Recombination in Semiconductors

This book is a graduate text devoted to the main aspects of the physics of recombination in semiconductors. It is the fi...

Author: Peter T. Landsberg

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This book is a graduate text devoted to the main aspects of the physics of recombination in semiconductors. It is the first book to deal exclusively and comprehensively with the subject, and as such is a self-contained volume, introducing the concepts and mechanisms of recombination from a fundamental point of view. Professor Landsberg is an internationally acknowledged expert in this field and, while not neglecting the occasional historical insights, he takes the reader to the frontiers of current research. Following initial chapters on semiconductor statistics and recombination statistics, the text moves on to examine the main recombination mechanisms: Auger effects, impact ionization, radiative recombination, defect and multiphonon recombination. A final chapter deals with the topical subject of quantum wells and low-dimensional structures. Altogether the book covers a remarkably wide area of semiconductor physics. The book will be of importance to scientists and engineers who are studying the physics and applications of semiconductors. Much of the book, particularly the first half, has been used in graduate courses and the explanations throughout are adequate for this purpose.

Recombination in semiconductors

1 2

3

4

The first International Conference on the Physics of Semiconductors, Reading, England, 1950. This is the official photograph which was taken at the time. It shows most of the participants.

1 John Hodgson (Keele). 9 R.G. Breckenridge, National Bureau of Standards, Washington, f4.9.86. 11 Dr Forsberg, Washington. 16 Dr Howard Etzel, US Naval Research Lab. (1950-6), Deputy Director National Science Foundation, 1971. 18 Dr E.W. (Ted) Elcock, Aberdeen. 20 Dr Ron Cooper, AEI Laboratories, Aldermaston. Later Professor of Electrical Engineering at the University of Manchester. 21 Dr Peter T. Landsberg, AEI Laboratories, Aldermaston. Later Professor of Applied Mathematics at Universities of Wales and Southampton. 22 Dr Trevor Moss, RRE, later RAE, 1953-78, then Deputy Director RSRE 1978-81. 23 Dr Dennis Sciama, Fellow, Trinity College, Cambridge 1952-6. Later Professor of Astrophysics, Trieste 1983, FRS (1983). 24 Dr A.F. Gibson, RRE. Later Professor of Physics at the University of Essex, FRS (1978). 11988(7). 25 Ron Bloomer, AEI Laboratories, Aldermaston (1947-59), Harlow (1959-63), CEGB (1964-86). 30 Dr P. Jutsum, University of Reading, later University of the West Indies. 33 O. Heavens, University of Reading. Later Professor of Physics at the University of York. 36 Dr T.B. Rymer, University of Reading. 38 Dr Pierre Baruch, Ecole Normale Superieure; later Professor at the University of Paris. 41 Mile M. Francois, Universities of Reading and Paris. 42 Dr Koop, Philips Research Laboratories f 1955(7). 44 Dr C.A. Hogafth, University of Reading. Later Professor of Physics at Brunei University. 45 Dr Park H. Miller, Professor, University of Pennsylvania, Philadelphia. Later at General Dynamics. 46 R.A. Smith, REE, later Principal Herriot-Watt University, FRS (1962). f 1980. 47 Dr J.W. Mitchell, Reader, University of Bristol (1945-59). Professor of Physics at the University of Virginia since 1959. FRS (1956). 48 Prof. Perruci. 49 Prof. A.E. Sandstrom, Sweden. 50 Mr T.R. Scott, Standard Telecommunications Laboratories, London. 51 Prof. K. Lark-Horowitz, University of Purdue. 52 Dr E.J.W. Verwey, Philips Research Laboratories. 53 Prof. N.F. Mott, FRS (1936), N.L. (1977). Professor of Physics at the Universities of Bristol, 1933-54, and Cambridge, 1955-66. 54 Prof. Fleury, Conservatoire National des Arts et Metiers. Later Director of LTnstitute d'Optique, Paris, f 1976. 55 Prof. R.W. Ditchburn, Professor of Physics at the University of Reading, FRS (1962). f 1987. 56 W. Shockley, N.L. (1956). Bell Laboratories (1936-^2, 1945-54). Professor at Stanford University (1963-75). f 13.8.89. 57 W.H. Brattain, N.L. (1956). Bell Laboratories (1929-67). f 13.10.87. 58 H.Y. Fan, Professor at MIT (1948-63) and Purdue University. (1963 + ). 59 Prof. R.L. SprouU, Professor at Cornell University (1946-63), President of University of Rochester (1975). 60 Prof. R.W. Pohl, University of Gottingen, 1916-53. 11976. 61 Prof. G. Busch, ETH Zurich (1949-78). 1954 supervisor of Nobel prize winner Alex Muller. 62 Dr L.P. Smith, Cornell University 1904. f 17.6.88. 64 Pierre Aigrain, Research & Study Centre, French Navy (1948-50). Later Secretary of State in charge of research (1978-81), Scientific Advisor Thomson Group (1983+). 65 Dr C.R. Dugas, Paris. 68 Dr Meltzer. 69 Dr W. Ehrenberg, Birkbeck College, University of London. He and R.E. Siday found in 1949 a precursor of the Aharanov-Bohm effect. 70 Dr P.C. Banbury, University of Reading. 71 Dr Vera Daniel, Electrical Research Association (ERA). 72 Dr Stella Mayne, later Rymer, University of Reading. 77 Dr W.W. Tyler. 87 Dr Ruth Warminsky, later Broser, Berlin. 88 H.K. Henisch, University of Reading. Later Professor at Pennsylvania State University. 89 Dr N. Mostovetch, Paris. 90 Dr O. Klemperer. 96 Dr H. Labhart. Later Professor of Physical Chemistry, University of Zurich (1964-77). f 1977. 97 Dr Tatjana Kousmine, University of Lausanne. 99 Dr J.B. Gunn (of the Gunn effect), Elliott Bros. Ltd (1948-53). Later Professor at the University of British Columbia (1956-71), IBM (1971). 101 Dr J.R. Drabble, GEC Research Laboratories. Later Professor of Physics at the University of Exeter. 106 E.W.J. Mitchell, University of Bristol. Later Professor at the Universities of Reading (1961-78) and Oxford (1978-88), Chairman SERC (1985-1990). FRS (1986). 108 Dr Torgesen, Norway. 109 Dr C.G. Kuper, later Prof. Technion, Haifa. 112 Donald Avery, RRE. Later Atomic Energy Establishment Director, Sellafield? 114 Dr K.W. Plessner, AEI Laboratories, Aldermaston. Later at Dielectrics Ltd. 119 Dr L. Pincherle, RRE. Later Reader Bedford College. 120 J.W. Granville, University of Reading. Later at RRE. 121 Dr E.H. Putley, RRE. 122 Dr B. Vodar, Paris? 123 Dr R.P. Chasmar, RRE. Later AEI? 124 Dr George G. McFarlane, TRE 1941-60. Director RRE 1962-7, Kt. 1971, Controller R&D MOD 1971-5, Board member British Telecom 1981^4, Corporate Director British Telecom 1984—7. 129 D. A. Wright, GEC Laboratories. Later Professor at Durham University. 134 Prof. F.H. Stieltjes, Philips'Research Laboratories. 11986. 140 Dr S.E. Mayer, Standard Telecommunications Research Laboratories, London, f 1986(7). 144 Dr Audrey Jones, University of Reading. Later Dr A. Hogarth. 149 H. Fritzsche, Purdue University (1954-63). Professor at Chicago University (1963). 150 Jacques Friedal, Ecole des Mines, Paris. Later Professor at the Univeristy of Paris-Sud (Orsay), Academie de France 1977. 151 Dr John Hirsch, Birkbeck College, London. 152 A.H. Gebbie, National Physical Laboratories. 158 Dr Fred Ansbacher, University of Aberdeen. 159 Dr George A.P. Wyllie, University of Glasgow. 160 Dr Charles W. McCombie, University of Aberdeen. Later Professor of Physics at the University of Reading. 161 Dr M.J.O. Strutt, Philips Research Laboratories. Professor at E.T.H. Zurich (1948-74). 163 Dr A. Fairweather, Post Office Research Laboratories. 165 Dr M. Wise, Philips Research Laboratories. 167 Dr P.W. Haayman, Philips Research Laboratories. 168 Dr W. Morton Jones, AEI Aldermaston (1949-61), Culham (1962), Saclay (1963-5), University of Strathclyde (1966 + ). 169 Dr J. Volger, Philips Research Laboratories. 11984. 171 Dr J. Ewels, University of Reading. 172 Dr H. Krebs, Germany. 173 Dr W. Meyer, Philips Research Laboratories, of the Meyer-Neldel rule. 175 Dr R.W.H. Stevenson, University of Aberdeen. 183 Dr H.J. Vink, Philips Research Laboratories. 190 Dr Philip Rhodes, University of Leeds. 198 Dr Richard Mansfield, Bedford College. Later at Brunei. 202 Dr R.W. Sillars, AEI Laboratories, Trafford Park, Manchester. 206 Dr Peter Myers, University of Aberdeen. Later Professor at the Chalmers University of Technology, Goteburg, Sweden.

Recombination in semiconductors PETER T. LANDSBERG University of Southampton, UK

The right of the University of Cambridge to print and sell all manner of books was granted by henry VIII in 1534. The University has printed and published continuously since 1584.

Cambridge University Press Cambridge New York

Port Chester

Melbourne

Sydney

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Victoria 3166, Australia © Cambridge University Press 1991 First published 1991 Printed in Great Britain at the University Press, Cambridge British Library cataloguing in publication data Landsberg, P. T. (Peter Theodore) 1922Recombination in semiconductors. 1. Semiconductors. Dynamics. Mathematics s. Mathe 537.622 Library of Congress cataloguing in publication data Landsberg, Peter Theodore. Recombination in semiconductors/P.T. Landsberg. p. cm. ISBN 0-521-36122-2 1. Semiconductors—Recombination. I. Title. QC611.6.R43L36 1991 537.6/226—dc20 90-45623 CIP ISBN 0 521 36122 2 hardback

up

Contents

List of main symbols Note on units Introduction 1 1.1 1.2

1.3 1.4 1.5

1.6

1.7

1.8

Semiconductor statistics The electrochemical potential A shorter statistical mechanics 1.2.1 The distribution laws 1.2.2 The Einstein diffusion-mobility ratio The classical gas theory Density-of-states expressions Appendix: The volume Vn of an ^-dimensional sphere Density-of-states expressions in the neighborhood of various points in k-space 1.5.1 General theory 1.5.2 Conditions near extrema and saddle points The number of electrons and holes in bands 1.6.1 Standard or parabolic bands 1.6.2 Nonparabolicity 1.6.3 The'Kane gas' 1.6.4 Additional remarks The number of electrons in localized states 1.7.1 General theory 1.7.2 Some special cases of occupation probabilities for localized states 1.7.3 Gibbs free energies and entropy factors 1.7.4 Points from the literature Interaction effects from impurities, including screening 1.8.1 Interaction effects (impurity bands, Mott and Anderson transitions)

xv xxi 1 7 7 10 10 14 16 20 23 24 24 27 30 30 33 34 39 40 40 43 47 51 51 51 ix

Contents

1.9 1.10

1.11

1.12

1.13 1.14 2 2.1

2.2

2.3

1.8.2 Concentration-dependent activation energies 1.8.3 Screening 1.8.4 Negative- U centers Saturation solubilities of impurities Appendix: Solubilities of multivalent impurities The equilibrium of simple lattice defects 1.10.1 General theory based on the grand canonical ensemble 1.10.2 Examples 1.10.3 The free energy argument Quasi-Fermi levels, reaction kinetics, activity coefficients and Einstein relation 1.11.1 Reaction constants for band-band processes 1.11.2 Reaction constants for processes involving traps 1.11.3 Einstein relation and activity coefficient Fermi level identifications, intrinsic carrier concentrations and band-gap shrinkage 1.12.1 Fermi level identifications 1.12.2 Heavy doping effects 1.12.3 A simple model of band-gap shrinkage 1.12.4 The relation between doping and solubility of defects 1.12.5 Points from the literature Quantities referred to the intrinsic level Junction currents as recombination currents Recombination statistics Basic assumptions for recombination statistics 2.1.1 General orientation 2.1.2 The assumptions needed 2.1.3 Example: Quantum efficiency in an intrinsic semiconductor The main recombination rates 2.2.1 Basic theory 2.2.2 Identification of the XdI 2.2.3 Recombination coefficients as state averages 2.2.4 Joint effects of distinct recombination channels 2.2.5 Special cases 2.2.6 Brief survey of selected literature on recombination (mainly to about 1980) Shockley-Read-Hall (SRH) statistics: additional topics 2.3.1 A derivation of the main formulae (any degeneracy, any density of states) 2.3.2 The addition of reciprocal lifetimes 2.3.3 Reduction of SRH to Fermi-Dirac occupation probabilities 2.3.4 An application of the formalism to Si and Hg^Cd^Te 2.3.5 The residual defect in Si

53 54 60 62 66 67 67 69 71 72 72 75 79 82 82 90 92 95 97 97 100 102 102 102 105 108 109 109 110 112 115 118 119 122 122 123 125 126 127

Contents

2.3.6 2.3.7

2.4

SRH recombination via a spectrum of trapping levels The joint action of band-band and SRH recombination

xi

134 137

Cascade recombination

144

2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.4.6 Field 2.5.1 2.5.2

144 146 150 157 162 163 167 167

2.8

Introduction The kinetics of the model The transient properties of the concentrations The steady-state recombination rate Generation lifetime Capture and emission coefficients in general dependence of capture and emission Introduction A simple intuitive treatment of the steady-state TCR model (with application to electric field effects) Appendix to section 2.5.2 2.5.3 Schottky effect and Poole-Frenkel effect 2.5.4 The field-dependence of the emission rate Appendix: The integrations of section 2.5.4 2.5.5 Thermionic and field emission 2.5.6 Geminate recombination Cascade capture 2.6.1 Some formulae for cascade capture 2.6.2 Implications of the classical conservation laws 2.6.3 Relations between cross sections and probabilities 2.6.4 Details of special cases covered by the exposition 2.6.5 The total capture cross section 2.6.6 The longitudinal acoustic phonon cascade capture cross section 2.6.7 Remarks on the relevant literature Surface recombination and grain boundary barrier heights 2.7.1 Recombination statistics for surface states 2.7.2 A schematic grain boundary model 2.7.3 The grain boundary barrier height Recombination at dislocations

168 172 172 175 181 183 189 193 193 196 198 201 202 207 208 208 208 210 212 216

3

Auger effects and impact ionization (mainly for bands)

220

2.5

2.6

2.7

3.1

Introduction

220

3.2

Fermi's golden rule

225 r°° i—COSJC

Appendix: Proof that / =

-2—dx = n

229

X

3.3

3.4

J -oo Reduction of the many-electron problem 3.3.1 Introduction 3.3.2 Determinantal eigenfunctions 3.3.3 Matrix elements between determinantal wavefunctions 3.3.4 Special cases Matrix elements of two-body and Coulomb potentials 3.4.1 A general argument

230 230 230 233 235 240 240

xii

3.5

3.6

3.7

3.8

4 4.1 4.2 4.3

Contents 3.4.2 The use of free-electron functions 3.4.3 The use of four Bloch functions 3.4.4 Localized states: the case of two Bloch functions 3.4.5 Four Bloch states: inclusion of Umklapp-type terms Threshold energies for impact ionization 3.5.1 Introduction 3.5.2 Two isotropic parabolic bands 3.5.3 Three iso tropic parabolic bands 3.5.4 One iso tropic parabolic, one iso tropic nonparabolic band 3.5.5 The relation between Auger recombination and impact ionization in bands 3.5.6 Some related relevant literature 3.5.7 General theory of threshold energies Bloch overlap integrals 3.6.1 A sum rule for the integrals 3.6.2 Overlap integrals for a scalar effective mass 3.6.3 The effect of the overlap integral on the matrix element of the Coulomb potential 3.6.4 The overlap integrals in the limit of a small periodic potential 3.6.5 Fourier analysis of the overlap integrals 3.6.6 Numerical estimates Auger and impact ionization probabilities 3.7.1 General (approximate) formulae for impact ionization probabilities per unit time and Auger recombination rates 3.7.2 The impact ionization rate for a simple case 3.7.3 Classification and general properties of results using the constant matrix element approximation Auger lifetimes 3.8.1 Basic results to be used 3.8.2 Two approximations: treatment of exchange terms, use of most probable transitions 3.8.3 The integration 3.8.4 The Auger lifetimes 3.8.5 Notes on experimental procedures Appendix: Proof of formula (3.8.11) Radiative recombination (mainly for bands) Introduction A review of the statistical formalism for single-mode radiative transitions The quantum mechanics of radiative transitions 4.3.1 Introduction 4.3.2 First-order time-dependent perturbation theory 4.3.3 Fourier analysis of the pure radiation field 4.3.4 The free field as a collection of oscillators

242 242 245 248 249 249 252 254 256 261 262 263 267 267 271 273 274 277 278 280 280 283 288 293 293 295 297 300 307 310 312 312 315 320 320 321 323 324

4.4

4.5

4.6

5 5.1 5.2

5.3

5.4

Contents

xiii

4.3.5 The number operator 4.3.6 The perturbation 4.3.7 The transition probability Interband emission rates 4.4.1 The rate of single-mode allowed radiative transitions 4.4.2 Multi-mode operation, spectral emission functions and k-selection rule for allowed transitions 4.4.3 Spectral emission functions with and without k-selection for multi-mode operation and direct transitions Interband absorption, emission and their relationship 4.5.1 Absorption-emission relationships and radiative recombination in terms of optical data 4.5.2 Interband indirect transitions and excitonic effects 4.5.3 Analytical interband emission rates via absorption rates Recombination-generation induced phase transitions and chaos 4.6.1 Nonequilibrium phase transitions due to generation-recombination 4.6.2 Recurrence relations for M-level systems 4.6.3 A recombination-generation model for chaos

327 327 328 329 329

343 346 353 356 356 361 364

Defects Introduction Radiative recombination involving traps

369 369 373

5.2.1 General theory; the hydrogenic ground state 5.2.2 Hydrogenic excited states 5.2.3 Tail states due to random potential fluctuations 5.2.4 Photoionization and photoneutralization 5.2.5 Some uses of detailed balance 5.2.6 Donor-acceptor pair radiation 5.2.7 Notes on excitons 5.2.8 Points from the literature Auger recombination and impact ionization involving traps without excitons 5.3.1 Construction of universal curves for the T± process 5.3.2 Introduction of four models 5.3.3 Zero-order approximation analytical trap-Auger coefficients in nondegenerate materials 5.3.4 Auger quenching 5.3.5 More involved arguments and effects More complex effects 5.4.1 Auger effects involving bound excitons 5.4.2 Donor-acceptor Auger effect 5.4.3 Shake effect 5.4.4 Recombination-enhanced reactions

373 381 382 389 391 398 405 410

333 337 343

411 411 417 418 426 433 436 436 440 441 442

xiv

Contents

6 6.1 6.2

Multiphonon recombination by T. Markvart Introduction Electron-lattice interaction in semiconductors 6.2.1 Electron states in the presence of electron-lattice interaction 6.2.2 Electron-lattice interaction in polar semiconductors Multiphonon transition rate at high temperatures 6.3.1 Landau-Zener formula 6.3.2 Relationship to the Born-Oppenheimer approximation 6.3.3 Multiphonon transitions at high temperatures Multiphonon transitions by tunneling Transitions involving free carriers. The role of the Coulomb interaction 6.5.1 Multiphonon capture and emission of free charge carriers 6.5.2 The role of the Coulomb interaction Quantum mechanical treatment of the multiphonon transition rate

468 468 474 476

Recombination in low-dimensional semiconductor by R.I. Taylor Introduction Energy levels and density of states Radiative recombination in QWs 7.3.1 Rough estimate of radiative recombination in a QW 7.3.2 Emission probabilities Nonradiative recombination in undoped QWs 7.4.1 Introduction 7.4.2 The theory of Auger recombination in a QW 7.4.3 Intervalence band absorption Ultra-low threshold current strained QW lasers

481 481 483 488 488 490 495 495 496 508 510

6.3

6.4 6.5

6.6 7 7.1 7.2 7.3

7.4

7.5

structures

Appendix A The delta function (section 3.2) Appendix B Useful identities arising from the periodic boundary condition (section 3.4) Appendix C Fourier expansions (section 3.4) Appendix D The effective mass sum rule and the dynamics of Bloch electrons (section 3.6) Appendix E Diagonalization and Jacobian for the threshold energy and impact ionization calculation (section 3.5.7) Appendix F The saddle-point method, and the method of stationary phase (section 6.3) Appendix G Evaluation of the integral (6.6.12) (section 6.6) References Name index Index of topics, concepts and materials

447 447 449 449 454 457 457 460 462 463

511 516 520 525 535 538 540 542 573 585

Main symbols

Latin A area A electromagnetic vector potential (4.3.1)* b impact parameter, Fig. 2.6.3 22s, B19 B2 band-band recombination coefficients, Fig. 2.1.1 c velocity of light cn, cp capture coefficients [L3!1"1], Fig. 2.4.1, cng for the ground state, cne for the excited state Cv, Cp heat capacities (1.3.18) d number of dimensions, section 1.3 D diffusion coefficient (1.2.11) D_p(z) parabolic cylinder function (5.2.38) e electron charge ej particle energy in state j (1.1.4) en, ev emission coefficient [T"1], Fig. 2.4.1, eng for the ground state, ene for the excited state e^(q) or ek^ unit vector for direction of polarization of a mode (q, X) or (k, X) (4.3.18) S electric field, Fig. 1.1.1 ( = — gradcp) Ec, Ev band edge energies (1.1.6) EG energy gap Ec-Ey, Fig. 1.5.5 £eff0* + i) effective energy, defined by Zr/Zr+1 = exp[Ee{f(r+±)/kT] (1.7.28) Ei system energies (1.2.1) Eo nonparabolicity parameter (1.6.16) * References are to equations unless otherwise stated. XV

xvi

List of main symbols

Et = (m*/m£2) t \me^/2h2) hydrogen-like ionization energy from orbit with principal quantum number t (1.8.4), me*/2h2 = 13.6 eV; EAt = fi2/2mAf2At acceptor ionization energy (5.2.19); Et = h2t2/2mr2t corresponding result for Bohr atom (5.2.20) eq suffix for equilibrium conditions; suffix 0 is also used / c , / v activity coefficients (1.11.4) f[E) Fermi-Dirac distribution function, section 1.6.3 FHelmholtz free energy (1.1.15) F = B? + B1n + B2p combination of band-band recombination parameters (2.2.11) F overlap integral (3.4.15) Fe9 Fh reduced quasi-Fermi levels (i.e. divided by kT) (1.12.25), (3.5.19); y e, yh are also sometimes used Fs(y) Fermi integral (1.6.4) g degeneracy due to spin (1.3.9) g momentum transfer, section 3.8.2 gn degeneracy of the ground state of an ^-electron defect, section 1.7.1, (1.7.15) G Gibbs free energy (1.1.10) G = r ^ + 7 ^ « + r 2 / ? = cn band-trap recombination parameters (2.2.11), (2.4.22) Ge, Gn, Gy dimensionless superposition function (3.4.24), (3.4.27), (5.2.1); Fourier coefficient of hydrogen-like function, e.g. (5.2.21) h momentum transfer, section 3.8.2 # enthalpy (1.1.15), (1.7.33) H = Tl + Tsn+Ttp = cp band-trap recombination parameters (2.2.11), (2.4.23) H Hamiltonian operator, sections 3.1, 6.2 70 = me*/2h2 ^ 13.6 eV ionization energy of the hydrogen atom (2.6.34), (3.7.17), (5.2.71) j current density, section 1.2.2, (2.5.38) k wavevector (1.3.9) K vectors in k-space, possibly extending beyond the first zone (3.6.24) Kn second modified Bessel function (1.6.23) Kjy equilibrium constant (1.12.15) / number of directions of polarization (4.4.13) /Deb Debye screening radius (1.8.22), (5.4.3) /j integers (1.4.1) L vector between lattice points in k-space (3.4.9) Ln diffusion length of electrons (2.2.24) m free electron rest mass mp mp m*, me9 mv, ms, ml9 mh, m* effective masses (1.4.2), (1.4.9), (1.4.11), (1.4.12), (1.8.2); m1 and mh refer to a light-hole and a heavy-hole band, respectively


—I

xvii

reciprocal effective mass tensor (1.5.6)

M maximum number of detachable electrons per defect (1.7.1) M D , M E general matrix element for direct and exchange transitions (3.3.21) Ms{s = 0,1,...) critical points in k-space, section 1.5.1, Table 3.7.1 n electron concentration (1.6.3) n0 or /2eq thermal equilibrium concentration, p. 104 n(r — |), n± = «(§) special electron concentration, section 2.2.4 ng = ejcng,ne = ene/cne (2.4.16) «j intrinsic concentration (1.12.10) nQ intrinsic concentration away from equilibrium (2.2.13) N number of particles (1.1.1); concentration of defects (1.8.27) N photon number (4.2.3) JV density of states of dimension (energy)" 1 (1.3.12), (1.4.4) Nc 'effective' number of states per unit volume (1.6.7) JVD, NA number of donors, acceptors (1.7.24) (1.11.8) normally per unit volume Ne mean number of trapped electrons (1.7.5), (1.7.17) Nt number of traps (1.7.24) Ny 'effective' number of states per unit volume (1.6.9) N(\i, 1, T) supplementary matrix element (3.4.30), (5.3.4) 0 suffix for equilibrium conditions, also denoted by 'eq' O12 overlap integral of vibrational wavefunctions (6.4.20) p pressure (1.1.1) p concentration of holes (1.6.9) p0 or /?eq thermal equilibrium concentration, p. 104 p momentum, section 1.3 P(r~ IX Pi = p(k) special hole concentration, section 2.2.4 pg = ep/cp(2AA7) /?! probability of occupation of state I, section 1.11 P probability (1.2.1), (1.6.1) P transition rate per unit volume (3.7.4), (4.3.15) q general charge (1.1.2) q normal coordinates (6.2.5) q magnitude of phonon wavevector (4.5.15) qd probability of vacancy of state / , section 1.11 Q constant pressure partition function, section 1.7.3 Q general charge (1.8.5) Q configuration coordinates (6.2.12) r position vector, section 1.2.2

xviii


r*lf(E), rfj(E) rates of spontaneous and stimulated emission per unit volume per unit energy range (4.4.15) rt = (em/m*) rx t2 radius of Bohr orbit of principal quantum number t in semiconductor (1.8.2), (4.5.18), (5.2.18) rt = t2rx radius of Bohr orbit of principal quantum number t, rx = h2/me2 = 0.528 A (1.8.2), (3.7.17), (4.5.18) R recombination rate, usually per unit volume (2.1.1) s density of states exponent (1.3.2) s velocity of sound, section 2.6.1 S entropy (1.1.1) S Huang-Rhys factor (6.2.3) SJJ transition probability per unit time (2.2.1) t principal quantum number (1.8.2) t0 = fts/e*m ~ 2.42 x 10"17 s, section 3.7.1, (3.7.17), (5.2.71) T absolute temperature (1.1.1) Tel, TY electron and lattice kinetic energy (6.2.4), (6.2.8) T\9 Tl, T19 T2, T3, TA band-trap recombination coefficients, Fig. 2.1.1, p. 103 T(E) transmission coefficient (2.5.35) Tit probability of a transition i^fin time t (3.7.1) u recombination rate per unit volume (2.2.1) u ion displacement (6.2.24) ua (k,r) dimensionless modulating part of Bloch function (3.4.9) wcr_i, wr_iv recombination rate into or out of traps (2.2.12b,c) wcv rate between bands (2.2.9) t/mean energy (1.1.1) Uel9 Ul electron and lattice potential energy (6.2.5), (6.2.7) v mobility (1.2.10) v saturation velocity (EQ/m$)* in a nonparabolic band (1.6.31) v thermal velocity of electrons r a (k, L) dimensionless Fourier coefficient of modulating part of Bloch function (3.4.10), (3.4.16), (3.4.22)

V volume (1.1.1) V electronic potential energy, section 2.5.3 ^combinatorial factor (1.7.11), (1.9.7) W transition rate (6.3.21) x coordinate (1.4.1) X entropy factor, exp(AS/k) (1.7.34) Xs, X±, X2 band-trap ionization coefficients, Fig. 2.1.1 *u = SuPiqj/Sjrfjqi

(2.2.2a,b)

7 s , Y19 Y2 band-band ionization coefficients, Fig. 2.1.1


xix

Z canonical partition function (1.2.9) Z d canonical partition function for a defect (1.12.34) Z D canonical partition function for a donor (1.12.37) Zs canonical partition function for an ^-electron defect (1.7.1) Greek a absorption coefficient (4.5.1) a 0 = e2/hc = 1/137 (5.2.71), (5.4.6) P covers uncertainty in value of like-spin matrix element (3.3.27) y EE (g/MHEt-iLykT (1.2.16), also \i/kT (1.7.24) ye, yh reduced quasi-Fermi levels [ie JkT{\. 11.1); Fe, Fh is also used (3.5.19). yn and yv are used in section 2.4 for Ye and yh, respectively, to release ye for the excited state F gamma function (1.3.4) T width of an energy level (6.5.4) 8n, dp electron and hole concentration departures from equilibrium (2.1.2) A split-off band gap, Fig. 1.6.3(a) A = AC + AV (1.11.5a) Ac, Av band edge shifts due to band-gap narrowing (1.11.5) Aeff effective band-gap narrowing (1.12.23) e particle kinetic energy (1.2.15) 8 dielectric constant (1.8.2) e(k,co) dielectric function (1.8.10) «x) = exp[eq>W/fcr] (2.7.10) TI reduced energy E/kT (1.7.23), (1.7.28) r| quantum efficiency (2.1.10) r| spin wavefunction (3.3.1) (ilD, n J == (EJkT, EJkT) K wavevector in tunneling region (6.4.1) X = exp(\i/kT) (1.7.1) X mean free path (2.5.17), (2.6.1) X screening parameter [L"1] of the Coulomb potential (3.4.19) X subscript specifying polarization (2.6.15), (4.3.17) A" 1 Debye screening radius (1.8.7), (1.8.22) ( = /Deb) [i electrochemical potential (1.1.3), Fermi level [i refractive index (4.3.32) |x mjmv (3.8.6) [i0 chemical potential (1.1.1) ^ e = yekT, \ih = yhkT quasi-electrochemical potentials (1.11.1)

xx


jij chemical potential for intrinsic sample (1.12.11) v general volume concentration of particles or defects (1.2.10), (1.8.16) v photon frequency (4.2.9) vr sometimes the number of r-electron defects (1.7.4) \ variance of random potential (5.2.36) S grand partition function (1.2.2) p(r, i) volume density of particles (1.8.5) a (recombination coefficient in cm 3 s" 1 /thermal velocity) cross section (2.3.31) dj = ± 1 parameters for energy surfaces (1.5.10) cix, aiy, aiz Pauli spin operators of particle / (3.3.1) xw, xp lifetime of electrons or holes (2.1.4) xr radiative life time, Table 4.1.1 (p electrostatic potential (1.1.2) (p wavefunction (3.1.7) cpk Fourier coefficient of electrostatic potential (1.8.8) ^ = en(£)/en(£ = 0) for ground and excited state (2.5.6), (2.5.23) XiiQ) vibrational wavefunction (6.4.1) \|/ wavefunction (electrons) (1.4.1), section 3.1 (op angular frequency of plasma oscillations (1.8.26) (o(k) normal mode frequencies (6.2.2), (6.2.11) Q volume of unit cell, Appendix B

Note on units

Some important books on solid-state physics use Gaussian units, and these units have also been adopted in this book in preference to SI units. Thus the dielectric constant 8 is a number just like the relative dielectric constant e r in the SI system, in which s 0 is the permittivity of the vacuum and is given by 8.854 x 10~12 farads per meter. Some examples in terms of the two systems of units are given in the table.

Gaussian units Poisson's equation for the electrostatic potential (p in terms of the charge density p Coulomb's law for the potential energy of two charges q19 q2 a distance r apart

4TC

qiq%

SI units

„_ qiq2

er

4ne0sTr

Poynting's vector when the displacement is proportional to the electric field

~8TT

2

For vacuum

8= 1

P_

xxii

Notes on units

Atomic units are employed occasionally (see pp. 280, 404): a0 = —2- = 0.529 x 10" em S

1 3 2ma\

= —

. 2/r6

e

= 2.419 x 10"

8

cm

V 17

s

a0 = e2/hc = 1/137 = (fine-structure constant)" 1 Atomic units for recombination: (
This yields, on using F(l/2) = \/n, Ej

(1.4.5) (1-4.6)

$

^

(1.4.7)

Here Fx is the length, V2 is the area and Vs the volume of the system considered. For g = 2 and m1 = m2 = mz = m, eq. (1.4.5) reproduces eq. (1.3.13), but we now have also the results (1.4.6) and (1.4.7) for a surface film (or a quantum well) and for a quantum wire. Applying eq. (1.4.5) to the conduction band in a semiconductor, Eo is the energy at the bottom of the band where J^(E) vanishes, and it can be interpreted as the

22

Semiconductor statistics

energy at the minimum E(k0) of the conduction band. There may be nc equivalent minima (which result from crystal symmetry), whence eq. (1.4.5) may be written: (1.4.8) It is usual to write Ec (instead of Eo) for the bottom of a conduction band. The surfaces of constant energy in &-space can be given the general form

if one is in the neighborhood of Ec by choosing appropriate (i.e. 'principal') axes in fc-space. In the case of Ge, Si and GaAs the symmetry of the crystal requires that at least two of the effective masses are equal. If for example m2 = m3 the surface is an ellipsoid of revolution about the A^-axis. Thus, m1 is called the longitudinal effective mass mp while m2 = m3 is called the transverse effective mass m±. It is associated with the plane at right angles to the axis. One then finds (1.4.9) The number ne is obtained from band theory. Thus the lowest conduction band of Si lies at roughly f of the distance from the zone center (usually labelled F) to the center (labelled X) of the square face perpendicular to the A^-axis. There are six (nc = 6) symmetrically placed such minima. In Ge there are eight symmetrically placed minima, but only half of each lies in the first Brillouin zone, so that One can write (1.4.8) as V(E-Ej

(1.4.10)

and mc is then called the density-of-states mass. Comparing with eq. (1.4.8) me = n\(m1 m2 ms)* or n\{m^ m])*

(1.4.11)

This gives the density-of-states mass in terms of the effective masses, for example, for Si and Ge. For GaAs one can put nc = 1, m1 = m2 = m3 near Ec since the constant energy surface is there nearly spherical. Hence mc = m1 in this case. The valence bands for these materials (diamond and zinc blend structures) have four subbands, three of which are degenerate at k = 0. The spin-orbit interaction causes a splitting of the bands which consist of a heavy-hole, a light-hole and a

1.4 Density-of-states expressions

23

' split-off' band. For further details a specialized treatment of band theory should be consulted. Assuming spherical surfaces for the light-hole and heavy-hole bands of masses m1 and mh, respectively, one finds

f ^§? V(EVE)* = £*L(EEf[ni

+ nft V

(1.4.12)

so that in that case

Some numerical values are given in Table 1.4.1. Appendix: The volume Vn of an n-dimensional sphere Suppose Vd{r) = Cd rd, where r is the radius. We then work out the integral of exp (— r2) over all space by two methods: (i) using Cartesian coordinates x19..., xd9 and (ii) using spherical coordinates C

L J

2 °fexp(-r2 )dx 1 ...dx d = f°°exp(-r )dF»

-oo J

(1.4.13)

Jo

The left-hand side is with r2 = x\ + . . . + x\

f T exp(-;c2)dxT = [2 P°exp(-jc2)d;cT = I" r ^ e ^ d ^ T = [T(|)]* = TC-/2

(1.4.14)

where eq. (1.3.5) has been used. The right-hand side of eq. (1.4.13) is with y = r2,

dr = dy/2&

r°° Cdd\ 0

D Equating eqs. (1.4.14) and (1.4.15),

•H)

d.4.,5,

24


Table 1.4.1. Some numerical data (Values are approximate and are liable to be adjusted at various times.) Ge

(1) Main energy gap at 300 K (eV) (2) Effective densities of states (cm~3) at 300 K (see eqs. (1.6.3) and (1.6.9)) (3) Effective electron masses in units of the electron rest mass (see eq. (1.4.9)) at4K Parabolic mass near band bottom

u

(4) Effective hole masses in units of the electron rest mass at 4 K heavy-hole band (mh) light-hole band (mj split-off band (ms) (5) Density-of-states effective masses at 4 K in units of the electron rest mass (see eqs. (1.4.11) and (1.4.12)) (6) Dielectric constant

0.66 1.03 x 10 19 5.35 x 10 18 1.59 0.082

Si

GaAs

1.12

1.42

3.22 x 10 19 1.83 x 10 19

4.21 x 10 17 9.52 x 10 18

0.92 0.19

0.067

0.54 0.15 0.23

0.51 0.082

0.55

1.06

0.067

0.36

0.59

0.53

0.35 0.043 0.077

m

16

11.8

0.15

13.2

For a hyper-ellipsoid Cd is retained and rd is replaced by the product of the d semiaxes. 1.5 Density-of-states expressions in the neighborhood of various points in k-space* 1.5.1 General theory

The result (1.4.4) can be seen in a yet more general setting, without appeal to wavefunctions, by integrating over a surface of constant energy E as follows. Let A and B be two points picked out by a common normal to two neighboring surfaces of constant energy (Fig. 1.5.1). The number of k-vectors in a volume element dk of k-space is for ^-dimensional vectors (cf. eq. (1.3.9)): (1.5.1) dv = [Fd/(27ir]dk Let Ak be the change in k on passing along AB. The rate of change of E normal to the surface being Vk£(k), the energy change is AE = V k £(k). Ak = |Vk£(k)| Ak * This section may be omitted at first reading.

1.5 Density-of-states expressions in k-space

25

E + AE

dS

Fig. 1.5.1. Neighboring elements of distinct surfaces of constant energy in k-space. Let dS be an incremental area around A on the surface of constant energy E in k-space. Then the number of states in an incremental volume dSAk between the surfaces is by (1.5.1)

Thus in an energy interval the number of states is given by an integral over the surface S of constant energy E = 2 fdv = ^

C

^

f

b[E-E(k)]dk

(1.5.3)

dS

}s \V E(k)\ s

k

We have multiplied by two to allow for spin. Also the Dirac 8-function is defined to have the property (see eq. (A.7) of Appendix A): J{x) 8(x) dx = /(O),

U(x) any continuous function; a, b > 0]

i

It looks after the confinement of the integration to the particular energy E. The expression (1.5.3) will be used, with the aid of a lemma. Lemma d

If y is a ^/-dimensional vector of length y = ( £ yfp, then

J i ] 8 (^)dy = { ^

(1.5.4)

Proof The surface area of a J-dimensional sphere of radius y has the form Cddyd~x, where 3C3 = 4TC and, more generally from the appendix to section 1.4,

26


Since the integrand in / is independent of angles, the volume element dy is dy multiplied by Cddya~x. Hence with y* = z

I = Cdd f°6(A -y2) / - 1 dy = ^ Cd f°6(A - z) z^r2,

y3->z,

dy^rdrdOdz

(1.5.14)

where 0 is defined by sinO = y2(y\+y22)~* and runs from 0 to 2TC. Then V (8m, m 4TC3

hz

-2n

w:

6(E0-E+r2-z2)dz

V (2m1 m2 n

•H: If E > EG, we have u — r2 — (E—E G) and a contribution arises only if r2 > E—E G. If E < EG, then r2 > E—E G and a contribution arises for all r values. A logarithmic divergence

1.5 Density-of-states expressions in k-space

29

Ec(k)

Fig. 1.5.5. A radiative transition. arises in both cases for r = oo. The r integration must therefore be limited to a value r R, beyond which the assumed band structure becomes a poor approximation. Hence (1.5.15) where

TlW

This type of curve is shown in Fig. 1.5.4 and is of importance in the theory of optical transitions (section 4.4.3). The simple example (1.5.13) arises from writing )

J where

+

= —

0J= 1,2,3)

(1.5.16)

In eq. (1.5.13) one is considering a band structure in which wv3, say, is negative but less in magnitude than mc3, while the other four masses are all positive. For instance if mc3 = 0.2m and mv3 = — 0.1m then o1 = G2= 1, a 3 = — 1, and m3 = 0.2m. The ^-conservation shown in Fig. 1.5.5 is characteristic of a certain type of radiative transition (see section 4.4.2). Use of the m/s in the expression for the density of states gives one the two-band (or 'joint') density of states.

30


1.6 The number of electrons and holes in bands 1.6.1 Standard or parabolic bands For the conduction band of a semiconductor, assumed to extend to infinite energy, the mean number of electrons in thermal equilibrium at temperature T and chemical potential |i is f00

N=

P{E)Jf{E)dE

(1.6.1)

EC

where Ec is the energy at the bottom of the band and P(E) is the occupation probability of a quantum state of energy E. This drops to zero exponentially for large energies, by eq. (1.2.8). Hence the finite extent of a band is well simulated by the limit E = oo in (1.6.1). Using eqs. (1.2.8) and (1.4.8): (E-E$dE

where, with mc given by eq. (1.4.11),

One finds for the electron concentration [1.6.1-1.6.3] jt

j

(1.6.3)

The parameter in the R function is, by eq. (1.1.8), in general a function of position. We define the Fermi-Dirac integrals by

where F^) is defined in section 1.3. The value of 7VC is, from eqs. (1.6.2) and (1.6.3),

Nc = ^Dlr(|) = i ( ^ ) W ) ^

d.6.5) (1.6.6)

Hence Nc = 2(2nmckT/h2)*

(1.6.7)

1.6 The number of electrons and holes in bands

31

3

I

1

built-in field

Fig. 1.6.1. A p-n junction in thermal equilibrium. The built-in potential has been denoted by V^ and the space charge has also been indicated. Fig. 1.13.1 gives additional details.

The valence band, being almost full of electrons, is usually better described in terms of electron vacancies [1.6.1]. The probability of an electron vacancy in a quantum state of energy E is 1-:

1

1

+ exp[0i-£)/*71

+exp [ ( £ -

(1.6.8)

Assume a valence band which extends from energy — oo to an energy Ev. Then the concentration of electron vacancies in a valence band is, in analogy with eq. (1.6.3), P = T,=

Here

=

(Ev-E)idE l+exp[(ji-£)/*71 NvFl[(Ev-[i)/kT]

Nv = 2(mv kT/2nh2%

(1.6.9)

mw = (n2y m1 m2

where rav is the density-of-states effective mass, based here on the valence band effective masses (not distinguished notationally from the conduction band effective masses). Also nw is the number of equivalent valence band maxima. The results (1.6.3) and (1.6.9) are fundamental in all simple semiconductor calculations of equilibrium properties like heat capacities, transport properties like electrical conductivity or generation-recombination processes. As seen in section 1.1, Ec, Ev and \i, and therefore n = N/V and p = P/V, can be functions of position. We omit the bars on n and p and assume that average concentrations are intended. Thus as one goes from left to right in a simple one-dimensional p-n junction in thermal equilibrium (Fig. 1.6.1), the hole concentration decreases as Ev{x)~ JJ, becomes more negative, while the electron concentration increases as ji — Ec(x) becomes less negative. On the left, where holes dominate, we have the/>type material, on the right we have the n-type material. Donors, which on losing

32


20

« 10 §

eq. (1.6.11)

5

I 8

I 0.2 8

0.1

I

.

1

Fig. 1.6.2. The concentration of electrons in a parabolic conduction band in units of 7VC.

electrons become positively charged, are the majority impurity on the right; acceptors, which become negatively charged on accepting electrons from the valence band, are the majority impurity on the left. We note two properties. The first is that for large and negative chemical potential, as discussed in section 1.3, l

f 00

xse~x+ydx ) Jo

=e

y

(all s)

(1.6.10)

Hence (with Ec = 0, s = |), eq. (1.6.3) yields the classical result (1.3.3), if eq. (1.6.7) is used: n = Ncexp[([i-j

,

N c = 2(mc kT/2nfi2f

(1.6.11)

Also eq. (1.6.9) yields (1.6.12) These approximations are called classical or nondegenerate limits or MaxwellBoltzmann statistics. If the Fermi integral has to be taken into account, one speaks of Fermi degeneracy. (This has to be distinguished from the degeneracy when several linearly independent wavefunctions belong to the same eigenvalue.) The electrochemical potential \i is more usually referred to as the Fermi level, and this will often be done in the sequel. Secondly, for highly (Fermi) degenerate material the parameter in the integrals (1.6.3) and (1.6.9) is positive, i.e. the Fermi level has risen into the conduction band or dropped into the valence band. In such situations a rough approximation is to


33

/}E0 sloped

slope - —^

(a) (b) Fig. 1.6.3. (a) A common band structure near k = 0 for III-V compound semiconductors, (b) Dispersion relation (1.6.16) showing also the asymptotes. replace the Fermi function by unity for energies up to \i and by zero for energies beyond \i. Then (1.6.13)

In particular (1.6.14) The accurate result and the classical and degenerate approximations are illustrated in Fig. 1.6.2. Other density-of-states functions are needed for disordered or amorphous materials, for nonparabolic bands, etc. As an illustration, a case of nonparabolic bands is considered below. 1.6.2 Nonparabolicity

We consider briefly the case of nonparabolic bands. Let m* be the effective mass at a conduction band minimum k = 0 and suppose that there is also a valence band and a split-off band as shown in Fig. 1.6.3(a). Then, with the notation of that figure, it can be shown from reference [1.6.4] that the relation between the energy E and the wavevector k is + 2A)(E+EG)(E+EG + A) 2m*

(1.6.15)

34


It is readily seen that both for large A and also for AÔ the above dispersion relation becomes (Fig. 1.6.3(b)) —

2m;

)E,

(2E0~EG)

(1.6.16)

In fact one can show from eq. (1.6.15) that

so that eq. (1.6.16) is often a fair approximation. The density of states based on eq. (1.6.16) is, from eq. (1.3.9)

JT{E) = f^*1-^ = ^(^j[E(l+E/2E0)]Hl

+E/E0)

(1.6.17)

For g = 2 and Eo = oo this goes over into the parabolic case of eq. (1.4.5). The result (1.6.16) has been applied quite widely, and we give here some relevant references. In silicon the piezo-resistance was studied in this way recently [1.6.5]; in InSb the metal-insulator-semiconductor (MIS) capacitance [1.6.6], surface waves [1.6.7] and the heat capacity of thin films [1.6.8] could be discussed with the aid of the dispersion relation (1.6.16); it was used for GaAs to study the Boltzmann equation for the interaction of energetic electrons with polar optical phonons [1.6.9]. One may also note studies of nonparabolicity by the measurement of thermoelectric power in a strong magnetic field, applied for example to lead compounds [1.6.10]. One can give discussions analogous to those based on eqs. (1.6.1) to (1.6.4) and Fig. 1.6.3, by using eq. (1.6.16). The integrals turn out to be more complicated. The statistical thermodynamics of such a ' Kane gas' has not been used in the past, but is developed below because of its intrinsic interest and for possible future use.

1.6.3 The 'Kane gas' We again apply the/?F-formula (1.3.7) for the pressure p of the gas. The sum is replaced by an integration, using eq. (1.3.11). The number of states in a small volume dk of phase space is Vg ^dk

Vz = 7 -fdp

(1.6.18)

where V is the volume of the material, g the spin degeneracy, and p = hk is the momentum. Hence


35

where

by a partial integration. Here pr denotes the magnitude of the momentum (the symbol p might cause confusion with the pressure here), and / is the Fermi-Dirac distribution function

The energy is now explicitly referred to the band edge energy Ec, so that x = (E—E c)/E0 is a measure of the kinetic energy of the electron. Defining a velocity v by m* v2 = Eo and using eq. (1.6.16)

The substitution x = cosh 0 - 1 leads to x2 + 2x = sinh2 0 so that the pressure of a Kane gas is

P = B[

,,

Sinh49de

,

(1.6.19)

Jo

where B = 4ng(m*yvb/3h3

(1.6.20)

One finds also the mean number of particles to be

ecoêde

The average energy is obtained by multiplying the integrand in N by E = xE0 = (cosh 9 — 1) m*v2. Hence

U = IBV\

sinh*9coshe(coshe-i)de 0

1 + exp Jjl[mV (cosh e-l)-(n-£„)]}

(l622)

36


1 -

Fig. 1.6.4. The functions eaKn{a) for n = 0,1,2 which occur in the expressions for AT and for U in eqs. (1.6.25) and (1.6.27). These general results can be evaluated in the nondegenerate limit. Consider e.g. (1.6.21) first. We have N

~

sinh 2 6coshe

exp

exp

/ m*v*

Now sinh2 G cosh 0 = Kcosh 36 - cosh 6) and

f

cosh «G exp (-zcoshO)de = Kn(z)

(1.6.23)

where Kn Jo is the second modified Bessel function. This gives for the integral l 4 k T

where the recurrence relation for the A^s has been used: (1.6.24) Hence (see Fig. 1.6.4) N_

4ngvVm*2kT

(1.6.25)


3.0

37

Cy/Nk

2.0

U/NkT 1.5 1.0

_l_ 0.2

_l_

I

I

1.2 0.8 1.0 0.6 kT/E0 Fig. 1.6.5. The ratios U/NkT and Cv/NKofa nondegenerate electron gas as a function of the nonparabolicity parameter is"1. The numerical data of Table 24 from [1.6.11] has been used. Arrows refer to the case of InSb.

0.4

One sees at once that in the nondegenerate limit pV=NkT and, using eq. (1.6.25), one can find an expression for/?. Next, consider U. We have

exp - ^ - c o s h 9 dO (1.6.26) The hyperbolic functions in the integrand reduce to K|cosh 46 - \ - cosh 36 + cosh 6} so that eq. (1.6.23) can be used to integrate them. Using the recurrence relation (1.6.24) twice, to eliminate K± and the sum of K2 and Ko, one finds TJ

Ang{m*vfkT

/n + < o » - £ e \ (1.6.27)

Writing a = m* vz/kT for brevity, U ~Nkf~

(1.6.28)

38


This quantity is shown in Fig. 1.6.5 and rises from 1.5 to 3 as the nonparabolicity is increased. The limiting behavior of eq. (1.6.28) is readily worked out as follows. For vanishing nonparabolicity we should obtain U = \NkT. This can be verified using (aM)

(1.6.29)

The other limit a -> 0 of very large nonparabolicity is not usually of interest, though future experiments may perhaps identify it. It corresponds to Kn(a) ~ 2n~\n-

\)\/an

(1.6.30)

It leads to U = 3NkT, as shown in Fig. 1.6.5, where the heat capacity per particle, CV/NK, is also shown. The case of InSb is also marked on the figure. Using 2E0 = EG = 0.18 eV this yields at room temperature kT/EQ ~ 0.278. Using also m* = 0.013m, where m is the normal electron rest mass, one finds v ~ 1.1 x 108 cm s"1 or 0.37% of the velocity of light. The velocity of light is relevant for the following reason. For an isotropic but nonparabolic energy band the electron velocity w is parallel to the wavevector k and is given by w = fi-ÊQs). Its magnitude is obtained by first solving the quadratic equation (1.6.16) for E:

The energy zero is at the band minimum and the positive sign is taken. Now, using eq. (1.6.16),

and, using (1.6.16) again,

Hence

v

\+EJE

0/

0/

This shows that as the kinetic energy is increased, the velocity saturates at v, rather like the velocity of a relativistically moving particle saturates at c. In fact, the momentum-mass is

w

Eo

m*

(1.6.31)

This is the familiar formula from special relativity, except that the velocity of light has been replaced by v [1.6.12]. Some verification of this analogy has been obtained by the study of


39

electrons in InSb in crossed electric and magnetic fields [1.6.13]. A key point is that the above effective mass (and others that could be defined) is increased as the electron gains energy. The density-of-states formula (1.6.17) has not been used explicitly here. However, it provides an alternative way of obtaining the thermodynamic functions. This will be illustrated by re-deriving eq. (1.6.21). If J{E) is the Fermi-Dirac distribution function

N 4ngV

Jo The integral is then identical with that which occurs in eq. (1.6.21) and so is the pre-factor. The electron concentration may be written in a form analogous to (1.6.3):

where r\ = E/kT and r\Q = EJkT. In the expression for JVC, mc is now the effective mass at the bottom of the conduction band. If one wants to work out the internal energy, one simply notes that in

U=

\Ejr(E)J[E)dE

one has now an additional factor, which is

E = Eox =

m*v\cQsh®-\)

This gives eq. (1.6.22). 1.6.4 Additional remarks The properties of bands can also be formulated in greater generality, allowing for both nonparabolicity and for Fermi degeneracy [1.6.14] and by allowing explicitly for electric fields [1.6.15] and screening [1.6.16], but the resulting formulae are too involved to be given here. The effect of electron-electron interactions have been neglected here, except in so far as an averaged effect is included in a band scheme. They lead to complications which can be treated only approximately. For full discussions see [1.6.17] and [1.6.18]. For a less detailed treatment see [1.6.19]. For simple introductions see [1.6.20] and [1.6.21]. The Kane model holds approximately for several III-V compounds (InP, InSb, GaAs, etc.) whose properties are widely reviewed. For example, see the review [1.6.22] of GaAs properties. A slow drop by a factor of about 2 x 10~5 K"1 in the conduction band edge effective mass ratio mjm of w-type GaAs, InSb and InP with rise in temperature between 70 K and 220 K has been noted in [1.6.23] by means of magnetophonon resonance measurements. This is due to dilational changes in the band gap, see also [1.6.24] and [1.6.25].

40


Fig. 1.7.1. Illustrating donors and acceptors. 1.7 The number of electrons in localized states 1.7.1 General theory

If the electron and hole concentrations are of the same order, the semiconductor is called intrinsic. This situation is encouraged by mechanical perfection (few vacancies, dislocations, interstitial atoms, etc. in the lattice) and chemical purity (few foreign atoms). However, some mechanical and chemical imperfections are always present, and if their effect is important the semiconductor is called extrinsic. Chemical impurities, called dopants, are often implanted or diffused to tailor the material appropriately. Electrons occupying these defects tend to have localized wavefunctions and they give rise to the localized states. Dopants can be neutral when inserted and then can give off electrons (donors), or they can be electrically neutral when inserted and then capture an electron, normally from the valence band where this leaves a hole (acceptors). Some chemical impurities can act either as donors or as acceptors; they are called amphoteric. The occupation probabilities for electrons in localized states are important throughout semiconductor physics and device design. They are therefore derived here by two alternative methods. The first is based on the grand partition function (1.2.2). However, for readers who have concentrated only on the results of section 1.2.1, rather than the derivations, we add an alternative argument at the end of this section, which is based on a simple free energy minimization and is independent of the grand canonical ensemble. The result of this work is, in the simplest case, that the occupation probability of a defect center is

where g0 and gx are the degeneracies of the ground states of the unoccupied center and of the occupied center, respectively, and E is an effective energy of the center. However, the precise meaning of E, the effect of the excited states of the unoccupied and the occupied center, and the modification of the formula for centers which can be in different charge conditions, have all to be explained. The result for the general case is given in (1.7.3), below, which is still basically a very simple result, as will be seen from the special cases studied in section 1.7.2. We now seek to derive the occupation probabilities of localized states, basing ourselves on eq. (1.2.2). Suppose a localized lattice defect or chemical impurity or

1.7 The number of electrons in localized states

41

vacancy is able to capture and lose electrons. The state that has the lowest positive (or largest negative) charge that is likely under usual conditions will be called 'AT. The change from 0 to M is effected by the capture of M electrons. For any charge state r (r = 0 , 1 , . . . , M) we sum over all states / of the center. By allowing different energy spectra for different values of r the effect of interaction between different electrons on the same center can be taken into account. Denote the appropriate sum of terms £ , exp [ — £(/, r)/kT] by Z r , the canonical partition function for an r-electron center. Thus from eq. (1.2.2), u being the electrochemical potential, M

S=£>rZr,

X = exp(ii/kT)

(1.7.1)

The probability of finding a center in a quantum state (/, r) whose energy is E(l9 r)

5=0

If this state is degenerate, this result has to be multiplied by the degeneracy of this state, and / then refers to the energy level of the center rather than the quantum state. Summing over /, one finds the probability of finding a center in charge state

(1.7.3)

VZs s=0

If there are N noninteracting centers, the number of centers in charge state r is vr = A^P(r)

(1.7.4)

The mean number of electrons trapped in these centers at temperature T and electrochemical potential u is M

V M

I M

"I

/ s=0

J

# e = L rNP(r) = £ rVZr \ £ X*ZS \N r=0

Lr=0

(1.7.5)

This matter has been reviewed in a number of books where experimental details may also be found. See [1.7.4]—[1.7.6]. A simplification may be made by neglecting the excited states of a center for each charge condition r. If gr is the appropriate degeneracy of the ground state, one can then put

where Er is the energy of the r-electron center in its ground state. The exponential involves therefore a difference between an s-electron and an r-electron energy, and

42


this is difficult to introduce into a one-electron energy scheme such as has been considered in sections 1.2.3 to 1.2.6. It is therefore more convenient to work with quantities [1.7.7] Eir-DÊr-E^

(r=l,2,...,M)

(1.7.7)

This is the energy needed to take the least strongly bound electron from an relectron center and to deposit it at infinity at the energy level which is taken to be the zero of energy. This is, as required, a single-electron energy. One can then use

d.7.8)

One can of course use eqs. (1.7.6) to (1.7.8) even if the excited states are not neglected. But the Er's are then effective energy levels, which one would not expect to see precisely in optical experiments. Also the 2sr's would become (at least weakly) temperature-dependent. The neglect of the excited states is normally satisfactory if they lie a few kT above the ground state. This matter has been studied in connection with acceptors in Ge [1.7.8]. A number of quite involved discussions exist in the literature designed to derive what turn out to be merely special cases of eq. (1.7.3) by means of free energy arguments. We here give an improved, brief and quite general argument of this type. Instead of deriving special cases, we shall again obtain the full result (1.7.3). Consider TV centers in charge states r = 0 , l , 2 , . . . , M . Hence £P(r)=l,

i.e. X > r = 7V

r=0

(1.7.9)

r=0

and 2 > v r = JVe

(1-7.10)

r=0

Now, as shown in statistical mechanics, the free energy corresponding to the canonical partition function Zr is — kT\nZr. Also the number of ways of choosing v0 empty centers, vx single-electron centers, etc., up to r = M, given that there are N centers, is, if one regards (v0, v 1 9 ..., vM) as the vector v 1\...vM\

(1.7.11)

v

Now -kT\n[W(y)Zl»Z\i...Z M] is the corresponding free energy F(y). Using Stirling's approximation, and denoting by ' e ' the base of the natural logarithm, v!~(v/e) v ,

(v^l)

(1.7.11a)

one finds for the free energy of the system F{\) = -kT^MZ^kTNXnN+kT^M^r

(1.7.12)


43

Allowing for the conservation conditions (1.7.9) and (1.7.10) by Lagrangian multipliers a'kT and ykT, we can minimize the free energy subject to these constraints, by minimizing in fact M f 1 L(v): = - kT\ NXnN-Y, vr[ln Zr - In vr + a' + ry] [

I

r=0

J

with respect to each of v0, v 1? ..., vM separately, treating N as a constant. This will yield the equilibrium expressions for the vr according to standard statistical mechanical procedures. Thus keeping all vr fixed, except for v;., 8L/8v;. = - kT[\n Zj - In v, + a' +/y] + kT = 0 The equilibrium expressions are, writing a = a' — 1, (1.7.13) In order to interpret the Lagrangian multipliers, note that by putting eq. (1.7.13) into eq. (1.7.9),

r=0

This enables one to write eq. (1.7.13) as

v.JN = VZ.I £ X'Zr (X = expy)

(1.7.14)

r=0

In order that this agrees with the results (1.7.3) and (1.7.4), one has to check that y = \i/kT. This can be done by noting from eqs. (1.7.12), (1.7.13) together with (1.7.9) and (1.7.10) that F(v0) = -kTNlnN+kTY,

vr0(a + ry) r=0

= kTN\nN+akTN+ykTNe However, from general thermodynamics, the chemical potential of an electron system is given by dF/dNe, whence ykT = [i as required. Note that the approach of section 1.2, based on the grand canonical ensemble, is exact, while the free energy approach utilizes Stirling's approximation. This situation is very much as already indicated at the end of section 1.2. In all these cases the Stirling approximation is rather spurious. Our procedure via the grand canonical ensemble is therefore more direct in yielding the desired mean, rather than most probable, values. 1.7.2 Some special cases of occupation probabilities for localized states

The simplest special case arises for M = 1, when eq. (1.7.4) gives

where the substitution (1.7.6) and the notation (1.7.7) have been used. For a hydrogen-like atom the (r = 0)-state might correspond to a proton-like particle

44


which has therefore g0 = 1. A captured electron may have either of two spins so that gx = 2 in this case. This is the case of the so-called 'unpaired' spin. If the (r = 0)-state has an electron in an estate, then the (r = l)-state must have the matching spin. In that case g0 = 2, gx = 1. This is the case of 'paired' spins. The number of trapped electrons is for r = 1

Nt = v, = JV-v0 = N \ \ + l2exp[^fa]}

(1.7.16)

where go/g1 = f if the spins are unpaired, and go/g1 = 2 if the spins are paired. If the center has excited states, then for this, and also for other reasons, £(f) can depend on temperature. Some additional examples can be based on eq. (1.7.4). Thus

Ne = Z>'Z, / £ VZr = — ^ p^ 1 "* 1 + E (M-j) VZ, \ E rVZr j=0

I

(1.7.17)

r=l

which leads to the following expressions for 7Ve: (M=l)

(1.7.18)

(M = 2)

(1.7.19)

These results have been reviewed (with additional details) elsewhere in [1.7.9] and [1.7.10]. They apply also in principle to vacancies or interstitials (see section 1.10). In the important case of Au in Si, r = 0,1,2 refer to the positively, neutral and negatively charged Au atoms, respectively. The three-component vector v is (VcVpVj = (Z0/XZ19 hXZJZJv,

(1.7.21)

where the substitution (1.7.6), (1.7.7) has been used. The energy E(\) is the donor energy level 2sD and E® is the acceptor energy level, both regarded as located on a single-electron energy level scheme. The energy zero is arbitrary and does not occur in eq. (1.7.22), which features only single-electron energy differences. Also v0 and v2 have been expressed in terms of the number of neutral centers. This interpretation of vx is, however, not essential and the argument is independent of which of the charge states is neutral. A careful analysis of carrier equilibrium effects as a function of temperature has been made for Cr-doped GaAs, for


45

example, the Cr levels follow the valence band as the temperature is changed [1.7.11]. As another simple example, consider NB singly-ionizable donors under neglect of excited states. Let g0 and g1 be the degeneracies of the ground states when no electron is trapped and when one electron is trapped, respectively, so that z.x

g±

kT

kT

Then is D is the energy level of the donor in the one-electron energy scheme. The number of trapped electrons is given, using eq. (1.7.16) with a slight change of notation, by TV

N, =

(1.7.24)

Suppose now 7VDu neutral donors with unpaired electrons and 7VDp neutral donors with paired electrons are introduced into a pure semiconductor at absolute temperature T and that they have ground state levels which lie at the same energy E^. Then the number of electrons in the conduction band which come from the paired sites is, with a = exp (t| D — y),

N

N

^

(L7 25)

S =

'

The number from unpaired sites is

(L726)

-

ff

Hence the fraction / of electrons in the conduction band from paired sites is given by

Suppose now that one does not know if the donors introduced into the sample were paired or unpaired and that we simply regard them as 7VDp + # D u donors with energy level

[

(z \

= 1— 1

1 .We wish to calculate g. Its equation

V>1/ effectiveJ

is given by N

i VV

D u Du

\+\a

••

N N

ii V VDp

Dp

\+2a

N

_

iV

D

l+ga

Solving for g, g

Substituting for N^JN^

in terms of/, finally gives

(L727)

46


Thus for 7VDp = 0 only unpaired electrons play a part, so t h a t / = 0 and g = \. If 7VDu = 0 only paired electrons play a part, so t h a t / = 1 and g = 2. Intermediate values are found in other cases. As 7VDp rises from zero, g rises from one-half to a maximum value of two. More generally, let E, Ef denote effective energy levels which incorporate the effect of degeneracy and excited states ( Ef ZfJZ'Q = exp I - —

( E\ ZJZQ = exp I - — 1 ,

Let the number of levels be TV at E and N' at E\ Then an effective energy level be denned by [a = exp(n-y), a' = exp ( i f - y ) , aeU = exp(n eff -y)] AT

N'

N+N'

\+a'

1+êff

-+-

\+a

Then one can show that the fraction of conduction band electrons/= n'/(n + n') which come from the N' levels is in the absence of other levels and bands exp(-n)-exp(-Ti e f f ) exp(-n)-exp(-r|') This can be established by first deriving _a(a'+\)(N/N') + a'(a Into this one can substitute the following expression derived from the /-equation: N N'

(\ V

\a'(a)a(a'-

This gives the desired relation for/. As a first application of the /-formula note that if E = E' (= Eo say) then EeU is also Eo, as one would expect. As a second application suppose that, when occupied, the N' levels are paired and the N levels are unpaired, both at the same energy Eo. Then exp(-ri) = 2exp(-T] 0 ),

e x p ( - n ' ) = |exp(-ri 0 )

and the/-relation gives at once exp(Tieff) = [ 2 / ( 4 - 3 / ) ] e x p ( r l o ) One sees from

that the effective energy level lies at Eo with effective degeneracy (1.7.27).

The effective energy level concept is useful when one wishes to ignore either degeneracies of energy levels or ignore excited states, or ignore both. For example, if excited states are neglected so that eq. (1.7.6) holds Z

l

gr+i

^

A

kl

«

gr+

= exp[neff(r + f)]

(1.7.28)


47

where r|eff incorporates the effect of degeneracies. This implies in general only a numerically small change because of the occurrence of the logarithm:

There is thus an entropy contribution k In (gr/gr+1) to Eetf(r + \) due to transitions between an r- and an (r+l)-electron centre. An additional term arises from the changes in atomic vibrations due to lattice relaxation. These terms must be regarded as formally included in our later results for Eett, AG, etc. in equations (1.7.29) to (1.7.36) and in Fig. 1.7.2. More generally one can put = exp[Tiett(r+|)]

(1.7.29)

to include the effect of degeneracy and excited states. Such effective energies are not solutions of the Schrodinger equation. A kind of averaging has occurred which brings in the Boltzmann factor, and hence a temperature-dependence of the effective energy. 1.7.3 Gibbsfree energies and entropy factors Let us consider a special case r = 0 of relation (1.7.28). It tells us that the degree of ionization of a donor is

=

0 7

=

30)

v0 + Vl 1 + XZJZ0 1 + exp [y - ileff(|)] Since \L = kTy can represent different thermodynamic functions depending on the variables which are kept constant (and put in brackets below): U(S,V), H(S,p), F(V,n

G(p,T)

(1.7.31)

one now has a choice already foreshadowed in eqs. (1.1.14) and (1.1.15). The last form is the most practical from the experimental point of view. It is therefore becoming accepted that AG = £err©-|i

(1-7.32)

can be regarded as a Gibbs free energy which is in this case associated with the ionization of a donor. If we go back to 1954 when J.A. Burton gave his key review of impurity centers in Ge and Si at the International Conference on Semiconductors in Amsterdam, the talk was about energies and not yet about free energies [1.7.12].

48


It is fortunate that different ensembles yield normally similar results for large systems, since the change to a basis of constant pressure from constant volume (the latter is implied by use of the grand canonical ensemble) really requires one to switch the basis of the theory to the so-called constant pressure ensemble. This is not greatly loved (even by statistical mechanicians), because it has a certain arbitrariness associated with it; see, for example, [1.7.13]. Fortunately this need not worry us here, and we shall often continue to speak about energy rather than free energy. The basic idea in the pressure ensemble is to suppose that the probability of finding a certain volume V and energy E when pressure and temperature are given is kT Here the number N of indistinguishable particles is also given, and Q is a normalizing factor - the constant pressure partition function. The suffix / specifies a state. Hence

= H+kT\nQ where H is the mean (thermodynamic) enthalpy. One sees that for consistency with thermodynamics

where HiN is the enthalpy EiN+pViN of state (i,N). Thus the ratio of probabilities yields an enthalpy difference, i.e. an enthalpy of activation: -exp rather than the energy difference, as expected from a canonical ensemble. We know from eq. (1.1.10) that G = H- TS so that AG = AH-TAS at constant temperature. Using eqs. (1.7.30), (1.7.32) and (1.7.33)

(1.7.33)


49

The 'entropy factor', X, for the interaction of the impurity and the conduction band can be estimated from

45

~m

V $T

0.7.35)

JP,N

which follows at once from eq. (1.1.13). Also

(m.

since H=G+TS

= G- T(dG/dT)p,N

The heat capacity at constant pressure is AC = (dAH/dT)v,N

(1.7.37)

For any energy gap AG, therefore, its temperature dependence furnishes the key ingredients, AS and AH, of equations like (1.7.34). Some results for the main energy gap (i.e. for the formation of electrons and holes) are shown in Fig. 1.7.2. If AS goes up to six Boltzmann constants, X rises to a value of 403. It has lower values for the interaction of a defect level with a band; see section 2.4.6, Table 2.4.6, where some physical interpretation of (1.7.35) is also given. For the main energy gap electron-hole pair creation can be represented by a reaction with an equilibrium condition: Q^±e + K

Heq + Hheq = 0

(1.7.38)

One can, in a chemical analogy, associate standard states with electrons and holes for which («,/?, \xe, \ih) = (n°,p°,\i% [i|J). With Boltzmann statistics one then has for low concentrations 'compositional fractions' fp so that the equilibrium product of concentrations is

The obvious standard states are those at the band extrema, given by (Ec, -Ey), whence from eqs. (1.6.11) and (1.6.12)

(,.7.39,


50

-

1.2

G

1.0

>

% 16 Ci 1-2 *" 0.8 _

y

^

*

~

^—— AS

0.4 f 200 400 600 800 10001200 1400

2.0 \-

y^ \

0

i 200

.

600

i

1

1000 T(K)

1400

0 1800

GaAs Si

\ff\r

GaP Ge

1.0 If/

GaAs ^^_ 1

0.5* i

0

i

I

1

200

600

1000

=^—

I

1400

1800

T(K) Fig. 1.7.2. The quantities (1.7.35)-(1.7.37) for the forbidden energy gap between the conduction and valence bands as a function of absolute temperature for four important semiconductors [1.7.14].

1.8 Interaction effects from impurities

51

The negative sign for Ev arises from the fact that on the usually adopted energy scale, electronic energies are measured upwards and hole energies downwards. 1.7.4 Points from the literature (1) An operator method of dealing with the statistics of localized impurities has also been presented [1.7.15]. (2) The theory has been applied in other areas, for example in the determination of capture cross sections by means of an amplitudemodulated electron beam [1.7.16]. (3) It should be noted that an overlap of localized states with a continuum of states, such as a conduction band, is not only theoretically possible, but has in fact been observed experimentally, for instance in CdF 2 [1.7.17]. (4) The experimental methods of characterizing the properties of semiconductors with special reference to impurities are reviewed in [1.7.18]. (5) Some additional early papers in which the subject of this section was explored are [1.7.19], [1.7.20], [1.7.21] and [1.7.22]. (6) The importance of Au in Si, referred to below eq. (1.7.20) is due to the fact that with its energy levels near midgap it can be an efficient recombination center. This, together with its large diffusion coefficient, makes it useful for controlling the lifetimes of Si devices. Au and Pt are used, for example, to decrease the switching time of transistors. The solubility of Au in Si was studied for instance in [1.7.23]. 1.8 Interaction effects from impurities, including screening 1.8.1 Interaction effects (impurity bands, Mott and Anderson transitions)

At low temperatures weakly doped semiconductors have a d.c. electrical conductivity which goes exponentially to zero, since the number of conduction band electrons does. This can be seen, for example, from eq. (1.7.26). We have \x > £ D so that all donor states are occupied near T = 0. Therefore a < 1 and

61

D

p(lD-Y)Ô

(1.8.1)

Electrical d.c. conductivity is thus of an activated type. At sufficient impurity concentration there is a significant overlap of electron wavefunctions for neighboring impurities so as to enable electrons to hop from impurity to impurity and so contribute significantly to the current. When this happens, one expects a

52


positive but small conductivity as 7 ^ 0 , just as in the case of metals (where it is larger). One can take the concentration JVD = AfDcrit ~ 107 to 3 x 1018 cm"3. One also talks of impurity-band conduction. This type of metal-insulator transition depends on the effect of electron-electron interactions and is often referred to as the Mott transition [1.8.1]; for a simple account see [1.8.2]. A large literature exists on this subject, but the nature of the transition is not fully understood; in some cases it can be due to a transition to a new crystal lattice, for example. What is certain is that the impurities form a band which covers an energy range and leads to a decrease in impurity ionization energy as doping increases (see section 1.8.2). A related transition occurs already in a model which neglects the Coulomb interactions among electrons. Instead of impurities consider a random arrangement of potential wells with an electron occupying the appropriate level in each well. Under what conditions are the electrons localized so that the conductivity again drops to zero as T-+01 This is the problem of Anderson localization [1.8.3], [1.8.4]. This is a many-scatterer wave coherence effect which leads to trapping of a wave excitation in a region with many scatterers or inhomogeneities. It applies to electron, optical and acoustic waves, and does not need to involve local trapping. It can also be shown that the density of states in an impurity band has a minimum at the Fermi level, should it lie in the band. This means that there is effectively an energy gap between the occupied and the empty states. It is brought about by the Coulomb interaction, and the energy gap is sometimes called the Coulomb gap. The calculation of the density of states in a heavily doped material is difficult. Several approximational schemes have been used (e.g. [1.8.5]—[1.8.8]), and we return to this problem in section 5.2.3, following the introductory remarks in this section. For a lightly doped semiconductor the resistivity, following (1.8.1), may be expected to behave as p oc - oc exp (r|c - n D ) = exp (E/kT) where E is the ionization energy of the relevant impurity. It was found early in the work on semiconductors that E depends both on doping and on compensation of donors by acceptors, or acceptors by donors ([1.8.9], [1.8.10]). Such effects have to be borne in mind when doping is varied. They are discussed semiquantitatively in the next subsection. For work in this area see [1.8.11] and [1.8.12] and for detailed reviews see [1.8.13] and [1.8.14].


53

1.8.2 Concentration-dependent activation energies A rough semiquantitative model will show that the activation energy of an impurity may be expected to decrease as doping increases. Consider a hydrogenlike impurity in a medium of dielectric constant e. The radius of the orbit with principal quantum number t is

ft~rS m*

k = - ^ = 0.528 A) \

me2

(1.8.2)

)

where m is the electron rest mass and m* is the effective mass. If 7VD is the number of donors per unit volume, then the rth orbits will just begin to overlap when j^3iVD=l

(1.8.3)

Let us define t by this condition. To be in the continuum of levels a bound electron need therefore be promoted only to the 'ionization' level t which comes down by (1.8.3) as iVD goes up. In fact, the ionization energy is for an initial electron state of principal quantum number s: E = A{s~2-r2\

A = (m*/me2)(rney2fi2)

= 13.6m*/me 2 eV

y

(1.8.4)

Using Table 1.4.1, this gives for germanium E = (0.029-0.72 x l O " 8 ^ ) eV

(7VD in cm" 3 )

which is in broad agreement with experiments on arsenic donors in germanium (see Fig. 1.8.1 [1.8.10]) E = (0.0125-2.35 x 10" 8 A^) e V

(1.8.4a)

Here A ^ is the concentration of ionized donors. The approach to E — 0 later gave rise to the suggestion [1.8.15] that E should vanish abruptly at a critical impurity concentration. This group of phenomena, in which activated electrical conduction goes over to metallic conduction as the impurity content is increased, is another aspect of the Mott transition. For more recent experiments on germanium see [1.8.16]. If both donor and acceptors are present in a material, the phenomenon of compensation occurs. Since donor electrons can fill the acceptors, only a small concentration of current carriers may result. The simple model (1.8.4) would have to be extended to allow for compensation.


54

0.015

0.010 -

0.005 -

Fig. 1.8.1. The ionization energy for As donors in Ge, as a function of the average density A^ of ionized donors at low temperature. The curve is given by eq. (1.8.4a). The experimental points are from [1.8.10].

1.8.3 Screening The Coulomb interactions among charge carriers, both fixed and mobile, in any given volume leads to very complicated correlations among the motions of the particles which present us with a truly many-body effect. Any simplified, if partial, view which one can gain of this situation is therefore welcome, and we present one such view here. The repulsion of like and the attraction of unlike charges in the neighborhood of a fixed charged impurity is represented by an electrostatic potential (p(r, i) which has in principle a long range, falling off only as r'1 with distance. However, mobile carriers of predominantly opposite charge tend to be near the fixed charge, and as a result the potential reaches a constant value in a shorter distance. The incredibly complicated dynamics thus leads to a comparatively simple law as a first approximation. This will now be derived. Suppose a charge Q, inserted at r = 0 at time t = 0 moves with velocity v. This produces a change in the charge concentration due to electrons by an amount qp(r, t) which is a function of space and time. If q = — \q\ is the charge on an electron, the electrostatic potential cp(r, i) due to the charge Q is related to p(r, i) by Poisson's equation V2(p(r, 0 = -47i[e5(r-vO + kexp (ik • r) = — ^ — v

k

£ exp (ik • r) k

It follows at once that for all k ^

i.e. cp(r) = g / | r |

(1.8.9)

It is from this argument that one can expect nonzero A(r, /) and nonzero velocity v to yield in the presence of polarizable material

It is now no longer possible to write down the expression for cp(r, i) in closed form: the dielectric function (it is hardly a 'constant') depends on the wavevector and the frequency co. The latter is introduced by a Fourier analysis in time. The quantity e(k, 0) is the static dielectric constant, and e(0, co) is the long-wavelength dielectric constant. The dielectric response function yields a great deal of insight into the behavior of many-body charged systems [1.8.17]. (ii) The form of eq. (1.8.7) suggests the next approximation of'linear screening' in which p(r, t) is proportional to cp(r, t). Again neglect the time dependence, put v = 0 and assume A(r, i) to be a constant. The Fourier component (1.8.10) is then 4nQ

™

Vk\l+A 2/k2)

— ~/1* m -

1

•

A2/ 2

k

(1.8.11)

In this case the potential can be found by (C.14) of Appendix C, and is

It is sometimes convenient to insert an effective dielectric constant to take account of the effect of interband transitions [1.8.17], the polarization of the ions, etc., in an ad hoc manner to find p ( ) Of

(1.8.12)

56


Effectively one has here a static charge Q inserted at r = 0 and finds that it is screened out at distances of order A" 1 . This is the case of static linear screening. It can give only a rough guide for the case of dynamic linear screening when all the interacting charges are in motion. It may be reasonable for slowly moving electrons, but if one considers a fast electron, shot through an electron gas, the remaining electrons have insufficient time to follow the motion. Accordingly the polarization cloud cannot form fully round the fast electron. In this case one would expect the screening effect to be slight. This apparent wavevector dependence of the screening parameter A merely reflects the fact that the passage from eq. (1.8.10) to eq. (1.8.11) is no longer justified. The detailed theory confirms this conclusion ([1.8.18] and [1.8.19]). (iii) Here it will be adequate to confine our attention to the approximations embodied in eq. (1.8.11). A slightly more general formulation considers groups of carriers distinguished by a subscript/ The linear screening approximation is then 0 = 1 , 2 , . . . ; ^ independent of r).

(1.8.13)

The Poisson equation for the disturbance due to a charge Q at r = 0 is, in extension of eq. (1.8.5),

where an effective dielectric constant has been introduced. Hence (V 2 -A 2 )(p(r) = - — g 5 ( r ) o

and one recovers eqs. (1.8.11) and (1.8.12). However, the screening parameter is now given by A2 It is the object of specific theories of linear screening to find expressions for the constants^, which occur in eq. (1.8.13) and hence to determine A via eq. (1.8.14). The above results show that the potential near a charged impurity falls off exponentially. In order to evaluate A, note from eqs. (1.2.16) and (1.1.7) that the parameter enabling us to treat electrons and holes by similar integrals is y

~gj

kT

~

qj

kT

V'*'1*'

Now let v^jj) be the carrier concentration (electrons or holes) in a group j of carriers when the quasi-Fermi level for the group has the value \ijm A disturbance


57

occurs and there is a small change from a constant electrostatic potential, taken to be zero, to the additional potential cp(r) produced by the charge. Then the change 8y; in yj is by eq. (1.8.15) - q^/kTp so that, assuming temperature and volume to be kept constant, (1.8.16) This leads by eq. (1.8.13) to (1.8.17) and finally by eq. (1.8.14) to ([1.8.20])

T,,V

e

j

D

J

This is our main result in this subsection showing that the screening radius is related to the Einstein ratio (1.2.18). One would indeed expect intuitively an easy response of carriers to an electric field (large v3) to lead to an efficient screening of the Coulomb field (large A). For parabolic bands eq. (1.6.3) enables us to put v, = v o,£(y,)

(1.8.19)

Hence one can verify by a partial integration that (1.8.20) The formula (1.8.18) then becomes

For example, for one conduction band and one valence band one finds [1.8.21]

As Fs(a) ~ e a for nondegenerate materials for all s > — 1, eq. (1.8.21) gives the socalled Debye screening length in that limit (with vOc = n):


58

1000

300 K limit for nondegeneracy

«-type limit for degeneracy

100

rate

10

1016

1017

10 18

1019

concentration (cm~~3) Fig. 1.8.2. Screening lengths in GaAs, according to eq. (1.8.21), for electrons and holes in GaAs at 300 K and 77 K using e = 12.6, mc = 0.072m, mv = 0.5m. It is believed to be a good approximation to the self-consistent screening length in the presence of band tails [1.8.22]. In the case of electrons, eq. (1.6.3) gives d-8.23) and in the degenerate limit we can use relation (1.6.14). Hence in that limit

(1 8 24)

- -

so that A~1 =

4q2m*

W

(1.8.25)

F o r G a A s at room temperature with n = 10 16 cm

3

a n d in c.g.s. units, eq.

(1.8.22) gives

10 2

4TC(4.8 x 10" ) 10

16

Fig. 1.8.2 gives a plot based on eq. (1.8.21) ([1.8.20]). Fig. 1.8.3 gives the results of a computation using calculated density of states curves for heavily doped specimens [1.8.23]. The screening length decreases with doping, i.e. as the potential due to a fixed or moving charge is more rapidly damped out, the more mobile current carriers are available.

59


1

1

100

1

77 K Vp NA = 0.2

80 300 K y

60 —

-

j ND

/

—

40 -

-

\ 1

1

1

1017

10 18

10 19

ND-NA

(cm" 3 )

Fig. 1.8.3. Dependence of the electron screening length in G a A s on net carrier —N A concentration

An imbalance of charge set up purposely or by a fluctuation causes mobile carrier of opposite charge to flow into the region, overshoot and be pulled back by the force field so induced, and then to overshoot again. This is the origin of the plasma oscillations whose angular frequency in the long-wavelength limit will be denoted by cop. This frequency can be estimated for one group of nondegenerate carriers by associating a travel distance AT 1 with the periodic time (o~f of the oscillation. Attributing to this oscillation also an energy kTp i.e. a velocity *)*, one finds

m

(1.8.26)

Using eqs. (1.8.25) and (1.8.26) one finds, if several groups of carriers are involved,

This is a key parameter entering the theory of plasma oscillations. The plasmon energy #oop is of order 10 meV at v ~ 1018 cm"3. For a modern introduction to this type of theory see [1.8.24]. Other theories of screening are available. That which is associated with the Thomas-Fermi method yields also eq. (1.8.25) and is discussed in many books.

60


The improved theory of screening in [1.8.25] is widely used. It has singularities for disturbances whose wavevectors have a length equal to the diameter of the Fermi sphere in k-space. The singularity is responsible for weak kinks in the phonon spectrum, an effect predicted by W. Kohn and called the Kohn anomaly. There is, however, no need to go into this matter here. The static dielectric screening was introduced fifty years ago in connection with the theory of electrolytes [1.8.26]. It was applied to the determination of the field surrounding a dissolved atom in a metal in [1.8.27] and [1.8.28], using the Thomas-Fermi approximation. The dynamic screening parameter was first used in calculations concerning soft X-ray spectra of metals [1.8.29] and was related to plasma oscillations in metals by [1.8.30]. Steadily improving theories of the dynamical screening effects in a wide variety of physical situations were developed in the last twenty years, [1.8.18] and [1.8.19]. However, the empirical and theoretical estimations of the screening parameters are still subject to some serious uncertainties [1.8.31]. 1.8.4 Negative-U centers The elegance of the partition function approach must not blind one to the fact that it holds for any interaction energy which arises when an electron is captured by a defect, whereas specific interactions occur in reality. Suppose as an illustration that a defect in its ground state has an unpaired valence electron or, equivalently, a partially occupied highest energy orbital. The defect energy is now expected to increase if an additional electron is captured. This is due to the repulsive nature of the Coulomb interaction energy which is however decreased somewhat by the lattice relaxation consequent upon capture. This converts the original interaction energy (associated with the name of J. Hubbard [1.8.32]) to an effective interaction energy (associated with P.W. Anderson [1.8.33]) here to be denoted by U. If U is negative due to a strong defect-lattice interaction, as seems to be the case in important examples, the two-electron state of the defect has a lower energy than the one-electron orbit. The latter is then never stable and the ground state of the defect cannot contain an odd number of spins. Thus no electron spin paramagnetism can arise from such defects. It is in fact this lack in amorphous chalcogenide glasses that led to the model in the 1970s in the first place. In the meantime it has been suggested that the self-interstitial in Si [1.8.34], the Si vacancy [1.8.35], interstitial B in Si [1.8.36] as well as certain thermal donors in Si [1.8.37] are negative-U centers (this list is not complete). These identifications depend to some extent on using the correct statistics. This turns out to be equivalent to analyses developed 15 years earlier since the partition function approach holds for positive- as well as negative- U centers. It is described in section 1.7.


61

As an example consider the case M = 2 for a /?-type semiconductor. Then the total concentration of defects is

o l 1

Pi

1

P

f

(1.8.27) \Pi

P

where p is the hole concentration, and for / = 1,2

The concentration of bound holes is, with eq. (1.8.27), v1 + 2 v 0 =

PJ

(\+2p/Pl)2N 2p/Pl + 2 + 2pJp

2N \+(\+2pJp)/(\+2p/Pl)

Let pT be the doping level, i.e. the concentration of free and bound holes distributed over the valence band and localized states. Then the neutrality condition is 2Ar

(1.8.28)

This gives the doping level as a function of /?, as required for the analysis of experiments. If excited states are neglected, as in eq. (1.7.22), then Pi =

fe/ft-i)êxp[-Ti(i-|)]

(1.8.29)

and into these relations one substitutes

so that the value of U and its sign will affect the relation between pT and p. Thus one obtains, as a special case, a key result, viz. equation (5) of a paper by Hoffmann [1.8.38]. See also [1.8.39]. Some additional temperature dependence from the excited states may be expected when the approximation (1.8.29) is not appropriate. However, the link with partition functions is here made for the first time. Note that the v0 defects can be doubly ionized donors D + + , singly ionized donors D + , neutral donors D x or acceptors, etc. If one were to change the value of U at will, one would find that as U became negative the defects with one electron would decompose into equal numbers of defects which had captured no electrons and defects which had captured two electrons :

A phase diagram could then be drawn with a phase transition at U = 0. Lattice relaxation effects are discussed further in section 5.4.4 and chapter 6.

62


The conventional notation in the negative- U literature is to write £(|) as £( + / + + ) suggesting, respectively, the capture or emission of a second hole by a vacancy and the capture or emission of a first hole by a vacancy. In the notation of this book + + is in this case the state of zero electron captured (which corresponds to a vacancy which has captured two holes), + refers to the state of one electron captured, and 0 to the state of two electrons captured by the vacancy. 1.9 Saturation solubilities of impurities Lifetimes of excess carriers in very pure materials are often limited by small residual impurity concentrations of order 1012 cm"3 or less, which are hard or impossible to measure. For cases like these it is useful to have an understanding of the principles governing the solubility of defects in semiconductors. Such theoretical considerations can provide clues as to the nature of these residual defects. The theory of section 1.7 can be applied to a situation which does depend on the charge state, s say, corresponding to electrically neutral centers. Its elucidation depends on two observations: (1) Electric fields, for example in a p-n junction, do not affect the distribution of such centers so that their concentration v s has a constant value throughout the material in thermal equilibrium. (2) The impurities will in general diffuse from a source and in thermal equilibrium diffusion has effectively ceased. In the case of mechanical defects such as vacancies the value of v s is determined by the history of the sample, the equilibrium temperature, etc. In any case there are important situations when the value of vs is independent of the Fermi level [1.9.1]. This holds, for example, for diffusion at sufficiently high temperature so that the impurity in its neutral charge state attains its maximum solubility, from which the maximum solubility of the defect in other charge states then follows by arguments based on eq. (1.7.6). In the theory which follows, ion pairing and the formation of new compounds are neglected. It is found that the concentration vs of electrically neutral centers can be determined experimentally and so furnishes an important base line for the other concentrations which can be obtained from eq. (1.7.3). Fig. 1.9.1 illustrates schematically the situation for M = 3 and s=\. The equations for the lines are:

Note that if \i/kT is varied at constant temperature, the lines drawn are exactly straight within the model. If T is allowed to vary, then the ratio of the partition functions becomes

1.9 Saturation solubilities of impurities

63

Fig. 1.9.1. Concentration of r-electron centers for r = 0,1,2,3 as a function of Fermi level divided by &r. Saturation conditions and thermal equilibrium at a fixed temperature have been assumed. temperature dependent and one ceases to have straight lines. If one uses effective energies as in eq. (1.7.28) with unit degeneracies, then the five values of \i/kT shown in the figure become kT'

2kT

' kT'

2kT

' kT

(1.9.1)

They are arranged correctly in the figure provided 2s(|) < £(§) < £(§). The variation of u in Fig. 1.9.1 can be assumed to be due to variable dopings by some other majority impurity. One sees that this affects the take-up by the material of the multicharge impurity under investigation, even if the source of this impurity is maintained at constant strength. This can be, seen from the expression for the total number of such impurities to be found in the material in thermal equilibrium. This is

; W./Z.

(1.9.2)

Note that

(1.9.3) is the average number of negative electronic charges on the impurities. The average charge is — (r — s) \q\. It is positive for low Fermi levels and becomes increasingly negative for higher Fermi levels as the impurities fill up. For small X the inverse powers of X dominate in eq. (1.9.2), provided s > 0, and the take-up (N) decreases as the Fermi level rises. For large enough X, N increases with the Fermi level. We then have an enhancement of solubility due to doping. This is illustrated in Fig. 1.9.2. N is also called the saturation solubility. There are many experiments on solubility effects. Note, for example, studies [1.9.2], [1.9.3] and [1.9.4] of the increased solubility of gold in silicon as a result of boron doping. If excited

64


T=1373K

80

120

160

200

240

280

320

360

X = Qxp(n/kT) Fig. 1.9.2. Saturation solubilities of Au in Si for different dopings. The curves are theoretical for case 4 of Table 1.9.2, using: (a) (ZJZJ vx = 30.6 x 1016 c m 3 , (Z2/Z1)v1 = 0.019 x 1016 cm" 3 ; (b) (ZJZ1)\1 = 50.0 x 1016 cm"3, (Z2/Z1)v1 = 0.107 x 1016 cm"3. The points are experimental as follows: (1) [1.9.2] for 9 x 1019 cm~3 B-doped Si; (2) [1.9.2] for intrinsic Si; (3) [1.9.11] for 4 x 1019 cm"3 /Mioped Si; (4) [1.9.11] for 6 x 1019 cm"3 /?-doped Si. states are neglected, these experiments yield some indication of the degeneracy factors which appear in eq. (1.7.6). A method of arriving at these factors [1.9.5] may be explained by using superfix / = 1, 2 , . . . , D to distinguish D different heavy shallow dopings which set the Fermi level at \i {l) for the temperature T. The additional dopings by centers which can be in different charge conditions are assumed small enough not to affect \i{l). Using eq. (1.9.2) one finds

v«> = W»Y-Z rv./Zt

(r = 0 , l , . . . , M ; / = l , 2 , . . . , Z > )

(1.9.4)

We have not attached a superscript / to vs because the neutral centers have the same concentration for all Fermi levels, as discussed above. We have known: N(l) (e.g. from radioactive tracer technique) and \i(l) from Hall effect and conductivity or from the charge balance equation;

unknown:

^,...,^,^,...,?f,vs

The Af + 1 unknowns can be identified for a given temperature Tfrom eqs. (1.9.4) provided D = M+l. Repeating the double-doping procedure for other temperatures furnishes curves of quantities (1.7.8) as a function of x = \/kT: (r<S)

y

n

z,

n

gs

l-l

(r>s)

(1.9.5)

1.9 Saturation solubilities of impurities

65

Table 1.9.1. Theoretical estimates of degeneracy factors r

e

p

gr given by eq. (1.9.7)

0 1 2

2 3 4

4 4 4

6 4 1

This yields the gjgs from the intercepts, and the energies from the slopes. The method is, however, complicated by the temperature dependence of energy levels which are often of the form E(T) = E(0)-aikT

(1.9.6)

whence the intercept is In (gr/gs) ± a and the slope yields the energy at the absolute zero of temperature. For more details see the appendix to this section. At the theoretical level one might argue that the degeneracies can be calculated from the number of ways W of filling p distinguishable places by e indistinguishable electrons: W = p\/e\(p-e)\

(1.9.7)

For trivalent Au in Si the results of Table 1.9.1 are possible. The values of e are due to the fact that the r = 0 state corresponds to two 5d electrons in the ions. The value p = 4 arises from the four positions of the unsatisfied valency. Of course, one could have monovalent Au in Ge and the degeneracy factors could then be g0 = 1, gx = 4, g2 = 6 [1.9.6]. The question of the correct degeneracy factors is hard to resolve [1.9.7] even for Au in Si. An attempt to identify experimentally degeneracy factors as well as energy levels has been made for Mg in Si [1.9.8] and for Cr in Si [1.9.9]. If one is not interested in separating energy levels and degeneracies and so lumps these together in the partition functions, eq. (1.9.2) gives the take up TV per unit volume of a minority impurity in terms of the concentration vg of the neutral impurity at a given temperature. It will be assumed that vs is not affected by the heavy majority doping. As already explained, vs is taken as independent of the amount of the heavy majority doping, which is assumed to exceed about 100 N. Thus vg is a constant for a given temperature. Table 1.9.2 gives a number of possibilities which can arise. We know of only one case where experimental points of TV as a function of X are available, and they are marked in Fig. 1.9.2. The theory which yields the curves in the figure is very simple. One just uses eq. (1.9.2) and puts

One then looks for the a and b values of best fit. This was done in Fig. 1.9.2 for each temperature. The minimum occurs at X2 = Z0/Z2 when v0 = v2. The agreement with the experimental points is remarkably good [1.9.10]. The concepts introduced in this section have had wide applications which are, however, largely outside the present scope, and only some will be briefly described, (a) For example,

66


Table 1.9.2. Saturation solubility of minority impurity as a function of X = exp ([i/kT), which is assumed fixed by heavy primary doping Case Number of electrons number M on neutral center 1

1

Equation for N/vs

0

xz

Z

Z

1+^X + pV Z 7

Z Z

the solubility of an acceptor in «-type extrinsic material should exceed its solubility in intrinsic material. For, in the simplest case, we then have M = 2 and s = 0. So in a suitably modified Fig. 1.9.1 the horizontal line represents r = s = 0, and the line rising at 45° represents In (v 1 /v 0 ) = In (Zx/Z0) + \i/kVindicating an increase in solubility with \i/kT. This has been applied for example to Si +-implanted InP [1.9.12]. (b) Conversely, donor-type defects should have an enhanced solubility in a strongly /?-doped crystal. One has two charge conditions r = 0, 1(M = 2) in the simplest case and one can use the lines marked r = 0 and r = 1 in Fig. 1.9.1. As the Fermi level is lowered by/7-type doping, the solubility of the donor increases by virtue of an increase in v 0. This consideration has been applied to Column III interstitials (donors) in AlGaAs heterostructures [1.9.13]. (c) An impurity or defect may be an acceptor in a given semiconductor when it is on one type of site (a), and it may act as a donor when it is on another type of site (P). Hence the Fermi level position in the band may decide which of the sites is stable in the sense that it leads to the lowest total energy of the system. An example is provided by Li in ZnSe where a is an interstitial site and (3 a Zn substitutional position [1.9.14]. For an introduction to this problem of the metastability of defects see [1.9.15]. Appendix: Solubilities of multivalent impurities A more formal presentation of the method of inferring solubilities of multivalent impurities in the presence of majority impurity doping is as follows. We use the notation of section 1.9. Let xi = Zj/Zs9 where the neutral impurity contains s detachable electrons, and let al} = [X{l)]j-S = exp [(j-s) \i{l)/kT] a, = W«>/v s =

£ v?> / vs

(assumed known at given T)

(Nw assumed known)

(1.9.8) (1.9.9)

1.10 The equilibrium of simple lattice defects

67

Then eq. (1.9.4) is the set of D(= M+ 1) equations, one for each doping, 2 X * . = az(/=l,2,...,Z>)

(1.9.10)

The unknowns are J C 0 , X V . . . , X M . In the matrix atJ strike out the ith row and the A:th column and denote (— l)i+k times the determinant of the matrix which has resulted by Ai1c. Then from the theory of linear equations the solution of eq. (1.9.10) is

M The case j = s is somewhat special as the left-hand side yields unity. So it is convenient to rewrite this particular solution as Als Na) + A2s N™ + . . . + A Ds

This equation thus furnishes the solution for vs. The ( M + l ) relations (1.9.11) give the M + 1 unknowns noted in the main text. From the xj one can then pass to the quantities yi = lnx^., as considered in eq. (1.9.5) and plot these for various temperatures. 1.10 The equilibrium of simple lattice defects

1.10.1 General theory based on the grand canonical ensemble

A lattice defect is typically a vacancy arising from the migration of an atom from a lattice site to the surface. Consider a compound of type CAr, where C denotes the (positively charged) cation and A the (negatively charged) anion. Then a Schottky defect consists of a C vacancy and r A vacancies, as shown in Fig. 1.10.1 for r = 1. The energy of formation (Ws) of such a defect relative to the ideal crystal is the energy required to extract the ions and to deposit them on the surface of the crystal. A Frenkel pair, on the other hand, consists of a vacant lattice site and a nearby ion in an interstitial position (Fig. 1.10.1 and Table 1.10.1). It exists in elements and does not require a compound. The energy of formation of a Frenkel defect will be denoted by W¥ . At the absolute zero there is ideally no such defect present in a crystal. But this number increases with temperature. With W s or WF ~ 1 eV one would expect 1010 cm"3 defects at room temperature and about 1016 cm"3 at 300 °C, as is readily calculated from the formulae below. As the creation of defects requires the supply of energy, they can be studied experimentally by looking for specific heat anomalies. It will be assumed in Example 1 that vacancies, in Example 2 that Schottky pairs, and in Example 3 that Frenkel pairs are the only defects present, and that they are present only in small numbers so that the interaction between them can be neglected. Furthermore the grand canonical partition function will be used in this context [1.10.1]. It has the merits of (a) expeditiousness and (b) absence of


68

Table 1.10.1. Elementary defects Energy range in alkali halides (eV) Schottky defect Frenkel defect Frenkel defect Unnamed defect

VcVa * VCIC * VaIa

1.8-2.5 2.0-4.3 2.6-4.6

* Illustrated in Fig. 1.10.1. V stands for vacancy, I stands for interstitial, c and a stand for cation and anion. C+

C+

Frenkel A" C+

A"

A"

C+

C+

A

A"

o C+

A"

Q

C+

A

A" C+

C+ A" C

Schottky

+

A"

C+A" Fig. 1.10.1. A Frenkel and a Schottky defect are illustrated at the top and bottom of the figure, respectively.

§

CD

Fig. 1.10.2. It requires less energy to remove an ion from a defect site to infinity than it does to remove an ion from a lattice site.

1.10 The equilibrium of simple lattice defects

69

mathematical approximations, over the free energy arguments normally employed. It presumes thermal equilibrium of the system at temperature T, volume V, and given chemical potential \i. One can imagine this situation to be brought about by regarding (for each lattice site) the surrounding crystal as a reservoir at temperature T and chemical potential u. Consider now the reaction L° + D# ^ L # + D °

(1.10.1)

where L° and L# are, respectively, vacant and occupied lattice sites, while D ° and D # refer to defect sites. The latter are surface sites for Schottky defects and interstitial sites for Frenkel defects. These sites have grand partition functions (1.10.2a) (1.10.2b) (1.10.2c) Here A = L or D and ZA0, Z A1 are the canonical partition functions for an empty site A° and an occupied site A*. Let NA, nA denote, respectively, the number of sites A and the number of occupied sites A, respectively. Then (A = L,D)

(1.10.3)

and this corresponds to the reaction A # Ax becomes

where the quasi-Fermi level of the trap divided by kT has been denoted by yx and kTr\(l) is an effective energy level (as explained on p. 42) incorporating of course the degeneracy factors. In equilibrium and for nondegeneracy we obtain a special case of eq. (1.11.11a) 't)N~ r

A1

\r ^.^.^ r~»

*,/l\l

AT ~,,,~ / * , \

Tlllllc^

where

is the effective trap excitation energy divided by kT. Equations such as (1.11.11) to (1.11.11c) are widely used in the analysis of data relating thermal carrier densities and temperature, see for example [1.11.6]. The reinterpretation (1.7.30)-* (1.7.34), i.e.

exp(y-n eff )-êxp^-~~kf) —J

(1.11.lid)

can also be applied to the above mass action ratios. For a center which has captured r detachable electrons one simply generalizes eqs. (1.11.6a) from r = 1 to general r. The concentration of such r-electron centers is then vr = [Z, exp (ryr) / £ Z s exp (sys)] nt /

(1.11.12)

5=0

where the yr are quasi-Fermi levels divided by kT and nt is the total concentration of centers: X?=ovr = nv ^n equilibrium one recovers eq. (1.7.3): »,.\

v

r

Zrexp(ryeq)

M

1.11 Quasi-Fermi levels and reaction kinetics

79

1.11.3 Einstein relation and activity coefficient We now establish a relation between the Einstein diffusion-mobility ratio (1.2.14) and the activity coefficients [1.11.7]. If the density of states for electrons is ê{E) and for holes oVh(E), the activity coefficients may be defined alternatively as /*

) d

EE fV e (20e*-dtf/ P° ^ JEC

f

=ê

^

=^ e ™

I J* c exp(Tl-Y e )+l

ft ^ f JrjJU&rÂEl f

^

n

(1.11.14)

J-oo / J-ooexp(Y h -Tl)+l p Here ye, yh are the reduced quasi-Fermi levels, r|c, T|V reduced band-gap energies, and

\

J Ec

J

jV

^

(1.11.15)

which has been elaborated in [1.11.8]. Thus, although the form of eqs. (1.11.4) is recovered, the density of states involved is seen to be arbitrary. AT, and 7VV are an effective number of levels for the conduction and valence band, respectively, and can be used for any density of states, as seen already in eq. (1.6.32). For parabolic bands, J/e(E) = AV(E—E C)*, where A is a constant, one recovers eq. (1.6.7): Nc = AV 1°°(E-Ec)i&°-*dE = JEC

—AV(kT)i

2

Observe now from eqs. (1.11.5a), (1.11.13) and (1.11.14) that ye = ln« + l n / c - l n JV c-Ac + r|c yh = - l n / ? - l n / so that eq. (1.2.14) gives d\nn dlnn)TV

\q\D

\d\npJTV

\d\np d\npJTV

\dlnn)

\d\npJ

Consider, as an example, a band with a constant density of states. Then eqs. (1.11.15) and (1.11.13) yield Nc = kTJTe, n = A:7Vreln[l H-e^-ic]

(1.11.17)

exp^=l+exp(y.-rie)

(1.11.18)

so that

and /\7-\r

/-\

i

(1.11.19)


80

constant

parabolic

1000

- nonparabolic 100 -

100

n/Nc Fig. 1.11.2. The activity coefficient/* is reduced by increased nonparabolicity (3 = kT/2E0 for given n/Nc. From eq. (1.2.14) for electrons in a band of constant density of states n/N c vekT

/*

(1.11.20)

Thus the activity coefficient, as well as the ratio (1.11.20), rises from unity as n increases from zero. One can also verify eq. (1.11.16) in this case [1.11.7], [1.11.9]. In the case of a nonparabolic band similar calculations can be made. Using eqs. (1.6.17) and (1.11.13), we show the activity coefficient for this more complicated case in Fig. 1.11.2. Using eqs. (1.2.14) and (1.6.17) the Einstein diffusion-mobility ratio has also been calculated; see Fig. 1.11.3 [1.11.9]. For other relevant calculations see [1.11.10]. Another simple case arises for a narrow band at energy Ev e.g. for an impurity band. This may be modelled by (1.11.21) where Nt is the number of levels. In this case from eq. (1.11.13) /* =

1

vekT

l-n/Nt

and these quantities rise as the Nt levels fill up [1.11.11].

(1.11.22)

1.11 Quasi-Fermi levels and reaction kinetics

81

constant

parabolic

nonparabolic

10

100

n/Nc Fig. 1.11.3. The Einstein ratio is decreased by increased nonparabolicity p = kT/2E0 for given n/Nc. For a normal parabolic band one finds / * = e?e-VFi( Ye - nc), /v* = e \q\De_ vekT

Fi(Ye-Tlc)

\g\ph= vhkT

- yh) K(

(1.11.23) (1.11.24)

[1.11.12]. Fig. 1.11.4 shows the ratios (1.11.24), allowance for some interaction effects having been made [1.11.13]. Some additional results are given in Table 1.11.1. We give a simple proof of the last entry in the Table. Since the Fermi function drops steeply to zero at low temperatures, a density of states A exp (E/EJ, where A and Ex are constants, leads at low temperatures to a concentration of electrons

n * A PexpfJ^W = AE.Lxp^-l] - ^ e x p ^ Thus eq. (1.2.14) gives

Qlnn )

T V

and \q\ D/v is again a typical carrier energy. Additional discussions of Einstein relations can be found as follows: heavily doped semiconductors [1.11.16], experimental justification [1.11.17], two-dimensional systems [1.11.18], [1.11.19], multi-band properties [1.11.20], the effect of writing the diffusion current as V(Dn) instead of DVn [1.11.21]—[1.11.23],


82

Table 1.11.1. Einstein relation for conduction bands of semiconductors under simple conditions [a = (\i — Ec)/kT]

1 \D Physical system

(1.11.16)

Parabolic band, extending from energy Ec

/'}*•**•

[1.11.12] [1.11.14]

(ii) K\i-Ec) \5 kT ^ | ( a ) + ~7 ^TT ^|( fl )

Z7T1 _i_ JT /O IT 1

p \{a)-\

(i) kT / C

•-.

« /c /2m* = ii[l +E/2E 0\ where E/2EQ < 1 Constant density of states jVe(E) = jVe

Reference

(i) kT

kTFi(a)/F_i(a)

As above but nonparabolic KiiJl

as given by eqs.

e

(i) Nondegenerate limit (ii) Degenerate limit

4 2

ô

F\{a)

kT(\ + e" a ) In (1 + e a )

u-£ (

l 2£0

•1 1

n 11 i n i [ 1.11.1U J

(i) A:r

(ii) H-ê Exponential band tail (with A exp (£•/£"!) as density of states) at low temperature

E1

Degenerate case only

[1.11.15]

quantum corrections [1.11.24], numerical estimates for the degenerate case [1.11.25], [1.11.26], nonparabolicity and superlattice [1.11.10], multi-band degenerate semiconductors [1.11.20], anisotropic parabolic bands and size quantization [1.11.27].

1.12 Fermi level identifications, intrinsic carrier concentrations and band-gap shrinkage 1.12.1 Fermi level identifications The Fermi level at temperature Tin a simple equilibrium semiconductor having iVD donor and NA acceptors can be identified from an equation which asserts electron conservation. Suppose that at T = 0, iVe electrons are available in localized states and in bands above the valence bands (V = 1,2,...), and that the valence bands are fully occupied. Then at an elevated temperature extra electrons are added to N e by the thermal creation of holes (pv in number) in the valence bands. The resulting 7Ve + YJVPV electrons are distributed over the conduction bands C = 1,2,... and the

1.12 Fermi level identification

0.01 1017 10 18

1019

1020

1021 1022

(a)

0.01 1017

1018

83

1019

1020

1021 10 22

(b)

Fig. 1.11.4. (a) Theoretical plots of the ratio of the electron diffusivity to mobility as a function of the net donor concentration in (1) Ge and (2) Si at 300 K including some heavy doping effects; NA = 1017 atoms cm"3, (b) Theoretical plots of the ratio of the hole diffusivity to mobility as a function of the net acceptor concentration in (1) Ge and (2) Si at 300 K including some heavy doping effects; 7VD = 1017 atoms cm"3.

10 2 2 cm - 3

100

200

300

400 500 600 700 T(K) Fig. 1.12.1. Relation between the degree of ionization of a donor and the temperature for different donor concentrations and energy levels. EC — EJ) (eV): 0.0001; 0.001; 0.01; 0.1.

84


localized states or traps, of which we assume that there are N(t) of type t (t = 1, 2,...). Hence from eqs. (1.6.3), (1.6.9) and (1.7.14)

)

- 7 H 0.12.1)

7Ve is thus the number of ionizable electrons in all the donor atoms plus the number of electrons in the conduction bands at T = 0. The relation links quasi-Fermi levels \i(eC)( = kTy(eC)) of electrons in conduction bands C, the quasi-Fermi levels |i{1F) of holes in valence bands V and the quasi-Fermi levels \i(t) of traps t = 1,2,.... The traps are assumed to be either occupied or empty, the modification to multiple charge conditions of traps being obvious. Assuming quasi-Fermi levels to exist, eq. (1.12.1) is a reasonable relation for a semiconductor in which equilibrium is disturbed by the injection of charge carriers or by incident radiation. In true thermal equilibrium, ^ ^ I T ^ ^ M

(C,F,f=l,2,...)

(1.12.2)

In that case eq. (1.12.1) determines the equilibrium Fermi level |i eq. Using an effective energy level E{t) of a trap (as in eq. (1.7.29)), Zf/Zf

= exp(-Tf>),

TIW = E(t)/kT

(1.12.3)

and suppressing the sums over bands, eq. (1.12.1) becomes \T(t) + e x p [ T f

>_Y 1, and the 7VD term attains a value of order N D /2, which is normally negligible. Hence eq. (1.12.8) yields (1.12.9) It follows that the concentration of conduction electrons, «, and the concentration of holes, /?, are equal: n = (NcNv)*exp(-r]G/2)=p(= nx)

(1.12.10)

The number (nc,ny) of extrema are by eqs. (1.4.11) and (1.6.9) included in the definition of mc and mv. The quantity (1.12.10) is called the intrinsic electron or hole concentration nv In this condition the impurity concentration has become unimportant (the intrinsic condition also holds good at lower temperatures if the impurity concentration is small enough). The temperature dependence of n^values


87

Inn , n=ND -NA by eq. (1.12.18)

by eq. (1.12.16)

\/T Fig. 1.12.4. The electron concentration in an n-type compensated semiconductor. for Ge, Si and GaAs is given in [1.12.15], p. 19. A few expressions for the Fermi level are given below. (i) Intrinsic Fermi level The intrinsic Fermi level lies by eq. (1.12.9) at p, value given by VL = Êe + Ev) + lkT\n(NJNc)(=iii)

(1.12.11)

This value is at each temperature taken as the origin in Fig. 1.12.3. The last term in eq. (1.12.11) can by eqs. (1.6.7) and (1.6.9) be rewritten as \kT\n(mJmQ) (if) Weak ionization of donors and acceptors, two nondegenerate bands In this case r|D ^ y N A)

(1.12.13)

The quadratic equation for ey can be solved in the form n = &NA + Kli)[(B+l)l-l]

(1.12.14)

where ^

2

*

=A

If NA, Nj> are constants, then the temperature dependence resides in ^ D , which can be interpreted as an equilibrium constant already familiar from eq. (1.11.11b): nN^

n(NA

= 7V c exp(y-ii c )

Zpo

z,D1 exp y

= Jy c exp[-(n c -Ti D )]

(1.12.17)


1U" =

IO 14

•

• • • " " 1

•

• •

•'•'•I

1

89

1

' ' "f'l 1 -

1 '-

— /

IO 1 3

/

/

Illll

IO 12

•" &&*

:

IO 1 1

—

•

:

=

\ \ 1 1 Mill

IO 1 0

1

IO9 IO

/

/

V

.

/

17

IO

18

1 1019

1 IO 20

1

1

1 1 1 1 1 1

io 21

ND (cm" 3 ) Fig. 1.12.5. The effective intrinsic concentration versus donor concentration in silicon at 300 K temperature [1.12.24]. Points are experimental values: # , [1.12.25]; O , [1.12.26]; x , [1.12.27]; + , [1.12.28]. Curves are theoretically derived: , [1.12.29]; , [1.12.26]; — — [1.12.30].

The increase in the conduction band electron concentration with temperature follows approximate exponential laws in two cases (Fig. 1.12.4). (a) Ku ^ JVA, favored by low temperatures; alternatively K^ > ATD-iVA. In these cases B — NA which is possible under neglect of the valence band, when (1.12.18)


90

1

.

I

. .,

.

i

.

i

1 ,

. . ._

m*/m = 1.45.

-

100 —

m*/m = 1.1

^y?

*

+ -4

D

j

~" -

^

—

10

1 10 17

.

.

, , 1 10 18

,

, ,i 10 19

10 20

10 21

n orp (cm 3 ) Fig. 1.12.6. Gap shrinkage A as inferred from transport measurements for ntype layers of Si from various sources at a mean temperature of ca. 340 K. The curves are based on eq. (1.12.24). The upper curve is for m*/m = 1.45 [1.12.31]; the lower curve is for m*/m = 1.1 [1.12.18]; and 8 = 11.7 (Si) has also been used. The horizontal axis is the majority-carrier concentration. Points are experimental, as follows: D, [1.12.32]; +, [1.12.33]; A, [1.12.27]; O, [1.12.34]; x, [1.12.25]; V, [1.12.28]; # , [1.12.35]; A, [1.12.18].

For a /?-type semiconductor the above results hold provided the following replacements are made:

Various ways of characterizing impurities starting from thermal carrier measurements have recently been compared, using as an example In and Hg acceptors in Ge [1.12.16]. 1.12.2 Heavy doping effects

There is considerable interest in heavy doping phenomena. They arise from the use of thin highly doped diffused and implanted layers in bipolar transistors, as used in Si integrated circuits, where a highly doped emitter is desirable. Heavy doping is also employed in other semiconductor devices such as solar cells. This brings into play the complicated phenomena outlined qualitatively in section 1.8.2. In particular, it brings about band-gap shrinkage already noted on p. 74.


91

As observed in section 1.8.2, the heavy doping effects arise from Coulomb interactions among current carriers, and between carriers and charged impurity ions, and also from the spatial fluctuations in the electrostatic potential. The latter is due to the random distribution of impurities and the electron-phonon interactions. These complicated effects can be assumed to give rise to the band-gap shrinkage parameter (1.11.5a). We shall now make use of this assumption. The equilibrium value of the 7i/?-product can be written in various ways (np)eq = ^ c 7V v /i( Y e q -Ti c )i|(Ti v -y e q )expA [parabolic = (7V c iV v // c / v )exp(-i lG )expA = {n\/fjv)

JV(E)]

exp A

(1.12.20) (1.12.21)

= (7V c 7V v // c */*)exp(-T lG )

(1.12.22)

EEHfexpAÊE^

(1.12.23)

The first three forms arise from eqs. (1.11.5). The form (1.12.23) is more conventional, although eq. (1.12.21) is also used [1.12.17]. Equation (1.12.23) lumps together the effect of degeneracy and density of states shifts and distortions into an effective band-gap narrowing reduced energy parameter Aeff. It then gives rise to an effective intrinsic carrier concentration « leff. Figs. 1.12.5 and 1.12.6 give numerical values of « ieff and A. In much of the literature Boltzmann statistics have been used to infer the band-gap narrowing even when the semiconductor was degenerate. These data have been recalculated so as to isolate band-gap shrinkage and remove the effect of degeneracy, assuming parabolic bands. Thus Fig. 1.12.6 gives A rather than Aeff. Note that eqs. (1.12.21) to (1.12.23) can be used for arbitrary density of states, but it is possible to separate degeneracy effects only if a density-of-states function is assumed. The curves in Fig. 1.12.6 [1.12.18] are based on the eq. e2 A=

j

where s is the dielectric constant. Reviews of the heavy doping effects are given in [1.12.19]—[1.12.23]. The interpretation of these experiments is not yet reliable enough to warrant a very detailed theory, and we give in section 1.12.3 merely a simple argument leading to the qualitative result (1.12.24). Away from equilibrium, but if quasi-Fermi levels exist, one has to replace in eq. (1.12.20) Yeq-^lc^ê-Tlo

^lv ~ Yeq "> ^v ~^h

(1.12.25)

and eq. (1.12.23) becomes np = Nc Nv Fi(ye - n c ) Efy\v - yh) exp A [parabolic JV*(E)] = n2enexp(Fe-Fh)

(1.12.26)

92


Fig. 1.12.7. Schematic diagram showing the decomposition of the band-gap energy into Wand the work done against attraction. More carriers are assumed present for curve (1) than for curve (2).

The experimental data is not easy to analyze, and the gap shrinkage can be plotted somewhat differently. For a precise definition of 'apparent band-gap shrinkage' the original papers [1.12.36] should be consulted. 1.12.3 A simple model of band-gap shrinkage

The experimental data on the reduced band-gap shrinkage A discussed in the last subsection depend on whether the experiments have been based on transport properties (typically electrical conductivity and Hall effect) or on optical measurements. The magnitude of A depends on temperature and doping. The explanations in terms of many-body theory involve the effects discussed in section 1.8, but they are complicated and beyond the present scope. These theories must ultimately furnish satisfactory results. For steps in this direction see [1.12.23] and [1.12.37]—[1.12.39]. A naive picture of some of the effects involved is based simply on the Coulomb interaction and leads to formula (1.12.24) which gives a fair account of the experimental results, and will now be established. In the present model the semiconductor is regarded to some extent as a neutral dielectric continuum in which positive and negative charges are smeared out. However, the particulate structure is not entirely neglected, as will be seen. The first step is to create an electron-hole pair which is in a bound state for a very short time. The distance, a say, between the maxima of their wave packets will be a few angstroms only, so that the effect of the smeared out electron and hole densities will not affect the energy, Wsay, to create the pair. The normal Coulomb potential acts between the particles for r > a and is cut off at r = a. Such cut-offs are often needed for small r as the Coulomb potential diverges as rÔ. Imagine now the hole to be trapped at a defect at r = 0 and the electron to be removed to infinity against the Coulomb attraction starting at the cut-off distance. Bearing in mind that the 'continuum' consists of particles which are in very complicated and


93

correlated motion, one cannot just use the bare Coulomb potential. The correlated motions of the many-body problem incorporate approximately the effect of the long-range part of the Coulomb potentials of the electrons, leaving short-range, or screened, potentials acting between largely independent particles. The screening parameter is approximated as a constant (it really depends on the wavevectors involved in the Coulombic collisions). With these approximations the total energy supplied to create the pair, and to separate it, is (Fig. 1.12.7) EG(n,p) = W+(e2/za)exp(-Aa)

(1.12.27)

This quantity is interpreted as the energy gap if n, p are the carrier concentrations. If the semiconductor is highly nondegenerate, then screening can be neglected and eq. (1.12.27) yields £G(0,0) = W+{e2/m)

(1.12.28)

As already explained, W and a are to be approximated as concentration independent. By subtraction, and with the notation (1.11.5a), A = EQ(090)-EQ(n,p) = (e2/ea)[l-exp(-Aa)]

(1.12.29)

Using Debye or Thomas-Fermi screening gives the same result in the limit of extreme degeneracy, as given for « +-material by eq. (1.8.25), so that

* 0.08 [ I £ i ] i A for Si with s = 11.7 and for m* = m, where m* is the density-of-states effective mass for electrons.) The approximation e x p ( - A a ) - l-Aa

(1.12.31)

requires the constraint Aa < z, where z is a number of order of \. From this and eq. (1.12.30) one has m*/m

rio i 8 ~i«.

(= 12.53z - ^ - A for Si). For n ~ 1020 cm"3 in Si, one has A = 0.17 A"1 and a < 1.45 A if z ~ \. Hence from eqs. (1.12.29) and (1.12.31) A = e 2 A/s

(1.12.32)

94


Numerically, eqs. (1.12.30) and (1.12.32) yield

A = 126.6 ™.!™

f - I L I meV

where n is in cm"3. For Si, with e ~ 11.7 and m* ~ m, this gives A ~ 215 meV at n ~ 1020 cm"3. In fact, one can put, using s = 11.7 for Si,

Although the band-gap shrinkage has been determined in eq. (1.12.29) in terms of a and A, a more sophisticated theory is needed to estimate a. The importance of a arises from the fact that it determines the relative contributions of the two terms in eq. (1.12.27). The beauty of the present treatment is that no commitment needs to be made as regards the numerical value of a. The reason is that we need only eq. (1.12.29) from which a cancels if eq. (1.12.31) holds, so that the actual value of a, which enters only eqs. (1.12.27) and (1.12.28), is not required. The above simple theory begins to fail unless by eq. (1.12.31) Aa < z where z ~ \ and also, by eq. (1.12.28) we need EG(0,0) > e2/ea, so that A, c"v Qc 1018 c m 3 , while nonradiative recombination is important below this value [2.2.64]. An analogous sharp drop of xp in GaP with increased impurity concentration has also been observed [2.2.58], [2.2.59], [2.2.65]. Some lifetime limitation in these materials is due to dislocations, see [2.2.66] and section 2.8. The recent improvement in epitaxial growth techniques has focused interest on the effect of thin layers on devices. This has led to the manufacture of multilayer and graded bandgap devices, and to electrons flowing in approximate confinement to two dimensions. In the case of semiconductor lasers, for example, it has raised the question of the improvements obtainable by the use of these so-called low-dimensional structures which include quantum wires. Can the threshold current and its temperature dependence be lowered by going to the quantum wells? Is the deleterious nonradiative transition rate as a fraction of the total

122

Recombination statistics

radiative plus nonradiative rate less for the low-dimensional structures? This matter has been discussed, and the present consensus is that the last question is to be answered in the negative [2.2.67], [2.2.68]; see also chapter 7.

2.3 Shockley-Read-Hall (SRH) statistics: additional topics 2.3.1 A derivation of the main formulae (any degeneracy, any density of states) Since the Shockley-Read-Hall statistics (2.2.20) for carrier recombination via traps has played an important role in comparisons with experiment, we shall return to it in this section. First we establish in more detail the result (2.2.17), using Fig. 2.1.1. Regard G(r — |), H(r — \) to be the mass action coefficients for the downward transitions conduction b a n d ^ (r— l)-electron trap and r-electron trap ^valence band. They are given by eqs. (2.2.11). Let XG(r — |), \iH(r — |) be the corresponding coefficients for the upward transitions; one then has the following transition rates per unit volume: ucr_{ = G{r-\)[nvr_x-Xvr]

Mv

=

(2.3.1)

(232)

^-})^-^J

In a steady state these two rates are equal and this leads to

where eqs. (1.11.12) and (2.2.5) have been used in the last step. The steady-state trap quasiFermi level for the transition {{r— l)-electron center «± r-electron center} has thus been identified:

This holds for r= 1,2, . . . , M , the most important case being r= 1. The steady-state recombination rate per unit volume involving these centers is

(2.3.5) which is of the form (2.2.17), (2.2.18). It remains to identify X and \i. The general theorem (2.2.10d) applied to eq. (2.3.1) yields -exp(Y r _.-y e )]

2.3 Shockley-Read-Hall statistics

123

yr_i and y(r —|) being equivalent notations. It follows that

= %i«exp(-

Ye)

= «(/•-!)

(2.3.6)

where eqs. (2.2.5) and (2.2.13) have been used. Similarly ^ P

V

( h- Y Yrr_i _i) = - ^ - p exp yh = p(r - | ) p (y

(2.3.7)

so that (2.3.8) The last step is justified if the bands are nondegenerate. Thus the important result (2.2.17) has been rederived. It is sometimes convenient to re-express it as * r

( -1)

X

r

p( ~ 2) J

C0Sn a

l + e X P [|(Yh ~ Ye)l C 0 S n P

where

(2.3.10) As regards the steady-state trap quasi-Fermi level (2.3.4) it is seen to be given by

2.3.2 The addition of reciprocal lifetimes Consider first band-band recombination. The electron lifetime may be defined, using eqs. (2.2.11), by

There is no straightforward way of writing this as an addition of reciprocal lifetimes. However, if the assumption of nondegeneracy is made, together with An = n-n0 = Ap=p-p0 the last factor is then

An

(2.3.12)

124


Hence the addition of reciprocal lifetimes is possible:

''rc.bb

j=l^n,bb,j

where 1

/I

1

\

\n9B2p)

(2.3.14)

The lifetimes refer respectively to direct band-band transitions (notably by photon emission), and band-band Auger transitions with the second electron in the conduction band or the valence band. Only tnXihl is independent of the nonequilibrium concentrations, and that only if An -type material and in «-type material, respectively, and for r = 1, (2.3.25) - = nF+ H(l) Nt = n

(2.3.26)

The underlined terms are clearly the most important ones in studies of minoritycarrier lifetimes. Consider Au-diffused Si at high carrier concentration for a number of different temperatures [2.3.2]. Using 7^, T2, B2 as fitting parameters for /?-type Si, and 7ij, Tz, Bx for «-type Si, the results given in Table 2.3.1 were found. It is seen that a trap-Auger effect had to be invoked only for «-type Si, though one might have expected /?-type samples to be similar in this respect. The experiment suggests T2 < 10"27 cm6 s-1 T3 ~ 5 x l(T 19 7i; These results are illustrated in Fig. 2.3.1.

(2.3.27)


127

A separability assumption has been made here, and it is often justified. It says that the dopant helps to set the Fermi level, but it does not participate in the recombination traffic which limits the lifetime. The recombination defects on the other hand, even if of low concentration, are included in the Fermi level equation. In the experiment under review, the validity of the assumption was established experimentally. Curves of lifetimes which fall as the equilibrium number of carriers (or the doping) is increased are shown in Fig. 2.2.3 («-type Ge) and 2.3.1 («-type Si). They occur in many other materials. Figure 2.3.2 shows an analogous curve for the material mercury cadmium telluride, Hg1-a. Cdx Te, which is widely used for photodetectors. The agreement with experiment is again good. The curves are theoretical and have been obtained not by the method of [2.3.2], which is favored here, but by the reciprocal addition of lifetimes as in eq. (2.3.18). xbb was calculated from — = - +—

(2.3.28)

which refers to radiative and Auger lifetimes for band-band transitions [2.3.3]. 2.3.5 The residual defect in Si If one looks at measured Si lifetimes as a function of doping, one finds the jumble of points shown in Fig. 2.3.3. However, one may consider only the best lifetimes for given doping on the argument that these crystals have attained some ideal lifetime, limited only by a particular, but unknown, defect. This defect could be mechanical (e.g. an interstitial), chemical, or an association of several of these. What are the characteristics of this 'residual' lifetime-limiting defect? To answer this question add to the separability assumption of section 2.3.4, secondly, the hypothesis that the defect has only one recombination level and that it is negatively charged when occupied; otherwise it is neutral. A third assumption is that the concentration of the neutral variant of the residual defect, 'frozen in' at temperature Tf9 has a concentration given by eq. (1.12.43) with M=5xl022cm"3

(2.3.29)

the concentration of host lattice sites of silicon. As explained in section 1.9, the maximum solubility 7Vd of the neutral defects at temperature T{ can be regarded as in equilibrium independent of position in the material and also as independent of Fermi level. Because of the statistical link (1.12.34) between N& and N% an increase in donor-doping raises JVd, while acceptor-doping decreases it, as already explained


128

£

fe

o

2

510 1 8 2

5 10 19 2

5 10 20

10 17 2

.5 1018 2

5 10 1 9 2

510 2 0

majority-carrier concentration (cm~ 3 ) Fig. 2.3.1. Minority-carrier lifetime as a function of the majority-carrier concentration in n-Si at different temperatures. The samples were diffused with Au at (a) 850 °C and (b) 920 °C, respectively. The lines are fits of eq. (2.3.26) to the data with B^ = B2 = T, = 0. Q, 400 K; A , 300 K; O, 77 K.

10 14

1015 extrinsic carrier density (cm" 3 )

Fig. 2.3.2. Mean minority-carrier lifetime in slices of «-type H g ^ Cd^.Te. The curves are theoretical, based on eqs. (2.3.18) and (2.3.28) [2.3.3]. T= 193 K; O , measured; , x = 0.30; , x = 0.35.


129

10

0.1

lO'

lO"

1015

1016

1017

1018

1019

1020

ND (cm- 3 ) 100 r

lot-

I

1

1

"1 :

(b)

• VST

_r :

•• • • •*

x

•v •

-i:

0.1 r-r * \ .

1

:

\

:

2

10"

10"3

^ lO1-4

\

r-

10"51 ' 1 , 1 101

,..,,1

10 18

1019

1020

A^D (cm" 3 ) Fig. 2.3.3. Some experimental minority-hole lifetimes in «-type Si. (a) From [2.3.4]; (b) from [2.3.5].


130

1000 -

3

10 15

10 16 NA,ND

10 17 (cm"

10 18

3

Fig. 2.3.4. Doping dependence of the best room temperature minority-carrier lifetime in Si according to experiments (points). Asterisks indicate that the band-Auger process is included along with the normal Shockley-Read process (unasterisked). Circles (for holes) and squares (for electrons) represent experimental points [2.3.6].

in connection with Fig. 1.12.7. This leads to lower minority lifetimes Tp(iVD) with doping, as expected, but to longer lifetimes Tn(NA) with doping, until these lifetimes are pulled down again by the band-band Auger effect as shown in Fig. 2.3.4 [2.3.6]. The second assumption concerning the charge on the unknown defect therefore enables the model to reproduce the asymmetric behavior observed experimentally as regards xn compared with xp. The lifetime curves for the correct concentration JVd of defects, as calculated at Tt, can be used at the lower measurement temperature T, assuming a generalized Shockley-Read mechanism [2.3.7]. They follow roughly the law (A^)"1 with doping, until they are both pulled down by band-band Auger effects which are strong at high concentrations. The details of the Fermi level calculation at the temperatures Tf and Tare given in section 1.12.4. One can neglect Tl9 T2, T3, T4 and B8 and adopt [2.3.5] Bx ~ 2B2 ~ 2 x 10~~31 cm6 s"1 from Table 2.3.1. As to 7*, T% one may regard them as fitting parameters, along with Tf and Ea. The inferred values from the best fit (Fig. 2.3.4) are then found to be: 2T\ = T* ~ 5 x 10' 5 cm 3 s"1 £a=

1.375 eV,

T; =

|

620K]

Position of defect level: 45 meV above mid-gap

(2.3.30)


131

Note that the SRH mechanism is invoked only for the calculation at the measurement temperature T < Tt when a relation of the type (1.12.46), of fourth power in the electron concentration n, has to be used. The defect concentration N a entering this calculation is derived from a cubic equation in N~ which holds at temperature Tt. We are left with two matters of interest: (1) What is T{7 (2) What is the nature of the inferred defect specified in eqs. (2.3.30)? As to the first question, recall the early quenching experiments on silicon which led to a relation of the type [2.3.8]

where the activation energy was found to be 0.6 eV, x was the minority-carrier lifetime and 7^ was the temperature from which the sample was quenched. Data enabling one to find C was given later for these thermally generated recombination centers:

([2.3.9]; note that the captions of Figures 8 and 10 should be interchanged.) More recently a thermally generated donor density v = C'exp(-EJkT(i) was found with C" ~ 8 x 1023 cm"3, Ea = 2.5 eV in 'pure' /?-type Si. The appropriate level was located 0.37 eV above the valence band edge [2.3.10]. These results suggest that eq. (2.3.29) is a reasonable assumption and that the freezingin temperature Tf may be identified as the quenching temperature for infinitely rapid cooling at least for some heat treatment histories. This corresponds to the 'perfect' quench. Departure from the perfect quench by slower cooling should lead to T{ < Tq. This relation between Tf and 7^ needs further study. The second question is made difficult by the variety of levels found by different methods in the forbidden gap of Si. In particular we cite nine relevant pre-1980 papers on thermally generated and/or quenched-in centers in Si [2.3.11]—[2.3.19]. Thus a donor level at Ev + 0A eV was found in/7-type Si in [2.3.10], [2.3.11] and in B-doped Si in [2.3.12], but not in [2.3.13], where the B concentration was heavier. It was again found in [2.3.15] as a complicated defect. The thermally generated defects were found to be hard to anneal out in [2.3.16] and in later work. In a series of later papers fast ('s') and slow ( ' r ^ r ' " ) thermal recombination centers were found and characterized. They have formation energies of 1.0 eV, 1.2 eV and 2.5 eV [2.3.17], the slower centers being less soluble. The high binding energy and the consequent difficulty of annealing out thermal centers was

132


confirmed [2.3.18], [2.3.19]. The slow centers were attributed to vacancy-copper complexes and later to vacancy-oxygen complexes [2.3.20]. The fast centers were attributed to native defects ([2.3.20], Figure 3). As regards energy level structure, many inconsistencies remain. Some of the discrepancies between the various experiments have been attributed to electrically active defects connected with traces of Fe in Si which may have been present in varying amounts [2.3.21]. They can be kept down to below 1014 cm" 3 by special treatment. Fe-related deep levels have, in fact, been studied separately [2.3.22] as has the level at 0.45 eV above the valence band edge [2.3.23]. Swirl defects (due to point defect agglomerates, presumably interstitial) of formation energy 1.3-1.4 eV were also noted in /?-type floating zone grown heattreated Si [2.3.24], and their annealing characteristics differ from those of divacancies of a similar formation energy (1.3 eV). Possible interpretations of the defect inferred and characterized in eqs. (2.3.30) will now be proposed. Note that a defect similar to the one inferred in relations (2.3.30) appears to have been found in swirl and dislocation-free float zone grown Si by deep level transient spectroscopy and derivative surface photovoltage [2.3.25]. This dominant recombination level was located at £ v + 0.56 eV with a capture cross section for holes equal to twice the capture cross section for electrons: GI = 2OI=

10" 14 cm 2

(2.3.31)

in fair agreement with the specification (2.3.30). If one puts 7* =fvos2 (v ~ 107 cm s"1 ~ thermal velocity) and inserts 7^ ~ 5 x 10"5 cm 3 s"1 into relation (2.3.31), the factor / (giving a recombination efficiency) turns out to be

/ = 0.05 The same result is found if 7^ and a\ are used. The defect may be a self-interstitial or a cluster of these [2.3.25] - this is a first interpretation of the residual defect. In view of the importance of controlling Si wafers for device work, two secondary suggestions can be made to interpret relations (2.3.30). The first suggestion is that it is a swirl. The A-type swirl, believed to consist of dislocation loops, loop clusters, etc., occurs in concentrations of typically 10 6-107 cm"3, and is therefore not a serious candidate. B-type swirls are smaller and are found in concentrations up to 1011 cm"3 or so [2.3.26]. This is of the order (10 n -10 1 3 cm"3) of defect densities implied by Fig. 1.12.8. The formation activation energy of 1.3-1.4 eV [2.3.24] is also of the right order. If such swirls can supply an acceptor level near mid-gap (their energy level structure does not seem to be well known


Al

Au

Cr

Fe

Mo

133

Ti

Fig. 2.3.5. Some deep levels in Si due to divacancies V2 [2.3.28] and due to metallic ions [2.3.29].

yet), the swirl B would be a serious candidate. This interpretation of the ' residual' defect in Si as used for semiconductor work, if correct, would be of importance for two reasons: In the first place swirl defects are known to have detrimental effects on Si, and secondly the elimination of swirl defects is under active study. One can use slow or fast crystal pulling rates, inert ambients during growth, or annealing after growth to reduce their occurrence. A second candidate is the ' s ' (native, fast) recombination center [2.3.20]. The slow centers ('r,r',r //v ) have levels which lie too close to the band edges, whereas the V center has a level near mid-gap. A recombination coefficient for minority carriers of ~ 10"7 cm3 s"1 has been suggested [2.3.14] which is 100 times larger than the inferred values of T\ ~ Tl ~ 10~9 cm3 s"1. This could, however, be understood in terms of different thermal histories. It is, of course possible that the 's '-center and the swirl B center are the same defect. Even a recent study [2.3.27] on the relation between recombination mechanisms and doping density leaves these matters unresolved. Deep level spectra are now well known, and some are shown in Fig. 2.3.5. It will be seen that they do not seem to apply to the residual defect. For a discussion of these topics see [2.3.30].

134


rp(ND)

rn(NA) OTNA

OTNA

Fig. 2.3.6. Schematic representation of the expected defect concentration N a and minority-carrier lifetime x on NB and NA. It is assumed that the residual defect is a donor. It has been assumed in the preceding discussion that the residual defect is of acceptor-type, i.e. is negatively charged when occupied. If it is positively charged when empty, it is a type of donor. The situation reverses relative to Figs. 1.12.8 and 2.3.4 as shown in Fig. 2.3.6. The lowering of the Fermi level increases the defect concentration Nd (because the solubility of N& increases) while raising of the Fermi level decreases Nd. In brief, X-doping (X = D or A) encourages the solubility of a non-X (A or D) type of defect while reducing the solubility of an X-type defect. The system is in a sense self-compensating. 2.3.6 SRH recombination via a spectrum of trapping levels Let us consider centers which can be only empty or singly occupied [r = 1 in eq. (2.3.5)], and let us replace «(|), /?(§) by n 19 px and G(|), //(§) by G and H, which is a more usual notation in this case. Assuming also nondegeneracy, the recombination rate per unit volume into a spectrum D(Et) of trapping levels per unit volume per unit energy range at energy Et is by eq. (2.2.20) (but see also eq. (2.7.4), below) u = GH(np — n\

D(Et)dEt

(2.3.32)

where the trap concentration is Nt = j D(Et) dEt. The integrations are performed over the trap energy Et. A more convenient variable is, by eqs. (2.2.13), n(r -1) -> «(i) ->«! = « exp (rjt - ye) = Nc exp (n t - r\c) Its lower and upper values will be denoted by na and ntu, respectively. Put d/i, = d(Nce".-"«) = -j+dEt,

Pl

=

ti t -n c = In K /

2.3 Shockley-Read-Hall so that with nu = Ncexp(r|tl-ric)

statistics

and nlu =

1 3 5

Neexp(Titu-r|c) (2-3.33)

w h e r e a = H n 2 , b = G n + Hp. T h i s g i v e s t h e volume into a spectrum of trapping states. compares with the rate into an equal number energy half-way between Etl and Ein. For a spectrum of constant D eq. (2.3.33) SP

G H k T D ( nnp-nf) p - i . 2 2 [(Gn + Hp) -4GHn f

SRH recombination rate per unit We wish to inquire how this rate of trapping levels located at a single can be integrated and

yields

[1-9(01+9(0] U-8(*n)l+e(*lu)J

where

a The total concentration

of trapping levels involved is

where

For recombination way in the band

v i a Nt t r a p s a t a s i n g l e e n e r g y

^ The SRH

recombination

^1^

^

rate per unit volume

GHN£np-n*)

to be placed

half-

( 2 . 3 . 3 4 )

through a single level is then

.

G H D k T ( n p - n 2 ) \ n X Gn + Hp + Gn u X* + Hn 2 /n u The ratio of interest is, after some

Et9 a s s u m e d

X*

algebra, )nu(Gn

+ Hp) + H n 2 2

n /nlt)nu

In A,

2C

l^ (2.3.35)

where \nu(X - l)[(Gn + Hp)2 \n u (X 4-1) (Gn + Hp) + X.G/ii

AGHn2f + Hn 2

136


3.0

n (cm 3 )

r

10

12

p (cm 10

3

)

12

10 14

10 10 "

1018

106

I

I

2.0 R 1.0

0

0.2

0.4 0.6 spread of spectrum/is G

0.8

1.0

(b) Fig. 2.3.7. (a) Dependence of R = uju&v on the energy Et of the single level. A constant density of states per unit energy range has been assumed, with this constant the same for each curve, and the spectrum extending throughout the energy gap. Energy independent transition probabilities have been assumed and eq. (2.3.35) has been used, (b) The dependence of R on the width of the spectrum, the latter being normalized by dividing it by EG. A constant density of states per unit energy range has been assumed, and cancels out of R; H = 10-11 cm 3 s"1.


137

The results of these calculations are illustrated in Figs. 2.3.7 and 2.3.8. Figure 2.3.7 assumes a uniform energy spectrum which extends throughout the energy gap. In Fig. 2.3.7 (b) the total number of states was kept fixed, the spectrum being centred on the mid-gap energy. Energy-independent transition probabilities have been assumed and eq. (2.3.35) has been used. In Fig. 2.3.8 this uniform spectrum has been cut off at 0.01 eV from the band edges which has only a negligible effect on wsp. Whereas an energy-independent transition probability per unit time between a level of energy E in one of the bands and the trapping level at energy Et has been assumed for Fig. 2.3.7, this transition probability has been taken as proportional to \E—E t\~x for Fig. 2.3.8. The cut-off mentioned above is therefore made for computational reasons. The energy dependence distorts the curves of R toward the valence band, for a strongly «-type material. If the probability of a transition between valence band and trap level decreased as the single trapping level moved toward the valence band edge, then we could expect a distortion toward the conduction band edge for strongly «-type material. As regards the shapes of the curves in Figs. 2.3.7(a) and 2.3.8, note that there always exists a level Et = £ t m a x at which R is a maximum. The spectrum of states is uniformly distributed over the gap and so includes levels which are far less recombination-efficient than the level at 2stmax. One would therefore expect the curve of R versus Et — Ev to have a maximum. Similarly if all the Nt levels are placed at a single level away from £ t m a x , then this single level will be recombinationinefficient whereas the spectrum will include levels which are more recombinationefficient. Hence R g» e)

(2.4.29)

Similar equations have been obtained by Rees et al. [2.4.15] by neglecting electron transitions to the ground state and hole transitions and by Pickin [2.4.16] for the electron-donor recombination. If one uses

then

2.4 Cascade recombination

151

Table 2.4.1. Expressions for constants Defining equation

Dimension

Expression for constant

1b

T1

«c ne +

(2.4.29)

rd \h [l P

L^T1 T"1 T" 1 T- 1

(2.4.29)

Q

T- 2

[ncng-\ ncng + ncng+ _ b +1 = n(cng + cj+pcp + eng + ene + ep + \/tn+\/fn (2.4.33) W-cA = crae cp np + [(« e + ng) cng cne + (c ng + c ne)

(2.4.29)

S'g

L 3 T2

(2.4.29)

Se

L" 3 T- 2

(2.4.29)

So

L" 3 T- 2

(2.4.25)

(2.4.26)

Constant

a

h — bd = —(nc ng+pg -^ne^n1 cd- al = - (ng cng +pcp) ncne Nt ~ (nCng +Pg Cp + nCne) NJt'n -Se-Sg-QNt

Table 2.4.2. Values of parameters used in the present section (Si) a ne a ng = ap

10-13 cm2 10-16 cm2

vth (for electrons and holes)

/ T \^ 1071 1 cm s"1

NT Ec-ETe Ec-ETg = ETg-Ev tn ATc = N v (at 300 K)

1013 c m 3 0.05 eV 0.55 eV 10- 10 s 2 x 10 19 cm" 3

Parameters inferred from the above at T = 300 K ne = Nc e x p [ - ( n c - n T e ) ]

2.89 x 1018 cm"3

(TI.-TI.J1

1.15xl0 10 cm- 3

= Arvexp[-(TiiTg Tg -Ti v)]

1.15 x 1010 cm-3 0.0251 s

t

(2.4.35) (2.4.36) (2.4.37)

152


Table 2.4.3. Steady-state occupation probabilities ft = vt/NT (a) for a depleted semiconductor and (b) under a uniform injection of electron concentration T (K)

P

Depletion

300 200 100

0 0 0

Uniform injection of electrons

300 200 100

0 0 0

Condition

n 0 0 0 1012 cm"3 1012 cm"3 1012 cm- 3

/e

/o

0.184 0.045 0.002

2.52 xlO" 12 2.29 x 20- 16 6.28 x 10"29

0.82 0.95 1.00

0.99 1.00 1.00

3.92 x l O 9 2.51 x 10-13 6.30 xlO- 2 6

1.13 xlO- 2 1.50 xlO- 7 0.00

This is a homogeneous linear differential equation with constant coefficients so that

where

Inserting some initial conditions by assuming vt(0) and \(0) to be given and, noting that for positive (o 12

vi(oo) =-SJQ,

(i = 0,g,e)

(2.4.31)

one finds the transient solutions for i = 0, g, e:

- {vf(0) + (O2[v,(0) - v,(oo)]} e ^

(2.4.32)

The explicit and exact expressions for TX, x2 and v.(oo) can be obtained from Table 2.4.1 using eqs. (2.4.25) and (2.4.31). For a numerical evaluation use the parameters given in Table 2.4.2. The capture cross section a n e has been assumed to be of the order of the largest measured cross sections [2.4.17], and more typical values have been chosen for a ng and cp. They have been assumed to be independent of temperature. The resulting effective electron capture cross section, which is defined below in eq. (2.4.49), does however show a temperature dependence. The excited states have been assumed to have an effective energy level at 50 meV below the conduction band edge. This is in accordance with the arguments given by Lax [2.4.2], Abakumov et al. [2.4.18] and others who assume that electrons are captured by levels with binding energy greater than kT. As to traps, Rees et al. [2.4.15] have obtained from the measured emission rates [2.4.19] for singly charged S and Se center in Si a value of tn ~ 0.5 x 10-10 s. We have kept to the same order of magnitude here (Table 2.4.2). In Table 2.4.3 typical values of the steady-state occupation probabilities/. = v ^ o o ) / ^ are given for two areas of practical interest. In the first case the semiconductor is depleted of carriers (p = n = 0), for example by reverse biasing a p-n junction, whereas in the second

2.4 Cascade recombination

153

Table 2.4.4A. Calculated values of parameters involved in a transient for the measurement of emission rates (n = p = 0),for the semiconductor parameters given in Table 2.4.2 T (K)

P (s"1)

Q (s-2)

Xl

300 260 220

2.90 xlO 12 1.62 xlO 12 7.79 xlO 11

1.81 xlO 14 4.33 xlO 12 3.11 xlO 10

3.44 xlO" 13 6.16 xlO" 13 1.28 xlO" 12

Table 2.4.4B. Calculated values of parameters measurement

= co-1 (s)

T2

=

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