The results obtained in the preceding section
POTENTIAL CALCULATIONS IN HALL PLATES
23
43-
E 4
2-
3 \
c
1-
0.5-
3
-4
5
7
9 11 13151719
n
FIG. 14. Comparison between the direct method and a first-order approximation.
have been redrawn in Fig. 14. One observes a slightly lower error due to the zeroth-order approximation. It is also remarkable that the same l/n law appears again. Similar results have also been found with higher order approximations of the source function (Stevens and De Mey, 1978). These results are in full agreement with the statements made at the end of the foregoing section. A l/n law always occurs, and the relative errors d o not decrease remarkably when higher order approximations are used for the source function. The integrals appearing in Eqs. (55)-(57) can easily be evaluated by complex integrations. By drawing an ( x , y ) coordinate system with the x axis directed tangentially and the origin at ri (Fig. 15), the integral appearing
t’
r z x Cx*yCy
*
Zzxejy
X
FIG. 15. Complex plane used to evaluate some integrals.
24
GILBERT DE MEY
sAc,
in Eq. (55) can be written as In ) r - r‘ I dC’ = Re
s
+ IAcJ1/2
- lAcJl/2
In(z - z’)dz’
(58)
Indeed, the real part of ln(z - z’) is nothing other than lnlr - r’I and dc’ = dz’ along the x axis (Fig. 15). The ( x , y ) plane is considered to be a complex plane, and the points z and z‘ correspond to r and r’. The complex integral appearing in Eq. (58) is easily evaluated analytically:
s
+ lAcj112
In(z - z’)dz’ = [-(z - z’)ln(z - z’) + (z - z ’ ) ] ~ : ~ ! l ~ ~ l \ ~
-IAcjI/Z
If one uses a computer with complex-number facilities, all integrals appearing in Eqs. (55)-(57) can be easily calculated as the real part of an analytic function. F. Numerical Calculation of the Current through a Contact
Using the constitutive relation (14) between the current density J and the electric field E, the total current through a contact AB (Fig. 16) is given by I = C J - u n d C= 0
E.u,dC
(59)
JAB
because the tangential component of E is zero at a metallic contact. From Eq. (46) the electric field in an arbitrary point r is found to be r
E(r) = -V$
=
-
p(r’) VG(r1r’)dC’ 9 C
The current I is then given by I = -Os~dCun.~~p(r’)VG(rIr.)dC.
(61)
If one intends to calculate Eq. (61) numerically, the electric field has to be found at a number of points on the contact AB, and then the summation along AB should be carried out. This procedure will require a lot of computation time, and the accuracy will be moderate because two numerical
FIG. 16. Use of the angle a to calculate current through AB
POTENTIAL CALCULATIONS IN HALL PLATES
25
approximations will be carried out. It is more convenient to interchange the sequence of the integrations in Eq. (61), which gives rise to I = -a~~p(r’)dC~~~’dCu..VG(rlr.)
(62)
The last integral appearing in Eq. (62) is nothing other than the flux of VG through AB. As G(r I r’) depends only upon the distance I r - r‘ 1, the field VG will show a radial symmetry around r’. Hence the flux of VG through AB is given by the angle a normalized to 2n (Fig. 16). The integral (62) reduces to I = -(a/2n)
h
dC’p(r’)a(r’,A, B )
where a(r‘,A, B) denotes the algebraic value of the angle under which AB is seen from the point r’. For the numerical calculations the evaluation of an angle poses no particular problems, and a single summation is sufficient. If r’ coincides with a point on AB, then a = n and the flux is then 4.This is another view of the fundamental discontinuities discussed in Section VII,B. The numerical calculation of contact currents is very important for the study of Hall generators because these devices are normally operated under a constant current through two opposite contacts generating a Hall voltage at the other contacts. G. Applicution to the Cross-Shaped Geometry
The BEM is now applied to the calculation of the geometry correction of the cross-shaped sample. The boundary conditions are shown in Fig. 17.
v0.+IH
q
8V@.
V@
.rn=,+,so@ ..;
--
qj=h/2
IH
IH
--
@ =-vH/2
v@..‘,=p,,
[email protected]
~#.u;l=~,
[email protected];
.
26
GILBERT DE MEY
FIG. 18. Hall voltage ( VH), supply current (I), and geometry correction (ApH/pH)as a function o f m : h / / = I ; p H B= 0.1.
Normally, the source current I is given and the Hall current is zero. It is difficult to impose the total current through a contact as a boundary condition because this often leads to an ill-conditioned algebraic set. Therefore the potentials at the contacts are given. The current-supplying contacts have potentials 4 = + 1 and - 1. The supply current I is then found from the potential distribution, as outlined in the preceeding section. The calculations are carried out twice, once with V , = 0 and once with V , = pHB.Each time the Hall current I , is also calculated. As V , varies linearly with I,, the correct Hall voltage is then found by putting I, = 0. Figure 18 shows some numerical results obtained for a geometry with h/l = 1. The Hall voltage V,, the supply current I, and the geometry correction ApH/pH have been drawn as a function of m where m is the total number of unknowns used during the numerical procedure. The geometry correction is far from accurate, as can be seen by comparing the numerical value of 12% with the exact value 4.5%. From this consideration one may not conclude that the BEM method is useless. One calculates the Hall voltage numerically and compares it with the theoretical value (25) according to Van Der Pauw's theory. In order to obtain the geometry correction these two values, which are normally close to each other, have to be substracted. This will increase the relative error so that the numerically calculated geometry correction cannot be used. Nevertheless, the potential and field calculations in the Hall plate are still accurate. If one wants to calculate ApH/pH with a 10% accuracy (for example), the Hall voltage and the supply current have to be known with much higher precision. In Section VII,H, a method is presented for finding
POTENTIAL CALCULATIONS IN HALL PLATES
27
t
- 0.86
-
5
60
21
120
m
FIG. 19. Difference between calculated and exact geometry correction versus m. Note that double logarithmic scale is used.
ApH/pH directly, without the intermediate calculation of the Hall voltage
VH or the supply current I .
The same calculations have also been carried out with other h/l values and other numerical approximations for the source function. The results and hence the conclusions remain identical. If the difference between the calculated and exact geometry correction is plotted on a double logarithmic scale as a function of m, one obtains linear behavior with a slope -0.86 (Fig. 19). This agrees with the results shown in Figs. 13 and 14. It also proves that the numerical values converge, although very slowly, to the exact value. H . Direct Calculation o j the Geometry Correction
The geometry correction can be calculated directly, and, as is proved later, this can be easily performed if one introduces the stream potential II/ defined by Eq. (16). The technique is outlined first for a Hall plate with two point-shaped Hall contacts, and a formula is then derived for an arbitrary geometry. Let us consider the rectangular Hall plate with two point-shaped Hall contacts P and Q (Fig. 20). A current 1 is supplied. Figure 20 indicates the boundary conditions if the stream potential t,b is to be used. From Eqs. (15) and (16) one finds the electric field
E = PP,BVt,b
+ P(U,
x V*)
(64)
28
GILBERT DE MEY
FIG.20. Boundary conditions for the stream potential on a rectangular plate.
where uz is the normal unit vector perpendicular to the Hall plate. In order to obtain the Hall voltage VH, the electric field (64) has to be integrated between the Hall contacts P and Q. One obtains VH
= j:E-dr
= pp.sj;V+-dr
= PPHB[$(P) -
(Q)]- P
IQ P
= pPHB(l/d) - P
J;
v$ ’(u:
+ psypC, x v+).dr v$.(U;
X
dr)
dr)
(65)
By Eq. (65),the Hall voltage VHcan be found from the stream potential $. It is remarkable that the first term in the right-hand member of Eq. (65)is nothing other than the theoretical Hall voltage (25) given by Van Der Pauw’s theory. Hence, the remaining term in Eq. (65)is the difference between the theoretical and the actual Hall voltage, or the so-called geometry correction. We remark
FIG.21. Stream potential used on an arbitrary shape.
POTENTIAL CALCULATIONS IN HALL PLATES
29
that the last integral in Eq. (65) is the flux of V+ through a line connecting P and Q. It has also been assumed that the flux V + * (u, x dr) is zero if PHB = 0, which is certainly the case for the symmetrical configuration of Fig. 20. The conclusion is that the geometry correction can be calculated directly as the flux of the V+ field. This can be calculated numerically by the method outlined in Section VII,F. We now consider an arbitrary Hall plate provided with four finite contacts (Fig. 21). The electric field is still given by Eq. (64). On the Hall contacts two points P and Q have been chosen. The integration of the electric field gives us the Hall voltage:
where A$(P) and A+(Q) denote the differences between the II/ values at P and Q, respectively, and the values at the ends of the contacts. For pointshaped contacts these differences are evidently zero and Eq. (66) reduces to Eq. (65).The first term in the right-hand member of Eq. (66) is the theoretical Hall voltage according to Van Der Pauw. The geometry correction is now given by a term containing the stream-potential differences A+(P) and A+(Q) and a second term involving the flux of V+ through PQ. If necessary, the ohmic potential drop occurring when pHB= 0 has to be subtracted from Eq. (66); however, most Hall-plate configurations show sufficient symmetry for this operation to be dropped.
I . Application to a Rectangular Hall Generator
The method outlined in Section VII,H is applied to the rectangular plate shown in Fig. 20. Because the Hall voltage has also been calculated with conformal mapping methods, the accuracy of the numerical values can be controlled. Figure 22 shows the relative error of the geometry correction for various values of PHB obtained by evaluating the last term of Eq. (65). In contrast to previous results, the accuracy is fairly high. It should be noted that a relative accuracy of 10% on the geometry correction means that the Hall voltage is known with a precision much better than 10%. We remark again that the
30
GILBERT DE MEY
20
-t
FIG.22. Relative error on the geometry correction as a function of IM for a square-shaped Hall plate: p H B= 0.1 ( 0 ) , 0 . 3 ( + ) , 0 . 5 ( A ) , a n d I ( 0 ) . IMisthenumberofunknownsperside.
error decreases in inverse proportion to the number of unknowns used in the numerical procedure. We may conclude that the accuracies are sufficient for practical purposes. J . Application to a Cross-Shaped Geometry Equation (66) is now used to analyze the cross-shaped geometry. Two points P and Q have been placed at the centers of the Hall contacts (Fig. 23).
FIG.23. Cross-shaped Hall plate analyzed with the stream potential.
POTENTIAL CALCULATIONS IN HALL PLATES
31
Figure 24 shows the relative error on the numerically calculated geometry correction for h/l = 0.5. For practical purposes these accuracies are more than satisfactory. For high values of pHB(>0.5), 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
POTENTIAL CALCULATIONS IN HALL PLATES
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
POTENTIAL CALCULATIONS IN HALL PLATES
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
POTENTIAL CALCULATIONS IN HALL PLATES
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.
POTENTIAL CALCULATIONS IN HALL PLATES
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.
POTENTIAL CALCULATIONS IN HALL PLATES
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.
POTENTIAL CALCULATIONS IN HALL PLATES
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
POTENTIAL CALCULATIONS IN HALL PLATES
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)
POTENTIAL CALCULATIONS IN HALL PLATES
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
POTENTIAL CALCULATIONS IN HALL PLATES
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.
POTENTIAL CALCULATIONS IN HALL PLATES
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,
POTENTIAL CALCULATIONS IN HALL PLATES
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
IMPURITY LEVELS IN GALLIUM ARSENIDE
~~
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.)
IMPURITY LEVELS IN GALLIUM ARSENIDE
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
IMPURITY LEVELS IN GALLIUM ARSENIDE
(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
IMPURITY LEVELS IN GALLIUM ARSENIDE
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)
IMPURITY LEVELS IN GALLIUM ARSENIDE
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.)
IMPURITY LEVELS IN GALLIUM ARSENIDE
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
IMPURITY LEVELS IN GALLIUM ARSENIDE
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.
-
IMPURITY LEVELS IN GALLIUM ARSENIDE
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
IMPURITY LEVELS IN GALLIUM ARSENIDE
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.
IMPURITY LEVELS IN GALLIUM ARSENIDE
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 >
<100>t110><111> Go
As
t110>
<111> 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
IMPURITY LEVELS IN GALLIUM ARSENIDE
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).
IMPURITY LEVELS IN GALLIUM ARSENIDE
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.)
IMPURITY LEVELS IN GALLIUM ARSENIDE
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
IMPURITY LEVELS IN GALLIUM ARSENIDE
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
IMPURITY LEVELS IN GALLIUM ARSENIDE
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
IMPURITY LEVELS IN GALLIUM ARSENIDE
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
IMPURITY LEVELS IN GALLIUM ARSENIDE
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.)
IMPURITY LEVELS IN GALLIUM ARSENIDE
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
IMPURITY LEVELS IN GALLIUM ARSENIDE
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
IMPURITY LEVELS IN GALLIUM ARSENIDE
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.
IMPURITY LEVELS IN GALLIUM ARSENIDE
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
IMPURITY LEVELS IN GALLIUM ARSENIDE
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).
IMPURITY LEVELS IN GALLIUM ARSENIDE
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
IMPURITY LEVELS IN GALLIUM ARSENIDE
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
<0.02 2
3
0.02 4 x 10-5 3 1-3 x lo-’
<0.02 0.11 0.17 0.025
0.048 0.36
0.1 0.3 0.3 0.03 0.1
0.44-0.55 0.14 0.5-1.0 0.054-0.16 0.27-0.9
0.016 0.30 0.14 0.30 0.059 0.08 0.40
0.5-1.0 1-2 x 10-4 1-3 x lo-’ 2-4 x lo-’ 2-4 x 10-3
a See Romanenko and Kheifets (1975) for other data on Cr, Si, Sn, Te, Zn, and Stringfellow (1974) for temperature dependence of the distribution coefficients of Se, Sn, and Te. Reproduced with permission of the Institute of Physics, UK.
A. Carbon
Carbon concentrations up to lOI7 cm-3 can be obtained as shallow acceptors by C doping of crystals during horizontal Bridgman growth. Undoped Bridgman crystals show C concentrations of 5 x 10’’ cmP3 or less. Measurements by Brozel et al. (1978)correlate the local vibration mode IR spectroscopy line with acceptor action and conclude that carbon is present, predominantly as C,, defects. In MBE layers, C at 1.6 K produces photoluminescence at 1.493 eV by a free-electron transition to the CAs acceptor (Kunzel and Ploog, 1980). Since the band gap is 1.519 eV a1 this temperature, this agrees with the carbon acceptor found to be 0.027 eV above the valence-band edge (Bimberg and Baldereschi, 1979; Bimberg, 1978;
IMPURITY LEVELS IN GALLIUM ARSENIDE
121
Bimberg et al., 1982). Evidence for the existence of low concentrations of CGa donors in epitaxal GaAs has been obtained by other methods (Wolfe et al., 1977).It seems therefore that carbon is an amphoteric impurity, although the ratio of the concentrations of the two species has yet to be determined for any type of material. Spark-source mass spectroscopy shows that the common etchant bromine methanol tends to leave residual surface carbon that is not readily removed. Etching with an H202:H,S0, solution leaves less carbon on the surface. Implantation of C into GaAs has been studied by Shin (1976) and Shin et al. (1978). B. Silicon
Silicon is an amphoteric impurity in gallium arsenide, and 28SiGadonors and 28SiAsacceptors give localized modes of vibration (LVM) absorption at 384 and 399 cm-l with linewidths of 1.4 and 2.0 cm-', respectively (Laithwaite et al., 1977).Samples with relatively low carrier concentrations ( 5lo'* ~ m - ~ ) contain predominantly Sic, defects, low concentrations of Si,, , and show negligible LVM absorption from other more complicated defects such as (SiGa-SiA,)or (B2-SiG,) pairs in LEC material (Morrison et al., 1974).When GaAs is grown from a melt of stoichiometric composition, silicon if present is incorporated on gallium sites as a shallow donor. However, if GaAs is solution grown from gallium, under conditions where there are few gallium vacancies, Si is a shallow acceptor. The effect of the Fermi level on Si-site distribution in GaAs:Si has been considered by Kung and Spitzer (1974). In solution growth, by careful control of the temperature range and cooling cycle, it is possible to grow np or pn amphoterically doped diodes (Kressel et al., 1968a; Moriizumi and Takahashi, 1969).The GaAs is usually n type if grown above 850°C and p type if grown below. At a temperature of about 7OO0C, mol fraction of Si in the Ga melt produces a net hole concentration of 1OI8 cm- '. In LED use, amphoterically doped junctions exhibit a broad emission bandwidth, perhaps involving transitions to three levels E, + 0.03, E, + 0.10, and E, + 0.22 eV. Kressel et al. (1969) suggest that the 0.03 level is due to Si occupying random As sites and that the 0.10 eV level is a complex involving Si. This might be (SiGa-SiA,)or (Sica-Vca) (Spitzer and Allred, 1968). The 0.22 eV level is attributed to an arsenicvacancy complex that is related to the presence of silicon. These attributions must be treated as hypothetical in the absence of direct supporting evidence. Antell (1965) has examined the diffusion of Si in GaAs by junction-depth studies. The diffusion rate is low and the profile is steep fronted and dependent on the arsenic overpressure. Silicon-ion implantation in GaAs has been studied by many workers (see, for instance, Nozaki, 1976; Tandon et al., 1979).
122
A. G . MILNES
If GaAs is grown in the presence of silicon, Si02, and/or oxygen, some workers believe that complexes of silicon and oxygen form. In particular, Weiner and Jordan (1972) infer that silicon atoms on gallium sites pair with interstitial oxygen atoms, forming a complex which behaves as an acceptor with energy levels about 0.2 and 0.4 eV below the conduction band. Wolfe et al. (1977) report that after adding G a 2 0 3to a Ga melt, the LPE material grown contained a lower concentration of the shallow silicon donor. This is in agreement with the observation of other workers that S i 0 2 dissociation is suppressed (Huber et al., 1979; Martin et al., 1981a). No shallow oxygen donor was seen and a search was not made for the deep-acceptor levels reported by Weiner and Jordan (1972). Silicon is an effective dopant in MBE (Stall, 1980). C . Germanium
Germanium on a gallium site introduces a shallow donor at about 6 meV, but on an As site it becomes a shallow acceptor at 40 meV. GaAs: Ge grown from a Bridgman melt is n type or closely compensated p type. Addition of Ge compounds during VPE gives n-type material. Hurle (1979d) has considered the compensation of this doping. In LPE, the Ge appears mainly on the As sites and so the layers are p type. Hole concentrations up to 10l9 cm-3 are obtainable. In LPE, Ge is a valuable shallow acceptor because its vapor pressure is much less than that of Zn or Cd and its diffusion coefficient in LPE material is low (Neumann et al., 1977; Shih and Petit, 1974; Springthorpe et al., 1975). In MBE, Ge is a useful n-type dopant. Its sticking coefficient is good, and under suitable growth conditions the tendency to give compensating acceptor centers is no worse than that of Si. However, it does exhibit some residual doping effects after the Ge effusion cell is shuttered, whereas Si does not. If the substrate temperature is high and the arsenic flux low, Ge acts as a p-type dopant (Ploog, 1981). The zero-phonon E, - GeAsphotoluminescence line is at 1.4778 for 4 K. Inplantation of Ge into GaAs has been studied by Kozeikin et al. (1977), Surridge et al. (1977), and Stoneham et al. (1980). D. Tin
Tin is predominantly a shallow donor (5.8 meV) in GaAs. With the electron concentration in the range 10'7-10'9 cmP3,VPE-grown GaAs-Sn is found to have less than 8% of the Sn in an acceptor state (at E, + 0.167 eV from photoluminescence measurement; Schairer et al., 1976;.Maier et al., 1981). Hall mobilities for electron concentrations between 10'5-10'8 cm-3 tend to lie on a curve that is independent of the growth procedure (LPE, VPE, or MBE) parameters such as the temperature or the As-Ga ratio in the gas
IMPURITY LEVELS IN GALLIUM ARSENIDE
123
phase. This is a further indication that the films are not heavily compensated. It is not known whether the acceptor that does form in limited quantities is a complex deep level or is simply due to Sn on isolated As sites. Hurle (1 979c) has a thermodynamic model that assumes the acceptor to be the complex Sn,,V,, . Photoluminescence of the deep Sn related level has been studied by Zemon et al. (1981). In MBE, Sn has been widely used as an n-type dopant, and there seems no evidence for acceptor formation, but it has been found to have a surface rate limitation to the incorporation of Sn atoms, so until a certain surface population, up to 0.1 monolayer, is formed the free-donor density increases exponentially. There has been some tendency, therefore, to prefer Si as an n-type dopant or to consider SnTe as a dopant source for MBE growth since this seems to be better behaved (Wood and Joyce, 1978; Ploog, 1981). Tin diffusion into GaAs from doped SiO films has been studied by Yamazaki et al. (1975) and by Gibbon and Ketchow (1971). Kachurin et al. (1975) conclude that Sn diffuses via gallium vacancies. In the diffusion of tin into GaAs through a window in a phosphosilicate glass mask, care must be taken to reduce interfacial stress between the mask and the substrate, otherwise a lateral diffusion of tin may extend up to 500 times the junction depth (Baliga and Ghandhi, 1974). Ion implantation of Sn in GaAs has been studied by Finstad et al. (1974). After a suitable anneal, about half of the Sn atoms (up to 1.7 x 10'' atoms cmP3)are on substitutional sites. Effects of surface states have been examined by Wolfe and Stillman (1973). Reflectivity measurements on Sn doped GaAs have been made by Rheinlander et al. (1969). E. Lead
Lead is also a shallow donor, but the low segregation coefficient (results in it being little used. The photoluminescence is broad band and suggests a compensated acceptor level at about E, + 0.12 eV (Kressel et al., 1968b). X. OXYGEN IN GaAs
Oxygen (0.J in GaAs was considered for many years to be responsible for the deep donor EL2 level seen at about E, - 0.75 to 0.83 eV and present in concentrations up to 10'7 ~ m - If~ net . shallow acceptors (NA- ND) are present in the crystal (at a lesser concentration), the deep donor provides the compensating electrons needed by the net acceptors. Hence the free carrier concentration becomes very small and the GaAs is semi-insulating ( lo6 R-cm or more). There is now considerable doubt about the role of oxygen in the
124
A. G . MILNES
formation of the EL2 level. One good reason for supposing that oxygen is involved is the study of Wolford et al. (1981, 1982) that commences with the oxygen substitutional on the phosphorus site in GaP with a donor level at E, - 0.89 eV and tracks this energy level through the alloy range of GaP,As,-, as x goes from 1 to 0.6. Extrapolation of this energy line to x = 0, as shown in Fig. 24, gives an energy level that nearly coincides with that of the EL2 trap; however, this does not allow us to presume that the EL2 level is substitutional oxygen on an arsenic site. Alternatively, one might accept that OAsdoes exist with an energy close to that predicted by the extrapolation in Fig. 24, but that its concentration is low and its effect sometimes obscured by the presence of a non-oxygen-related complex, the
2.2
-
0
0.5
GaAs
X
FIG.24. The 0 levels for GaP and GaP,As, - * fall on a line that may be projected to GaAs, with a result in the vicinity of the EL2 level; T = 5 K. (After Wolford et al., 1981, 1982 and S. Modesti, D. J. Wolford, B. G. Streetman, and P. Williams, private communication; reproduced with permission of the authors.)
IMPURITY LEVELS IN GALLIUM ARSENIDE
125
EL2 electron trap, that just happens to have an almost identical energy level. Photoluminescence at 0.64 eV has been attributed by Taniguchi and Ikoma (1982)to an oxygen-related trap that is distinct from EL2. The possibility also exists that oxygen-related complexes may also be present in the GaAs that are quite different in energy than the EL2 position. For studies of GaAs, - x P x see Sankey et al. (1980). Huber et al. (1979) state that oxygen is not involved, either directly or as part of a complex defect, in the origin of the main electron trap (EL2) at E, - 0.75 eV. From secondary-ion mass spectrometry (SIMS) they determine the oxygen content of certain crystals to be in the low lOI5 ~ m - ~ , whereas the EL2 concentration is one order of magnitude higher. Attempts to create the E, - 0.75 eV level in LPE GaAs or boat-grown GaAs by the addition of G a 2 0 3to the Ga melt are unsuccessful, although the residual Si content may be reduced in the crystal by such addition (Kaminska et al., 1981). Consider other observations of the effects of oxygen on trap levels in GaAs. Weiner and Jordan (1972) suggest that silicon atoms on gallium sites pair with interstitial oxygen atoms, forming a complex which behaves as an acceptor with energy levels near E, - 0.2 eV and E, - 0.4 eV. It is assumed that this complex can dissociate upon annealing at 800°C by the reaction 2(siG,0,)= (si,,O,)O
+ Siba + 3e-
Gallium arsenide grown in quartz under a B20, layer converts from semi-insulating to n type. This is accompanied by disappearance of the 0.2 and 0.4 levels. Annealing at much higher temperatures (1 100°C)reconverted the material to semi-insulating type. Gallium arsenide grown in A1203 did not show these conversion effects so, clearly, Si is involved. Weiner and Jordan are uncertain whether the E, - 0.2 and E, - 0.4 eV levels are different charge states of the Si,,Oi level or represent different configurations of the complex (such as spacing of the Si and Oi atoms). The Weiner and Jordan specimens were Bridgman-type material. However, ion-beam implantation of oxygen in n-type GaAs also results in levels at E, - 0.2 eV and E, - 0.4 eV from conductivity versus temperature measurements and from photovoltage measurements (Abdalla et al., 1975). Other studies of the effects of oxygen implantation in GaAs have been made by Deveaud et al. (1975) and Favennec (1976). Introduction of O2 into the gas stream during VPE growth of GaAs does not affect the EL2 concentration or produce any other electron traps (Wallis et al., 1981b). However, if Al,Ga,-,As is grown, even in trace quantities O 2 produces an acceptor level at E, - 0.41 eV for x c 5%, and this level seems to deepen with increasing x. Hall studies by Akita et al. (1971) of oxygendoped boat-grown GaAs and VPE layers grown in the presence of oxygen provide support for levels between E, - 0.176 eV and E, - 0.21 eV. Look (1981) also has studied levels in this region of the band gap.
126
A. G . MILNES
Monemar and Blum (1977) in a study of O-implanted GaAs find a broad photoluminescence that peaks at 1.23 eV, but they attribute this to a VGadonor transition. Bound exciton-donor luminescence has been studied by White (1974).A survey of photoluminescence has been presented by Williams and Bebb (1972). Vasudev and Bube (1978) have studied photocapacitance effects in bulk n-type GaAs grown by the horizontal Bridgman technique at 1200°C with G a 2 0 3added. They see dominant deep levels at E, - 0.46 eV, E, - 0.79 eV, and E, - 1.10 eV. The E, - 1.10 eV level is considered likely to be trace contaminated by iron (the possibility of Cu or Ni contamination might be considered). The E, - 0.46 eV level (present in lower concentration than the 0.79 eV level) is unidentified, but it is considered that this may be an oxygenassociated level. The range of specimens available did not permit a systematic study of the trap concentration versus the Ga,O, added or versus the actual oxygen content of the material. Photocapacitance studies have also been made by White (1976). Aleksandrova et al. (1973) in studies of VPE layers (Ga-AsC1,-H, system) grown in the presence of oxygen find deep centers at E, - 0.35 eV, E, - 0.56 to 0.62 eV, E, - 0.8 and 1.01 eV. The acceptor level at 0.35 f 0.03 eV is considered the most clearly related to oxygen. The 0.56-0.62 eV levels are stated to be donor-like in nature and to be related to lattice defects. The 0.8 eV level is attributed to Cr acceptors. Further evidence that there is a significant level at about E, - 0.4 eV in n-type boat-grown GaAs: 0 occurs in the studies of Hatch and Ridley (1979). The concentration of the center was about 7 x lo', cm-,. The photoionization threshold was 0.41 eV, and the spectral dependence agreed with the Lucovsky model (suggesting that the center involved was donor-like and neutral in the n-type material). In other work, with similar n-type GaAs: 0, levels at E, - 0.15 eV and E, - 0.23 eV have also been observed (Leach and Ridley, 1978). Some evidence that oxygen can be present in an interstitial position in GaAs comes from the IR absorption band near 840 cm-' that is confirmed to be oxygen related by the introduction of the O'* isotope (Akkerman et al., 1976). The relatively high vibration frequency suggests interstitial position for the oxygen atoms and covalent binding to the neighbors. The vibrations of substitutional impurities (for instance, C") lie at longer wavelengths (582 cm- ). Studies of 0,-doped n-type GaAs by Li and Huang (1973) conclude that photoinjected excess carrier recombination at 20 K is dominated by band-toband radiative recombination and that any deep-level oxygen impurities that may be present are neither acting as recombination centers nor as trapping centers for the excess carriers.
'
IMPURITY LEVELS IN GALLIUM ARSENIDE
127
The effective diffusion coefficient of oxygen in GaAs from mass spectroexp( - 1.1 eV/kT) cmz metric data has been stated to be DcR = 2 x sec- between 700" and 900°C (Rachmann and Biermann, 1969); however, this work reports oxygen contents of 10'9-1020 atoms cm-3 instead of the 10I6 atoms cm-3 typically measured when considerable precautions are taken in the measurement (Blackmore et al., 1976). High solubility of oxygen in GaAs ( > 10" atoms cm-3) has also been reported by Borisova et al. (1977) in systematic doping studies. On the other hand, a study of oxygen by charged-particle activation with the 160(3He,p)l'F reaction in which "F is created by 3He bombardment suggests the oxygen concentration to be 4 x lo1' atoms cmP3in LEC material and 2 x 10I6 cm-3 in other material (Emori et al., 1981). In summary, then, the oxygen-related levels in GaAs are not well known. There are a number of observations that suggest that levels at about E, 0.2 eV and E, - 0.4 eV should be considered oxygen related, and there is growing evidence that the E, - 0.75 to 0.83 eV level is not oxygen related. Evidence that the "oxygen level" when filled with an electron may have a metastable state (Bois and Pinard, 1974; Bois and Vincent, 1977; Vincent and Bois, 1978; Bois et al., 1978) is accumulating (there is good evidence for similar effects associated with 0 in Gap). See also Mitonneau and Mircea (1979).
'
XI. GROUPVI SHALLOW DONORS: S, Se, Te Sulfur, selenium, and tellurium are shallow n-type dopants in GaAs. The activation energies, as discussed in Section IX, are all quite small. At high doping concentrations, the behavior of Se and Te in GaAs is disturbed by impurity complexing or precipitation (Abrahams, 1974; Casey et al., 1971; DiLorenzo, 1971; Greene, 1971; Hutchinson and Dobson, 1975; Morgulis et al., 1974; Sealy, 1975; Shifrin et al., 1974; Takahashi, 1980). A broad emission band centered at 1.2 eV is observed in GaAs doped with group VI elements, whether melt or vapor grown, or grown by LPE with gallium as the solvent. This photoluminescence increases in intensity, relative to the band-gap radiation, with increasing dopant concentration in the lo1' cmP3range. The luminescing recombination centers responsible for this 1.2 eV band may be complexes such as V, + 3Se or V,, + 3Te. These respresent the solid solution of Ga,Se, or Ga,Te, in GaAs; VGaTeAscomplexes also have been surmised. With increasing dopant content, the solubility limit is eventually exceeded and precipitates of Ga,Se3 or Ga,Te, are formed. ~ , With the dopant concentration increased beyond 2 to 3 x 10'' ~ m - the radiative efficiency falls off sharply. Kressel et al. (1968b) suggest that the
128
A . G. MILNES
falloff is partly due to nonradiative recombination via small precipitates, each containing on the order of 1000 dopant atoms. (The radiative efficiency of n-type silicon-doped GaAs prepared by LPE does not fall off up to 4.2 x lo1' cmP3)The formation of complex centers is to be expected since the strain energy of the crystal is reduced when a vacancy relocates next to an impurity atom that is somewhat large for its site position. Nishizawa et al. (1974)have studied Te-doped GaAs crystals heated in the presence of arsenic vapor. Complexes associated both with the donor impurity (Te) and with the defects due to the deviation from the stiochiometric composition are formed. These complexes are responsible for both the 0.18 eV acceptor level and the 1.30 eV emission band at 77 K. Furthermore, defects with acceptor levels of 0.12 and 0.15 eV are found. These defects are unstable and disappear by reannealing at comparatively low temperatures (300-400°C), leaving acceptor levels at 0.1 8 eV. Precision lattice measurements on 4-8 x 10" cmP3 n-doped GaAs:Te show superdilation effects (Dobson et al., 1978). A possible defect causing superdilation could be an interstitial complex of Ga and Te on a split substitutional site, VGaGaiTei,which could either be a donor or a neutrally charged complex in equilibrium with the Te substitutional donor TeAs.Annealing at 880°C may remove this complex and form interstitial (GaiAsi)loops and the complex TeAsVGamay form. Tellurium is thought to interact with other impurities in GaAs such as Zn (Blashku et al., 1971). (Te-Li complexes were mentioned earlier in the discussion of lithium doping.) The diffusion of Te in GaAs has been studied by Karelina et al. (1974), who conclude that arsenic vacancies play an important role in the diffusion process. Ion implantation of Te in GaAs has been studied by Pashley and Welch (1975) and Eisen and Welch (1977). Selenium implantation has been studied by Gamo et al. (1977) and Lidow et al. (1978).Tellurium is not a suitable dopant in MBE of GaAs because of a pronounced surface segregation behavior (Cho and Arthur, 1975). Sulfur is not usually used as a dopant in melt or LPE growth of GaAs because of its high vapor pressure; however, it may be added to VPE-grown layers because of the convenience of handling in the form of sulfur gas compounds such as H,S, SCl, and SC1, (Savva, 1976). Diffusion of S into GaAs is best accomplished by the use of a ternary diffusion source of Gas :As (Matino, 1974). Sulfur-ion implanted profiles in GaAs (200 keV) have been studied by Comas and Plew (1976). A review of ion-implantation in compound semiconductors has been made by Degen (1973). Implantation is a technology that is rapidly increasing in importance in GaAs processing.
IMPURITY LEVELS IN GALLIUM ARSENIDE XII.
129
OTHERIMPURITIES:Mo, Ru, Pd, W, Pt, Tm, Nd A. Molybdenum
Molybdenum has been studied as a possible barrier layer in ohmic contacts to n-type GaAs to impede the tendency for the metals of the contact structure such as gold to cause loss of arsenic and gallium and inward diffusion of gold at temperatures of a few hundred degrees. In earlier work, Mo did not seem to be a very effective barrier layer, but more recent studies have suggested that barrier action is observed if the layer is sputtered in the presence of nitrogen (Nowicki and Wang, 1978). Studies of the diffusion of Mo into GaAs have not been found in the literature. The photoluminescence peaks of GaAs: Mo have been reported as at 1.34 and 1.43 eV (Kornilov et al., 1974), but no activation energy with respect to a band edge has been reported. B. Ruthenium
Chemisorbed Ru ions on the surface of n-type GaAs decrease the surface recombination velocity of electrons and holes from 5 x 10' to 3.5 x lo4 cm sec-'. The ions in a one-third monolayer thickness are confined to the surface and do not form a new junction by diffusing into the GaAs, even after a 300°C treatment for 1 hr. This is the first known example of simple chemisorption reducing the surface recombination of GaAs. The use of an overlayer of Al,Ga, -xAs, however, on GaAs results in a recombination velocity of 450 100 cm sec-'. (Nelson et al., 1980). C . Palladium
Palladium is a part of certain recipes that make moderately satisfactory contacts to n- and p-type GaAs. For contacts to n-type GaAs, Pd-Ge is used and typical anneals are at 350°C or more for 10 min. The specific contact to 5 x R-cmZfor dopings between resistances may vary from 5 x 1OI6 and 1 O I 8 electrons ~ m - respectively. ~ , For contacts to p-type GaAs, the metallization may be Au-Pd, Ag-Pd, A1-Pd, and specific contact resistances are high, about R-cm2, even on 1 O I 8 cm-3 material. The surface metallurgy of the reaction of Pd with GaAs has been studied by Olowolafe et al. (1979). The kinetics of Pd penetration into GaAs was found to be controlled by diffusion with an activation energy of 1.4 eV, corresponding to a characteristic (lle) distance of 160 A after 1 hr
130
A. G. MILNES
at 250°C and 730 8, after 10 min at 350°C. This considerable penetration may be harmful in thin junction devices. The As and Ga compounds formed as a result of contact reactions were identified to be PdAs, and PdGa at 250°C; PdAs,, PdGa, and Pd,Ga at 350°C; and PdGa at 500°C. In a study of Pd-Ge-n-type GaAs contacts, Grinolds and Robinson (1980) find that the Pd-Ga and Pd-As interactions provide a situation favorable for greater occupation of Ga sites by Ge than As sites and so the n-type doping of the GaAs is increased in the sintering process. Their results confirm the specific resistance of 3-4 x R-cm2 for 1OI6 cm-3 n-type GaAs, reported by Sinha et al. (1975), with sintering temperatures of 500°C or more. There appear to have been no studies of the energy levels of Pd in GaAs. D. Tungsten Tungsten (about 3000 A thick) offers an excellent barrier to diffusion of Au and Pt atoms at temperatures up to 500°C. A structure such as Au-WPtGa-PtAs,-n-type GaAs exhibits a good Schottky barrier (& = 0.89 eV) due to the formation of the PtAs, layer (Sinha, 1975). Tungsten directly on the n-type GaAs has a barrier height of about 0.69 eV. Tungsten or W: 10% Ti layers may also be used as barrier layers in ohmic contact technology of the kind Au-Pt-W-Au: Ni: Ge/n-type GaAs. The W serves as a diffusion barrier and also as a stress compensator (Rozgonyi et al., 1974). The Pt layer (2000 A) helps improve the adhesion between the thick outer Au layer and the W. Tungsten-doped GaAs exhibits photoluminescence bands at 1.13 and 0.83 eV (Kornilov et al., 1974) and 0.707-0.651 eV (Ushakov and Gippius, 1980). Attempts have been made to dope GaAs with W by Czochralski growth from a W-doped melt and by addition of a tungsten compound to a chloride-gas transport system (750°C). Thermally stimulated current measurements then show activation energies of 0.39-0.40 eV (Aleksandrova et al., 1976).Since this level is similar to those seen in GaAs and attributed to other causes (such as Cu or Ni), confirmation of a direct W relationship is needed. E. Platinum The metallurgy of Pt layers (up to 9000 A thick) on GaAs when heat treated in the range 250-500°C has been studied by Chang et al. (1975).When Pt-GaAs was treated in vacuum, the Pt-GaAs interaction was initiated with a rapid migration of Ga into Pt and simultaneous formation of an As-rich layer at the Pt-GaAs interface. Gallium eventually traveled the entire thickness of even the 9000-A Pt film, whereas As stopped abruptly at a distance ~ 2 / of 3 the way into the Pt. Little or no Au was detected (< 1 at. %)
IMPURITY LEVELS I N GALLIUM ARSENIDE
131
in the Pt or GaAs after extensive Pt-GaAs reaction in Au-Pt-GaAs. The reacted Pt film displayed a multilayered structure with each layer containing different amounts of Ga, As, and 0.Pt-GaAs heated in air behaved similarly with two added features: a surface layer containing mostly Ga and 0 formed over the Pt, and 0 diffused through the (reacted) Pt to form an 0-rich layer at the Pt-GaAs interface. Attempts to dope GaAs with Pt have not been found in the literature and the energy levels (if any) are unknown. Platinum is an effective recombination center and lifetime controller in Si; however, a layer of Pt heated at 600°C for 5 min on the surface of VPE n-type GaAs has been found to raise the minority-carrier (hole) diffusion length (Partin et al., 1979d). This is presumably because of gettering to the surface of a lifetime killing center rather than the entry of Pt deep into the GaAs layer. F . Rare-Earth Studies in GaAs:Tm and GaAs:Nd Rare-earth elements are of interest because their incomplete 4f subshells give spectral properties that are useful, such as stimulated emission when incorporated in various crystal hosts. Relatively little work has been done on the incorporation of rare earths in covalent semiconductors such as Si, Ge, and GaAs since the solubilities are expected to be low. The solubility and diffusion of thulium in GaAs have been studied by Casey and Pearson (1964).Thulium was chosen because of its relatively small atomic size (1.746 A) and the availability of a suitable radioactive isotope, l7'Tm. The maximum solubility was found to be 4 x 10l7 cm-3 at 1150°C. This low solubility was attributed to the fact that the Tm atom is 25% larger than the Ga atom that it replaces substitutionally. Richman (1964) has made initial studies of Nd doping of GaAs. The pumping band of Nd closely matches the emission obtained from GaAs luminescent diodes. Vapor-phase epitaxy was attempted and some inclusion of Nd in the grown crystal was observed.
XIII. MINORITY-CARRIER RECOMBINATION, GENERATION, LIFETIME, A N D DIFFUSION LENGTH Since GaAs is a direct band-gap material, non-phonon-assisted recombination processes set an upper limit to the excess minority-carrier lifetime. However, because of the high recombination center content in GaAs (bulk, VPE, and LPE), the upper limiting lifetimes are hardly ever approached (Ryan and Eberhardt, 1972; Acket and Scheer, 1971; Alferov et al., 1973; Garbuzov et al., 1978).
132
A. G . MILNES
Mayburg (1961) has considered the direct recombination expected from the recombination theory of Hall (1959). The expression obtained for the probability of direct radiative recombination is
where n is the refractive index, m is the free electron mass, m, and m, are the density of states effective masses of electrons and holes, and Eg is the band gap. For GaAs at 300 K, with n = 3.6, m, = 0.068, m, = 0.5, and Eg = 1.43 eV, Bd, = 1.7 x lo-’’ cm3 sec-’. The direct recombination lifetime therefore for an electron concentration n cm-3 should be given by 1
-
5.9 x lo-* sec for n n
td, N Bdrn - {5.9 x 10-6 sec for
= =
cm-3 101s cm-3
(25)
For n = 1 O I 7 and 10” cm-3, the 300 K hole mobilities tend to be about 300 and 400 cmz V-’ sec-’, respectively. Hence the hole diffusion lengths from L, = and 0,= pkT/q may be expected to have upper limits of 6.8 and 78 pm, respectively. Measurements of actual hole diffusion lengths show considerable scatter from specimen to specimen (Sekela et al., 1974). The largest values are often about one third of the L, values estimated above. So recombination centers typically lower the minority carrier lifetimes by an order of magnitude from the direct recombination lifetimes. The scatter of diffusion-length values versus doping concentration indicates that in the early 1970s, many laboratories were unable to control the recombination levels present, either because trace impurities were in every crystal or because the processing used led to the formation of electrically important native defect levels or defect-impurity complexes (Baliga, 1978; Bergmann et al., 1976; Biswas, 1971; Bludau and Wagner, 1976; Chakravarti and Parui, 1973; Kamm, 1976; Jastrzebski et al., 1980; Leheny et al., 1973; Lender et al., 1979; Miki et al., 1979; Lomako et al., 1976; Matare and Wolff, 1978; Mathur and Rogers, 1977; Pilkuhn, 1976; Papaioannou and Euthymiou, 1978; Peka et al., 1975; Seki et al., 1978; Kushiro et al., 1977; Goodman, 1961; Kamejima et al., 1979; Chan and Sah, 1979; Mircea et al., 1976; Hwang, 1971, 1972; Kravchenko et al., 1977). Since the recombination processes are affected by the growth processes used, effects in LPE, VPE, bulk (Bridgman or LEC) materials have to be considered separately. The technique employed to determine GaAs diffusion lengths is usually measurement of the variation of electron-beam-induced current (EBIC) as a beam of electrons ( - 25 keV) in a scanning electron microscope is moved across the surface of a pn junction or metal n Schottkybarrier junction (Wittry and Kyser, 1965).At 25 keV beam energy, the beam
,/m’/’
IMPURITY LEVELS IN GALLIUM ARSENIDE
133
FIG.25. Carrier-capture cross sections versus inverse temperature for various deep levels in GaAs. Subscript n denotes electron capture; subscript p denotes hole capture. Levels shown for GaAs are due to Cu, Fe, Cr, 0, the E3 radiation-damage defect, and two unidentified, but commonly occurring, levels A and B. Dashed lines show temperature dependence of cross sections extrapolated to T = a.The value of urnpredicted by the theoretical model is indicated on the ordinate. (After Henry and Lang, 1977.)
produces a carrier generation region roughly 0.5 pm in diameter and this limits the accuracy of the diffusion-length measurement to about 0.2 pm at best (Martinelli and Wang, 1973; Rao-Sahib and Wittry, 1969; Van Opdorp et al., 1974; Vitale et al., 1979; Wu and Wittry, 1978). Other factors that limit accuracy have been considered by Berz and Kuiken (1976), Chi and Gatos (1977), Epifanov et al. (1976), Flat and Milnes (1978), Hu (1978), Hu and Drowley (1978), Kuiken et al. (1976), LastrasMartinez (1979), Partin et al. (1979b), Rossin et al. (1979), von Roos (1978, 1979), Wight et al. (1981), and others. A. In LPE-Grown GaAs Layers
Liquid-phase epitaxial-grown GaAs layers are relatively free of traps, except for levels (A and B) at E, + 0.40 eV and E, + 0.71 eV, present in concentrations of 10'4-1015 cmP3. The electron- and hole-capture cross sections for these traps vary with temperature as shown in Fig. 25. From such data and from other measurements, Partin et al. (1979a) arrived at the cross sections shown in Table XIV for a specimen of an LPE GaAs (ND- NAabout ~ ) the A and B levels in the concentrations shown. 2.8 x lo'' ~ m - containing The expression for the excess hole lifetime may be expected to be of the form
134
A. G . MILNES
TABLE XIV CONCENTRATIONS, CAPTURE CROSSSECTIONS, AND EMISSION RATES FOR THE A AND B LEVELS AT 300 K" Level
N , (cm- 3,
A B
2.4 x 1015 2.7 x l o i 5
0"
(cm')
1.4 x 10-17 4.2 x
up (cm')
ep (sec-')
4.2 x 1 0 - 1 5 2.1 x 1 0 - l ~
1.2 x 106 1.7 x lo6
Reproduced with permission of the American Physical Society.
With hole and electron velocities up = 1.6 x lo7 cm sec-' and 0, = 4.5 x lo7 cm sec-' and the data from Table XIV, we have for level A: u p o p p = lo5 sec-'; u,a,n = 1.8 x lo6 sec-'; e p = 1.2 x lo6 sec-'; and vpopN,= 1.6 x lo8 sec-'. Thus, upopp is negligible, but e p is comparable to sec for level A at 300 K. v,a,n. Thus, z p is 1.0 x Level B is much deeper than level A, so ep is negligible. Also, for p = 2 x 10l2 cmV3,upapp = 7 x lo4 sec-', and n,a,n = 5 x lo3 sec-', and so upopp dominates the numerator of Eq. (26). Therefore, Eq. (26) reduces to zp = [p/(v,a,N,)] = 1.5 x sec. Thus, zp determined by level B is injection-level dependent and is much larger than zp determined by level A. The large value of z p for level B is caused by its very small electron-capture cross section, which makes electron capture the rate-limiting recombination step. Therefore, zp is determined only by level A and is independent of holeinjection level. The value of Dp in p-type GaAs at NA - N,, = 3 x 10l5 is 9 cm2 sec- '. If we assume that Dp is the same in n-type GaAs with ND - NA = 3 x lo", then for L, = (Dpzp)'/2 a diffusion length of 3.0 pm is obtained. The hole diffusion length measured by Partin et al. was 3.4 pm, so, they conclude that the A level in LPE GaAs is the minority-carrier (hole) lifetime controller. The A- and B-level concentrations, which tend to be closely similar, depend on the details of the LPE process in some way that has not been well characterized. To see these levels by DLTS, it is necessary to form a p n junction unless optical injection of holes is used. The p region preferably should be grown by LPE. If an LPE n region is grown and followed by a Zn diffusion to create the p layer needed, the junction tends not to exhibit the A and B levels. The Zn diffusion process in some way getters or destroys the A and B levels. The physical natures of the A and B levels are still the subject of speculation, as mentioned in Section 111 (Ashley and Beal, 1978; Casey et al., 1973; Kressel, 1974). Electron diffusion lengths in p-type GaAs grown from Ga solution with Ge as the dopant are quite large. Values in the range 23-18 pm have been observed for doping levels of 1.1 x 10l8 to 5 x 10l8 cm-3 by optical microprobing of beveled junctions (Ashley et al., 1973). These values correspond
135
IMPURITY LEVELS IN GALLIUM ARSENIDE
TABLE XV ELECTRON DIFFUSION LENGTHS IN LPE p-TYPEGaAs :Ge" Ga/Ge atomic ratio in solution 10 100 1000 10
Estimated Pn D* S N A - N, (cm' V - ' (cm' (lo6 cm T,,,, Calc' ( ~ m - ~ ) sec-') sec-l) Lc (pm) Ld (pm) sec-') (nsec) L (pm) 1.1 x 10'' 2.0 x 10'' 6 x 10l6 1.2 1019
1500 3500 6500
40 90 165
-
-
5.5 10.5 20.0 5.9
4.6 9.0 26.0 -
1.1 0.74
11 12.5 38
6.6 10.6 25
-
-
-
-
From Ettenburg et al. (1973). Reproduced with permission of the American Physical Society. Calculated from estimate of electron mobility. ' Determined from laser beam scan measurement. Determined from particle-scan measurement. Calculated from luminescent decay time (T) measurement.
to electron lifetimes of about 5 x l o w 8sec. The nature of the recombination process (band-band or band-recombination center) was not studied. In similar studies of LPE-grown GaAs: Ge, the electron diffusion lengths were examined by two direct methods: (1) He-Ne laser beam scan on beveled pn junctions, and (2) an a-particle scan on cleaved samples; and by an indirect method, that of studing the electroluminescent decay time. The results obtained for the three methods were in good agreement (see Table XV). The diffusion length for 2 x l o r 8 cm3 doping is seen to be about 10 pm or about half of that observed in the Ashley et al. study. The literature shows lower diffusion lengths (3-6 pm) for p-type LPE material doped with Si, Zn, or Cd; however, it is not certain that this must be generally true. In p + n solar cells grown by LPE, the hole diffusion lengths for the n-type GaAs (NDtypically 8 x 1 O I 6 cm-,) tend to be about 3 pm. After irradiation with 10" protons (of energies 100-290 keV), the diffusion lengths become less than 1 pm, and the recombination is attributed to deep-level defects at E, 0.57 and 0.7 eV, and E, - 0.52 and 0.71 eV.
+
B. I n VPE-Grown GaAs Layers
In studies of the Ga-AsC1,-H, VPE growth process for GaAs, Humbert et al. (1976) find levels at E, 0.4 eV and E, 0.75 eV predominant. These are present in about equal concentrations, and they consider these two levels to correspond to two ionization states of a unique center involving a native defect (perhaps a Ga vacancy), a very common impurity, or a complex of
+
+
A. G. MILNES
136
both, rather than believing that these two levels are the typical “Cu” and EL2 levels. See also Iwasaki and Sugibuchi (1971). One commonly used VPE growth process for GaAs involves the reaction of ASH, and GaCl at about 700°C. The dominant deep impurity seen is then at E, - 0.82 eV (which has been termed the EL2 level, see Section 111).In one study, by Miller et al. (1977), the ratio of ASH, to GaCI was varied from 3/1 to 1/1 to 1/3 and the layers grown were n type and about 6 x 10’’ cm-, in the absence of intentional n doping. The density of the deep level (Table XVI) was found to correlate with variations of the ratio AsH,/GaCl, and the hole lifetime varied in agreement. Since the smallest deep-level densities and the largest lifetimes are obtained under Ga-rich conditions, the recombination level is probably related to gallium vacancies or As,,-related defects. Mitonneau et al. (1979) believe, however, that this is not related to the EL2 trap, which they show is not a significant hole recombination center since the hole-capture cross section is only 2 x lo-’’ cm’. They find a hole trap cm2 and to be the main HLlO at E, + 0.83 eV to have cr = 5 x recombination level. See also Ettenberg et al. (1976) and Ozeki (1979). A growth study has been performed for Zn-doped ( p = 10’’ cm-,) VPE-grown GaAs, also with AsH,/GaCl ratios of 3/1, 1/1, and 1/3. For electroluminescence decay studies this yielded electron lifetimes of 1.3, 3.2, and 3.4 nsec. For an electron mobility of 1500 cm2 V-’ sec- the diffusion lengths are 2.3, 3.6, and 3.6 pm. Direct measurement of diffusion lengths by laser probes gave 2.5, 4.8, and 4.8 pm. So, Ga-rich conditions during VPE growth minimize recombination actions. Vapor-phase GaAs may also be grown from the organometallic compound trimethyl gallium and arsine. The dominant (electron) trap in n-type MOVPE GaAs was again at E, - 0.82 eV, and the density of this trap varied with As/Ga ratio, as shown in Fig. 26 (Bhattacharya et al., 1980). Hole traps were also observed at E, + 0.36 eV and E, + 0.31 eV and ascribed to
’,
TABLE XVI THEEFFECTS OF GAS-PHASE STOICHIOMETRY ON DEEP-LEVEL CONCENTRATION AND MINORITY-CARRIER LIFETIME^ ~~
Sample
Ratio of ASH, to GaCl in gas phase
Minority-carrier lifetime 5 (nsec)
Density of dominant deep level (cm-3)
1471 1470 1500
31 1 111 113
5 11 15
9 x 1013 4 x 1013 2 x 1013
From Miller et al. (1977). Reproduced with permission of the American Physical Society.
137
IMPURITY LEVELS I N GALLIUM ARSENIDE
2
4
6 8 As/Ga
10
12
FIG.26. Dependence of the concentration of the dominant 0.82 thP A o / G *
mtin in
thn
pnit-vial
I ~ v P ~ .The Ant-
. & t o
0.02 eV electron trap on
h . , ,.lr\oeA
;nA;,.otnA
,.irrlnr
..,PWP
Antar-
mined by Miller et al. (1977). The bars indicate the dispersion in the value of NT.(After Bhattacharya er al., 1980; reproduced with permission of American Physical Society.)
unknown impurities. In other studies of M OVPE-grown GaAs, Samuelson a concentration in the range 5 x 10I3 to 3 x 1014 cm-3 that varies in proportion to P& . This result is compatible with the EL2 level being gallium vacancy or AsGaVGa related. However, the results of Zhu et al. (1981) are less decisive. Considerable care is needed in processing GaAs. For instance, Ni has been shown to degrade L, in VPE n-type GaAs when diffused inward from the surface at moderate temperatures (600-700°C) (Fig. 27). Since epilayers are usually only a few micrometers thick, it is conceivable that a tenth or hundredth of a surface monolayer of Ni (1014-1013cm-2) could produce very high concentrations of recombination centers if it all entered the layer. The trap levels that are clearly Ni related are at E, 0.39 eV (believed to be Ni,,, although see the discussion in Section VI) and at E, - 0.39 eV. The trap concentrations measured in one VPE specimen after a 7OO0C, 20-min treatment are shown in Fig. 28 together with the estimated effective lifetimes associated with the possible recombination paths. The dominant recombination action is seen to be through the E, - 0.39 eV level which has a capture cm2. This level is present in low concentracross section op of about tion (typically, rnid-10I3 crnp3) in as-grown VPE n-type GaAs because of inadvertent contamination, but the concentration is greatly increased by the Ni diffusion. The as-grown VPE n-type GaAs (Te doped 5 x 10'' ~ m - also ~ ) contained an energy level at E, - 0.75 eV (the EL2 level at E, - 0.82 eV of ' ~ and was other studies). This was in substantial quantity ( ~ 1 0 cmP3) considered to be a complex of a gallium vacancy with a presently unidentified
+
138
A. G. MILNES I
I
I
I
I
I
STRIPE 5 0 0 i N i ; H E A T E D 600°C,5 M i N .
V P E ( N , = ~ ~ I O ern-31 ' ~
+
+
+ t
0
+
0 0
0
0
0
0
0 0 0 0 0
0
0
24OC
DATA
+ Il0"C DATA 0
0.3688 k
0
I
I
9
10
N
i I
4
I
I
li P O S l T i O N A L O N G T H E W A F E R (mrn)
I 12
I
13
E F F E C T O F N I C K E L I N DEGRADING Lp IN n V P E G o A s
FIG.27. Effect ofnickel in degrading L, in VPE GaAs. The dip in the diffusion length occurs at the location of the applied Ni stripe. (After Partin e l al., 1979a; reproduced with permission of the American Physical Society.)
impurity (or defect), namely (VGa+ X). When Ni was diffused into VPE n-type GaAs, the concentration of the complex decreased and the E, - 0.39 eV-level concentration increased, and in equal quantity a level at E, 0.39 eV appeared. This E, + 0.39 eV level is considered to be Ni substitutional on a Ga site. (It has parameters quite different from those of the A level in LPE material.) The model tentatively proposed for the interaction of Ni with the E, - 0.75 eV level is that inward-diffusing interstitial Ni dissociates the gallium complex (VGa+ X) to form Ni,, and X. These become the levels E, 0.39 eV and E, - 0.39 eV, respectively. A Zn-diffused layer is found to getter both of these levels and to improve the hole diffusion length. An Ni layer heated to 600-700°C is not effective in altering the characteristics of LPE layers, presumably because the absence of gallium defects impedes its entry. It is found that surface treatments of VPE GaAs with many other metals (for instance, Au, Cu, W, and Pt), followed by 600°C 5-min heat treatments had the effect of increasing the hole diffusion length in the GaAs (say, from 3 pm to 5 or 6 pm). Partin et al. (1979d) believe that these metals form shallow surface regions and that the electric fields created have the effect of gettering to the surface the recombination center that is controlling the hole diffusion length in the n-type side of the junction. It is noted that in these experiments Au had no effect on L, in GaAs, even when heated at 700°C. Gold has acceptor levels in GaAs at E, + 0.405 eV and possibly at 0.05 eV. However, it is also known that an nf layer forms under a heated Au
+
+
139
IMPURITY LEVELS IN GALLIUM ARSENIDE
----- -------0.39eV
0.75 eV
I
I I I
I I I
1I
I
I
I I
J III
21I
b"l
I I I
I
I
I
I
JI
I
I
I
I
I
I -------I ------- T--------'--
2; "b"'
I I
I
I
I I
i-
I
Ec EF"
I
I I
II
+ I I
1.43 eV
0.63 eV
i
0.39 eV
a
IP
b
'I
1
FIG.28. Band diagram showing the concentration and value of zp for each defect level in (Ni, Zn)-diffused VPE n GaAs; T = 700°C. The recombination rate-limiting defect parameter is also indicated: L, = (Dp~p)l’* = 1.4 pm; L, = 1.6 pm (SEM). (After Partin et al., 1979a; reproduced with permission of the American Physical Society.)
contact on GaAs. This effect is related to outward diffusion of Ga and can be suppressed by deposition of a Au-Ga alloy of eutectic composition. If it is assumed that the hole lifetime controlling center at E, - 0.39 eV is in some way related to Ni in the bulk of the crystals in a positively charged and somewhat mobile form, then the electric field of a pn junction would tend to getter the centers into the p region; however, an n’n junction would cause repulsion of Ni-related centers from the surface. Further independent studies are needed to confirm these effects and to explore other possible interpretations. If leakage currents in GaAs pn junctions are involved in such studies, care must be taken to control or eliminate surface-generated recombination currents (Henry et al., 1978).
C. DifSusion Lengths in Bridgman-Grown GaAs The studies of Sekela et al. (1974) of hole diffusion lengths in Bridgmangrown n-type GaAs resulted in the data points in Fig. 29 that are not prefixed with a symbol V for VPE or L for LPE. Most of the specimens were in the 5 x 10’6-10’8 cm-3 doping range, and very few of these exhibited hole diffusion lengths in excess of 2 pm. No correlation of diffusion lengths with etch-pit counts (which tended to be in the 103-105 cm-* range) or with the dopant type (Si, Sn, Te, Se) was observed. Deep-level transient spectroscopy equipment was not available at the time of the studies and so the trap levels
140
A. G . MILNES
v7 L17
v 11 VlOV9 V8 L18 v4
v12
“g3
54 ES
V2 52 Vl 51
E3V6 LlS L16
vs
E6
58 ’413
v5
4 25 2&422
11 59
63
L2
I8
42
64
18
2 29
61
65
0.2
kZom
’8
67 62
1615 E4
;y
48 395$28~5 S634275 65 66 4348 1 0 5 0 4:t:44,38 31
L14 L13 5780 El L12 17
3532
L7 49
Ll
1
1 10”
10“
10 1 7
lOl8
10”
No- NA ( ~ r n - ~ )
FIG.29. Hole diffusion length as a function of net electron concentrationfor n-type gallium arsenide samples. Sample numbers prefixed with “V” are VPE material, “L’ samples are LPE, “ E ’ are unspecified epitaxial samples, and those with no prefix are from single-crystal boules prepared by either the LEC or the horizontal Bridgman process. (After Sekela et al., 1974; reproduced with permission of the Institute of Physics, UK.)
in the specimens were not studied. Electron diffusion lengths in Zn- and Cd-doped ingot material were found to be larger than L,, mainly in excess of 5 pm, as might be expected from the fact that the electron mobility is much larger than the hole mobility. Experiments relating measured diffusion lengths to trap determinations in bulk GaAs have not been found in quantity in the literature. Workers have tended to concentrate instead on diffusion lengths and traps in epitaxial layers since the device implications are clearer here. Also, suppliers of bulk GaAs tend to provide the material only on two forms: doped in excess of 10” cm-3 or semi-insulating, and this limits the supply of material for trap studies. However, as Tables V and VI show, electron traps at E, - 0.85 eV and E, - 0.3 eV and a hole trap at E, + 0.35 eV have been observed. In studies by Li (1978) and Li et al. (1980a), the following levels are reported in their n-type GaAs (2 x 10l6 cm-3 doped): (1) A donor-type defect center at E, - 0.38 eV with an electron-capture cross section of about 3 x cm2 and no field dependence. It is possible that this has association with oxygen. Okumura et al. (1976)have seen a level in this vicinity in many n-type GaAs specimens.
IMPURITY LEVELS IN GALLIUM ARSENIDE
141
(2) A donor-type defect level at E, - 0.65 eV with a thermal capture cm2 with no field dependence. cross section of about 3.4 x (3) A donor-type level at E, - 0.78 eV with electron-capture cross section of 1.4 x cm2 that is not field dependent. This is presumably the EL2 center of Table (V), although the capture cross section is smaller than in some previous studies. In recent work with Bridgman-grown material, the EL2 trap concentration has been found to be inhibited by shallow donors (Si or Te) and enhanced by shallow-acceptor doping (Zn). This is compatible with EL2 being related to the AsGa antisite defect and perhaps AsGaVAsor AsGaVGa.Addition of oxygen to the melt during the growth decreases the concentration of Si shallow donors and thus it increases the concentration of EL2. Oxygen itself, however, is not considered to be a part of EL2 (Lagowski et al., 1982). Systematic studies seem desirable of Bridgman- and LEC-grown n- and p-type GaAs with dopings in the range 2 x 1016-10'8 cm-3 to establish the nature of the dominant traps, their roles in minority carrier recombination, and the extent to which they match up with traps seen in typical VPE, MBE, and LPE expitaxial material. Studies of GaP have been made by Hamilton et al. (1979) and Scheffler et al. (1981). XIV. CONCLUDING DISCUSSION Experimental energy levels in the GaAs band gap have been reviewed. A number of factors make it difficult to attribute particular crystal complexes to the observed energy levels. Since GaAs is a compound semiconductor, the number of defects possible is larger than for Si, where antisite defects do not have to be considered. The energy conditions in the crystal (although only partially known) seem to favor complexing of defects, and the formation of antisite defects. Another problem is that the surface of GaAs reacts with many metals, and this can be significant even though the metal is present in a submonolayer quantity. Defects may propagate into the first few microns of the crystal as a result of such reactions at temperatures of a few hundred degrees. Alternatively, gettering to the surface of important defect levels may occur in the course of an experiment. Since experiments involving depletionlayer capacitance transients interrogate regions close to the surface, the potential for error and misinterpretation of results is substantial. A further difficulty is that electron paramagnetic spin resonance experiments can be applied only to a limited set of impurities and defects in GaAs. Experiments for deciphering successfully the physical or chemical nature of defect levels in GaAs tend therefore to be hard to devise and execute successfully. Some progress, however, has been made. Certain chemical species have been identified with particular levels. This is particularly true for the transition elements. The mid-band-gap level (at E, - 0.75 to 0.83 eV) seen in most
142
A. G. MILNES
GaAs (except LPE layers) is still unidentified. Evidence exists that it is not oxygen related but that it involves an antisite defect AsG, or a gallium vacancy. Candidates for this level are the complexes AsGaVAs, AsGaVGa,or AsGaVGaAsGa,but this identification is very tentative. The evidence that the level at E , - 0.33 eV, produced by irradiation, is the gallium vacancy V, has recently been challenged and the suggestion made that arsenic vacancies are involved. Also recently an acceptor at E, + 78 meV has been ascribed to the GaAsantisite defect. With respect to oxygen, an accumulation of measurements seem to indicate that the 0.64 eV luminescence band is involved and that the oxygen may also be associated with an energy level at about E, 0.40 eV. In VPE studies an energy level at about this position, E, - 0.39 eV, is a strong recombination center. These levels need further study, as do the E, + 0.41 eV and E, + 0.71 eV A and B levels in LPE GaAs. The one prediction that may be confidently made is that it will be many years still before all of the defect energy levels in GaAs are identified and understood. ACKNOWLEDGMENTS The support of the Solid State Division, Westinghouse Research and Development Center, Churchill, Pennsylvania, during the preparation of this article is gratefully acknowledged.
REFERENCES Abdalla, M. I. el al. (1975). Transport properties of O f implanted in GaAs. Conf. Ser.-Inst. Phys. 24, 341. Abrahams, M. S . et al. (1974). Interdependence of strain, precipitation, and dislocation formation in epitaxial Se-doped GaAs. J . Appl. Phys. 45, 3277. Acket, G. A,, and Scheer, J. J. (1971). Relaxation oscillations and recombination in epitaxial n-type gallium arsenide. Solid-state Electron. 14, 167. Akita, K. et al. (1971). Electrical properties of n-type gallium arsenide at high temperatures. Jpn. J . Appl. Phys. 10, 392. Akkerman, Z. L. et al. (1976). Infrared absorption spectra of oxygen-doped gallium arsenide. Sou. Phys.-Semicond. (Engl. Transl.) 10(5), 590. Aleksandrova, G. A. et al. (1973). Dominant contribution of oxygen to the compensation of high resistivity GaAs films. Sou. Phys.-Semicond. (Engl. Transl.)6(7), 1170. Aleksandrova, G. A. et al. (1976). Investigation of deep trapping centers in epitaxial GaAs: Cr:W by the method of thermally stimulated discharge of a capacitor. Sou. Phys.-Semicond. (Engl. Transl.) 10, 1162. Alferov, Zh. I. et al. (1973). Radiative recombination in epitaxial compensated gallium arsenide. Sou. Phys.-Semicond. (Engl. Transl.) 6, 1718. Allen, G. A. (1968). The activation energies of chromium, iron and nickel in gallium arsenide. Br. J . Appl. Phys. [2] 1, 593. Allen, J. W. (1968). Energy levels of nitrogen-nitrogen pairs in gallium phosphide. J . Phys. C 1, 1136. Allred, W. et al. (1968). Site distribution of silicon in silicon-doped gallium arsenide. Int. Symp. Gallium Arsenide, 1968 [Conf. Ser.-Inst. Phys.].
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Sinha, A. K. (1975). Metallization scheme for n-GaAs Schottky diodes incorporating sintered contacts and a W diffusion barrier. Appl. Phys. Lett. 26(4), 171. Sinha, A. K. et ul. (1975). Sintered ohmic contacts to n- and p-type GaAs. IEEE Truns. Electron Devices ED-22,2 18. Sokolov, V. I., and Shishyanu, F. S. (1964). Diffusion and solubility of gold in gallium arsenide. Sou. Phys.-Solid Stare (Engl. Trans/.) 6,265. Spicer, W. E. et ui. (1980). Unified defect model and beyond. J . Vuc. Sci. Techno/. 17(5), 1019. Spitzer, W. G., and Allred, W. (1968). Local-mode absorption and defects in compensated silicon-doped gallium arsenide. J . Appl. Phys. 39, 4999. Springthorpe, A. J. et al. (1975). Te and Ge-doping studies in Ga,-xAI,As. J. Electron. Muter. 4, 101. Stall, R.A. et al. (1980). Growth parameter dependence of deep levels in molecular beam epitaxial GaAs. Electron. Lett. 16, 171. Straws, G. H. et al. (1979). EPR determination of the concentration of chromium charge states in semi-insulating GaAs: Cr. J. Appl. Phys. 50(10), 6251. Stein, H. J. (1969). Electrical studies of low-temperature neutron- and electron-irradiated epitaxial n-type GaAs. J . Appl. Phys. 40,5300. Stillman, G. E. et al. (1976). Donor impurities in GaAs: Evidence for a stoichiometric hydrogenic shallow donor level. Proc. Int. Con$, I3th, I976 p. 623. Stoneham, E. B. et ul. (1980). Formation of heavily doped n-type layers GaAs by multiple ion implantation. J . Electron. Muter. 9(2), 37 1. Strack, H. (1966). Iron-doped gallium arsenide transistors. Conf Ser.-Inst. Phys. 3, 206. Stringfellow, G. B. (1974). Calculation of distribution coefficients of donors in Ill-V semiconductors. J. Phys. Chem. Solids 35,775. Su, J. L. et al. (1971). A deep level in Zn-doped liquid phase epitaxial GaAs. Solid-Slate Electron. 14, 262. Suchkova, N . I. et ul. (1975). Properties ofnickel-doped gallium arsenide. Sou. Phys.-Semicond. (Engl. Trunsl.) 9, 469. Suchkova, N . I. et ul. (1977). Behavior of the nickel impurity in the 3d8 state in gallium arsenide. Sou. Php-Semicond. (Engl. Transl.) 11, 1022. Surridge, R. K., and Sealy, B. J. (1977). A comparison of Sn-, Ge-, Se and Te-ion-implanted GaAs. J. Phys. D 10, 91 I , Surridge, R. K. et al. (1977). Annealing kinetics of donor ions implanted into GaAs. Con{ Ser.-Inst. Phys. 33a, 161. Ta, L. B. et ul. (1982a). Reproducibility and uniformity considerations in LEC growth of undoped semi-insulating GaAs for large-area direct-implant technology. Conf. Ser.-Inst. Phys. 65, 31. Ta, L. B., Hobgood, H. M., and Thomas, R. N . (1982b). Evidence of the role of boron in undoped GaAs grown by liquid-phase Czochralski. Appl. Phys. Lett. 41, 1091. Takahashi, K. (1980). Structural evaluation of GaAs crystals from etch pits. Jpn. J . Appl. Phys. 19, 1947. Takai, M. et al. (1975). Lattice site location of cadmium and tellurium implanted in gallium arsenide. Jpn. J. Appl. Phys. 14, 1935. Tdndon, J. L. et al. (1979). Silicon implantation in GaAs. Appl. Phys. Lett. 34(2), 165. Taniguchi, M., and Ikoma, T. (1982). Re-examination of the mid-gap electron trap (EL2) in different GaAs wafers from photocapacitance spectra. COP$Srr.-Inst. Phys. 65. Taylor, P. D., and Morgan, D. V. (1976). The effects of radiation damage on the properties of Ni-n GaAs Schottky diodes. 11. Solid-State Electron. 19,481. Thomas, R.N. et al. (1981). Growth and characterization of largediameter undoped semi-insulating GaAs for direct ion implanted FET technology. Solid-Stute Electron. 24,387.
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ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS, VOL. 61
Quantitative Auger Electron Spectroscopy M. CAILLER Laboratoire de Physique du Metal Ecole Mationale Supkrieure de MPcanique Mantes, France
J. P. GANACHAUD Laboratoire de Physique du Solide Institut de Physique de I' Universitk de Nantes Nantes, France
D. ROPTIN Laboratoire de Physique du Mkzal Ecole Nationale SupPrieure de MPcanique Nantes, Frunce
I. Introduction . . . . . ....... .................... A. Purpose ............................................... B. Basic Experiment.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
. . . . . . . . . . . . . . . . . . . 164 . . . . . . . 167
....................
A. Cross Sections ...................................................... B. Mean Free Path, Attenuation Length, and Escape Depth of the Electrons. . . . Ill. Dielectric Theory of Inelastic Collisions of Electrons in a Solid . . . A. General Equati B. Normal Metals
167 168
D. Semiconductors and Insulators ........................................ E. Depth Dependence and Anisotropy of the MFP ...
181
IV. Elastic Collisions V. Auger Transitions A. Introduction .................... B. Classification of Auger Transitions .
................... ...................
VI. Quantitative Description of Auger Emission A. General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Theoretical Analysis of Auger Emission.. ............................... ....... V11. Auger Quantitative Analysis ...... ................................. A. Introduction . . . . ................................. B. Operating Modes
187 188
2 I3 222 244 244 245
161 Copyright rC' 1983 by Acadcmic Press, Inc. All rights of reproduction in any form rcseerved. ISBN n - i m 4 6 6 1 - 4
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C. Quantification of Auger Analysis ............... D. Sample Preparation ........................... E. Effects of Sputtering on the Surface Composition of Multicomponent Materia . F. Electron-Beam-InducedEffects in AES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. Conclusion . . ...... ............. References.. . ...... .
273 282 289 289
I. INTRODUCTION
A . Purpose
This article reviews Auger electron emission from a target subjected to electron bombardment. The mechanisms involved in generation of Auger electrons by ion or photon bombardment are therefore not considered here. Despite this restriction, we shall use the term Auger electron spectroscopy (AES) without reference to the excitation mode throughout in what follows to denote the analysis technique developed for electron-induced Auger electron emission. An Auger transition can occur when a vacancy is produced in an inner subshell of an atom. This atom, being in an unstable state, will try to revert to a lower energy state by filling up its vacancy with an outer shell electron. There are two ways for the atom to get rid of its energy excess: (1) by emitting a photon (X-ray fluorescence) or (2) by emitting a second outer shell electron (Auger transition). During the past decade AES has become one of the most widely used techniques for analysis. Its two major features are the following:
(1) The energy of the Auger electron characterizes the nature of the emitter atom and eventually its chemical environment. (2) The electrons emitted from the target with this characteristic energy can only emerge from the outermost atom layers at the surface. Accordingly, AES is an analysis technique for elements a t surfaces (eventually, a chemical analysis technique). However, by combining ion sputtering with AES it is possible to probe the bulk of a sample and to plot depth profiles of the element concentrations in it. Because of its practical importance a number ofbooks and general papers have been devoted to the AES technique. Sevier’s book (1972), although having a wider scope, devotes an important part to AES, and this book is very useful to experimentalists working on Auger transitions. Chattarji (1976) has described the theory of Auger transitions, following the earlier book by Burhop (1952).Although these books have different objectives, together they
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constitute a rather complete reference on the physics of the Auger effect considered as an elementary mechanism. In the case of a complex sample, however, there are no direct means of anticipating the results of a quantitative analysis in terms of first principles, although some obstacles have been removed. Most of the general papers on AES (Chang, 1971; Palmberg, 1973; Bouwman et al., 1973a,b,c; Chang, 1974; Miiller, 1975; Defrance, 1976; Le Gressus, 1978;Holloway, 1978,1980; Powell, 1978; Van Oostrom, 1979) are intended for researchers confronted with analysis problems of multicomponent targets which are frequently of a technological nature. They review more or less extensively the Auger analysis method and stress the physical fundamentals, possibilities, limits, and applications of the method. In these papers the quantitative description of the creation and emission mechanisms of Auger electrons is phenomenological. During the pa st decade some papers appeared that were devoted to establishing the relation between the phenomenological description and the more fundamental description of the elementary mechanisms. Usually this relation is shown via a simulation model on a computer, and Shimizu and Ichimura (1981) have published a very important contribution on this question. The present review article describes, as far as possible, all the steps from first principles to the real sample analysis, but, as mentioned above, the path is far from complete. B. Basic Experiment A very short general description of a typical experiment in AES is given first, to introduce some basic vocabulary, after which we outline the content of this article. In this experiment, an electron beam emitted by an electron gun collides with the sample to be analyzed. The electrons of this beam are called primary electrons, They are characterized by their energies and their angles of incidence on the target (Fig. 1). Quite frequently, the incident beam can be considered monoenergetic and monodirectional. Some of the electrons in the primary beam are reflected by the outermost atom layers without appreciable
electron
baam
\
=by the
target
Target
FIG.1. Basic diagram of secondary-electron emission.
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M. CAILLER, J . P. GANACHAUD, AND D. ROPTIN
energy loss. They are called elastically reflected electrons. The others penetrate to different depths into the sample, interact with it, and thus are deflected and undergo energy loss. In classical language the primary electrons undergo elastic and inelastic collisions in the solid. In an inelastic process the energy and momentum are transferred to the target and contribute directly or indirectly to the excitation of electrons in the solid. Before excitation these electrons have an energy lower than the Fermi level. After excitation they join the incident beam, and can therefore undergo elastic and inelastic collisions themselves. This is known as the cascade process. At some point in their paths these electrons may reach the free surface of the target and escape into the vacuum. If the escaping electrons belonged initially to the primary beam they are called inelastically backscattered electrons (or, simply, backscattered electrons). If they were dislodged from the solid they are called secondary electrons. Since electrons are indistinguishable particles, this twofold classification is rather arbitrary. However, it is possible to a certain extent to separate these two contributions experimentally (Gerlach et al., 1970). The fundamental datum that we desire from the experiment is the angular energy distribution of the electrons emitted from the target; however, experimental results about it are very scarce. The most frequently used result is, undoubtedly, the energy distribution or its derivative curves. With grid analyzers the electrons are captured in a large solid angle which can be of the order of 2.n sr. When other spectrometers such as the cylindrical mirror analyzer (CMA) are used, the electrons are collected in a rather small acceptance angle. In the energy spectrum of the electrons emitted by the target, several fine structures emerge from the background. The elementary mechanisms that create them are very diverse in nature (characteristic energy losses caused by creation of elementary excitations in the solid, interband transitions, plasmon decays, Auger transitions, etc). Because they represent specific responses of the solid to an external excitation, they are fundamental clues to the nature of the sample. In this article, only the information provided by Auger transitions are considered, although some of the other elementary mechanisms are included to give a general description of the secondary-electron emission process (especially in computer-simulation models).
C . Contents This article is divided into seven major sections in addition to the introduction (Section I). In Section 11, some general definitions, such as cross sections, mean free paths (MFPs), attenuation lengths, and escape
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
165
depths, are given. Stress is placed on what constitutes, according to the authors, the links and the differences between these different quantities. Then we present quantitative theoretical evaluations of the inelastic (Section 111) and elastic (Section IV) MFP for a randiun-jellium model of the solid. In this model the ions (randium) are randomly distributed in a free electron gas (jellium). The jellium contribution to the inelastic MFP can be calculated by means of the dielectric function scheme, and the randium contribution can be obtained using the classical binary-encounter theory of Gryzinski (1965a,b,c) or with the help of the generalized oscillator strengths. The elastic collisions that scatter the excited electrons result from their interaction with the ions, which may be regarded as scattering centers. Calculation of the elastic MFP by expanding the electron wave function in partial waves is discussed only very briefly. The various cross sections and MFPs defined in Section I1 are used as fundamental data in computer-simulation models (Section VI) of the secondary- and Auger electron emissions. For the authors of the present article, that justifies the emphasis on discussion of methods of calculation of these quantities. Section V is devoted to a consideration of Auger transitions regarded as elementary mechanisms. Classification of these transitions is considered in Section V,B and evaluation of their energies in Section V,C. Special attention is given to line-shape analyses of Auger transitions involving the valence band because of the interest generated by this topic, especially in view of quantitative measurements (Gaaren st room, 1 98 1 ). Section VI gives a quantitative but approximate description of the creation and emission of Auger electrons. In Section VI,A the presentation is made on a phenomenological basis. This description is quite usual, and the equations which are obtained constitute the fundamental expressions for a further development of quantitative AES (cf. Section VI1,C). One of the most important factors introduced in such a development is the backscattering factor, and it is defined in Section VI,A. Section VI,B is devoted to a theoretical quantitative analysis of Auger emission by using computer-simulation models. In the first part, several original results obtained on a pure A1 target are shown in order to test the validity of the principal assumptions made in the phenomenological approach. Moreover, evaluations of the backscattering factor are given and discussed. In the second part of Section VI,B we extend to the case of more complex samples the evaluations of the backscattering factor. The major part of this work was done by Shimizu and his co-workers. Here it is important to stress the fact that results obtained by a simulation method are only as good as the model. Therefore, the use of simple models,
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M. CAILLER, J. P. GANACHAUD, AND D . ROPTIN
necessary to reduce the time and cost of a simulation, could lead to signhcant errors: Thus such models must be checked. In Section VII the different problems experienced in experimental quantitative AES are reviewed. The adjustment of all the measurement devices and the correct use of the apparatus must be carefully checked. The results of the round robin realized by the ASTM 42 Commission show (Section VI1,A) that in the absence of such checks, very important differences can be observed between the experimental values. Several operating procedures have been developed in AES. For instance, it is usual in AES for the secondary-electron signal to be measured as a function of the secondary-electron energy. Then, by derivation it is possible to make more apparent the fine structure of this secondary-electron current. Other operating modes have been used, such as the measurement, as functions of the primary-electron energy of either the total secondary-electron current [appearance potential spectroscopy (APS)] or the elastically reflected primary-electron current [disappearance potential spectroscopy (DAPS)] . These diverse operating modes are briefly sketched in Section VI1,B. The various ways of quantifying experimental Auger analysis are presented in Section VI1,C. There have been few attempts to correlate the experimental and theoretical values on an absolute basis. In most cases, the investigators use a relative calibration technique. In the internal calibration technique, the bulk composition of the sample is known. Assuming from the start that the surface elemental concentrations are the same as in the bulk, it is possible to define a sensitivity factor for each element of the sample. For a given element, this sensitivity factor is equal to the ratio of the intensity of one of its Auger peaks to the concentration of the element. Assuming then that a variation of the surface concentrations has occured, the new Auger peak intensity will give the new surface concentration of the element. In the external calibration method, the sensitivity factors are defined for the pure elements. In such a case, application of the elemental sensitivity factor, without correction for the so-called matrix effects, to the quantitative elemental surface analysis of a multicomponent sample will lead to errors. A procedure for correction of the matrix effects was developed by Hall and Morabito (1979) on the basis of the phenomenological description of Section V1,A. ' Before analysis, the samples have to be prepared. In any case the mode of preparation raises some questions, at least about its influence on the results of the analysis. These questions are reviewed in Section VI1,D. Because of its practical importance, preparation by sputtering is considered separately in Section VI1,E. During analysis, the sample is submitted to the action of an electron
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beam, which can induce a modification of the composition of the target surface. The principal disturbances, associated with stimulation by electron beam, are the following processes (Section VI1,F): (1) adsorption or desorption of species at the surface; (2) dissociation of species at the surface; (3) dissociation of the solid sample itself; and (4) diffusion along the surface or within the bulk. In some extreme cases, therefore, AES could possibly destroy the system it hopes to analyze and thus would not work. More usually, there is a risk that contamination of the sample can occur. Although this article is rather lengthy, it omits such very important questions as theoretical evaluation of Auger line intensities or crystallographic angular effects in Auger emission. There were no special reasons for these omissions; it was simply a matter of choice. If the reader is interested in further information on theoretical evaluation of Auger line intensities, he will find much to interest him in the original McGuire papers on this question. 11. GENERAL DEFINITIONS A . Cross Sections
Let us consider the interaction between a parallel beam of No monoenergetic particles per unit area and a scattering center (e.g., a charged particle). If dn is the number of particles scattered without energy loss per unit time into the solid angle d o = sin 8 d 0 d 4 around the direction (0,4) (Fig. 2), the elastic differential cross section a(0,4)is defined by
dn
=
N0a(8, 4) d o
(1)
Similarly, it is possible to associate a particular differential cross section with each type of reaction between the parallel beam and the target particle. If dn is the number of incident particles that reacted with the target in a particular way characterized by initial parameters ( a , ,pi, . . .) and final parameters (af,&, . . .), the differential cross section or(cli,pi, . . .; orf, fit, . ..) for this reaction is defined by dn = Noa,.(ai, pi, . . .;u,.,pf, . . .) dor, dp,
(2)
where the subscript r denotes the type of reaction (or channel). The energy of the incident particles can be considered to be one of the parameters of the reaction.
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M . CAILLER, J. P. GANACHAUD, AND D. ROPTIN
t
FIG.2. Geometry of particle scattering.
For N , scattering centers, if we neglect the coherence effects between the waves scattered by each center, the above differential cross sections have to be multiplied by N , . For the reaction considered, the total cross section or is deduced from the differential one by integrating over all final parameters or(Mi,
Pi,
* *
*) =
s
br(Ni, P i , .
*
*; ut > P i > ..
dP,
(3)
B . Mean Free Puth, Attenuation Length, and Escape Depth of the Electrons There is some difficulty about properly defining these different quantities. We shall thus consider the mean free path (MFP) to be essentially a theoretical notion which can form the subject of calculations, whereas attenuation lengths and escape depths are considered only a s experimental quantities. In some idealized cases, the MFP and the attenuation length could be identified, but more generally, the determination of an MFP from attenuation-length data is possible only in an approximate way, by introducing a physical model. 1. Mean Free Path
An MFP is generally associated with a given reaction by considering a Poisson process. For that, one can consider a flow of N particles having identical dynamic parameters (energy, direction of propagation) impinging normally on an elementary layer of thickness dz a t a depth z in the target. In this layer, these particles can undergo a reaction characterized by the total cross section or(i),where i denotes the set of initial parameters of Eq. (3).
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
169
Along the infinitesimal path d z , according to Poisson's law, an incident particle can undergo only one or no reaction with the sample. So the number of particles which have undergone this reaction by crossing the layer is dN'
= NN,o,(i)dz
= -dN
(4)
where N, is the number of scattering centers per unit volume of the target. Equation (4) can be written d N = - N dz/A,.(i)
(5)
where &(z)
= [N,o,(i)]-
is the M F P of the identical incident particles for the channel labeled by subscript r. Now, one can consider a flow of No particles, characterized by the initial parameters i, impinging normally on a target of thickness t , in which they can undergo only reactions of type r. The residual amplitude of the beam which is transmitted through the target but which conserves its initial dynamic parameters is N = No exp[ - t/A,(i)] Between two collisions, the mean distance traveled by an incident particle is
( z ) = N,'
J:
z dN = A,(i)
(8)
Usually, different channels are open to the incident particles, and between two collisions of type r , collisions of another type may occur and cause the trajectories to be different from straight lines. Equation (7) can be extended to this case if the direction of propagation is no longer included in the set of conserved dynamic parameters. For instance, if this set is restricted to the initial energy E , only, the probability that a particle will run path 1 in the target without undergoing the reaction r is
It is important to note that the definitions leading to Eqs. (7) or (9) apply to the case of a homogeneous target, whereas Eq. (5) could be used even in the case of a heterogeneous sample or for the surface layer of a solid. In these cases N , may vary with the depth z and & may become z dependent. To the contrary, attenuation lengths and escape depths are considered to be experimental data where surface and volume effects are not distinguished but can be defined and measured for any target.
170
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
Some authors prefer using the concept of inverse mean free paths (IMFPs) A;'(i) = N p , ( i ) owing to their additive character. However, in what follows we promulgate the MFP notion, which is, in our opinion, a more direct one. 2. Total M F P If we consider the set of possible reactions which can take place in the target, a total MFP AT can be associated with an incident particle of energy E , through the relation
G W O )
= Cn;'Wo)
(10)
r
From Eq. (4) and owing to the additivity of the IMFPs, it is possible to evaluate the probability (1 1) pT(EO) = exP[ -l/AT(EO)] of an incident particle traveling a path 1 in the target without undergoing any collision. The total MFP includes both elastic (noenergy loss) and inelastic effects. However, the additivity of the IMFPs can be used to separate a total inelastic MFP effect from the elastic one. 3. Attenuation Length
Quite frequently, the attenuation length is assimilated into the total inelastic MFP. In fact, this identification is always approximative. Thus, differences in the interpretation of attenuation lengths for hot electrons have led to some controversy (Oxley and Thurstans, 1975a,b; Collins and Gould, 1975). Reviews on measurement techniques and data have been given by Powell (1974) and by Seah and Dench (1979). The principal experimental methods are (1) the transmission technique; (2) the overlayer technique; and (3) the semi-infinite-target technique, although low-energy electron diffraction (LEED) and extended X-ray absorption-edge fine structure (EXAFS) experiments can also give some informations on MFPs. The first method can be considered the basic technique for measuring attenuation lengths. In these experiments (Kanter, 1970; Stein, 1976), the number of electrons transmitted with no energy loss through a film (selfsupported or not) is related to the number of incident electrons by NT
= NO exP[ - t/&]
(12)
where A, is the attenuation length of the target. Consequently, the quantity A, is obtained by measuring the dependence of the transmission coefficient N T / N o on the thickness of the target. If the aperture of the analyzer is ideally
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
171
narrow, the measured attenuation length can be assimilated into the total (elastic and inelastic) MFP. However, this condition is never realized. The so-obtained apparent attenuation length can be interpreted as the MFP between collisions in which the electrons are scattered either elastically at large angles (i.e., out of the aperture) or inelastically. Interpretation of the measurements also raises some questions related to the variation of the thickness and of the morphology in thin films. According to Stein (1976), it appears that the shortest apparent attenuation lengths are the most representative of the ideal situation and can be accounted for by inelastic scattering alone. Some special information can be given about partial MFPs, e.g., by varying the temperature of the target. Thus by separating temperaturedependent effects from the others, Kanter (1970) obtained an evaluation of the electron-phonon MFP. The overlayer technique resembles the transmission method to the extent that it consists of measuring the attenuation in a thin overlayer of an electron beam of given energy created in the substrate by an excitation mechanism (UV or X rays in general), either directly (photoemission) or indirectly (Auger emission). Basically, the electrons created in a direction 8 are supposed to have a straight-line path, traveling over a distance t/cos 8 in the overlayer, according to an attenuation law given by NT(@= No(0) exp[ - t/(A, cos O)]
(1 3)
As for the transmission method, the dimension of the aperture of the analyzer plays an important role in the measurement. This question concerning energy analyzers has been investigated by Shelton (1974) and by Norman and Woodruff (1978). A rather complete bibliography on the overlayer technique can be found in papers by Powell (1974) and Seah and Dench (1979). The principal types of measurements concern (1) X-ray photoemission (Battye et al., 1974, 1976a,b; Szajman et al., 1978; Evans et al., 1977; Hattori, 1977) (2) UV photoemission (Peisner et al., 1976) ( 3 ) Synchroton radiation photoemission (Norman and Woodruff, 1977, 1978) (4) Auger electron emission (Palmberg and Rhodin, 1968; Seah, 1972, 1973; Gallon, 1969; Jackson et al., 1973). Using this method, Auger electron emission of both the overlayer and the substrate is measured. In the latter two papers, an analysis of the dependence of Auger intensities on the thickness of the overlayer is proposed that assumes a model of layer-by-layer growth of the deposit on the substrate.
172
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
The use of spin-polarization measurements to distinguish the electrons emitted by the substrate from those emitted by the overlayer must also be cited (Pierce and Siegmann, 1974). The semi-infinite evaporated-target technique has been used by Powell et al. (1977) to measure electron attenuation lengths in beryllium and aluminum. The principle of this method is the creation of electrons in the target by an excitation source and the measurement of the emission yield. In this paper, the excitation source was a proton beam and the induced emission that of Auger electrons. It is still supposed that until they escape into the vacuum, the electrons follow a straight-line path. Thus, the current collected by the analyzer is proportional to the attenuation length. Of course, the estimation of these attenuation lengths requires a knowledge of the instrumental characteristics and of some other physical parameters such as the density of atoms in the solid, the ionization cross section for a given proton energy, and the fluorescence yield. The method used by Koval et al. (1978) is intermediary to the semi-infinite and the overlayer techniques. It consists in measuring the intensity of the peak of the elastically reflected electrons from semi-infinite metals, say 1 and 2, and from thin layers of metal 1 deposited on a substrate 2. The depth dependence of the reflection coefficient yields information on the attenuation length if we suppose that the electrons suffer a very localized reflection either in the overlayer or in the substrate but otherwise have a straight-line trajectory. Along their paths the electrons can undergo inelastic collisions, and the beam is attenuated according to an exponential law versus the effective path traveled in the overlayer. Some measurements of the electron-plasmon interaction MFPs can also be realized by determining the relative intensities of the characteristic loss peaks and of the elastic one. It is possible to evaluate inelastic effects from LEED experiments by studying either the widths of the Bragg peaks or the LEED intensities. In layer-by-layer techniques of LEED theories the attenuation effects are usually incorporated as an imaginary part for the component of the electron wave vector perpendicular to the surface via a complex optical potential (Stern and Sinharoy, 1972; Payling and Ramsay, 1977) or via a complex electron self-energy (Duke and Tucker, 1969). Inelastic effects have also been introduced by a spherical attenuation coefficient in the interpretation of EXAFS experiments (Ashley and Doniach, 1975; Lee and Beni, 1977). This wave-mechanical aspect makes somewhat difficult the interpretation of these attenuation coefficients in terms of MFP or of attenuation lengths. This is the type of difficulty pointed out by Feibelman (1974) in his comments on theories of photoemission.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
173
4. Mean Escape Depths
Let us now consider an inner distribution of monoenergetic electron sources in a solid (e.g., Auger electron sources). Some of these electrons can pass through the surface potential barrier and escape into the vacuum if their energy and normal momentum are sufficient. Their mean escape depth he is defined as the depth for which the escape probability, without energy loss, is lle. An evaluation of the escape depth necessarily makes use of a model. Usually, the monoenergetic electron sources (of energy E) are supposed to be uniformly distributed up to a maximum depth z, and to create n electrons per unit volume and per unit time. Moreover, electrons are supposed to follow straight-line paths up to the surface and to be attenuated according to an exponential law. These are precisely the assumptions introduced to define the attenuation length. Thus, the number of electrons escaping into the vacuum is, per unit area and per unit time, N = (n/47c)m J :
Joz^
Jr
exp[ - ~ / ( A ~ cos ( E 0)] ) sin 0 d0 d+ dz
(14)
where 0, is the half-angle of the emergence cone
cos 0, = (W/E)"' (15) and W is the work function of the solid. After angular integration, one obtains (Fitting et al., 1978) N = fn J
0
W J Z / A ~ E, ,
w)dz
where we is the escape probability. If z , goes to infinity and if W/E goes to zero, the depth dependence of w e is well described by the relation we = exp( - 2z/Aa)
(17)
In this case, the mean escape depth is approximately equal to half the attenuation length. 111. DIELECTRIC THEORYOF
INELASTIC COLLISIONS OF
ELECTRONS IN A SOLID
For the electron energies used in Auger spectroscopy, a rigorous description of the inelastic interactions would be rather sophistocated. In
174
M. CAILLER, J. P. GANACHAUD, AND D . ROPTIN
fact, it is a many-body problem that includes all the electrons of the incident beam and of the target. An essential simplification of this problem is to consider that the solid reacts as a whole to an external probe, which is here the incident-electron beam. With this in mind, the dielectric theory of the response of the solid can be considered as having a sufficiently general frame. However, from a practical point of view, it is advisable to break the dielectric response function into several separate contributions, which makes description easier. Afterward, we shall consider principally the case of metals and very briefly that of semiconductors and insulators. Normal and “d-electron” metals are distinguished. By normal metals we mean those for which the energy states are well separated into a valence (or conduction) band and core levels which can be considered as nearly atomic states. The d-electron metals include the transition and noble metals, for which a separation of the electrons, as in the case of the normal metals, is difficult if not impossible. Indeed, the electrons coming from the d atomic states, called d-electrons, are rather localized on the ionic cores but have energies in the range of the valence band. A . General Equations
Let @&) be the initial wave-packet state for a particle incident on a many-particle system in its ground state. Here K represents the four-vector [K, iE(K)] and x is (r, it). If the single-particle state predominantly characterized by the fourmomentum K is well defined, its lifetime z is high and can be evaluated from the relation (Engelsberg, 1961) T-’ = (2/h)Im M ( K )
(18)
In this relation M ( K ) is the four-dimensional transform of the self-energy operator M ( x , x’)of the incident particle. For a periodic system, @&) and E(K) are the wave function and the energy of the band-structure theory (Adler, 1963, 1966), respectively. For a translationally invariant system, OK(x)is a simple plane wave E ( K ) = E,(K) M ( K ) = iA
=
h2K2/2m
s4
W(q)G(K- q ) r ( K ,q) exp( - iw6)
(19)
(20)
where q represents (a; io).In Equation (20)6 is a positive infinitesimal; W(q ) and G ( K - q ) are the Fourier transforms of the effective interaction operator
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
175
and of the electron Green’s function; and T(K, q) is the vertex function which can be considered a sum of the functionals of G and W (Minnhagen, 1974, 1975). At the lowest order of development, T ( K ,q) = 1. For a screened Coulombic interaction, W(q)= (4ne2/q216- ’(q, 0)
(21)
where t-l(q, o)is the dielectric response function of Martin and Schwinger (1959; cf. also Falk, 1960). If the incident particle is described in the free-electron approximation,
G(K
- q) =
[E,(K) - ho - Eo(K - q)
+ id’sgn(lK - q1 - k , ) ] - ’
(22)
Taking T(K, q) = 1 and letting 6 and 6’ approach 0, the imaginary part of the self-energy becomes
a19J0
Im M[K, Eo(K)]= -
x
a[,
4
-
do Im[ -t-’(q, o)]
EOB) - Eo(K - 9) h
In this case, the inverse MFP can be written 2-1
m 2 Im M[K, Eo(K)] --me 2 Im V[K, Eo(K)] h IKI h hlKl h
=-
(24)
where V is the optical potential. Eqs. (23) and (24) correspond to the first Born approximation and were developed by Quinn (1958, 1962) for the free-electron gas. The response of the solid is then entirely incorporated into the dielectric function, a very general expression of which is (Kittel, 1967) the following: e-1(q7m) =
4xe2
-~
~ ~ ( @ b ~ ) q I @ a ) ) 2 ( -k ( m Oba
hq2
b
-(o- m b a
+
iYb)-’]
+ %)-’ (25)
where 3, is the density fluctuation operator for the particles, and yb is a small positive quantity. In Eq. (25), hwba= (Eb - E,) represents the difference of energies between excited states I @ b ) and the ground state 1 Oa). The frequencies w b a depend on q and tend to be grouped in more or less extended bands, every one giving a contribution to Im M (Ritchie et al., 1975). In the case where 6 - depends only on the modulus Iq 1 of q, which is later written q except where otherwise stated, the angular integration of Eq. (23) is direct.
176
M . CAILLER, J . P . GANACHAUD, AND D . ROPTIN
For simple (or normal) metals we can consider two independent contributions to the imaginary part of the inverse dielectric function. The first comes from the valence electrons and is well described by a free-electron gas model (cf. Section 111,B). The second comes from the core electrons and can be described in terms of the Bethe’s oscillator strengths [Bethe, 1930; lnokuti, 1971; Inokuti et al., 1978; (cf. Section lII,B)]. It can also be accounted for by a classical description such as that developed by Grysinski (1 965a,b,c). For transition or noble metals, the d electrons add their contribution to that of the s-p electrons of the conduction band. Thus the free-electron gas model is no longer suitable. Moreover, local field effects must be considered in the calculation of the d-electron contribution (Wiser, 1963; Nagel and Witten, Jr., 1975). Possible ways to approximate the sum of these two contributions are given in Section II1,C. The ionization of the inner shells can still be described as in the case of normal metals. B. Normal Metals
As indicated above, the response of the conduction electrons is well described by a free-electron gas model and is given, for instance, by the Lindhard (1954) dielectric function EL(q, w). Some more refined dielectric functions taking into account exchange and correlation effects in the electron gas or finite lifetime of the elementary excitations have been developed (Hubbard, 1955; Singwi et al., 1968, 1970; Vashista and Singwi, 1972; Mermin, 1970).Results on the M F P obtained by using these dielectric functions can be found in papers by Ashley and Ritchie (1 974), Ganachaud ( 1 977), Tung and Ritchie (1977),and Ashley et al. (1979).The corrections brought to the results obtained with the Lindhard dielectric function are rather small (nearly 10%). A description of the many-body effects in an interacting electron gas has been given by Hedin and Lundqvist (1 969). Numerical results using their model can be found in several papers (Lundqvist, 1967, 1968, 1969a,b; Bauer, 1972; Lezuo, 1972; Shelton, 1974; Penn, 1976a). 1. Separation of the Contributions of the Plasmon Mode and of the Valence-Electron Gas Individual Transistions
A remarkable feature of normal metals is that they show a well-characterized collective oscillation (or plasmon) mode. It is thus possible to consider for the valence electrons two different contributions to the M F P : the plasmon mode and the individual transitions. This is now discussed in terms of Eqs. (23) and (24) and Lindhard’s dielectric function. Near the plasmon-dispeflion curve defined by the condition el [q, w,(q)] = 0, t , can be expanded in powers of the difference [ w - w,(q)], and c2 is
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
177
FIG.3. Energy- and momentum-transfer domain in the Lindhard approximation. Domain of individual excitations in the solid: I and 11. 1
2
--f
+ 2qk,),
hw
=
(h2/2m)(qZ
hw
=
(h2/2m)(q2- 2gk,),
3
-
hw = (fiZ/2m)(2Kq- 42)
4 + h o = (h2/2m)(2gk,- 42)
an infinitesimal number. Then, to a first approximation of the expansion, the electron-plasmon inverse MFP is written
A:;,
=
(26)
Here we have written K for I K 1, and fl(x) = 1 for x > 0 and 0 for x < 0. Except near the plasmon threshold, qmax is equal to the cut-off wave vector qc for which the plasmon line enters the individual transition domain (Fig. 3). The individual transitions of the valence-electron gas are restricted in Lindhard's approximation to the domain of the (0,q) plane limited by the parabolas o = (h/2rn)(q2+ 2qkF) and o = (A/2rn)(q2 - 2qkF) and by the restrictions 0) > 0 and o < h-'[E,(K) - E F ] . The electron-electron inverse MFP can be obtained from Eqs. (23) and (24) by integrating Im[ - E ; (4, a)] on the individual transition domain. Figure 4 shows the results obtained for both contributions to the MFP in aluminum. A more detailed analysis of the inelastic collisions can be made; for instance, the energy-loss function associated with the individual processes is given by
4(4=
(27)
178
M. CAILLER, J . P. GANACHAUD, AND D . ROPTIN
I . . . . . 100 200
0
E (eV)
FIG. 4. Mean free paths in aluminum: (1) between two individual excitations Ae-e; (2) between two collective excitations 2e-p,. Reprinted with permission from North-Holland Publishing Co., Amsterdam.
In the assumption of an infinite lifetime of the elementary excitation, 4(0) presents a singularity when the plasmon line enters the region of the individual collisions. In a more realistic description this singularity is softened by plasmon damping (Fig. 5). For a given value of O, several momentum transfers can occur. The probability for a primary electron to transfer a momentum q to an electron of the solid is 4(q, 0) = 4 - Im( - [&
4- 1
(28)
2. Collisions on the Inner Shell Electrons
If in Eq. (25) the values of w are restricted to positive values and if y b goes to zero, one obtains
Equation (29) has to be integrated over q and w as indicated in Eq. (23). For low values of q, Eq. (29) does not depend on the direction of the wave vector. Because the inner subshell binding energies are well separated, it is possible to associate with each subshell an oscillator strength I,&) according to Eq. (30):
(30) where Z,, and En,are, respectively, the occupation number and the binding energy of the band (nl)supposed flat; ER is the Rydberg energy; and a. is the radius of the first Bohr orbit.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
179
hw (ev)
FIG. 5. Energy-loss function 4(0) for an individual collision in Al; dielectric function: 1, Lindhard (1954); 2, Singwi et al. (1968): 3, Mermin (1970) (y = 0.1).
By neglecting the q dependence of the oscillator strengths, the simple ionization inverse M F P is written
where qminand qmaxare the roots of Eq. (32): q2 - 2Kq
+ (2mEnI/h2)= 0
(32)
Expression (31) could be compared with the usual Bethe total cross section (Bethe, 1930)
one obtains
where cn,is in cm2 and En, in electron volts. In the validity domain of the Born approximation the agreement with experimental values is obtained by fitting b,, and C,, (Burhop, 1940; Arthurs and Moiseiwitch, 1958).
180
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
Some modifications to the Bethe formulation have also been proposed, but they are still in the general line of the Born approximation (Worthington and Tomlin, 1956; Rudge and Schwartz, 1966; Kolbenstvedt, 1967; McGuire 1971a,b). Note: By using in Eq. (31) the boundaries given by Eq. (32), the result for a primary electron of high energy will be exactly twice that of Eq. This factor could be representative of the (33) if B,, is taken as equal to En[. order of magnitude of the uncertainty in this cross section. The difficulty of the quantum mechanical calculation and the uncertainty mentioned above led frequently to the use of the classical expression of Gryzinski (1965a,b,c)for the total ionization cross section, that is, onl= 6.51 x 10-’4(Z,l/E~,)g(U,,) [cm2]
(35)
with g(U) = U - ’ [ ( U - 1)/(V + 1)]3’2{1 +$[1 - ( 2 U ) - ’ ] ln[e
+ ( U - 1)1’2]}
(36)
where e cz 2.718 is the base of the natural logarithm. This approximation considers that the collision between the incident electron and an electron of the p21 subshell is binary. Equation (36) is interesting because it gives very simply and directly a reasonable evaluation of cross sections. In form, expression (35) can be compared with Eq. (33’). Some semiempirical expressions have been proposed to account for the energy dependence of the total ionization cross section. In a general way they are empirical modifications of the expressions obtained in the Born approximation or in the binary collision approximation. Their aim is a better agreement with experimental results (Drawin, 1961, 1963; Lotz, 1967, 1968, 1969, 1970; and Pessa and Newell, 1971).
C. &Electron Metals A particular requirement in the study of the response function of rather highly localized states, related to the polarization of these states, is to take into account the existence of a local field differing from the mean macroscopic field. The effects of this local field on the dielectric function have been considered by Adler (1963)and Wiser (1963)and more recently by Nagel and Witten (1975) who have concentrated on the local field effects in the dependence of the dielectric function on q. By assuming that the localized electrons behave as polarizable charges that are strongly concentrated on the lattice sites, Nagel and Witten have deduced a dipole approximation expression for c(q, w). Ritchie and Howie (1977) have analyzed the requirements coming from the sum rules for the extension of the “optical” dielectric function to nonzero values of q. Assuming that the experimental function Im{ -[t(O, o ) ] - ~ }
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
181
obeys the sum rules, then using a procedure similar to that of Ritchie and Howie, Cailler et al. (1981) have approximated the loss function by Im { - [&, o)]- } = M," Im { - [@,M ~ O )-] ]
(37)
with
~f
= (1
+ hq)-'
(38)
where b is a constant to be determined (in copper, h N 0.05 - 0.1 A). The shape of the curve is the same for each value of b and is in agreement with the behavior generally assumed for the MFP curve. Two theoretical evaluations of the contribution of the d-like states to the self-energy of the energetic electron have been made (Ing and Pendry, 1975; Beni et al., 1976). Between the two estimations there is a factor of 2, which can result from differences in the treatment of the screening of the electron interaction. These results are not directly comparable with those deduced from Eqs. (23) and (37) because the experimental dielectric function includes the responses of the delocalized electron and of the 3d electrons. But it is possible to combine them with a free-electron gas model MFP to obtain a total inelastic MFP for delocalized and d electrons. With these conditions a general agreement is obtained for copper by taking b N 0.05 - 0.1 A.
D. Semiconductors and Znsulators In this section we shall give only a few indications of some calculations made for semiconductors and insulators. Emerson et al. (1973) studied the electron slowing-down spectrum in silicon and used the dielectric function of Callaway (1959) and Tosatti and Parravicini (1971) to describe the valenceband contribution. Tung et al. (1977) also studied the electron slowing-down spectrum in Al,O, by using, to describe the response of valence electrons, a model similar to that used by Fry (1969) in which the valence electrons are represented in the tight-binding approximation in their ground state and by orthogonalized plane waves when they are in excited states. Each atom of aluminum and of oxygen gives, respectively, three and six electrons to the valence bands. So, per molecule, there are 24 electrons shared out between two bands, one occupied by 15 electrons (mean binding energy ho, N 9 eV) and the other by nine electrons (ho,N 29 eV). The contributions of the inner shell electrons were described by generalized oscillator strengths. Penn (1976b) has suggested describing the 24 valence electrons of Al,O, as if they were free and proposed for them the MFP
2,
=
E,(K)/{a,[ln Eo(K)+ b,]}
(39)
182
M. CAILLER, J. P. GANACHAUD, AND D . ROPTIN
where E,(K) is the electron energy expressed in electron volts and a, and b, are two numerical constants. E. Depth Dependence and Anisotropy of the M F P
Because a large proportion of the Auger electrons come from the first atom layers of the surface, the question arises whether there is a depth dependence of the MFP in this surface region. This aspect is usually neglected in quantitative theories of Auger emission. Gersten (1970) and Gersten and Tzoar (1973) introduced the limit assumption of a S-type optical potential strictly localized at the surface. They concluded that its effect was only noticeable in the range between surfaceand bulk-plasmon threshold energies for photoemission calculations. In calculations for aluminum, Feibelman (1973)has shown that near the surface, the increase of the total IMFP due to surface plasmon excitations is compensated by the reduction of the bulk-plasmon creation probability. So, at least within a simple model of surface effects, the assumption of a spatially nonvarying IMFP is not in serious error, but is was suggested that this constant IMFP had to persist some distance beyond the solid surface in an improved theoretical description. Rasolt and Davis (1980) evaluated the anisotropic contributions to the imaginary part of the optical potential caused by the broken symmetry due to the presence of surface. They concluded that these anisotropic effects could be of the same order of magnitude as the isotropic effects. However, as shown by these authors, these modifications do not bring major changes in, for instance, LEED I-V theoretical profiles. The importance of these effects has not yet been stated for the Auger domain. However, one may hope that they do not represent the major cause of uncertainty in a quantitative description of the Auger emission. Additional work is requested to confirm this assumption. In what follows, they are ignored.
F . Some Comparisons of Theoretical Evaluations and Experimental Data All of the inelastic mechanisms are important for a quantitative interpretation of the Auger emission. In the previous sections, the stress has been essentially on the MFP concept; however, energy-dissipation effects also play an important role. For instance, the collisions with the inner shells contribute an important part to the stopping power of the primary beam and to the spatial variation of the Auger source function.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
183
Ab initio MFP calculations must represent some standard references. This is the reason for the previous, rather extensive, theoretical discussion. However, most of the inelastic MFP evaluations have been made within a jellium model, and more empirical estimations are obviously very useful whenever theoretical values are lacking. A substantial amount of MFP data have been published, for instance, by Powell (1974), Penn (1975), and Seah and Dench (1979). Ionization cross sections have been reviewed by Powell (1974, 1976) for K an L subshells. The theoretical evaluations obtained by Gryzinski's expressions show good agreement with other estimations using different formulations (Drawin, 1961) or Lotz (1967, 1968, 1969, 1970), i.e., within about 10%. These estimations also compare rather well with the experimental determinations (cf. Hink and Ziegler, 1969, for the K shell of Al, for instance). The collisions with the inner shell electrons can essentially play a role in the higher energy domain of the MFP curve. At low energies individual and collective excitations of the delocalized electrons determine its shape. There is apparently a somewhat universal behavior for the inelastic MFP in quite different materials. For a metal, the arguments are easy to understand. Near the Fermi level, the Pauli exclusion principle models this shape. The decrease of the MFP is approximately given by E-' (the zero-energy reference being the Fermi level). The oscillator strengths for the excitation mechanisms are then gradually exhausted as the energy grows. Thus, after a flat minimum for E values ranging in the 50-100-eV region, the MFP increases according to an E/ln E approximate law. In fact, most of the collision mechanisms can be described by partial cross sections involving an E-'(AE)- ln(4EIAE) general term whenever a mean excitation energy AE can be defined for the corresponding process (Powell 1974). For Penn (1976b) the general behavior of the total inelastic MFP can be described by
I,
= E/[a(ln E
+ b)]
(40)
where a and b depend on the solid investigated. Two contributions can be separated, one coming from the valence electrons and the other from the core inner shells:
A,'
=
A;'
+ A,'
(41)
where A,, for a free-electron-likematerial, can be written in a form very similar to that of AT but with suitable parameters a, and b,, which have been evaluated by Penn for various electron concentrations; A, also has the same form, and the corresponding parameters a, and b, have been given by Powell.
184
M . CAILLER, J . P . GANACHAUD, AND D. ROPTIN TABLE I
COMPARISON OF MFP ESTIMATIONS FOR ALUMINUM E - E, (ev) 25 50 100 250
Penn Seahand Cailler ez al. (1976b) Dench (1979) E - E , (4 (‘4 (4 (eV) 4.1 3.4 4.4 1.4
-
7.3
500 150 1000
4.8 4.3 5.4 8.4
Cailler et al.
Penn
Seah and Dench
(4
(4
(4
12.1
11.8 14.5 16.7
12.0 16.2 20.3
16.5 20.6
Another approach, which is a rule of thumb, has been given by Seah and Dench. From an important compilation of all the published data on the inelastic MFP, these authors propose a general relation
L = ( A / E 2 ) + BE”2
(42)
Different values for the parameters A and B pertaining to different groups of materials (elements, inorganic compounds, organic compounds, adsorbed gases) have been proposed. It appears that the most reliable way to present a universal law is to express the MFP in monolayers. This quantity A,,, is related to the usual MFP 1 by
L
=
aL,
(43)
where a is the monolayer thickness for the solid considered. We shall only present some comparisons for elements for which the universal law proposed by Seah and Dench is
A,,, = 538E-’
+ 0.41(aE)”2
(44)
where the energies are expressed in electron volts and the distances in nanometers. We shall consider two examples: A1 and Cu. In the first case, several theoretical evaluations have been proposed. Let us just mention, for instance, since the work by Quinn (1962), calculations made by Kleinman (1971), Bauer (1972), Shelton (1974), Penn (1976), Ganachaud (1977), Tung and Ritchie (1977), and Ashley et al. (1979). This list is far from being exhaustive. Starting from Lindhard’s dielectric function, several ameliorations have been proposed. These aspects have already been discussed in Section II,B. The general conclusion is that all of these theories agree (within ~ 1 0 % ) for MFP-value predictions. Table I gives a comparison between our own estimations, Penn’s calculations expressed by the general law [Eq. (40)], and Seah and Dench’s
185
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
TABLE 11 COMPARISON OF MFP ESTIMATIONS FOR COPPER E - EF.
Cailler et nl. Penn Seah and (b = 0.05 (1976b) Dench (1979) E
W)
A)
(4
(A)
25 50 100
2.8 2.3 2.6 4. I
-
4.2 3.7 4.6 7.1
250
~
~
4.6
- EF W 500 750 1000
Seah and
Cailler et (I/. ( b = 0.005
Penn
A)
(A)
(A)
6.9 10.2 12.8
7.0 9.3 11.5
10.0 12.2 14.1
Dench
universal curve for aluminum for typical values of the energy. These figures include the inner shell contributions. We can conclude that there is in this case rather good agreement between the theoretical values and the experimental results, except perhaps at low energies, but this is not in fact the important domain for Auger transitions. However, the universal law, based on an E1j2variation for E > 150 eV has not the same shape as that given by the Elln E law generally predicted by the electron gas theory. For Cu, some elements of comparison have already been given in a previous paper by Cailler et al. (1981), in which theoretical estimations and experimental values were reviewed. We shall only present here some brief comments. Table I1 shows the inelastic MFPs that we 'have obtained by adopting an intermediate value of 0.05 A for the parameter b in Eq. (38). These figures are compared once more with Penn's values and those of Seah and Dench. Our theoretical predictions are not far from Penn's values but differ appreciably from the Seah and Dench universal law. Tung et al. (1979) have also calculated the inelastic MFP for several solids including copper by using an electron gas statistical model. Their values are substantially higher than ours, especially at high energies ( 15 for E = 1 keV). Although a universal law is certainly a very useful tool for immediate investigations, it appears that some additional theoretical and experimental work is still needed before a complete conclusion can be reached. For instance, one could be tempted to reject experimental MFP values which depart too strongly from the experimental law although they are, in fact, correct. N
a
IV. ELASTIC COLLISIONS The very weak energy losses (<0.2 ev) caused by electron-phonon interactions can contribute significantly to the attenuation length of very slow
186
M . CAILLER, J. P. GANACHAUD, AND D . ROPTIN
electrons; however, in the energy range of Auger spectrometry these quasielastic effects will play a minor role. We shall now consider the purely elastic interaction of the electrons with the real part of the potential surrounding the ionic cores in a solid. In quantitative Auger analysis calculations (for instance, by Monte Carlo methods), the ionic cores are assumed to be randomly distributed in a highly delocalized electron medium. This randium model (Bauer, 1970) is evidently a crude approximation which cannot account for coherent diffraction effects. The fine structures appearing in the angular distribution of the Auger electrons have to be studied by multiple-scattering approaches, that is, by LEED theories. However, the randium model is apparently sufficient for describing more integrated properties of Auger emission such as the global energy distribution of the electrons or the emission yield. Bauer (1972) discussed the problem of the proper choice of the ionic core potential qr). The charge redistribution coming from the delocalization of the outer shell electrons when an atom is brought in a solid-state environment causes the failure of the free-atom approximation, especially at low energies (c100 eV). Self-consistent potentials are then needed. The most commonly used approximation is the muffin-tin picture. In this model the spherically symmetrical potential V ( r ) varies inside spheres of radius R centered on each ionic site. In the surrounding medium the true potential is replaced by a mean term with a constant value V,. Thus, the differential elastic cross section is given by a partial wave analysis:
h K / m is the velocity of the incident particle; P,(cos 0) is the l-order Legendre polynomial; and dl is the phase shift suffered by the lth partial wave. The total elastic cross section is m
o(E) = (4.1r/K2)
C (21 + I) sin’ 6,
l=O
(46)
and the elastic MFP A,(E) can be written Figures 6 and 7 show results obtained for aluminum by Ganachaud (1977) with a muffin-tin potential evaluated by Smrcka (1970), in complete agreement with Baudoing’s calculations ( 1971). It should be mentioned that the values of the phase shifts are very sensitive to the choice of the potential. This explains the differences in 6, evaluations by various workers [see, for instance, Pendry’s values (1974) obtained with a Hartree-Fock exchange potential]. A simple Slater exchange term can, in
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
187
FIG.6 . Principal phase shifts in Al. Reprinted with permission from North-Holland Publishing Co., Amsterdam.
-
50
E
- V&V)
100
150
FIG.7. Elastic MFP in Al. Reprinted with permission from North-Holland Publishing Co., Amsterdam.
fact, incorporate some part of the correlation effects which are omitted from the Hartree-Fock exchange operator and can give better MFP values. In any case, the order of magnitude of the MFP for elastic collisions clearly indicates that this is a strong mechanism which can play an important role in the backscattering of the primary electrons and, consequently, in the source function of the Auger electrons as well as in their escape process. V. AUGER TRANSITIONS IN A SOLID A. Introduction
An Auger transition is part of a global process involving the excitation and relaxation of a solid. The excitation step leads to the ionization of a core
188
M . CAILLER, J. P. GANACHAUD, AND D. ROPTIN
electron. The vacancy created in an inner shell is then filled by another electron which can come from (I) an outer shell of the same ionic core or (2) from the valence band. The excess energy is brought to another electron of the solid, which can itself originate from these various states. It is usual to assume that excitation and relaxation steps are independent. So, the probability of an Auger emission is just the product of the vacancycreation probability and that of the relaxation process. This assumption has been recently questioned (Abraham-Ibrahim et al., 1978, 1979). In fact, some coherence effects may occur in the creation of the deep holes. This question is certainly of basic importance. However, we shall adopt hereafter the simpler decoupling scheme, and we shall consider the relaxation step of the Auger transition in particular. Section V is devoted to classification of Auger transitions (Section V,B); theoretical evaluation of the Auger electron energies (Section V,C); and study of the line shapes, when at least one of the electrons involved in the Auger transition comes from a valence-band state (Section V,D). B. Classification of Auger Transitions
Classification of purely intra atomic Auger transitions makes reference to the atomic multiplet structure, and this is considered first. Transitions in which at least one of the jumping electrons is in the valence band are considered in Section V,B,2. 1. Purely lntraatomic Auger Transitions
The initial and the final state of an intraatomic Auger transition present one and two holes in otherwise filled subshells. However, an equivalent classification of these one- or two-hole states can be obtained by considering, instead of holes, one or two electrons in an otherwise empty subshell. This makes classification easier. The atom, in a neutral or in an ionized state, can be described by a manywhich is an eigenstate of the Hamiltonian particle wave function operator I9 H $ = E+ (48)
+,
Different A's are associated with the different states of ionization of the atom. A rather complete description of the Hamiltonian has been given by Larkins (1975), but is not reproduced here. Let us just consider the nonrelativistic form -
Ze2 ri
-
'' e2 + -21 1 + [(ri)li.si rij -
j=l j#i
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
189
For each of the Z' electrons (Z' = Z for a neutral atom), four contributions have been separated. They represent, respectively, the kinetic energy operator, the Coulomb attraction of the nucleus, the Coulomb interaction with the other electrons, and the spin-orbit coupling. Determination of Auger electron energy entails the resolution of Eq. (48) for both the initial and final states of the atom. This is, in fact, a very intricate problem, and it is necessary to proceed by steps. A basic simplification consists in neglecting the spin-orbit coupling and in replacing the mutually antagonistic second and third terms of Eq. (49)by a potential having a spherical symmetry around the nucleus. This central field approximation is now briefly discussed. a. Central Field Approximation. In this approximation, I? is replaced by a sum of one-particle decoupled contributions
where U ( r i )is a spherically symmetric potential. For closed subshells, this assumption is strictly true; for open subshells, this is only a very convenient approximation. In this scheme, the functions II/ can be represented by antisymmetrized products of the eigenfunctions of (Slater determinants), each one-electron state being characterized by the usual five quantum numbers n, 1, m,, s, and m,.However, the total energy of the atom, described by the Hamiltonian of Eq. (50),does not depend on the two numbers m, and m,. Thus, several Slater determinants correspond to the same eigenvalue of 8'. In the general case, the order of the degeneracy is given by
D
=
n
c,+z
(51)
1
where z, is the occupation number of the subshell of orbital quantum number 1. Only incomplete subshells contribute to the degeneracy. For instance, for an ( n p 2 )configuration, two electrons occupy the subshell (n, 1 = 1) and all the other subshells, including (n, I = 0) are complete. Thus, the degeneracy is C; = 15. b. Coupling Modes. The approximate Hamiltonian I?' of Eq. (50) differs from the original I? by two contributions:
190
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
’
where <(Ti) = (h2/2m2c2)r; [dU(ri)/dri] (if in this corrective term the potential is evaluated in the central field approximation). The perturbation theory is usually brought into play to include the effects of the corrections H’ and H ” . In extreme coupling schemes, only one corrective term is taken into account, either only 8’(L-S or Russell-Saunders coupling) or only 8” ( j - j coupling), whereas, in the intermediate coupling mode, the corrections coming from both 8’and fir’ are introduced. Intermediate coupling is apparently the most reliable, but it is well known that for light atoms Russell-Saunders coupling is sufficient, whereas j - j coupling works well for heavy atoms. c. Russell-Saunders Coupling. Let 2, 3 be, respectively, the orbital, spin, and total angular momentum operators for the atom (with, for instance, 2= The Hamiltonian operators fil and fitseparately commute with E,, S2, and Thus, the eigenstates of this set of four operators can be used to construct the eigenfunctions of 8’+ B‘,which are characterized by the four corresponding quantum numbers L, M , , S, M , . They are represented by the notation 2s+1X. The multiplicity is 2S + 1 and the symbolic letter X is one of the sequence: S, P, D, F, G, H, I,. . ., according to the value of L: 0,1,2,3,4,5,6,. ... The degeneracy associated with the incomplete Hamiltonian fil is partially raised in this new basis. Let us return to our example of an ( n p 2 )configuration. It is easy to prove that the closed subshells make a zero contribution to L, M , , S , and M s . Letting I , = 1, = 1 and s1 = s2 = 1/2 be the quantum numbers for the two electrons of the open subshell, L, M , , S, Ms can vary between the following limits (by a unit increment):
s,
z2,
ziJi).
s,.
11, IS1
121
- s2(
+ < S < SI +
< L < 1,
< M,
12,
-L
~ 2 ,
-S < Ms < S
So, the 15 degenerate states associated with
(53)
8’are split into
5 degenerate states ‘D ( S = 0, L = 2)
9 degenerate states 3P ( S
s
1
1 state
=
1,
L = 1)
(S = 0, L = 0)
the degeneracy being given by (2L + 1)(2S + 1). d. j - j Coupling. In this coupling scheme, the Hamiltonian is i
i
This operator remains separable, and the corresponding eigenfunctions can be represented by Slater determinants involving the one-electron
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
191
functions which are now the eigenstates of &. This latter Hamiltonian [with corresponding eigenvalues h2ji(ji + 1) commutes with j^z and and hrnji]. For an (np2)configuration, the electrons of the closed subshells still play no role in the classification,The two remaining electrons are characterized by the two quantum numbers j , and j , , which can vary (by unit increment) between the limits
TZi
11, - slJ < j , < 1,
+ sl,
11, - s 2 J < j 2 < 1,
+ s2
(55)
The 15 degenerate states of the original Hamiltonian 8' can now be split into 1 state with j , = j , = 112 8 states with j , # j , ,
6 states with j , = j ,
j , , j 2 = 112 or =
312
312
Each one-electron state is denoted by naj where a is a symbolic letter asso-
ciated with the 1 value. For instance, np,,, represents a state of the n shell with 1 = 1 and j = 112. The corresponding subshell is denoted by X , where the label z is 1,2,3,4, , d5,,, f 5 / 2 , f7,,, ..., respectively. 5,6, 7,. .., for the states slI2,pIl2,k l 2d3/2, For example, the states 2sIl,, 2plI2, 2p3,,, 3d3,,, 3d,,, belong, respectively, to the L,, L,, L,, M,, M, subshells. e. Intermediate Coupling. By applying both correction 8'and 8'' to A', the degeneracy can be more completely raised. When 8''is applied after H', 2 and $ no longer commute with 8. On the other hand, application of 8' after a'' sets up an electrostatic coupling between the electrons of the atom, which can no longer be described as individual particles with a given total angular momentum. However, in this domain, corresponding to the intermediate coupling, the total angular momentum for the atom f still commutes with A. These two operators have the same eigenstates, which are denoted by using a double notation coming from the two extreme coupling modes, the L, S designation being completed by a right-hand subscript label equal to the value of J . The term J varies by unit increment from L - S ] to J L + S J ,and for a 2p2 configuration, the 15 states are now broken into 1 state L2L, ( , S o )
I
3 states L,L3
('PI)
5 states L2L3 (ID,) 1 state
L3L3 (3P0)
5 states L3L3 (3P2)
where the multiplicity is now 25
+ 1.
192
M. CAILLER, J . P . GANACHAUD, AND D . ROPTIN
f. Designation of Auger Transitions. This designation concerns the initial and final states of the atom. We have already mentioned that from the point of view of classification, the states involving holes in otherwise closed subshells could be replaced by states with electrons in otherwise empty subshells. This is not true for energy calculations. The corresponding modifications have been indicated by Condon and Shortley (1970). Following Larkins's notation, an electron configuration such as (2s2)(2p4) corresponds to a [2p2] hole configuration. Accordingly, an Auger transition in which a vacancy in the L, subshell is replaced by a double hole in the final state M,M, ['Po] is denoted by L,M3M3 ['Po] in the intermediate coupling scheme. However, more imprecise designations often refer to the extreme modes of coupling and, for example, the notation L,M,M, ['PI or L2M3M3or even L23M23M23 can be used. In the latter case, all transitions from L, or L, states to M,M,, M2M3, and M3M, states are supposed to be included. We should mention that the L, S designation is sometimes completed by indicating the parity of the atomic state, which is the parity of the sum of the Ii values. An odd state is denoted by the symbol ''0." For instance, a multiplet arising from a (2s, 2p) configuration has an odd parity. This is indicated by the notation 3P0(Slater, 1960).In an Auger process, the parity of the atomic state does not change. Thus, some transitions can be forbidden [e.g., the L,L3( ,PI) line of the KLL spectrum] by this parity rule. On the other hand, collective excitations can take place, for instance, plasmon effects in a metal. They can lead to an extra satellite structure which will obscure the purely atomic spectrum. Good examples are aluminum (Dufour et al., 1976) and sodium and magnesium (Steiner et al., 1978). Another possibility is that of satellite peaks due to a double ionization of the atom, the Auger transition taking place in an atom with two holes initially present. This kind of transition has been presented in a paper by Melles et al. (1974), for instance, for the L2,, Auger emission bands of Al, Si, and P. It is a bit difficult for a noninitiate to the multiplet-structure theory to understand the rather complex Auger spectrum notation. Table I11 indicates the possible Auger transitions for given final configurations with their complete designations. For example, the ninth column corresponds to [np,n'p] final configurations ([pp'] for the sake of brevity); that is, to [ X , , Xi], [ X , , Xi], and [ X , , X i ] states where X and X ' are symbolic letters (L, M, . . .) representing the subshells. When n = n', the two holes of the final state are said to be equivalent, and the configuration is denoted by [p' ] (eighth column). The rows correspond to the intermediate coupling notation, and the numerals in the table give the degeneracy of the final states. To illustrate how Table I11 can be used, let us mention, for instance, that the [L2, L2], [L2, L3], [L3, L3], [M,, M,], [M,, M3], [M3, M31r "2,
193
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
TABLE 111 DESIGNATION AND DEGENERACY OF AUGER-TRANSITION FINAL STATES
l1 l1
12 13
s2 ss'
sp sd
14 15
16 17
18 23 19 33
sf
sg
p2 pp'
~
IS, 1
1
3s,
3
'PI 3p0 3p, 3p* 'D2 3D1 3D2 'D3
1
3 5 5 3 5 7 5 7
9
7 9 11
I 3 5 5
3 5
5 3 5 1 1
dd'
5
3 5 7 1
1
5 7 9 9 7 9
5 7 9 9 7 9
11
11
5 7 9 9
f2
df
~
~~
1
1
3 3
1
9 9
d2
~
3
I
3G4 3G,
'16
1 3 5 5
pg
1
5
3G3
3H4 3H5 3H6
1
3 3 1 3 5 5 3 5 7
3F2
pf ~
1
3
7
'H,
pd
~~
IF3
3F3 3F4 'G4
23 33
3
1
1
1
3 5 5 3 5 7 7 5 7 9 9 7 9 11
3 5
3 5 5
11 9 11
13 13
N2], . . . final states correspond to the eighth column and to the 'So, 3P,, 3P,, 3P,, ID,, . . . lines. A good example of what appears in the true Auger spectrum is given by Fig. 8a for the [L3M23M23]lines of copper according to Mariot (1978). Some additional examples are also given (Fig. 8b-d) for M,,,], [N,,,, N,,,] final configurations. [MI, M z , ~ ] , Determination of Auger line energies and calculation of these Auger transition probabilities have been the object of numerous papers. It is beyond the scope of the present article to give a full account of such a vast theoretical field. The interested reader will find detailed information in the books by Chattarji (1976) or by Sevier (1972) and also in the original papers (Asaad and Burhop, 1958; Asaad, 1963a,b, 1965a,b; Rao and Crasemann, 1965; McGuire, 1970, 1971a,b, 1974, 1977, 1978; Larkins, 1971a,b, 1973a,b, 1974, 1976a,b, 1977a,b,c, 1978a,b; Crotty and Larkins, 1976,1977; Bambynek et al.,
194
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
t
L-.
:
a
0.
.Experiment
?
L
780
t
FIG. 8. Auger transition line shapes. (a) Cu (L3-Mz.3M2,3); (b) Cu (L3-M1M2,3);(c) Cu (L3-Mz.3M4,5);(d) Ag (M4,5-N4,5N4,5). Reprinted from Mariot (1978) by courtesy of the author.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
+Ms-Nes
N+
+M
4
195
+%-
-N
(d)
FIG. 8. (Continued)
1972; Price and Rao, 1972; Shirley, 1973; Yin et al., 1972, 1974; Kowalczyk et al., 1973a,b, 1974; Ley et al., 1973; Kim et al., 1976a,b;Mariot and Dufour, 1977a,b; Aksela et al., 1980; Vayrynen et al., 1980; Chen et al., 1980a,b,c). However, some general considerations about Auger transition energies are given in the next section. g . Conjiguration Interaction. Let us return to the second step of the calculation leading from the Russell-Saunders coupling to the intermediate coupling. The corrections introduced to account for the A’’ perturbation were evaluated by using eigenfunctions of the four operators E2, t,, and However, as indicated above, the full Hamiltonian r? still commutes with J2 and Jz. In fact, in the former discussion the analysis was made configuration by configuration. This is an unnecessary supplementary restriction. For instance, a given configuration was associated with a given J value. Other configurations corresponding to the same value of J have to be introduced to describe the many-particle wave function of the atom. This interaction of configurations a priori brings into play many Slater determinants. In practice, only functions corresponding to nearly the same energy are mixed. Let us consider the example of the KLL spectrum, or more precisely, the function corresponding to J = 0. In practice, the associated subspace was separated into two subspaces related to either a [2s2] or a [2p2] configuration. The first subspace was spanned by the [L,L,, ‘ S ] Russell-Saunders eigenstate, whereas the [L2L2,‘S] and [L,L,, ,PP] states gave a basis for the
s,.
s2,
196
M . CAILLER, J . P. GANACHAUD, AND D . ROPTIN
second subspace. (In the intermediate coupling these three terms became
(L,L,,'So]; [L2L2, 'So]; and [L3L3, 3Po], respectively.) When the different configurations are treated separately, two distinct secular equations have to be solved, the secular determinants being of rank 1 in the [2s2] case and of rank 2 for the [2p2] configuration. Taking account of the configuration interaction (CI) consists now in introducing a three-dimepsional subspace spanned by the three basis wave functions of the Russell-Saunders scheme. So this leads to consideration of a secular determinant of rank 3. Returning to Table 111, we may remark that the [2s2p] configuration yields a 3P0term. A priori, this term would also mix with the above basis functions. However, this 3P0function does not have the same parity as the three other eigenstates and cannot be brought into play in the above CI calculation. 2. Lines Involving the Valence Band
A designation with respect to the multiplet structure is not suitable for this case. When the two holes of the final state are in the valence band, the transition is simply denoted by CVV, where C indicates, in the j-j coupling mode, the subshell in which the initial hole has been created and V represents valence (or conduction). When only one hole is present in the valence band in the final state, the other hole appearing in a C inner subshell, the line will be designated CC'V (KL,V or KL,,,V, for example). Owing to the information it can give about the chemical environment of atoms on solid surfaces, the shape of these lines has been the object of several studies. These are reviewed in Section V,D. Although in the present study we are mainly concerned with metals, we should first like to examine the special case of ionic crystals. The ionic crystals can be described by their band structure (see Chelikowski and Schliiter, 1977, for SO,), but they have also been depicted in terms of molecular orbitals (Fisher et al., 1977). These molecular states are designated by the irreducible representations of the symmetry group of the molecule. However, some of these molecular states are very close to atomic states of a given component of the compound. They are rather localized, and their energy level is nearly equal to atomic levels of this component. They can be called quasi-atomic molecular states. The CVV Auger lines of the compounds have been often denoted by referring to the molecular orbital (MO) designation (Bernett et al., 1977; Ramaker et al., 1979b). Some were also considered to be interatomic or crossover transitions (Citrin et al., 1976)if the initial vacancy was in a core of a given component and the final vacancies were in the quasi-atomic molecular states of another component.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
197
It could therefore be conjectured that there are different physical processes leading to two holes in the solid valence band (CVV Auger transitions) or in molecular valence states or yet in quasi-atomic states (interatomic transitions). In fact, for solids they are simply different pictures of the same mechanism. From this point of view, a comparison between the densities of states calculated by Chelikowski and Schliiter (1977) and the XPS valence-band assignation in terms of M 0 s is enlightening. The case of Si02 is used to illustrate the terminology of the ionic compound CVV Auger transition. First, it is necessary to indicate the nature of the atom and of the subshell in which the initial vacancy occurred. Considering the case of S O z ,it is, for example, the subshell L2,3of a Si atom. The central atom is bound to four oxygen atoms in a tetrahedral configuration. The corresponding symmetry group is denoted Td. This symmetry group has 5 irreducible representations : 2 one-dimensional representations,
A and A ,
1 two-dimensional representation,
E
2 three-dimensional representations,
T , and T2
The MOs centered on the Si atom have the symmetries of these five representations; for instance, the various orbitals of the representation Al are denoted la,, 2a,, 3a1, . . . in the order of the increasing energies. Similar notation is used for the MOs of other representations. In the particular case of SiO,, the MOs la,, 2a1, . .., It,, 2tz, . . . are nearly identical to the Si(ls), O(ls), Si(2s), Si(2p) atomic orbitals. In the absence of s, p hybridization for the oxygen atoms (which should be weak according to Yip and Fowler, 1974),the MOs of the valence band correspond to the Si and 0 atomic orbitals, as indicated in the accompanying tabulation, Number of orbitals
4a, +-+ O(2s) and Si(3s) 3t, t* O(2s) and Si(3p) 5a1 tt 0 ( 2 p f ) and Si(3s) 4t1 0(2p+) and Si(3p) le, 5t,, I t
-
1 3 1
3 8
where le, 5t2,and It are antibonding orbitals corresponding to a 2p oxygen orbital, with some mixing with the Si orbitals. The ( 2 ~ ' )orbitals of the 0 atoms are those pointing towards the central Si atom. The other 2p orbitals give the eight orbitals of the le, 5t2, and Itl configurations. On the whole, ther? are 14 MOs in the valence band: that is, the number of the atomic orbitals from which they are issued. To be more rigorous, there
198
M. CAILLER, J. P. GANACHAUD, AND D. ROPTJN
is in fact some hybridization of the s, p orbital of oxygen, and each of the 4al, 3t, states is in fact a linear combination of 0(2s), Si(3s),and Si(3p). Auger transitions involving the valence band are thus designated by Si(Ll)Si(L,,,)4a, or Si(L,,,)4a13t,. Each oxygen atom is bound with two silicon atoms. Its environment has the symmetry of the CZygroup; A,, A,, B,, B , are the irreducible representations of this group. The KVV lines starting from a vacancy in the 0 atom are denoted by K(0)6a,6al or K(0)6a15b,(S), where S indicates that it is a singlet state. As far as the energy level of at least one of the MOs playing a role in the Auger transition is essentially determined by that of an atomic orbital (AO), one can speak of interatomic or crossover transitions and designate them by A(C)A(C')B(C") or A(C)B(C')B(C)(cf. Citrin et al., 1976). C. Kinetic Energy of Auger Electrons: Purely Atomic Transitions
To evaluate this kinetic energy we shall assume that, the initial hole being created, the Auger transition follows two successive and independent steps : (1) annihilation of the initial hole in the C subshell of atom A; and (2) creation of the two final-state holes. As indicated earlier, these two holes can appear in C' and C" subshells of the same atom A or in the valence band. To make the formulation more explicit, we shall adopt notation corresponding to the first case. An extension to a more general situation would not be too difficult. Moreover, we shall assume that a classification of the states adapted to the intermediate coupling scheme in the central field approximation can be used. The first step of the Auger transition in an isolated atom can be described by the equations: A(Z - 1;ATC) + eAEl
= E [ A ( Z ) ] - E[A(Z
+
A(2)
- 1; A,C)] = -Ek(Z+Z
(56) - 1;
ATC) (57)
Equation (56) indicates that the A atom has initially one hole in the C subshell (A,C)and that by collecting an electron, it returns to its ground state; Z is the atomic number of the neutral atom. Equation (57) relates to the energies and indicates that the energy difference AE, between the initial and the final states described by Eq. (56) is the opposite of the ionization energy for an electron in the C subshell (the arrows indicate the dynamical nature of the transition). An energy equal to this binding energy is thus liberated, a fraction of which is carried by the Auger electron. Here we shall
199
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
consider that the electron by which the hole is annihilated was initially at the Fermi level or, in other words, that the binding energy is referred to that level. The second step of the Auger transition obeys the following equations: A(Z)+A(Z - 2 ; A,C'; A,C", x)
+ 2e-
(58)
which specify that the two holes of the state have taken place in the atom A, one in the C' subshell, the other in the C" subshell. These two holes interact and, assuming that the central field approximation still holds for this twohole situation, one can make reference to the multiplet structure in the intermediate coupling scheme to designate the final state of the atom; here x represents the more complete notation "+lX If the holes were created in the valence band, A,C' and A,C" would be replaced by P. Similarly, for interatomic transitions, -and would become B,C'and B,C". The kinetic energy of an Auger electron, measured from the Fermi level, is given by J ' -
E(A; CC'C";
X)
= - AEI
- AE,
(60)
Asaad and Burhop have used the formalism leading to Eq. (57), (59),and (60) to evaluate the kinetic energy of the Auger electrons of the KLL atomic lines. Within the frozen orbital (or sudden) approximation, they obtained the result E"(A,KLL';x)=E;(Z+Z-
l;K)-E$(Z+Z-
1;L')
- Ei(Z*Z - 1 ; p)- F(A,L'L";x)
(61)
In Eq. (61) the quantities E; represent the binding energies in the atom measured from the vacuum level, whereas the last term F represents the interaction energy of the two holes in the final state. This interaction energy has been evaluated by Asaad and Burhop in terms of Slater integrals and the spin-orbit coupling constant. The binding energies were set equal to the first ionization energies, measured for the neutral atom. We should note that these binding energies differ from the electronic orbital energies, the creation of a hole (or its annihilation in the reverse transition) causing a perturbation in the electronic cloud. The most important effect is that for the subshell in which the hole is created and for the more external subshells. This perturbation has been called dynamic relaxation by Shirley (1 973). The relation between the binding energy and the orbital energy for the C subshell of atom A is written E,A(Z * Z
-
1 ;ATC)
= ~ ~ A,C) ( 2 ;
Edr(Z --t Z - 1; C C )
(62)
200
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
where dr is the dynamic relaxation (Fig. 9). For a metallic environment this relation must be modified to account for the screening of the holes by the electrons of the valence band. Kowalczyk et al. (1973a,b) and Ley et a/. (1973) supposed that for this environment (denoted by M in what follows), the screening of a hole was due to the occupation of an additional quasi-atomic state appearing around the vacancy in the atom, according to Friedel’s model (1952). Equation (57) has to be replaced (Hoogewijs et al., 1977; Larkins, 1977a,b) by
AEl
=
Ef(Z-tZ
=
E[M(Z - 1
- 1
+ n ; A,C, AC,)
+ n ;A T ; AC,)]
-
E[M(Z)]
(63)
In Eq. (63), C, is the atomic orbital of the atom, ionized in the C subshell in which the screening electron takes place. Larkins (1 977a,b) considered that this screening did not involve a unit electronic charge but a fractional charge n set equal to 0.6 per hole, whereas Hoogewijs et al. chose n = 1. In both cases, the screening charge for the two holes was set equal to twice the screening charge of one hole. In these conditions, Eq. (59) is replaced by AE,
= E[M(Z
-
2
+ 2n; A,C’; A,C”; X ; AC,, AC,)]
- E[M(Z)]
(64)
and the kinetic energy of the Auger electron can be, at least in principle, calculated from Eqs. (62)-(64), provided that the energies of the right-hand members of Eqs. (63) and (64) are evaluated in the adiabatic approximation, for instance, by a self-consistent field method. However, it is possible to follow a different approach initially proposed by Shirley (1973) for an atom. He introduced the concept of a static relaxation energy which was adapted to the case of metals by Kowalczyk et al. (1973a,b) and Ley et al. (1973)and was then ameliorated by Kim et al. (1976a,b)to take
Reference
level
T ------
--L-
75,(=-1,A7) -c (Z,A,C)
9. Relation between the binding and orbital energies: for an atom, the reference level is the vacuum level; in a metal, it is the Fermi level. FIG.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
20 1
into account the dynamic relaxation corrections for the energies. More recently, Hoogewijs et al. (1977) and Larkins (1977a,b) returned to this approach with different calculation procedures but equivalent points of view, as shown by Larkins (1978a,b). By adopting the main features of the Shirley and Kim et ul. scheme, but with slight adaptations, one may consider the second step of the Auger process to have three stages. (1) Creation of a first hole in the C' subshell, with screening in the C, subshell, according to the energy relation
+ n ; A,C'; AC,)] - E [ M ( Z ) ] - 1 + n ; p, AC,)
6EI = E [ M ( Z - 1 =
E,M(Z+Z
(65)
(2) Creation of a second hole in the C" subshell, with screening but with no repulsion between the two holes:
+ 2n; A T ; c; ACI; AC:')] - E [ M ( Z - 1 + n; A T ; AC,]
6E2 = E [ M ( Z - 2
(3) Intrgduction of the repulsion between the two holes: 6E,
= E[M(Z
-
-2
+ 2n; A,C' ; A,C" ; x;ACI ; AC;)] 2 + 2n; A T ; A,C"; ACI; AC:')]
E[M(Z -
= F(mA,C"; ; x)
(67)
This last energy term is assumed to be independent of the screening. Conversely, one assumes that the screening does not depend on the multiplet structure. Adding Eqs. (65)-(67) evidently gives back Eq. (64). As first noted by Shirley, stage (2) does not exactly correspond to the ionization of the C" subshell, which itself obeys the relation 6E;
+ n; A T ; AC,)]
= E[M(Z -
1
=Ez(Z+Z
-1
-
E[M(Z)]
+ n ; A T ; AC,)
(68)
where E f is related to the orbital energy by EF(Z+Z - 1
+ n ; A T , AC,) = - e M ( Z ; AC") - Ed,(Z+Z
-
1
+ n ; A T ; AC,)
(69)
whereas
6El
=
+ n ; A T , AC"; ACI) - 1 + n - 2 - 2 + 212;n, A,C";
-P(Z - 1
- &,(Z
AC;; AC:)
(70)
202
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
So, from Eqs. (66) and (68) we obtain
+ n ; A T , AC,) + E [ M ( Z - 2 + 2n; A T , A T , ACA; AC:)] - E [ M ( Z 1 + n ; A X AC,)] - E[M(Z - 1 + n ; A,C"; AC,)] + E[M(Z)]
6E2 = E F ( Z + Z - 1
-
(71)
which, from Eqs. (69) and (70), can also be written
6E2 = E F ( Z + Z - 1 + n ; F,AC,) with
E,"r = c'(Z - 1 + n ; A,C'; AC"; AC,) AE:, = Ed,@ - 1 + n + Z - 2
- AEE
(72)
tM(Z;A,C")
(73)
- EZ
-
+ 2n; A T , A T , AC,'; AC:)
1 +n;AF,AC,) (74) Equation (73) defines the static relaxation energy, according to Shirley, and Eq. (74) defines the dynamic relaxation-energy correction. In the atomic case, the sum E;r + AE& has been called the cross-relaxation energy R; by Hoogewijs et al. It is identical with the adiabatic relaxation correction K(C'C") of Larkins (1976a,b). From Eqs. (60),(63), (65)-(67), and (72) we obtain -Ed,(Z+Z-
E'(A,CCC";
X)
= Ef(Z
--t
Z - 1 + n ; ATC; AC,)
+ n ; A,C'; AC,) - E ; ( Z + Z - 1 + n ; A,C", AC,) + F(A,C', A,C"; x) + EZ + AEYr
-
E!(Z
+
Z-1
(75)
The sum of the last three terms is identical with the term UM(A,C', A T , x) from Larkins. This author sets Ci = Cf = C, and, for that case,
UM(A,C';A,C"; X)
=
E[M(Z - 2
+ 2n; AC';
+ E[M(Z)]
- E[M(Z - 1 + n ; AC',AC,)]
AC";
X; AC,,
AC,)]
- E[M(Z - 1 + n ; Ac",AC,)]
(76)
For an isolated atom this relation would become
U a ( A T ; AC";X) = E[A(Z
-
2; AC'; AC"; x)]
+ E[A(Z)]
- E[A(Z - 1 ; AC')]- E[A(Z - 1; AC")] (77) and could be evaluated by energy self-consistent field (SCF) calculations. It could also take the form:
U a ( A T A T , X) = F a ( E , AC",X)
-
Ka(ACT;A T )
(78)
203
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
and U'(AC', A T , x) can be calculated by referring to U " ( E ,AC", x) according to UM(AC', AC",X) = U"(AT, C, x) + GU(AC'; A T , X)
(79)
where the 6U corrective term is defined by substracting Eq. (77)from Eq. (76). Various estimations of this corrective term (assumed to depend on the nature of the element but not on the particular transition considered) have been given by Larkins. Thus, for instance, in the equivalent core approximation, 6U = 2n(CC,)Z+' - n y C , C , ) Z + '
(80)
where the terms between angle brackets represent pair interactions for the electrons of the C and C, or C, and C, subshells in an atom having 2 + 1 as its atomic number. From the above theory, it is possible to reproduce with great precision the experimental results for KLL, LMM, MNN, N O 0 lines of a large number of elements with an approximation of 1-2 eV when the binding energies are known. Larkins supposed that the (CC, ) pair-interaction term was relatively independent of the subshell C and proposed to take for it the subshell of the penultimate occupied shell in the atom. The choice of n and C, raises some questions. For Larkins a good approximation is obtained by giving n the value 0.6 by taking for the screening orbital the first available orbital in an atom having atomic nuqber 2 + 1 and by using results obtained by the nonrelativistic Hartree-Fock method for the pair-interaction values. For example, the screening orbital he chooses for zinc is 4p. Hoogewijs et al. take the same orbital for zinc but use n = 1. On the other hand, Kim et al. (1976)suggest that the initial configuration of solid nickel has more 3d charge than the free atom in its ground configuration. Then the screening of a first hole comes from the mixed occupation of 3d and 4s orbitals, whereas the screening of the second hole takes place nearly exclusively through the occupation of the 4s orbital. 1. Computational Aspects From Eq. (75) the kinetic energy of the Auger electrons emitted by an atom labeled A is EM(A; CC'C; x) = ~ f ( z _ + z - i +~;A,C;AC,)-E,M(Z+Z-~ ~
-
where
E?(Z
+
Z
-
1
_
+~;A,C';AC,) _
+ n ; A , C ; AC,) + UM(A,C';A , C ; X)
~- --
U'(A,C'; A,C"; X) = F(A,C'; A,C";x) + EZ
+ AEZ
_
(81)
(82)
204
M. CAILLER, J. P. GANACHAUD, AND D . ROPTIN
For an isolated atom, Eq. (82) is written E"(A; CC'C"; x) =
E ~ ( z - 2- 1; ATC) - pB(z+z - 1; A,C') -
F~(z-2- 1; A,C") + u a ( mA T 7x )
(83)
+ g, + AE&
(84)
with U"(A7, A T ,
X)
= F ( A 7 , A,C"; x )
Let us consider this case first and neglect the relaxation corrective terms E:, AGr.It is the Asaad and Burhop approximation.
+
2. Asaad and Burhop Approximation Basically these calculations necessitate making an evaluation of the energies of the initial and final states involved in Auger transitions and subtracting them. Let us consider the case of the KLL lines. For example, using the notations given by Eqs. (57) and (59), we obtain
E ( A ;KL,L,; 'Po) = E[A(Z
-
1; AK)]
E[A(Z - 2;
-
c; 6; 3Po)]
(85)
which we shall write in the simplified form E(KL,L,;
3p0)=
E[R]
-
E[L,L,; 3p0l
(86) --
Neglecting spin-orbit coupling, the energy of the final state is E(L,, L, ; 3P)and can be expressed via the average energy of the [2sp] hole configuration associated with it, plus some additional terms. For the three [2s2], [2sp], and [2p2] configurations, the average energies are the following (Slater, Vol. 11, Appendix 21): Ea,(2S2)= 2ELl + (2s, 2s), E&P2)
=
E,,(2SP) = E L 1
+ EL2,3 +(&
2EL,*, + (2P7 2P)
2P) (87)
are the binding energies for the L, and L2,3subshells, where ELIand EL2,3 respectively. In our notation, EL,
E i ( Z + Z - 1 ; AL, )
(88)
The (2s, 2s), (2s, 2p), and (2p, 2p) terms represent the interaction energies for two 2s, one 2s and one 2p, and two 2p electrons, respectively. The interaction energies for pairs have been tabulated by Slater for
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
205
equivalent and for nonequivalent electrons. For example, (2s, 2s) = F0(2s, 2s) (2P, 2P)
=
F0(2P, 2P) - M 2 ( 2 P ,2P)
( 2 ~ 2p) , = F0(2s, 2p) - &G1(2s,2p) where F and G are Slater integrals defined by Rk(ab,cd) = e2 x
J; J:
Rala(Tl
Rnal&Z
)Rflbl&21
r( pi Rnclc(r* 1
k3.;d r , dr2
(90)
where Fk(i,j ) = Rk(&ij) and Gk(i,j ) = Rk(ij,j i ) . The additional terms have also been listed by Slater. Thus, for a [2sp] configuration one obtains a triplet state for which the corrective term is - $G1(2s, 2p) and a singlet state associated with a $G1(2s, 2p) term. The multiplet energies, without spin-orbit coupling corrections, are E(LILl; 'S) = 2EL, + F0(2s, 2s) E(L,L2,3; 'P)
= EL1
E(L,L2,3; 'P)
= ELI
+ + F0(2s, 2p) - $G'(~s, 2p) + EL2,3+ F0(2s, 2p) + $G1(2s, 2p) EL2,a
+ F0(2p, 2p) - iF2(2p, 2p) E(L2,3L2,3 ; ID) = 2EL2,, + F0(2p, 2p) + &F2(2p, 2p) E(L,,,LZ,, ; '9= 2EL2,, + F0(2P, 2P) + 5F2(2P,2P) E(L2,3L2,3
; "1
= 2EL2,,
(91)
The spin-orbit interaction corrections can be obtained from the above results and from the spin-orbit interaction matrices given by Condon and Shortley (1970).For example, their sl spin-orbit interaction matrix gives the following result for the [2sp] configuration:
In matrix (92)the rows have been ordered according to the sequence 'P2, 3P1, 'PI, 3P0,and the signs have been changed from the original Condon and Shortley formulation because we are concerned here with holes, not electrons. The coupling parameter izl= $(EL,- EL3)and can be obtained from experimental measurements of the binding energies.
206
M . CAILLER, J. P. GANACHAUD, A N D D . ROPTIN
The matrix obtained by adding to the diagonal terms of matrix (92) the energies of the corresponding multiplets (without spin-orbit coupling), now has to be diagonalized. For instance, for the 3P1 and 'P, multiplets the matrix to be diagonalized is
We obtain
[
3p)f
E(L2,3L2,3;
E(L1L3;
(121/2)
-121ld5
-[21/&
E(L1,
]
(92')
L2,3;
3p1)= ELI+ EL2,3 + F0(2s, 2 ~ +) +C2] -
{ [fG1(2s, 2 ~ -) $1211
+ i1$1])1'2
+ E L 2 . 3 + F0(2s, 2P) + $ 1 2 , + {[+G1(2s,2p) - $[,,I2 + $1$1}li2
(934
E(L1L2; lP1) = E L ,
(93b)
The final expressions for the energies of the (KL,L, ; 3P1)and (KLlL3; 'PI) Auger lines can be obtained by introducing in Eqs. (93a) and (93b), respectively, the relations = EL2,3 - $ 1 2 1 (944 EL2
= EL2,3
+ 121
(94W
and by subtracting from E(K) the obtained results. For more complete tabulation of the KLL spectrum the reader is referred to Chattarji (1976). Larkins (1976a, 1977c) has tabulated the Slater integrals [F0(2s, 2s); F0(2s, 2p); . . .; G1(2s, 2p); . . .] for all the elements with atomic numbers between 10 and 110; For that he has used a semiempirical method. These tables also give adiabatic relaxation cohections [K(2, 2); K(3, 3); . . .] associated with the different atomic shells (L, M, . . .) and also solid-state corrections as calculated from relation (SO). The agreement between calculated and experimental values is, in most cases, of the order of 1 or 2 eV and never exceeds 5 eV.
D. Auger Line-Shape Analysis Auger transitions arc part of a global process where the atom (or the solid) suffers an ionization-relaxation mechanism via an intermediary unstable state. If we separate the relaxation mechanism but retain for it the name of Auger transition, we can associate with this step the following characteristics :
(1) It is a many-body dynamical process (interaction between the two final-state holes, dynamic relaxation-energy corrections). (2) It is a local phenomenon. This means that, except perhaps for ionic
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
207
compounds, the two final-state holes are created near the ionic core which contains the initial hole. Thus it might be important to take into account the local modifications of the charge density caused by the screening of the holes. (3) It obeys parity-selection rules and kinetic momentum conservation (Feibelman et a!., 1977). Considerable progress has been obtained in line-shape analysis. Interest in such analysis stems from the possibility of using Auger spectroscopy as a local microprobe for the chemical environment of the atomic species present in a traget. It does not seem that a complete theory that accounts for all the above-mentioned aspects yet exists in a form comparable with what has been done, for instance, by Nozieres and De Dominicis (1969) for X-ray absorption and emission. For simple metals, Von Barth and Grossmann (1979) have shown that, except for the edge singularities, most of the results of the many-body theory could be recovered by a static one-body analogy, provided that the final state was conveniently described. When evaluating matrix elements, local densities of states are quite different in initial and in final states. For instance, compared to the ground state, the presence of one hole gives rise to a strong screening charge with an-s symmetry in the low-energy part of the valence band. This screening charge represents the onset of a bound state which could separate from the low-energy tail of the valence band if the potential due to the hole were sufficiently strong. No effect coming from this screening charge appears in the experimental emission XL,,, spectra for sodium. However these spectra are in good agreement, at least for their shape, with thedensities of states of the atom in its ground state. This observation is the basic confirmation for the final-state rule given by Von Barth and Grossmann (1979). The same rule would lead, for Auger emission, to the search for a correspondence between the line shapes and the two-hole local density of states of the final configuration. In the general case, for CVV lines, this local density does not result from a simple self-convolution of the one-hole valence-band local density of states due to the correlations between the two holes and to the matrix-element effects. The many-body aspect of the correlations between the two holes has been very well analyzed by Cini (1976, 1977, 1978a,b) and by Sawatzky (1977) and Sawatzky and Lenselink (1980). To present this analysis, let us start from the energy relation
E [ M ( C , C’,C ” ;x)] = €[M(C)] - E[M(C‘, C”; x)]
(95)
which can be written
E [ M ( C , C’,C”; x)] = &(C) - &(CT - EB(C”) + U ( C’, C ” ;X) (96)
208
M . CAILLER, J. P. GANACHAUD, AND D. ROPTIN
u
I
i band states
bound states
Energy (arbitrary units)
(a )
(b)
FIG.10. (a) Two-hole density of states; (b) local two-hole density of states. From Sawdtzky and Lenselink ( I 980); reprinted with permission of the American Institute of Physics.
using simplified notations with respect to Section V,C. In the last expression, U represents the effective interaction energy of the two holes in the final state of the x configuration. The importance of the correlation effects can be checked by comparing the effective interaction energy U to the width W of the x line in the valence band. The narrower the bands (small W ) ,the stronger is this effect. On the other hand, it is weak for crystals with strong covalent bonds such as silicon, for which Coulombic correlations would not be so important, the screening charges being rather delocalized. For U >> 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 << W ) .The shape of the lines is a result of the self-convolution of the one-hole local densities of states (including screening effects) and of the Auger matrix-element effect. For normal metals the screening charge has essentially an s symmetry. It is the same for noble metals. However, for transition metals, the screening could come from an s and a d charge mixing. This s-screening charge reinforces the s partial local density of states in comparison with densities having other symmetries. This can lead to an enhanced contribution from the final s-s, s-p, and s-d configurations. However, the contribution of the s-s configuration remains negligible. According to Jennison, the high-energy-side shoulder of the M,VV line of copper could result from the s-d enhanced contribution, the interaction between the two holes being weak and giving rise to a bandlike line. We should note that Jennison describes the screening of an initial state [here, the 3s-'(d'Os2) configuration], whereas Von Barth postulates a final-state rule. The initial-state screening effect has been reexamined by Jennison et al. (1 980) for beryllium. Simplifying their model somewhat, the screening charge effect was introduced via a scaling factor for the partial densities of states (PDOS). The best fit with experiment was obtained by taking a screening density of a charge of pure s symmetry, which reinforces the s-s and s-p contributions compared to the unscreened case. For lithium, the screening
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
21 1
effect might be important; however, it has little chance to be observed because s-p and p-p contributions have similar shapes and also because they are positioned at the same energy. By considering the final-state rule, Lasser and Fuggle (1980) were led to study the KLV Auger spectra of Na, Mg, Al, and Si. Indeed, the KL,,,V Auger transition of sodium led to the same final state as L2,3V X-ray absorption. In considering a final-state rule, one should find in both lines the same manifestation of screening. For the KL,,,V line it explains the presence of a peak in the low-energy part of the Auger spectrum. This effect is even more pronounced for the KL,V line. The difference in the importance of these peaks can be understood from theoretical considerations involving the angular part of the Auger matrix elements. These considerations indicate that, for the Auger KL,V transition, the final hole in the valence band has an equal probability of having either an s symmetry or a p symmetry. On the contrary, for a KL,,V line, the probability of occurrence of an s-symmetry hole in the valence band is only one-third of that of a p-symmetry hole. Thus, enhancement by screening of the s-symmetry charge in the valence band leads to a stronger effect for the KL,V line than for the KL,,,V line. The case of silicon seems slightly different. Indeed, if for Na, Mg, and Al, the s-symmetry screening charge reinforces the density of states at the bottom of the valence band, in Si it is due to the creation of a bound state below the valence band (at about 5 eV under the bottom of this valence band according to Lasser and Fuggle). It also appears from all the measurements made by Gsser and Fuggle that the concept of a screening resulting from the occupation of one atomic orbital, which was used in Section V,C to evaluate the kinetic energy of the Auger electrons, could be a relatively rough approximation. In fact, the screening effect leads to important modifications of the local densities of states in the entire valence band. Before the Esser and Fuggle observations it was admitted that, for covalent crystals, the screening was rather delocalized and that this effect might not be very important. However, in this case, the shape of the Auger line still does not result from a simple self-convolution of the one-hole local density of states because the Auger matrix element depends on the kinetic momentum, especially on its angular component. Feibelman et al. (1977a,b) advanced this important effect and presented calculations of the L2,VV and L,L,,3V line shapes in silicon by using a Fermi golden rule. The two wave functions of the valence band involved by the calculation of the matrix elements were described by a LCAO approximation, with a basis composed of (3s) and (3p) atomic orbitals centered on the various atomic sites. Owing to the strong spatial localization of the initial hole, it was admitted that the main contributions to the Auger matrix element was due to the very neighborhood
212
M. CAILLER, J. P. GANACHAUD, AND D . ROPTIN
of the hole that is composed of (3s) and (3p) orbitals centered on the initially ionized core (valence local density of states). A study of the selection rules for parity and for angular mementum conservation shows that for the damping of an L2,3hole, 46 channels are open, leading to a final p-p configuration, whereas there are 24 channels for an s-p configuration and only 3 for the s-s configuration. The angular parts of the various matrix elements are proportional to these numbers of channels. Thus, the damping mode leading to the creation of two (3p) holes in the valence band is strongly favored with respect to the others in a L,,,VV transition. However, the “angular momentum” character of the Si valence band varies all along this band. It has an s type near the bottom of the band and a p type near its top. So, the dependence of the Auger matrix elements on the angular momentum acts as an energy-dependent weighting factor on the (ss, sp, pp) convolutions of the s and p local densities of states. Consequently, this causes the presence of an intense peak on the high-energy side of the lines. In the calculations by Feibelman et al. (1977), the ppp nature of the vacancy final states in the Auger transition was favored, which led to a much better agreement between the calculated and the experimental lines. However, the s-p character of the L,,,VV line was still overestimated. In fact, experiments seemed to indicate that the LVV line of silicon in practice arises only from a self-convolution of the p component of the optical density of states. The same observation can be made for lithium and aluminum. Jennison (1978a,b) pointed out some inconsistencies in the theory of Feibelman et al., who used atomic orbitals to evaluate matrix elements, whereas Wannier functions were introduced to calculate the partial local densities of states. Consequently, the reduction of the local atomic charge due to the contribution of part of the charges to the covalent bond was not accounted for. This is due to the fact that the Wannier functions have no binding charge. By reevaluating the partial local density of states for the initial state with an atomic orbital basis, Jennison could show that the (3s) electrons contribute more than the (3p) electrons to the binding charge. The direct consequence was that the contribution of the (3s) states to the local atomic density (which acts, together with matrix elements, to determine the Auger line shape) was more reduced than that of (3p) states. Additional calculations using the Von Barth and Grossmann final-state rule, the screening effects, and the Auger matrix elements effects need to be developed. However, it seems that the problem is now well outlined. Analysis of the lines shapes for X0:- ions (where X represents the elements of the third row of the periodic table) shows that the LVV Auger lines of the X central atom in a tetrahedral environment present a characteristic doublet separated by about 14 eV (Bernett et al., 1977).The origin of this doublet is the splitting of the (3p)states of the central atom into two molecular
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
213
orbitals 4t2 and 3t,. The energy levels of these orbitals are essentially determined by the (2s) and (2p) levels of oxygen. The two lines of the doublet are associated with (2p4t24t2) and (2p4t23t2) transitions. The (2p3t,3t2) transition is also possible, but it has a low intensity. These results have been discussed by Losch (1979) from experimental Auger spectra of sulfur obtained from Cu,FeS, surfaces by segregating the sulfur at the surface of Cu-Au alloys. A method for the quantitative analysis of the surface composition from LVV spectra of sulfur in SO:-, SO23 9 S2anions has been proposed by Turner et al. (1980). They compared the lines measured for compounds of these anions with a combination of the lines calculated for pure materials. The precision obtained was 5% for SO;-, S2compounds, but only 30% for SO:-, SO;- compounds. The lines shapes of silicon and oxygen in silica has also been studied in detail (Ramaker and Murday, 1979; Ramaker et al., 1979a,b; Ramaker, 1980) from the experimental and theoretical points of view. Charge effects and some dissociation due to the incident beam take place in these kinds of targets (Carriere and Lang, 1977; Ichimura and Shimizu, 1979). Grant and Hooker (1976)proposed to analyze the line shape of carbon to distinguish its different chemical states at a surface (graphite overlayers on metallic surfaces and carbides). Gaarenstroom (198 1) adapted the principalcomponent analysis technique to take advantage of the modifications of the line shapes in compounds (when the spectra of the pure components overlap). For such cases, our own opinion is that this method could be very efficient and quite comparable to self-deconvolution techniques (Tagle et al., 1978), although their mathematical bases have not been completely and clearly detailed. We shall return to this aspect in Section VI1,C. VI. QUANTITATIVE DE~CRIPTION OF AUGER EMISSION A. General Description
Let us consider a flow of N p electrons per second impinging upon the surface of the target. A detailed analysis of the incident beam would indicate that just before reaching the surface of the target it is spatially distributed and is shows some dispersion in energy and angle. If no(0) = no(xo,y o , Eo, no) the differential distribution of the primary electrons, one has
214
M . CAILLER, J . P. GANACHAUD, AND D. ROPTIN
In these relations, CEO = (cos do, 4,) where 8, is the incidence angle and c$o the azimuthal angle measured around the normal to the surface. By crossing the surface region, these primary electrons are slightly accelerated and deflected because of the surface barrier of potential. Some of them will be elastically reflected in the very first atom layers, but most of them will penetrate the solid. Owing to the collisions in the solid, the electron beam is scattered and slowed down. Moreover, its intensity varies because of the cascade process. At a mean depth z,, the distribution of the energetic electrons is given by NflzI(l)where n,,(l) is related to n,(O) by r
As a rule, the h( 1,O) function is quite complicated. It must account for the slowing down and scattering of the primary beam. It must also account for the existence of a backscattered electron beam and for the creation of the secondary electrons in the cascade process. Finally, it must account for the roughness of the surface. However, such a description is valuable only if the height of the roughness is small in comparison with the electron MFPs. Clearly, and analytical evaluation of nzl(1) is quite impossible. Only simulation or numerical methods, carried out on a computer, can bring some information about this function. However, a rough approximation of h( 1 , O ) and similar functions, which are introduced in this section in a general way, will be given later. Focusing attention on the ionization process giving rise to the Auger effect, the number of electrons scattered by ionization of atoms A at the depth zl, is, per unit path length,
with VJl)
= N A ( Z 1 )[n,,(l)/cos
61 1 4 C ( E , )
(103)
In Eq. (103),oX,,(E,)is the total ionization cross section for the subshell C of an atom A. This cross section depends on the energy of the ionizing electron. The d z , cos-' 8, term accounts for the effective distance covered in a layer of thickness dz, by the electrons propagating in the direction n,(cos el, 4, ). The N , term is the numerical density of atoms A. For the sake of simplicity, it has been tacitly assumed that NAonly depends on the depth z1 (the sample is laterally homogeneous).
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
215
Now, it is assumed that after ionization, the atoms A can create Auger electrons via a transition denoted as ( A , x). At a depth z , , the creation rate of these Auger electrons is, per unit length, dNt;"/dz,
=
J d(2)vtiX(2)
Np
where vti"(2) can be deduced from v,, (1) by using a general equation similar to Eq. (101) but extended in this case to five variables: I-
I-
Considering the whole generality, the function h ( z , , 2 I z l , 1) is certainly very complex, but approximations of it are given later. Among the N,vt;"(2) dz, Auger electrons created in a layer of thickness dz, at depth z 2 , only some of them reach the free surface and escape into the vacuum. Thus, their distribution just after they enter the vacuum would be given by rw r where the subscript zero recalls that this distribution is obtained near the plane z = 0. It should be noted that in the distribution n$9"(3) most of the Auger electrons created in the solid have lost their characteristic energy in inelastic collisions. In fact, there are three domains in the energy distributions of these Auger electrons. The first domain corresponds to the characteristic Auger energy and is occupied by the true Auger electrons. The second domain ranges several electron volts under the characteristic energy. It corresponds to the energy-loss spectrum of the Auger electrons. Finally, the third domain is very far below the characteristic Auger energy. The energy distribution of the Auger electrons in this domain is flat and is hardly separated from the background of the secondary electrons. Depending on the analyzer, all or only a part of these electrons enter the collector, and at the spectrometer output they have a distribution nAv"(4) given by nA3"(4)=
s
d(3)h(4, 3)n$*"(3)
(107)
The function h(4, 3) accounts for the capture of the electrons by the collector, but also accounts for the transmission function of the spectrometer and for
216
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
the broadening due to the voltage modulation (in the derivative mode). At the spectrometer output the energy distribution of Auger electrons is NA3"(E4) =
s
dx4 dy4 da4nA*"(4)
(108)
For practical analyses, one has to simplify the preceding general description. These principal simplifications are now reviewed. Throughout the presentation, all unnecessary subscripts are dropped; they are retrieved in the final result. (1) Let us suppose that all the primary electrons which make up the incident beam have the same energy Epand the same direction of propagation so that
(2) For simplicity, we neglect the acceleration and the deviation caused by the potential barrier. For the case where the primary electrons are of high energy, this approximation is good. The elastic reflection can be taken into account with the help of a transmission coefficient T < 1 which depends on the surface topography and on the parameters Epand $2, of the primary beam. (3) The effect of the roughness of the surface is important at both the entrance and the exit and is usually taken into account by a multiplying factor on the intensity of the Auger peak. This effect is not well understood and deserves much further research. Here we consider it to be included in the factor of transmission T , at least concerning its effect on the primary beam. Therefore, 8, is just an average incident angle. (4) Inside the target, it can be assumed that there are two electron beams: the primary beam and the backscattered electron beam, whose contributions to the creation of Auger electrons can be added. In the first step, the contribution of the backscattered electrons is omitted. Then, at the end of the calculation it is introduced in the form of a correction factor to the result. If we neglect the effects of coherent diffraction, of incoherent scattering, and of slowing down of the primary beam in the target, h(1, 0) becomes a simple product of Dirac functions, thus,
217
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
with P
Nevertheless, by introducing Eq. (111) into Eq. (103) and then introducing the result into Eq. (102), we have (1 13)
dNz,/dzl = NA(zl)(c/cOs d,,)N,,T
( 5 ) After ionization in the subshell C, the probability of the reorganization of atom A taking place by Auger transition, denoted x, is yA,c,x. Generally, yA,c,x is not the complement to unity of the fluorescence yield. The difference is due to the contribution of the Coster-Kronig transitions and of the possible cascading and filling up of the holes in the ionic core. If several simultaneously possible channels exist, then their contributions to the creation of the Auger electron can be summed up to give Ec c ~ ~ & , ) y , . , ~ , ~ . However, for simplicity we drop the summation symbol in what follows. Usually, we consider that the Auger transition is local; in other words, it takes place where the ionization has occurred. For transitions involving the valence band, this assumption can be questioned. We can assume that an overall compensation occurs in the bulk of the solid, giving validity to the hypothesis of locality. However, at the surface, because of the loss of symmetry, this assumption is more doubtful. However, this reservation is disregarded. Thus, using the hypothesis of locality and also separating the spatial and the angle-energy parts, we can write
w, 1) = with
s
Consequently,
dN$;"ldz2
Y*,C,xw.2
- rl )C(E,,
a,)
dE2 da2C(E,, 0,)= 1
=
YA,C,XC(~~ n2)NA(z2)[Tnz,(x2 Y2)/c0s
=
N,
j
3
d(W;;"(2)
5
ep]aC
eP)%
=Y A , C , X ~ A ( ~ Z ) ( ~ ~ ~ / ~ O ~
(6) By integrating Eq. (116) over x2 and y,, one obtains V 3 E 27
a,) = $;"C(E, a,) 9
with = ~ A , C , X N A ( ~ep)aC ~)(~/~~~
218
M. CAILLER, J . P. GANACHAUD, AND D. ROPTIN
In the usual quantitative description of Auger electron emission, it is tacitly assumed that the distinction between the true Auger electrons (i.e., those that have kept their characteristic creation energy upon reaching the sample surface) and the inelastically scattered Auger electrons can be made unambiguously. Then, assuming that these true Auger electrons emitted in the direction of the solid surface reach it after following a straight-line path along which they are exponentially attenuated, their angle-energy distribution just before the surface is
In this expression, no,= refers to an internal distribution (denoted by the subscript i) in the neighborhood of z = 0 and considered for one primary electron. (7) If one takes into account the effect of the potential barrier, Eq. (120) represents the internal distribution of the true Auger electrons. From this, one can derive the external distribution with the help of nA,x
o,T
a,1 =
(Ee,
f
dEi dQh(e, i)n2,(Ei, Q )
(121)
where the subscripts e and i represent external and internal, respectively. For a rough surface the function h(e,i) is, a priori, very complex. For a smooth surface, for simplification we have h(e, i) =
-
$i)b{cos 8, - [ ( E cos’ ~ Oi - w ) / E , ] ~ ’ ’ }
x 6(Ei - E, - W )
(122)
We usually neglect this correction and assume that Eq. (120) also represents the external distribution [denoted by the subscript 3 in Eq. (106)l of Auger electrons. Nevertheless, this omission is less justified for Auger electrons of low energy than for primary electrons. For a rough surface the result is usually corrected by a simple multiplying factor as indicated above. Logic dictates that the correction factor must be a function of the surface roughness and of the Auger electron energies and their angular distributions: We denote it as T ’. (8) Frequently, the MFPs involved in the energy range of Auger electrons are very small so that the exponential term in Eq. (120)decreases from 1 (for z2 = 0) to negligible values within a thickness so narrow that the variation of the concentration NA(zZ)can be neglected in this domain.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
219
Calling NA the density of atoms A near the surface, one obtains Npn>”(E3,03)
=
y(NpT/cos 8,)aT‘C(E3, n,)NAn(E3) cos 83 (123)
Guglielmacci and Gillet (1980) evaluated the corrections to the atomic density NAfor the case of a binary alloy A-B when the concentration gradient in z cannot be neglected. If N g is the mean density of the atom A in the escape zone of the Auger electrons, and N,(z) and C,(Z) are the actual density and concentration, respectively, then N , and N,(z) are related by
NZ
- NA(z)
= (NgLA,”
+ Ng&,y)(dCA/dz)
( 124)
where jlA,”, J.B,y are the escape depths associated with the Auger peaks (x for A atoms and y for B atoms). In the case of a linear variation of the concentration with z and if LA = A,, the correction would be simply written as NA(Z) = N T ( z - A)
(125)
(9) The total primary-electron contribution to the true Auger electrons emitted into the vacuum is obtained by integrating Eq. (123)over R, and E, . It is recalled here that the total number of Auger electrons leaving the target is obtained from Eq. (123) after correction by a backscattering factor. In principle, Eq. (123) would give the angle-energy distribution of the Auger electrons at the surface of the solid. However, it is necessary to keep in mind that this equation was derived from the assumption of an exponential attenuation of the Auger electrons along their straight-line escape paths. In other words, it is assumed that there is no broadening of the natural Auger line in the sample. Under these conditions, the natural line shape of the Auger transition is obtained by integrating Eq. (123) over R,. Clearly, the omission of the sample broadening is not realistic in studies of line shapes, and Eq. (123) must be corrected to take account of this broadening. Deconvolution procedures have been proposed to find the original natural line shapes from the experimental shapes. They rest on estimations of the so-called sample transmission function and of the spectrometer function (Ramaker et al., 1979) and are reviewed in what follows. (10) Another point that is disregarded in Eq. (123) is the spatial distribution of the Auger electrons. The effect of such a spatial distribution could be important in an absolute measurement of the Auger peak intensity in relation to the capture of the Auger electrons by the collector. For relative measurements, the problem is certainly less important. However, Holloway and Holloway (1977) have shown that for a target of pure gold the ratio of the intensities of the low- and high-energy Auger peaks depends on the size of the primary beam. For a scanning analysis the effect was found to be as large
220
M . CAILLER, J. P. GANACHAUD, AND D. ROPTIN
as 30%. They also mentioned that a misalignment of the primary beam spot and the analyzer focal point affects the peak intensity ratio. It is well known that compensation of the electromagnetic fields is indispensable for measuring well the low-energy electron distributions. Holloway and Holloway have suggested that the variations of the peak intensity ratios that they observed were due to the electromagnetic field effects. The question of the influence of the axial position of the sample on Auger measurements was considered by Sickafus and Holloway (1 975). They reported that the apparent energy and the intensity of an Auger peak are sensitive functions of the sample position. They also showed that this dependence is basically the result of the finite size of the exit aperture of the CMA. Despite these instrumental difficulties, we suppose that the angle-energy distribution of the true Auger electrons is given by Eq. (123). (1 1) Moreover, it is assumed that this distribution is not perturbed along the electron path from the target to the collector. Therefore, the number of Auger electrons which are collected by the analyzer is
s
N;,” = Np dE3 dQ3n$7”(E3,Q,)
where R, is the acceptance angle of the collector. Incidently, it can be seen that the anisotropy effects can considerably affect the measurements. In the present simplified description, these effects Then, they arise only from the are included in the function C(E,, a3). creation process. However, it is clear that anisotropy effects can originate from the transport process of the Auger electrons from their place of creation to the surface. These effects are important for monocrystalline targets. (12) For polycrystalline samples, the anisotropy effects can be neglected so that w
39 Q 3
1 = (474- C(E31
(127)
If, moreover, the Auger line is sufficiently narrow to allow any variation of the MFP within this energy domain to be neglected, practically, F
where EA,” is considered the characteristic energy of the infinitely narrow Auger line.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
22 1
Then, Eq. (126) is written
(13) For a four-grid analyzer, integration over $2, concerns a cone with an aperture 0, and N?'x
=
(4)-'yA,C,xNP(~'/cosOp)$A,,(E,)NAn(EA,xX1 - cosz e C )
(130)
For a CMA the integration now concerns a cone of revolution with an aperture AO, for 0, and the integral in Eq. (129) becomes
s
s
Bc + W c / 2 )
cos 8, d$2,
= 2a
8c - (ABcIZ)
cos 0, d(cos 0,)
(131)
For a narrow aperture A@-, Eq. (131) becomes
s
cos 8, d$2, = 7c AO, sin 20,
Dejinition of the Backscattering Factor
In the very simplified description developed above, the number of Auger electrons N;," of the (A, x) line collected by the detector is given by Eq. (129). An actual measurement of this number gives a very different result that can be written N;,"(exp). In fact, this difference comes from several origins. However, it is quite generally admitted that the main reason lies in the fact that we have neglected the contribution of the backscattered electrons to th? creation of the Auger electrons. To make this description more realistic, after Eq. (131) has been evaluated, one multiplies the right-hand member of this relation by a multiplicative I~,,~,(E~,,, called the backscattering factor. To estimate factor [l this backscattering factor, let us return to the general formalism. The combined action of the penetrating primary beam and of the backscattered electrons (among which we shall include the secondary electrons of the cascade) results in the substitution, for the various functions n and v used in the above calculations, of new functions in which the backscattering effects are included. We denote them by a prime symbol. Thus, for instance, Eq. (1 11) becomes
+
&)I,
nk1(1) = nz,(x1,Yl)Wl- Ep)6($2,- a,)+ n:,(l)
(133)
where n,: (1) is the distribution (for one primary electron) of the backscattered electrons.
222
M. CAILLER, J. P. GANACHAUD, A N D D. ROPTIN
A calculation very similar to that developed above leads to the replacement in Eq. (126) of TT'olcos 0, by
where P
is the integral of the second term of Eq. (1 33) evaluated near the free surface of the solid. Practically, nB(E,,0,) is the angle and energy distribution of the emitted backscattered electrons (for one primary electron) and can either be experimentally determined or estimated, for instance, by a simulation method. The backscattering factor is defined by the relation 1
+r =N~A9"/N~~"
( 136)
and has the value
B. Theoretical Analysis of'Auger Emission 1. Introduction
The purpose of this section is to link together the theoretical aspects developed in the previous sections and some more practical considerations necessary for a quantitative analysis of the Auger emission of a solid target. A quantitative Auger analysis is obviously based on interpretation of experimental results via a manageable model for the creation of Auger electrons in the target and for their transport toward the free surface. Accordingly, one needs a theory that allows (1) an estimation of the parameters which appear in the usual formulations of the quantitative analysis (MFP, backscattering factor, etc.); (2) a possibility of gaining some insight into the validity of some assumptions currently admitted for the "hidden" variables of Auger emission (depth dependence of the Auger source functions, escape process of the Auger electrons, etc.).
This is the type of problem more generally encountered in secondaryelectron emission. When one considers the emission of a pure metal, in some
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
223
special instances, the elementary processes are sufficiently well known to allow either an analytical treatment or a Monte Carlo simulation method. By analytical treatment we mean the resolution of a Boltzmann-type equation, including (in principle) all the collision effects. This kind of approach has been used by several authors, and has given very interesting results (see, for instance, Chung and Everhart, 1977; Bindi et al., 1980; Schou, 1980; Rosler and Brauer, 1981a,b for the most recent accounts for this approach and all the references to previous work on the same subject countained therein). However, it seems that analytical treatments do not represent the best way to obtain all the important information one must expect from a sufficiently precise model of Auger emission if one wants to have reasonable computation times. In our opinion, the complexity of the calculations required by practical problems (such as the emission of alloys or of compound materials) is such that the Monte Carlo method represents the most efficient technique. For instance, in one calculation it allows detailed statistics to be obtained about the individual events which occur to cause Auger emission, by suitably marking their separate interventions in a computer program. This kind of flexibility is difficult to obtain with an analytical method. 2. Some General Features of’ a Monte Carlo Simulution Method A primary beam impinging in a solid target suffers elastic and inelastic collisions with the components of the solid. The effect of the inelastic collisions is to bring the electrons of the solid to the upper levels so that they can themselves take part in the transport process. This is the cascade effect. In a study of Auger emission, one must describe more precisely the ionizing collisions that are caused by the primary electrons which penetrate the target or those that are elastically turned back towards the free surface. In addition, we must follow those which are inelastically backscattered and, more generally, all the electrons of the cascade which maintain an energy above the ionization threshold of the atomic subshell in which an Auger hole can be created. All these processes can be incorporated a priori into a Monte Carlo simulation, and the main aspects of this method are now briefly recalled. The Monte Carlo method is based upon the statistical concept of a trajectory for the particles which take part in the transport process in the solid. A trajectory is a sequence of free paths si separated by interaction “points.” During a free path, the particle propagates along a straight line, keeping its classical parameters (energy and velocity) unchanged. This behavior is modified by collisions with other components of the solid which restrict the amplitude of this free path. In practical calculations, one admits that the effects of these interactions can be located at some points where the
224
M . CAILLER, J . P. GANACHAUD, AND D. ROPTIN
angle and energy characteristics of the particle suffer their accidents. Most frequently, these are the end points of the free paths, and the accidents determine completely the random concatenation of the rectilinear portions of the full trajectory. With the assumption of a linear response of the solid to the excitation provoked by a primary electron, each primary particle is followed individually, as well as the electrons of the cascade which it generates. In principle, calculations with a Monte Carlo simulation method require theoretical knowledge of all the cross sections associated with the various types of collisions. Whenever these requirements are fullfilled, a direct simulation is possible. However, for practical problems, the complexity is such that one has to accept less elaborate but more efficient descriptions. Except in a close neighborhood of the surface, the solid is considered to be a homogeneous medium described by a randium-jellium model. The delocalized electrons of the jellium are assimilated to an electron gas. The other component of the solid is thus composed of the ionic cores. For the sake of simplicity, one assumes that they are randomly distributed, according to a uniform background model (randium). This, of course, precludes any direct treatment of the diffraction effects. Under these conditions, the propagation of an electron in the solid becomes a stochastic process, and one generally admits that the collisions it suffers can be described by a Poisson distribution law. Except near the surface, the total MFP 12, depends only on the energy E of the electron but not on its position or its direction in the solid. This MFP &(E) is deduced obtained by summing over from the total cross section (oT(E)= (Nat,IT(E))-' the elementary cross sections associated with the elastic and the inelastic processes a, = oel+ Gin. The probability of an electron having a free path in the [s, s + ds] domain between two collisions is given by P(s)ds = 12;le-sfATds
(138) This, in principle, allows the determination of the extent of the different straight-line portions si which constitute the zig-zag path of an electron in the target. As already indicated, ,IT is most frequently imprecisely known. However, the universal curve described in Section JII,F indicates that in the energy domain concerned by Auger analysis, AT is small, of the order of a few angstroms. This means that a primary electron can suffer an important number of collisions before its energy falls below the ionization threshold of many Auger lines of practical interest. Because the description of an individual collision is itself a rather complex problem, the computational times may quickly become prohibitive.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
225
Another approach is to describe some condensed histories for the particles and to make use of the so-called continuous slowing-down approximation. It has been extensively used in electron microprobe analysis (see, for instance, Bishop, 1976, or Shimizu et al., 1972, for reviews on this subject). The energy lost by an electron per unit path length in a pure element solid target is most frequently represented by the Bethe expression: dE/ds = -2ae4(N,,Z/A)E-'
ln[1.166(E/J)]
(1 39)
where N,, is the atomic density of the target, 2 its atomic number, A its atomic weight, E the energy of the electron at a given stage of its history, and J is a mean ionization potential for which several analytical expressions have been proposed (for example, Shimizu et al., 1972). To describe a condensed history, a simple solution could be to choose for each si an arbitrary constant value, not too small, however, in order to reduce the computation times. During si, the electron thus generally undergoes several collisions, the mean number of collisions being
Pi
= (pNat/A)dE)si
(140)
The individual atom cross section has often been taken to be of the Rutherford type and it may include, through simple approximate expressions, both elastic and inelastic collisions (Reimer, 1968). Under such conditions, the angular deviation aspect must be treated by a Lewis-type multiplescattering theory, that is, by the relation
where a, is itself a Legendre coefficient of the individual cross section. Additionally, one may assume that the energy transfers, deduced from Eqs. (139) and (141),and the angular deviations are in a practical simulation simply located at the end point of each si segment. Some improvements have been proposed (for instance, Shimizu et al. 1972,1975)either by choosing si in order to keep pi approximately constant for each segment, or by locating the angular deviation at an intermediary point of si. For several reasons, which will become clearer after the analysis of the Auger emission of aluminum presented in Section IV,B,3, the condensedhistory approach is more suited to the analysis of the penetration of the primary beam than to the intricate problem of Auger emission. Thus, the end of the present section is devoted to the presentation of the direct simulation method. This method can be used mainly for standard reference calculations. Beyond its ability to give much data about the emission of a few solids, its secondary role is to allow a judgment as to the validity of other methods, less
226
M. CAILLER, J . P. GANACHAUD, AND D . ROPTIN
precise but much more efficient for complex targets, by applying both methods to the standard cases. This is discussed in Section IV,B,4. In a direct simulation method, the free path s(E)followed by an electron between two successive collisions can be calculated from Eq. (138) by where ys is a random number uniformly distributed between 0 and 1. An elastic collision represents a global interaction process with the potential field surrounding the ionic cores. An inelastic collision can result from various effects (individual collisions with the electrons of the jellium, collective excitations of the electron gas, ionization of the inner shells), for which the energy losses and the angular deflections follow too dissimilar laws to remain, a priori, undifferentiated. The cross sections o,,,,cre,pl, and correspond to these various inelastic effects. By drawing a random number ys, the free path is determined according to Eq. (142). Knowing the initial position and direction of a simulated electron, this fixes the location of the collision. To know which type of collision occurs, one must use a new set of random numbers. For instance, if the number y1 selected is such that Oel/OT
< Y1 <
(143)
the collision will be an elastic one. The next step is to analyze in more detail the angular deflection 8 suffered by the electron and thus to fix the new direction of propagation of this particle after the collision. This is done by drawing a new random number y2 ; 8 is determined from the differential elastic scattering cross section o,,(E, 0) by using the relation
and by evaluating for which angle 0 this relation is satisfied. For an inelastic collision which can put into play several elementary mechanisms, the occurrence of a particular process is determined by a method analogous to that of Eq. (143), that is to say, in proportion to the probabilities associated with the various mechanisms. By a method similar to that described via Eq. (144), new sets of random numbers allow us to define the energy ho transferred in the collision and the corresponding momentum transfer kq by using the differential cross sections o(E,q, 0). In inelastic collisions, electrons of the solid reach excited states and have to be followed themselves by the same simulation method. For a study of Auger emission, all electrons which have an energy lower than the ionization threshold of the Auger line investigated can be forgotten. This allows us to
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
227
set aside many important aspects in a general secondary-emission study (plasmon damping, for instance) and to gain a substantial amount of computation time. The next section is devoted to the application of the direct simulation method to the LzJVVemission of aluminum. This is done for the purpose of checking some assumptions and of evaluating some hidden parameters. More precisely, we shall try to give some information about the source function of Auger electrons by delimiting the respective contributions of the primary and the backscattered electrons. The depth dependences of these contributions are studied. Auger-electron escape is also investigated. This should clarify the participation of the backscattered electrons in the Auger yield (backscattering factor and variations of this quantity with the depth). In order to keep computation times reasonable, the Auger process itself has been extremely simplified. In our model, Auger electrons are created at one energy (that of the main Auger line, ignoring its linewidth), and the creation has been assumed to be isotropic. The same kind of simplification has been retained for the angular distribution of the electrons ejected in the initial stage of the ionization process. Because Auger emission is, in fact, located in the near subsurface of the solid, we shall briefly recall the assumptions that we have retained for the surface region. As discussed in Section 111,a simple assumption is to consider that in a thin layer of the solid (with an extension a, of typically 1 A), the bulk collective excitations are replaced by surface excitations, all the other types of collisions remaining unchanged. This surface layer has also been extended outside the solid (its thickness being a, 5 a i )to account for the possibility of surface plasmon excitations outside the solid. In this spatial domain it was assumed that this type of excitation was the only possibility for a particle to suffer energy losses. 3. Results Obtained f o r Aluminum by the Simulation Method
a. Elastically ReJected Primary Electrons. As indicated above, diffraction effects have been ignored in the present model. Thus, the elastically reflected primary electrons are those which have suffered one or more individual elastic collisions but no energy loss. The total angular deflection due to these processes must be sufficient to turn them back towards the free surface. The number of these particles per primary electron is defined as the elastic reflection coefficient. In the energy range considered in the present section (0.1-1 keV), this coefficient is of the order of a few percent. It decreases as the primary energy E, increases. This tendency is all the more pronounced as the primary electrons reach a normal incidence. The overall behavior of
228
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
-w
n
F
0.025
L
500
1000
Ep(aV)
FIG.12. Elastically reflected primary beam, influence of the primary energy and incidence angle on the reflection coefficient: tIp = 60" (V),45"( a),0" (m).
this reflection coefficient can be approximated by a (cos tl,)-a law, where 0, is the incidence angle and ci is a positive exponent which depends on E,. It increases with E, and is nearly zero at low primary energies. These results are illustrated in Fig. 12. The inelastic collisions are frequent for all the energies considered, so the
FIG.13. (a) Differential contribution n(E,, z) of the different atom layers to the reflection coefficient ( E , = 700 eV); (b) integrated contribution dz n ( E , , z ) ( E , = 700 eV).
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
229
primary electrons which are elastically reflected originate at the near subsurface of the solid. This maximum-penetration depth results from a balance between two processes which are strongly energy dependent. As indicated in Fig. 13, for a normal incidence, 50% of the elastically reflected primary electrons originate from the first three angstroms for Ep = 700 eV, whereas for Ep = 150 eV, it is 80% (Fig. 14). Let us also remark that for Ep = 700 eV, 40% of the elastically reflected electrons have suffered one collision, 25% two collisions,207; three collisions, and 15% more than three collisions; whereas at Ep = 150 eV, these percentages become 60, 25, 7 and S%, respectively. b. Creation of Auger Electrons in a Solid. In the present calculations, only electrons coming from the L,,,VV line of aluminum have been considered. These Auger electrons have been created by primary incident electrons and by those which have been backscattered in the solid. In this latter class we include all the electrons with an energy between (E, - 1) eV and the ionization threshold of the L2,3level. The separate contributions of these two groups of electrons to the dependence of the Auger source function on the depth z have been presented in Fig. 15a for Ep = 700 eV and a normal primary incidence. Primary electrons clearly do not represent the main contribution to the Auger current. The corresponding source function decreases rather strongly 0.0 :
OD 2
0.5
0.0 1
FIG.14. (a) Differential contribution n(E,,, z ) for E,, = 150 eV; (b) integrated contribution for E, = 150 eV.
230
0.0 2
1
M. CAILLER, J. P. GANACHAUD, AND D . ROPTIN
(Ep> z ,
0
50
0
50
100
100
ztii,
(b)
FIG.15. Creation of Auger electrons in Al: (P) contribution of the primary electrons; (B) contribution of the backscattered electrons; (T)total; (a) E,, = 700 eV, 8, = 0";(b) E,, = 700 eV, ep = 45";(c) E,, = 150 ev, e,, = 0".
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
23 1
FIG.15. (Continued)
with the depth z due to the inelastic collisions and it is in fact rather concentrated near the surface, with a quasi-exponential decrease. Assuming a true exponential law of the form S,(z) = Sp(0)e-'/L;L is found to be about 15 A for this set of primary parameters. For an incidence angle 0, = 45" (see Fig. 15b), the general behavior of the Auger source function is the same. The contribution of the primary electrons can still be roughly approximated by an exponential law with an exponential parameter of about 10 A. This is quite consistent with an overall exp( - z / L cos 0,) behavior. Creation of Auger electrons by the backscattered electrons takes place for the most part in the bulk. For Ep = 700 eV, the high-energy electrons can suffer several elastic and inelastic collisions (including ionizations) before they fall under the Auger ionization threshold. The corresponding source function presents a maximum which for 0, = 0" is located at about 30 8, from the surface. For 0, = 45", this value becomes 20 A. This seems to indicate that the position of the maximum still varies roughly like (cos 0,)-'. Thus, at least in the domain of sufficiently high values of E,, the backscattered-electron contribution is representative of a rather general law for the stopping power in the bulk of the solid. For Ep = 700 eV, this contribution of the backscattered electrons is by far the strongest (90% of the total Auger electron creation) and dominates the shape of the Auger source function. However, the Auger electrons (which are those of the L,,,VV transition) have a low initial energy, and the major part of them can not be emitted unless they originate at moderate depths.
232
M. CAILLER, J . P . GANACHAUD, AND D . ROPTIN
Consequently, in Auger emission, the primary-electron contribution is approximately equal to that of the backscattered electrons. At lower energies (E, = 150 eV for Fig. 15c), the contribution of the inelastically backscattered electrons is strongly reduced. That of the primary electrons also decreases but relatively less. Thus, at low energies, there is an equilibrium between these two contributions. So, the total source function is strongly decreasing near the surface. This shape is principally due to the primary-electron contribution. For the emission, both groups of electrons, coming from similar depths, will still have rather comparable contributions. c. Emission of Auger Electrons. In this section, we shall focus our attention upon Auger electrons emitted at the free surface without having lost too large a fraction of their initial energy along their path in the solid (less than 5 eV, for instance). They can still contribute to the shape of the L,,,VV line located at about 67 eV above the vacuum level. The other electrons of the Auger source function which are emitted in the “background” have not been considered.
iiOOli
FIG. 16. Contribution nA(z) of the different atom layers to the emission of the Auger electrons in A1 ( E , = 700 eV, 0, = 0”): (a) total emission n,(z); (b) contribution of the primary and backscattered electrons; (c) relative integrated contribution [nA = S$dzn,,(z)]; P, B, T as in Fig. 15.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
233
Because the absorption law is nearly exponential, only Auger electrons created near the surface can reach it with no energy loss. In our model, we have neglected the initial Auger linewidth as well as the electron-phonon collision mechanisms, so that only a very small fraction of the internal Auger electrons can reach the surface with an energy loss lower than 5 eV. Thus the width of the corresponding external Auger line obtained by a simulation method is in fact very small. This approximation is not very severe if we focus our interest on the intensity of an Auger line and not on its precise shape. Figure 16 shows that for a primary energy E, = 700 eV and at normal incidence, 90% of the emitted Auger electrons come from the first five angstroms. The primary- and backscattered-electron contributions have nearly the same values. The total Auger yield is 0.75%. It becomes 1% for the same primary energy but for an incidence angle of 45" because the creation mechanism has then been reinforced near the free surface of the solid. As shown in Fig. 17, for a primary energy Ep = 150 eV, the Auger yield is nearly 0.56%, but in this case, 70%of the emitted Auger electrons have been created by the primary beam and only 30% by the backscattered electrons. The dependence of the total Auger yield on the primary energy is represented in Fig. 18 where reduced variables have been used. The reducedenergy variable is (E, + 6)fj L2,3 where Epis the kinetic energy of the primary
0.00'
1
c
z(X) 45
40005
0
(b)
FIG. 17. Contribution nA(z) of the different atom layers to the emission of the Auger electrons in Al (E, = 150 eV,8, = 0"): (a) total emission nA(z);(b) contribution of the primary and backscattered electrons; (c) relative integrated contributions [nA = J;dzn,(z)]; P, B, T as in Fig. 15.
234
M . CAILLER, J. P. GANACHAUD, AND D . ROPTIN
electrons in the vacuum, Cp = W - EF is the work function of the target, and EL,,3is the ionization energy of the L2,3 level (73 eV). This Auger yield is compared with the ionization cross-section curve of Gryzinski plotted versus the same reduced-energy variable. These two curves have been fitted at a common reduced energy ( ~ 4 . 8 5 )corresponding to Ep = 350 eV, the Gryzinski curve itself being normalized in such a way that its maximum amplitude becomes unity. One can remark that the reduced Auger yield estimated by the simulation method for several primary energies shows rather good agreement with the Gryzinski curve. There are no a priori evident arguments to predict this; however, some reasons can be found from the fact that the creation rate of the Auger electrons is not strongly z dependent near the free surface of the solid and that the true backscattering factor varies rather moderately with Ep,at least in the energy range considered in our study. By a true backscattering factor we mean the ratio of the true number of Auger electrons escaping from the solid and the number of Auger electrons emitted after they have been excited by a primary electron. This definition differs from the usual definition of backscattering factor, which one can denote as the effective backscattering factor. This latter concept comes from several assumptions (for instance, a uniform spatial distribution for the Auger electrons created by the primary beam and also by the backscattered elec-
0
5
10
15
Epi 1 En1
FIG.18. Dependence of the normalized total Auger yield nA/nA,msx on the primary energy: ( A ) primary electron contribution; ( 0 )total.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
235
trons, at least in the region of the solid which can contribute to the Auger emission). If instead of the total Auger yield we consider the partial yield due to the primary electrons only, and use the same normalization conditions as indicated above, then agreement with the Gryzinski curve is expected to be better still. Thisis precisely the case at low primary energies.For higher values of E , ,some slight discrepanciesappear, but the overall agreement is still good. This discussion seems to indicate that an experimental study of the variation of Auger yield with E, might make it possible to measure the ionization cross sections of a target and to compare them with their theoretical estimations. The variation of the Auger yield with the incidence angle of the primary electrons has been represented in Fig. 19a for several values of E,. For relatively high energies, this yield varies as (cos O J l . However, for lower Ep values, its variation is not so rapid. This distortion from the (cos O,)-' law has its origin in the contribution of the primary electrons (See Fig. 19b), whereas that of the backscattered electrons seems to fit this ideal law rather well whatever the primary energy (Fig. 19c). The reason for the distortion from the ideal law which appears in the primary-electron contributions has to be found in the reduced penetration of these electrons. In fact, a more detailed analysis of the influence of the incidence angle on Auger electron creation rate, made layer by layer, reveals a more subtle behavior. For instance, at E, = 150 eV, in the first two or three angstroms, the creation rate of the Auger electrons by the primary electrons increases with e,, and the ideal (cos OP)-' law is well followed. On the contrary, beyond a depth of 5 or 6 A, this creation rate becomes weaker at a 45" incidence angle than at normal incidence. Because this spatial domain is still within the escape depth of the Auger electrons, the variation of the Auger yield with 0, is not so rapid as that for high energies. These results, obtained by the simulation method can be compared, at least qualitatively, with the experimental observations made by Matsudaira and Onchi (1978) for nickel. In order to make such a comparison, one has to think in terms of reduced energies. This can approximately be done by comparing results corresponding to identical values of the ratio EA,../E,, where EA,x is the Auger electron energy and E, the primary-electron energy. Matsudaira and Onchi (1978) presented results for EA,.. = 61 and 850 eV and for a primary energy Ep = 2000 eV. This leads to the values 0.0305 and 0.425 for EA,../E,. For the L,,,W line of aluminum, EA,.. = 67.5 eV, and for the primary energies, Ep = lo00 and 150 eV, the ratio is EA,/E, = 0.0675 and 0.45. The results presented in Fig. 19d correspond to the curve n(d,)/n(O), obtained for E,,,/E, = 0.45. This curve, plotted versus O,, increases much less rapidly than (cos O P ) - l , For instance, at 0, = 60", the value 1.52is reached,
236
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
0.5
cos
e,
0.3
1
cos
0
(C)
ep
(b)
(0)
cos 8,
1
I
3 0 '
S'O"
9'0"
8, (d1
FIG. 19. Dependence of the Auger yield on the incidence angle: (a) total emission; (b) contribution of the primary electrons; (c) contribution of the backscattered electrons; (d) curve n(O,)/n(Oo)for the total emission; E, = (A)150 eV, (v)350 eV, ( 0 )700 eV, (*) 1000 eV.
237
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
'
whereas (cos gP)- = 2. The same behavior appears for the experimental results on nickel for EA,,/Ep = 0.425, whereas for 9, = 60°, the function takes approximately the value 1.6. On the other hand, for low EA,,/Ep values, the (cos OP)- ' law is followed well in both cases up to 0, = 60". Some disparities appear in the case of nickel, but they correspond to grazing incidences which have not been simulated for aluminum. The fact that in the first few angstroms the ideal law is satisfactorily followed, even when at low primary energies important distortions appear for greater depths, suggests that the same behavior could occur for an adsorbed layer. Bui Minh Duc (198la) envisioned this hypothesis to explain the results of Matsudaira and Onchi (1978). These authors have experimentally observed that a monolayer of sulfur deposited on nickel reinforces Auger emission according to a law which is very close to (cos 6; ), 0, being the incidence angle. An Auger transition characterizing the sulfur monolayer can be induced by electrons backscattered in this layer or in the bulk of the nickel substrate, but for these groups of electrons, the ideal law is correctly followed. The primary-electron signal, itself confined to moderate depths, presents in this case quite identical variations. Thus the simulation method seems to confirm the set of experimental results obtained by Matsudaira and Onchi, at least qualitatively, because the material that we have simulated is not the same as those they have experimentally studied. d. Buckscattering Factor. One of the most important parameters in the quantitative analysis of Auger emission is the backscattering factor or, more precisely, what we have called the effective backscattering factor. This quantity is frequently expressed as a function of the ratio U-' = En,/Ep, where En, is the binding energy of the nl subshell. The results do not differ significantly from one author to the other (Bishop, 1967; Jablonski, 1978; our own results are presented in Fig. 20). Additional results obtained by the present authors (but which have not been reported here), indicate that the backscattering factor is nearly independent of the incidence angle, at least at high energies. However, as can be seen from Fig. 20, the backscattering factor is not simply a function of U - but it also depends separately on the value of the primary energy E p ,although the differences between the curves obtained for different Ep values never exceed 10% in the energy range studied. We should also remark that this backscattering factor differs more strongly from the true backscattering factor defined as the ratio of the total Auger yield to the partial yield due to the primary electrons only. This factor cannot be rigorously defined, for instance, by an experiment. However, a simulation method allows one to estimate it, provided one can distinguish
'
',
238
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
2
-+ l.5
I ’
U-’
-
FIG.20. Dependence of the backscattering factor on the primary energy and on the ratio
U-’= E,,/E,, (Epi= Ep + W - E F ) . ‘B’ refers to Bishop’s (1967) results, and ‘J’ refers to Jablonski’s (1978) results.
between the contribution of the primary electrons and that of the backscattered electrons. Such an estimation is presented in Fig. 20. It is compared with various values obtained for the effective backscattering factor. One notices that, on the average, the true backscattering factor exceeds the effective one by 20-30”/:, depending on which estimation and which value for U are considered. In Fig. 2 1, we have indicated how the backscattered electrons of different energies contribute to the effective backscattering factor (1 I ) . This evaluation has been made by varying the lower limit of integration Einf
+
Eh-EI
FIG.21. Contribution of the backscattering electrons of different energies to the effective backscattering factor (1 + r ) : E,, = Ep - 1 ; El is variable.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
239
in the expression
where the value of Einfranges from the ionization threshold to the value Esup [taken as ( E , - 1) eV]. These results are presented for various values of E,. e. Angular Distribution of Emitted Auger Electrons. It is generally admitted that Auger electrons are absorbed according to an exponential law along their path towards the free surface of the solid. If one assumes that they have been uniformly created in the target, with an isotropic angular distribution, the law describing their angular emission is necessarily of a cosine type. In our simulation model, Auger electron creation has also been assumed to be isotropic; however, it is no longer uniform in z. As can be seen Fig. 22, Auger electron angular distribution obtained by simulation follows a cosine law very well for all the values of E,, at least for normal primary incidence. One should note that the experimental results obtained by Matsudaira and Onchi (1978) in fact show some deviation from the simple cosine law. Bui Minh Duc (1981a) considered, from a theoretical point of view, a method for generalizing the form of the angular distribution by including several physical parameters such as the MFPs of the Auger and primary electrons, the incidence angle, and the mean parameter of the lattice in order
WJ
9.
(a)
con 9.
(b)
FIG.22. Dependence of Auger electron emission on the emergence angle.
240
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
to interpret the results of Matsudaira and Onchi. Although his generalized expression still contained some simplifying assumptions, he was able to reproduce the shape of the experimental distributions obtained for nickel, either pure or with a sulfur overlayer, by simply adjusting one parameter called the lateral stopping power of the layer. 4. Extensions to Complex Materials
The direct-simulation method was presented in Section IV,B,2 and applied to the Auger emission of aluminum in Section IV,B,3. The present stage of our knowledge of individual cross sections is such that this method can be applied only in exceptional instances. A second drawback is that the direct-simulation technique involves large computational times. Thus, to go beyond the description of a pure target, one has to look for a simplified simulation method in which the most basic aspects of Auger emission can nevertheless be incorporated. In this spirit, one has to keep in mind several conclusions resulting from the above discussion and from the theoretical analysis which preceded it. (1) Auger emission is located near the surface of the solid, owing to the reduced escape depth of the Auger electrons, whereas their source function can extend rather deeply into the solid. This indicates that some simplifications which could have been useful for describing the penetration of the primary beam into the target, for instance, are now inadequate. (2) The Auger source function is itself composed of various contributions. The primary electrons penetrating the target or turned back towards the free surface play an important role. However, the backscattered electrons, which include the primary particles which have lost part of their initial energy and the high-energy secondary electrons created during the cascade process, give rise to comparable contributions. (3) The elastic and inelastic collisions correspond to rather different angular deflection laws. The various types of inelastic collisions often have themselves well-separated energy-loss spectra. Although a sharp line can yield important information about the validity of a model, effects associated with a broad spectrum are much more difficult to observe. (4) Many important Auger lines can be excited at moderate primary energies (in the keV domain) where the usual stopping-power formulas may become partly inadequate. These remarks seem to indicate that a multiple-scattering approach is not useful in the study of Auger emission. One basic question would be how to
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
24 1
describe correctly the cascade process within this method. However, even for the penetration aspect of the primary beam (which models the source function of the Auger electrons), a pure continuous slowing-down approach would certainly be too crude. It would lead to an oversimplified statistical description. Due to various competing scattering processes, some portions of an electron path can in fact be quasi-elastic in character, whereas other portions can be mainly governed by inelastic events. A simulation technique based upon the “use of some condensed histories’ of an electron (in order to save computational times) may lack some important aspects (for instance, a description of the elastic peak). In this connection, inelastic processes having comparatively small probabilities of occurence can give rise to important individual energy losses. These losses can be quite different from a mean energy-loss value. This is precisely the case for the ionizing collisions by which Auger emission is initiated. Moreover, the validity of a multiple-scattering theory is certainly questionable near the surface, that is, in the region probed by Auger analysis. The basic idea is, consequently, to reduce the step length of the si portions to an extent comparable with the global MFP so that individual collisions could in fact be simulated. Elastic and inelastic effects have to be sharply distinguished. The ionizing collisions themselves have to be treated separately. Without returning to the pure direct-simulation concept, a suitable compromise has to be found. A very satisfying answer, in view of the large variety of problems it was able to solve, has been given by Shimizu and co-workers (Ichimura et ~ l . , 1981; Shimizu and Ichimura, 1981). It seems important to retrace, at least briefly, the main arguments of these authors. A clear separation is made between elastic collisions and inelastic effects. A good solution is to break the elastic effects by using a partial wave analysis of the scattering cross section, starting from a suitable potential. These authors adopt the Thomas-Dirac-Fermi formulation for the potential for which Bonham and Strand (1963) have given general analytical expressions. The partial wave-expansion method, which has also been used by the present authors (see, for instance, Ganachaud, 1977) was reviewed in Section IV, where the choice of the potential was discussed. The inelastic part has to account for the slowing-down effects and for the central problem in the present context, that is, the creation of Auger electrons. In general, we remark that all the events in a simulation method which are associated with a well-defined energy (the creation of an Auger electron in an ionization process, for instance) or which keep unchanged the energy of a colliding particle (elastic collisions) have to be described with much care. On
242
M. CAILLER, J. P. GANACHAUD, AND D . ROPTIN
the contrary, collisions giving rise to broad energy-transfer spectra contribute mainly to the “uniform background.” For these events, less precise descriptions can be sufficient in practice. It has been recognized by Ritchie et al. (1969) that the result obtained by summing over the different inelastic contributions to the stopping power (which can be theoretically evaluated in simple cases) is close to that given by the general Bethe law. Moreover, the Gryzinski formulation seems to have a very large domain of validity. Starting from these conclusions, and following the prescriptions of Krefting and Reimer (1973), one can define a stopping power, due to the valence electrons, by the difference ( 146) (dElds)va, = (dE/dS)Bethe- (dE/ds)core where the first term of the right-hand side (RHS) is given by Eq. (139), and the second term is evaluated by Gryzinski’s theory. The contribution (dE/ds),,,,includes all the inelastic collisions, either individual or collective, with the delocalized electrons. The above treatment can be improved, particularly for low energies, (where the Bethe law becomes questionable when E 5 J ) . According to Shimizu and Ichimura (1981)a better choice is to adopt the following scheme:
(dElds) = (dE1dsXore + (dE/dsXa,
(147)
with
(dEldsk, = (dE/dS)Bethe - (dE/ds)core (dElds) = (dE/ds)core
for (dEldShethe > (dE/ds)core (148) otherwise
(147’)
This seems to be a more general approach than that suggested by Shimizu and Everhart (1978),who proposed an extension of the Gryzinski formalism to valence electrons. The choice of s can be accomplished by finding a suitable MFP 1, and by using the usual random-number-generation procedure described by Eq. (142). Following Shimizu and Ichimura, the MFP 4to be used is given by
AT1 = 2,’
+ 1-’ core
(149)
where ,Ie1 denotes the elastic MFP and Lcor, represents the inelastic effects of all the deep levels j that can be excited by the simulated electron. Thus, in practice, the separation of elastic and inelastic collisions is first done according to the standard procedure of Eq. (143): (1) An elastic collision is described by the corresponding differential cross section cre,(E,0) which fixes the angular deviation.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
243
(2) An ionizing collision (of which j is first determined) gives rise to an energy loss AE. Gryzinski’s formula can be used under the differential form
x {g(l -
2) +
;ln[2.7
+
rTy’2]} (150)
where n j is the electron population of the subshell considered and E j its binding energy. The corresponding angular deflection 0 can itself be approximated by the simple formula AE = E sin2 0 (151) The cascade Auger effect following this ionization can be described by the standard procedure of Section VI,B,%. In fact, the MFP A, used in Eq. (149) does not include the valenceelectron contribution to the stopping power. This can be recovered by incorporating an additional energy-loss term
(A%,, =
l:
(dE/dsX,,ds
(152)
for the estimated path s. An important point to note is the great flexibility of this method, which can be extended to compound materials and alloys by using generalized relations (Philibert and Texier, 1968; Matsukawa et al., 1973) such as
where p is the density of the compound material where the constituent labeled k is present with a mass concentration c k and
where NAv is the Avogadro number and njk is the population of the j t h subshell of the kth constituent. Calculations made by Ichimura et d.(1980) for a pure aluminum target can be considered as standard reference calculations. They did this by three methods : (1) using a direct-simulation technique; (2) describing the contribution of the conduction electrons by the relation proposed by Shimizu and Everhart (1978); and (3) using the method analyzed in this section. No
244
M . CAILLER, J. P. GANACHAUD, AND D . ROPTIN
marked differences could be found among the results. Method (3) has subsequently been applied to a large variety of solids (pure materials, alloys, compound targets) to give, among other results, an evaluation of the backscattering factor in view of an Auger quantitative analysis. These results have been extensively presented by Shimizu and Ichimura (1981) in their review article.
VII. AUGERQUANTITATIVE ANALYSIS
A. Zntroduction Auger quantitative analysis aims to determine the various concentrations in the elements existing at the “surface”of a sample. It is understood here that “surface” can not and must not be considered in its proper geometrical meaning; rather, it means the few outermost atom layers under the electron beam. There is still a long way to go before being able to relate the detailed theoretical descriptions presented in Sections I l l and IV with the experimental observations made on complex targets; even the possibility of realizing this project completely is open to question. In order to attain this goal it is nevertheless necessary to follow various simultaneous complementary approaches. Thus, from a theoretical point of view, one must be able to give not only a complete quantitative description of the simplest targets, i.e., pure samples, but also to develop realistic models of less simple targets such as binary alloys and compounds. In fact, the description of complex targets seems possible only phenomenologically as initiated in Section VI. The same progression must be followed in the experimental studies. Indeed, it seems fundamental that for the pure targets playing the role of standards a complete quantitative agreement could be achieved between theoretical and experimental results. Quite evidently, this agreement must first be achieved among the results from various experimenters. This implies that a rigorous and indisputable procedure must be defined not only to control the quality of the conception, fabrication, and adjustment of the whole apparatus of analysis, but to ensure correct usage of this apparatus as well. In this regard, the round robin led by the ASTM Commission 42 (Powell et al., 1981) revealed the ticklish problem of controlling the validity of measurements in AES. The targets used in the round robin were very simple (pure copper and
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
245
gold), and a general experimental procedure was defined. However, the values of the ratios between the intensities of the high- and low-energy lines for the same element differed by an important factor (120 for gold) among the extreme measured values. In fact, these intensities were really measured by the peak-to-peak heights H in the derivative mode. The disagreements noted can particularly arise from some error in manipulation. However, they point the direction to follow in order to achieve good quantitative agreement even in the case of simple targets and, aJortiori, for complex samples of a technological nature. The definition and general acceptance of standard procedures for the adjustment of analysis devices (such as that proposed by Duraud and Le Gressus, 1981) is a sine qua non condition of progress. Studies of binary and ternary alloys and compounds are essential to knowledge of the spurious effects which occur in the preparation of the sample and in Auger analysis itself. A serious disadvantage of AES results from the use of an ultrahigh vacuum (UHV) chamber. Apart from questions of experimental uncertainties or manipulation errors, the problem of the interest of the AES is even posed for technological samples such as those encountered in the study of catalysis, aqueous corrosion, and lubrication. Indeed, when a technological sample is introduced in the UHV system, some disturbances come from the vacuum itself and from the decontamination treatments carried out before the analysis. Another means of studying technological problems is to attempt to reproduce in the UHV system some of the most basic physical processes. This leads to proposing an idealized situation instead of the reality and poses the problem of making a correct link between them. The above questions are beyond the scope of the present review article. There are additional problems in making an AES experiment quantitative. They can arise, for instance, from the calibration aspects, from the preparation in situ (i.e., in the UHV system),or from the electron-beam effects during the analysis. Some of these problems are reviewed in the following sections.
B. Operating Modes Assuming that all the conditions required to achieve a good analysis of the target are realized from the point of view of the sample preparation and the general setup of the apparatus, various operating modes can be used. The choice depends on the desired information. For a given direction of incidence of the primary electron beam and for a given acceptance angle of the collector, the secondary-electron distribution is n(E,, E,, Zp), This is a function of the secondary-electron energy E,, the energy Ep,and the intensity I, of the primary beam.
246
M. CAILLER, J . P. GANACHAUD, AND D. ROPTIN
The collector always selects the electrons from between two energy limits: E,, and E,, . For a high-pass filter such as a grid-retarding analyzer, E,, is in principle infinite, but in fact E,, = E,. Thus, all the electrons with an energy higher than the retarding potential E,, = E, are caught by the collector. Therefore the current detected by the analyzer is
Ep,1,) =
[:
n(& E p ,Ip)dEs
(155)
It is possible to modulate any of the three variables E,, E,, I, and to detect the modulation induced on N . The brightness (or primary-electronbeam intensity) modulation technique has been proposed and developed by Le Gressus et al. (1975). In the normal AES operating mode, Ep and I , are kept constant, and a small alternating component e, sin ot is superposed on the retarding voltage E, . The secondary current can thus be approximated by N(E,
+ e, sin at) n ( E $ )dE$ - dE, 2 ! - n(Es)e,sin ot
By using a lock-in amplifier, it is possible, for instance, to select the o component of the signal [third term on the RHS of Eq. (1 Its amplitude is n(E,)e, and it gives the secondary-electron energy distribution. It is phase shifted by .n with respect to the grid potential modulation. In practice, it is generally desirable to use the second derivative of the current N , i.e., dn(E,)/dE,. This is achieved by detecting the second harmonic of the modulation frequency [fourth term in Eq. (156)l. This operating mode was developed by Harris (1968), and this began the rapid progress in the field of AES. When E, and I , are kept constant by superposing a small sinusoidal voltage e, sin wt onto the accelerating voltage Ep,one obtains as a first-order expansion:
%)I.
N(E,, Ep + e, sin cot, I , )
(157)
The signal corresponding to the second term of the RHS can be detected by the lock-in amplifier on the collector.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
247
In the integral of the RHS member of Eq. (157) [later termed 6 N ( E , ) ] , the lower limit E, must only be lower than Ep.Two particularly interesting cases are E, = 0 and E, = Ep - 6Ep; 6Epis a positive quantity which has to be made as small as possible, but it must exceed epin order to keep Eq. (157) valid. The first case, E, = 0, corresponds to Auger electron appearance potential spectroscopy (AEAPS) (Gerlach, 1971), and the second case corresponds to disappearance potential spectroscopy (DAPS) (Kirschner and Staib, 1973, 1975). In AEAPS, the detected signal is the sum of the two terms which have opposite variations when Ep increases: (1) N ( E , = 0) is the variation of the total secondary-electron current. Except at high primary-electron energies, 6 N increases with Ep. (2) n ( E p ,E p ,Ip) represents the elastically reflected beam. It decreases smoothly with increasing Ep except when its derivative dn/dE, presents a singularity. Such a singularity can result from the opening of an ionization channel in the ionic cores or from coherent diffraction effects. In DAPS, 6 N is nearly zero, and only the variations of the elastically reflected beam n are detected. In the case of a polycrystalline target, the diffraction effects are negligible. The rapid variation of n can only come from the opening of new inelastic channels when the primary-electron energy becomes equal to a core-ionization threshold. In DAPS, the decrease of n is not compensated for by an opposite variation of the secondary-electron current, hence the name disappearance spectroscopy given to this operating mode. In AEAPS, the decrease of n is overcompensated for by the appearance of newly emitted low-energy secondary electrons. These secondary electrons are created via the cascade process by the Auger electrons which are generated by reorganization of the excited atoms. The measurement of the variations of the total X-ray emission of a target as a function of Ep has led to a similar operating mode called soft X-ray appearance potential spectroscopy (SXAPS) (Park et al., 1970). In the case of single-crystal targets, diffraction effects can completely obscure some of the appearance potential features (Gerlach, 1971; Den Boer et al., 1978). To separate the diffraction effects from the appearance potential features, Den Boer et al. proposed a double-modulation (in E, and E p )technique. In such a case the RHS members of Eqs. (156) and (157) have to be added. The results obtained by AEAPS and DAPS are not analyzed in detail in this review, which is devoted more to the quantitative aspects of Auger analysis. However, one must cite the important pioneering work done by Park et al. (1970) who showed that it was possible to measure accurately the binding energies of the core electrons and the work functions of metals by AEAPS and DAPS.
248
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
The accuracy obtained in AEAPS ( 20.2 eV for the binding energies, f0.04 eV for the work functions) is quite excellent. Therefore for this type of experiment this technique can compete with much more expensive ones such as synchrotron radiation photoemission. Park‘s group has also shown (Cohen et al., 1978; Park, 1979) that the target current presents an extended fine structure over a few hundreds of electron volts above the ionization threshold of the inner shells. This extended appearance potential fine structure (EAPFS) is quite similar to the extended X-ray absorption fine structure (EXAFS) above the absorption edge of the X rays in solids. The EAPFS results from interference effects between the outgoing spherical wave associated with the Auger electron leaving the atom source and the waves coherently scattered by the neighboring atoms. For a given outgoing wave, the phase shifts of the reflected waves depend on the distances between the central atom and the atoms playing the role of scattering centers. The results of both EAPFS and EXAFS therefore depend on the interatomic distances and give quantitative information on these distances via a Fourier transform of the spectra. However, whereas EXAFS characterizes the bulk of the sample, EAPFS is especially sensitive to the surface (Elam et al., 1979). Thus, in spite of the complications resulting from the presence of a background in the spectra, it has been possible to obtain information on the surface interatomic distances in the cases of clean and gas-covered metallic surfaces (Elam et al., 1978, 1979;Fukuda et al., 1978; Den Boer et al., 1980a,b; Laramore et al., 1980). Park has devoted some review papers (1979, 1980a,b) to the core-level spectroscopies by AEAPS and to the studies of surface spacings. Readers interested in these topics will find much of value in these papers. Finally, the surface sensitivity of the appearance and disappearance spectroscopies are now briefly considered. Nishimori et al. (1980) reported that DAPS is more surface sensitive than classical AES. This would seem to be intuitively foreseeable by taking into account the physics of these operational modes. However, to the authors’ knowledge there are no theoretical determinations of the thickness of the information zone in DAPS, but some information might be obtained with the help of a simulation model. In this way, the results presented in Section VI,B concerning the depth at which the elastically reflected primary electrons penetrate aluminum give an indirect measurement of the DAPS surface sensitivity. Diferential or Direct-Energy Spectra in Quantitative AES ?
It was mentioned earlier that the rapid growth in the importance of AES was started by Harris’ idea (1968) of using a derivative of the secondary-
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
249
electron energy spectra. Thus, it was possible to suppress the background and then to amplify the Auger lines. A quantitative technique was built on these differentiated spectra. It consists in using the peak-to-peak height H of an Auger line in the differentiated mode to measure the intensity of the same line in the nondifferentiated mode. The differential mode has so far been the most frequently used technique of quantification, but recently there has been a strong tendency to return to the use of the energy (or direct) spectra, especially with the development of automatic data processing. Two major physical arguments favor the use of the direct-energy spectra. First, the intensity I of an Auger peak is directly proportional to its area in the energy spectra, whereas the peak-to-peak height in the derivative spectra only indirectly measures this intensity. Consequently, it is possible to proceed to absolute measurements from the energy spectra. Thus, comparisons between theoretical and experimental values of Auger line intensities can be attempted. Second, the accuracies resulting from the influence of chemical effects on Auger line shapes are less important to the energy spectra than to their derivatives. For instance, the peak-to-peak height H in the derivative mode can be strongly disturbed by a change in the shape of the Auger line even if the area under this line does not change. For an Auger line having a Gaussian line shape, the proportionality of H to the intensity I of the line, as measured on the energy spectrum, has been shown by Hall et al. (1977), who obtained
H = 16 In 2(2.ne)-'l2l/o2 where e is the base of the natural logarithm and w is the half-height width of the Auger line as measured on the energy spectrum. In the derivative mode, the Auger line presents a maximum and a minimum which, in the case of a Gaussian line shape, are separated by the distance It thus appears from Eqs. (158) and (159) that if the Auger line changes in width but retains its Gaussian shape, the intensity of the line is better described by H d 2 than by H only. In fact, the relation of proportionality between I and H d 2 is not limited to the case of Gaussian line shapes, but as noticed by Hall and Morabito (1979), it can be extended to other line shapes. This extension is possible because the origin of the proportionality relation between I and H d 2 is in the abscissa and ordinate scale factors and not in the Gaussian line shape itself. In practice, the energy spectra are recorded either directly, generally by an electron-counting technique, or indirectly. In the latter case the recorded signal is obtained by a derivation-integration procedure (the dynamical
250
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
background subtraction). In this procedure the secondary-electron current is differentiated as many times as necessary for a complete elimination of the background. These derivations are then followed by successive integrations until the energy spectrum is obtained. If the derivative mode gives a better contrast of the signal, the use of the direct-energy spectrum gives, at least for low beam currents, a better signalto-noise ratio (Seah, 1979a,b).But the use of the direct-energy spectra in view of quantitative AES raises the essential question of separating the selected Auger line from the background. The major difficulty arises, however, from the creation by the energy-loss spectrum of the Auger electrons of a long tail on the low-energy side of the Auger line. For a quantitative Auger analysis, it seems desirable if not vital to separate the Auger line from its energy-loss spectrum. The question is all the more important because computer facilities will undoubtedly lead to a rapid development of direct-spectrum analyses. Some progress has been made in this field, as we shall see in what follows. It was shown in the section devoted to the simulation of Auger emission that only a very small number of the Auger electrons created in the solid can reach the free surface with their characteristic energy. The others either penetrate more deeply into the sample and are lost or undergo inelastic collisions before reaching the free surface of the target. Taking into account the total Auger electron creation (obtained by summing both the primaryand the backscattered-electron contributions), the energy distribution NA(E3) of the Auger electrons created in the solid when they reach the free surface is obtained by integrating n$3”(3) (see Section V1,A) over x,, y,, and $2,. In writing the energy distribution, the superscript prime and x has been dropped for simplification in what follows. In principle, NA(E,) is extended over a very large energy range. Indeed, Auger electrons created in the solid can escape into the vacuum with any energy value lower than their creation energy. Over the major part of the energy range, N A ( E ,) is superimposed on the secondary-electron background B from which it cannot be rigorously separated (see Fig. 23). The secondaryelectron background would be the electron background if the channel of the Auger emission which is considered here is not open. In fact, far under the Auger peak NA&) is considered so small that the secondary-electron background and the total background can be confused. In Section VI,A it was supposed that true Auger electrons, i.e., those emitted with a characteristic energy, could be unambiguously separated from other electrons. In practice, no exact separation is possible and approximate procedures have been proposed. They consist of extracting from N ( E , ) = B + N A ( E 3 )the unknown true Auger electron distribution which is denoted here by A,. The problem is complicated further because the energy distribution N ( E , ) is itself partially unknown. Indeed, the experimental energy
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
25 1
distributions are measured not at the output of the sample but at the output of the spectrometer. In a general formulation, the distribution of the electrons at the output of the spectrometer are given by (see Section VI,A)
nd4) =
s
43)h(4,3)43)
f 160)
where h(4,3) is the total transmission function of the spectrometer, taking into account the capture of the electrons by the collector as well as their transmission by the analyzer itself. In the simplest case it is assumed that h(4, 3) depends only on the energy variables E3 and E4. The integration over x,, y,, and R3is therefore direct and WE,)
=
s
d E 3 W 4 , E3 ) N ( E 3 )
(161)
The experimental energy distribution of the electrons at the output of the spectromer can be written as NS(E4) = BS(E4) + A,@,)
( 162)
where B,(E,) is the experimental background and A,(E,) is the experimental Auger line with its energy-loss tail on the low-energy side. To separate the two contributions B, and A, on the RHS of Eq. (162), Staib and Kirschner (1974) suggested that the experimental background could be described by a fitted spline polynomial. The interpolation is performed by fitting the polynomial to both low- and high-energy wings of N,(E,) without accounting for the energy-loss contributions on the low-energy side. In the dynamical background subtraction, the background disappears provided that sufficient derivatives are taken. However, the rapid variations of the Auger signal are not suppressed by derivation, and they can be retrieved by successive integrations (Grant et al. 1974).
Background
0 Energy (arbitrary units)
FIG.23. Background and Auger electron peak.
252
M. CAILLER, J. P. GANACHAUD, AND D . ROPTIN
Ramaker et al. (1979a,b) have criticized both these procedures: the dynamic background subtraction because it is equivalent to subtracting a background fitted only to the high-energy wing of the energy distribution, and the Staib and Kirchner procedure because it does not account for the energy losses. They proposed the following approximative scheme. The experimental Auger line at the output of the spectrometer is connected to the unknown true Auger line A,(E) at the output of the sample by the relation As(E4) =
1
dE3H(E4, E 3 ) A T ( E 3 )
(163)
In practice, the function H ( E , , E 3 ) is very complex because it has to take into account the following: (1) the broadening of the Auger line in the sample; (2) the capture by the collector of the electrons emitted by the target; (3) the transmission of the analyzer; (4) the modulation broadening (in the derivative mode); (5) the efficiency of the detector. The sample position effects on the energy shifts and on the peak-intensity changes have been considered by Sickafus and Holloway (1975). The other factors have been analyzed by Powell (1978) and by Bui Minh Duc (1981a). Ramaker et al. assumed that H(E4,E 3 )could be obtained by convolution of the target and spectrometer transmission functions. The target transmission function HT transforms the true Auger line AT at the output of the sample into N A ( E ; )according to the relation NA(r3)
=
s
dE3H'I'(E!3
9
E3
( 164)
The spectrometer transmission function H , can be experimentally studied. It transforms N A ( E ; ) into the experimental Auger line A,(E4) according to ME41 =
f
dE3HdE4, E W A ( E 3 )
Therefore, from Eqs. (163)-( 165), we obtain
For HT, Ramaker et d.proposed the following approximation:
(165)
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
253
in which each Auger electron source in the solid is assimilated to a primaryelectron source. In this expression Hi and He are the inelastic (energy-loss spectrum) and the elastic (elastic peak) parts of the experimental energy spectrum as measured with a primary beam of energy E A , x and of intensity equal to unity. The factor (1 - E), where t is a numerical constant, takes account of the differences between the geometries of the Auger electron beam and the primary beam. To extract the true Auger electron peak from N,(E,) as given by Eq. (162), Ramaker et al. have proposed the following iterative procedure: (1) A solution A,(E) is guessed. (2) The lower energy side of the experimental Auger peak at the output of the spectromer can be identified with the experimental energy-loss spectrum of the true Auger electrons. Therefore in this energy range,
&(E
-
EL) N
(1 - C)Hi(E, EA,X)C
( 168)
where C is obtained by dividing the area of the true Auger peak by the area of the experimental elastic peak. ( 3 ) The background on the lower energy side of the experimental Auger peak can be estimated by subtracting A,(E EL)from the experimental N,(E,) according to the relation
-
B(E
-
EL) = N,(E
-
EL) - A,(E
-
(169)
EL)
-
(4) A functional form including adjustable parameters is fitted to N,(E) above the Auger peak and to B(E EL) below the Auger transition. The functional form chosen by Ramaker et al. is
B(E) = AE(E - Eo)-l(E
+ B ln[(E,
+ q5)-m
- E)/E,] [(E, - E)/Eb]- n
+C
170)
where A, B, C, Eo, 4, Eb are adjustable parameters. (5) The fitted background is then subtracted from N,(E), giving
A d a = N,(E) - B(E)
171)
(6) A new estimation ofA,(E) is obtained from A,@) by a deconvolution procedure, i-e., by solving the integral equation (163).
The entire process is then repeated until the solution for A,(E) converges to an unchanging value. Sickafus and Kukla (1979) have used the functional form (Sickafus, 1977a,b) B(E) = BE-"
(172)
to describe the background. They used nearly the same analysis as Ramaker et al.; however, they determined the adjustable parameters B and m by
254
M. CAILLER, J. P . GANACHAUD, AND D. ROPTIN
fitting Eq. (1 71) only to the higher energy wing of the Auger transition. Sickafus (1980) has simplified Eq. (163) by using the approximate relation
JE*,x E+€
A,(E) = AT(E) - BE-"
AT(E')dE'
(173)
where the constant t represents the minimum energy that can be lost by an electron in an inelastic collision. In metals t = 0. In studying the Auger line shapes, a solution to Eq. (163) is preferable, but according to Sickafus, Eq. (173) is acceptable in the case of Auger-yield measurements. In this latter case the algorithm used to evaluate the Auger yield is much simpler and faster than that for the deconvolution in Eq. (163). Moreover, the yield measurements made in this manner show greater modulation stability than do peak-to-peak measurements in the derivative mode (Sickafus and Winder, 1981). By eliminating the long, lower energy tail, the deconvolution procedure leads to a narrow true Auger line. In principle, the width of the experimental Auger line would be equal to the natural linewidth of the transition. In view of the difficulties inherent in the deconvolution procedure, Seah (1979a) has proposed an alternative scheme to estimate the actual intensity of an Auger peak without the losses. To obtain this intensity, the true Auger line is presumed to have a symmetrical shape which is obtained by mirroring the higher energy part of the experimental line. Such a scheme seems quite valuable for a single narrow Auger peak, but its use for the multiple lines such as those obtained in the study of Li,SO, and Li,PO, by Ramaker et al. (1979a) is inadequate. Theoretically, the broadening of the natural Auger line during the transport of the Auger electrons to the free surface of the sample could be simulated on a computer. The comparison between the so-obtained theoretical results and the experimental line shapes would be very interesting; however making the simulation implies considerable effort to take into account the natural width of the Auger line and all the individual mechanisms of inelastic and elastic collisions. Until there is general agreement concerning the operating conditions and methods of calculation, Seah (1979a) proposes that the differential spectrum be used for a quantitative AES. However, the energy-loss disymmetry on the direct Auger spectra leads to a similar disymmetry on the derivative spectrum. Giving a symmetrical shape to the direct spectrum by suppressing the energy losses is equivalent to symmetrizing the negative peak in the derivative spectrum. Therefore, Seah proposes to quantify the Auger line by measuring the peak-to-background height of the negative peak in the derivative mode. To conclude, the question of the use of the direct or derivative spectra is still not settled.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
255
C. QuantiJication of Auger Analysis
A quantification in absolute values leading, at least for simple targets, to an agreement between theory and experiment has yet to be accomplished. The way lies through a theoretical and numerical evaluation of the Auger yield, from first principles; for instance, by using a simulation model on a computer. Absolute measurements of the intensity of the Auger lines are also necessqry. To the knowledge of the authors, the most remarkable experimental approach in that sense is that of Staib and Kirschner (1974).They evaluated the absolute values of the Auger currents for different simple targets from the measurement of the area under the Auger peak after subtraction of the background. For this they used a conversion factor obtained by comparing the area under the energy distribution curve of the electrons emitted by the target with more than 1 keV and the corresponding measured current. The number of the Auger electrons, at the output of the spectrometer, is given by a relation similar to the simplified formula [Eq. (130)l after correction for the backscattering factor and for the transmission factor H S ( E A , J of the spectrometer NA'x = ?A,C,xNP
x
(471)-1
[TP/cOs
OPIT
, x d ,C (EP
( E A ,x ) N A
IQccos0 3 d f i 3 H S ( E A , x ) [ 1
+ rP(EA,x,
(174)
where the dependence on the primary variables of T and r has just been recalled by the subscript P. Here H s is presumed to depend only on E A , x . Therefore it is only a multiplying factor. This simplification is valuable only if the number of the Auger electrons is studied. It does not work for studies of line shapes. From Eq. (174), Staib and Kirschner could go back to the numerical atomic density N A . They worked at normal incidence (cos Op = 1) and set yA,c,x T and T' equal to unity. The collection and transmission factor of the spectrometer C(474-l jQ,cos e3 d f i L , H , ( ~ , , , ) ]was evaluated experimentally by measuring in the same experiment the primary, target, and collected currents. The cross section ohc was evaluated with Drawin's formula (1961), the backscattering factor from the results of Gallon (1972) and Meyer and Vrakking (1972), and the escape depth from the compilation given by Tracy (1972). The values N A deduced from the measurements by using Eq. (1 74) agree with the values usually obtained for Ag, Cu, and C within better than 50%. The disagreements mainly arise from uncertainties in the evaluation of the various parameters involved in Eq. (174). They also result from uncertainties concerning the validity of Eq. (174) itself, and as pointed out by Ramaker
256
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
et al. (1979a,b), from an incorrect subtraction of the background and of the loss peaks. In spite of the difficultiesencountered, this procedure is strongly requested by theoreticians, although it is not absolutely necessary for the development of a quantitative Auger analysis. A more important number of measurements of this type are necessary in order to set up realistic models. The most widespread method, in the absence of absolute measurements, is always the calibration of the apparatus. Once the problems of adjustment of the apparatus and preparation of the sample are completely solved, one can distinguish, according to the case, external calibration methods from internal. In external calibration methods, the Auger line intensities of either pure elements or alloys are measured under experimental conditions (primary current density, accelerating voltage, etc.) as close as possible to those used during the analysis. The concentrations of the analyzed target in its various elements are then evaluated by making a comparison with the intensities of the lines of the reference targets. If the external calibration is performed by using a series of alloys of various but known compositions, there is generally a systematic variation of the line intensities with the concentrations. Thanks to this correspondance, the measurement of the line intensities allows in return the evaluation of the composition of the unknown target, provided that the preparation conditions of the sample and the adjustment of the apparatus are the same in both cases. Because it is somewhat tedious, this method is not often used. Generally, one prefers the method of the relative elemental sensitivity factors evaluated for pure targets (Palmberg, 1973; Davis et al., 1976). This method is presented in some detail later. For a long time it was criticized because it neglected matrix effects. These criticisms led Bouwman et al. (1973a,b; 1976a,b)to propose the internal calibration method. This method is well suited to the study of a physical or a chemical problem for a given sample. For instance, it was extensively used to study the elemental surface enrichment produced by sputtering or by segregation. The principle of the internal calibration lines in obtaining a virgin surface (for instance, by fracture of a prenotched test sample) and in measuring just after the fracture (i.e., before the physical phenomenon one has to study takes place) the Auger line intensities of the various components. In general, one uses for a control several preparation modes of the virgin surface (fracture in DUCUO, scribing, for instance). If the Auger line intensities do not depend on the preparation mode, one can assume that they characterize well the bulk composition. When this composition can be determined by other experimental methods [by an electron-probe microanalysis (Ho et al., 1976,1979)or by absorption spectroscopy (Goto et al., 1978), or by any
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
257
other method], it is possible to calibrate the line intensities. It seems that this method is quite quantitative for compounds and for binary alloys. In this case, one obtains a linear relation between the surface concentrations measured by AES and the bulk concentrations. This was pointed out by Bouwman et al. (1973a) for the three compounds Pt,Sn, PtSn, and PtSn, ;by Van Santen et al. (1975) for Cu,Au and Au,Cu alloys; by Mathieu and Landolt (1975) for Ag-Pd and Ni-Pd alloys; by Bouwman et al. (1976b)for Ag-Au alloys; and by Betz (1980)and Betz et al. (1980a,b) for a large number of solid solutions. Similar observations have been reported by Mathieu and Landolt (1979) for the ratios of the Auger line intensities of Cr and Fe and of Mo and Fe in Fe-Cr-Mo alloys. Some different observations have been noted for nonmixable systems such as Ag-Cu and Ag-Au-Cu. If one assumes that the surface concentrations remain proportional to the line intensities, by measuring the variations of these intensities one can express the modifications of the elemental concentrations at the surface which result from the physical process being studied (Betz, 1980; Betz et al., 1980a,b). Another method is connected with the internal calibration method of Bouwman, but it only allows estimation of the variations of the surface concentrations. It consists in measuring the high- and low-energy peaks for a given element. The low-energy peak represents well the surface composition, whereas the high-energy peak gives information about a much thicker layer and is thus much more connected with the bulk composition (Van Oostrom, 1979; Watanabe et al., 1976, 1977). Whatever the physical process being brought into play, the low-energy peak increases more than the high-energy peak (relative variations), and one might conclude that there is a surface enrichment in this element. The results obtained by AES could be made more complete with a probe much more surface sensible than the low-energy Auger peak itself, such as SIMS (secondary-ion mass spectrometry) or ISS (ion-scattering spectrometry). These techniques are respectively based on the study of the mass distribution of the sputtered particles created by the primary-ion bombardment and on the measurement of the energy of the primary ions elastically scattered by the solid surface (Yabumoto et al., 1979). The basic advantage of the internal calibration method compared to that of external calibration is that matrix effects are automatically taken into account, at least when the concentration variations remain located close to the surface. This advantage is largely suppressed by the improvements brought by Hall and Morabito (1979) to the relative elemental sensibility factor method to account for the matrix effects in dilute alloys. In this case, the compensation is no longer automatic, but it can be calculated. If one assumes that this corrected elemental sensibility factor is exact, no difference
258
M. CAILLER, J. P. GANACHAUD, AND D . ROPTIN
can be found, in principle, between the results obtained by this way and those obtained by the internal calibration method. Holloway (1977) gives an example of such an equivalence. The general formalism for this calibration technique can logically be applied to the intensities of the Auger currents obtained by integrating the n(E) spectrum; however, it has been most frequently used together with derivative spectra. Under these conditions its use can be justified as long as the peak-to-peak height of the derivative spectrum actually represents a measure of the Auger line intensity. In the present context, it is not necessary to show how this intensity is measured. One should admit that the experimental method is adequate. For instance, if the measurement is performed on the n(E) spectrum, it is assumed that the subtraction of the continuous background and the loss tail has been exactly achieved. The relative elemental sensitivity factor of element A compared to a reference element R evaluated from the Auger lines (x for the element A and r for R) of the target T is
with CA(T) = NA/NT, CB(T) = NB/NT,
.-
- 9
CR(T) = NR/NT
(176)
where N , is the average number of atoms per unit volume of the analyzed target and NA,NB,. .., NRare the numerical atomic densities of the elements A , B,..., R. In the case of a multicomponent target, the sum of the concentrations CDCDmust be equal to unity. From the definition relation, one notes that Pre,(R,r ; A, x ; T) = P,;;(A, x ; R, r ; T)
(177)
P,,,(A, x ; B, Y ;T) = P,,,(A, x ; R, r ; T)P,,,(R,r ; B, Y ;T)
(178)
In other words, any element can be chosen as a reference. The elemental concentrations can be expressed in terms of the intensities N and the relative elemental sensitivity factors according to the relation
1
CA(T) = NA9”(T)/ [ND,z(T)Pr,,(A, X ; D, Z ; T)]
( 179)
DET
where the sum in the denominator contains all the elements D constituting the target T, and z is a label identifying the Auger line used to characterize the D element. The simplest approximation for the measurement of the relative elemental sensitivity factors consists in supposing that they are independent of the composition of the analyzed target. Thus, they can be evaluated from the Auger line intensities of the pure elements measured under the same experi-
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
259
mental conditions as those for the unknown sample. This factor is therefore approximated by Pr,,(A, x ; R, r ) = NA~"(A)/NR*'(R)
(180)
where the labels A or R indicate that the measurements have been performed on pure A or R targets. The major disadvantage of this definition is that matrix effects connected with the composition of the target are neglected. These effects manifest themselves at least in the backscattering factor and in the escape depth of the Auger electrons. For the case of dilute alloys, Hall and Morabito (1979)have proposed evaluating the corrective factor, which allows the calculation of PreI(A,x ; R, r ; T) whenever Prel(A,x ;R, r) is known, according to the relation Pre,(A, x ;R, r ; T) = Pre,(A,x ;R, r)F(A, x ;R,
T)
(181)
For this purpose they made use of Eq. (174), which expresses the number of Auger electrons in the A,x line, at the output of the spectrometer. This number can be written In a given target, the backscattering factor r depends not only on the composition of the target but also on the parameters E p , Rp, Ea,,, and R,. kA,.+ stands for
and, in practice, depends on the Auger line considered. It depends also on the parameters of the primary beam, the roughness of the target surface, the ionization cross section of subshell C, and the characteristics of the spectrometer. If one assumes that all the parameters determining the value of k are kept fixed (except the choice of a particular A,x Auger line), kA,xdepends only on the selected line, as indicated by the A and x subscripts. In the case of a pure target A, the number of Auger electrons in the (A,x) line is
where LA,,(A) is the inelastic MFP in the elemental target A of the Auger electrons of the Auger line A,x which have an energy E A , x , and 1 + rA,JA) is the backscattering factor for the Auger line A,x in the same target A. For an arbitrary target T, the number of Auger electrons in the Auger line B,y is NB'y(T) = ~ B , Y ~ B ( ~ ) ~+BrB,y(T)]NP ,~(~)[~
(185)
260
M. CAILLER, J. P. GANACHAUD, AND D . ROPTIN
If this target is a dilute alloy of B into A, one has NB9y(T)
NB’y(A) =
kB,yNB(T)AB,y(A)[l+ ‘B,Y(~)]~P
( 186)
In Eq. (186), NB(T)is the number of B atoms per unit volume of the target and N(T) is the average number of atoms, per unit volume, independent of their nature. The concentration in element B within the explored zone is cB
= NB
(T)/N(T)
(187)
From Eqs. (179, (1 SO), (1 8 l), (1 84), and (1 86), one can write
It must be recalled here that the backscattering factors depend on the acceleration energy of the primary electrons. Hall and Morabito have tabulated, the values of the corrective factor F for 55 different elements and for two different choices of MFP values (Penn, 1976; Seah and Dench, 1979). For the atomic densities, they used the values tabulated by Kittel (1953) and for the backscattering factors, the values deduced from the backscattering coefficients q given by Reuter (1 972) from the relations ‘B,y
=
12.8[1 - 0*9(EB,C/Ek’)]Y
q = -0.0254
+ 0.0162 - 0.000186Z2 + 8.3 x
(189) 10-’Z3
(190)
where E B , c is the energy of the ionization threshold for the subshell C of the atom B, and 2 is the mean atomic number of the sample (for an A based dilute alloy 2 = ZA). A comparative study of the results obtained from the internal (Bouwman et al.) and external (Hall and Morabito) calibration techniques would be particularly interesting. This question is particularly important if, as suggested by Hall and Morabito, their relative elemental sensitivity factors depend only slightly (or even not at all) on the composition of the binary alloy. An example of such a comparison has been given by Holloway (1977) in the case of Cr-Au alloys for which the concentration in Cu varied from 0 to 20%. From results communicated by Hall and Morabito, before they were published, they could correct the elemental concentrations measured by AES on a scribed surface. After correction ( 1 5% in the concentration domain studied), Holloway obtained good agreement between the surface and the bulk concentrations. A method of quantification based on a layer model of the volume analyzed has been developed by Pons et al. (1977a,b,c)and has been applied to Fe-Cr
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
26 1
ferritic steels to the study of passivation films on Fe-Cr and Fe-Cr-Mo steels (Da Cunha Belo et ul., 1977)and also to Ti and its alloys (Bouquet et al. 1977). By reference to Eq. (117), the number of Auger electrons created by the primary beam in the ith atom layer is [Np(i)/(cos 6 > 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:
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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
M. CAILLER, J. P. GANACHAUD, AND D. ROPTIN
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
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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%),
266
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
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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
268
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;
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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
M . CAILLER, J . P. GANACHAUD, AND D . ROPTIN
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 )
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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
272
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.
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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
274
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
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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
M . CAILLER, J . P. GANACHAUD, AND D . ROPTIN
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;
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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-
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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)
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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
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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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
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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
QUANTITATIVE AUGER ELECTRON SPECTROSCOPY
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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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IN ELECTRONICS AND ELECTRON PHYSICS. VOL.
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The Wigner Distribution Matrix for the Electric Field in a Stochastic Dielectric with Computer Simulation D. S. BUGNOLO* Department of’ Electrical and Computer Engineering Florida Institute of’ Technoloyy Melbourne. Florida
H. BREMMER 31 Bosuillaan, Flargebouw Houdringe Bilthoven, The Netherlands I. Introduction
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11. The Differential Equation for the Electric Field Correlations
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A. Reductionof(EijCjk) ............................... B. Evaluation of the Functional Derivative in Eq. (9). ...... C. The Ensemble Average of eijCjk,a Most Relevant Parameter.. . . . . . . . . . . . . . D. The Equation for the Ensemble Average of the Electric Field Correlations . . , 111. Derivation of the Equations for the Wigner Distribution Functions . . . . . . . . . . . . A. Introduction of the Wigner Functions and Derivation of Their Equations. ... B. Preparation for Further Analysis.
IV. Related Equations for the Wigner Distribution Function
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V. Asymptotic Equations for the Wigner Distribution Function . . . . . . . . . . . . . . . . . . A. The Forward-Scattering Approximation ................................ VI. Equations for Some Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Case of a Plane Wave Incident on an Isotropic Stochastic Dielectric Half Space.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Monochromatic Waves in an Isotropic Time-Invariant Stochastic Dielectric. . C. Monochromatic Waves in an Anisotropic Stochastic Dielectric ............ VII. A Brief Review of Other Theoretical Methods . . . . . . . . VIII. The Coherent Wigner Function .................... A First Approximation for the Coherent Wigner Function.. . . . . . . . . . . . . . . . . . . IX. Computer Simulation of the Stochastic Transport Equation for the Wigner Function in a Time-Invariant Stochastic Dielectric. . . . . . . . . . . . . . A. An Integral Equation for a Very Narrow Beam B. An Algorithm for the Computer Simulation of Equation for the Case of a Very Narrow Beam C. Variables Used in the FORTRAN Programs . ............ D. Experimental Program Number One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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* Present address: Department of Electrical and Computer Engineering, University of Alabama in Huntsville, Huntsville, Alabama 35899. 299
Copyright 0 1983 by Academic Precs, Inc. All rights of reproduction in any form reserved.
ISBN 0-12-014661-4
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E. Experimental Program Number Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Conclusions and Suggestions for Future Work ........................... 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I. INTRODUCTION In this work we shall make extensive use of the Wigner distribution function. This function was first defined by E. Wigner in 1932 in connection with a problem in quantum mechanical systems. Using present-day notation, the one-particle Wigner function is defined by Reickl(l980) as
In effect, the Wigner function is the Fourier transform of a special kind of quantum mechanical density. Applications of the Wigner function to quantum theory have been discussed by many authors, including Mori et al. (1962). Because our applications are limited to stochastic but classically posed propagation problems, we do not attempt to present a complete summary of the applications to quantum theory. We begin by noting that the Wigner function of quantum theory is defined as the Fourier transform of the expectation of a density matrix. Our definition is somewhat more general. Let the Wigner distribution function of the time function h(t)be defined by
We stress that this is a “distribution” since the function is not yet ensemble averaged. It is also evident that the ensemble average of Eq. (2) defines at most a nonstationary spectral density function; i.e., a spectral density function that may be a function of the time of observation. It is also evident that for the general nonstationary case given by
(W, 0))=
-
dt‘ exp(iwt’) (h*
(t + g) (t - I;)) h
(3)
a time-averaged determination of the time correlation function of h(t) may not be exchanged or substituted for the ensemble average. This point may be of importance when theoretical results are to be compared to experimental results.
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
30 1
Wigner functions as defined by Eq. (3) have proved useful in propagation problems and have thus occurred in Tatarskii’s (1961, 1971) theories, although the reference to Wigner was not mentioned. We further observe that the Wigner function has been introduced with great success by Bastianns (1978, 1979a,b) for the case of optical signals and optical imaging considerations (Bartelt et al., 1980). Proceeding in a straightforward manner by way of the Wigner function, Bremmer (1979) has used these assumptions to obtain many of his results. However, the analysis was concerned with the scalar Helmholtz equation. Nevertheless, Bremmer (1979) was able to obtain a stochastic transport equation for wave propagation in a stochastic medium in a rigorous manner. In effect, this has served to give a more solid foundation to an earlier and more heuristic analysis by Bugnolo (1960a,b, 1972a). In this article we extend the method first proposed by Bremmer (1979) to the complete vectorial Maxwell’s equations. This is achieved by first deriving the wave equation in the stochastic medium in terms of a tensor of the second rank, the elements of which consist of correlation functions associated with the three components of the electric field. At a later stage it will be possible with the aid of a simple Fourier transform to pass from the matrix of these correlation functions to that of the Wigner distribution functions. The latter are of special physical interest because they are connected with local (running, i.e., nonstationary) energy distributions over the various wave-number vectors and over different frequencies. We also obtain a set of stochastic transport equations for the Wigner function in the tensor formulation. We have made every effort to apply our most general result to a number of special cases. The most general of these special cases is that of forward scattering by a space-time variable stochastic dielectric. Our Eqs. (119) and (120) are transport-like equations for the real and imaginary parts of the ensemble-averaged Wigner tensor. Since the Wigner tensor is Hermitian, it follows that these equations contain nine independent equations from which it is theoretically possible to obtain a complete discription for the Wigner tensor. In view of the complexity of our general results for forward scattering, we have also obtained some approximate transport equations for the case of a plane wave incident on an isotropic stochastic dielectric, and for a monochromatic plane wave incident on a time-invariant stochastic dielectric. We have compared this last result to some other results previously reported in the literature by Howe (1973). Since the stochastic transport theory is just one of many methods used to address this class of propagation problems, we have also included a brief review of the parabolic equation method, comparing the physical assumptions of this method to our own. We conclude that the parabolic equation
302
D. S. BUGNOLO AND H. BREMMER
method and our method do not differ a great deal in their physical assumptions, although differences do exist of perhaps fundamental importance. Any application of stochastic transport theory must eventually address the problems associated with the physics of the fluctuations of the medium. When these fluctuations are due to turbulence, it is then necessary for completness to address the problem of the space-time correlation or spectra of the additives that determine the statistics of the dielectric. In the lower atmosphere, passive additives such as temperature and humidity are of importance. In a weakly ionized gas or in a plasma, reacting additives such as the electron and ion densities are of importance. In view of the present fundamental limitations of the theory of turbulence and the theory of additives, caution is required, particularly if the dissipation regions are of importance to the problem at hand. This is particularly true at optical frequencies where the field is affected by the dissipation region under certain conditions, since the inner scale size ld is large compared with the wavelength of the radiation. Some problems may also exist at millimeter wavelengths. In conclusion, if the theory of propagation in stochastic media is to be placed in a solid theoretical framework, it is essential that the theory of turbulence be closely examined, particularly in wave-number regions affected by the dissipation of reacting and nonreacting additives. We have also made a considerable effort to present the results of many computer simulations using a VAX-11 system and have developed an algorithm for the special case of a narrow beam of monochromatic radiation normally incident on a time-invariant stochastic dielectric. The computer simulation is a somewhat idealized example for propagation in the wavelength range of 1 cm to 1 mm in an atmosphere characterized by very weak to strong turbulence. We have presented our results for the ensemble average of the principal diagonal component of the Wigner tensor, both as a function of range to about 20 km and as a function of wavelength for various fixed ranges. Since our algorithm is limited to a single component of the Wigner tensor, the range of propagation is limited. These limitations are also discussed in detail. We have also included a number of our computer programs, written in the FORTRAN language, for the special case of our narrow-beam algorithm. Our programs have been written to call integration subroutines, using the cautious adaptive Romberg extrapolation, by way of IMSL (1979). Doubleprecision computations have been used throughout the programs, thus ensuring good numerical accuracy. Our programs may be used to simulate the effects of any valid, self-consistent, spacewise spectrum for the dielectric fluctuations. Of the various wavelengths of interest for propagation in the earth’s atmosphere, we should like to suggest that the millimeter-to-optical regions are of the greatest interest. The recent activities of other authors, particularly
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303
Bastiaans (1979b), have suggested a connection between the Wigner scalar function and the ray concept of geometrical optics for the case of an inhomogeneous but nonstochastic medium It should indeed be possible to find a simular connection for the case of a stochastic dielectric described by way of our general results as given by Eqs. (119) and (120); however, it is self-evident that the connection is not simple for the case of a complete tensor Wigner function. Indeed, the Wigner tensor may at most have a constant value along a geometrical optical light ray, in a nonstochastic medium. In our conclusions we have made a number of suggestionsfor future work, such as the problem of very long range propagation at both millimeter and optical frequencies. In this case we believe that a single component of the Wigner tensor is insufficient and a more complete discription is required. The complete tensor description is also required for the case of strong turbulence with strong fluctuations in the dielectric and for the case of a stochastic anisotropic dielectric. Some examples are atmospheric turbulence, ionospheric turbulence, or wake turbulence. It is evident that a great deal of work remains to be done in this relatively new field of stochastic transport equations for the ensemble-averaged Wigner tensor in a stochastic spacetime-variable dielectric. Some thought should also be given to the performance of well-controlled laboratory experiments, since the statistics of the atmosphere are usually imperfectly known in the large, and may indeed be quite variable from point to point.
11. THEDIPFERENTTAL EQUATION
FOR THE
ELECTRIC FIELDCORRELATIONS
In order to ensure that the results of our theory may be applied to one of the most general discriptions of the media, we shall characterize the stochastic dielectric by the nondispersive tensor to(l + i). The fluctuations of the dielectric from its constant background value of co may be a function of both position r and time t ; however, we shall assume that (Z) = 0. Although this may at first appear restrictive, media such as partially or totally ionized plasmas may still be included by a simple extension in the anisotropic case and by including any average background in co . In such a general medium, Maxwell's equations are
V xE
+ PO- aH = 0, at
V x H-
a
€0 -
at
((1 + B)E)
=J
with the following wave equation for the electric field intensity:
(4)
304
D. S. BUGNOLO AND H. BREMMER
We now define a space-time “correlation function” for the electric field by Cik(T,t ; I’, t ‘ )= ET(r
+ r’/2, t + t’/2)Ek(r
-
d/2, t - t’/2)
(5b)
where the asterisk denotes the complex conjugate and i, k = 1,2,3. By using a systematic derivation which parallels that previously introduced for the Helmholtz equation by Bremmer (1979), we can show after considerable algebra that the “correlation function” c i k satisfies the two independent wave equations, each defined by either the upper or lower notation throughout Eq. (6):
Given a deterministic, tensor-like dielectric, Eq. (6) could in principle be solved for a likewise deterministic “correlation function.” However, given a stochastic dielectric tensor, we must ensemble average Eq. (6) over all possible realizations of the tensor E. Defining the correlation function of the electric field by such an average, we may write (Cik(r, t ; r’, t ’ ) )
= (ET(r + r’/2, t + t’/2)Ek(r- r‘/2, t - t’/2))
(7)
We then obtain the wave equation for the correlation function (Cik) in a stochastic tensor dielectric
Ensemble averages such as (&ijCik)present us with some difficulty. This, or a similar difficulty, seems to exist in all formulations of wave propagation in
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
305
stochastic media. In the following we shall introduce a number of simple assumptions which may be used to relate terms such as (cijCjk)to the various elements of (C i k ) . A. Reduction OJ’(&ijCjk) 1. First Physical Assumption: A Gaussian Distribution for E In order to reduce the cross correlation (cijCjk),we shall limit our discussion to media fluctuations that may be characterized by a joint Gaussian stochastic process. By this we mean that the elements of Eik(r,t ) observed at a point in space and time and the product of their values at different points in space and time may both be characterized by a joint Gaussian probability density function. Although this may appear restrictive, we note that the Gaussian process is characterized by the condition of maximum entropy for a fixed standard deviation and as such is found to occur repeatedly in nature. This is true in ionized as well as nonionized media and has been observed experimentally. With regard to plasmas, we note that whereas the electron density may be Gaussian in its fluctuations under the turbulent stress of the gas, the tensor B will only be Gaussian when the frequency of the probing electromagnetic wave is well under the critical frequency of the plasma. The Gaussian restriction permits us to apply the Novikov (1965) relation previously used by Bremmer (1979) in connection with the Helmoltz equation. We may write (E(r, t)F{E(r,t ) } )=
s
dr’ dt’(E(r, t)E(r’,t’))
where the last term on the right-hand side (RHS) is a functional derivative of F with respect to changes in the stochastic function E. Functional derivatives are to the calculus of variations what partial derivatives are to the ordinary calculus (see Gelford and Formin, 1963). Equation (9) also constitutes an extension of the Novikov relation as previously used by Bremmer (1979) without the time dependence. The inclusion of both t and t‘ will later permit us to apply this result to a much larger class of problems.
2. Second Physical Assumption
In order to calculate 6Ej(r, t)/6Eik(r’, t’), we must begin with Eq. ( 5 ) for the jth component of the electric field:
306
D. S. BUGNOLO AND H. BREMMER
Introducing variations in both E j and E and subtracting the resulting equation from Eq. (10a) yields
(8" - $);
6 E j - axj a (V.6E)
We shall proceed by neglecting second-order products in the 6 terms, a reasonable assumption at all frequencies. However, if we are to proceed in a simple manner, we must also neglect the first term on the RHS, E~~ 6Er. We shall address the assumption in a moment. With this final assumption, Eq. (lob) reduces to
This form of the variational equation may be addressed directly. By neglecting terms of the type 1
a2
-c2 a t 2
ejr 6Er
we have, in effect, neglected the effect of the stochastic medium on the field E itself, as is apparent from Eq. (11). This amounts to the use of a first Born approximation for the evaluation of the functional derivative. Such a procedure is similar to that used by Bugnolo (1960b) in evaluating one of the integrals of a proposed stochastic transport equation for the spectral density of the electromagnetic field. The first Born approximation for the effects produced by a local perturbation is useful when the volume involved is small compared with a mean free path (MFP)for scattering by the stochastic dielectric. Over such a bound volume the probability of more than a single scattering is very small. Using models for the spacewise spectrum of the dielectric fluctuations originally introduced by Norton (1960), we obtain the following expression for the MFP d,:
A result for the MFP of this form was first obtained by Bugnolo (1960a) in connection with the solution of the approximate stochastic transport equation for the spectral density of the field which was proposed at that time. It should also be noted that the MFP d, should be on the order of ( k & , ( ~ ~ ) ) - ' , independent of the model used to characterize the stochastic dielectric. This
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
307
latter point was first obtained by Bremmer (1964), who also later noted the same in Bremmer (1974). Returning to Eq. (12), we note that the parameter p = 1/3 for the Kolmogoroff spectrum and that such a spectral region occurs frequently in both ionized and nonionized media when the fluctuations are the result of a fully developed turbulence of the gas (see Bugnolo, 1972b). If we limit the range of the spacewise variation of the 6 E terms to a distance on the order of the mean scale size lo, and if the wavelength A0 is such that d, >> l o , then it follows that we may use Eq. (11) provided that we make a second physical assumption: Although this places a physical lower bound on the wavelength, or an upper bound on the frequency, we note that our overall theoretical model may be on weak ground when Eq. (13) is not satisfied. This is not due to the approximations made in obtaining Eq. (ll), but to other reasons related to the models used to characterize the spacewise spectrum of the dielectric fluctuations. In cases where the dielectric fluctuations result from turbulence in the medium, the higher frequencies will interact with scale sizes beyond those controlled by the viscous subrange of the turbulent velocity spectrum. We must at these higher frequencies deal with the inertial diffusion region or perhaps even with the dissipation region. A typical example of an isotropic Kolmogoroff spectrum is illustrated in Fig. 1 for the very simple case of an A
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)
D. S. BUGNOLO AND H. BREMMER
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’))
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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
D. S. BUGNOLO AND H. BREMMER
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
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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
D. S. BUGNOLO AND H. BREMMER
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
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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) = $6(r). We then obtain, after some reordering,
+ r ’ ~ + t’;
(Wjkcr
a2
d2
ky
ij
(d.:a.;
-
7s
1 +-64n5c2
s
k
dk’ do‘ dr exp[fi(o’
6(t’
7 p)
-
o)z]
s
dr’
+ Ir’l/c) lril
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
exp( f 2ik r’) ds hij(s,z) exp[ k(k - k’) s] S
k
( Wjj(r+ r’, t
exp[i(k.h - op)] ($(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,
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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
D. S. BUGNOLO AND H. BREMMER
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
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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
D. S. BUGNOLO AND H. BREMMER
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$r, t; k , a> V
This concludes our present reduction of the radiation term.
D. Reduction of the InterJerence Contribution R, The second term on the RHS of Eq. (37) was identified as the interferen term. It shall be evident later that this term, in the limit of single scatterir yields a first Born approximation for the Wigner function. Again, using Ec (38),(39),and (57), we obtain 1 R, = 647csc2 x
1
fdk’do’ f
dz exp[ fi(o’- o ) z ]
dr’ exp[ f2ik.r’l
- + o”z)]Pij(k”,w”)
d k do” exp[ - i(k” s
x exp[ k i(k - k ‘ ) s]
f
s
dt‘
k
exp(T 2iot‘)
-
>’
+ 2io
(it-
I
ds
323
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
x [dh dp Fjj(I,p ; k’, w ’ )
The t’ integration yields 2nS[o” T (w‘ - w)], enabling the This yields
7J
1 R2
=
dk’ do‘
J
J dr‘ exp(k2ik-r’)
ds
x jdk”P$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
D. S. BUGNOLO AND H. BREMMER
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
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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
D. S. BUGNOLO AND H. BREMMER
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
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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).
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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
D. S. BUGNOLO AND H. BREMMER
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).
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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)-' << 1
(110)
We are now in a position to proceed with a forward-scattering approximation for Eq. (69). Our formulation for the operational equation for the Wigner function [Eq. (69)] should prove to be most useful because of the occurrence of the operators d/dr and a/&, which lead to a set of reasonably well-defined conditions for conventional approximations. We only need observe that noticeable changes in ( W ) are to be expected over distances of the order of the outer scale size 1,. This may be used as a further justification for explicitly using this scale size in the propagation theory. As a consequence, the order of magnitude of the effect of the operator a/ar may be given by
a/ar z 1;'
(111)
Similarly, the time dependence of the fluctuations of the medium will be associated with a characteristic time scale, to be denoted by z. In practice, this is due to the motion of the irregularities with the mean wind and to their internal decay across the wave-number spectrum. The order of magnitude of this effect is connected with the operator a/&. Hence it follows that a/at
z
(112)
7-1
If we now heuristically invoke a disturbance velocity
'Vdist
defined by
I/dist
then it may be shown that the operator R of Eq. (69) reduces to
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
333
provided that the following inequalities are satisfied:
Since the phase velocity of the waves vph is on the order of c, and since Vdis, as defined by Eq. (113) is very small compared to the velocity of light, it follows that Eq. (114) will be the forward-scattering approximation for the operator R. This follows from Eq. (110). Theoretically, from inspection of Eq. (108) we not only deduce the condition of Eq. (110), but also note that the eddies sampled by the wave as it proceeds through the stochastic medium will depend on the scattering angle 8, the size of the sampled eddy decreasing as 8 increases. If the above is now applied to the RHS of Eq. (81), we obtain
Returning to Eq. (114) and comparing the denominator with that occurring in Eq. (56) enables us to write
Qi’,(k, U ;a/&,
a/&)
-+
(- kikk
+ ($/c2)U2)
Substituting this result into the expression for R$ on the RHS of Eq. (69) yields our result:
{(L
-
2ikT
-
-$(:
+ 2 i w Y ] ( Wik(r,t ; k, u))
x ( WIk(r,t ; k, w ) )
x
dk‘ do’Pj,(k - k , w - w’)
1
+
1
]
x ( Wll(r, t ; k’, w ’ ) ) - __ S + Jk]]
334
D. S. BUGNOLO AND H. BREMMER
As before, the independent second equation may be obtained from the complex conjugate of the above, taking into account the Hermitian property of the Wigner function.
2. Transport Equations for the Wigner Function
Since according to Eq. (96), the operator a k . -a + o_2 _ dr c2 at
is characteristic of a “stochastic transport equation,” we may obtain such an equation for the Wigner function by writing in succession the real and imaginary part of Eq. (118), taking into account the fact that P and W,, are real. We obtain Re( wik(r, t ; k, 0 ) )
+4
( ir + k.-
Im( Wik(r,t ; k, w ) )
-
it)
dk‘ ddPj,(k
)[
-k
, co - 0‘)
Re( Wlk(r,t ; k, 0)) a 6(k‘ - ] o ’ ( / c ) k‘2 - 012/c2 2 w’/c
”
x {(-k>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
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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
D. S. BUGNOLO AND H. BREMMER
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
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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
D. S. BUGNOLO AND H. BREMMER
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.
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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
D . S. BUGNOLO AND H. BREMMER
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
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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
D. S. BUGNOLO AND H. BREMMER
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)
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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
D. S. BUGNOLO AND H. BREMMER
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’)“’ << 1
(157)
The third physical assumption of the parabolic method (assumption 3) is that the stochastic dielectric is 6 correlated in the direction of propagation of the wave, which in the case of our example is the x direction: Assumption 3 <EI(X?
P)&’,
P’)) = &- X ’ M P - P’)
(158)
[See Eq. (20.9) of Ishimaru (1978) and Eq. (1.2) of Dashen (1979).] Dashen’s condition (ii) is really a fourth assumption and is similar to our condition on the velocity of propagation of a disturbance within the stochastic dielectric. Assumption 4
His kL << T
is similar to our
Vdist= l,/t << c
(159)
It is not our purpose here to criticize the parabolic method. Each method has its advantages and disadvantages, and the reader is certainly encouraged
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
347
to read the literature, which for the case of the parabolic method is extensive. In the Soviet literature, this is often referred to as the Markov approximation. Our primary purpose here is to indicate that the parabolic method and our method for the Wigner transport equations do not differ a great deal in their physical assumptions. Dashen’s criterion (iii), that the RMS multiple-scattering angle, i.e.,
KP2)(WW’*
<< 1
( 160)
is small, differs from any of our physical assumptions; however, the method used to solve the resulting transport equations of our method is certainly affected by this condition. Yet the stochastic transport equations can be used when the RMS scattering angle is not small, as is the case in ordinary transport theory. Of greater interest is Dashen’s alternative for the Gaussian assumption for I. This alternative is that kL( p 2 ) 1/2 is small. This is the equivalent of our condition on the stochastic MFP dB as given by our Eq. (13). In view of this, we may write this condition as
dB >> 1, (161) This concludes our discussion of the physical assumptions of the parabolic method. We now return to the principal purpose of this work.
VIII. THECOHERENT WIGNER FUNCTION Of all the field quantities that may be calculated by any of the existing theories, the coherent part of the field is the most straightforward. Physically, we here define the coherent part of the Wigner function as that part of the original coherent incident wave remaining after interaction with the stochastic dielectric. In this section we shall address this from the point of view of our stochastic transport equation for the ensemble-averaged Wigner function. We shall also compare our results to the results of some of the other methods. In keeping with our interest in the problem of multiple scattering, we limit our discussions to path lengths greater than one mean-free part in the medium; i.e., greater then our dB, as defined by Eqs. (13), (123), and (124). For a discussion of the coherent field at short distances, the reader is referred to the early work of Wheelon (1959). Our most general result for the coherent part of the Wigner function may be obtained from Eq. (120)by neglecting that part of the result obtained
348
D. S. BUGNOLO AND H. BREMMER
by way of the term R,. Following this procedure, we obtain
=
o2
C ( F Sj - kikj
dk’ do’Pjl(k- k’, o - 0‘)
J
This result contains the effect of the blurred resonances and all of the crosspolarization terms. Equation (162) must be used where the dielectric is anisotropic or where the fluctuations of the dielectric are strong. We now proceed to a simple case. Let us assume that the dielectric fluctuations are sufficiently weak such that we may use our self-consistent approximation and that the effects of the blurred resonances are removed by our usual arguments. For this case, Eq. (162) reduces to the following:
a nk3 k . - ( Wii(r,k ) ) = - 2( Wii(r,k)) ar 2
s
dk’P(IK1)G(ko- k’)
(163)
with IKI = Ik’ - kl. The above could also have been obtained by way of Eq. (151). This follows from a consideration of the details of the second term on the RHS of Eq. (151), which is the term obtained by way of R,. A First Approximation for the Coherent Wigner Function
We may proceed with our solution for the coherent part of the Wigner function by obtaining a solution of Eq. (163). The integral on the RHS may be written as follows:
The spacewise spectrum of the dielectric fluctuations for this case may be written as
4n
dR exp(iK.R)(c(r)E(r
+ R))
(165)
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
349
Historically, our function P ( K ) was first defined as S(K), where these are related by way of P(IK1) = (8n3)-’S(K)
(166)
This should be noted in comparing our results to those presented previously by Wheelon (1959) for the case of a single scattered short path, or by Bugnolo (1960a,b) for the case of multiple scattering on a long path. The vector length IKI in the above is related to the vectors k and k by way of IKI = Ik’ - kl. We shall approximate the length of this vector by
K
E 2k
sin 38
(167)
It is evident that by so relating K to k, we have neglected magnitude changes in the wave number k resulting from the effects of the stochastic dielectric. Since we are dealing here with our Eq. (151), which is valid in the timeinvariant case, we may neglect all doppler effects as well as all shifts in k due to the internal decay of the turbulence; however, by neglecting the effects of the dielectric on the magnitude, we have indeed made an approximation. It is self-evident that we have included the effects of the stochastic dielectric on the direction of k. The differential scattering cross section per unit volume per unit solid angle may be obtained by way of a first Born approximation as discussed previously. This is here related to our P ( K ) by o ( K ) = *(nki)P(K)
(168)
We put Eq. (164) into a more familiar form by noting that
dR’ = sin 8’ do‘ dcp‘
(169)
Integrating Eq. (164) over k‘ and using Eqs. (168) and (169), we obtain
k
.-dra (Wii(r,k))
= - ko( Wii(r,k))
f
dQ’a(K)
We next define a first approximation for the total stochastic cross section for scattering by the stochastic dielectric in a manner similar to that first introduced by Bugnolo (1960a,b), namely, Q,(ko) =
f
dR’o(K)
with K = 2k0 sin @‘, so as to obtain the result
350
D . S. BUGNOLO AND H. BREMMER
Y
FIG.
5. Geometry for a plane wave normally incident on the stochastic dielectric half
space.
Before proceding we note that we have here three equations for the three diagonal components of the coherent Wigner function. To this approximation, the equations are independent of each other. In view of this we may search for a solution for any given component without any loss of generality. Let use consider the case of a monochromatic plane wave normally incident on a stochastic half space as illustrated by Fig. 5. It is evident that the only component of the Wigner function that does not vanish at the boundary is given by Eq. (136), which we repeat here for convenience: W,,(O>t ; k, 0)= 6(0
-
o,)W, - k x ) 6 ( k , ) W , ) ( G )
(136)
We propose that the solution of Eq. (172) for this geometry and boundary condition is (173) (wY,(x,ko)) = wyy(O, t ; k, o)exp[:-Q,(ko)x] We suggest that this is the correct self-consistent result for the coherent component of the Wigner function to a first approximation in that we have neglected the effects of the dielectric fluctuations on the magnitude of k (hence the use of k , in our result). [A detailed derivation of Eq. (173) may be found in Bugnolo (1983).] We may specialize our result so as to make numerical evaluation possible by beginning with Eq. (108) for P ( K ) and by using the definition for Q,(k,) as given by Eq. (171). Hence we obtain
35 1
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
This may be integrated directly without difficulty:
2n k 0 -- -
(175)
10
We note that within the constraints of our first approximation and our model for P ( K ) , Eq. (175) is exact and may be used to determine Q,(ko) for any wavelength. There are two models which we believe to be of particular interest. These are the exponential model for the spacewise spectrum and the Kolmogoroff model. The exponential model may be obtained from the above by taking the parameter p equal to 1/2, whereas we obtain the result for the Kolmogoroff model by taking p equal to 1/3 in Eq. (175). For the exponential case we may write
whereas for the Kolmogoroff case, we may write
+
Q,(ko) = A ( 2 ~ ~ ~ l o / i l ~1) ( ~[I~ ) { 4(2do/&)2]-5’6}
(177)
with A = 0.7468342005. In obtaining this value for A we have used the numerical data for the r function given in the Handbook of Mathematical Functions (Abramowitz and Stegun, 1964). These tables are given to 11 significant figures and should be adequate for our present task. Our numerical examples of Eqs. (176) and (177) have been plotted as Figs. 6 and 7. At very high frequencies we may neglect the factor within the brackets. At low frequencies this factor, which is labeled 2 in Figs. 6b and 7b, may be used to define a correction factor which is presented as a function From these results we may conclude that the high-freof the ratio lo/l0. quency approximation may be used with little error when the ratio Zo/lo 2 10. Later in this work we present the results of computer simulations of the stochastic transport equation in wavelength ranges of 1 cm to 1 mm, with lo values of 10 to 1 m and ( E ’ ) from 1 x lo-’’ to 1 x lo-”. In Figs. 6a and 7a we have plotted the function exp[ - Q,(ko)R] for the exponential and Kolmogoroff space-correlation functions. We have done this for four different intensities of the turbulent fluctuations, ranging from weak to very strong. We note from these results that the coherent component will be larger in an atmosphere discribable by way of a Kolmogoroff spectrum for the dielectric fluctuations. Although this is evident from the equations, Figs. 6 and 7 provide us with examples of these differences at a wavelength of 3 mm.
t
Range (km) (a)
FIG.6 . (a) Plot of exp( - Q,R)for exponential case: A,, = 3 mm, .I exponential correlation: Q, = [ ( 2 d o ) / ~ ~ ] ( & (m-'). *)z
Curve
Strength
A B
Weak Mildly strong Stronger Very strong
C D
=
(EZ)
1 x 10-12 2 x lo-'* 5 x 10-12 1 x lo-"
10 m. (b) Low-frequency correction factor,
MFP (km) 45.59 22.195 9.118 4.559
1 .I
O.! 0.:
t
0.
0.0
0.0
0.c
0.02
1.o
0.1
10
*
Ickfko
b) FIG.7. (a) Plot of exp(-Q,R) for Kolmogoroff case: 1, = 3 mm; 1, factor, Kolmogoroff correlation: Q, = 2.344[(2n1,,)/A~](e2)z(m-'). Curve
Strength
A B C D
Weak Mildly strong Stronger Very strong
(4 1 x 10-'2
2 x lo-'' 5 x 1 x lo-"
=
10 m. (b) Low-frequency correction
MFP (km) 61.6 30.55 12.22 6.11
354
D . S. BUGNOLO AND H . BREMMER
This concludes our discussion of the first approximation for the total stochastic cross section for scattering by an isotropic dielectric contained in the half space defined by x > 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.
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
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
D . S. BUGNOLO AND H. BREMMER
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)
(180)
We now proceed with the integration over k'. Let this integral be denoted by CD where
s
CD = *(zkg) dk'P(IKl)G(k' - ko)(Fxx(r',k'))
(181)
dk' = 'k d k sin 8' do' d4'
Integrating over k' by way of the 6 function yields 0 = $(zk:)
s
dQLfYIK/)(f'xx(r',ko))
(182)
where d o ; = sin 8' do' d4'. Using Eq. (180) and our definition for the differential cross section per unit volume per unit solid angle o, as well as Eq. (182), it follows that the integral equation may be written as
(183)
For the stochastic differential cross section o ( K ) ,we shall use the models
Once again two models are of interest. The exponential space correlation for the stochastic dielectric, which may be obtained from the above by taking
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
351
the parameter p = 1/2, and the Kolmogoroff space correlation function, which may be obtained from the above by taking p = 113. We shall obtain numerical results for these two cases for an effective aperture A,, = 1 m. B. An Algorithm for the Computer Simulation of the Stochastic Integral Equation for the Case of a Very Narrow Beam
Before entering into a discussion of our computer algorithm or method for programming the computer, it is necessary to call the reader’s attention to a number of interesting and important properties of our integral equation for the Wigner function. First, we note that the integrals on the RHS contain the solution itself in the form of (F,,(r’, ko)). Because of this, the integral equation cannot be integrated directly because (Fx,.(r’, k,)) is unknown for all r > 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, << L = RIM
(185)
On the other hand, the extent of each section must be bounded in the large such that the wave within any given section does not undergo multiple scattering with any degree of probability. We therefore propose to bound
FIG.9. The cone of volume V partitioned into M sections of linear extent RIM.
358
D. S. BUGNOLO AND H. BREMMER
the linear extent of each section, denoted by the length L, by the condition 1, << L << Qs-'(ko) = dB1
(186)
where, once again, dB is the stochastic MFP in the right half space z 2 0. We next address the details of our simulation. Let us consider the ith section. The function (Fxx(r',k,)) within the integral on the RHS of Eq. (183) must be estimated by way of a numerical simulation. We propose to do so by summing the contributions from all previous sections, i.e., from m = 0 to i - 1. In summing these incoherent contributions, we scatter by way of the empty space characterized by the dielectric constant z0. These contributions add directly because they are statistically independent by virtue of their separation in space. (We recall that the extent of each section must be large compared to lo, the mean scale size.) To the sum of the incoherent components we propose to add the coherent component, calculated by way of Eq. (173). The first integration on the RHS of Eq. (183) is over the variable dQ;. Let us consider this integration by way of the coordinate point at the center of the ith section. We recall that dQL = sin 8' dd' d4'. It is self-evident that the angle 4' varies from 0 to 2n in this geometry. On the other hand, the range for the angle 8' is not as obvious. A number of possibilities suggested by the integral are to bound 8' by the angle to the receiving device or by the initial angle of the beam BETA/2 or by, say, the 10- or 20-dB points in the forwardscattering angle of o(K).We propose to use both the 20-dB point and the angle of the beam BETA/2 as our choice for what will be denoted by the angle B in our programs. The details of the conditions for the angles in the case of the 10- or 20-dB-point bounds will depend on the spectrum used for the dielectric fluctuations. For the case of the exponential spectrum we obtain
8' = 3i1,/4nlO 10-dB point
(187)
8' = 3L0/2n10 20-dB point
(188)
whereas for the case of a Kolmogoroff space correlation function, we obtain 8' = 3.17L0/4w10
10-dB point
(189)
8' = 3.366~,/2d0 20-dB point
(190)
A study of the integral equation (184) will indicate to the reader that the effects of our choice for the angle B will be frequency dependent. At low frequencies, where the angle of scattering may be large, the 20-dB point could easily exceed the angle BETA/2. On the other hand, at very high frequencies, BETA/2 will exceed the 20-dB angle for any antenna beam widths of practical interest. In view of this, we expect that our results will be of greater practical significance. However, we note that any reduction of
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
3 59
our numerical results in practice must consider the effects of the receiving antenna. The last integration in the RHS of Eq. (184) is over r'. This has the effect of taking the results of the integration over dQ; section by section and transferring the results of each section to the end of the path. We propose to approximate this part of the integration by multiplying the results of the integration over dokby the volume of the ith section V(i) and the function ~XPC-QAR- Y - W2)I 16nZ(R- y - H/2)2 This result must be summed over all sections by way of integrations (DO loops in FORTRAN). There remains the problem of the first section. We begin by noting that the length of each section L is small compared with the stochastic MFP d,. We propose to estimate the magnitude of the incoherent component in section zero (see Fig. 11) by way of a first Born approximation. The magnitude of the coherent field is to be estimated in the usual manner. In all coherent field estimations, we use the center of each section as our point of calculation. It is our expressed hope that the above-detailed discussions of our method of approximation for all components of the integrals will enable our readers to follow the computer programs. Before proceeding to the first program, we list all computer variables in alphabetical order. C . Variables Used in the FORTRAN Programs
B BETA CH DELTA FREE F(N) WN)
WN) LAMBDA LO MFP RANGE QS
The angle for integration over 8' in the subroutine INT(IS, B) The angle of the narrow beam The coherent field at the end of the path
(4
The free-space attenuation from the source to the end of the path: 1/16n2R2 A subscripted variable used to evaluate the double integral The coherent field in the Nth section The incoherent field in the Nth section The wavelength in meters The mean scale size in meters The stochastic mean free path in meters: MFP = dB The range in meters from the source to the path end The total stochastic cross section in meters-'
The programs contain some other notation as well, but that notation should be obvious after a study of Section IX,B and the programs.
360
D. S. BUGNOLO AND H. BREMMER
D. Experimental Program Number One
Our first program was written in an effort to understand how the number of sections M , used to evaluate the double integrals, affects the magnitude of the Wigner function at the end of the path. In view of the condition given by Eq. (186), we expect that our results for the Wigner function will depend on the number of sections. For small M , we may violate the condition RIM << d,, whereas for large M , we may violate the condition RIM >> I , . Intermediate between these two bounds, we expect to obtain a result that approximates the correct solution for each simulation. Although we have performed numerous computer simulations in order to gain an understanding of these effects, the results are always similar. We next present a listing of the FORTRAN program for Experiment Number One. Note that the program contains a number of WRITE commands that have been c'd out. These may be used to trace the program should the user wish to do so. The program was written using double-precision Real *8 in an effort to minimize roundoff error in summing the incoherent components. The integration subroutine was written using available subroutines on the VAX-11 obtained from the IMSL (1979). Subroutine DCARDE: numerical integration of a function using cautious adaptive Romberg extrapolation was used. We note that although our program was written using the FUNC above, any other model may be used for P ( K ) , including future models based on more complete and rigorous theories of turbulence. C C
I N P L I C I T KEALtH(A-2) EXPERIUENTAL PROGRAU NUUBER 1 YRITTEN BY DR. D.S. BUGNOLO INTEGER NrPrLpUrS DIUENSION 1 1 ~ 0 ~ 1 0 0 0 ~ r I Z ~ O ~ 1 0 0 0 ~ r I % S ~ O ~ 1 0 0 0 ~ r V ~ O ~ 1 0 0 0 COUUON /FUNARG/ PIrPHvOELiA-LAUBDA TYPE Sr'EXPERIUENTAL PROGRAM NUNI3f.R 1:' TYPE tr' NAWHUW B E A M ALGORITHM' TYPE tr' KOLUOGOROFF CORRELAIION' TYPE LOr'ENTER RANGE' ACCEPl t r R TYPE I O r ' E N T E R BETA' ACCEY i * I BETA TYPE I O r ' E N T E R UAX U ' ACCEPT t r M A X TYPE 1 0 r ' E N T E R M I N N' ACCEPT S r M I N TYPF I O r ' E N T E R STEP SlZE I N M ' ACCEPT * * S T E P TYPE 10s'CNTER ( E P S l L U N t t 2 ) ' ACCEPi **DELTA PI~3*1415926535897932 TYPE 1 0 r ' E N T E R UAUELCNCilH I N METERS' ACCEPT trLANBDA TYPE 1 0 r ' E N T E R UEAN SCALE S I Z E ' ACCEPT t r L O PH=(2SPISLO/LAMBUA) U R I T E ( 5 r S ) 'EXPERINENIAL PROGRAM NUMBER ONE'
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
URITE(5rt) ' NAKROU BEAM ALGOHITHH' URITC(5rt) ' KOLHUGORUFF CORNFLAllON' URI'TE(Sv I)'RANGE.: ' t R URIlE(5rX) 'BETA= ' , B E T A URITE(5rt) 'EPSILON**S= ',DELTA URITE(Sr8) 'LAHBDAr 'rLAHBDA URITE(Srt) 'LO= 'rLO BEPI C4LL INTl(1SrB) 0S=IS URITE(5rt) 'OS= ' 9 0 6 URITE(5rt) 'HFPs 'rl/RS *
C C
C C C C
ANGLE FOR INCOHERENT FOWUAKU SCATTER B=BETA/2 URITE(5tt) 'BETA OF ci= 'PB CALL INTl(KSt8) G=IS YRITF.C5rIl 'G= '90 DO 500 S=HINrHAXrSTEY URITE(5t40) S H=R/S I=O u=s-1 DO 400 N l O r U URITE (5rt) 'N= ' t N Y=N*H CALCULATE THE COHERFNT COHPClNhNT
C C C
C
C
100 C
150 C 200
250
300
C C
C C
C
IF
361
362 400 C C C
D. S. BUGNOLO AND H. BREMMER CONTINUE PRINT THE RESULTS CH=(EXP(-UStR))/((4tPI*~)$*~) WRITE ( S p Y I ) 'COHERENT COMPUNENT='rCH U R l T E ( 5 v X ) 'INCOHERENI CUMf'UNENT~'v1 SUM-ItCH U R I T E ( 5 9 8 ) 'SUM='vSUH FREE=1/(14tPI*R)**2) WR1TE ( S 9 ) 'FREESPACE=: ' r FREE ATTEN=lOXLOGlO (SUM/FREE) U R I T E ( S v X ) 'ATTENUATION I N OB RELATXVE TO 1 FREESPACE UIGNER I N T E N S I T Y = ' r A T T E N CONTINUE STOP FORMAT ( ' S ' ? A ? ' ' ) FORHAT(/,' n= ' 9 13 ) FORHAT(' ' v A 2 r 1 3 v A Z r l P E 2 4 . 1 6 ) END
*
500 10 40 50
SUXKOUTINE I N T l ( 1 S v L ) I M P L I C I T REhLtB(A-Z) FUNC E X T E RNAL INTEGER IERR COMMON /FUNARG/ PIvPHvUELTAvLAMHUA ACCl=lE-S ACCZ=O LOW=O HIGH-8 IS=llCADRE( FUNCILOW r H I G H ?A C C l PACCZIEKKOR~ I F ( I E R H .NE. 0 ) THEN U R I l E ( 5 v X ) 'ERROR NUMBER ' v I E R R 'IS= ' r ~ ~ WRITE(J,YI) U R I T E ( 5 v t ) 'ERROR BOUND I S ',ERROR END I F RETURN END
IERK)
FUNCTION FUNC(THETA) I M P L I C I T REALXB(A-2) COMHON /FUNARG/ P I ~ P H ~ D E L T A I L A M B I M A=.6217870772 FUNC=(AtZ. $PI/LAMBIIA) t ( P H S X 3 )*IIEL'I A * ( DSIN( 1HETA)
)/
1 ((1~+4.S(PH1$2)$(~SIN(THETA/2.))1$2)X1(11~/6~)) RETURN END
1. Results of a Computer Simulation for the Case of a KolmogorofS Space Correlation Function
This program has been run at many different wavelengths, but we have elected to present some typical results for 3 and 1 mm. These are within the range of wavelengths used later in this work, namely, 1 cm to 1 mm. We have used two values for the mean scale size lo, i.e., 10 and 1 m, whereas ( E ~ ) ranged from to This range is typical of weak to very strong fluctuations of the dielectric in the earth's troposphere. Very strong fluctuations may occur as a result of some natural or unnatural phenomena. For
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
363
our propagation range in these examples, we have used the interesting case of 10 km. Our numerical results for these eight sample computer runs have been plotted as Fig. 10a and b. It is essential that these results be interpreted with an understanding of the approximations. We begin by noting that the condition as given by Eq. (186) must be satisfied. However, these bounds are “weak,” in the sense that << or >> 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 << 10,00O/M << MFP
(192)
where the MFPs are given for each run in the Fig. 10a. In order to understand our proposed procedure, some reference must be made to the character of Eqs. (183) and (184).The final integration is over the variable r’. It is over this integration that the parameter M is used to subdivide the integration over the volume of the cone of angle BETA. As is the case for ordinary numerical integration, we expect the numerical result to approach its correct value as the number of subdivisions increases. However, since this is a stochastic problem we are also faced with the weak upper bound of Eq. (191). We have used the double-integration subroutines in the IMSL (1979) to study the effects of a finite volume of integration on our estimates for the differential scattering cross section o(K)and have found that this effect is weak for the ranges in question. By this we mean that an integration volume of ten mean scale sizes in extent will not greatly affect the numerical value. In view of the weak bounds on the parameter M and the fact that all subsequent runs of our programs will take I, = 10 and 1 m, we have elected to vary the parameter M in all runs such that the LENGTH variable of Experimental Program Two is always equal to 100 m. This will ensure that the length of any section will always be very small compared to the applicable MFP, whereas the lower weak bound is still satisfied, even for the cases when I , = 10 m. It should be noted that this choice for the variable LENGTH is not to be taken lightly, since the final values obtained for the attenuation will be weakly dependent on this choice for propagation ranges of 20 km or less. This very weak dependence is apparent from the study of our two samples, Fig. 10a and b. This conclusion is based on the results of over 100 computer runs; however, we should like to note at this time that our choice of
364
D. S. BUGNOLO AND H. BREMMER
-2O [
L 0
-
A B
I
10
20
30
40 50 60 Number of sections, M
70
80
90
100
70
80
90
100
t
(a)
O r
-m -2
-
-
A B
C
-0
5-4-
+-
3
D
E -6 -
2
I 0
I
10
20
30
40 50 60 Number of sections, M
(b)
FIG.10. Variation in attenuation of the Wigner function relative to the free-space Wigner function as a function of the number of sections M (Kolmogoroff case):
MFP (km)
10-12
A B
C D I, I,
= =
2 x lo-'' 5 x 10-12 1 x lo-"
61.1
30.553 12.221 6.11
10 m ; I,, = 3 m m ;range, 10 km. 1 m ; I , = 1 m ;range, 10 km.
67.896 33.948 13.576 6.789
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
365
LENGTH equal to 100 m does lead to difficulties at very long ranges of, say, 100 km for reasons that are related to other assumptions inherent in our narrow-beam algorithm. These effects at very long ranges are discussed in Section IX,F of this work since that section is devoted to problems worthy of future study. In conclusion, our extensive numerical study of the validity of our narrow-beam algorithm by way of over 100 computer runs indicates that our choice for the new variable LENGTH equal to 100 m is good for propagation ranges of 20 km or less in the earth‘s atmosphere.
E . Experimental Program Number Two This is the principal program for our computer simulations of Eq. (183). In this program we seek to develop an understanding of how the Wigner function varies with the range of the propagation path for the case of a very narrow beam. We have used the same algorithm as that used for Program Number One, modifying the first program so as to input our new variable LENGTH. This variable is used within the program to fix the parameter M , given any particular range, by way of the Eq. (193): M = S = JIDNNT(R/LENGTH)
(193)
This takes the integer value of the division of the two double-precision numbers. The variable LENGTH will be taken as 100 m in all subsequent runs, with the exception of some further work reported in Section IX,F. This is justified once again by the fact that the mean scale size 1, will always be taken equal to or less than 10m. It follows that the MFP, d g , must be at least 1000 m if this choice for LENGTH is to be justified by way of Eq. (186). This is the case for all of our sample runs in this section. We again present a listing for the case of a Kolmogoroff spacewise spectrum for the dielectric fluctuations. This is given by the variable FUNC in the subroutine. However, we note that our program has been written so as to call an integration subroutine of great “power” and that it may be used for any other spectrum in the future. Suggestions for future work are contained in Section IX,F of this work. We again choose the angle B equal to half the antenna beamwidth. This will ensure that our result includes all scattered components that remain within the main beam of half-angle BETA/%. Returning to the program itself, we should like to call the reader’s attention to the PRINT THE RESULTS section, beginning at the bottom of p. 367. It should be noted that this section of the program contains both
366
D. S. BUGNOLO AND H. BREMMER
TYPE and WRITE commands. When running this program, the TYPE commands are printed to the screen, while all WRITE commands are printed to a file on the disk. In this way, we may observe the results of a program run on the screen and delay any decision to print a hard copy on the printer until the run is complete. The program also contains a number of WRITE commands that have been c'd out. These may be used to TRACE the program. A listing of Experimental Program Number Two is now presented (a listing of this program for the case of an exponential space correlation function may be found in Appendix 1).
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
c
ALC1,F F O R
I N C O H E R E N T FORNARD S C A T T E R
C C
C C C
C
c t
r
100
r
150
t 200 250 300
C
c
c C
C 400
c
c c
CNEh
367
368
D. S. BUGNOLO AND H. BREMMER
1. Results of the Computer-Range Simulations for the Case of an Exponential Space Correlation Function
We have performed a number of simulations using the exponential space correlation function for the stochastic dielectric because this correlation function has received a great deal of attention in the literature. Our Experimental Program Number Two is based on our narrow-beam algorithm. In
369
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
1 10
0
20 Range (km)
30
40
*
(a)
a -20
1
MFP (km) Fig. part
Curve A ( E ~ )= 1 x 1 0 - l ~
(a) (b)
506.563 45.594 455.934 5.066 50.661
(c)
(4 (e) a (EZ)
(EZ)
= 2 x 10-12. = 5 x 10-12.
Curve B
(2)= 1 x lo-'' 50.662 4.559 45.593 2.533" 10.132*
1.
10
(m)
(mm)
10 10
1 (cm) 3
1 10 1
3 1 1
370
D. S. BUGNOLO AND H. BREMMER
0
m
x
.U.
6 -1
._ 4-
W
C 3 c
a
-2
10
0
30
20
t
40
Range (km) (C)
-25
I
0
2
4
6
10
8
12
14
16
18
20
*
Range (km)
(d)
I 0
I
2
*
I
4
6
8
10 12 Range (km)
(e)
FIG.11. (continued)
14
16
18
20
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
37 1
view of this we have taken the beamwidth BETA equal to 0.2 rad for all simulations. This corresponds to a beamwidth of 11.4591559", a value easily obtained by antennas in this frequency range, which ranged from 1 cm to 1 mm. For these simulations we let the mean scale size I , range from 10 to 1 m, whereas ( E ' ) ranged from 1 x to 1 x 10-l'. The above conditions are typical of those to be found in the earths troposphere. A ( E ~ )of 1 x lo-'' corresponds to the case of a weak turbulence, whereas a ( E ~ )of 1 x lo-'' may be used as an example of strong turbulence. We may expect to find strong turbulence in regions where the normal atmosphere is disturbed by some natural or unnatural phenomena. It is of importance to note that our simulations do not include the effects of molecular absorption. Since this does occur in the earth's atmosphere in the frequency range used for our simulations, this effect must be included in any practical application. This was not included at this time because we are primarily interested in studying the effects of the stochastic dielectric on the propagation path. We note that the effects of molecular absorption may easily be added to our program should the need to do so arise. The absorption will affect both the coherent and incoherent components of the Wigner function. Our numerical results for the exponential correlation function have been plotted as Fig. 1la-e. We have plotted the attenuation of the Wigner function relative to the free-space Wigner intensity in decibels. The latter is calculated by way of 1/16x2R2.From a study of the results, it is self-evident that the attenuation increases with frequency and the mean scale size 1,. We have also included our numerical result for the MFP dB. We shall not dwell on this case further because our principal results are the computer simulations for the Kolmogoroff space correlation function. 2. Results of the Computer-Range Simulations for the Case of a KolmogoroffSpace Correlation Function We have performed a number of simulations using the Kolmogorof space correlation function for the stochastic dielectric since this function i: a useful model for cases where the turbulence causing the dielectric fluctu ations is fully developed. This model for o ( K ) may be obtained from ou Eq. (184) by taking the parameter p equal to 1/3. We note again that ou model does not include the effects of dissipation because the wave-numbe dependence in this region is still open to question. Since our Experimental Program Number Two is based on our narrow beam algorithm, we have taken the beamwidth equal to 0.2 rad for all ou simulations. This corresponds to a beamwidth of 11.4591559",a value easil obtained by antennas in this frequency range. Once again the frequenc
D . S. BUGNOLO AND H.BREMMER
372 0
-
-1
m
E
0
8
4
12
16
.-cE
20
24
28
32
36
14
16
18
40
(a)
a c a2
3
0
-5
-
-10
I
a
I
0
2
4
6
8
I
10 12 Range (km)
20
*
(b)
FIG. 12. Attenuation of the Wigner function relative to the free-space Wigner intensity (Kolmogoroff case) for BETA = 0.2 rad; p = 1/3; B = BETA/2; length 100 m:
Fig. part
Curve A (E2)
=
Curve B
( 2 ) = 1 x lo-"
10
20
(m)
(mm)
10 10 1 10 1
1 (a) 3 3 I 1
~~
(4
678.973 61.107 61 1 6.789 67.899
(4 (c)
(4 (e) ~
~~
(&2)
* (&*)
= 2 x 1o-I2. =5 x
10-12.
67.899 6.110 61.1 3.395" 13.579b
373
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
0
30
20 Range (km)
10
40
(C)
I 0
2
4
6
8
10
Range (km)
(d)
Range (km)
(el
12
14
16
18
20
w
374
D. S. BUGNOLO AND H . BREMMER
range of the simulations will be bounded by 1 cm and 1 mm. We have taken the mean scale size 1, = 10 and 1 m, whereas ( E ' ) ranged from 1 x to 1 x lo-". The above conditions are typical of those found in the earth's troposphere. A ( e 2 ) of 1 x corresponds to the case ofweak turbulence, whereas a (c2) of 1 x lo-' may be used as an example of strong turbulence. We may expect to find strong turbulence in regions where the normal atmosphere is disturbed by some natural or unnatural phenomena. It is important to note that our simulations do not include the effects of molecular absorption. Since this does occur in the earth's atmosphere in the frequency range used for our simulations, these effects must be included in any practical application. This was not included at this time because we are primarily interested in studying the effects of the stochastic dielectric on the propagation path. We note that the effects of molecular absorption may easily be added to our simulation program should the need to do so arise. The absorption will affect both the coherent and incoherent components of the Wigner function. We have performed many simulations in the frequency range of 1 cm to 1 mm. From these we have selected ten range runs which have been plotted. These may be used to study the effects of propagation range on the attenuation of the Wigner function (see Fig. 12a-e). In Fig. 12a-e we have included our result for the MFP d, in kilometers. From a study of these results and those of other computer simulations, we propose to conclude the following with regard to the attenuation of the Wigner function relative to its freespace value of 1/16z2R2for propagation paths of 20 km or less. (1) Weak turbulence: (c2) = 1 x 10-l2. For 1, = 1 m (Fig. 13a), the attenuation will be less than 1 dB in the wavelength range of 1 mm or greater. For 1, = 10 m (Fig. 13b), the attenuation can be appreciable at wavelengths of about 2 mm or less. For 1, = 1 m (Fig. (2) Mildly strong turbulence: (c2) = 2 x 13c), the attenuation will be less than about 2 dB in the frequency range of interest; wavelengths of greater than 1 mm. For 1, = 10 m (Fig. 13d), the attenuation will be appreciable and exceed 3 dB at wavelengths of less than 3 mm. For 1, = 1 m (Fig. 13e), (3) Stronger turbulence: (c2) = 5 x the attenuation will be less than 3 dB for wavelengths greater than about 1 mm. For 1, = 10 m (Fig. 13f),the attenuation will be appreciable and exceed 3 dB at wavelengths of less than about 5 mm at a range of only 10 km. For 1, = 1 m(Fig. 13a), (4) Verystrong turbulence: (c2) = 1 x lo-". the attenuation will be appreciable and exceed 3 dB at wavelengths of less than 4 mm. For 1, = 10 m (Fig. 13h), this example is the most severe of those
375
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
(a1
-
m 9
-1.2
1.o
2
3
I
5
10
1a
2
3
5
10
Wavelength (mm)
Wavetength (km)
C
.-c a
p
(dl 1 .o Wavelength (mml
2
3
5
10
*
Wavelength (mml
FIG.13. Attenuation as a function of wavelength for frequency ranges of 10 and 20 km and 5 and 10 km [see pp. 376 and 377 for parts (e)-(h)]:
376
D . S. BUGNOLO AND H. BREMMER
v
1 .o
10
Wavelength (km)
(el
Wavelength (mm)
(f)
FIG.13. (continued)
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
1 .o
2
3
5
Wavelength (mm)
(9)
0-
-5-
-m
-10
-
XI C
._ g -15C 3
c
2
FIG.13. (continued)
10
377
378
D. S. BUGNOLO AND H. BREMMER
included in this work. The attenuation will exceed 3 dB at wavelengths of about 5 mm or less. The attenuation at 2 mm will exceed 10 dB at 20 km. From the results of our simulation it is possible to conclude that attenuation due to scattering by a time-invariant stochastic dielectric is of some consequence in the wavelength range of 1 cm to 1 mm. At some wavelengths this can easily exceed the attenuation due to molecular absorption, which has not been included in this work. For example, at a wavelength of 1 mm, the molecular absorption will result in an attenuation of about 1.0 dB/km, or 10 dB in a path of 10 km. This is exceeded by the attenuation due to scattering when the turbulence is only mildly strong (Fig. 13d) or stronger (Fig. 13f-h). This concludes our discussion of the results of the simulations by way of our experimental programs. F. Conclusions and Suggestions for Future Work
In this Section IX we have presented our results for the computer simulation of our Eq. (184) in an effort to prove that computer simulation may be used to provide us with a solution of the stochastic transport equation. In order to proceed we have written an algorithm for the special case of a very narrow beam launched into a time-invariant isotropic stochastic dielectric free of other sources. The results of numerous simulations, some of which were presented in this work, indicate that computer simulation is indeed feasible on a VAX-11 computer system, given the availability of the system CPU time required for the runs. This amounts to some 2 min per range run. Our results indicate that the attenuation due to scattering is of some consequence in the wavelength range of 1 cm to 1 mm. Our simulations have been limited to those considered by way of our narrow-beam algorithm and any improvement would require a much larger amount of computational time. It is evident that there is ample opportunity for further work in this area. We should like to suggest the following as worthy of further study by way of the large digital computer. In this work, we have limited our simulations to an approximate form of the stochastic transport equation, namely, that given by our Eqs. (1 53) and (154), for the case of a time-invariant isotropic stochastic dielectric. At very long ranges, or in regions of very strong scattering, we expect that it may be necessary to consider Eq. (149). This equation includes the effects of the interactions between the various components of the Wigner tensor. We note that although our simulations are valid at ranges of about 20 km or less, the assumption that the LENGTH variable may always be taken as equal to
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
379
100 m can lead to difficulties at very large ranges. In Figs. 14a-f we have presented the results of simulations to very large ranges at a wavelength of 3 mm, a mean scale size of 10 m, a beamwidth of 0.2 rad, and a Kolmogoroff spectrum with mean squared fluctuation levels of 1 x 10-l2, 5 x 10-l2, and 1 x lo-". In order to demonstrate the effects of changes in the length parameter on attenuation, we have performed all simulations for a LENGTH of 100, 150, and 200 m. From an inspection of Fig. 14a,c, and e, it is evident that changes in the LENGTH parameter do not have a large effect on the attenuation for ranges of 20 km or less; however, the effects can be appreciable at large ranges. In these figures, we have also included the coherent component so that the reader may study the differences between the magnitude of the coherent and incoherent components of the normalized Wigner function. We attribute the apparent failure of our narrow-beam algorithm at very large ranges to our neglect of two effects. As the range increases, the width of the beam becomes appreciable and requires partitioning. At large ranges the coupling between the various components of the Wigner tensor may not be negligible. This is worthy of further study; however, the amount of computational time will increase significantly. Another interesting case is that given by our Eq. (155) for the case of a time-invariant but anisotropic dielectric. Since anisotropic dielectrics are to be found in nature, the solution of this case could be of some practical interest. However, we have reason to believe that an equation of the form of Eq. (149) must be used, since the anisotropic dielectric will scatter energy into other Wigner components. We again note that our computer programs have been written so as to call integration subroutines for the integrals involving o(K),i.e., P ( K ) , the spacewise spectrum of the dielectric fluctuations. Another problem of interest is that of the effects of the dissipation region of the dielectric fluctuations on the results of our simulations. These effects will most likely be of significance at optical frequenciesand, perhaps, to a lesser degree in the millimeter region. This, of course, will depend on the wave number of the eddies characterizing the dissipation region (see Fig. 1). In the millimeter range, we expect to find an effect when the dielectric fluctuations are strong, or perhaps when the turbulence is not fully developed, but strong. Such fluctuations may be expected to occur when the environment is stressed in times of conflict or by some natural phenomenon. This concludes our discussion of the possibilities for future work by way of the large digital computer. Perhaps the reader can easily formulate plans for some additional work of practical as well as theoretical significance.
380
D. S. BUGNOLO AND H. BREMMER 1 .o
LENGTH 0.5 -
03
0.05
-
0.03-
0
20
40
60
80
100
120
140
la)
c
a
O h -
-2.5
LENGTH
-5-
-10 -
-7.5
-12.5 -15
-
-
10
20
30
40
60
50
(C)
FIG.14. (a, c, e) Effects of variation in length parameter on the attenuation; (b) normalized coherent and incoherent components; (d) effects of variation in length parameter on the magnitude of the incoherent component; ( f ) effects of variation in length parameter on the magnitude of the coherent and incoherent components: p = 1/3; BETA = 0.2 rad; 1, = 3 mm; I, = 10 m.
(b)" (c) a
1 x lo-'* 1x 5 x 10-12
fIiol6= 1.607 x
61,106 61,107 (km) 12,221
5 x 10-12 1 x
(f)
rad; Kolmogoroff case.
1 x lo-"
12,221 6,110 6,110
38 1
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
0
20
10
\I
L ~ G T H 100 rn
-Y Range (km) (f)
LENGTH
40
50
382
D . S. BUGNOLO AND H . BREMMER
X. CONCLUSIONS In this work we have attempted to present the derivations of our principal results [Eqs. (119) and (120)] for the stochastic transport equations for the Wigner function, in a manner that may be followed by the reader. These equations are obtained for the most general case of a time-variable anisotropic stochastic dielectric which may contain sources. Equation (119) is a transport equation for the real part of the Wigner function, whereas Eq. (120) is our result for the imaginary part. At present, we are of the opinion that the Im( Wik)part is of less importance than the Re( Wik) part. We have also obtained an integral equation of the second kind for the Wigner function in a general stochastic dielectric [Eq. (121)]. In addition, we have tried to obtain equations for the Wigner tensor function in a number of interesting special cases, namely, that of a plane wave incident on an isotropic dielectric [Eq. (141)] and that of a monochromatic plane wave in an isotropic time-invariant dielectric [Eq. (151)]. We have compared our fundamental physical assumptions with those of the parabolic equation method and found that there are many simularities, as well as some differences. In order to obtain some estimate of the magnitude of the effects of scattering as predicted by our transport equations, we have developed an algorithm and written a number of programs in the FORTRAN language. These have been written for the special case of a very narrow beam normally incident on a half space containing an isotropic time-invariant stochastic dielectric. In our effrot to model the stochastic dielectric we have used a function based on the exponential model and one based on the Kolmogoroff model for the spacewise correlation function of the stochastic dielectric. We have presented the results of many computer runs for the, hopefully, interesting wavelength range of 1 cm to 1 mm, taking (&’) ranging from the case of weak turbulence, i.e., (2)= 1 x lo-”, to the case of very strong turbulence, i.e., ( 6 ’ ) = 1 x 10-l1.The latter could occur in an atmosphere stressed by some natural or unnatural phenomenon. In our simulations, we have taken the mean scale size of the dielectric fluctuations 1, (sometimes referred to as the outer scale size in the literature) as either 10 or 1 m This range is expected to cover any typical value found in the earth’s troposphere. We have performed the majority of our simulations for propagation ranges equal to or less than 20 km. As a result of these simulations for the cases under study, we have concluded that multiple scattering is significant in the wavelength range of 1 cm to 1 mm, and that this may exceed the attenuation due to molecular absorption at the shorter wavelengths. However, we also note that our algorithm, based as it is on a single component of the Wigner tensor, is probably not
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
383
very useful at very large ranges such as, say, 100 km. This is due to our neglect of the other components of the Wigner tensor and to the width of the narrow beam itself at very large ranges. We have also made a number of suggestions for future work which we hope to pursue given the support that such extensive computer simulations would require. We note that the CPU time requirements for the simulation of stochastic transport equations can be very large, depending on the approximations used in writing the algorithm and the number of Wigner components required for the simulation. It is our hope that this work will be of some value to the scientific and engineering communities.
APPENDIX 1 . A LISTING OF EXPERIMENTAL PROGRAM NUMBER Two FOR THE CASE OF AN EXPONENTIAL SPACECORRELATION FUNCTION C C
TMPLTCTT k E A L * 8 t A - Z ) F I P F R l M F N f A L PROGRAM NUMBFR T h o M R T T T E N a y DR. D.s.. BUGNOLC TNTFGVR N,P,L,H,RH,RN ~IMFNSTI)~ 11(o:1000),12r0:1000)rr2S~o:lOoo~,v~o:looO~, I Frn:1nOO) C m w m / F U N A R C / PI,PH,DELTA,LAMBDA T Y P F *,*LXPEI?I*EWTAL PRnGRAM NUMRFR THO' TYPF NARRr)W REAN A I . G O R I T H M ' TYPF EXPOWENTIAL CO RRE LA TI ON ' T Y P F 10,'LNTCR YAXIMUM RANGE' n r r F P T *,HHAX TYPF IO.'ENTER M INIM UM R A N G E ' ACCFFT * r R Y I N T V P F 10,'ENTER NUMBER OF STEPS I N RANGE' PCPFPT *,HN
*,'
*,'
I)Fl,74RtfRMAX-RPIIN)/RN TYPF in,'CNTER BFTA'
ACPFPT *,BETA TYPF I0,'ENTFR bCCCPT *,DELTA
(EPSILON**?)'
P1=~.141502h5)5X979J2 T V P C i n . * m ~ wU~AVELENCTHN I CIETFRS' ACrFPT *,LAMRnA TVPV 10,'F.NTFD MEAN SCALE SIZE' DCPFPT * , L O 'rvpF i n , * E N T F R SECTION LENGTH' ACPFFT * , L E W G T t { p H = f 2* P T *LO/LAMBDA ) W P T T F(5,* ) ' E X P CHIM E Y TA L PROGRAM NUMRER T N O' NARROW PPAM ALGORITHM' HRlTFr5.*) WRTTFlS,*) *EXPOVENTIAL CO RRE LAT ION ' ' R MAX='+RMAX WQTTF(5,*) WRTTFf5,*) ' R YIN=',PMIN WQlTF(S,*) 'NO OF S TE P S IL RANGE=',RN W ~ T T F ( 5 , 1 ' RFTA Z ,@ETA WWTTFf5,*) 'FPSILON**Zr *.GELTA IVFTTF(5, 1 'l,Af+RnA= ,LPHBOA 'LO= '.LO WRTTF(S,*) ! d P T TF(S .* ) 'LENCTH=',LENGTH
'
* *
'
'
384
D. S. BUGNOLO AND H. BREMMER
r
c
C
C C
C
C C C
c c
TF 100
c
150
C 200 250 300 C
C C C C
STOCHASTIC DIELECTRIC WITH COMPUTER SlMULATION
C
400 C C
C
son 1v 40 50
C C
385
386
D. S. BUGNOLO AND H. BREMMER
APPENDIX 2. A SAMPLE OF A COMPUTER SIMULATION
387
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
1G Nk H
I
I NI EN SI 7 Y
=
388
D. S. BUGNOLO AND H. BREMMER
REFERENCES Abramowitz, M., and Stegun, 1. A., eds. (1964). “Handbook of Mathematical Functions,” Appl. Math. Ser. 55, June. Natl. Bur. Std., Washington, D.C. Barabanenkov, Yu. N., Vigorradov, A. G., Kravtsov, Yu. A,, and Tatarskii, V. I. (1971). Sou. Phys.- Usp. (Engl. Transl.) 13,551. Bartelt, H. O., Brenner, K. H., and Lohmann, A. W. (1980). The Wigner distribution and its optical production. Opt. Commm. 32,32-38. Bastianns, M. J. (1978). Wigner distribution function applied to optical signals and systems. Opt. Commun. 25,26-30. Bastianns, M. J. (1979a). Transport equations for the Wigner distribution function. Opt. Acta 26,1265-1272. Bastianns, M. J. (1979b). Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium. Opt. Acta 26, 1333- 1344. Bremmer, H. (1964). Semi-geometrical-optical approaches to scattering phenomena. M . R. L. Symp. Proc. 14,415-437. Bremmer, H. (1974). Propagation through inhomogeneous and stochastic media. AGARD Conf. Proc. 144 (March). Bremmer, H. (1979). The Wigner distribution function and transport equations in radiation problems. J. Appl. Sci. Eng. A 3,251-160. Bugnolo, D. S. (1960a). Transport equation for the spectral density of a multiple scattered electromagnetic field. J. Appl. Phys. 31, 1176-1 182. Bugnolo, D. S . (1960b). Dissertation, Columbia Univ., School of Engineering and Applied Science, New York. Bugnolo, D. S. (1960~).On the question of multiple scattering in the troposphere. J . Geophys. Res. 65,879-884. Bugnolo, D. S. (1972a). Phys. Rev. A 6,477-484. Bugnolo, D. S. (1972b). Turbulence in weakly ionoized plasma. J . Plasma Phys. 8, 143-158. Bugnolo, D. S. (1978). Multiple scattering of pulsar-signals by turbulent interstellar plasma. Sou. Astron. (Engl. Transl.) 22(1), 38-42. [Astron. Zh. 5569-75, Jan.-Feb. (1978).] Bugnolo, D. S. (1983). The coherent electromagnetic field in a stochastic dielectric. Trans. IEEE Southeastcon, Orlando 1983. (In press.) Chandrasekhar, S. (1 950). “Radiative Transfer.” Oxford Univ. Press, London and New York. Dashen, R. (1979). Path integrals for waves in random media. J. Math. Phys. (NY)20,894-920. Frisch, A. (1968). In “Probabilistic Methods in Applied Mathematics” (A. T. Bharucha-Reid, ed.), Vol. 1. Academic Press, New York. Gelford, I. M., and Formin, S. V. (1963). “Calculus of Variations.” Prentice-Hall, New York. Howe, M. S. (1973). O n the kinetic theory of wave propagation in random media. Philos. Trans. R . SOC.London 274% 523-549. IMSL (1979). “FORTRAN Subroutines in the Areas of Mathematics and Statistics.” International Mathematical & Statistical Libraries, Inc., Houston, Texas. Ishimaru, A. (1978). “Wave Propagation and Scattering in Random Media,” Vols. 1 and 2. Academic Press, New York. Mori, H., Oppenheim, J., and Ross, J. (1962). Some topics in quantum statistics: The Wigner function and transport theory. In “Studies in Statistical Mechanics” (J. de Boer and G. E. Uhlenbeck, eds.), Vol. 1, pp. 262-298. North-Holland Publ., Amsterdam. Norton, K. (1960). “Carrier Frequency Dependence of the Basic Transmission Loss in Tropospheric Forward-Scatter Propagation.” Mem. Rept. PM-83-21, Natl. Bur. Stand., Washington, D.C.
STOCHASTIC DIELECTRIC WITH COMPUTER SIMULATION
389
Novikov, E. A. (1965). Functionals and random-force methods in turbulence theory. Sou. Phys.-JETP (Engl. Transl.) 20, 1290-1294. Reickl, L. E. (1960). “A Modern Course in Statistical Physics.” Univ. of Texas Press, Austin, Texas. Tatarskii, V. J. (1961). “Wave Propagation in Turbulent Media.” Tatarskii, V. J. (1971). “The Effects of the Turbulent Atmosphere on Wave Propagation.” Natl. Tech. Inf. Serv., Springfield, Virginia. Whelon, A. D. (1959). Radio-wave scattering by tropospheric irregularities. J . Res. Nutl. Bur. Stand. Sect. D-Radio Propagation 63D. Whittaker, E. T., and Watson, G. N . (1952). “Modern Analysis.” Cambridge Univ. Press, London and New York. Wigner, E. (1932). On the quantum corrections for thermodynamic equalibrium. Phys. Reu. 40, 749.
This Page Intentionally Left Blank
Author Index Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although his name is not mentioned in the text. Numbers in italics indicate the pages on which the complete references are given. Ashley, K. L.,134, 135,143 Aszodi, G., 171, 296
A Abdalla, M. I., 125, 142 Abraham-lbrahim, S., 188,289 Abrahams, M. S., 127, 142 Abramowitz, M., 14,59, 351,388 Achete, C., 289, 297 Acket, G . A,. 131,142 Adler, I., 195, 298 Adler, J., 195, 298 Adler, S. L., 174, 180, 289 Adler, R., 5,61 Agaev, A. M., 106, 107, 147 Ahn, J., 288,289 Akita, K., 125, 142 Akkerman, Z. L., 126, 142 Aksela, H., 195,289 Aksela, S., 195,289, 298 Aksela, U., 195, 298 Aleksandrova, G. A., 126, 130,142 Alferov, Zh, I., 131, 142 Allen, G. A,, 101, 142 Allen, J. W., 142 Allred, W., 109, 121, 142, 157 Allred, W. P., 113, 119, 159 Amato, M. A., 94,143 Amy, J. W., 273, 281, 293 Andersen, V. E., 181, 185,292,296,298 Anderson, C. L., 116,143 Andrews, A. M., 115,143 Andrianov, D. G., 106, 107,143 Antell, G. R., 121, 143 Antoniewicz, P. R., 290 Aoki, K., 116, 143 Aratama, M., 243,293 Arthur, J. R., 128, 145 Arthurs, A. M., 179,290 Asaad, W. N., 193,290 Ashen, D. J., 116, 117, 143 Ashirov, T. K., 112, 143 Ashley, C. A,, 172, 290 Ashley, J. C., 176, 184, 290,296, 298
B Bachelet, G. B., 143 Bacon, F., 265,297 Badawi, M. H., 116, 143 Bahraman, A,, I 1 3, 143 Baittinger, W. E., 195, 200, 273, 281, 282,293 Baldereschi, A., 144 Baliga, B. J., 123, 132, 143 Bambynek, W., 193, 195,290 Baneji, S. K., 264,290,295 Barabanenkov, Yu. N., 344,388 Bartelt, H. O., 301, 388 Bastianns, M. J., 301, 303,388 Bastow, B. D., 149 Battye, F. L., 171, 290 Baudoing, R., 186,290 Bauer, E., 176, 184, 186, 290 Bazhenov, V. K., 102, 107,143 Beal, S. W., 143 Bebb, H. B., 126,159 Bekmuratov, M. F., 103,143 Benbow, R. L., 297 Beni, G., 172, 18 I , 290,294 Benninghoven, A., 267,271,28 I , 290 Berg, N. J., 82, 90, 143, 152 Berger, R. A., 83, 149 Bergmann, Ya. V., 132,143 Bernett, M. K., 196, 212, 290 Berz, F., 133, 143 Bethe, H. A., 176, 179,290 Betz, G., 257, 266, 273, 276, 290 Bhattacharya, P. K., 75,80,136,137,143,152 Biermann, R., 126, 155 Bimberg, D., 121, 144 Bindell, J. R.,271, 290 Bindi, R.,223,290 Birkhoff, R. D., 181, 185,242,292,296, 298 391
392
AUTHOR INDEX
Bishop, H. E., 225, 237, 238,286, 290, 291 Bishop, S. G., 92, 116,144 Biswas, S. N., 132, 144 Blackmore, G. W., 127, 144 Blatte, M., 113, 114, 144 Blakemore, J. S.,77,91, 106, 144 Blashku, A. I., 128, 144 Bludau, W., 132, 144 Blurn, J. M., 126,154 Boborykina, E. N., 112, 144 Bois, D., 81, 127, 144, 145, 158 Boltaks, B. I., 104, 105, 110, 113, 144 Bonfig, K. W., 2,59 Bonham, R. A., 241,290 Borisova, L. A., 127, 144 Bouquet, S.,261, 290 Bourdin, J. P., 181, 185, 291 Bourgoin, J., 75, 83, 85, 88,155 Bouwman, R., 163,256,257,264,281,290,298 Brauer, W., 223, 297 Braun, P., 257, 266,290 Brebbia, C., 18,59 Brehm, G. E., 84,144 Bremmer, H., 301, 304, 305, 307, 388 Brenner, K. H., 301,388 Briant, C. L., 264,265,266,290,295,297 Brillson, L. J., 75, 145 Brown, I. C., 18, 59 Brown, M., 114,155 Brown, W. J., 106, 145 Brown, W.L., 272,294 Brozel, M. R., 120,145 Bube, R. H., 94, 126, 158 Bublik, V. T., 67, 145 Buczek, D., 281,290 Bugnolo, D. S., 301, 306, 307, 308, 327, 331, 336,337,341,345,349,350,354,356,388 Bui Minh DUC,237,239,252,290 Bulman, W. E., 2,59 Burhop, E. H. S., 162, 179, 193,290,291 Bykovskii, V. A., 105,145 C
Cailler, M., 181, 185,291,292 Callaway, J., 181, 291 Caroli, B.,188,289 Caroli, C., 188, 289 Carriere, B., 213, 288, 291 Casey, H. C., Jr., 127, 131, 134, 145
Chakravarti, A. N., 132, I45 Chambers, A., 171,293 Chan, P. C. H., 132, 145 Chandrasekhar, S., 329, 388 Chang, C. C., 130,145 Chang, C. G., 163,291 Chang, L. L., 69, 145 Chantre, A,, 81, 113, 145 Chapman, R. A,, 104, I45 Chattarji, D., 162, 193,291 Chelikowski, J. R., 196, 197, 291 Chen, M. H., 195,291,298 Cheng, L. J., 72, 145 Chi, J.-Y., 133, 145 Chiang, S. Y., 68, 145 Chiao, S.-H. S., 11 1, 145 Cho, A. Y., 100, 128,145,154 Chollet, L., 267, 298 Chou, N. J., 279,291 Christman, S. B., 196, 198, 291 Chu, W. K., 256, 274,276,278,291,293 Chung, M.S., 223,291 Chwang, R., 15, 36,59 Cini, M., 207, 291 Citrin, P. M., 196, 198,291 Clark, D. J., 91,159 Clerjaud, B., 92, 93, 145 Coad, J. P., 286, 291 Coates, R., 86, 88, 145 Coburn, J. W., 288,289 Cohen, P. I., 247,248,291,292 Colby, J. W., 271,290 Collins, R. A,, 170,291 Comas, J., 128, 145 Condon, E. U., 192,205,291 Conley, D. K., 248, 271, 288, 290,292 Conway, E. J., 86, 158 Cook, Jr., C. F., 271,291 Cooke, C. J., 114,145 Corelli, J. C., 84, 154 Courant, R., 52,60 Crasemann, B., 193, 195, 291,296, 298 Crawford, J. H., Jr., 113, 114, 146 Cronin, G. R., 106, 120, 148 Crotty, J. M., 193,291 Crowell. C., 15, 36,59 D
Da Cunha Belo, M., 261,291
393
AUTHOR INDEX
Damestani, A., 107, 146 Das, B. K., 149 Dashen, R., 346,388 Davidson, R. S., 3, 60 Davies, D. E., 91, 146 Davis. H. L., 182, 296 Davis, L. E., 192, 256, 291,293,295 Dawson, P. T., 288,291 De Dominicis, C. T., 207, 295 Defrance, J. E., 163,291 Degen, P. L., 91, 128, 146 Degreve, F., 271,294 Delord, J. F., 287,291 De Mey, G., 5, 11, 12, 13, 18, 21, 23, 39, 49, 57, 59,60,61 Demidov, E. S.,105, 146 Den Boer, M. L., 247,248,291 Dench, W. A., 170, 183,260,297 Desjonqueres, M. C., 209,298 Deveaud, B., 93, 125, 146 de Visschere, P., 18, 60 Dieball, J. W., 286, 293 DiLorenzo, J. V., 127, I46 Distefano, T. H., 196, 292 Dobson, P. A., 72, 128,146 Dobson, P. S., 113, 127,149 Doniach, S., 172,290 Donnelly, J. P., 91, 146 Dove, D. B., 288,295,296 Drawin, H. W., 180, 183,255,291 Driscoll, C. M. H., 75, 146 Driscoll, T. J., 172,296 Drowley, C., 133, I48 Ducastelle, F., 209, 298 Dufour, G., 192, 195,291, 295 Duke, C. B., 172,291 Dunlap, H. L., 116, 143 Duraud, J. P., 245,282, 286, 292 Dzhafarov, T. D., 112,146
E Eaves, L., 92,93, I46 Eberhardt, J. E., 131, 156 Edmond, J. T., 120, 146 Edwards, T. W., 18,60 Einstein, P. A,, 282,298 Einstein, T. L., 248, 291, 292, 294 Eisen, F. H., 73, 91, 128, 146 Elam, W. T., 248,291,292
Elliott, C. R., 91, 146 Elliott, K. R., 86, 146 El Toukhy, A. W., 277,297 Emerson, L. C., 181, 185, 292,298 Emori, H., 127, 146 Engelsberg, S.,292 Ennen, H., 106, 107, 146, 130 Epifanov, M. S., 133, 147 Erickson, N. E., 244,296 Erlewein, J., 270, 293 Ettenberg, M., 135, 136, 147 Euthymiou, P. C., 82, 132, 150,155 Evans, S., 171, 292 Everhart, T. E., 223, 242,243, 291,297 Ewan, J., 90, I47
F Fain, Jr., J. C., 287, 291 Falk, D. S., 175, 292 Fan, J. C. C., 275,292 Farmer, J. W., 82, 86, 88, 147 Farrell, H. H., 297 Faulconnier, P., 287,295 Favennec, P. N., 91, 125,147 Feibelman, P. J., 172, 182, 207, 212, 285, 292 Ferrel, R. A,, 296 Fiermans, L., 200,293 Fink, R. W., 193, 195,290 Finstad, T. G . , 123, 147 Fisher, B., 196,292 Fisher, D. G., 267,298 Fistul’, V. I., 105, 106, 107, 147 Fitting, H. J., 173, 292 Flat, A., 133, 147 Fontaine, J. M., 282, 286, 292 Forbes, L., 107, 146 Forgacs, G., 171,296 Formin, S. V., 305,388 Fowler, W. B., 197, 298 Fox, D. C., 271,291 Fransen, F., 18,60 Frederick, P. J., 287, 292 Freund, H. U., 193, 195,290 Friedel, J., 200, 292 Frisch, A., 327,388 Fry, J. L., 181, 292 Fuggle, J. C., 21 1,294 Fujimoto, M., 116, I47
394
AUTHOR INDEX
Fukuda, Y., 248,291,292 Fuller, C. S., 108, 147
G Gaarenstroom, S. W., 165,195,200,213,262, 282,292,293 Gallon, T. E., 171,255,292, 293 Gamo, K., 128, 147 Ganachaud, J.P., 176, 181, 184,185,186,241, 291,292 Ganapol’skii, E. M., 105, 147 Garber, E. W., 242,296 Garbuzov, D. Z., 131, 147 Garrison, B. J., 272, 292 Gatos, H. C., 133, 145 Gelford, I. M., 305, 388 Gerlach, R. L., 164,247,292 Gersten, J. I., 182, 292 Ghandhi, S. K., 123, 143 Ghosh, S., 9,60 Gibbon, C. F., 123, 147 Gibbons, J. R., 91, I56 Gillet, M., 219, 264, 292 Gippius, A. A., 92, 130, 147, 158 Glaefeke, H., 173,292 Glinchuk, K. D., 92, I47 Goldstein, Y., 4,61 Gomer, R., 284,295 Goodman, A. M., 132,147 Goodwin, A. R., 70,147 Goto, K., 256, 264, 279, 292 Could, R. D., 170, 291 Gourlay, R. D.. 3, 60 Grant, J. T., 213, 251, 293 Gray, R. M., 60 Greene, J. E., 277, 297 Greene, P. D., 127, 147 Grimm, A., 108, 147 Grobman, W. D., 196,292 Grossmann, G., 207,298 Grover, N. B., 4.61 Grutzmann, S., 15, 60 Gryzinski, M., 165, 176, 180,292 Guglielmacci, J. M., 219, 264, 292 Guinolds, H. R., 129, 148 Gutkin, A. A., 103, 148
H Haas, T. W., 251,292
Haase, S., 155 Haeusler, J., 2, 11, 33, 34, 60, 61 Haff, P. K., 275,292 Haisty, R. W., 106, 120, I 4 8 Hall, D. D., 267, 298 Hall, J. T., 248, 249, 288,292 Hall, P. M., 166, 248, 249, 257, 259, 292 Hall, R. N., 132, 148 Hamilton, B., 141, 148 Hammer, C., 297 Hansma, P. K., 248,292 Harris, L. A,, 246, 248, 292 Harrison, Jr., D. E., 272, 292 Hasegawa, F., 75, 110, 148, 156 Hasegawa, H., 106, 148 Hashiba, M., 257, 273, 278, 298 Hashimoto, H., 225,279, 297 Hatch, C. B., 126, 148 Hattori, T., 171, 292 Heavens, 0.S., 288,291 Hedin, L., 176,292 Helms, C. R.,271,291 Hemment, P. L. F., 91,148 Hennel, A. M., 92,94,148 Henrich, V. E., 275,292 Henry, C. H., 133, 139, I48 Hiesinger, P., 115, 148 Hilbert, D., 52,60 Hilsum, C., I48 Hink, W., 183,293 Hinterman, H. E., 267, 298 Hjalmarson, H. P., 103, 148 Ho, P. S., 256, 213, 276, 278, 293 Hobgood, H. M., 157 Hochstadt, H., 52, 61 Hochst, H., 192, 298 Hofmann, S., 267, 270, 271, 272, 273, 279, 2Y3,2Y7 Holloway, D. M., 219, 220, 252, 293; 297 Holloway, P. H., 163,219,258,266,293 Holonyak, N., Jr., 115, 143 Holscher, A. A., 163,256,257,264,281,290 Hoogewijs, R., 200,293 Hooker, M. P., 213,293 Houston, J., 196,213,252,254,255,256,2% Houston, J . E., 164, 247,251, 292, 296 Howard, J. K., 256,273,274,278,291,293 Howe, M. S., 301,344,345,388 Howie, A,, 180, 296 Hruska, S. J., 287, 292 Hu, C., 133, 148
AUTHOR INDEX Huang, C. I., 100, 126, 152 Hubbard, J., 176, 293 Huber, A. M., 122, 125, 148 Hiifner, S., 192,298 Humbert, A,, 135, I48 Hume-Rothery, W., 114, I45 Hunsperger, R. G., 91, I48 Hurle, D. T. J., 66, 67, 69, 70, 72, 122, 123, 149,152 Hurych, Z., 297 Hutchinson, P. W., 113, 127,149 Hutchinson, W. G., 104, I45 Huth, F., 118,149 Hwang, C. J., 132, 149 I Ichimura, S., 163,213,241,242,243,244,288, 293, 297 Ignatiev, A,, 281, 298 Ikoma, T., 75, 80, 106, 125, 149, 156, 157 Ikuta, T., 225, 241, 279, 293, 297 Ilic, G., 91, 149 Il’in, N. P., 67, 103, 149 Ing, B. S., 181,293 Inokuti, M., 176, 293 Ishikawa, K., 256, 263, 278, 292 Ishimaru, A., 310, 330, 331, 345, 346,388 Itikawa, Y., 176, 293 Ito, S., 15, 61 Iwasaki, H., 136,149 J
Jablonski, A,, 237,238,293 Jackson, D. C., 171,293 Jacobi, K., 287,296 Jacobs, B., 18,60 Jain, G . C., 114, I49 Janak, J., 5,61 Jardin, C., 209,210,293 Jaros, M., 103,149 Jastrzebski, L., 132, I49 Jaswon, E., 18,59 Jaswon, M. A,, 18, 54,61 Jenkin, J. G., 171,290,298 Jennison, D. R., 210, 212, 293 Jesper, T.. 96, 149 Johnson, P. D., 297 Jordan, A. S., 122, 125, 158 Joshi, A., 293
395
Joyce, B. A,, 123,159,286,293 Joyner, R. W., 287,293
K Kachurin, G. A,, 123,149 Kadhim, M. A. H., 116, I49 Kakibayashi, H., 257,279,298 Kalma, A. H., 83, I49 Kamath, G. S.,90, 147 Kamejima, T., 132, I49 Kaminska, M., 125, 149 Kaminsky, M., 279,293 Kamm, J. D., 132, I49 Kanter, H., 170, 171,293 Karamalikis, A., 2, 59 Karelina, T. A,, 128, I50 Karnatak, R.C., 192,291 Kataoka, Y., 225, 279, 297 Kaufmann,U.,86,100,101,103,107,150 Kawasaki, Y., 116, 154 Kellokumpu, M., 195,298 Kelly, P. W., 272, 292 Kelly, R., 275, 280, 293 Kendal1,D. L., 113, 114, 118, I50 Kennedy, T. A,, 86, 150, 159 Ketchow, D. R., 123, 147 Kheifets, V. S.,120, 155 Kim, H. B., 115, 150 Kim, K. S., 195,200,273,281,282,293 Kimerling, L. C., 83, 84, 151 King, D. A., 297 Kirby, R. E., 286, 287,293,295 Kirschner, J., 247, 255, 293, 297 Kitahara, K., 106, I50 Kittel, C., 175, 260, 294 Kladis, D. L., 82, 150 Klein, P. B., 92, 104, 150 Kleinman, L., 184, 294 Klerk, M., I58 Knechtli, R. C., 90, 147 Knotek, M. L., 292 Kobus, A., 2, 3, 61 Kolbenstvedt, H., 180,294 Kolesov, B. A., 150 Kornilov, B. V., 103, 107, 129, 130, 150 Koshikawa, T., 256, 263, 278, 292 Koval, 1. P., 172, 294 Kowalczyk, S. P., 195, 200, 294 Kozeikin, B. V., 122, 150 Kravchenko, A. F., 132, 150
396
AUTHOR INDEX
Kravtsov, Yu. A., 344, 388 Krebs, J. J., 92, 150 Krefting, E. F., 242,294 Kressel, H., 121, 123, 127, 134, 150 Kroger, F. A., 66, I50 Krynko, Yu. N., 172,294 Kudo, H., 73, I50 Kiinzel, H., 120, 151 Kuhrt, F., 11,21,61 Kuiken, H. K., 133,143, I51 Kukla, C., 253,297 Kumar, V., 107, 113, 115,151 Kung, J. K., 121,151 Kushiro, Y., 132, 151 Kwun, S. I., 116, I51 Kyser, D. F., 132, 159 L Lasser, R., 211,294 Lagowski, J., 86, 94, 141, 151, 287, 294 Laithwaite, K., 118, 121, I51 Lancaster, G. M., 281,298 Land, R. H., 176,297 Landolt, D., 257,270,272,273,295 Lang, B., 213,288,291,294 Lang, D. V., 76,82,83,84,85, 87,88,90, 133, 148, 1-71 Langeron, J. P., 260, 261, 262, 290, 291, 294, 296 Lanteri, H., 223, 290 Laramore, G. E., 248,291,294 Larkins, F. P., 188, 193, 200, 201, 202, 206, 291,294 Lastras-Martinez, A., 133, 151 Laty, P., 271, 294 Laver, R. F., 274, 278, 291 Lea, C., 267,270,294,297 Leach, M. F., 126, 151 Lebedev, A. A., 106, I51 Leckey, R. C. G., 171,290,296,298 Ledebo,L.-A., 107, 113, 115,150 Lee, K. S., 116,151 Lee, P. A., 172, 181,290,294 Leedy, H. M., 109, I51 Legally, M. G., 196, 213, 252, 254, 255, 256, 296 Le Gressus, C., 163, 245, 246, 282, 286, 292, 294
Leheny, R. F., 132, I51 Le Hericy, J., 260, 261, 262,290, 291, 294, 296 Lehovec, K., 82, 160 Lender, R. J., 132, I51 Lenselink, A., 207, 208,296 Leung, P. C., 109,151 Levenson, 192,295 Lewis, J. E., 256,273, 276, 293 Ley, L., 195,200,294 Lezuo, K. L., 176,294 Li, G. P., 94,151 Li, S. S.,82, 89, 100, 126, 140, 152 Liau, Z. L., 272,277,294 Lichtensteiger, M., 287, 294 Lichtman, D., 286,293 Lidow, A., 128, 152 Lieberman, A. G., 143 Liesegang, J., 171, 290, 298 Lindhard, J., 176, 177, 184, 294 Lindquist, P. F., 95, 152 Lippmann, H., 2, 11, 21, 33, 34,61 Littlejohn, M. A., 91, I52 Logan, R. A., 76,151 Logan, R. M.,66,72,152 Lohmann, A. W., 301,388 Lomako, V. M., 132,152 Look, D. C., 82, 86, 88, 91, 94, 95, 125, 147, 152 Lorimor, 0. G., 109,152 Los, J., 287,298 Losch, W., 213, 282,283,289,294,297 Lotz, W., 180, 183, 294, 295 Loudliche, s., 89, 152 Luberfeld, A., 294 Ludman, J. E., 90, 152 Lundquist, S., 292 Lundqvist, B. I., 176, 295
M Ma, Y. R., 94, 151 MacDonald, N. C., 291 Madden, H. H., 209,210,293,295 Madelung, O., 5,151 Madey, T. E., 244,296 Magee, T. J., 115, 116,152 Maier, M., 122, 152 Majerfeld, A., 80, 152 Makram-Ebeid, S., 81, 94, 153
397
AUTHOR INDEX
Malm, D. L., 288,295 Many, A., 4,61 Margoninski, Y., 287,295 Mariot, J. M., 192, 193, 194, 195, 291, 295 Mark, H., 193, 195,291 Marsh, 0. J., 91, I48 Martin, G. M., 77, 78, 93, 94, 95, 96, 97, 98, 99, 100, 103, 122, 153 Martin, P. C., 175,295 Martinelli, R. U., 133, 153 Martinez, G., 92, 93,94, 146, 148, 153 Martin62 Saez, V., 213,298 Marton, J., 257, 266,290 Massignon, D., 246, 294 Masterov, V. F., 67, 103, 149 Matart, H. F., 132, 153 Mathieu, M. J., 257, 270, 272, 273, 295 Mathur, V. K., 132, 153 Matino, H., 128, I53 Matsudaira, T., 235, 237,239,295 Matsukawa, T., 225, 243,279,295,297 Matveenko, Yu A., 107,153 Mayadas, A. F., 2,62 Mayburg, S., 132, 153 Mayer, J. W., 272, 277,294 McFeely, F. R., 195, 200, 294 McGlure, D. E., 270,272,295 McGuire, E. J., 180, 193, 207, 209, 212, 292, 295
McGuire, G. E., 295 McLevige, W. V., 116, 152 McMahon, Jr., C. J., 264,295 McNichols, J. L., 90, 152 Mei, K., 18, 61 Melles, J. J., 192, 295 Melnik, P. V., 172,294 Menzel, D., 284,295 Mermin, N. D., 176, 179,295 Meyer, F., 255,295 Miki, H., 132,153 Miller, M. D., 75, 136, 153 Milnes, A. G., 114, 115, 133, 147, 159 Mimizuka, T., 15,61 Minnhagen, P., 295 Mircea, A., 77, 81, 127, 132, 153, 156 Mitchell, E. W. J., 86, 88, I45 Mitonneau, A., 77, 78, 80, 81, 102, 127, 136, 153,154
Monch, W., 94,154
Mohri, M., 257,279,298 Moiseiwitch, B. L., 179,290 Monemar, B., 126,154 Moore, G., 196, 213, 252, 254, 255, 256, 2% Moore, J. A,, 91, 154 Morabito, J. M., 166, 248, 249, 257, 259, 288, 292
Morgan, D. V., 82, 157 Morgulis, L. M., 113, 127, 154 Mori, H., 300, 388 Morkoc, H., 100, 154 Morrison, S. R., 118, 121, 154 Morrizumi, T., 121, 154 Miiller, K., 163, 295 Mulford, R. A., 265,266, 295 Mullin, J. B., 95, 154 Murata, K., 225, 243, 219, 295, 297 Murday, J. S., 196,212,213,262,290,298 Murdey, J. J., 196,213,252,254,255,256,296 Murygin, R. I., 107, 154 Murygin, V. I., 103, 143 N
Nagel, D. R., 176, 180, 295 Nahory, R. E., 132, 151 Nakai, K.,106, 154 Nakai, M. Y., 242,296 Nakayama, K., 279,295,2Y7 Nakhodkin, N. G., 172,294 Nalecz, W., 3,61 Narayanan, G. H., 108, 154 Neave, J. H., 286,293 Needham, Jr., P. B., 172,296 Nelson, R. J., 129, 154 Nemeth-Sallay, M., 171,296 Neumann, H., 122, I54 Newell, W. R., 180,296 Newsome, J. P., 1 5 6 1 Nilson-Jatko, P. E., 192,291 Nishimori, K,, 295 Nishizawa, J. I., 71, 128, 154 Nojima, S., 116, I54 Noonan, J. R., 209, 210,295 Norman, D., 171,295 Norris, B., 108,154 Norton, K., 306,331,388 Novikov, E. A., 305,389 Nowak, W. B., 90,152
398
AUTHOR INDEX
Nowicki, R. S., 129, 154 Nozaki, T., 122,154 Nozieres, P., 207, 295 0
OBrien, J. K., 84, 154 Okajima, Y . ,295 Okurnura, T., 140, 154 Okutani, T., 279,295 Oldham, W. G., 113,143 Olivier, J., 287, 295 Olowolafe, J. O., 129, 154 Omel'yanovskii, E. M., 106, 154 Onchi, M., 235,237, 239,295 Ono, M., 295,297 Onoda, Jr., G. Y . ,288,2Y6 Oppenheirn, J., 300,388 Oxley, D. P., 170, 295 Ozeki, M., 75, 119, 136, 154
P Padden, F. J., 288,295 Palfrey, H. D., 114. 155 Palmberg, P. W., 163, 171, 256, 291, 295,296 Pandey, K. C., 207,212,292 Panish, M. B., 113, 114, 155 Pantano, Jr., C. G., 288, 295,296 Papagno, L., 265,296 Papaioannou, G. I., 132,155 Parikh, M., 248, 292 Park, R. L., 164, 247. 248, 291,292, 294,296 Park, Y. S., 116, 156, 160 Parravicini, G. P., 181, 298 Partin, D. L., 102,108,131,132,133,138,139, 155
Parui, D. P., 132, I45 Pashley, R. D., 128, 155 Payling, R., 172,296 Pearson, G. L., 68, 84, 131, 144, 145 Peisner, .I.171. , 296 Peka, G. P., 132, 155 Pendry, J. B., 181, 186,293, 296 Peng, J., 115, 152 Penn, D. R., 176, 183, 184, 260, 296 Perlberg, C. R., 288,289 Pessa, V. M., 180, 296 Peters, R. C., 158
Petit, G. D., 122, 156 Philibert, J., 243, 296 Picoli, G., 93, 155 Pierce, D. T., 172,296 Pilkuhn, M. H., 132, 155 Pinard, P., 127, 144 Pitfield, R. A., 18, 61 Platzman, P. M., 181,290 Plew, L., 128, 145 Ploog, K., 120, 122, 123, 151, 155 Poate, J. M., 271, 272,290,294 Poirier, R., 287, 295 Pollack, M. A., 132, I51 Pollak, R. A,, 195, 196,200,292,294 Pollard, A. M.,288,291 Pons, D., 75, 77, 83, 84, 85,86,88, 155 Pons, F., 260, 261,290,291,296 Powell, C. J., 163, 170, 172, 183, 244, 252, 296
Price, R. E., 193, 195,290,296 Pritchard, R. G., 171, 292 Putley, E. H., 59,61
Q Quichaud, G., 2, 3,61 Quinn, J. J., 175, 184,296
R Rabalais, J . W., 281, 298 Rachmann, J., 126, 155 Ramaker, D. E., 196,213,252,254,255,256, 262,296,298
Ramsey, J. A., 172,296 Ranke, W., 287,296 Rao, P. V., 193, 195, 290,296 Rao-Sahib, T. S., 133, 155 Rasolt, M., 182, 296 Redhead, P. A., 284,296 Rehn, L. E., 265,279,280,296 Reickl, L. E., 300,389 Reirner, L., 225, 242,294,296 Reiter, F. J . , 192, 298 Reuter, W., 260, 266, 281,296,298 Rheinlander, B., 123, 155 Rhodin, T. N., 171,295 Riachi, G. E., 291 Richman, D., 131, 155
AUTHOR INDEX
Rickman, J., 287, 293 Ridley, B. K., 126,148, I51 Ringers, D. A., 195, 298 Ritchie, R. H., 176, 180, 181, 184, 185, 242, 290,292,296,298 Riuaud, L., 277,297 Riviere, J. C., 286, 291 Robinson, G. Y., 129, 148 Roll, K., 282, 283, 289, 297 Roelofs, L. D., 248,291,292,294 Rogers, S., 153 Rogers, S. C., 132, 159 Rojo, J. M., 213,298 Romanenko, V. N., 120, 255 Romankiw, L. T., 2,62 Rondot, B., 261,291 Rose-lnnes, A. C., 148 Rosler, M., 223, 297 Ross, J., 300, 388 Rossin, V. V., 133, I56 Rostaing, P., 223, 290 Roulet, B., 188, 289 Roussel, A,, 81, I56 Rowe, J. E., 196, 198,291 Rowe, R . G., 265,266,295,297 Rozgonyi, G. A,, 130,156 Rubin, V. S., 107, 154 Rudge, M. R. H., 180,297 Ryan, R. D., 131, 156
S Sadana, D. K., 149 Saeki, N., 274, 297 Sah, C. T., 132, 145, 156 Saito, T., 75, I56 Sakai, K., 80, 156 Salmeron, M., 213,298 Samuelson, L., 137, 156 Sankey, 0. F., 103, 125,156 Sansbury, J. D., 91, 156 Sanz, J. M., 267,271,297 Sastri, S., 281, 290 Savua, M.A., 128,156 Sawatzky, G. A., 207,208,297 Scarmozzino, R., 265,296 Schairer, W., 122, 156 Scheer, J. J., 131, 142 Schemer, M., 141, 156
399
Schlachetzki, A., 106, 156 Schliiter, M., 196, 197,291 Schneider, J., 75, 86, 100, 101, 103, 150, 158 Schou, J., 223, 297 Schreiner, D. G., 247, 296 Schrott, A. G., 287,291 Schwartz, S. B., 180,297 Schwinger, J., 175, 295 Seah, M. P., 170, 171, 183,260,267,270,271, 275,294,297 Sealy, B. J., 91, 127, 156, 157 Segal, D., 287, 295 Sekela, A., 132, 139, 140, 156 Seki, Y., 132, 156 Seltzer, M. S., 104,156 Sevier, K. D., 162, 193,297 Shafer, M. W., 279,291 Shaldervan, A. I., 172, 294 Share, S., 107, 156 Shaw, R., 18,61 Shelton, J. C., 171, 176, 184, 297 Shifrin, S. S., 127, 156 Shih, K. K., 122, 156 Shikata, M., 279,295, 297 Shimizu, H., 279,295,297 Shimizu, R., 163, 213, 225, 241, 242, 243,244, 256,263,274,278,279,288,293,294,295, 297 Shin, B. K., 121, 156 Shirley, D. A., 195, 199, 200, 295,297 Shishyanu, F. S., 113, 115, 144,157 Shortley, G. H., 192,205,291 Sickafus, E. N., 220, 252, 253, 254, 297 Siegmann, H. C., 172,296 Sigmund, P., 275,276,297 Simoni, F., 265, 2% Singwi, K. S., 176, 298 Sinha, A. K., 130, 157 Sinharoy, S., 172,298 Sjolander, A., 176,297 Slater, J. C., 192, 297 Slifkin, L. M., 113, 114, 146 Sliisser, G. J., 266, 277, 297 Smith, A., 5,61 Smith, B., 15, 36, 59 Smith, N. V., 297 Smith, T., 297 Smrcka, L., 186,297 Sokolov, V. I., 115, 157
400
AUTHOR INDEX
Solov, H., 106, 156 Solov’ev, N. N., 102, 143 Sopizet, R., 245,294 Sorvina, F. P., 105, 158 Spanjaard, D., 209,298 Spicer, W. E., 73,74,75,157 Spitzer, W. G., 109, 121, 150, 152, 157 Springthorpe, A. J., 122, 157 Staib, P., 247,255,293,298 Stall, R. A., 122, 157 Stegun, I. A., 14, 351, 59, 388 Stein, H. J., 90, 157 Stein, R. J., 170, 171, 172,296,298 Steiner, P., 192, 298 Stern, R. M., 172,298 Stevens, K., 23,61 Stillman, G. E., 70, 119, 123, 157, 159 Stoneham, E. B., 122,157 Strack, H., 106, 157 Strand, T. G., 241,290 Strauss, G. H., 86,92, 150, 157 Stringfellow, G. B., 120, 157 Su, J. L., 118, 157 Suchkova, N. I., 107,157 Sugibuchi, K., 136, 149 Surridge, R. K., 91, 122, 157 Swift, C. D., 193, 195,290 Switkowski, Z. E.,275,292 Sykes, D. E., 267,298 Symm, G., 18,52,54,61 Szajman, S., 171,298
Thomas, S., 281,298 Thomas, T. D., 195,289 Thompson, D. A., 2,62 Thompson, V., 267,298 Thurmond, C. D., 69,158 Thurstans, R. E., 170,295 Tisone, T. C., 271,290,297 Tokutaka, H., 295 Tomlin, S. G., 180,298 Tompkins, H. G., 287,298 Toneman, L. H., 163,256,257,290 Tongman, L.H., 257,298 Tosatti, E., 181,298 Tosi, M. P., 176,297 Tottenham, H., 52,62 Tracy, J. C., 255,298 Traum, M. M., 297 Treglia, G., 209,298 Tsang, T., 195,298 Tsaur, B. Y.,272,277,294 Tuck, B., 116,149 Tucker, Jr., C. W.,172,291 Tung, C. J., 176, 181, 184, 185,290,296,298 Turner, J. E., 176,293 Turner, N. H., 196, 212, 213, 252, 254, 255, 256,262,290,296, 298 Tzoar, N., 182,292
U Ushakov, V. V., 92,130, 147, 158 Uskov, V. A., 105, 158
T
Ta, L. B., 86, 157 Tagle, J. A., 213, 298 Takahashi, K., 121, 127, 154, 157 Takai, M., 91, 157 Takashima, K., 295 Takikawa, M., 75, 149 Tandon, J. L., 122, 157 Taniguchi, M., 125, 157 Tarng, M. L.,267, 298 Tatarskii, V., 301, 331,344,388,389 Taylor, J. A., 281,298 Taylor, P. D., 82, 157 Texier, R., 243,296 Thomas, J. M.,171,292 Thomas, R. N., 91, 157
v Vayrynen, J., 195,289,298 Vaitkus, J., 94, 158 Vallin, J. T., 92, 158 Van Blade], J., 18,60,61 Van Der Pauw, L. J., 7, 12,36,62 Van Mechelen, J. B., 264, 281, 290 Van Oostrom, A., 163,257,265,298 Van Opdorp, C., 137,158 Van Santen, R. A., 257,298 Van Vechten, J. A., 67, 68, 69, 72, 158 Vashista, P., 176, 298 Vasile, M. J., 288, 295 Vasil’ev, A. V., 103, 158 Vasudev, P. K., 81,94, 126, 158
401
AUTHOR INDEX
Vennik, J., 200, 293 Verhoeven, J., 287,298 Vigorradov, A. G., 344,388 Vincent, G., 80, 127, 144, 158 Vine, J., 282, 298 Vitale, G., 133, 158 Von Barth, U., 207,298 von Roos, O., 133, 158 Vrakking, J. I., 255,295 W Wada, O., 76,158 Wagner, E., 132,144 Wagner, M., 280,2% Walker, G. H., 86, 158 Wallis, R. H., 89, 125, 158 Walls, J. M.,267, 298 Wang, C. C., 133,153 Wang, I., 129, 154 Wang, K. L., 94, I51 Ward, I. D., 217,297 Warsza, Z. L., 3, 61 Watanabe, K., 257,273.278,279,298 Watkins, G. D., 89,158 Watson, G. N., 339,389 Webb, C., 287,294 Weber, E. R., 75,88, 158 Weber, R. E., 291 Weiner, M. E., 122, 125, 158 Weinert, H., 155 Weiser, K., 92, 150 Welch, B. M., 128, 146, 155 Whelon, A. D., 347, 349,389 White, A. M., 93, 126,159 Whittaker, E. T., 339, 389 Wick, R. F., 11, 34,62 Wieder, H. H., 59,62 Wiedersich, H., 265,279, 280,296 Wight, D. R., 133, 159 Wigner, E., 300,389 Wilcox, D. L., 288,289 Wild, W., 173,292 Wildman, H. S., 256, 273,293 Willardson, R. K., 113, 119, 159 Williams, E. W., 126,159 Williams, P. J., 92, 93, 159 Williamson, K. R., 116,159
Willmann, F., 113,144,159 Willoughby, A. F. W., 75, 114,146, 155 Wilsey, N. D., 86, I59 Wilson, R. G., 105, 159 Winder, D. R., 254,297 Winograd, N., 195, 200, 266, 272, 273, 277, 281,282,292,293,297 Winters, H. F.. 288, 289 Wirth, J. L.,82, 159 Wiser, N., 176, 180, 298 Witten, Jr., T. A., 176, 180,295 Wittry, D. B., 133,155,159 Womer, R., 86, 159 Wolfe, C. M., 70, 119, 120, 121, 122,123,159 Wolff, G. A., 132,153 Wolford, D. J., 124, 159 Wolfstirn, K. B., 108, 147 Wonsidler, D. R., 271,290 Wood, C. E. C., 123, 159 Woodcock, J. M., 91,159 Woodruff, D. P., 171,295 297 Worthingon, C. R., 180,298 Wu, C. J., 133, 159
Y Yabumoto, M., 251,219,29 Yamashina, T., 257,273,278, 219,29 Yamazaki, H., 123,159 Yan, Z.-x., 114, 115, 159 Yellin, I., 195, 298 Yin, L. I., 195, 298 Yip, K. L., 197,298 Yu, M. L., 266,281, 298 Yu, P. W., 91, 116,160 Z
Zakharova, G. A., 112,160 Zalar, A,, 270,273, 279,293 Zehner, D. M., 209, 210, 293, 295 Zelevinskaya, V. M., 91, 160 Zemon, S., 123, 160 Zhu, H.-z., 137,160 Ziegler, A., 183, 293 ZOU,Y-x., 75, 76,160 Zucca, R., 95,160 Zuleeg, R., 82,160
Subject Index
A
AEAPS, see Auger electron appearance potential spectroscopy AES, see Auger electron spectroscopy Aluminum Auger emission, 227-240 elastic collisions, 186-187 inelastic collisions, 177-179, 184 Aluminum-copper alloys, sputtering, 273, 278 Aluminum oxide, inelastic collisions, 18I Anistropic stochastic medium, propagation through, 336,345,379,382 Appearance potential spectroscopy (APS), in Auger electron spectroscopy, 166 Arsenide vapor, behavior of GaAs in contact with, 66-67 Attenuation length, measurement of, 170- 172 Auger electron appearance potential spectroscopy (AEAPS), 247-248 Auger electron spectroscopy (AES), 161-298 attenuation length, 170-172 Auger emission, quantitative description of, 213-244 theoretical analysis, 222-244 Auger transition in a solid, 187-213 classification, 188- 198 kinetic energy of Auger electrons, 198206 line-shape analysis, 206-21 3 basic experiment, 163-164 calibration techniques, 166, 256-258, 260 cross sections, 167-168 definitions, 167-1 73 elastic collisions, 185-187 inelastic collisions, 173- 185 d-electron metals, 174, 180-181 depth dependence and anisotropy of mean free path, 182 general equations, 174- 176 normal metals, 176-180
semiconductors and insulators, 13 1- 182 theoretical evaluations and experimental data, comparison of, 182-185 introduction, 162-167 mean escape depths, 173 mean free path, 168- 170 quantitative analysis, 244-289 electron-beam-induced effects, 282-289 operating modes, 245-254 quantification of Auger analysis, 255-263 sample preparation, 263-273 sputtering, effects of, 273-282 Auger transition, 162 in a solid, 187-213 classification, 188-198 intraatomic, 188-196 kinetic energy of Auger electrons, 198206 line-shape analysis, 206-2 13 lines involving valance bands, 196-1 98
B Backscattered electron, in Auger electron spectroscopy, 164, 221-222, 227, 231 232,234-235,238 Backscattering factor, in Auger electron spectroscopy, 221-222, 234, 237-239, 259260 BEM, see Boundary-element method Beryllium Auger line shape analysis, 210 in gallium arsenide, 116-117, 120 Boron, in gallium arsenide, 118 Boundary-element method (BEM), Hall-plate potential calculations, 18-38 cross-shape geometry, 25-27, 30-33,45-46 direct calculation of geometry correction, 27-29 functions, calculation of, 40-43 improvement, 38-48
402
403
SUBJECT INDEX
integral equations, 18-20, 39-40 numerical solution, 20-21 numerical calculation of current through a contact, 24-25 octagonal Hall plate, 34-36, 46-48 rectangular Hall plate, 21-22, 29-30, 3335,43-49 unsymmetrical cross-shaped form, 37-38 zeroth-order approximation, 22-24 Bridgman-grown gallium arsenide, diffusion lengths in, 139-141 C
Cadmium, in gallium arsenide, 116-1 18, 139 Cadmium sulfide, electron-beam-induced adsorption of oxygen in H,O, 287 Carbon Auger line shape analysis, 2 13 Auger quantitative analysis, 255,286 in gallium arsenide, 116-1 17, 120-121 Carbon dioxide, adsorption enhancement on semiconductors, 287 Carbon monoxide, adsorption enhancement on semiconductors, 286-287 Cascade process, in Auger electron spectroscopy, 164,241,243 Cassette recorder, Hall-effect applications, 2 Chromium Auger quantitative analysis, 257, 260-261, 270,213 in gallium arsenide, 91-100, 102, 120 CMA, see Cylindrical mirror analyzer Cobalt, in gallium arsenide, 100, 102-103, 106-107,120 Coherent Wigner function, 347-354 Computer simulation, stochastic transport equation for the Wigner function, 302, 354-381 algorithm for integral equation for very narrow beam, 357-359, 378, 382 exponential space correlation function, 368-37 1 FORTRAN program, 359,382 integral equation for very narrow beam, 355-357 Kolmogoroff space correlation function, 362-365,371-378,382 listing of experimental program, 383-385 sample, 386-387
Conformal mapping, Hall-plate problems, 10-15 basic ideas, 10-1 1 cross-shaped geometry, 1 1 15 Copper Auger quantitative analysis, 255, 257, 260, 270 Auger transition line shapes, 193-194, 209 in gallium arsenide, 100, 102-103, 106, 109-113, 120 inelastic collisions, 184-185 Copper-indium alloys, sputtering, 277 Copper-nickel alloys Auger quantitative analysis, 263-264 diffusion, by increasing temperature, 289 sputtering, 278-281 Cross section, in Auger electron spectroscopy, 167- 168 Cylindrical mirror analyzer (CMA), in Auger electron spectroscopy, 164 -
D DAPS, see Disappearance potential spectroscopy Deep-level optical spectroscopy (DLOS), gallium arsenide studies, 81 Deep-level transient spectroscopy (DLTS), gallium arsenide studies, 76-81, 90, 96, 98, 100 Desorption, in Auger quantitative analysis, 284-286 Disappearance potential spectroscopy (DAPS), in Auger electron spectroscopy, 166,241-248 DLOS, see Deep-level optical spectroscopy DLTS, see Deep-level transient spectroscopy
E EAPFS, see Extended appearance potential fine structure Elastic collision, in Auger electron spectroscopy, 164-165, 185-187, 226-229, 231,241-242 Electric field in a stochastic dielectric, Wigner distribution matrix for, see Wigner distribution matrix for an electric field in a stochastic dielectric
404
SUBJECT INDEX
Electron-beam-induced effects in Auger electron spectroscopy, 282-289 charge effects, 283-284 dissociation, 288 electron-impact-stimulated adsorption, 286-288 migration and diffusion, 288-289 physicochemical effects, 284-286 temperature increase, 282-283 Electron-beam stimulated description (ESD), in surface studies, 284-285 Electron-irradiated gallium arsenide, 82-90 Electron paramagnetic resonance (EPR), GaAs studies, 73, 86 ESD, see Electron-beam stimulated description EXAFS, see Extended X-ray absorption-edge fine structure experiments Extended appearance potential fine structure (EAPFS), in Auger quantitative analysis, 248 Extended X-ray absorption-edge fine structure (EXAFS) experiments attenuation lengths, measurement of, 170, 172 in Auger quantitative analysis, 248
irradiation, levels produced by, 8 1-91 minority-carrier recombination, generation, lifetime and diffusion length, 13l 141 molybdenum, 128- 129 oxygen, 123-127 palladium, 129- 130 platinum, 130-131 possible native defects and complexes, 6516 rare-earth studies, 13I ruthenium, 129 semi-insulating GaAs with and without chromium, 91-100 shallow acceptors, 116-1 18 transition metals, 100- 108 traps and nomenclature, from DLTS studies, 76-8 1 tungsten, 130 Gaussian stochastic process, 305 Germanium, in gallium arsenide, 116-1 17, 119-120, 122 Gold Auger quantitative analysis, 270 in gallium arsenide, 115- 116, 139 H
F Fast-neutron-irradiated gallium arsenide, 86 FEM, see Finite-element method Field-effect transistor (FET),effects of radiation on, 82 Finite-element method (FEM),potential calculations in Hall plates, 18 Forward scattering by stochastic dielectrics, 30 1 asymptotic equations for the Wigner distribution function, 330-335 G Gallium arsenide, impurity and defect levels, 63- 160 concluding discussion, 141 142 group I impurities, 108- I 16 group IV elements as dopants, 118-123 group VI shallow donors, 127-128 introduction, 64 -
Hall-effect applications, 2-3 photovoltaic cell, 57-58 Hall-effect measurements, gallium arsenide with and without chromium, 94-99 Hall plates, potential calculations in, 1-62 boundary-element method, 18-38 cross-shaped geometry, application to, 25-27,30-32,45-46 functions, calculations of, 40-43 geometry correction, direct calculation Of, 27-29 improvement, 38-48 integral equation, 18-20, 39-40 integral equations, numerical solutions of, 20-21 numerical calculation of current through a contact, 24-25 octagonal Hall plate, application to, 3436,46-48 rectangular Hall plate, 21-22,29-30,3335,43-41
405
SUBJECT INDEX
unsymmetrical cross-shaped form, application to, 37-38 zeroth-order approximation, 22-25 conclusions, 48-49 conformal mapping techniques, 10-1 5 basic ideas, 10-11 cross-shaped geometry, 11- 15 current’s contribution to magnetic field, 58-59 finite-element method, 18 Fredholm integral equations, 52-54 fundamental equations, 3-7 approximations for thin semiconducting layer, 4-5 externally applied magnetic field, boundary conditions with, 6-7 externally applied magnetic field, constitutive relations, 5-6 general equations for semiconductors, 3-4 geometry, influence of, 9- 10 Green’s theorem, 54-57 introduction, 2-3 relaxation methods, 15-17 three-dimensional Hall effect, 49-52 Van Der Pauw method, 7-9 Hall probe, applications of, 2 Hall voltage, 2 I IMFP, see Inverse mean free path Indium phosphide, electron-beam-induced oxidation, 287 Inelastic collision, in Auger electron spectrosCOPY,164-165, 173-185, 226, 228, 231, 241 -242 d-electron metals, 174, 180-181 depth dependence and anisotropy of the mean free path, 182 general equations, 174-176 normal metals, 176-180 inner shell electrons, collisions on, 178180 plasmon mode and individual transitions, 176-178 semiconductors and insulators, 181- 182 theoretical evaluations and experimental data, comparisons of, 182-1 85
Insulator charge effects in Auger electron spectroscopy, 283 dielectric theory of inelastic collisions, 181 182 Interstellar plasma, 308 Inverse mean free path (IMFP), in Auger electron spectroscopy, 170, 182 Ion-scattering spectrometry (ISS), 257 Iron Auger quantitative analysis, 257,260-261 in gallium arsenide, 100-102, 104-106, 120 Iron-silicon alloys, Auger quantitative analysis, 265 Irradiation, effect on GaAs and GaAs devices, 81-91 Isotropic stochastic dielectric half space, plane wave incident on, 337342, 382 scattering effects, 336-337 time invariant, monochromatic waves in, 342-345, 378, 382 ISS, see Ion-scattering spectrometry
K Kinetic energy, Auger electrons, 198-206 Kolmogoroff space correlation function, computer simulation for, 362-365, 371-378, 382 Kolmogoroff spectrum, 307,351
L Lead, in gallium arsenide, 1 19-120, 123 LEED, see Low-energy electron diffraction Liquid-phase epitaxy (LPE) gallium arsenide, 64,76, 132-135, 140 Lithium Auger line shape analysis, 210-21 1 in gallium arsenide, 108-109 Low-energy electron diffraction (LEED), attenuation length, measurement of, 170, 172 LPE gallium arsenide, see Liquid-phase epitaxy gallium arsenide
406
SUBJECT INDEX
M Magnesium. ingalliumarsenide, 116-1 17,120 Magnetic bubble memory, Hall-effect applications, 2 Magnetic field, measurement with Hall-effect components, 2 Manganese, in gallium arsenide, 100, 102104, 120 Mean free path (MFP) in Auger electron spectroscopy, 165, 168170,218 elastic collisions, 186- 187 inelastic collisions, 182- 185 stochastic dielectric, 306 Mercury, in gallium arsenide, 118 MFP, see Mean free path Minority carrier, recombination, generation, lifetime, and diffusion length in gallium arsenide, 131-141 Molybdenum Auger quantitative analysis, 257, 261 in gallium arsenide, 128-129 Monte Carlo simulation method, in Auger electron spectroscopy, 223-227
Auger quantitative analysis, 285 in gallium arsenide, 123-127
P Palladium, in gallium arsenide, 129- 130 Parabolic equation method, for propagation problems, 301, 345-347 Photoconductivity, gallium arsenide: chromium, 94 Plasma, stochastic transport equations for Wigner function, 303, 305, 308 Platinum, in gallium arsenide, 130- 131 Potassium, electron-beam-induced migration of atoms, 288 Potential calculations in Hall plates, see Hall plates, potential calculations in Preferential sputtering, 271-272, 275-278 reactive and nonreactive, 281 Proton-irradiated gallium arsenide, 89-90 Push buttons, Hall-effect applications, 2
Q Quantitative analysis, Auger, see Auger electron spectroscopy, quantitative analysis
N
Neodymium, in gallium arsenide, I3 1 Neutron-irradiated gallium arsenide, 86, 8990 Nickel Auger emission, 237 Auger quantitative analysis, 270, 273 electron-beam-induced oxidation, 287 in gallium arsenide, 100, 102, 106-108, 120, 137-139 Nickel-chromium system, sputtering, 279 Nickel-palladium alloys, Auger quantitative analysis, 257 0 Optical imaging, Wigner functions, 301 Optical signals, Wigner functions, 301 Oxidation of semiconductors, 286-287 Oxygen adsorption enhancement on semiconductors, 286 Auger line shape analysis, 2 13
R Randium-jellium model, in Auger electron spectroscopy, 165, 186,224 Rare-earth elements, in gallium arsenide, 131 Relaxation method, Hall-plate potential calculations, 15-17 Russell-Saunders coupling, 190 Ruthenium, in gallium arsenide, 129 S Scandium, in gallium arsenide, 102 Secondary electron, in Auger electron spectroscopy, 164 Secondary-ion mass spectrometry (SIMS), 257 Selenium, in gallium arsenide, 119-120, 127 Semiconductors, see also Gallium arsenide; Hall plates dielectric theory of inelastic collisions, 181 I82 electron-beam-induced oxidation, 286-287 n-type, general equation for, 3-4 ~
407
SUBJECT INDEX
size effects, 9 thin-film, approximations for, 4-5 Sequential-layer sputtering (SLS), 267 Silicon Auger line shape analysis, 21 1-2 13 Auger quantitative analysis, 265, 286 electron-beam-induced oxidation, 287 in gallium arsenide, 116-122, 141 inelastic collisions, 181 sputtering, 281 Silicon devices, effects of radiation on, 82 Silicon dioxide Auger quantitative analysis, 271 Auger transition, 197-198 electron-beam induced dissociation, 288 Silver Auger quantitative analysis, 255, 263, 270 in gallium arsenide, 1 13-1 15 Silver-palladium alloys Auger quantitative analysis, 257 sputtering, 277,281 SIMS, see Secondary-ion mass spectrometry SLS, see Sequential-layer sputtering Sodium Auger line shape analysis, 21 1 electron-beam-induced migration of atoms, 288 Soft X-ray appearance potential spectroscopy (SXAPS), 247 Solar cell, radiation damage, 82 Sputtering, in quantitative Auger analysis, 268-273 multicomponent materials, 273-282 Steel, Auger quantitative analysis, 261, 265266 Stochastic dielectric, Wigner distribution matrix for electric field, see Wigner distribution matrix for an electric field in a stochastic dielectric Sulfur Auger spectra, 213 in gallium arsenide, 120, 127-128 SXAPS, see Soft X-ray appearance potential spectroscopy T Tellurium, in gallium arsenide, 69-72, 127128, 141 Thulium, in gallium arsenide, 13 1
Tin, in gallium arsenide, 69-70, 120, 122-123 Titanium Auger quantitative analysis, 261-262 in gallium arsenide, 102-103 Transition metals, effects in gallium arsenide, 100- 108 Troposphere. stochastic propagation, 308,382 Tungsten, in gallium arsenide, 130 Turbulence theory computer-range simulations for Kolmogoroff space correlation function, 371, 374, 378, 382 Wigner function, 302-303, 307-308
U Ultrahigh vacuum (UHV) system, in Auger electron spectroscopy, 245,264
V Vanadium, in gallium arsenide, 100, 102-103 Van Der Pauw method, Hall-mobility measurements, 7-9 Vapor-phase epitaxy (VPE), gallium arsenide, 64,76, 135 - 139
W Wave propagation, Wigner functions, 301 303, 308 Wiener- Khintchine theory, 317 Wigner distribution matrix for an electric field in a stochastic dielectric, 299-389 asymptotic equations for the Wigner distribution function, 330-335 coherent Wigner function, 347-354 computer simulation, 354-38 I algorithm for stochastic integral equation for very narrow beam, 357-359,378, 382 exponential space correlation function, 368-371 FORTRAN program, 302, 359, 382 integral equation for very narrow beam, 355-357 Kolmogoroff space correlation function, 362-365,371-378,382 listing of experimental program, 383-385 sample, 386-387
408
SUBJECT INDEX
conclusions, 382-383 derivation of the equations for the Wigner distribution functions, 313- 325 differential equation for the electric field correlations, 303-313 equations for special cases, 335-345 forward-scatteringapproximation, 330335 introduction, 300-303
related equations for the Wigner distribution function, 325-329 theoretical methods, review of, 345-347
Z
Zinc, in gallium arsenide, 1 16- 1 18, 120, 139, 141
