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Theory of the phonon broadening of impurity spectral lines. Nishikawa, Kyoji 1962

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THEORY OF THE PHONON BROADENING OF IMPURITY SPECTRAL LINES by KYOJI NISHIKAWA B.Sc, University of Tokyo, Japan. 1957 M.Sc, University of Tokyo. Japan, 1959 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October. 1962 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study. I further agree that permission for extensive copying of t h i s thesis f o r scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department The University of B r i t i s h Columbia, Vancouver 8, Canada. Date :<•••••.•.- . . PUBLICATIONS •.. 1. Approach to Equilibrium of a Large Fermion System, K. Nishikawa, J. Phys,, Soc, (Japan), 15, 78 (1.960), ---bu; 2. Feynman Diagram Methodxyg§d-.4.-n • t h e ^ , xDeasivatioii of the.Transport Equation,^ K. Nishikawa, B u l l Ami .-Physio Soc , -Series I I 5 , 3793(1960). ' . H ••. :• ix- • . . . . . . . . . .:. " 3. C r i t i c a l Phenomerra in Thin Films using the Bragg~Wil.li.ams Approximation, K= Nishikawa, D, Patterson and G. Delmas. J, Phys. Chem. 65, 1.226 (1961). 4. On the Optical Absorption by Impurities in Semiconductors, K, Nishikawa, Physics Letters 1, 140 (1.962) , 5. Phonon Broadening of Impurity Spectral Lines, K. Nishikawa and R. Barrie. B u l l , Am. Phys, Soc. Series;- I I . 7 , -.485 (1962). The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of KYOJI NISHIKAWA B.Sc, University of Tokyo, 1957 M.Sc, University of Tokyo, 1959 THURSDAY, NOVEMBER 8, 1962, AT 3:30 P.M. IN ROOM 303, PHYSICS BUILDING COMMITTEE IN CHARGE Chairman: F.H. Soward R. Barrie W. Opechowski J.A.R. Coope R,F. Snider F.A. Kaempffer L.G. de Sobrino G.M. Volkoff External Examiner: D, ter Harr Clarendon Laboratory Oxford University THEORY OF THE PHONON BROADENING OF IMPURITY SPECTRAL LINES ABSTRACT The theory of the phonon broadening of impurity spectral lines i n homopolar semi-conductors i s .„• discussed within the framework of a Kubo-type formu-l a t i o n of the adiabatic d i e l e c t r i c s u s c e p t i b i l i t y and the subsequent calculation of this using the double-time Green's function method. The basic assumption i s the smallness of the interaction of the electrons (or holes) bound to impurity sites with the l a t t i c e vibrations. This interaction i s then treated as a small perturbation of the indepen-dent systems of electron and vibrating l a t t i c e ; the use of the adiabatic approximation i s thereby avoided. The so-called decoupling of the i n f i n i t e hierarchy of equations for the relevant Green's functions i s discussed i n d e t a i l and is given i t s j u s t i f i c a t i o n i n the present problem. In the case of nondegenerate electronic levels, the line-shape function i s obtained e x p l i c i t l y i n terms of the matrix elements of the electron-phonon interaction. I t is found that the absorption line consists of a sharp peak with a width a r i s i n g from a f i n i t e life-time of the unperturbed states due to the electron-phonon interaction and of a continuous background arising from the multi-phonon processes which accompany the optical absorption. In the degenerate case, a general method of obtaining the line-shape function i s discussed and is i l l u s t r a t e d i n an example. The results are compared with those obtained by previous workers in the f i e l d . The general theory i s applied to shallow impurity ievels i n s i l i c o n with the use of a modified hydro-genic model and a deformation potential description of the electron-phonon interaction; numerical estimates' are made for t y p i c a l contributions to the widths of the lines i n both acceptor and donor cases. GRADUATE STUDIES F i e l d of Study: Theoretical Physics Special R e l a t i v i t y Theory P. Rastall Low Temperature Physics J.B. Brown Physics of the Solid State R. Barrie Advanced Quantum Mechanics . F.A. Kaempffer Related Studies: Nonlinear D i f f e r e n t i a l Equations E. Leimanis Transport Properties of Gases R.F. Snider Surface Chemistry J . Halpern ABSTRACT The theory of the phonon broadening of impurity s p e c t r a l l i n e s i n homopolar semi-conductors i s discussed within the frame-work of a Kubo-type formulation of the adiabatic d i e l e c t r i c sus-c e p t i b i l i t y and the subsequent c a l c u l a t i o n of t h i s using the double-time Green's function method. The basic assumption i s the smallness of the i n t e r a c t i o n of the electrons (or. holes) bound to impurity s i t e s with the l a t t i c e v i brations. This i n t e r a c t i o n i s then treated as a small perturbation of the independent systems of electron and.vibrating l a t t i c e ; the use of the adiabatic approximation i s thereby avoided. The so-called decoupling of the i n f i n i t e hierarchy of equations for the relevant Green's func-tions i s discussed i n d e t a i l and i s given i t s j u s t i f i c a t i o n i n the present problem. In the case of nondegenerate ele c t r o n i c l e v e l s , the line-shape function i s obtained e x p l i c i t l y i n terms of the matrix elements of the electron-phonon i n t e r a c t i o n . It i s found that the absorption l i n e consists of a sharp peak with a width a r i s i n g from a f i n i t e l i f e - t i m e of the unperturbed states due to the electron-phonon i n t e r a c t i o n and of a continuous background a r i s i n g from the multi-phonon processes which accompany the o p t i c a l absorption. In the degenerate case,,a general method of obtaining the line-shape function i s discussed and i s i l l u s t r a t e d i n an example. The r e s u l t s are compared with those obtained by previous workers i n the f i e l d . - i i i -The general theory i s applied to shallow impurity l e v e l s i n s i l i c o n with the use of a modified hydrogenic model and a deform-ation p o t e n t i a l description of the electron-phonon interaction*, n u m e r i c a l estimates are made for t y p i c a l contributions to the widths of the l i n e s i n both acceptor and donor cases. ACKNOWLEDGEMENTS My sincere thanks are due to Dr. R. Barrie who suggested t h i s problem and under whose continuous guidance t h i s thesis has been written. I also wish to express my gratitude to Professor W. Opechowski for c r i t i c a l l y reading the manuscript of t h i s thesis. Thanks are also due to Dr. J.W. Bichard, Dr. J.C. G i l e s and Mr. K. Colbow for making t h e i r experimental data available before publication and to Dr. J. Grindlay for the use of unpublished c a l c u l a t i o n s . This research has been supported f i n a n c i a l l y through the award of a National Research Council of Canada Scholarship. -iv -TABLE OF CONTENTS Page At)s ti*fl-c *t • • • • • • ••• • • • e • • ••• • • • A A Table of Contents iv Acknowledgements ... ... ... ... ... ... ... v i i Chapter I - Introduction and Summary 1 Chapter II - General Theory ... 8 A b s t i r A c i * • • • • • • • • • • • • • • • • • o 8 Section I : Introduction 9 Section 2 : General Formalism 12 Section 3 : General Outline of the Green's Function Method ... 19 Section 4 : Calculation of the Green's Functions - Nondegenerate. ... ... 23 Case Section 5 : Discussion of the Line-Shape -Nondegenerate 40 I Case Section 6 : Degenerate Case 4? Section 7 : Summary and Discussions ... 56 Acknowledgements ... ... ... ... ... ...58 References ... ... ... ... ... ... ... 59 Appendix I : Relation to the Bethe-Sommerfeld Formula 60 Appendix I I : Order Estimation for Equation (4-2).. 62 Appendix III:Proof of (4-9) and (4-10) 64 Chapter III - Application to S i l i c o n ... • • • • • Section I Section II Section III Section IV Section V Acknowledgements References . Tables.. Introduction.. ... Hydrogenic Model ... Acceptors Donors Comparison with Experiment • • • • • • • • # Page . 71 . 71 . 72 . 75 . 81 . 83 . 90 . 92 . 93 . 95 Appendix to the Thesis.I. Relation of the Present Method to Previous Ones. 39 Appendix to the Thesis.II. Mathematical Appendix to Chapter II Appendix to the Thesis .III.Mathematical Appendix to Chapter III Bibliography o • • • • • « • • * 103 ..!»8 1. CHAPTER I INTRODUCTION AND SUMMARY The ele c t r o n i c structures of pure s i l i c o n and germanium c r y s t a l s are such that at zero temperature the electrons f i l l the valence band leaving the conduction band empty. When a small concentration of Group III or Group V elements i s added, a set of energy l e v e l s , l o c a l i z e d at the impurity atoms, i s formed. These l o c a l i s e d states are c a l l e d the impurity states. In the case of Group V elements, the impurity l e v e l s l i e j u s t below the bottom of the conduction band and serve as electron "donors" to the conduction band. In the case of Group III elements, the impurity le v e l s l i e j u s t above the top of the valence band and serve as electron "acceptors" from the valence band, giving r i s e to a "hole" i n the valence band. The electrons i n the donor states are c a l l e d the bound electrons and the holes i n the acceptor states are c a l l e d the bound holes. Since these bound electrons or holes can work as c a r r i e r s of e l e c t r i c charge when they are excited from the impurity states to nonlocalised states i n the con-duction or valence band, we s h a l l often r e f e r to them as "bound c a r r i e r s " . The properties of the bound c a r r i e r s i n s i l i c o n and germanium have been extensively studied both experimentally and t h e o r e t i c a l l y i n recent years (Kohn, 1957). It has been shown that the bound c a r r i e r states can roughly be described by a hydrogenic modelj i . e . these states can be considered to s a t i s f y roughly the same 2. Schrodinger equation as the states of an electron i n a hydrogen atom, but with the Coulomb pote n t i a l modified by the presence of the d i e l e c t r i c medium and with the mass replaced by an e f f e c t i v e mass. This simple description of the bound c a r r i e r states i s made possible due to the fact that the bound impurity lev e l s i n s i l i c o n or germanium are s u f f i c i e n t l y shallow and that the orbits of the bound c a r r i e r s extend over many unit c e l l s . At low temp-eratures, the bound c a r r i e r s are mostly i n the ground state, namely the IS state i n the hydrogenic scheme. If we shine l i g h t on them, they can be excited to one of the excited bound states, namely one of the p-states i n the hydrogenic model, by absorbing photons. The line-spectrum of these o p t i c a l absorptions occur i n the inf r a - r e d region and has been observed by several authors (Burstein, B e l l , Davisson, and Lax, 1953, Picus, Burstein and Henvis, 1956, Newman, 1955, 1956, Brostowski and Kaiser, 1958, Colbow, Bichard and G i l e s , 1962). If the bound c a r r i e r states were unaffected by the l a t t i c e vibrations and could be exactly described by a hydrogenic model, these s p e c t r a l l i n e s would have no widths other than th e i r natural l i n e widths. However, the c a r r i e r s do interact with the l a t t i c e vibrations and t h i s causes an important broadening of the l i n e s . The main object of the present thesis i s to study t h i s so-c a l l e d phonon broadening of impurity s p e c t r a l l i n e s . Other sources of broadening of the l i n e s w i l l not be considered here. Two types of broadening mechanism can be attributed to the i n t e r a c t i o n of 3. the bound c a r r i e r s with the l a t t i c e v i brations. (We s h a l l r e f e r to t h i s i n t e r a c t i o n as the electron-phonon i n t e r a c t i o n ) . One i s due to the fact that the bound, c a r r i e r states described above are not stationary states of the system because of the presence of the electron-phonon interaction) any bound c a r r i e r state has a f i n i t e l i f e - t i m e and can decay into other impurity states through absorption or emission of phonons. This mechanism has been c a l l e d the " l i f e - t i m e " e f f e c t (Kane, 1960). The other mechanism i s due to the fact that the e x c i t a t i o n of a bound c a r r i e r by absorption of l i g h t can be accompanied by a simultaneous absorption or emission of phonons. This process i s c a l l e d the"one-phonon process", "two-phonon process", etc. depending on the number of phonons which accompany the o p t i c a l absorption considered. This i s i n e f f e c t a photon absorption with multi-phonon absorption and emission process. The f i r s t mechanism i s due to the mixing of various impurity states, while the second exists even i n the absence of t h i s mixing. These two types of broadening have the following q u a l i t a t i v e l y d i f f e r e n t features. In the l i f e - t i m e broadening, the widths of the absorption l i n e s are mostly deter-mined by those of the excited l e v e l s . Then the widths of the l i n e s w i l l c r i t i c a l l y depend on the energy larel structure. Moreover, d i f f e r e n t l i n e s w i l l have d i f f e r e n t widths. On the other hand, i n the multiphonon processes, the widths of the l i n e s are mostly determined by the width of the ground state and hence the l i n e s have almost a l l the same width (Lax and Burstein, 1955). The reason that the width of the ground state i s larger than that of an excited state i s as follows: the s p a t i a l extension of the 4. excited state i s so large that the i n t e r a c t i o n with the l a t t i c e vibrations i s by interference e f f e c t r e s t r i c t e d to the very long wave-length phonons of which there are few. In addition to the above, there are other d i s t i n c t differences between the two types of broadening. They w i l l be discussed in d e t a i l ±& zhe text (Chapter I I , §5). Early work on the theory of the phonon broadening of the impurity spectral l i n e s i n s i l i c o n or germanium has been based on the following physical model. (a) The bound c a r r i e r interacts only w i t h the lo n g wave-length a c o u s t i c phonons. (b) The in t e r a c t i o n between the bound c a r r i e r s i s ignored. In the present work, t h i s same physical model i s accepted. The mathematical method used i n the present work i s , however, e n t i r e l y d i f f e r e n t from the previous ones. The method of previous workers (Lax and Burstela, 1955, Kubo and Toyozawa, 1955) may be summarised as follows ( c . f . Appendix to the Thesis, I ) . (1) The s t a r t i n g formula i s the Bethe-Sommerfeld formula for the absorption constant (Sommerfeld and Bethe, 1933). (2) To f i n d the approximate eigenstates of the system, the Born-Oppenheimer adiabatic approximation i s used (Born and Oppenheimer, 1927). (3) The Condon approximation (Dexter, 1958) i s made i n using the Bethe-Sommerfeld formula.* * This approximation was not made by Kubo and Toyozawa. 5 (4) To evaluate the width at half maximum, the moment method i s used. The r e s u l t s of this theory are as follows. ( i ) The peak po s i t i o n of the l i n e , which i 3 evaluated from the f i r s t moment, i s approximately independent of the temperature. ( i i ) The half widths of the l i n e s , evaluated from the second and fourth moments, are almost the same for a l l l i n e s . ( i i i ) The widths of the l i n e s s t a r t increasing with temperature at 60°K for s i l i c o n and increase as jT ( T i s the temperature) at high temperatures. The r e s u l t ( i i ) shows that the main source of the width considered by these authors i s the second mechanism, i . e . the multiphonon processes. At the time when these theories were presented, the experimental r e s u l t s seemed to show f a i r l y good agreement with the theory. Recently, by improvement of the experimental techniques, new r e s u l t s which do not f i t the above theories have been obtained (Colbow, Bichard and G i l e s , 1962). The present work was inspired by these new experimental r e s u l t s . In the present work, the mathematical method consists i n the following. (1) The s t a r t i n g formula i s the Kubo-type formula (Kubo, 1957, O'Rourke, 1957) for the absorption constant. This formula i s shown to be equivalent to the Bethe-Sommerfeld formula (Chapter I I , Appendix I ) . 6. (2) The double-time Green's function method (Bogoliubov, and Tyablikov, 1959, Zubarev, 1960) i s used i n the c a l c u l a t i o n of the absorption constant. (3) In the c a l c u l a t i o n , the electron-phonon i n t e r a c t i o n i s treated as a small perturbation. In t h i s method, the adiabatic approximation,* the Condon approximation* and the moment method are a l l avoided. The great advantage of the present method l i e s i n the fact that i t gives an e x p l i c i t expression for the line-shape function. As a r e s u l t , the author has succeeded i n evaluating the half-width d i r e c t l y without using the moments. The p r i n c i p a l r e s u l t s obtained by t h i s method are as follows. ( i ) The line-shape i s approximately Lorentzian near the peak. ( i i ) The peak pos i t i o n i s again approximately temperature independent. ( i i i ) The half-width i s due to the l i f e - t i m e e f f e c t . Hence d i f f e r e n t l i n e s have d i f f e r e n t widths. (i v ) The half-width s t a r t s increasing at a temperature which depends on the l i n e , and increases as T at high temperatures. (v) The multiphonon processes contribute to the continuous back-ground, but not to the sharp peak. (We see that Lax and Burstein and Kubo and Toyozawa were i n fact discussing the background.) * The v a l i d i t y of these two approximations i s discussed i n Appendix to the Thesis, I. 7. Some of the r e s u l t s obtained i n t h i s theory, have already been suggested by Kane i n 1960. In t h i s sense, the present work i s an extension of Kane's work. In the present thesis, emphasis has been put on the mathemat-i c a l rigour of the theory. In p a r t i c u l a r , the method of double-time Green's functions has been discussed i n d e t a i l . This method has become one of the most powerful methods i n the f i e l d of s t a t i s t i c a l mechanics. However, there has been an inherent d i f f i c u l t y i n t h i s method, v i z . the so-called decoupling of the i n f i n i t e hierarchy of coupled equations. This problem i s discussed i n d e t a i l i n Chapter I I . The author believes that i n the present p a r t i c u l a r problem the decoupling of the hierarchy i s given i t s j u s t i f i c a t i o n i n a s a t i s f a c t o r y fashion. In Chapter II of t h i s thesis we consider a f a i r l y general system. The Green's function technique and the general r e s u l t s of the c a l c u l a t i o n are discussed. In Chapter I I I , t h i s general theory i s applied to the shallow impurity l e v e l s i n s i l i c o n . These two chapters consist of the manuscripts of two papers to be submitted for publication i n conjunction with Dr. R. Barrie who has supervised the work. Three appendices are added at the end to bring out some points more c l e a r l y and to give derivations of some of the formulas used i n the text. 8. CHAPTER JC Phonon Broadening of Impurity Lines I. General Theory Kyoji Nishikawa* and Robert Barrie Department of Physics, University of B r i t i s h Columbia, Vancouver 8, B.C. A b s t r a c t The theory of the phonon broadening of impurity spectral l i n e s i s discussed within the framework of a Kubo-type formulation of the adiabatic d i e l e c t r i c s u s c e p t i b i l i t y and the subsequent c a l c u l a t i o n of t h i s using double-time Green's functions. The i n t e r a c t i o n of the bound electron (or hole) with the l a t t i c e vibrations i s assumed to be weak and i s treated as a small perturbation of the independent systems of electron and v i b r a t i n g l a t t i c e j the use of the adiabatic approximation i s thereby avoided. The cases i n which the bound c a r r i e r states are degenerate and non-degenerate are separately d i s -cussed. The r e s u l t s are compared with previous work i n the f i e l d . Present address: Research Inst i t u t e for Fundamental Physics, Kyoto University, Kyoto, Japan. §1. Introduction The presence of unionised impurities i n s o l i d s i n general gives r i s e to an o p t i c a l absorption l i n e spectrum i n a wave-length region i n which the pure s o l i d i s usually transparent. The line-broadening caused by the i n t e r a c t i o n of the electron or hole bound to the impurity s i t e with the l a t t i c e vibrations has been studied by Lax and Burstein (1955) and by Kubo and Toyozawa (1955). (References to e a r l i e r work are given by these authors). Both sets of authors worked within the framework of the adiabatic approximation and used the Bethe-Sommerfeld expression for the absorption c o e f f i c i e n t . The difference between these two works l i e s i n the method of evaluating the sums involved. As a r e s u l t of using the Bethe-Sommerfeld expression as a s t a r t i n g point, they were not able to obtain the line-shape function e x p l i c i t l y and r e s t r i c t e d themselves to a c a l c u l a t i o n of the moments of the l i n e shape. Lax and Burstein calculated the f i r s t four moments and Kubo and Toyozawa obtained a general expression for the nth. moment. Sampson and Margenau (1956), using the same model as the above authors, have discussed the problem i n the framework of the o r i g i n a l Lorentz broadening treatment. This automatically gave a Lorentzian l i n e shape with a half-width related to the inverse of the time between c o l l i s i o n s of the bound c a r r i e r with the l a t t i c e v i b r a t i o n s . Deigen (1956) has repeated the Lax and Burstein c a l c u l a t i o n using what he c a l l s the "condenson " i n t e r a c t i o n of the electron with the l a t t i c e vibrations rather than the deformation pot e n t i a l i n t e r a c t i o n . Lax and Burstein considered the important broadening mechanism i n the photon absorption to a r i s e from the simultaneous absorption or emission of a phonon. Kane (1960) has shown that when the electron-10. phonon in t e r a c t i o n i s weak the optical absorption unaccompanied by phonon emission or absorption i s much stronger than the one-phonon process. This leads to a £-function type of absorption peak with the multiphonon processes giving a continuous background. To obtain a non-zero width at half-maximum, Kane suggested that the width arose from a f i n i t e l i f e - t i m e of the excited state due to the electron-phonon i n t e r a c t i o n . We report here an extension of Kane's work, using a d i f f e r e n t technique. The method consists of using Kubo's formulation (Kubo, 1957) of the adiabatic d i e l e c t r i c s u s c e p t i b i l i t y and of c a l -c u l a t i n g t h i s quantity using the double-time Green's function method. (See, for instance, Zubarev 1960). This method i s use-f u l i n that i t usually gives the line-shape function i t s e l f rather than j u s t i t s moments. However, there are inherent d i f f i c u l t i e s i n the method, i n p a r t i c u l a r the s o - c a l l e d decoupling technique, these w i l l be discussed i n d e t a i l . The j u s t i f i c a t i o n of the de-coupling w i l l be given i n the appendices to the paper. In t h i s paper we s h a l l deal with a f a i r l y general system. The application of the r e s u l t s to shallow impurity l e v e l s i n s i l i c o n w i l l be given i n the following paper,. Preliminary reports of t h i s work (Nishikawa, 1962aj Nishikawa and Barrie, 1962) contained some incorrect statements which w i l l be corrected i n the present paper. In §2 we specify the system to be studied and a r r i v e at the r e -lation between the line-shape function and the appropriate Green's function. In §3 we discuss the method of c a l c u l a t i o n of the 11. Green's function and some of the associated d i f f i c u l t i e s , and i n §4 we perform t h i s c a l c u l a t i o n for the non-degenerate case showing how the d i f f i c u l t i e s have been met. Then i n §5, the line-shape function and the r e l a t i o n s h i p of t h i s work to previous work i n the f i e l d are discussed. The degenerate case i s discussed i n §6. The l a s t section consists of a b r i e f summary and discussion of the important r e s u l t s . 12. §2 General Formalism We s h a l l s t a r t from the well-known r e l a t i o n between the absorption constant •Jtw) and the adiabatic d i e l e c t r i c s u s c e p t i b i l i t y : In the Kubo-type formulation (Kubo, 1957, 0'Rourke,1957) £&o) i s written as where we have used the following notations: ^ : the e l e c t r i c dipole moment operator of an electron bound to an impurity s i t e } M t t ) . ttHt tj eHHt , a - » ) ; H : Hamiltonian of the system; |?B : Boltzmann's constant; 7* J temperature of the system at the time t> . The following assumptions were involved i n these formulas. (1) The system i s i s o t r o p i c . (2) The external e l e c t r i c f i e l d has been applied a d i a b a t i c a l l y s t a r t i n g from the distant past: jTwi Fiti =0 . (2.1) (3) The system was i n thermal equilibrium at T « • (4) The Hamiltonian associated with the external f i e l d i s of the form In t h i s form the l i g h t quanta can be absorbed only by the electrons trapped at impurity s i t e s and the interactions between these 13. electrons i s ignored. (5) The external f i e l d i s s u f f i c i e n t l y weak that i t s e f f e c t can be treated i n f i r s t order perturbation theory. The above formula for the absorption constant can be shown to be exactly equivalent to the well-known Bethe-Sommerfeld formula. (O'Rourke, 1957$. see also Appendix I ) . Now, the function <^£>) as given by (2-2) i s c l o s e l y r e l a t e d to the temperature dependent double-time Green's function of two operators A and B f i r s t introduced by Bogoliubov et. a l . (BogJiiubov and Tyablikov, 1959, Zubarev, 1960), The Green's function | B , ) ) F i s a two-branch anal y t i c function of E defined everywhere outside the r e a l axis by «A |e» E j= ^ L V t e ^ ^ B ] ) ; X , E < O ^ By comparison of (2-2) with (2-5), one can immediately see the r e l a t i o n One can show that the Green's function ^ A / 8 ^ B a s d e f i n e d by (2-5) i s a n a l y t i c both i n the upper and lower half planes and that i t obeys the following equation E « A I 6 > f i = & < C A ,B3> + < | > , H ] 1 5 > g , (7.7) Our problem i s then to solve t h i s equation with the condition that the s o l u t i o n be a n a l y t i c i n the domain fo r which the Green's function i s defined, that i s , outside the r e a l axis. The actual procedure of solving t h i s equation approximately w i l l be i l l u s t r a t e d i n §§3,4,6. For further d e t a i l s of the Green's function method, we r e f e r to Zubarev's a r t i c l e . 14. Next we specify our system and i t s Hamiltonian. We assume that our system consists of one electron* bound to one impurity atom and of the l a t t i c e vibrations and that the i n t e r a c t i o n of the bound electron with the l a t t i c e vibrations i s weak. This s i m p l i f i e d model w i l l be applicable to a c r y s t a l which contains a small concentration of impurity atoms whose associated energy lev e l s are s u f f i c i e n t l y shallow. S i l i c o n and germanium, doped with a small concentration of group III acceptors or group V donors, have these properties. We write the Hamiltonian of our model system i n the second quantized form: H s H e + H p + Hep (2 .1) H p - I t *>t *»t*t (2.<*b) ^ * v*t % I **** - c^c) The f i r s t term i s the Hamiltonian of the bound electron when the l a t t i c e i s f i x e d at the p o s i t i o n which corresponds to the equilibrium p o s i t i o n i n the absence of the bound electron, and the second term Hp i s that of the l a t t i c e vibrations centred at the above-mentioned equilibrium p o s i t i o n . (The zero point energy associated with the second term has been omitted. No error i s introduced by t h i s . ) We s h a l l ignore the spin of the electron. The s u f f i x e s X a n d £ specify the complete e l e c t r o n i c and phonon states, r e s p e c t i v e l y . The &*and ^ are the creation and the a n n i h i l a t i o n operators of the bound electron and the and hf are those of phonons, s a t i s f y i n g the ordinary commutation and anti-commutation r e l a t i o n s : * This should be understood as a "hole" i n the case of an acceptor impurity. 15. Ux,^] - c ^ / b j ] - r«J,fcf]- CflM4J-o (2.1!) where ' The l a s t term Hep i n equation (2-8) describes the electron-phonon interaction, with a coupling parameter x which measures the strength of the i n t e r a c t i o n . By assumption We e x p l i c i t l y mention that only the terms l i n e a r i n the phonon coordinates are considered i n the electron-phonon i n t e r a c t i o n . We s h a l l assume throughout th i s paper that the matrix elements VAA'<£ a r e non-zero over a wide range of (Of. The s i g n i f i c a n c e of t h i s assumption w i l l become clear l a t e r . We s h a l l also assume that the properties of VxV^  are such that any sum appearing i n t h i s paper and involving these matrix elements i s convergent. In using the Hamiltonian (2-8), we have to keep i n mind that we are dealing with a si n g l e electron system, i . e . , our whole argument is,confined to the H i l b e r t space of a single electron. This i s equivalent to saying that we introduce the following a d d i t i o n a l conditions! 16. We remark that we have made no other assumptions i n regard to the Hamiltonian and the commutation r e l a t i o n s of the operators. More s p e c i f i c a l l y , we stress that the adiabatic approximation, which has almost i n v a r i a b l y been used i n t h i s type of problem, i s not used i n the derivation of (2-8), (2-9), (2-l0) and (2-12). The adiabatic approximation i s the next step made i n finding approx-imate eigenstates of the Hamiltonian. An advantage of our method then l i e s i n the fact that t h i s second step - the finding of the eigenstates of H- i s made unnecessary i n the discussion of the absorption line-shape G~(eo) . With the a i d of the creation and a n n i h i l a t i o n operators de-fin e d above, we also write the dipole-moment operator i n the second quantized form: \A rr £ 2 , M M > fit*/ C 2 . I S ) where P ^ > f i s the matrix element of the dipole moment operator of the electron between the two ele c t r o n i c states X A N D X ' . Since M depends only on the elec t r o n i c coordinate, i s a number and i s independent of the phonon operators. Substituting (2-13) int o (2-6) and then into (2-1), we get Using the re l a t i o n s { « A | B » W - : £ - « B * I A * » < O + * W where the ast e r i s k denotes the complex conjugate, we get (W.4Z:I2 F y U o [ f r S ' C E ^ t t <2.ii) (7. with and In t h i s form our problem i s reduced to the c a l c u l a t i o n of the Green's function G"j»p» In general, <T*(u>)consists of several peaks superimposed on a continuous background. For the purpose of studying the shapes of these various peaks, i t i s convenient to think of or(Co) as the sum of various l i n e s peaked at the positions of the maxima. More s p e c i f i c a l l y , we s h a l l consider a normalized line-shape function ^^(c*>) which i s defined for the p a r t i c u l a r l i n e peaked at 6da&)Z". S t r i c t l y speaking, there remains some ar b i t r a r i n e s s i n the d e f i n i t i o n of Q^O**); since i t i s not quite clear which part of the background should be considered to be the contribution to O'Cco") from the l i n e peaked at CO^COt. Our method of treating the back-ground w i l l become clear l a t e r i n the paper. In addition, i t often occurs that several peaks overlap with each other. We s h a l l treat these as co n s t i t u t i n g a single absorption l i n e ; t h i s case w i l l be c a l l e d the degenerate case and w i l l be discussed i n d e t a i l i n §6. With the above a r b i t r a r i n e s s we write ( T M ^ to 2 ?c?V<o) ( 2 . 2 0 ) and take IS. where the bracket C indicates that only the contribution to the l i n e peaked at tO^CJe and part of the background should be picked out and C i s the normalization constant defined by where Let us write (2-20) as the sum of two terms: We s h a l l c a l l the f i r s t term Jp* (CJJ the " d i r e c t term" as i t comes from the d i r e c t t r a n s i t i o n s between the two e l e c t r o n i c states, and the second term $^Gu>\ "the "interference term" as i t comes from the interference of t r a n s i t i o n s among three or four e l e c t r o n i c states, produced by the electron-phonon i n t e r a c t i o n . The f i r s t term has a rather simple form and has been studied i n considerable d e t a i l by other methods, while the second term has not been studied i n d e t a i l i n previous work although i t i s important when some of the e l e c t r o n i c states are degenerate. §3 General Outline of the Green's Function Method §3-1 The Calculation f o r 0 To i l l u s t r a t e the Green's function method, we s h a l l f i r s t con-sid e r the case i n which the electron-phonon i n t e r a c t i o n i s absent. In t h i s case, the Hamiltonian of the system i s simply given by (cf. equation (2-8)). H ( 6 > = He + H p , (3-/) which describes a system of two independent types of excitations; free electrons and free phonons. The equation for the Green's function 6fyj/(E) defined by (2-18) can then be obtained, by choosing A- Apflp' 8 m tft'^A a n d using H ^ H < 0 ) i n equation (2-7);ac In t h i s 1 r ^ V . ^ , :.' ' " .• V. \-j • . • c \h- *V7* ;1 T i t - < « * * X * • > • e • /If € B T } . (3.3) As shown by Zubarev (1960), the soluti o n of (3-2) compatible with the rPM' d e f i n i t i o n of GrtuXS), i s given by Substituting (3-4) i n t o (2-21) and using the asymptotic formula where p denotes the Cauchy's p r i n c i p a l value, we f i n d The r e s t r i c t e d summation [ 2^JA' - « • J £ m a v * n t h i s case be written as 20. Z A I * ( « t - s ( o <T7A'- T; < tit + } ( - 3 - g ) where J ^ j i s related to the power of resolution determined by the experimental condition. If the e l e c t r o n i c states are nondegenerate and the e x c i t a t i o n energies of any two d i f f e r e n t t r a n s i t i o n s are well separated from each other, then the line-shape function becomes a set of separated delta-functions. In t h i s case the observed absorption l i n e s cannot have any width other than the instrumental width. On the other hand, i f the e l e c t r o n i c states are approximately degenerate, the line-shape function may appear to have a width greater than the instrumental width. In any case, there i s no width due to the l a t t i c e vibrations when there i s no electron-phonon i n t e r a c t i o n . Our c a l c u l a t i o n of the line-shape function i s exact i n the absence of the electron-phonon i n t e r a c t i o n . In p a r t i c u l a r , the i n t e r -ference term s t r i c t l y vanishes i n t h i s case * §3-2 E f f e c t of the Electron-Phonon Interaction When the electron-phonon i n t e r a c t i o n i s present, i t i s no longer easy to calculate the exact line-shape function. Indeed, i f we write the equation for the Green's function Gr^^C5) using the Hamiltonian (2-8), we get where J""]^^ (E) and JT^f^tB) a r e t n e Green's functions defined by 2). r, (S) and »»W£ C E ) a n d hence i s not self-contained. To solve t h i s equation, we have to write the equations for these higher order Green's functions and then combine them with (3-9). Equations for these higher order Green's functions, however, involve s t i l l higher order Green's functions. In t h i s way, we w i l l get an i n f i n i t e hierarchy of coupled equations. To solve t h i s hierarchy of equations i t i s usual to break o f f the i n f i n i t e set of coupled equations somewhat a r b i t r a r i l y to obtain a set of equations which contains only lower order Green's functions, which i s self-contained and which i s solvable by elementary means. This procedure i s referred to as the decoupling of the hierarchy. Naturally, the g i s t of the Green's function method l i e s |n a proper decoupling. Fortunately, i n most cases, a very simple decoupling gives a f a i r l y reasonable r e s u l t . It i s i n t h i s s i m p l i c i t y that the great advantage of the Green's function method usually resides. The problem s t i l l remains, however, of j u s t i f y i n g t h i s simple decoupling. One often t r i e s to do t h i s by comparing i t s r e s u l t with that obtained by some other method. This i n i t s e l f does not j u s t i f y the decoupling. This d i f f i c u l t y i s c h a r a c t e r i s t i c of the Green's function method and w i l l be discussed i n d e t a i l . In our p a r t i c u l a r problem, we have assumed that the coupling con-stant i s very small and i t i s natural to ask whether the decoupling used i s related to standard perturbation theory. Ter Haar (1961) has shown that the Green's function technique leads to a perturbation expansion expression for the perturbed energy l e v e l s of a system, but 22. i n the Brillouin-Wigner form, i . e . the perturbed energy of the l e v e l being considered appears i n the energy denominators. In our p a r t i c u l a r problem, we are concerned with s i n g u l a r i t i e s due to energy denominators since we are interested i n the resonant absorption of photons. We need then to pay p a r t i c u l a r attention to the energy denominators. It transpires that as a r e s u l t of a simple decoupling, proper energy denominators are obtained ( c . f . ter Haar and Wetgeland (1961)). 23. § 4. Calculation of the Green's Functions - Nondegenerate Case Let us assume that the absorption l i n e under consideration i s peaked at to =CO;and that there exists only a single pair of (non-degenerate) ele c t r o n i c states, (a,B), for which the unperturbed energy difference i s close to (x>i : We s h a l l also assume that the o p t i c a l t r a n s i t i o n between these two states i s allowed i n zero-order, i . e . ? Mo^ ^ 0. We s h a l l c a l l t h i s case the nondegenerate case. § 4 and § 5 are devoted to this case only.. We s h a l l f i r s t calculate the Green's functions ^ T A N ' C E ) (the d i r e c t term) and then ( X A A ' ^ ) (the interference term). Combining the two r e s u l t s , we f i n a l l y derive the complete line-shape function J^'Cto) . We s h a l l be interested i n the lowest order nonvanishing contribution from the electron-phonon i n t e r a c t i o n . Higher order ca l c u l a t i o n s are s i m i l a r and are only roughly described i n §4-1 (C). § 4-1 The Direct Term We s t a r t by writing the f i r s t few equations of the i n f i n i t e hierarchy of coupled equations we have mentioned before. These are 'i A/A (4 .3) ( E - T , . t T » + Wt)ri J ( e ) = § • { V A ' N & J - r , A N„V j - V,*ty<Af ^ / «f v | At « A > 6 + w f ? , <t£1»/ a* A T I ^ « A ^ > E where ) 25. In deriving (4-3) and (4-4), we used the r e l a t i o n s (2-12) as well as (2-10). The equations for the higher order Green's functions can be obtained i n s i m i l a r forms and w i l l not be given here. (a) Decoupling at zero-order The simplest decoupling w i l l be to ignore a l l the higher order Green's functions i n the f i r s t equation (4-2). The so l u t i o n then becomes which i s e s s e n t i a l l y the same as the K—0 solution (3-4). This s o l u t i o n w i l l be appropriate i n the lowest order, provided that £ i s not very close to (7A'~TA). In the v i c i n i t y of E - "TV ~ Tx j however, t h i s i s no longer the v a l i d lowest order s o l u t i o n . Indeed, i t can be shown from the equations (4-3) and (4-4)* that the second term i n the l.h.s. of (4-2) i s of order X?(j*K'(£). This implies that, for E = (.TV"*Tx)£ 0(j&); t h i s second term becomes comparable to the f i r s t term and hence i s not n e g l i g i b l e . Now we are interested i n the values of the Green's functions G"AA' (E) near £ s &)<il€ which l i e i n the range E = T>-Tw ± 7 € ± 0 0 e * ) . * See Appendix II for a proof. A crude way of checking t h i s order estimation i s to put V-\ and £- = A/ i n equation (4-3) and sub-s t i t u t e i t into (4-2). 26. Our zero-order decoupling i s then not appropriate i f and On the other hand, i t i s t h i s Green's function G^CEJ which gives the most important contribution to the absorption l i n e under consideration. This brings us to the conclusion that the decoupling at zero-order i s inappropriate i n the present problem. (b) Decoupling i n the equations (4-3) and (4-4) We s t a r t from the following order estimations, proofs of which are given i n Appendix I I I : « bth v / * x » e - »i b $ w + 0 (ft) <*« « K V V I At- « A > > E = 0(*Xi)G-xx C 6 ) ( » * ? ' ; < < 4 ^ a ? V K ' A * & = bl%)b**W («•«*) where ^ i s the number of phonon states for which V^'^ % 0 a n d J ^ - <fc|^> - f e * ( $ ) - / ] ' ' + 0(A). U.n) S t r i c t l y speaking, the order estimations (4-10) are not v a l i d for c e r t a i n p a r t i c u l a r values of £ 4 However, for these p a r t i c u l a r values of l i t t l e error i s introduced to the c a l c u l a t i o n of fr^'Li)^ provided that the phonon?of importance are not confined to a narrow range of (Of (see Appendices II and I I I ) . Using these order estimations and ignoring the terms of order */My X * ( e ) a i » d X^&^tffi) we get the following lowest lis order equations for V^CE) and J~f (E) ; Equations (4-12) and (4-13) combined with (4-2) are now s e l f - c o n -tained and do not involve any higher order Green's functions. Thus we have succeeded i n decoupling the hierarchy. Solving (4-2), (4-12) and (4-13), we get where 1 1 ) + i v l^r & + o+'fao i I L E-T„ + T,,-Wt E - T v + TX, + U l J J (4.15) r — 1 + — 1 — 1 + V*t x y t + N ^ T - 1 1 f 4 l A ) 28. In deriving (4-14), the following r e l a t i o n s have been used: For the purpose of studying the line-shape function, i t i s more convenient i f we take a l l the E-dependence to the denominator of (4-14). With the assumed property of |^ y f^M' L&) i s a non-singular function of E i n both upper and lower half planes. One can then write (j+ "X ^ N x x 'correct to the second order i n X j as (|- K.iKlxA.'(E))~' * from which we get where Putting E c £ 0 ± r % and using the asymptotic r e l a t i o n (3-5) and a s i m i l a r r e l a t i o n ( X * T r - T x ) H - *o), we get In* ^xvC«o±r6.) - Jcu^w) T [ Ju'Ctf) (4.-21), Note that we are interested i n the values of t ~ W * | E and not the values on the r e a l axis. " 29. where T /^tto) - f A V M + at, { to-Ty+lWe) WxA/ (tt-rc) J 0.23), with • « K I M •> z=#tt\ "** ^ s O - T p + Tx + e0t] } + | V v A o J 2 { )^ttto-T A /+-T^-fO t ] and s i m i l a r expressions f o r the r e a l and imaginary parts of f T V + T A - rs) NjAX' ( « - r c ) . Substituting (4-21) into (4-18), we get 30. where and £T A V i s the solution of the equation fix/ - T w - T x + ^ K ^ / C S V ) . ^ » » ) Since, with the assumed property of V^|'£, Kxx't&O i s a n o n° singular function of CO ^  one can approximate by y ^ ' C t o ) . Moreover, the slowly varying nature of \(A\'(LO) ensures the replacement of K x X ' ( £ ^ X V J b y ^AX' C Tx/ ~ Tk ) Equation (4-26) can then be written as 1 1 IT XX' LB M ~ " where Since ^f(eo) i s also a slowly varying function of £0 , the function (4-29) has a sharp maximum at 60 « 6JXk' * - Tx + 0 C * 0 . We no?recall that we are interested i n the absorption l i n e ' which has i t s maximum at ; The only nonvanishing lowest order contribution of the d i r e c t term to t h i s absorption l i n e i s from the Green's function LTV|Q (&) » a l l the other Green's functions contribute to other absorption l i n e s i n the lowest order and hence w i l l be truncated i n the discussion of ^ ^ ( e d ) . Then putting A=X and A'=|S i n (4-29) and sub-s t i t u t i n g i t into (2-24-), we f i n a l l y obtain 31. For the values of not near the line-shape function 'l9> Cto) can be approximated by On the other hand, near the sharp maximum at C<0—60*^ o n e c a a n o longer ignore the term of order x.*" i n the denominator. In thi s region, the function tf^jjCeo) nay he approximated by " f l ^ tip-"!**), since t h i s function i s slowly varying and nonvanishing at 60= T p - To{ (see §5). Then the line-shape function becomes Lorentzian: where AO) ^ a x ^ K i p C T ^ - T O . (4,3*) The peak po s i t i o n and the f u l l width at half power of t h i s Lorentzian l i n e are given by C0otp> and respectively. We end t h i s subsection (b) with three remarks. F i r s t , we note that the reason we have rewritten (4-14) i n the form (4-18), by taking a l l the E-dependence into the denominator, i s simply because we wanted to approximate the line-shape by a (modified) Lorentzian curve, i n which case the peak position and the f u l l width at h a l f power can e a s i l y be estimated. In our perturbation approach, 32.. we were not interested i n the perturbation expansion of the l i n e -shape function i t s e l f , but were interested i n the expansions of the peak po s i t i o n and the f u l l width at half power; these are r e a d i l y obtained by writing the Green's function i n the form of (4-18). Secondly, we note that the approximation of the line-shape by a (modified) Lorentzian curve i s not always possible. We r e c a l l that we have used the fact that NA^CE") and LAX/1^) are non-singular and slowly varying (except when crossing the r e a l axis) and that Glp'Tct) i s nonvanishing. A l l these properties were necessary i n approximating the line-shape by a Lorentzian curve. For instance, i f we had approximated VAX'<£ by as was done by Lax and Burstein, V^^CIJj-T )^ would have vanished (see equation (5-5)). Then no approximation of the line-shape by a Lorentzian curve would have been possible. F i n a l l y , we remark on the r e l a t i o n to a straight-forward perturbation approach. In a straightforward perturbation approach, we would have written the order estimation (4-9), for the case of A- K- > V« K > as T E-Tx* + T> J . If we had used (4-9a) instead of (4-9), we would have also succeeded i n decoupling the hierarchy by ignoring the terms of order 1 (_ - > V N i a (4-3) and (4-4). However, i f we had written ^ ^ E-Tx' + TV / the s o l u t i o n i n the form of (4-14), the new function Nx\'(ErJ would 33. no longer have been a slowly varying function, since i t would have contained a term with the factor (E-Tx'+Tx)'. Then one would have no longer been able to write the solu t i o n i n the form of ( 4 - 1 8 ) . This d i f f i c u l t y i s c h a r a c t e r i s t i c of the simple second order perturb-a t i o n theory and arises from the fact that (4-9a) with X s O i s not a correct order estimation i n the v i c i n i t y of E « Tx' - T\ - In t h i s v i c i n i t y , one of the terms ignored from (4-9a) becomes corn-arable to the term retained. On the other hand, the order estimation (4-9) i s correct for a l l values of E. (c) Higher order decoupling As i n ordinary perturbation theory, i t i s possible, although f a i r l y complicated,,to proceed to a higher order c a l c u l a t i o n systematically. Indeed, i f we want to proceed to the next higher order c a l c u l a t i o n , we may write the equations for a l l the Green's functions appearing In the l.h.s. of the order estimations ( 4 - 8 ) , (4-9) and (4-10) correct to the lowest order i n > C f instead of simply ignoring the terms of order X ? G * A V C E ) i n the equations for f * ^ ^ ( E ) and T " ^ y / ^ • In doing t h i s , we may use the order estimations s i m i l a r to ( 4 - 8 ) - (4-10)J for instance, «b\ \>b^ tfj &yj C& * hi *w (*) + *tt* r^, &} << b%\>v J * * <*t *#'| 4x' = 0 , etc. IMi'b). 34. Then we can again solve the r e s u l t i n g equations i n the form of (4-14). In general, the Green's function Gr^' CE) can be found i n the form (4-14) with N ^ W - N#Ce) +tfN&Ce) + --- 0 - 3 7 ) LAVCE) * uJ LB) + x^Liv Ce> + ~-where a l l the c o e f f i c i e n t s are nonsingular and slowly varying i n the upper and lower half planes. We then consider the function Obviously, i n the lowest order, i t i s equal to jF^p | 2 G*g<|3tE), In general order, we can write where (.B) are a l l slowly varying i n the upper and lower half c'i planes. We then write ( f ) i n the form of (4-18); i . e . , we take a l l the E-dependence i n the denominator: where K«p(.fc) = K $ ce)+ x ' ^ t o * — K"a CE) = Ll^C&) + (.*-> +U)Nw,J LB) = (*.Hi O - * < 0 »- — a. — » *Note that the d e f i n i t i o n s of L ^ i E ) and N\ytE) are d i f f e r e n t from (4-15) and (4-16). In the present notation, L *>*(£) = \$) and KI^)(E)*(<klO.The same holds for (4-43) and (4-44). 3 5 . P utting £ s GO±T£ a n d writing we get the line-shape function ^p'c<o) i n the form where the notations are s i m i l a r to those i n (4-26). The line-shape function (4-46) i s correct to any order i n VC, provided that a l l the perturbation s e r i e s used i n the derivation of (4-46) are uniformly convergent. 2 **** Again, for 60*^ 1 ^ )C ^ ^ w ) y w e can approximate the l i n e -shape function by On the other hand, the approximation of by a Lorentzian curve i n the v i c i n i t y of the peak i s no longer possible i n higher orders, since the CO-dependence of C*o) cannot be ignored i n general orders, However, i t i s s t i l l possible to obtain the peak pos i t i o n and the f u l l width at half power i n the perturbation expansionj i . e . , the peak p o s i t i o n i s given by and the f u l l width at half power by AW- * X * l % L t % ) V ^ ] ^ ^ 2. 36 These r e s u l t s show that the decoupling made i n (b) was the appropriate lowest order approximation i n studying the line«shape function ^ ( w ) . §4-2 The Interference Term The c a l c u l a t i o n of the interference term i s s i m i l a r to that of the d i r e c t term. We s t a r t from equations (3-9), (4-3) and (4-4). Since the f i r s t term on the r.h.s. of (3-9) i s , f o r C\rA')* (K K ) ; of order x\ a l l terms on the r.h.s. of (3-9) have to be retained. As i n the case of the d i r e c t term, we approximate the equations (4-3) and (4-4) by t h e i r lowest order ones. Equations (4-12) and (4-13) are, however, not the v a l i d lowest order equations i n t h i s case. Indeed, we have to r e t a i n the Green's function £r^/C&) i n the equations (4-3) and (4«4), since we are interested i n the A'A s i n g u l a r i t i e s of Grf*^L^) and not of LTA*' * I f w e d o t h i s , we get the following lowest order equations f o r (^) Old fy^/^te)** (E-T».*T>-wOr y ^ C C ) » & K W * S»x Kf A V * } A'A * We note that these equations are v a l i d only for obtaining Q-^^^E) AfA \ and not f o r other Green's functions, such as fy + ^ jLE) ( tf, I'J $ (M»K)y 37. A* A If we combine (4-50) and (4-51) with (3-9) and then use (4-14), we obtain the lowest order expression for G r ^ / IE) i n the form where CE) and CE") are c e r t a i n slowly varying functions of E ( i n the upper and lower half planes) whose e x p l i c i t expressions w i l l not be given here. Using (4-18), we can write Substituting (4-53) into (4-52) we get T — ilfc — < r A A , CE) (4.5T4-) 38. The Green's function Grp|»> l£) as given by (4-54) has two sharp maxima, one at E^Tx'-lx, which comes from GTAA*^)>AND T H E other at E T Th' ~ V > which comes from (j^ /^ (e). Now we are con-s i d e r i n g the absorption l i n e which has the peak at E =• T Tp-Td. In other words, we want to pick out only those terms which contribute to the peak at E 8 WL> This can be done by reta i n i n g only the terms involving Gf^C9) i n the expressions (4-54). i f we do t h i s , we get the following f i n a l expression for the interference term where , §4-3 The Complete Line-Shape Function Combining (4-55) with the d i r e c t term we get the complete line-shape function ^\tj) i n the form S u b s t i t u t i n g (4-18) i n t o (4-57) and u s i n g t h e f a c t t h a t RoqjtE) i s n o n -s i n g u l a r i n t h e u p p e r and l o w e r h a l f p l a n e s o f E ? we g e t c o r r e c t t o t h e l o w e s t o r d e r i n K: f\o) = c I > W ' f c £ ^ Irm r — r T " ' % (*-^ 39. where J ^ C E ) = K « ( i C E ) + C E - T p + T o O l ^ l E ) Using the r e l a t i o n s (3-5) and (4-20), we can write F i n a l l y , noting that ^(A>) and H ^ l o l are slowly varying functions of cO > we obtain where The line-shape function as given by (4-61) i s v a l i d for a l l values of CO i n the lowest order i n K . In p a r t i c u l a r , i f (pJ-G^) i s of order unity, ^ceo) c a t t be written as S Cm) % L . | h £ p | 0 * - > y ) fo.^)* > whereas i f CO i s very close to Woya^ i t can be approximated by a Lorentzian shape where the f u l l width at half power ACQ' i s given by As we s h a l l see l a t e r , both the peak po s i t i o n flj^ and the f u l l width at h a l f power AtO are unaffected by the interference term i n the lowest order; 40. § 5. Discussion of the Line-shape - Nondegenerate Case (a) Peak Posit i o n To calculate the peak position (4-62), we f i r s t note that JtfphjrTSO i s simply given by L«^("^-T*). Indeed, i f we put W-lp-Tol—X i n equation (4-20), the r e a l part of t h i s equation j u s t vanishes. Hence we have no contribution from The quantity Lt^lp-Tx) can e a s i l y be evaluated from (4-24) as Now, i n most cases the diagonal term | V / x ^ | J i s much greater than the nondiagonal one / V j ^ t l * C ^  h) . If we ignore the nondiagonal terms, we get the following simple expression f o r the peak position: where * . - „ .T i ! £ L * CM) i The chief c h a r a c t e r i s t i c of t h i s simple expression i s i t s temperature independence. On the other hand, the neglected nondiagonal terms have a temperature dependence through the phonon occupation number Ug • The r e l a t i v e importance of the nondiagonal term could then be checked experimentally by observing the temperature dependence of 4!. the peak pos i t i o n , (b) The F u l l Width at Half Power As i n the case of the peak po s i t i o n , we s t a r t by pointing out the r e l a t i o n Indeed, i f we put i s equation (4-20) and then integrate over y ) we get no contribution from The quantity J^Cp-Ocaa be obtained from (4-25). In doing t h i s we note the r e l a t i o n which follows from the fa c t that the density of phonon states vanishes at &J<£ = 0. Then the f u l l width at half power i s given by 42 The f u l l width at half power as given by (5-6) i s of e s s e n t i a l l y the same form as the natural l i n e width f i r s t given by Weisskopf and Wigner, except that % i s not a photon state, but a phonon state. Hence we may say that t h i s width arises from l i f e - t i m e broadening. The c h a r a c t e r i s t i c s of this l i f e - t i m e broadening may be summarised as follows: ( 1 ) At zero temperature, the line-width i s non-vanishing and i s given by £ o O o ) = f J3jJ V * 1 ' s l T r Ta'°^ (2) In p a r t i c u l a r , i f a i s the ground state, the second term i n (5 -7 ) vanishes, so that the broadening of the absorption l i n e i s due to that of the excited state only. This implies t h a t the absorption line-widths at zero temperature depend c r i t i c a l l y on the positions of the absorption peaks. In general, i f two absorption l i n e s are f a i r l y close to each other, the one corres-ponding to the t r a n s i t i o n to the higher excited state w i l l have a greater width than that to the lower excited state. (3) As d i s t i n c t from the peak po s i t i o n , the line-wdith i s strongly temperature dependent. With increasing temperature, the con-t r i b u t i o n to the width from a p a r t i c u l a r state A stays almost constant u n t i l the. c h a r a c t e r i s t i c temperature Tk and then increases l i n e a r l y with T 43. (c) Moments As we have already obtained the most important" informatics. y the line-shape, the peak position and the f u l l width at half power, there i s l i t t l e point i n c a l c u l a t i n g the moments. However, since most of the previous authors have calculated only the moments and have t r i e d to estimate the peak position and the half-width from the moments, i t i s o f some int e r e s t to calculate the moments using our line-shape function and to compare the r e s u l t s with those obtained by previous authors. We s h a l l f i r s t determine the normalization constant, which can be done by c a l c u l a t i n g the zero-th. moment. Ignoring the con-t r i b u t i o n of the t a i l , which i s of order 7C% ^ and using (4-64j)^we get the normalization constant to be equal to In c a l c u l a t i n g the higher order moments, we can no lonpr ignore the contribution of the t a i l . In f a c t , one can show that for the l i n e -shape i n question, the contribution of the sharp peak i s less important than that of t a i l , even for the f i r s t moment. We may therefore use the line-shape function (4-63) i n c a l c u l a t i n g these higher moments. Substituting (5-8) into (4-63) and using (4-66) we have The moments are then given by 4 4 . In t h i s form we have a nonvanishing contribution from both j^C*)) and J w ^ C ^ l i t T o l - r * ) (^WpCw-fN^Cu-ra))}h To obtain a ' more e x p l i c i t expression, we have to know the e x p l i c i t expression for R*(j&(&) which i s , however, quite complicated. Here we s h a l l make the same approximation as we did i n the derivation of (5-2) for the peak po s i t i o n ; i . e . , we r e t a i n only the diagonal terms \/\\<^  and ignore a l l the nondiagonal terms This approximation i s usually very good and moreover only i n th i s approximation did the previous authors obtain the moments in a r e l i a b l e way*. In t h i s approximation one can show that J^s^-) vanishes and that the only nonvanishing contribution i s from T^(p (w) which i s i n the present approximation given by Although Kubo and Toyozawa (1955) have obtained an expression for the moments i n a more general case, these authors used the adiabatic approximation. It i s , however, then inconsistent to r e t a i n the o f f -diagonal matrix elements V | | ' ^ (See K.Nishikawa, 1962 b) To zero-order i n "X , we have Substituting (5-13) into (5-12) and then into (5-11), we get Expressions (5-14) and (5-15) for the moments are i d e n t i c a l with those obtained by Kubo and Toyozawa (1955) i n the l i m i t of low temperatures, i . e . , i n the l i m i t If we r e s t r i c t our model to the more s p e c i f i c one used by Lax amd Burstein (1955), .our f i r s t four moments agree with t h e i r s . It i s now obvious that the moments obtained by previous authors have no d i r e c t r e l a t i o n to the peak position and the f u l l width at half power. Indeed, to f i n d these" i t i s s u f f i c i e n t to use (4-64), v a l i d i n the v i c i n i t y of the peak, whereas to f i n d the moments i t i s s u f f i c i e n t to use (5-9), v a l i d only i n the t a i l . According to the c l a s s i f i c a t i o n by Lax and Burstein, the peak corresponds to the zero-phonon process, the t a i l to the multiphonon processes. The above points were already pointed out by Kane i n 1960. He made an evaluation of the r e l a t i v e i n t e n s i t y of the multiphonon processes compared with the zero-phonon process and arrived at the conclusion that the strong absorption l i n e observed for doped s i l i c o n corresponds to the zero-phonon process, the raultiphonom processes being observed only as a broad t a i l at high temperatures. He then suggested that any broadening due to the electron-phonon in t e r a c t i o n would have to be a l i f e - t i m e e f f e c t . The above r e s u l t s confirm the v a l i d i t y of h i s speculation. (d) Summary of the Results F i n a l l y , we s h a l l summarize the r e s u l t s . (1) Both the peak pos i t i o n and the f i r s t moment are temperature independent i n the approximation of the neglect of the o f f -diagonal matrix elements I/AA'£ £A^\'). (2) The root mean square width i s of order K while the f u l l width at half power i s of order K * . (3) At zero-temperature, the root mean square widths for various l i n e s are approximately the same since they are mostly deter-mined by the width of the ground state. On the other hand, the f u l l widths at half power are mostly determined by the widths of the excited states and are d i f f e r e n t f or d i f f e r e n t absorption l i n e s . (4) At high temperatures, the root mean square width increases with temperature as \lT^ the f u l l width at half power as T* (Cf. Sampson and Margenau, 1956). 47. § 6. Degenerate Case §6-1. Introduction The discussion i n the previous section was based on the assumption that there i s only one pair of states which s a t i s f i e s the condition (4-1). It i s under t h i s assumption that the absorption l i n e has a single sharp maximum at 60s&)c with a Lorentzian shape i n the v i c i n i t y of the maximum and with no appreciable contribution to t h i s sharp peak from the interference term. Any one of these r e s u l t s , however, may be incorr e c t , i f there i s more than one pair of states s a t i s f y i n g condition (4-1). From equations (4-52) and (4-53), i t i s seen that one cannot say that £ A ? £E) i s of order X*<$)^{£) near E S 7^'"^ / i * (7*x'-Tx) and C"^'""7/A) a r e very close to each other. This implies that our fundamental order estimation (4-8) i s no longer v a l i d i n t h i s case. In such a case, the interference term becomes comparable to the d i r e c t term near the sharp peak and we cannot calculate the interference term separately from the d i r e c t term. In t h i s section, we s h a l l assume that the states associated with the energy eigenvalues Tat and Tjz are a-fold and b-fold degenerate (or approximately degenerate), respectively, and that there i s no other pair of states that s a t i s f i e s the condition (4-1), Such i s the case which often occurs i n actual systems with shallow impurity l e v e l s . We s h a l l use the following notations: ±j fcj *-*• w i l l be used to l a b e l the a-fold (approximately) degenerate states which correspond to the energy l e v e l Tat / t>J^ i^*** w i l l l a b e l the b-fold (approximately) degenerate states which correspond to the energy l e v e l lj$ } V 4 T V * — * T i ' 4 — * T V 4- > ; )i/fAt)j) w i l l denote the e l e c t r o n i c states, including C,/, — and C', J ' , fc', . Throughout our c a l c u l a t i o n we s h a l l be interested only i n the lowest order c a l c u l a t i o n of the line-shape function i n the v i c i n i t y of the sharp peak at <o ^ tOt. Contribution to the t a i l of the absorption l i n e , as well as higher order corrections to the sharp peak, w i l l always be ignored. As can be seen from the d i s -cussion for the nondegenerate case, i t i s s u f f i c i e n t i f we consider only those Green's functions that are of the form r ( & 4 k>* = A l l other Green's functions give only higher order contributions to the sharp peak at fo±(OC j although they may contribute to the t a i l i n the lowest order. §6-2. Calculation of the Green's functions The equation for Gtfa' (£) i s The equations for the Green's functions f^tf*^ C^) a n d /"^A^^^ involve higher order Green's functions of the form $ j^kq^l ftp'j as well as the Green's functions of the form $0LjfcClftfij An order estimation s i m i l a r to those made i n the nondegenerate case shows, however, that only those Green's functions that are ©f the form Jbg < y / ^ ^ £ a n d / A? fy»£r give a lowest order contribution to the equations. Moreover, as i n the nondegenerate case, one can approximate the higher order Green's function $hcjivj oXa**! as ( s e e Append TX W) Making use of these order estimations and ignoring the higher order \ . .v. terms, we get the following equations for /"Vx'^  tEy and /~^V<£ CB) Z If we substitute (6-5) and (6-6) into (6-3), we get a set of non-homogeneous l i n e a r simultaneous equations for the ( y ^ ^ t L&) ' (e-Tr+TK)Grti Ce, - ««£ £ CM) SD. where + — — J . + J \ I VX'*l I f - T J + T ^ (Ot + E-T^TkMt) and a s i m i l a r expression for N CSJ , As i n the nondegenerate case, the contribution of t^fcgt i s unimportant i n the v of the sharp peak. Hence we s h a l l ignore t h i s term as compa "Plug* i n the r e s t of the discussion. The solu t i o n of the simultaneous equations (6-7) can be obtained i n the form where and X|»»n', k £ / ^ ) i s t n e c o ~ * a c t o r o f the (ff£^ M f l ' J element of the above determinant. We write P ( E J i n the form In the cases we have considered the slowly varying nature of £ ^ W n / (B) ensures the same property for A ^ g / C E ) , Then the Green's f u n c t i o n (6-10) w i l l usually be expressible i n the form where iA 1* . i s a c e r t a i n constant. Putting fsft&PE and writing tr* Amn,(o±ii) 4**/ to)? to), we get £'^ .s The line-shape function can then be written as where til sa The line-shape function as given by (6-17) i s a l i n e a r super-p o s i t i o n of several Lorentzian curves with the superposition co° e f f i c i e n t s In p a r t i c u l a r , i f a l l the CAtf and are the same, the line-shape function becomes a simple Lorentzian shape. Although i t i s not easy to f i n d e x p l i c i t general expressions f o r the peak positions and the f u l l widths at half power of the constituent Lorentzian l i n e s , the q u a l i t a t i v e statements made about these quantities i n §5 for the nondegenerate case s t i l l hold i n the degenerate case. §6-3. Example As a s p e c i f i c example, we s h a l l consider the case i n which the ground state p( i s nondegenerate and the excited states f& a r e (approximately) doubly degenerate. In t h i s case, the determinant P( jE) takes the form where and i t s co-factors are S3. where ( 3 and | 3 ' denote the pair of excited states we are considering and the d e f i n i t i o n of JL^(E) i s the same as i n (4-15). The Green's functions can then be written as where we have ignored rt^ and W /^ as compared with W v  Case I In t h i s case, we have Then we get r ^ / c » - ! / * 64. If we ignore the cross terms ^ a n < * (jo!^/ > the r e s u l t turns out to be exactly what we would have expected from the non-degenerate case; that i s the absorption l i n e i s a l i n e a r super-p o s i t i o n of two Lorentzian curves each of which i s given by the formula obtained i n the nondegenerate case. Conversely, the i n e q u a l i t y (6-29) w i l l give the c r i t e r i o n that the two states be treated as nondegenerate. Case II and In t h i s case, + X*(lty<6)-liftL*))sO and The Green's functions can then be written as + i I (l-St) 55. The line-shape function i s , near O^lja-Ttf j where a 7 r + I B ^ P ] and we have written ? ^ C e o ) f f y ( J y y ^ CM) ) , The line-shape function (6-40) i s a l i n e a r super-position of two Lorentzian curves with the peak separation being equal to ZX^Lefp/jpiP* t An i n t e r e s t i n g feature i s that t h i s peak separation i s temperature dependent. The r e l a t i v e height of the two peaks i s a n d i s also temperature dependent. If the degeneracy arises simply from the c r y s t a l symmetry and the perturbation has the same symmetry, there w i l l be no r e -moval of the degeneracy, Low/Jf/j' 0 (cff. £4*Cb)y following paper). 56. § 7. Summary and Discussion of Results The double-time Green's function technique has been used to obtain the shape of the absorption peaks a r i s i n g from o p t i c a l absorption by electrons (or holes) bound to impurity s i t e s i n semi-conductors. We have assumed that these electrons interact weakly with the phonons i n the system and have studied the broadening a r i s i n g only from t h i s i n t e r a c t i o n . We have shown that i n thi s case there need be no arb i t r a r i n e s s i n the decoupling of the hierarchy of equations occurring i n the method. By taking the electron-phonon i n t e r a c t i o n as a small perturbation one can show with c l a r i t y the v a l i d i t y of the decoupling technique. The great advantage of the use of the Green's function method i s that we have been able to f i n d the line-shape function i t s e l f , whereas previous authors, s t a r t i n g from the Bethe-Sommerfeld formula, have had to be content with finding the moments of the l i n e . Their d i f f i c u l t y lay i n the performance of the sums involved; ours l a y i n the j u s t i f i c a t i o n of the decoupling of the equations. More-over, i n our method there i s no need to f i n d exact eigenstates of the Hamiltonian of the electron-phonon system. Indeed, i f we had used exact eigenstates of the Hamiltonian as our representation, the Green's function method would have had no advantage. The de-coupling would have been t r i v i a l , but i t would have led to the Bethe-Sommerfeld formula (see Appendix I ) . The shape obtained i s a modified Lorentzian one (or a super-p o s i t i o n of such) With the neglect of the off-diagonal matrix elements VAA'<£ ( A + V ) the frequency-dependent damping terra Stfj^* vanishes near the peak. This leads to an i n f i n i t e l y sharp peak corresponding to the zero-phonon process (Kane 1960). To obtain a non-zero width we have to include the off-diagonal matrix elements. (This i s what Kane c a l l e d the l i f e - t i m e e f f e c t s . ) Away from the peak, the off-diagonal matrix elements are unimportant and the absorption l i n e corresponds to multi-phonon processes. These give an important contribution to the moments and these moments agree with those found by previous authors. We see that the moments are of l i t t l e use i n obtaining the f u l l width at half power. The d i f f e r e n t l i n e s w i l l have d i f f e r e n t widths, contrary to the statement of Lax and Burstein. The f u l l widths at half power increase l i n e a r l y with temperature above ce r t a i n character-i s t i c temperatures. These c h a r a c t e r i s t i c temperatures are also d i f f e r e n t f o r d i f f e r e n t l i n e s . In the non-degenerate case, the interference term contributes to the sharp peak only i n higher order i n ><. In the degenerate case, th i s interference term i s important and may cause a temperature dependent s p l i t t i n g of the peaks. Acknowledgements One of the authors (K.N.) wishes to thank the National Research Council of Canada for the award of a studentship. Thanks are also due to Professor W. Opechowski for valuable discussions and help i n preparing the manuscript. 51 References Bloch, C. and de Dominicis, C. 1958. Nuclear Physics, 2,459. Bogoliubov, N.N. and Tyablikov, S.V. 1959. Dokl. Akad. Nauk, (USSR), 126. 53; t r a n s l a t i o n Soviet Phys, Doklady, £,604. Deigen, M.F. 1956. J.E.T.P. (USSR), 31, 504. Kane, E.O. 1960. Phys, Rev,, 119, 40, KUDO, R, 1957. J . Phys. Soc. (Japan), 12, 570. Kubo, R. and Toyozawa, Y. 1955. Prog. Theor. Phys., 13. 160. Lax, M. and Burstein, E. 1955. Phys. Rev, 100. 592, Nishikawa, K. 1962a. Physics Letters, JL, 140. Nishikawa, K. 1962b. Ph.D. Thesis, University of B r i t i s h Columbia, Vancouver, B.C. Nishikawa, K. and Ba r r i e , R. 1962. B u l l . Am. Phys. Soc. Series I I , 7, 485. O'Rourke, R.C. 1957. U.S. Naval Research Laboratory Report No. 4975, p.207. Sampson, D. and Margenau, H. 1956. Phys. Rev,, 103. 879. Ter Haar, D. 1961. Proc. Roy. Norwegian Acad. Sciences. Ter Haar, D. and Wergeland, H. 1961. Proc. Roy. Norwegian Acad. Sciences. Zubarev, D.N. 1960. Usp. F i z . Nauk. (USSR), 71, 71; tr a n s l a t i o n : Soviet Phys. USPEKHI 3_,320, 60. APPENDIX I RELATION TO THE BETHE-SOMMERFELD FORMULA (A-3) We s t a r t from (2-1) and (2-6): where Let us denote the complete orthonormal set of the elgenstates of the Hamiltonian H by { ff>i ; Hlf> - E*|f>. One can write the operator M as where Substituting (A-4) into ( A - l ) , we get TCcO) « C W - C O Ofm ^ ^ ^ / I S w ' l 1 - ' where we have used the r e l a t i o n « i » < f i I /*»><*•"'I > B which follows from the fact that the states \0 |^>y . are i elgenstates of the Hamiltonian H . Now, the equation f o r the Green's function ^ IfXf'l | I f ' X f l ^ Lven by E« IfXfl | lf'*+l>e " - >*'><*'!> SI where we have used the r e l a t i o n C I fX f ' l . H] - CE^-S F ) I fX f ' l . CM) In (A-8), therefore, only one Green's function occurs and there i s no decoupling problem. As i n §3, the required solution of (A-9) i s given by where *i* ' < *><f I > = e" E * / h 8 T / ( r e ~ C f / r » t ), ft-..) Putting £ eW*tg and using the asymptotic formula (3-5), we get which i s the Bethe-Sommerfeld formula. Naturally i t i s also possible to pass d i r e c t l y from the Kubo-formula for the adiabatic d i e l e c t r i c s u s c e p t i b i l i t y to the Bethe-Sommerfeld formula without using Green's functions. The basic step involved i s that which follows immediately from (A-9). 62 APPENDIX II ORDER ESTIMATION FOR EQUATION (4-2) We s h a l l prove the required r e l a t i o n by showing that the term i s of order K ^ < T A X ' ^ ) » w i l 1 be obvious that the other terms i n the second term of the l.h.s. of (4-2) are of the same order of magnitude as t h i s term. We write the equation for T\j>^ ^ *-n * n e f ° r m A'A A » * t + K ^ i » t f r A V ^ ) + — • ^ and consider only the terms written here e x p l i c i t l y . As f a r as the order estimation i s concerned, t h i s i s s u f f i c i e n t since the other terms w i l l not change the order of magnitude of the r.h.s. of (A-14). Let us consider the region of £ for which CTAX' (E\ i s of order X^, We s h a l l assume to be zero or negative; i n the range i n which we are interested, t h i s assumption i s s a t i s f i e d (see (4-2)). In making an order estimation of (A-13), we s p l i t the summation over % i n the following way: t i where denotes that the summation should be taken only for those values of \ for which 63. Now, for the values of $ which s a t i s f y (A-16 }y /Ay^<.*/as given by (A-14) i s of order X . 0 n t n e other hand, the number of phonon states for which (A-16) i s s a t i s f i e d i s of order X .* Hence the following order estimation follows: The r.h.s. of (A-17) i s independent of a. Therefore, provided that 2 - i s convergent, which i s i n fact true since there are only a f i n i t e number of phonon states, we have For any dispersion r e l a t i o n t h i s w i l l always hold for s u f f i c i e n t l y small X * For s i l i c o n and germanium, )£ i s such that i t holds for the long wave-length acoustic phonons» for the o p t i c a l phonons, i t does not hold. 6<f. APPENDIX III PROOF OF (4-9) AND (4-10) In the f i r s t part of thi s appendix, we consider an order estimation i n N of the Green's functions appearing on the l.h.s. of (4-9) and (4-10). We do thi s by using the r e l a t i o n * If we use t h i s r e l a t i o n , our problem i s reduced to the order estimation of the c o r r e l a t i o n function C$ACt)^ , The second part of thi s appendix i s devoted to the order estimation i n X> i n doing th i s we make an argument s i m i l a r to that used i n Appendix I I . The order estimations we f i r s t want to prove are As can be seen from equation (A-19), i t i s s u f f i c i e n t i f we prove See Zubarev, 1960, 65. Here we s h a l l prove only the f i r s t of these equations (A-22) and (A-23)j the other two can be proved i n an e n t i r e l y analogous way. We write the l.h.s. of (A-22) more e x p l i c i t l y as 3 where 1 then use the well-known expansion of the exponential function: We -tffr a f t * F / > where He/><:^  = e " n Hep e " H r T $6. The expression (A-27) can then be written as a sum of products of creation and a n n i h i l a t i o n operators. It i s then possible to apply the contraction theorem of Bloch and de Dominicis (1958) to the term According to t h i s theorem, the s t a t i s t i c a l average of a product of creation and a n n i h i l a t i o n operators: i s equal to the sum of a l l the possible systems of complete contrac-tions taken among the creation and a n n i h i l a t i o n operators fj 1j /"""'> where the contraction of two operators ^ and i s defined by (0) ii S Q ? Tr U~'H 1 1 } . C A - V ) Now, i f a contraction i s taken between the operator h^or bqt appearing e x p l i c i t l y i n (A-31) and one of the phonon operators ap-pearing i n the expansions of the exponential functions, i t i s , i n the expression (A-30), always accompanied by a factor of the form KV)>i/'£ or KVvfa%/, which i s of order Vjht . Then, out of a l l the possible combinations of contractions which contribute to (A-30), those which do not contain the contraction between and b^f ( and (fey'are appearing e x p l i c i t l y i n (A-27)) are always of order Ylst ^  since they are accompanied by a factor of the form Now, from the d e f i n i t i o n of the contraction, we have from which we f i n d + OC/H ) = t o . T r i e 1 e e <A*«j,e e «v?v!)y Substituting (A-33) into (A-27), we get The order estimation (A-20) then follows from (A-19). We remark that t h i s theorem can e a s i l y be extended to a more general case} i . e . , we can show, for instance, - V {s* . r ' ^ M - h W r (B)} a+oti)) 68. We can show that these order estimations are consistent with the equations f o r the Green's functions. Next we make an order estimation i n K » that i s , we want to derive the three equations (4-10). Again we s h a l l consider only (4-10a). We f i r s t use the same expansions of the expoential functions and the contraction theorem as above. Since each V#j)'p or i s accompanied by a factor X one can immediately see Then from (A-19), we have for most values of £ This i s , however, not quite correct near the s i n g u l a r i t i e s of th i s Green's function. There are two types of s i n g u l a r i t y , one occurring at values of £ which depend on $ and $^and the other at values of £ which are independent of t and The f i r s t type of s i n g u l a r i t y can, however, be disregarded, since t h e i r e f f e c t i s unimportant aft e r the summation over % and (We r e c a l l that we are primarily interested i n lty^> £ e ) and not i n {4j ^ / 4 > ^ S )# The only important s i n g u l a r i t i e s are those of the second type, i . e . , those which are common to a l l b\hc\i Gjttif*' / 4$^A/^g f o r d i f f e r e n t ^ and We s h a l l show that near these s i n g u l a r i t i e s the Green's function < by fl/Tfy' \ dp #A i s of order tf*CTAA'te), To show t h i s , we s h a l l write the equation for < T ^ ^ / j C^ jjf AA/^r: We can see from t h i s equation that the possible s i n g u l a r i t i e s of 4 kth Afty' (4$ ^ A » S O C C U R A T S * (Tp ~rh~ti<i<* and also at the s i n g u l a r i t i e s of the Green's functions appearing on the r.h.s. Obviously, the f i r s t one i s 9 " and 9 * dependent, so that we can ignore t h i s s i n g u l a r i t y . To f i n d the s i n g u l a r i t i e s of the Green's functions appearing on the r.h.s. of (A-38), we have to write the equations for these Green's functions. The equations for Typ^f^/ ^fil and are given i n (4-3) and (4-4). As can be seen, from these equations, these tireen's f u n c -tions have s i n g u l a r i t i e s at the s i n g u l a r i t i e s of Q'^j/C^)j which are obviously independent of £ and One can i n general show that the s i n g u l a r i t i e s independent of £ and occur o n l y at the same values of £ as do these o£_ ( J T ^ / ( E ) 0 Out of these s i n g u l a r i t i e s , the one which contributes i n the lowest order i n x i s the s i n g u l a r i t y of G v s K (tr) ( c f. (4-8 )). As can be seen from (4-3) and (4-4), T ^ C f i l and P^fe) are of order CB) * *This i s not quite correct for some p a r t i c u l a r values of % or can't. f o r J/sA and jJ^Xj respectively, and are of higher order for L*^A/A' . Hence the second term on the r.h.s. of (A-38) i s of order H^Ct^C^)* The same i s true for "the t h i r d term i f ^" s . ^ (fy \!s It can be shown, by an order estimation s i m i l a r to that used i n Appendix I I , that the other terms are also of order X^^X^p) or l e s s . In t h i s way, we f i n d that ^ b ^ ^ ^ ^ * j A \ f ^ A ^ i s of order ,X*fi>W*J near the sing-u l a r i t y of (zTHfttr)- Near other s i n g u l a r i t i e s , the contribution i s of higher order i n *< , F i n a l l y , we remark that our order estimation i n f\f i s v a l i d even i n the degenerate case, but the order estimation i n X i s not. This i s because the order estimation (4-8) i s no longer v a l i d i n the degenerate case. In the degenerate case, one can write (4-10) as where the summation runs over a l l pairs of states for which •cont. However, t h i s can again be disregarded since a f t e r the sum o v e r ^ and ^/ the e f f e c t of such p a r t i c u l a r values of 1 and % i s unimportant (see Appendix I I ) . 7 1 . CHAPTER 311 Phonon Broadening Of Impurity Spectral Lines II Application to S i l i c o n Robert Barrie and Kyoji Nishikawa* Department of Physics, University of B r i t i s h Columbia, Vancouver 8, B.C. Abstract The general theory of the phonon broadening of impurity s p e c t r a l l i n e s discussed i n the preceding paper i s applied to shallow impurity l e v e l s i n s i l i c o n . With the use of a modified hydrogenic model and a deformation pot e n t i a l description of the electron-phonon i n t e r a c t i o n , expressions are obtained for t y p i c a l contributions to the half-widths. Some numerical estimations are made for both acceptor and donor cases and are compared with experiment. Present address: Research I n s t i t u t e for Fundamental Physics, Kyoto University, Kyoto, Japan. 72. §1 Introduction The absorption spectra due to the presence of impurities i n s o l i d s have received considerable attention. I t has been stated (Kohn, 1957) that the widths of the l i n e s observed for the so-c a l l e d shallow impurity l e v e l s i n homopolar semi-conductors are well understood i n terms of a broadening mechanism a r i s i n g from the i n t e r a c t i o n of the bound electron (or hole) with the l a t t i c e v i b r a t i o n s . This statement was based on an experiment with boron-doped s i l i c o n (Burstein et. a l . , .1953) and on a t h e o r e t i c a l treatment by Lax and Burstein (1955). One of the. l i n e s had an observed width at 4.2°K of 10 e.v., increasing to 1.4 x 10 e.v. at 77°K. The theory predicted a width of 3 x 10~ 3e.v. at 4.2°K increasing by the appropriate 40% at 77°K. The agreement with experiment seemed reasonable. Experiment and theory also seemed to agree i n that the widths of the three observed l i n e s lg to 2p, 3p and 4p were roughly equal. This l a b e l l i n g of the states i s now known to be incorrect and the widths are known to be less than 10~ 3ev at 4.2°K (Hrostowski and Kaiser, 1958). In a recent experiment, Colbbw et. a l . (1962) made allowances for instrumental broadening and found that the true half-width crib one of the l i n e s i n boron-doped s i l i c o n was 0.2 x 10° 3e.v. at 4.2°K and i t s increase with temperature was very d i f f e r e n t from that predicted by Lax i and Burstein. In addition d i f f e r e n t l i n e s had d i f f e r e n t widths. In the l i g h t of the above, we have performed a detailed general analysis of the phonon-broadening of impurity spectral l i n e s f or the case of weak int e r a c t i o n between the bound electron and the l a t t i c e vibrations ( p r e c e d i n g paper). In addition to the 73, weakness of the i n t e r a c t i o n another approximation i n the theory was that the i n t e r a c t i o n be e f f e c t i v e over a wide range of phonon energies. For a given phonon dispersion r e l a t i o n , t h i s would always be s a t i s f i e d by making the electron-phonon i n t e r a c t i o n s u f f i c i e n t l y weak. In s i l i c o n , however, the magnitude of the i n t e r a c t i o n and the phonon dispersion law are such that the two approximations are s a t i s f i e d only for the long wave-length a<g,oustic phonons (Lax and Burstein (1955): Kane 0-960).) We s h a l l i n t h i s paper then discuss the application of the general theory to the shallow impurity l e v e l s i n s i l i c o n with the i n t e r a c t i o n of the electron (or hole) i n these l e v e l s being with the long wave-length a c oustic phonons. The energies of other modes of l a t t i c e v i b r a t i o n s are comparable with or greater than the i o n i z a t i o n energies of the impurities being considered and they would not anyway a f f e c t the spectrum i n the region i n which we are interested. In the preceding paper i t was shown that for weak electron-phonon i n t e r a c t i o n , the observed width at half maximum due to t h i s i n t e r a c t i o n arises from a " l i f e t i m e e f f e c t " ( c . f . Kane, 1960). The process discussed by Lax and Burstein, the emission or absorp-t i o n of phonons accompanying the o p t i c a l t r a n s i t i o n , contributes a background to the peak. The conduction a<nd valence bands of s i l i c o n are rather com-p l i c a t e d and t h i s leads to complexity i n the discussion of the impurity states (Kphn, 1957). To avoid having the d e t a i l s of t h i s complexity obscure the general properties of the o p t i c a l absorption we have devoted §2 of t h i s paper to an application of the general theory to a modified simple hydrogenic model of the impurity states. The modification of the usual hydrogenic model 74. was made i n order to make the model hear a closer resemblance to the actual case of impurities i n s i l i c o n . In §3, the case of acceptor impurities i n s i l i c o n i s discussed. Our understanding of the detailed nature of these states i s f a r from complete and we have not f e l t i t to be worth while to proceed beyond the model of §2 . In the discussion of §4 on donor states we have considered various corrections to the model of §2 to bring i t much more into l i n e with our knowledge of these states. The •f f i n a l section contains a discussion of the r e s u l t s and comparison with experiment. The preceding paper on the general theory w i l l be referred to as I and equations taken from that paper w i l l have t h e i r number preceded by I. The notation used here i s taken d i r e c t l y from I. 75. §2 Hydrogenic Model In t h i s section, we s h a l l i l l u s t r a t e the c a l c u l a t i o n i n the modified hydrogenic model s p e c i f i e d by the following. ( i ) The eigenfunctions* of the unperturbed electronic Hamiltonian He are given by the simple hydrogenic ones with e f f e c t i v e Bohr radius equal to a. There i s , however, a s l i g h t devia-ti o n of He from a true hydrogenic Hamiltonian, so that the degeneracy i n the excited states, which exists i n the hydro-genic model, i s p a r t l y removed. M©re s p e c i f i c a l l y , we assume that only remaining degeneracy i s that of the states. Transitions w i l l be considered between the ground state (the ) S state) and the ^-st a t e s . ( i i ) Those phonons which in t e r a c t nonvanishingly with the electron are the l o n g i t u d i n a l long wave-length ! acoustic phonons. The energy spectrum of these phonons i s given by the i s o t r o p i c dispersion r e l a t i o n ; CO* = V ? ; t = l& \ , <2J) where £ i s tne wave number vector of the phonon and i f i s the sound v e l o c i t y which i s assumed to be independent of the d i r e c t i o n of the wave propagation. The density of these phonon states i s assumed to be unaffected by the presence of the impurity atom. The summation over the phonon states can then be approximated by These are a c t u a l l y the envelope functions discussed by Slater (see Kohn, 1957). 76 where V i s the volume of the system and denotes the integration over the angle variables, ( i i i ) The electron-phonon i n t e r a c t i o n can be treated i n the is o t r o p i c deformation pote n t i a l approximation of Bardeen and Shockley (1950). In t h i s approximation the matrix elements (see I(2-9C)) can be written as (Lax and Burstein. 1955): where £ i s the deformation pote n t i a l constant. N i s the number of unit c e l l s . ^ i s the mass per unit c e l l and e^esj » fwc F^)*^,(r ) e l J " c &-« ^ C C ) being the eigenfunctions of the unperturbed e l e c t r o n i c Hamiltonian and being taken to be the simple hydrogenic functions i n the present model. Now. the quantities which often occur i n our c a l c u l a t i o n are of the form where ^Sjfi^ depends on £ only through . Using (2-2) and (2-3), one can write ^ P J ^Wt ^ A M V where X and £ are dimensionless quantities defined by X* at 17 = S * f . N*1 (2.8) and the bar denotes the average over the angle variables: The approximation involved in(2-6) i s the replacement of the upper l i m i t of the i n t e g r a l by i n f i n i t y . This can be j u s t i f i e d under the assumption that the matrix elements VAA'£ are nan-'s* vanishing only for the long wave-length phonons. Some of the functions |0A/tC&)|4 evaluated using simple hydrogenic wave functions are l i s t e d i n table I. (a) The Peak Po s i t i o n We calculate the peak position i n the approximation 1(5 - 2 ) , (uo) i . e . COolp = Tp - Tot ; where fx = T„ - x*Z , V * X * - «-") Using (2-6), we get From table I, we have [Butt*! 1 = y 4 ( < * = , s ) and From these we f i n d 78, This i s i n general true for a l l ^ 's. Then we f i n d cOotp - T<* A Thus the peak s h i f t due to the electron-phonon i n t e r a c t i o n i s mostly determined by the s h i f t of the ground state. (b) Peak Separation of the Doublet Since the electron-phonon i n t e r a c t i o n has spherical symmetry, there i s no removal of degeneracy by the electron-phonon i n t e r a c t i o n . Then there! i s no peak separation. (c J The F u l l Width at Half Power For the same reason as i n (b), (see I (6-41) J. Then i t i s s u f f i c i e n t i f we calculate ¥*i^Tp-T«). (see I (5-6)). The contribution of state A to the f u l l width at half power AC*) can be written as IS. -where #/3A In (2-19J, the contribution of the width of the ground state was ignored. The numerical values of are given i n table II for the cases of (5 = 2 f and A,= 2P cmd 3S. The c h a r a c t e r i s t i c temperature above which s t a r t s increasing with temperature i s given by (d) Approximate- Line-Shape Since there i s no s p l i t t i n g of the doublets, the line-shape near the peak i s purely Lorentzian for a l l l i n e s . Away from the peak, the line-shape i s approximately given by 1(5-9). If we ignore the off-diagonal elements VxfA? (A$|^this line-shape function can be written at zero-temperature as 80. where <aoO = ^  $dA { e*At%) - G P P c i ) i p (2.24), For 0< l^S and (3 = 2po ; QOO can be written as The function (2-28) describes the line-shape of the one-phonon process discussed by Lax and Burstein (1955). The c h a r a c t e r i s t i c s of t h i s l i n e (©UiS, ^e2.P0) are l i s t e d i n table III and are compared with those of the Lorentzian l i n e which corresponds to the zero-phonon process for the same t r a n s i t i o n . It i s obvious from t h i s table that the one-phonon process contributes only to the back-ground and not to the sharp peak. At higher temperatures, the i n t e n s i t y of the one-phonon pro-cess increases by a factor of about ( 1 + 2 ^ ) , the c h a r a c t e r i s t i c temperature being of the order A. Tie * The line-shape of the one-phonon process at high temperatures has two maxima near OO - ^ jg - To< y the r e l a t i v e i n t e n s i t y of the two maxima being about / C ^4 ) 81. §3 Acceptors The nature of the acceptor impurity states has been discussed i n a review a r t i c l e by Kohn (1957). At the centre of the B r i l l o u i n zone, where the valence band has i t s maximum energy, the l e v e l i s , including the spin degeneracy and neglecting the spin-orbit s p l i t -t i n g , s i x - f o l d degenerate. This introduces an additional degeneracy into the impurity states. The uncertainties i n the values of the mass parameters occurring i n the electron energy-wavenumber r e l a t i o n and i n the value of the spin-orbit coupling make an accurate c a l c u l a t i o n of the impurity state eigenfunctions and energies impossible. Schechter (1962) has, however, c a r r i e d out a c a l c u l a t i o n of these using the best available values for these parameters. The ground state of an acceptor impurity i s l a b e l l e d "IS" and the lowest o p t i c a l l y excited states are l a b e l l e d "2P"j these would be the nature of the states i n the hydrogenic l i m i t . A l l of these states carry a degeneracy greater than that for the hydrogenic l i m i t . There i s as yet no clear-cut l a b e l l i n g of the observed states. The agreement between theory and experiment i s not s u f f i c i e n t to warrant the use of Schechter*s calculated eigenfunctions i n a detailed c a l c u l a t i o n of the o p t i c a l absorption c o e f f i c i e n t . We s h a l l therefore base our discussion of the acceptor impurity states on the model of §2. We use the following numerical values for the parameters: f = 3 . 3 3 3/W 82. These numerical values are the same as were used by Lax and Burstein (1955J* The peak s h i f t (see (2-17)) i s then calculated to be S A J X | t f * e * ( 3 . 2 ) a /iff T y p i c a l contributions to the f u l l width at half power can be estimated by using table II and (3-2) i n (2-19)j they are of order 10~*eV which agrees with the observed half-width at zero-temperature for boron-doped s i l i c o n (Colbow, et a l . , 1962). The c h a r a c t e r i s t i c temperature can be estimated from (2-22) with CL ke * Our numerical r e s u l t d i f f e r s from that of Kane (I960) Drimarilv through the use of d i f f e r e n t numerical values. 83. §4. Donors The nature of the donor states has also been discussed i n the review a r t i c l e by Kohn (1957). The conduction band has s i x equivalent minima at positions j££ i n the B r i l l o u i n zone. In the neighbourhood of one of the minima, say for ~(.Q,0, K© )} the electron dispersion r e l a t i o n i s If we consider our impurity eigenfunctions to be made up of Bloch functions from states near one of these minima, then the p-states are such that the p± states are s t i l l degenerate, but the degeneracy with P© i s removed. In the subsection (a) of t h i s section we consider t h i s case. If we now, i n forming the impurity eigenfunctions, use Bloch functions from states near a l l s i x minima, then an ad d i t i o n a l s i x - f o l d degeneracy i s introduced into the impurity states. This degeneracy i s par t l y l i f t e d when corrections to the e f f e c t i v e mass formalism are taken into consideration. . The s i x - f o l d degeneracy of the ground "IS" set i s , for instance, s p l i t into states of one-, two-, and three-fold degeneracy. The nondegen-erate "IS" state has the lowest energy. In subsection (b) of, th i s section we include t h i s degeneracy a r i s i n g from the many-v a l l e y nature of the conduction band. In subsection ( c ) , we rela x the i s o t r o p i c approximation f o r the deformation potential and consider some of the e f f e c t s a r i s i n g from the shear waves. 84. ( a) Single Valley Approximation We f i r s t apply the hydrogenic model we have discussed i n §2, but with a s l i g h t modification. The modification consists of using the following envelope functions 4 1: Fa sC£. = ==t=r e" f / a (a-f) C <^ J * C " G T7 € where We also make the approximation Then we can use a l l the formulas obtained i n §2 with the following changes: '"These functions tend to the simple hydrogenic ones i n the l i m i t of K^y. 85. ( i ) replace by Cu\ ( i i ) multiply the expressions for given i n table I (except when \^ = s: 2 } by a factor where rA = 1 < A « S rfate ) We use the numerical values given below: a** o CJ — T a p * "~ T2P0 Tap© = £ met/ *#; T * p e - T i p ± = / wiey T 3 p ± -The peak s h i f t i s then estimated to be equal to 0.25 meV. Some t y p i c a l contributions to the f u l l widths at ha l f power are also estimated and are given i n table IV. *'Kohn, 195«|. **)Bichard, and G i l e s , 1962. 36 < (b) Many Valley E f f e c t We now consider the e f f e c t of the degeneracy of the states which ari s e s from the many-valley nature of the conduction band. In the e f f e c t i v e mass approximation, which i s good f o r the p-states (Luttinger and Kohn, 1955), the 2fo states are s i x - f o l d degenerate and the 2Rk states are twelve-fold degenerate. Of these only three i n each set are o p t i c a l l y excitable from the nondegenerate ground state . These three o p t i c a l l y excitable states i n each set belong to the same i r r e d u c i b l e representation of the complete tedra-hedral group. Since the electron-phonon i n t e r a c t i o n has the same tedrahedral symmetry of the c r y s t a l , there w i l l be no s p l i t t i n g of these l e v e l s due to the electron-phonon i n t e r a c t i o n . Then the absorption l i n e corresponding to tr a n s i t i o n s IS to or 2Rt w i l l be a super-p o s i t i o n of three i d e n t i c a l Lorentzian l i n e s . In p a r t i c u l a r , i n the i s o t r o p i c deformation p o t e n t i a l approximation, the r e s u l t turns out to be exactly the same as i n subsection (a). (c) E f f e c t of Shear Waves In the i s o t r o p i c deformation p o t e n t i a l approximation, only the l o n g i t u d i n a l phonons give a nonvanishing contribution to the electron-phonon i n t e r a c t i o n . In t h i s subsection, we s h a l l consider the e f f e c t of the transverse phonons by using the Herring and Vogt (1956) generalized deformation p o t e n t i a l theory. Accoring to Herring and Vogt, the s h i f t of the electron energy of the v a l l e y due to a s t r a i n tensor Ptc' (^ l '»xy»2) * J . Grindlay, private communication. 87. i s given by Srf (txtf + ^ /; ) + C u d + ! ? * ) ^ ^ ^ where and g « a r e t h e Herring and Vogt deformation p o t e n t i a l constants and i s taken along the d i r e c t i o n of the conduction band minimum from the o r i g i n . If we use t h i s deformation p o t e n t i a l , we have a non-vanishing contribution from both the long i t u d i n a l and transverse phonons. We than specify the phonon state by the mode of o s c i l l a t i o n <T~ C * 1 / -2 > 3) and the wave-number vector t and also assume the phonon dispersion r e l a t i o n i n the form 6 0 ^ = IT*-? r<u*; •ft where y^. i s taken to be independent of Since the deformation pote n t i a l (4-13) depends on 4, we have to consider the many v a l l e y e f f e c t on the wave functions. According to Luttinger and Kohn, the donor wave functions i n s i l i c o n can be written as ^ where ( r j is the wave function associated with the conduction band minimum. (The s u f f i x A i s used to specify the complete donor state including the many v a l l e y e f f e c t and f i€ used to specify the state near one of the conduction band minima.) In the e f f e c t i v e mass approximation, t h i s function (.1) can be written as a product of the Bloch wave at the conduction band minimum and the envelope function F-^ C t ) . For we s h a l l use the modified hydrogenic functions (*f.2j t o (A-6) with C * , /*, 2j being replaced by CXj,Yjt  T h e c o e f f i c l e n t s o(. £y can be determined by elementary Group theory (Luttinger and Kohn, 1955). 88. Using (4-13) and (4-15), one can write X Vx/u9 <4-» as where Ug, and ^ are the longitudinal and transverse sound v e l o c i t i e s , &j the angle between and the Zj-QxTs and N i # >' -or-6jf> ( £ > = f < C Fj (£) Fy ID e ' 2 £ f4w'7; Since we have at present l i t t l e information on the numerical value of E e l , i t i s n°t possible to estimate the r e l a t i v e importance of the transverse phonons i n general. In the case of the ground state, however, an important mixing occurs with the excited IS states only through the u n i a x i a l component uu °* the deformation pot e n t i a l (Hawegawa, 1960, Roth, 1960, Kondo, 1960). The width of the ground state i s given by (see I (5-6))' Using (4-|£) and considering only the contribution of the excited IS states, we obtain the following expression:* A , ,(*) I - _ L f A p . X** i where A i s the e x c i t a t i o n energy of the excited IS states, ft*, £i,X^ and are the dimensionless quantities given by * See, for instance, Ha$sgawa (1960) for the method of the c a l c u l a t i o n . 89. If we use the numerical values?** 0 we obtain . with the contribution of the transverse modes being about 10%. The r e s u l t shows that the width of the ground state i s n e g l i g i b l e at least at low temperatures and that the contribution of the transverse phonons i s comparatively small. ** 1) Hasegawa, 1960. 2) Wilson and Feher, 1961. ' v 3) This i s an approximate value for the phosphorous'doped s i l i c o n (Long and Myers, 1959. Castner, 1962). 90. §5 Comparison with Experiment There i s very l i t t l e information on the shapes of the impurity s p e c t r a l l i n e s i n s i l i c o n . However, ce r t a i n points appear to be d e f i n i t e l y established and i n these a comparison between experiment and theory can be made. F i r s t l e t us discuss the r e s u l t s of experiment made at 4.2°K. Here there seems to be general agreement between theory and experiment. Experimental studies have been made on two donor impurities, phosphorous and arsenic, (Bichard and G i l e s , 1962) and on one acceptor, boron (Colbow, Bichard and G i l e s , 1962). For both donors and acceptors the shapes of the l i n e s are Lorentzian near the peaks and t h i s i s as predicted by theory. A The measured half-widths for the boron l i n e s were 10 ev and theory predicts t h i s order of magnitude. The t h e o r e t i c a l value i s uncertain by at least a factor of 2 and there i s a v a r i a t i o n of t h i s order between half-widths of d i f f e r e n t l i n e s . The fac t that there i s a v a r i a t i o n i n half-widths i s predicted by a l i f e -time e f f e c t . The theory predicts much smaller half-widths f o r the donor impurity l i n e s and thi s again seems to agree with experiment. For phosphorous l i n e s , the broadening was > almost e n t i r e l y instrumental and only an upper l i m i t of order 10 ev could be claimed. As fa r as the actual numerical values of half-widths at 4.2°K are concerned, the present position might be summarised by saying that there i s no disagreement between theory and experiment. At higher temperatures, however, a d e f i n i t e disagreement appears. Only one l i n e has been studied above 4,2°K (Colbow, et. 91. Although the actual values of the measured half-widths become more uncertain at higher temperatures, i t i s quite clear that above 60°K the half-width of t h i s l i n e increases much more r a p i d l y than the l i n e a r increase predicted by the theory. In t h i s temperature range some other broadening mechanism must come into e f f e c t , a possible one being the Stark broadening due to the ionized impurities. This.might be expected to give the observed sharp r i s e i n half-width. It may also be possible that a strong i n t e r a c t i o n of the electron or hole with a l o c a l i s e d mode of o s c i l l a t i o n i s playing a r o l e . The general theory of paper I was confined to in t e r a c t i o n with nonlocalised modes of the l a t t i c e v i b r a t i o n s . One of the authors (K.N.) wishes to thank the National Research Council of Canada f o r the award of a studentship. The authors are g r a t e f u l to Dr. J.W. Bichard. Dr. J.G. Gi l e s and Mr. K. Colbow for making available t h e i r experimental data p r i o r to publication and to Dr. J . Grindlay for some unpublished c a l c u l a t i o n . References Bafdeen, J . and Shockley, W., 1950. Phys. Rev. 80, 72. Bichard, J.W. and G i l e s , J.C., 1962. to be published. Burstein, E., B e l l , E.E., Davisson, J.W., and Lax, M., 1953. J . Phys. Chem., 57, 849. Castner, T.G. J r . 1962, Phys, Rev. Lett. j3, 13, Colbow, K., Bichard, J.W., and G i l e s , J.C., 1962. to be published, Hasegawa, H., 1960. Phys. Rev. 118, 1523. Herring, C. and Vogt, E., 1956, Phys. Rev. 101. 944. Hrostowski, H.J. and Kaiser, R.H., 1958. J . Phys. Chem. (Solids) 4, 148. Kane, E.O., 1960. Phys. Rev. 119. 40. Kohn, W., 1957. S o l i d State Physics (Seitz and Turnbull), Vol. 5, p.257. Kondo, J . , 1960. Prog; Theoret. Phys. 24, 161. Lax M., and Burstein,, E., 1955. Phys. Rev. 100. 592. Long, D., and Myers, J . , 1959. Phys. Rev. 115. 1119. i Luttinger, J.M., and Kohn, W., 1955. Phys. Rev. £8, 915. Nishikawa, K., and Barrie, R., 1962. Preceding paper. Roth, L.M., 1960. Phys. Rev., 118. 1534. Schechter, D., 1962. J . Phys. Chera. (Solids) 23, 237. Wilson, D.K., and Feher, G., 1961. Phys. Rev. 124. 1068. 95. Table I Table of |Qkfi C%,) | A i n the simple hydrogenic model. In t h i s table, X stands for X, y^  2 ^ and t&l/s , X IS 2Pr 2S 2Pc 3S 3Pf 4* * V a + x a 5 * ; CT*T') 5 9*. T a b l e I I Numerical v a l u e s of {B^C%)\Z i n the simple hydrogenic model. ( X = 2?% } X^CLl t a= 4 ^ ) . 2Pr 2Pc/ V * . * N 3$ 0*2 $,2* JO* 0& ft a * 7 D R OA 0,3 #0 0.€ 6,5 x id* 2 1,23X10' 2,3X{0"2 A & X I O 1 0,5 ! of&xio1 1,0 0 - / L O O X / o ' 9.7 X/O"2 2 , 7X/D 2 . V 7 X / D 2 fjxto1 V*x/o2 / . 7 X / 0 2 U UXfOV wxio2 J.StX/O* ix 2,2 X / o " 1 /»7x*o2 O^tf/p" 2 2.0 tfX.101 .3JX/0 3 n Table III Comparison of the one-phonon process with the zero-phonon process f o r d(sJS > ^ S - 2 P 0 at zero-temperature. Zero-phonon Process One-phonon Process F u l l width at s lrf Bf?fa£ | J 2 X Half Power Z ^  C L Height at the v Q n h J_ JL n h<%z Q J L maximum Z * R tV* °*0S^ K i l T 98. Table IV Typical contributions to the f u l l widths at half power of the donor impurity l i n e s i n s i l i c o n . In t h i s table jS i s the o p t i c a l l y excited state and A U ) ^ i s the contribution of the state A to the width of the state P» 25 1 <6**lD"5ev IP* — — 3P. ~ * o \ 2P± 2S 1 - <1D*tv 2P± 4ft , M X 10 «.v 2,ox io6ev 3S < I05*v ^P± aPi 3P± 3P0 2P± 3.7XI0*ev 3P± 1 2P± 37°fC 5.5XI06cv -5 K4XID «.v ] I APPENDIX TO THE THESIS.I Relation of the Present Method to the Previous Ones t In Appendix I of Chapter I I , we have shown the equivalence of our s t a r t i n g formula I (2-1) to the Bethe-Sommerfeld formula. The approximations involved i n thi s formula were c l e a r l y stated i n §2. In using t h i s formula, we have introduced a s p e c i f i c model for the system (Chapter I I , §2). The main features of t h i s model system and of the calculations i n the subsequent sections are; ( i ) the in t e r a c t i o n between electrons bound to impurity s i t e s i s completely ignored; ( i i ) the electron-phonon int e r a c t i o n i s taken to be s u f f i c i e n t l y weak that i t can be treated as a small perturbation; ( i i i ) the matrix elements of the electron-phonon i n t e r a c t i o n , VAN'?? are nonvanishing for a wide range of cog • I t should be stressed that the following two approximations were not made i n our method: (1) The adiabatic approximation; (2) The Condon approximation. These approximations were almost always used i n previous work i n t h i s f i e l d (Dexter, 1958). However, l i t t l e attempt has so fa r been made to give a mathematical j u s t i f i c a t i o n for the use of these approximations i n t h i s problem. The purpose of t h i s appendix i s to formulate mathematically the above two approximations, (1) and (2), within the framework of the present theory and to c l a r i f y the nature of these approximations 100. (a) The Adiabatic Approximation (Born and Oppenheimer, 1929) The adiabatic approximation may be stated as follows: ( i ) To f i n d the approximate eigenstates of H given by 1(2-8), we f i r s t treat the phonon operators and Jbg as parameters and solve the following eigenvalue problem: (He + H e R)|X ( ^ ^ t > > = V / J ^ > ^ / A O ^ ) ) , «M) where <f(£f/fe{) denotes that the function depends on a l l and by . ( i i ) We then solve the eigenvalue problem I H p + W A 0 > « , b\)} M x > = E A * J-0x>, «W) ( i i i ) Then we approximate the eigenstates of the t o t a l Hamiltonian H by Let us write the eigenstates of (a-1) as where ( 0 ^ i s the e l e c t r o n i c vacuum state defined by aX /0>€ * 0 for *JI A., f4-&) and S As an antihermitian operator which depends on a l l operators^ CL^, tip. , b^cw*e( b^ and which vanishes i n the absence of the electron-phonon i n t e r a c t i o n . We then have the following theorem. Theorem The eigenstates of the Hamiltonian defined by H = e * t y f S 4 H e t H e p (a-*) are given by (a-3). Proof We s h a l l prove t h i s theorem by showing that the matrix elements of n with respect to the states (a-3) are diagonal. From (a-4), we have - S M ' < & ' * \ H P I 4 X > . ^ From (a-1) Using (a-7) and (a-8) and noting (a-2), we get <**! Of l*at,if>>M*> The above theorem implies that the adiabatic approximation i s equivalent to approximating \r\ by £f. We s h a l l now check the v a l i d i t y of t h i s approximation i n our s p e c i f i c model. We s h a l l do t h i s by c a l c u l a t i n g *S*by the perturbation method. For s i m p l i c i t y , we s h a l l assume that there i s no degeneracy i n the unperturbed e l e c t r o n i c states and r e s t r i c t ourselves to a c a l c u l a t i o n to the lowest order i n X • In finding S / we f i r s t note that 1 0 2 . we i s diagonal i n the representation ^l/O/fe* Indeed, i f multiply (a-1) by (A'tef/bf )| from the l e f t and then substitute (a-4), we get This shows that fi^Ht+'Hfcp)^ has the above-mentioned property. . Secondly, we note S O as H a p 0 • We then write H e p and S* as x H e p and X S ^ respectively, and use the following expansion: e " x s ' C « e + *Hef>)e * He + X C + C He, S ' J ) + 0l#). From the requirement that (a-11) be diagonal i n the representation At I t o a n y o r d e r * M * > we f incf F i n a l l y subsituting (a-12) into (a-6), we get + O C X ' J y fa-13.) 103. or Equation (a-13) or (a-14) shows that the v a l i d i t y of the adiabatic approximation depends on either the smallness of the t y p i c a l phonon energy as compared with the t y p i c a l energy separation of the unperturbed e l e c t r o n i c l e v e l s , or on the smallness of the o f f -diagonal elements VAA'^ (A^A'^. NOW, i n our model, we have assumed that the t y p i c a l phonon energy i s comparable to the t y p i c a l l e v e l separation of the e l e c t r o n i c states. Moreover, we have seen i n §5 that the width of the absorption l i n e i s due to a l i f e - t i m e e f f e c t i n which the contribution of the off-diagonal terms i s most important. This shows that i t i s inconsistent to use the adiabatic approximation i n our p a r t i c u l a r problem. In previous works, the adiabatic approximation was used. In the c a l c u l a t i o n by Lax and Burstein, the contribution of the off-diagonal elements VAA't CA4A') W a s omitted. In t h i s sense, t h e i r c a l c u l a t i o n i s consistent. However, as we have seen i n §5, the line-shape function i n t h i s approximation consists of a simple delta-function (the zero-phonon process) superimposed on a continuous background (the multiphonon processes). They should then have obtained zerowidth at half maximum. They obtained non-zero half-width by i n c o r r e c t l y assuming that the half-width can be obtained from the moments. (Kane i960). 1 0 4 . Kubo and Toyozawa, on the other hand, retained the o f f -diagonal elements VAA'^  ( A ^ / ) ^ i n t h e i r higher order c a l c u l a t i o n of the moments. This can be j u s t i f i e d only when the t y p i c a l phonon energy i s s u f f i c i e n t l y small compared to the t y p i c a l e l e c -t r o n i c energy l e v e l separation. (Their lowest order c a l c u l a t i o n i s e s s e n t i a l l y the same as that of Lax and Burstein). (b) The Condon Approximation (Dexter(1958) p.367). If we replace f-f by t( and the exact eigenstates of H by the states (a-3), the Bethe-Sommerfeld formula I(A-12) becomes (rCw^wizzz K & K * * * . * t ) | M |x«,,i|>>lix>|*. A A The quantity i n general depends on ^ and ^ and hence i s an operator. The Condon approximation consists of ignoring t h i s bq ~ and h>\-dependence of (a-16). More pre c i s e l y , the Condon approximation corresponds to writing and i s taken to be independent of b^ and 6fc (see Dexter, 1958). 105. Since the Condon approximation as formulated above has meaning only within the framework of the adiabatic approximation, there i s not much point of discussing the v a l i d i t y of t h i s approximation i n the present problem. However, i n the case of weak electron-phonon i n t e r a c t i o n , i t i s possible to extend the d e f i n i t i o n of the Condon approximation to the case i n which the adiabatic approx-imation i s not v a l i d . Indeed, by treating the electron-phonon i n t e r a c t i o n Hep as a small perturbation of one can f i n d the exact eigenstates of H i n the form |X"««,i>$>>l&> n w v where / A ^ O H / ^ ) ^ a n d f ^ A ^ tend to the free electron state A iff ftnd the free phonon state \k^^ respectively^ i n the l i m i t of vanishing H « p • W e t h e n replace l A ^ b f ) ) by I Ax} b y l&x) a n d toy Bxt j the exact energy eigen-value of H > i n the formulas (a-15)-(a-18), We now want to examine the v a l i d i t y of thi s generalized Condon approximation. Since, however, the meaning of the average ^ | M A > ' ) i s n o t Quite c l e a r , we s h a l l replace t h i s average toy I fRPI* ( c« f. 1(2-13)) and write the generalised Condon approximation as ^ ^ l o t w ^ T O P $2,tf(*Pe"- e " ^ ' ): (*«•»•) We discuss the v a l i d i t y of t h i s approximation by comparing with the r e s u l t obtained from the exact Bethe-Sommerfeld formula: log. We f i r s t subsitute the following expansion (c . f . 1(2-13)) into (a-21): The r e s u l t i n g expression for (a-21) then consists of the sum of the d i r e c t term (the term with a factor J*) a Q d the inter= ference term (the term with a factor bfa'pwith xfe If we assume that the e l e c t r o n i c states are nondegenerate, we can ignore the contribution of the interference term near the sharp peak. The r e s u l t i n g expression then becomes of the following form: The Condon approximation as given by (a-20) can then be j u s t i f i e d , i f we can approximate ( ^ ' / A j J ^ / f ^ b y i n the equation (a-23). To examine the v a l i d i t y of (a-24), l e t us f i n d the e x p l i c i t expression for the state The state i s obtained from a unitary transformation of Ci^tOy^ and s a t i s f i e s the eigenvalue equation (07, Such a state can be obtained by writing where as Hep"*0 f and by deterraing ft by the condition that be diagonal i n the representation Ajf/o)^ ( c . f . (a-lJ,(a-4) and (a-10)). By using an expansion s i m i l a r to (a-11), we obtain R i n the f o r m where to the lowest order i n X Substituting (a-27) into (a-26) and then into (a-24), we get »t<o| Of/ tR e ' «A '°>e + ? { ? ( R " * ^  + *** ^  r x V ^" ^ V **** We then have, noting fcyy = 0 j I O S . + — Thus we have obtained a correction to (a-20) i n the second order i n X * In f a c t , i t can be shown that the second term i n (a-30) has an important contribution to the sharp peak near £j a 7V "7x. It i s th i s term which contributes to the breadth of the absorption l i n e (the l i f e - t i m e broadening}. Since i n the Condon approximation (a-20) we would have dropped t h i s term, we may conclude that i n the approach from the Bethe-Sommerfeld formula we would not have been able to use the Condon approximation i n the form used here. 109. APPENj)^X TO THE THESES, JJ, Mathematical Appendix to Chapter II (a) Derivation of the Equation for the Green's Function Substituting 1(2-18) into 1(2-7), we get where |-| i s given by (2-8) and (2-9). We calculate the commutation- r e l a t i o n s with the a i d of the following general r e l a t i o n s : IX Y, 2W] = XC^J w + Cx/?Jrw/ + 2C*w]Y + *X CY, wj Using (a-32) and the commutation r e l a t i o n s 1(2-10), we obtain - CTh-Th')*fty' H O » +vjf t 44 - ^ f c | « j f v i . Substitution of (a-33)-(a-36) into (a-31) d i r e c t l y gives the formula 1(3-9). The derivation of the equations for CE) and Pyy^ C£) i s completely analogous. Here we have to calculate the com-mutation r e l a t i o n s Usi-nj 1(2.10) > ore can rwmediately show tt|*J**',-H«+ Hp] « (T*-V " ^ f ) 4 ^ < V . The c a l c u l a t i o n of Zb^dfAp^ H*p3 i s s l i g h t l y involved. The basic r e l a t i o n s are D* a? av, bJ 4 - t ^ *£l ait a£ a t / H I . where i n the second equation i n (a-39) (and also i n (a-40)) we have used the r e l a t i o n did)' <ty - 0$ ( Si>\ - tn\ <£»>) the l a s t step being obtained from 1(2-12). Using (a-37)-(a~4l3, one can e a s i l y derive the equations 1(^-3) and 1(4-4). (h) Calculation of blktft (1(4-6)), From the d e f i n i t i o n , we can write We use' the expansion I (A-28a): It i s obvious from the d e f i n i t i o n of trace that •Tr ie - *"" It*!**} « 0 . ' . . 1 I 2 . Then we can write X we can wrixe Now from (a-37) we get Then we have On the other hand, from the property of trace, we have Substituting (a-48) into ( a - 4 6 ) , using (a-49) and then integrating over we get K W « r - * A t f * l J t a « , , ( r , r R * - , ) T r f « ^ A « « f « f J I 13. R e p l a c i n g Q* by f l f A - ^ ) ) and u s i n g we get W t W » + » t ftpMt-l)U'"•:<J In p a r t i c u l a r , fort\9)lf/ we have N* t« " "if ^ .V,-: • I APPENDIX TO THE THESIS.Ill Mathematical Appendix to Chapter III Ca l c u l a t i o n of 0^(1) In the simple hydrogenic model Let F\l£) be the simple hydrogenic functions. We expand (^£)Fju.C£) i n t h e spherical harmonics: where ( f, 8,90 a r e t h e spherical co-ordinates of £ and a, being the e f f e c t i v e Bohr radius. The function 0ty(%) can then be written as where d| i s the angle between T and { . We use the expansion z where a r e *^e spherical co-ordinates of £ , Substituting (a-57) into (a-56) and using the orthogonality M5\ r e l a t i o n s r T J r t k "lC»>t|A)<P ~ ^ ^ _ .£ Ci+M)' we f i n d I (fit*, * ) (°-*<1) where In p a r t i c u l a r Examples fO F^o - j ^ j f £~f£ i f t - i) I $ + | p;a-»)j We then f i n d ( * * &f „ <y= } The c a l c u l a t i o n of l©AK<A>)*can be pfcirtoirtasd by using the r e l a t i o n (a-58). ri8-BIBLIOGRAPHY Bardeen, J, and Shockley, W., 1950. Phys. Rev. 80, 72. Bichard, J.W., and G i l e s , J.C., 1962. to be published. Block, C. and de Dominicis, C., 1958. Nuclear Physics, T ^ 4 5 9 ' Bogoliuboy, N.N.., and Tyablikov, S.V., 1959. Dokl. Akad. Nauk. (USSRJ 126, 53. Born, M., an<i( Oppenheimer, J.R., 1927. Assn. Physifc, 84,457. i Castner, T.G. J r . , 1962. Phys. Rev. Lett. j|, 13. Colbow, K.., Bichard, J.W., and G i l e s , .J.C.., 1962. .to be published. Deigen, M.F., 1956." J.E.T.P. (USSR), 3_1, 504. 1957. Optika i Spektrosk (USSR) £, 587 Dexter, D.L., 1958. S o l i d State Physics (Seitz and Turnbull), Vol. 6, p.355. Hasegawa, H., 1960. Phys. Rev. .118. 1523. Herring, C. and Vogt, E., 1956. Phys. Rev. 101. 944. Hrostowski, H.J. and Kaiser, R.H., ,1958. J . Phys. Che®. (Solids) 4, 148. Kane, E.O., 1960. Phys. Rev. ,119. 40. 119. Kohn, W., .1957. S o l i d State Phys. (Seitz amd Ttarmbull) Vol. 5,p.257. Kondo, J . , 1960. Prog. Theoret. Phys. 24, 161. Kubo, R.f 1957. J . Phys. Soc. (Japan) 12, 570. Kubo, R. and Toyozawa, Y. 1955. Prog. Theoret. Phys.-L3, 160. Lax, M. and Burstein, E., 1955. Phys. Rev. -100, 592. Long, D. and Myers, J . , 1959. Phys. Rev. 115, ,1119 Luttinger, J.M. and Kohn, W., 1955. Phys. Rev. £8, 915. Newman, R., 1955 Phys. Rev. 99, 465 1956 Phys. Rev. 103. 103 Nishikawa, K. 1962a. Physics Letters, .1, 140. Nishikawa, K., and Barrie, R., 1962. B u l l . Am. Phys. Soc. Ser i e s . I I , £, 485. O'Rourke, R.C., .1957. U.S. Naval Research Laboratory Report No. 4975, p.207. Roth, L.M., 1960. Phys. Rev. ' 118. 1534 Sampson, D., and Margenan, H., 1956. Phys. Rev. 103, 879. Schechter, D., 1962. J.Phys. Chem. (Solids) .23, 237. 1 2 0 . Sommerfeld, A., and Bethe, H., 1933. Handbuch der Physik, Vol. 24, Part 2, p. 333. Ter Haar, D., 1961. Proc. Boy. Norwegian Acad. Sciences. Ter Haar, D. and Wergeland, H., 1961. Proc. Roy. Norwegian Acad. Sciences. Wilson, D.K., and Feher, G., 1961. Phys. Rev. 124. 1068. Zubarev, D.N., 1960. Usp. F i z . Nauk. (USSR), 71, 71. 

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