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Some higher-order processes in the optical absorption and emission by impurities in nonmetals Chow, Hau-Cheung 1977

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SOME HIGHER-ORDER PROCESSES IN THE OPTICAL ABSORPTION AND EMISSION BY IMPURITIES IN NONMETALS by HAU-CHEUNG CHOW B.S., University of Texas, Austin, 1968 M.S., University of California, Los Angeles, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA in the Department of PHYSICS A p r i l , 1977 Hau-Cheung Chow, 1977. 11 In present ing th is thes is in p a r t i a l fu l f i lment o f the requirements f o r an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I a g r e e that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying o f th is thes is for scho la r ly purposes may be granted by the Head o f my Department or by his representat ives . It is understood that copying o r p u b l i c a t ion o f th is thes is fo r f i nanc ia l gain sha l l not be allowed without my wri t ten permission. H. C. Chow Department of P h y s i c s The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 ABSTRACT The effects arising from quadratic electron-phonon interaction and anharmonicity on the optical spectra of imperfections in nonmetals are examined. The absorption or emission line-shape function is expressed in terms of Kubo's formula and is reduced to a trace over phonon states only via the application of the Barrie-Sharpe method. This trace is evaluated with the aid of quantum field-theoretic techniques. It is shown that both the quadratic electron-phonon interaction and anharmonicity modify the phonon spectrum, and the influence of this modification on the optical spectra is studied in detail with regard to position, intensity, width, Stokes' shift and mirror reflection symmetry " , iv TABLE OF CONTENTS Page Abstract . . . . '.. . . . . . . i i i Table of Contents . . . . . . . . . . . . iv List of Figures . . . . . . . . . . . . v i Acknowledgements . . . . . . . . . . . . v i i Chapter I : Introduction . . . . . . . . . 1 Chapter II : General Formalism . . . . . . 9 Chapter III : The Line-shape Function and the Higher-Order Electron-Phonon Interaction . . . 1 6 1 . Introductory Remarks . . . . . . 1 6 2 . The Use of a Canonical Transformation 18 3 . Simplification of ^ f / , * 1 ^ / ^ • The Contraction Method . . . . . . . . . 2 2 4 . The Line-shape Function . . . . . . 2 8 Chapter IV : The Line-shape Function and the Anharmonic Interaction . . . . . . . . . 3 1 1 . Introductory Remarks . . . . . . 3 1 2 . The Use of a.Canonical.Transformation 3 4 3 . Approximations. Green's Function Method 3 8 4 . The Line-shape' Function . . . . . . 5 1 Chapter V : The Impurity Absorption and Emission Spectra 5 2 1 . The Line-shape Function for the Basic Hamiltonian . . . . . . . . . 5 2 2 . The Weak Electron-Phonon Coupling Case 5 4 V 3. 4, Chapter VI 2. 3. Chapter VII 1. 2. 3. Chapter VIII Appendix A Appendix B Appendix C Appendix D Appendix E Appendix F Appendix G Bibliography The Strong Electron-Phonon Coupling Case The Mirror Reflection Symmetry . . . The Effects of Higher-Order Electron-Phonon Interaction . . . . . . . . . The Weakly Coupled Electron-Phonon Systems The Strongly Coupled Electron-Phonon Systems Discussion . . . . . . . . . The Effects of Anharmonic Interaction The Weakly Coupled Electron-Phonon Systems The Strongly Coupled Electron-Phonon Systems Discussion- . . . . . . . . . Summary . . . . . . . . . Justification of the Use of the Reduced Hamiltonian (4.28) . . . . . . Justification of the Decoupling Scheme Derivation of (6.13) . . . . . . The Time-Dependence of the Dipole Moment Operator . . . . . . . . . On the Diagonalization of the Hamiltonian as an I n i t i a l Step in Calculating the Line-shape Function . . . . . . . . . On the Significance of the Neglected Quadratic Electron-Phonon Interaction Terms . . . Comparison with Silsbee's Results . . . 57 58 60 60 66 69 72 72 77 79 82 84 86 89 92 94 96 102 104 v i LIST OF FIGURES Fig. 1 Main features of impurity absorption- and emission spectra 59 Fig. 2 Asymmetric broadening effects due to higher-order electron-phonon interaction (Case of Wy* > ) . . . . . . 68 Fig...3 Effects of anharmonicity on the weakly coupled electron-phonon systems . . . . . . . . . . . . 76 V l l ACKNOWLEDGEMENTS I wish to express my deep appreciation to Professor R. Barrie, who suggested this research topic. His continued interest and advice in connection with this work have been most valuable. I benefited from discussing my research projects with Professor B. Bergersen, Professor J. Eldridge and Professor L. de Sobrino, who sat in the Committee that oversaw my graduate program. I profited from discussions with Professor J. Bichard, Professor M. Bloom, Professor F. W. Dalby, Dr. G. Kirczenow and Dr. J . Rostworoski. For the time they spent on my behalf and for the knowledge and insight they shared with me, I am grateful. I also thank a l l of those whose help made i t possible for me to come to U.B.C. and complete studies here. Financial support by the National Research Council of Canada is also gratefully acknowledged. CHAPTER I : INTRODUCTION This work deals with higher-order processes in the optical absorption and emission by impurities in nonmetals. The physical systems under consi-deration may be illustrated by the following two specific examples: f i r s t , a Group V impurity such as phosphorus in an elemental Group IV semiconductor such as si l i c o n ; second, an excess electron bound to a vacant negative ion site in an al k a l i halide (an F-center). When their concentration i s sufficiently low (say less than 10*'' per c c ) , these impurities can exist in discrete electronic states and optical transitions can take place among these states in frequency regions where the perfect crystals are normally transparent. The reason for- giving these examples is that they i l l u s t r a t e two extreme cases that have marked differences in their optical spectra. Take the absorp-tion as an example. The observed absorption spectrum of phosphorus-doped si l i c o n is characterized mainly by a series of narrow lines (widths~meV) in the infrared, the spectrum of an F-center is usually a broad band in the vis i b l e with a bandwidth as large as a fraction of an electron volt. This difference is explained by the extent to which the process of absorption is influenced by the lattice vibrations. The impurity electron of phosphorus-doped si l i c o n is essentially subject to the potential due to the phosphorus ion, the roles of the lattice being the modification of the electronic mass to some effective value and the provision of a dielectric medium. The result is that the electron moves in a large orbit and interacts weakly with the lattice vibrations. Hence the absorption spectrum is similar to that of a hydrogen atom, namely a series of lines. For an F-center, the bound electron being highly localized in the v i c i n i t y of the vacant site exerts a large coulombic force on i t s neighbors and, therefore, a transition from one bound state to another is accompanied by a large number of lattice excitations, hence the broadening of the absorption spectrum. Summarizing, we deal with optical transitions in coupled electron-phonon systems and we make a distinction whether the system is weakly coupled or strongly coupled. The study of optical properties of a coupled electron-lattice system can be made from a number of different . approaches. By far the most popular is the so-called adiabatic approach. (Dexter, 1958; Pryce, 1966) This approach rests on two major premises: one makes assumptions on the wave functions of the coupled electron-lattice system (the Born-Oppenheimer approximation) and on the nature of the electronic dipole transition (the Condon approximation). The outstanding feature of this approach is that i t recognizes at a very early stage the most basic dynamic nature of the system. One disadvantage of this approach is that the calculational scheme i t proposes is d i f f i c u l t to perform, as may be seen from the sketch now given. For the system under consideration, the hamiltonian i s d . l ) H = K£ + V ( r , S ) * K; t V ( £ ) where 1^ , K; are the kinetic energy operators for the electron and ions, respectively, V(X,R) is the interaction between the electron (position r ) and the ions (positions R) , and U(R) ^ s t j i e potential energy of the ions, According to the Born-Oppenheimer approximation, the wavefunction of the system is taken as a product of an electronic part ^f^Li*.) and a lat t i c e part ^j|(£) , which are determined, respectively, by the following eigenvalue equations (1.2) C K e t Vftr,&) + Wl) 3 fylr,*) - ^ ( 5 ) ^ ( ^ ( 5 ) and (1.3) r r t ; t E ^ R ) 3 ^ ( * ) = Equation (1.2) is to be solved for every configuration of lattice coordi-nates. After the eigenvalues ^ *$) are determined for a l l possible confi-gurations, they are to be used to determine the lattice states via (1.3). To study the optical properties in the dipole approximation, one deals with matrix elements such as The so-called Condon approximation corresponds to the assumption that in (1.4) the dipole moment matrix element between the electronic states may be taken as independent of R. The next step is to perform a thermal average >on (1.4) and sum over the fin a l states subject to some energy requirement, to o b t a i n say the absorption s p e c t r a l d i s t r i b u t i o n . Quite apart from the d i f f i c u l t i e s a s s o c i a t e d w i t h the determination of the a d i a b a t i c s t a t e s , as o u t l i n e d above, f o r r e a l i s t i c s i t u a t i o n s , the o p t i c a l problem i s not very s u s c e p t i b l e to a n a l y s i s i n the a d i a b a t i c approach. Granting the Condon approximation and t a k i n g simple i n t e r a c t i o n s such as l i n e a r e l e c t r o n - l a t t i c e i n t e r a c t i o n and harmonic l a t t i c e p o t e n t i a l , the problem c a l l s f o r consi d e r a b l e mathematical i n g e n u i t y (Dexter, 1958; Pryce, 1966) . When higher-order processes a r i s i n g from anharmonicity or qu a d r a t i c e l e c t r o n - l a t t i c e i n t e r a c t i o n are to be i n c o r p o r a t e d , t h i s approach becomes unwieldy. I t has been recognized f o r many years ( f o r examples, S i l s b e e , 1962, 1963; K r i v o g l a z , 1964) th a t higher-order processes i n o p t i c a l t r a n s i t i o n s can best be handled i n the language o f second q u a n t i z a t i o n . These authors i n v e s t i g a t e d the problem o f o p t i c a l p r o p e r t i e s o f imp e r f e c t i o n s by combining the a d i a b a t i c approach and the use of c r e a t i o n and d e s t r u c t i o n o p e r a t o r s ; the c a l c u l a t i o n o f the moments or of the line-shape f u n c t i o n s o f the spec-t r a l l i n e s was then f a c i l i t a t e d by the method of ordered operators (Feynman, 1951), a method that had been found u s e f u l i n the work o f Lax (1952) on the same su b j e c t . One f i n d s , i n t h i s approach, t h a t i t i s necessary to use some e f f e c t i v e l a t t i c e hamiltonians i n v o l v i n g parameters not c l e a r l y r e l a t e d to or definable, by the o r i g i n a l h a m i l t o n i a n (1.1). Moreover, s i n c e two e f f e c t i v e l a t t i c e hamiltonians are g e n e r a l l y used i n p r a c t i c e (one as s o c i a t e d w i t h the ground e l e c t r o n i c s t a t e and the oth e r , the e x c i t e d electronic state) i t is not clear how results obtained in this approach can be extended to cover the cases where more than two, electronic levels are present. Perhaps even more important is the fact that one is constra-ined to work in the representation given by the adiabatic wave functions and this means that certain techniques such as canonical transformations have to be employed with a great deal of c a r e ^ . More recently Barrie and Sharpe (1972) adopted s t i l l another approach to the problem, in which no adiabatic approximation was made. Consider the hamiltonian (1.1). If an expansion is made on V(r,£) and "W?) about Ro , RQ being the equilibrium l a t t i c e positions in the absence of the impurity electron, the hamiltonian takes the following schematic form, (i.5) +||^-s*><- i i H r L ( » K H » ) + -- v ^ , K„ ~ ~ ~ >Q _ By very standard techniques (Schiff, 1968; Ziman, 1960), (1.5) may be transcribed into occupation-number representation of electrons and phonons, d.6) H - + •<- H f where Hc t and ^ are obtained," respectively, from the f i r s t , second and third lines of (1.5) and are ^ T h i s is illustrated by the controversies concerning the second-order corrections to electronic energy due to electron-phonon interaction. See Koehler and Nesbet (1964) and references cited therein. (1.7) . He = (i.8) fi«r = + + ^ ( w ^ U K ^ + Z111 K$ i j t t & j J L t v ^ j t b j + vi j iUI j - ) < f o • • • • CA+A") where the Greek index X labels the electronic state, the Latin index i labels the phonon state with momentum"!; and polarization , i means (""& > )» * is a coupling constant; the hamiltonian is e x p l i c i t l y given only up to quadratic electron-phonon interaction and cubic anharmonic terms. It is to be noted that , ti , b* , Vijk are la t t i c e vibrational elements defined in the absence of the bound electron, and , &A , &A are defined in the absence of lattice vibrations. Barrie and Sharpe (1972) used the Kubo formula (Kubo, 1957) for the adiabatic dielectric susceptibility and were able to calculate exactly the absorption and emission line-shape functions for a hamiltonian consisting of selected components of (1.6). Specifically, they took the hamiltonian to be d.io) H 0 - XT;-A\ t * Z I ( U * „ b ; ) a X + Z I^ki Their study had the same starting point as the work of Nishikawa and Barrie (1963) on the broadening of impurity spectral line in the weakly coupled electron-phonon systems, which topic necessitates the considera-« « -f the o „ . d l a g o n a l e l e c t r o n . p h o n o n i M m t i m i n a d d i t i o n t o Ho a s given by (1.10). However, by t r e a t i n g the problem i n v o l v i n g (1.10) e x a c t l y , B a r r i e and Sharpe were able t o p r o v i d e a u n i f i e d theory f o r both the weak and s t r o n g electron-phonon systems. I t can be shown ( B a r r i e , Sharpe and Jones, 1972) t h a t the study o f the s i m p l i f i e d h a m i l t o n i a n (1.10) a l r e a d y takes i n t o c o n s i d e r a t i o n the e l e c t r o n -l a t t i c e c o r r e l a t i o n s t h a t are covered i n the a d i a b a t i c approach. The purpose o f the present work i s t o extend the treatment o f B a r r i e and Sharpe (1972) by i n c o r p o r a t i n g h i g h e r - o r d e r processes. More s p e c i f i c a l l y , i t i s intended t o study the e f f e c t s o f q u a d r a t i c electron-phonon i n t e r a c t i o n and anharmonicity on the a b s o r p t i o n and emission s p e c t r a . Because o f the r a t h e r formidable complexity of these problems, i t i s found necessary to adopt a number of s i m p l i f i c a t i o n s and approximations, as w i l l be d e t a i l e d i n the l a t e r chapters.. B r i e f l y , we use s e l e c t e d terms i n the case o f q u a d r a t i c electron-phonon i n t e r a c t i o n and a model h a m i l t o n i a n i n the case o f anharmonicity. We then a v a i l o u r s e l v e s o f one v a r i a n t or another o f quantum f i e l d - t h e o r e t i c techniques (Bogolyubov, 1967; F e t t e r and Walecka, 1971; Zubarev, 1960) t o s o l v e the f i r s t problem e x a c t l y , and the second, approximately. I t i s our c l a i m t h a t d e s p i t e these approximations, the present study has captured the essence o f the problems. 8. In Chapter II, we present a general formalism that specifies the system and relates the Kubo formula to the line-shape function. It is further shown there that the method of Barrie and Sharpe forms a suitable starting point even for the present problems and that i t s application renders the calculation of the line-shape function to an evaluation of traces over the phonon states only. In Chapter III and Chapter IV, we r e s t r i c t ourselves exclusively to the detailed mathematical derivation of the line-shape function, including the quadratic electron-phonon intera-ction and anharmonic interaction, respectively. Chapter V describes the absorption and emission spectra for systems in the absence of quadratic electron-phonon interaction and anharmonic interaction. The effects of the latter, higher-order interactions on the spectra are then examined in Chapter VI and Chapter V I I , respectively, where comparison with other, related works is also presented. It w i l l be shown that the present study yields a number of new results which have not been obtained from either of the two other approaches described earlier. These results are summarized in Chapter VIII. CHAPTER II : GENERAL FORMALISM We begin with the relat ion between the absorption constant <T"(to) and the adiabatic d ie lec t r ic suscept ibi l i ty : (2.1) <T{o) = const co In, XM From Kubo's linear response theory, X(w) is given by •Kb (2.2) • /Ciw) = 4 , f t n \ **• e " W t " £ t <LMtt) M J> The notation is the same as in Nishikawa and Barrie (1963) and Barrie and Sharpe (1972) and is recapitulated below: V\ : the impurity electronic dipole moment operator ft) - e H e H : the hamiltonian of the system C Hit) M] = I (M. l t iH A -M.M ; W) , i = x, y, z < • • • > = - T V * e * H - • J / T V U " P H J ft = T ' ^B ' ^ ° l t z m a n n ' s constant, T : temperature We may also introduce the line-shape function L^ ) v ia (2.3) 0~(w) = const u) L (w) where, from (2.1) and (2.2), L(co) is given by 10. (2.4) L W » = 4 | i t e w t <CM10,MJ> -«> We now specify the physical system. The hamiltonian of the system is (2.5) H = H0 + H* with (2.6) U 0 = H E + H F + H E P He - l TA^ Hp = I ^ b U , and (2.7a) H' - I ? d W ^ t l4t0<^  or (2.7b) H' = H i %{\*+\>l)(M})^*^) ' j k J • Here He is the hamiltonian of the bound impurity electron when the la t t i c e is fixed at the equilibrium position in the absence of the bound electron; 6t* and CL^ are the creation and destruction operator of the bound electron in the state 7v and satisfy the fermion anticommutation relation. Hp is the hamiltonian of the lattice vibrations in the harmon approximation, in the absence'of the bound electron; b\\ and t; are the creation and destruction operators of phonons in the state «- and obey the boson commutation relations.. Hep is the linear electron-phonon interaction with the neglect of the off-diagonal terms which describe the 11. lifetime effects of the bound electron (Nishikawa and Barrie, 1963). As is shown in Barrie and Sharpe (1972) and discussed in Chapter V, the retained terms of W0 , which wi l l be designated as the basic hamiltonian, are adequate to reproduce the main features of the impurity absorption emission spectra. U as given by (2.7a) is the selected higher-order electron-phonon interaction terms that are bilinear in the phonon operators. It w i l l be argued in Chapter VI that these selected terms simulate the most significant effects expected from the quadratic electron--phonon coupling. As given by (2.7b) W , together with Wp t gives a model hamiltonian for the lattice that takes into account anharmonicity to the'cubic order. In the following we shall deal with (2.7a) and (2.7b) separately. The bound electron is assumed to be endowed with a dipole moment with the corresponding operator given by (2.8) M = IT ^4ax where , a c-number, is the matrix element of the electronic dipole moment operator between the electronic states X and A We further assume that only one electron may be bound at the impurity site, This means that in dealing with the electronic part of the problem, i t is necessary to consider only the subsp.ace spanned by the vacuum and one electron state, which may be done elegantly by introducing an appropriate projection operator ?. (Barrie and Cheung, 1966). For example, the line-shape function L(w) may now be written as +00 (2.9) Lcun = JyUt eiu)t <CMW, M]> -to * where (2.10) <A> « Tr ( P e~^H A } / Tr ( P e"^ } £ For our purposes we require the knowledge of only one"property enjoyed by £ , namely (2.11)' £ a j a + r The restriction to one bound electron offers a decisive advantage in the calculation, which has been exploited by Barrie and Sharpe (1972). We shall now recapitulate their method which reduces the calculation of l« (.<*») to a problem of evaluating traces over the phonon states only. One .'begins . with the observation that the hamiltonian (2.5) of the system is of the form (2.12) H « A t I B^ X where A and are functions of phonon operators o n l y . ^ With the notation (2.13) = A + ^Throughout this work similar notation to that of .Barrie'and -Sharpe (1972) w i l l be used as much as is. practical. 13. one expands, for an arbitrary c-number <*• , € into a series: (2.14) £" = . " . n . 4 - ^ , t n . • The precise form of is n o t given since i t w i l l not be needed. Upon using (2.14) in evaluating ensemble averages such as (2.10), the electronic part can be handled exactly because of (2.11), leaving only a trace to perform over the phonon states. In particular, (2.15) <wm> = u \\\ T v [ e * A - e  (2.15) may be further simplified by introducing the impurity distribution function (Barrie and Cheung, 1966) ^ which is given by (2-16) * A = <<«>> = T ^ C ^  3 £ T,LYPAJt IT r le'? Hs] r» 5 R -yielding (2.i7) <WM]>£ - n ny i^yt) - ^ i - o ] where (2.18) • Tyt) . £ - ^ ] Substituting (2.17) into (2.9) and making use of (2.19) P ^ M - ' i j ^ e i u t y t ) -to one obtains the line-shape function (2.20) U«0) = ^ i ^ f l ^ i - ^ r ^ N ] Eq. (2.20) has a simple interpretation. In the summand, F^M is the normalized line-shape function for absorption of a photon (energy u) ) leading to an electronic transition from state 7\ to and ^ ( - w ) is the normalized line-shape function for emission of a photon tO resulting in an electronic transition from state j*- to \. , or in more descriptive symbols, (2.21) L ^ M =: fyM , L ^ A ( - ) = ^ ( - ^ The total line-shape function L(UJ) is simply the sum of these individual normalized line-shape functions, weighted with the appropriate distribu-tion functions and the interaction strengths | M ^ 1 . In the next two chapters we shall be concerned with evaluating the function f^tt) defined by (2.18), which shall also be called the line-shape function since the knowledge of this function is equivalent to the knowledge of the normalized line-shape function l / y - ^ » o n e D e i n g t n e 15. Fourier transform of the other. In calculating (2.18) we shall from now on omit the subscript "ph" in the trace to avoid redundancy, as only traces over phonon states w i l l henceforth be encountered. 16. CHAPTER III : THE LINE-SHAPE FUNCTION AND THE HIGHER-ORDER ELECTRON-PHONON INTERACTION 1. Introductory Remarks In this chapter we restri c t ourselves exclusively to the derivation of the line-shape function ^ii) of (2.18) including selected quadratic terms in the electron-phonon interaction. To be specific, we take the hamiltonian to consist of (2.6) and (2.7a) , i.e., (3.1) H = He + H' It i s immediately observed that the quadratic electron-phonon interaction is incomplete in the sense that terms nondiagonal in phonons (those containing \>. ) or terms involving two creation (destruction) phonon operators (those containing b l ^ or b;\>j ) are not included. Inclusion of these terms would make the already complicated c a l c u l a t i o n ^ much more d i f f i c u l t to perform and, as w i l l be discussed in Chapter VI, is not expected to yield any significant effects that are not obtainable in the In essence the complication originates from the noncommutability of the linear and quadratic electron-phonon interaction terms in (3.1). This complication prevails both in calculating the line-shape function and in extracting physical information from that function. It is easy to show that the problem of including one and neglecting the other interac-tion can be tackled in a f a i r l y straightforward fashion. 17. present case. The restriction to the hamiltonian (3.1) enables one to solve for the line-shape function exactly, and is sufficient to simulate the situation wherein the frequency of a phonon mode depends on which electronic state is occupied. The problem of quadratic electron-phonon interaction was f i r s t discussed by Silsbee (1962, 1963), who did not include the linear part of the electron-phonon interaction. However, as w i l l be shown in Chapter V, the presence of the linear terms is crucial to endow the so-called zero-phonon line with i t s temperature-dependent factor and is also the origin of the so-called Stokes' sh i f t . As such, Silsbee's treatment is more relevant to the study of the Mossbauer line than to the study of optical spectra of imperfections, with which this work is concerned. The same problem was also studied using the adiabatic approach by Krivoglaz (1964), who considered a more complete hamiltonian than (3.1). Nonetheless, his treatment does not seem to contain the principal finding of the present study, namely, that the presence of higher-order electron-phonon inter-action causes an asymmetric broadening of both the line and band spectra. These points w i l l be discussed again in Chapter VI, where comparison with other studies can more appropriately be made. In this chapter we carry out the mathematical analysis to obtain the line-shape function. Having done the similar thing in the next chapter for anharmonicity, we return in Chapter VI and Chapter VII to a detailed discussion of the physical significance of the results. 18. 2. The Use of a Ganoical Transformation Turning now to the calculation of l^lt) , one f i r s t makes the identi-fication that, from (3.1) , (2.12) and (2.13), Now i t is a simple matter to see that from the requirement that the hamiltonian (3.1) be hermitean i t follows that the matrix elements "U^ii and "W^U must both be real and hence equal. Making use of this property and the boson commutation relations, one gets for (3.2) (5.3) H ^ = T A + 2U A i l + KZCftk + Vtf-bJ) + 1 ^ , 2 ^ ) bX-The line-shape function (2.18) is now seen to be formally equivalent to an ensemble average of e taken with respect to a f i c t i t i o u s system whose hamiltonian is . Such an average is most easily performed i f H A is diagonal. The diagonalization can be achieved by (2) introducing the following transformation v J (3.4) tK = L + 4L . B l = b ? t where (3.5) wL = u>. * 2U>;L (2) It is also advantageous to regard (3.4) as a change of variables, the use of which puts the electronic state JV in a more preferential position, i.e., a l l the phonon operators w i l l henceforth be expressed in terms of 6,* and only. 19. It is readily verified that the operators and themselves satisfy the boson commutation rules: and hence the transformation is canonical. The use of this transformation leads to (3.6) H A = \ t I ^ 6 t A 6 ^ r r i r A * i r i o ^ I J, / ^ % where (3.8) t A = Ta - k * z _ y _ v I^L 1 U J „ *• " A Substitution of (3.6) and (3.7) into (.2.18) leads to (3.9) y t ) = v ^ i t C v ^ - ^ p ^ l ^ - ^ f a } < f ^ f t ) > where (3.10) ^ - e e. r t ' (3.11) K M = ^ * * X B * (3.12) = I ^ M ; 20. and (3.14) <-•> = T r l e ^ ••• \ It is convenient to introduce a new operator 'f^W) via One readily obtains the differential equation satisfied by. f^^ >W The solution to (3.16) is simply (3.17) f It) « exf [-1 f v«) ax i where •• » b 0 and where T is an ordering operator which puts ^t* i )^.^ x »V- in ascending order of * i in going from right to l e f t . 21. It follows from (3.15) and (3.17) that (3.19) < y t ) > > = < e ' f e - V H ^ > K ) s , ^ <V»>, where one uses the shorthand notation = <e It w i l l be shown in the next section that <f (t)> has a very simple representation. 22. 3. Simplification of ^f^))^ . The Contraction Method. The purpose of this section i s to derive a much simpler version of <•? li)> > defined by (3.17) and (3.20) and given ex p l i c i t l y here, (3.21) Fi r s t , i t is desirable to establish that (3.22a) < T^(^V r J" 'V' t"^ r = ° ' l f n 1 5 ° d d ; and the sum of a l l possible products of /^-averages of pairs of , i = 1, 2, .... n, i f n is even. The above statements are reminiscent of the contraction theorem in the finite-temperature diagrammatic formalism (Fetter and Walecka, 1971). It is to be noted that the conditions under which the contraction theorem is valid are not the same as those encountered in the present case and that i t i s -therefore necessary to establish (3.22) ab i n i t i o . For n = odd, the proof is simple. In taking the trace of (3.22a), one f i r s t uses a representation in which ^^x^ is diagonal. The general state in that representation would be (3.22b) <TV, (3.23) 23. in which MAS is the eigenvalue of QjJ$iK • From (3.18), one sees that the traces of <T V L*) YuJ*0 • - involve in general a sum of terms such as T (3.24) <••• %-l e * v r * T B^BjV- |-^-> where (3.25) C = B; ^ - 6^ The matrix element (3.24) clearly vanishes when an odd combination of '" ^ S ^- n v°l v e cl - o^w t n e t r a c e is invariant under a change of representation and one is thus led to (3.22a) For n = even, i t is easily shown by virtue- of (3.18) that to establish (3.22b) is tantamount to proving the vali d i t y of the contraction theorem applied to { f LBv3 , . . ••• 6 „ "> - I t is expedient to f i r s t differen-* % / A y-. f-\ T \ / tiate with respect to a complex variable JLC*,^*-) the following expression: obtaining (3.26) 5 «fC X £ « f E- ? & B * 3 > 24 . where I- i 3 is a commutator. Solving the differential equation ( 3 . 2 6 ) , one obtains ( 3 . 27 ) - f C - I W ^ f P = e * W l & or, what is i t s equivalent, With the use of identity ( 3 . 2 8 ) the desired proof can be constructed in a manner similar to Goudin's proof of the generalized Wick's theorem (Fetter and Walecka, 1971 ). Let o ( 3 . 2 9 ) J l ; 0 / r ) = +rt(S^-«3, ) and ( 3 . 3 0 ) ^ to = ^ c - i ^ B ^ V T r [ « f [ - ? w ^ ^ : l Thus, clearly, where an operator without a superscript (±) sign can be either a creation or destruction operator. (This convention is used in this section only.) 25. One can rewrite (3.31) as The last term on the R.H.S. of (3.32) i s , as a result of the invariance of trace and property (3.28), T (3.33) M(to^ ~ a X ] = T r i 8 * t f t 3 ^ -Upon substitution of (3.33) into (3.32), one obtains Equation (3.34) follows from (3.32) because the commutators under the trace signs are c-numbers. In particular, the only non-vanishing c o e f f i -cients in (3.34) are of the following type: (3.35) - J j L _ M a „ ' S'» = >B+ f i- . 26. I t i s thus seen t h a t what (3.34) has achieved i s none other than o b t a i n i n g a l l the p o s s i b l e c o n t r a c t i o n s of w i t h the remaining o p e r a t o r s , i . e . , (3.37) <B^BJA„-OT = <£^<W*~B*>i <^<SA'"B-V One can repeat the argument f o r the averages such as B^^Bp.-- w i t h the ensuing r e s u l t that ^^^t^y*."" ^ i s the sum of a l l the p o s s i b l e products of >j*-averages of p a i r s of B ^ . Hence i t f o l l o w s t h a t (3.22b) i s t r u e . For n = 2m, m being a p o s i t i v e i n t e g e r , the enumeration of a l l terms i n (3.22b) when combined w i t h the proper i n t e g r a l signs i n (3.21) leads t o a product o f m i d e n t i c a l f u n c t i o n s , namely, W <TIUSIIUO> In a d d i t i o n , the number of p o s s i b l e ways of p a i r i n g 2m obj e c t s i s These c o n s i d e r a t i o n s and the-use of (3.22a) render a much simp l e r form f o r (3.21): 27. = « x p [ - A J A « , f i x k < T V (x,)V ( « . ] > ] The explicit evaluation of (3.38) w i l l be carried out in the next section. 28. 4. The Line-shape Function Upon combining (3.9), (3.19) and (3.38), one obtains (3.39) It remains to carry out the averages and the integration. From (3.18) and the circumstance that the reason for which is the same as encountered in establishing (3.22a), one gets (3.41) The integration is most easily performed using the transformation of variables: (3.42) U = , V* = *> + * i . 29. and the recipe (3.43) yielding (3.44) where use of (3.35) and (3.36) has been made. The average is evaluated, without loss of generality, in the representation in which §X§Ah is diagonal. In the notation of (3.23), the average becomes (3.45) ^jTl?w.-Kr>K _ — _ * IT L^ e Substitution of (3.44) and (3.45) into H vn i , i. ^ into (3.39) leads to the desired line-shape function: 30. 46) 31. CHAPTER IV : THE LINE-SHAPE FUNCTION AND THE ANHARMONIC INTERACTION 1. Introductory Remarks In this chapter we again re s t r i c t ourselves exclusively to the math matical derivation of the line-shape function ^U) when anharmonic interaction is included. The hamiltonian dealt with in this chapter now consists of the basic hamiltonian (2.6) and the anharmonic term (2.7b), that i s , (4.1) H = H e + H' vV - i l l v ^ a ^ K ^ ^ ^ C ) -The anharmonic part of the hamiltonian is not the most general possible. As i s well known, anharmonic terms appear when the lattice potential i s expanded in powers of ion displacements from equilibrium positions. Such an expansion can be carried out to any order, but in (4.1) only the cubi terms are retained. Further, when the ion displacement i s expressed in terms of creation and destruction phonon operators, the combinations generated are s t r i c t l y of the type ( ^ + b.^  ^  ) , where -| and <r are labels for the phonon crystal momentum and polarization branch, respectively. The hamiltonian H ' is somewhat a simplification of thi situation. Nevertheless, the hamiltonian (4.1) does incorporate the 32. essential feature of the coupling between the harmonic modes, a most important character known to be associated with anharmonicity. In view of our attempt to perform only an approximate calculation in the present case, i t appears that the simplifications adopted here do not entail serious errors in the subsequent discussion. The anharmonic hamiltonian (2.7b) is in fact the same one as has been used in a number of Raman scattering studies (for example, Hizhyakov and Tehver, 1967). It i s also well known (Maradudin and Fein, 1962; Cowley and Cowley, 1965) that , . at the microscopic level, the effects of anharmonicity are to change the frequency of the harmonic normal mode (the self-energy effect) and to impart to i t a f i n i t e decay time (the life-time effect). As such, the connection between anharmonicity and impurity optical processes-, which involve electronic transitions, may at f i r s t sight seem to be tenuous. This consideration and the realization that few problems dealing with anharmonicity can be solved exactly must undoubtedly have contributed to the circumstance that anharmonicity is generally neglected in the study of impurity absorption and emission, even though no crystal is harmonic. A notable exception is the paper by Krivoglaz (1964), which purports to have included anharmonicity while dealing with higher-order electron- phonon interaction. The present work considers the case wherein the anharmonic term H' in (4.1) is sufficiently weak so that i t s effects can be regarded as a perturbation on the system described by the basic hamiltonian H . The 33. line-shape function is calculated here to second order in the strength of the anharmonic interaction V^k , via the use of a number of approximations stated in Section 3. It w i l l be shown that the line-shape function contains e x p l i c i t l y those features that describe the phonon self-energy and life-time effects and that i t leads to some observable effects in the optical spectra. These points w i l l be examined in Chapter VII, where comparison with Krivoglaz's work w i l l also be made. 34. 2. The Use of a .Canonical Transformation Beginning with the general formalism set up in Chapter II, we shall in this section calculate exactly to as far a stage as the hamiltonian (4.1) permits. In the notation of (2.12) and (2.13), one has for the present case, (4.2) H, - T, , + ^ ^ v ^ ^ i - J V ^ a ^ j In (4.2) only those terms that contain a single phonon operator depend on the linear electron-phonon interaction. It is a cardinal feature of this calculation that the linear electron-phonon interaction is handled exactly. This can be achieved by the use of the same canonical trans-formation as in Barrie and Sharpe (1972). One.introduces A A As may be readily verified, these new operators satisfy the usual boson commutation rules. The use of this transformation renders (4.2) a new form: (4.4) H A = -T; + ^ + H' where (4-5) f A = T,- K : I 35. (4.7) H = III V, j kC(6^B;)(B^Bp^Bjj + (V*->>V$)N* C 4 ; 8 ) = - * ^ C ) M Similarly, ' I I / ' t o . v;here ( 4 . i o ) y = kT(vr;"v^B^ +Upon substituting ( 4 . 4 ) and ( 4 . 9 ) into ( 2 . 18 ) one obtains ( 4 . 1 1 ) or ( 4 . 1 2 ) where (4.13) pt) = ^ ( ^ • f i ' l ^ U ^ H ' t V ^ and <4.14> .<..-> , - . T . ^ ^ ' - J / T,K^ 8''] 36. The' basic problem of finding ^M) is n o w seen to be that of evaluating the thermal average of f it") in a f i c t i t i o u s system speci-fied by the hamiltonian Vy+Vt . Comparison with the i n i t i a l problem may give the i l l u s i o n that the requisite average is actually more complex than the original problem. It must, however, be remembered that the use of the transformation (4.3) has effected an apparent disappearance of the linear electron-phonon terms, leaving the transformed hamiltonian to consist of an easily manageable part, , and a remaining part, h , each term of which is proportional to the strength of the anharmonic interaction , so that the p o s s i b i l i t y of a perturbation calculation now exists. This is clearly in line with the viewpoint enunciated .in the preceding section. It is now expedient to find an alternative version of " j ^ ^ Differentiating (4.13) and introducing one finds (4.16) -h^fjk) = ~l fa) f r ( l ) The solution to (4.16) is obtainable by iteration. In fact, i t is (4.i7) = r ^]4<,J^---]<K v ^ y t o - y * . . ) 0 0 0 and, hence, after taking the thermal average, one gets 37. (4.18) - Z where one has again used the time-ordering operator T 38. 3. Approximations. The Green's Function Method. In order to complete the calculation for , i t is necessary to evaluate the multiple-time correlation function as indicated in (4.18). Unfortunately, an exact calculation of (4.18) is out of the question and one must resort to a number of approximations for further progress. At this point some remarks regarding the behavior of the correspond-ing multiple-time correlation function for the basic system H q are in order. For the basic hamiltonian, one has (4.19) <frw> = | ! i " f l , f v - k a ^ w v ^ - v ^ 0 0 0 where Ij^lt) and ^ t t ) are obtained from (4.13) and (4.15), respectively, by taking the limit V-jk ='0 and where ^ is an ensemble average taken with respect to K A , as defined in (3.14). The multiple-time correlation functions in (4.19) satisfy the p r o p e r t y ^ : ; . "* r 0 , i f n is odd; (4.20) <TV^K)Ox t).-V' W > = j the sum of a l l possible products of / ' ' ' \ I A. -averages of pairs of , ^ i = 1, 2, .... n , i f n is even. This result is shown to be true in Sec. 3 of Chapter I I I , for the system including selected higher-order electron-phonon interaction terms, and is therefore true for the basic hamiltonian, a f o r t i o r i . 39. According to (4 . 2 0 ) , the phonon part of the coupled electron-phonon system H q may be viewed as propagating under the influence of a time C x n )-dependent force w (4.21) « e V ^ c The fact that the multiple-time correlation function can be expressed in terms of two-time contractions corresponds to the physical circumstance that the propagations of phonon states from one time to another are s t a t i s t i c a l l y correlated but propagations between one pair of times and another pair of times are uncorrelated, or, in the language of stochastic theory, the processes of propagation are Gaussian. This is not surprising since Gaussian processes are generally associated with long-ranged fluctuation forces (Martin and Schwinger, 1959) and here the "fluctua-tion forces" are provided by the independent la t t i c e oscillations and are clearly long-ranged in the time domain. Continuing in this language, one finds that i n the presence of anharmonic interaction the phonon propagations are subject to the fluctuation forces (4-22) ^ .^I^.^^M") Now, the effects of anharmonic interaction are to cause a complex frequency sh i f t a^iT in the time-dependent factors of (4.21), where A l4 and Vj , 40. both much smaller than ^  , describe the self-energy'and life-time effects proclaimed in Section 1 . Since the long-ranged nature of the fluctua-tion forces i s preserved in the present case., we expect the correlations most predominant to be those of the Gaussian type. That means the contraction theorem (4.20) should be valid, at least in the lowest-order approximation, i f one replaces V^L**) by y ( x „ ) . i f , in addition, the reasoning immediately preceding (3.38) i s repeated here, one obtains in place of (4.18) ( 3 ) t x, (4.23) < J J « > ' * <*p\- J^jix, < Y>,)^K)> } . 0 0 The remaining part of this section w i l l be devoted to the calculation of the two-time correlation function K.Y^(x,)V^(*0^ via the use of double-time Green's function method (Zubarev, 1960). The gist of the above-mentioned method is to make use of the so-called Zubarev's identity for the two-time correlation function (4.24) < B W A ^ > = U ( £M^A\ .•[«A18» -^.«AlBfcJ ^ y where for any two operators A and B, A q and B q are the diagonal parts of (^This result may also be surmised from the calculation in the latter part of this section. (3) v 'Compare Krivoglaz (1964) A and B with respect to the hamiltonian under consideration and where the Green's function ^ A l B ^ i s a two-branch analytic function of complex E defined everywhere outside the real axis by ( 4 . 2 5 ) . «A \ 6 » = ) r#i E i f . . J t t and obeys the equation (4.26) E « A \ B > > = S5 -<[A ,B3) + « r A , 3 t 3 | 6 » E Several remarks w i l l row be made before one continues the calcula-t ion. F i r s t of a l l , comparison of (4.23) and (4.24) reveals that one needs the Green's function ^^.J^)) which,.by virtue of (4.10), i s a l inear combination of Green's functions «^ .> lBj N >^ , «6<£lBj>^l , «Bi>l8jA^ > and <<6^IBjA'^ . In order to be consistent with the approximations leading to (4.23), only those Green's functions should be considered that are nonvanishing in the lowest order; a simple calculation shows that these are tfCB^Ig^>> and << 8^ .Secondly, the hamiltonian involved in solving (4.26) is that of the f i c t i t ious system and consists of KA+ fj' , as given by (4.6) and (4.7). However, as shown in Appendix A, under very broad conditions a perturbation calculation need only be performed with respect to the reduced hamiltonian given by (4.27) H,d ^ KA + H" 42. with t j k J » With these s i m p l i f y i n g remarks out of the way the c a l c u l a t i o n may (4) be resumed. From (4.26), one obtains It follows a f t e r some algebraic manipulation that (4.29) (E-W,)«W**»s + V M J M « W C n The twelve second-order Green's functions i n (4.29) themselves s a t i s f y equations, of which only the f i r s t one i s exhibited: (4.30) " £«UX> = i<^>$J> + «W^)%^\t >> or (4.31) ^ T h e s u f f i x E w i l l be dropped from the Green's function <JAI6>^ whenever no confusion can a r i s e . 43. One can write down the equations for the third-order Green's functions that appear in (4.31) and continue the process to obtain a hierarchy of equations. Instead, the standard decoupling method w i l l be applied at this stage. The procedure of decoupling is illustrated by the manner in which the Green's functions attached with ^ r in (4.31) are handled. These are (4.32) <<B-MX» + « + « B . X B J O + « M X I C» where one has adhered to the approximation scheme stated below equation (4.26) and made use of the relations (4.33) with (4.34) It is shown in Appendix B that the decoupling involved in obtaining (4.32) 4 4 . can be j u s t i f i e d i f terms of the order N or lower are neglected, where N is the number of the phonon states. Calculating a l l the Green's functions in (4.31) in the same manner and taking into account that ( 4 . 3 5 ) ' « y u i B + » = zii V^{iUAM0 + U^M-tf Following the procedure beginning with (4.30) and ending with (4.35), one calculates the other eleven Green's functions that appear in (4.29). When results such as (4.35) and i t s counterparts are substituted into (4.29), one obtains a f i r s t order algebraic equation, the solution to which is readily obtained: in the harmonic approximation, one gets for (4.31) + 1 1 ^ , ^ ( 1 + * ^ ) fW ry R + ^ ^ 3 (4.36) « JTT-where 45. (4.37) M ; ( w ) = I I I I I I \tJrit( - ^ i f f x S , x r P , „ r P , , 7 + L r [ ^ t ^ i ( i + ^ ) t < r J t ^ ( i t z D , ) t 5 ,^ . t ( i t » i ) ] * *it C J r 5U (>^ )^+ C£> v - + y~3 + C U s ^ O + ^ U ^ s ^ v ^ r ^ ^ 3 j + ur i^ , i - C t $Stk n+vj + £ t $ a + LitWl + ^[ >U + $»x l\At t»+yA) Jitter v J L 3 CSflr ^c t l l + ^ a) t <5*t £ i r One' then makes use of the identity (4.38) x i i > | * "f • where S stands for the principal value, to write (4.39) * 4 » T ^ >>> wherein 47. ( 4 . 4 0 ) ^ u n = yllllll ^Xt(^~l U U > > ^ t k U ' ^ + M t ^ l * h C 4 <L(i+zy„) t V W 1- ^  3 t £w C^ i^t V* t 5t ^  ^ 3 . + ^ JX<^ VN + 3 - £ ? j i 0+ i V r ) * £ t <T.y y t <£r £ , vTc 3 ?tfe £ C $ s («• ^ ) + ^ (i <- yIM) 3 ] K> + t j x^ wi r. Sit t &,r<^ ^ 3 + ^ C Snr Sxtiit)>n) + <Tmr £rtfm3 * t «£r Sib c i t v ) ) + ^ £ r ^ j 4^£^^(«t>) t)^<r ( 1^(i +vi)] } 48. + % [ 2 « £ m df-2>>) t £v(0 t & r £ , y„3 - Su* L cvs £.s (it ivy) * <L ve + & y4 3 49. Thus one gets (4.42> iVB*l$» 4-With the use of (4.42) and the Zubarev's identity (4.24) applied to (5) the present case o 1* ' s i (4.43) « W » - "-.( ^  t***> ^ M f o , , - * ^ 1 -Co C the correlation function <^>fcoB,^,60^  can be calculated by closing the contour in the lower half of the complex *>-plane i f > or in the upper half plane i f x^< x . The result is (4.44) < B j A K ) P ^ 0 > = 4j V,e where is the zero of A and >^  is given by (4.45) = v The diagonal part is here taken as zero as is the case when the hamiltonian is .A f i n i t e value of would imply the presence of a component in the two-time correlation function which i s proportional to £(*,-*»•) 50. In o b t a i n i n g (4.44) the usual assumption i s made that i s slowly-v a r y i n g i n the neighborhood of 10=0)^  . I n (4.44), as elsewhere i n t h i s work, (4.46) = - 1 where the c o n d i t i o n (4.47) ^ « ^ has been used. S i m i l a r l y , one gets and (4.49) < B j > ( ^ 8 ^ o > = = o With the a i d of (4.44), (4.48), (4.49) and the d e f i n i t i o n s (4.15) and (4.10), one gets The i n t e g r a l i n (4.50) can be evaluated a f t e r i n t r o d u c i n g the same chang of v a r i a b l e s as i n connection w i t h c a l c u l a t i n g (3.41), v i z . , u = x^ - x 51. and v = + x 2 . Then i t follows from (4.23) that (4.51) < f (« > a «f i" (5*A- < > J 4. The Line-Shape Function Upon combining (4.12) and (4.51) and allowing for the circumstance that t may take on negative as well as positive values, one gets the approximate expression for the line-shape function: (4.52) FT(t) = * r { i u V £ ) * f " »v. +o2 IM 52. CHAPTER V THE IMPURITY ABSORPTION AND EMISSION SPECTRA. 1. The Line-shape Function for the Basic Hamiltonian In this chapter we study the absorption and emission spectra for the basic system as described by the hamiltonian H q of (2.6), which includes only the harmonic part of the lattice and the diagonal parts of the linear electron-phonon interaction. As w i l l be shown, the study of this simplified system can already reproduce many of the observed features' of the impurity optical spectra. The line-shape function for the present cas e. c a n ] 3 e : obtained from (3.46) by taking the limit T^ML = 0 , and.noting that, from (3.5) and (3.8), CO.,. •= U). a (J Atlernatively, i t may be obtained from (4.52) by tak i n g the l i m i t Vjk = 0, and noting that, from (4.40) and (4.41), % = o) ( t y = 0 ) 53. Either limit leads to •i This expression is identical to the line-shape function obtained by Barrie and Sharpe (1972) obtained in a somewhat different way, and w i l l form the basis from which the main features of impurity spectra are derived. 54 . 2. Weak Electron-Phonon Coupling Case For systems such as s i l i c o n doped w i t h phosphorus, the i n t e r a c t i o n between the im p u r i t y e l e c t r o n and the l a t t i c e may be regarded as a weak p e r t u r b a t i o n . For such cases, one f a c t o r s out the time-independent p a r t of (5.1) and expands the remainder i n powers of the electron-phonon c o u p l i n g constant K . One then makes use of (2.21) and (2.19) t o o b t a i n the normalized a b s o r p t i o n line-shape f u n c t i o n as a s e r i e s : (5.2) L ^ u ) ) + A^C). + -where the bracketed s u p e r s c r i p t denotes the order of expansion i n H*~ . The f i r s t few terms of the expansion are e x p l i c i t l y : ( 5 .3 ) A " h M = «tI-|KIMill(2Vl«,)J J"(«VIf) For the case of a s i n g l e phonon mode the absorption spectrum comprises, according to (5.2) - (5.5), a prominent l i n e l o c a t e d at u) = L - T » and weaker l i n e s at frequencies , 2 ^ , ... away from e i t h e r s i d e 55. of the main line. , the latter corresponding to absorption or emission of one, two, ... phonons. These are generally referred to as the zero-phonon, one-phonon, two-phonon, ... lines. When the continuous phonon spectrum case is considered, the phonon lines merge to form one-phonon and two-phonon sidebands, as given by (5.4) and (5.5). In n^w) terms attached with 6 (^ >+T^ -T^  •* -dj ) or iiWr^ -T^ -w,-+a) have been omitted, since their contributions are of lower order of magnitude than those in (5.3) and (5.4), In exactly the same way one obtains the normalized emission line-shape function from (2.21), (2.19) and (5.1), (5.6) with (5.7) (5.8) • E r x M (5.9) £^ xl*) which describe the emission zero-phonon line, one-phonon, two-phonon sidebands, respectively. 56. It is of interest to note that the intensity of the most outstandi zero-phonon absorption or emission line depends on the linear electron-phonon interaction via the exponential factor (5.10) *^\-l*JkVil u > » + o } In.addition to i t s dependence on the linear electron-phonon interaction the factor diminishes as the temperature is elevated, i t being also dependent on phonon population via >^  . It transpires that this so-called "background" or "Debye-Waller" factor has a rather interesting origin, which cannot be discerned at this level of approximation. This factor w i l l be discussed again in the next chapter. 5 7 . 3. The Strong Electron-Phonon Coupling Case For systems such as an F-center in an alkali-halide, the interaction between the impurity electron and the lattice vibrations must be regarded as strong. For such cases, the expansion method used in the preceding section is no longer legitimate. Instead, o n e uses Toyozawa's method (Toyozawa, 1967) which amounts to expanding e in (5.1) and keeping terms quadratic in t. Within this approximation i t is readily found, for the normalized absorption line-shape function, that and, for the normalized emission line-shape function, that ( 5 . 1 2 ) ll ( „ , = [ ^ f u . j . ( ^ V V t ^ l f c - f c l V ) ' - 1 i Expressions (5.11) and (5.12) are two Gaussian curves with halfwidth ( f u l l width at half maximum) Z^Kl Z ^ l ^ - ^ ; ! * " l'*" a n d peaked at (5.13) <0 = y% t**%ViW and y\ - X K ^ l V for absorption and emission, respectively. The separation between the absorption and emission peaks, I Z^I^-'VM f -^*' in magnitude, is the so-called Stokes' shift. Clearly, the Stokes' shift depends on the electron-phonon interaction. 58. 4. The M i r r o r R e f l e c t i o n Symmetry I t i s f i n a l l y of i n t e r e s t to observe the s o - c a l l e d m i r r o r r e f l e c t i o n symmetry ( B a r r i e , Sharpe and Jones, 1972). This property s t a t e s that the absorption spectrum i s the exact m i r r o r image r e p l i c a of the emission spectrum r e f l e c t e d about the zero-phonon l i n e l o c a t e d at WsT^-T^ This may be e a s i l y v e r i f i e d , f o r both the weak and strong electron-phonon coupling cases, v i a comparison between (5.3) - (5.5) and (5.7) - (5.9), and between (5.11) and (5.12). The r e s u l t s of the present chapter are s c h e m a t i c a l l y summarized i n F i g . 1. Taken together, they a f f i r m t h a t the main observed fe a t u r e s of the i m p u r i t y a b s o r p t i o n and eimssion s p e c t r a may i n f a c t be reproduced by c o n s i d e r i n g the b a s i c h a m i l t o n i a n H q . The purpose of the f o l l o w i n g chapters i s t o examine the e f f e c t s of higher-order processes on the s p e c t r a , w i t h regard to p o s i t i o n , shape, i n t e n s i t y , widths, Stokes' s h i f t and m i r r o r r e f l e c t i o n symmetry. 59. (a) 1. Main Features of i m p u r i t y absorption and emission s p e c t r a . (a) Weak electron-phonon absorption spectrum ( s 0 ] ^ d - l i n e , case o f a s i n g l e phonon mode; dotted l i n e , continuous phonon spectrum) (b) Weak electron-phonon emission spectrum ( s o l i d . • l i n e , case of a s i n g l e phonon mode; dotted l i n e , continuous phonon spectrum) (c) Strong electron-phonon absorption band ( s o l i d . . l i n e ) and emission band (dotted l i n e ) 60. CHAPTER VI : THE EFFECTS OF HIGHER-ORDER ELECTRON-PHONON INTERACTION 1. Weakly Coupled Electron-Phonon Systems In t h i s chapter the e f f e c t s of higher-order electron-phonon i n t e r a c -t i o n on the i m p u r i t y a b s o r p t i o n and emission s p e c t r a are examined. One begins w i t h the case of weak electron-phonon c o u p l i n g , making use of (3.46). The r e s u l t s are more transparent i f one f i r s t s t u d i e s the case of a s i n g l e phonon mode, f o r which (3.46) reads (6.1) i>) « ^ p J ! t ( r - t ) - ^ - ^ - | V i - - j •> ,[ ' , ] As i n Chapter V, the normalized absorption line-shape f u n c t i o n i s expanded as a power s e r i e s i n K l and w r i t t e n as The zero-phonon c o n t r i b u t i o n /l^(w) may be obtained, as i n the previous case, by t a k i n g the F o u r i e r transform of the "background" f a c t o r of i; it) , i . e . , few (6.3) = \fj^ ^ \M^r)-,f4-^[-~l + •__ I, « [ — l - y ^ -1 61. Upon the use of (6.4) * K» and where (6.7) (6.5) ! + ! = | + (6.6) ' - e r 2. g ' C A 2 a £ (6.3) becomes (6.8) A M « e ' 4 - y - j ^ « C P [ - - 3 C U ] With the use of the notation (6.9) Jl = to a closed form for (6.8) is possible i f the values of the following integrals are noted: 62. (6.10a) [%4at - ^ (6.10b) +09 (6 10c) ( ^ = t ^ / ^ ^ V ^ + 3 e 3 ^ ^ * 5 i 3 i A - 3 H . O 7 ' 4- jeT^  <T(^  + -.it (6.10f) S u b s t i t u t i n g (6.9) and (6.10) i n t o (6.8), one obtains a f t e r c o l l e c t i n g terms, -U-to) py<K { *• e ¥ w-> CI t i u t k ) + ir (-2^«) X 3 cT(^vy t i S ^ - a w ^ ) . . . } 63. o r (6.12) K'(W) = e'^'Z' £ ± ± . Ln(*^) £Cu+vV*V^V W where L n (211^ ) is the Laguerre polynomial 1^(21^.) with = 0. The explicit forms of the few lowest order L r (21^ ) are given in the square brackets of (6.11). In Appendix C i t is shown that the one-phonon contribution to the spectrum is (6.13) A^M = v. | £ L. u-u;,) y) + f s^fe [ (I: LjfccLA) f k*yy »*• Z A j ; * ' Aw , (u) can also be obtained after some labor but w i l l not be given e x p l i c i t l y here because i t s weak intensity makes i t of l i t t l e practical significance. In much the same way the normalized emission line-shape function may be obtained: (6.14) M - E ( \ M t + + ••• /.^ X f*iS where 64. (6.15) £ > ) = e ^ I L B U & r i f t« * V V * V * V r net) /;„ ' (6.16) and *-(£ L j ( ^ v O ) t o y y "V J I t i s now p o s s i b l e to s t a t e i n words the e f f e c t s o f the in c l u d e d q u a d r a t i c electron-phonon i n t e r a c t i o n terms on the o p t i c a l s p e c t r a o f weakly coupled electron-phonon systems. The presence o f these terms simulates the s i t u a t i o n i n which the frequency of a phonon mode v a r i e s according t o which e l e c t r o n i c s t a t e o f the im p u r i t y i s occupied. In an o p t i c a l t r a n s i t i o n from s t a t e 'A to s t a t e ^ , f o r example, t h i s d i f f e r e n c e i n phonon frequencies manifests i t s e l f i n the appearance of a d d i t i o n a l l i n e s at i n t e g r a l m u l t i p l e s of -ui^") away from the zero-phonon and one-phonon l i n e s , f o r the case o f a s i n g l e phonon mode. The p o s i t i o n (to the r i g h t or l e f t of the o r i g i n a l l i n e ) o f these l i n e s depends on the r e l a t i v e magnitudes o f and Ktx , and the i n t e n s i t y f a l l s as £ L ^ u ^ ; ) . In the case of continuous phonon modes, the l i n e s i n the v i c i n i t y of the one-phonon l i n e coalesce i n t o the one-phonon side-band, but those • adjacent to the zero-phonon l i n e r e s u l t i n an asymmetric broadening of that l i n e . The reason f o r the above r e s u l t becomes c l e a r i f one takes (6.11) as a concrete case and considers the l i m i t of v a n i s h i n g q u a d r a t i c e l e c t r o n -65. phonon interaction. The f i n a l result of such a limiting process is of course to yield the zero-phonon absorption line (5.3), as i t must. It is , however, more illuminating to leave the result as a sum similar to (6.11) or (6.12). The result is (1(^ = 0) (6.17) fy. . e ^ * * £ | g - L ^ k M ) , According to (6.17),..the so-called zero-phonon line is made up of a l l those transitions between states A and , in which an arbitrary number of phonons may participate so long as the number of phonons before and after the transition is preserved. It is significant to note that i t is the t o t a l i t y of these transitions to which must be traced the origin of the so-called "background" or "Debye-Waller" factor (5.10),( and CO hence the latter is- somewhat a misnomer.) In the event that phonon frequencies associated with differenct electronic states are different, the aforementioned transitions, while conserving the number of phonons, w i l l require different amounts of energy and show up as the broadening of the zero-phonon line. By comparison between (6.12) and (6.15) and between (6.13) and (6.16), i t is clear that the mirror reflection symmetry discussed in Chapter V is no longer preserved since the skewness of the additional contributions from quadratic interaction is emphasized on the same side for both absorption and emission. As a secondary cause of the removal of this symmetry, one may note that the intensity of the absorption spectrum is now different from the emission counterpart. (^The equivalence between (6.17) and (5.10) in the case of a single phonon mode may be readily demonstrated by making use of the generating function for the Laguerre polynomials, (l-^)* 1 e*p[-xjt/(h£n = 2 L Wt"" > a n d letting 66. 2. Strongly Coupled Electron-Phonon Systems Turning now to the case of strong electron-phonon coupling, one makes use of (6.1) and (6.6) to get the normalized absorption line-shape function, for the case of one.phonon mode, V e*f*> Tat r r , ' " e y _ " 2 ^ — 'r exf - ^ L e^'^'V - l t -One may then employ a straight forward generalization of Toyozawa's method, as discussed in Chapter V, to expand the second exponent in (6.18) in a power series of t up to the second order. The ensuing integration may then be evaluated, yielding (6.19) where a. r i i r - A 6 4 - m £ u ) . v , _ _ _ _ _ t (6.20) - i According to (6.19) the absorption line-shape is a superposition of Gaussian curves peaked at ~£v"Tx t ^ H^+^w^c^O with exponentially decreasing intensities for increasing m. 67. Similarly, the normalized emission line-shape may be obtained with the result (6.21) L' M = f ±31 f 7 ^ h j J ^ y y t ^ i ^ ^ - ^ ] that i s , i t is a superposition of Gaussian curves peaked at ^ ~ ^ ~ ^ ; ^ + ^ t ^ - ^ ) . It i s not possible to obtain the exact shapes and the location of peaks of the absorption and emission bands without the use of physical parameters to perform a numerical calculation. Nevertheless two features clearly emerge in the present consideration. The peak positions are in general dependent on temperature in the present case. In the approximation where only linear electron-phonon interaction-is taken into account, the peak positions are independent of temperature (See Eq. (5.13)). The present result i s closer to the experimentally observed feature associated with F-centers. Secondly, since both (6.19) and (6.21) are superposition of a number of Gaussian curves with uniformly displaced peaks, the absorption and emission bands are asymmetric, with the asymmetry emphasized on the same side. The last property, together with change in intensity and half-width, causes a removal of the mirror-reflection symmetry. In Figure 2 the asymmetric broadening of the line and band spectra is schematically exhibited. 68. (a) I (b) * * V \ 4 f * # \ \ V * V 1 1 / * 1 / 1? (c) , / ' \ / \ ! ! \ F i g . 2. Asymmetric broadening e f f e c t s due to higher-order e l e c t r o n -phonon i n t e r a c t i o n (case of ^ > uj.^ ) (a) Weak electron-phonon absorption spectrum ( s o l i d l i n e , case of a s i n g l e phonon mode; dotted l i n e , continuous phonon spectrum) (b) Weak electron-phonon emission spectrum ('solid ~ l i n e , case of a s i n g l e phonon mode; dotted l i n e , continuous phonon spectrum) (c) Strong electron-phonon a b s o r p t i o n band ( " s o l i d l i n e ) and emission band (dotted l i n e ) 69. 3. Discussion Several remarks w i l l now be made in connection with the present study of the effects of quadratic electron-phonon interaction on the impurity spectra. First of a l l , the preceding calculation clearly indicates that the asymmetric broadening effect arises from a multitude of transitions involving phonons of varying frequencies. That this i s a . natural and necessary consequence of the quadratic electron-phonon interaction can be ascertained by studying the time-dependence of the dipole moment operator, for a coupled electron-phonon system with only the quadratic coupling terms, as is carried out in Appendix D. It should be emphasized that while that calculation affords further evidence with regard to the validity of the present conclusion, the method is not useful in calculating the line-shape function for the general case including the linear electron-phonon interaction, because i t i s not possible to obtain an explicit form for the time-dependent dipole moment operator in general. A related d i f f i c u l t y is met in calculating the line-shape function when one f i r s t diagonalizes the hamiltonian. This is because in the representation in which the hamiltonian i s diagonal, the dipole moment operator becomes very complicated. This i s demonstrated in Appendix E. Second of a l l , the considerations of the preceding sections are based on a rather restricted set of quadratic electron-phonon interaction terms (namely those of (2.7a) ). It is shown in Appendix F, that the only addi-tional effect that can be el i c i t e d by including the omitted higher-order 70. terms is' the generation of two-phonon processes in a single absorption or emission act, without any significant consequences as far as the spectra are concerned. For, these processes in principle contribute, in the case of weak electron-phonon systems,to the two-phonon sidebands, which are generally too weak to be of practical interest. In the case of strongly coupled systems, these two-phonon processes can alter the amount of Stokes' sh i f t , which fact by i t s e l f , however, can hardly be used in practice to indicate the significance of quadratic electron-phonon interaction. For these scanty additional items of information, the calcu-lation with the f u l l quadratic electron-phonon interaction becomes prohi-b i t i v e l y cumbersome; i t is neither solvable exactly nor instructive in elucidating physical consequences. It appears, therefore, that the advan-tages of restricting to those selected terms are unequivocal. Finally, i t is necessary to compare brie f l y other works bearing on the same subject. Silsbee (1962, 1963) considered an adiabatic hamiltonian for a two-level system, taking the difference of " the ground and excited (electronic) states to depend quadratically on the phonon operators. This situation is realized in dealing with the second-order Doppler shift in the Mossbauer problem. With regard to the optical problem of imperfections, his case roughly corresponds to the neglect of the linear electron-phonon interaction which, as detailed in Chapter V, is responsible for the temperature-dependent intensity of the zero-phonon line, for the occurrence of the one-phonon sidebands, and for the magnitude of Stokes' sh i f t . Despite this circumstance, i t is interesting to demonstrate the equivalence of the present results with v>. Silsbee's, in the appropriate limit. This demonstration is carried out in Appendix G. Krivoglaz (1964) also considered an adiabatic-type hamiltonian for a two-level system, taking the difference of the two electronic states to consist of a f u l l quadratic electron-phonon interaction in addition to a linear electron-phonon interaction. The calculation of the line-shape function in that work made use of the method of ordered operators (Feynman, 1951) and included up to the second order in the electron-phonon interaction terms. Krivoglaz's results, however, embodied nothing whatsoever about the variation of phonon frequencies with electronic states nor the asymmetric broadening. In addition, his results, in the .limit of the neglect of linear electron-phonon interaction and off-diagonal quadratic electron-phonon terms, disagree with those of Silsbee or the present study, as they should. 72. CHAPTER VII THE EFFECTS OF ANHARMONIC INTERACTION 1. Weakly Coupled Electron-Phonon Systems In this chapter the effects of anharmonic interaction on the optical spectra of impurities w i l l be examined. The treatment here is parallel to the preceding chapter and is based on (4.52). One begins with weak electron-phonon coupling limit, considering f i r s t the case of a single phonon mode. For this case, (4.52) reads (7.1) )3 where (7.2) (7.3) M / I I 1 A One develops a power series in K1" for the normalized absorption line-shape function L A - f : (7.4) L L ^ = 'O') + A^rM + It is not d i f f i c u l t to obtain the expression for the general term in (7.4), but of practical interest are only the f i r s t two which can be calculated 73. via the method used in Chapter V. The results are and J . j S , C i + % )  Similarly, one has for the normalized emission line-shape function (7.7) L^(w) = E^(u») + E^M 1; E^U0 + ... with (7.8) Erxl«) - ^ L - ^ ^ , , ) ] ±  and (7.9) ir > By comparison with the results of Chapter V, i t is seen that the 74. the zero-phonon line and the one-phonon lines each suffer a displacement from the original position by an amount ^ K ^ T > with the displacement being opposite for absorption and emission spectra. A l l the lines are now broadened; the zero-phonon line now has a width «- 1£*MU^)^ and the one-phonon line twice that width. The shape of a l l the spectral lines are Lorentzian within the approximation that 2^ and X< are slowly varying functions of W For the case of continuous phonon modes, similar calculation leads to the counterparts of (7.5) - (7.9): (7 . 10 ) £LM = fl*fC-2*jtfl$+i)] I ^ (7 .11 ) <>) = ^ r[-lV^ -3 ] | i^i;y, + ^uv,+o?,] \ } (7 . 12 ) E^luO = ^ C - I ^ ( ^ - t ' ) ] i & and (7.13) E*' M = w p t - X ^ M ^ - H ) ] i r ^ [ X . ^ ^ i U « t i ) ^ ] [ i where (7.14) = I U(WkiA. 75. and (7.15) ^ = I tj^  (*£+.)£ It is clear from (7.10) - (7.15) that for continuous phonon spectrum, the absorption and emission zero-phonon lines suffer an opposite displace-ment from y ^ by A^y. . The spectral shapes are Lorentzian with width y * . From (7.14) and (7.15), both the shift of peak position and the width are the sum of the corresponding quantities obtained for individual phonon.mode Cases. The one-phonon lines now merge into phonon sidebands so that i t is not possible in the present case to discern the broadening of the individual one-phonon lines. Finally, i t is evident from the expression given above that the absorption and emission spectra enjoy the mirror reflection symmetry with respect to , as in the case without anharmonicity (Chapter V). The effects of anharmonicity on the spectra in the weakly coupled electron-phonon systems are shown schematically in Figure 3. (a) t-T-3. y * r V T„-T«yvJ. 00 /I 1 1 1 1 1 \ j 1 >\ t 1 \ ; VV _« AV 3. Effects of anharmonicity on weakly coupled electron-phonon systems. (Solid line, case of a single phonon mode; dotted line, continuous phonon spectrum.) (a) Absorption; (b) Emission. 77. 2. Strongly Coupled Electron-Phonon Systems In the strong electron-phonon interaction limit, one applies Toyozawa's approximation to (4.52),as was done in Chapter V. The resulting expression reads (7-16) r It) & ^ i i t ( v f r - Z K ' l ^ l V ) " 2hltl ' i?Ztyr*t(**+» } From (7 .16) , (2.21) and (2 .19) , one obtains the absorption band after an integration where (7,18) u f c ~ u> r%-fr - p ' l ^ l * ' * (7.19) o( = AVl x (7.20) J * K^-Vfcftefc + i) and (7 .2 i ) g,fc^) = j= j e ^ t = i - Erf (fr) 3 78. The normalized emission spectrum ly^(w) may be similarly obtained and in fact differs from (7.17) only by a replacement of J^*. by U)e , which is (7.21) K)c = " ^ f - f . - 2 ^ / ^ ' Since the numerical value of oi is typically small compared with other parameters in strongly coupled systems, the absorption and emission spectra are s t i l l Gaussian in shape, with their peaks determined by the zeroes of ^ and , respectively, and with a halfwidth given by ^.^2.2*12. £K*H1;-Vx;1 U^+^l^ . By comparison with the results in Chapter V, i t may be concluded that the effects of anharmonicity on the spectra are very minute indeed. 79. 3. Dicsussion From the preceding exposition, one sees that the primary effects of anharmonicity are to renormalize the phonon frequency of a given phonon mode and to impart to i t a f i n i t e lifetime or, what is equivalent from the uncertainty principle, a phonon energy width. That one can make this identification in the present calculation is due to the fact that, within the framework of the approximations set forth in Chapter IV, the effective hamiltonian (4.27) for the ficticious system i s identical to a lattice hamiltonian with the original cubic anharmonic terms, so that the self-energy and = lifetime effects derived for the f i c t i t i o u s system are in fact the very same ones for the original l a t t i c e . Looked from mother viewpoint, this circumstance is equivalent to the statement that the approximations adopted in Section 3 of Chapter IV stipulate that the presence of the linear electron-phonon interaction does not alter the phonon spectrum. This is in agreement with the knowledge (Barrie, Sharpe and Jones, 1972) that the role of the linear electron-phonon interaction is to change the equilibrium positions of the lattice potentials and not the energy spectrum i t s e l f . In terms of the optical spectra, the only nontrivialVeffeet-arising from anharmonicity is the broadening of the zero-phonon line for the weak electron-phonon systems. This otherwise surprising result can now be understood easily in light of the discussion following Eq. (6.17). It is shown there that the zero-phonon line arises from a l l those transitions 80. in which- an arbitrary number of phonons may participate, subject only to the requirement that the phonon number before and after a transition remains the same. When the phonon energy is uncertain to some width, as is the case when anharmonicity is present, there is then some latitude in the frequency value about which these transitions can take place, hence the broadening. It is not surprising, on the other hand, that anharmonicity does not produce marked effects on the spectra in strongly coupled electron-phonon systems. The characteristics features of these systems, the band widths, Stokes' shift etc., are generally many times the magnitude of phonon energy, whereas the self-energy and lifetime corrections are typically small fractions of the phonon energy involved. Finally, i t is relevant to comment br i e f l y on the work of Krivoglaz (1964). Two cases were considered in that work. Fi r s t , the effects of anharmonicity were studied hand in hand with the quadratic electron-phonon interaction; here, the conclusion was that the presence of quadratic electron-phonon interaction accounts for the major broadening effect of the zero-phonon l i n e ^ . In fact, according to that author the line-shape function should be the same with or without anharmonicity. This result is already refuted in Chapter VI, so w i l l not be discussed further. Second, the effects of anharmonicity were studied along with linear electron-The effects on strongly coupled electron-phonon systems were not considered in that work. 8 1 . phonon interaction. The line-shape function obtained in this case is quite different from (4.52) but nevertheless led that author to conclude that anharmonicity broadens the zero-phonon line by an amount, which is comparable in structure to (7.15). His treatment in this case seems to suffer from two drawbacks. It i s not clear how j u s t i f i c a t i o n can be made of the neglect of most terms in the line-shape function, a step that is essential for his conclusion. In addition, the line-shape function derived by Krivoglaz does not have the same behavior in the limit of vanishing anharmonicity as the line-shape he obtained in the f i r s t case in the corresponding limit of vanishing quadratic electron-phonon interaction. Since the derivation in both cases was given only in sketch, i t is not possible at present to decide whether this unfortunate situation was due to some typographical error, mathematical s l i p , or the nature of approximations used in that work. 82. CHAPTER VIII : SUMMARY Without making the adiabatic approach,'.- ... the problem of optical absorption and emission by imperfections in nonmetals is studied with regard to the effects arising from quadratic electron-phonon interaction and anharmonicity;- . \; . The line-shape function for absorption or emission i s expressed in terms of Kubo's formula and is reduced to a trace over phonon states via the application of the Barrie-Sharpe method. For selected terms of quadratic electron-phonon interaction, the line-shape function is determined exactly (while the effects of including the neglected terms are examined); the principal method involved is similar to the contraction theorem in quantum f i e l d theory. In the case of anhar-monic interaction, the line-shape function is determined approximately) the primary step taken here is analogous to the Gaussian approximation in stochastic theory, followed by a Green's function approximation on the remaining two-time correlation functions. It i s shown, in the process of deriving the line-shape functions, that both quadratic electron-phonon and anharmonic interactions modify the phonon spectrum of the system. The former changes the phonon frequency value according to which electronic state i s occupied. The latter gives a f i n i t e width in addition to renormalizing the phonon frequency. In terms of the optical spectra, the effects may be summarized as follows. For weakly coupled electron-phonon systems (such as phosphorus i 83. doped silicon) the quadratic electron-phonon interaction gives rise to an asymmetric broadening of the so-called.zero-phonon line, and anharmonic interaction, a symmetrical broadening. This is because the zero-phonon line originates, as demonstrated e x p l i c i t l y in the present work, from a l l the phonon number-conserving transitions, whose energy conservation requirements are relaxed when the phonon spectrum is modified. For strongly coupled electron-phonon systems (such as F-centers) the absorption and emission bands are asymmetrically broadened, and have temperature-dependent peak positions, both owing to the quadratic electron-phonon interaction. Anharmonicity does not have pronounced effects on the spectra. 84. APPENDIX A : JUSTIFICATION OF THE USE OF THE REDUCED HAMILTONIAN (4.28) The purpose of this appendix is to show that the reduced hamiltonian (4.28) can be used in lieu of Kx. + H defined by (4.6) and (4.7), ( A - I D K» = lr ^fcXV CA'2) s' \ ^ J ^ * ^ ^ + ( * B :k f f) $in) x. where the phonon crystal momentum and polarization indices •kl° m are restored, A^-suffixes in the B-operators are suppressed, and where The terms independent of phonon operators do not have any effect on the line-shape function since they contribute the same factor to the denominator and numerator of'the ensemble average. The remaining terms of H treated as a first-order perturbation on in general have the effect of causing transitions among the eigenstates • l " - r i ( ? ( ^ - " ^ of KN . I 85. Now, the' contributions of the various terms of H' .to the transition  amplitude are additive so that their effects may be examined individually The terms linear in 6^ and 6^ give zero contribution because the transition matrix involves <--\<r —\^k<A"'\<r/-) o r <" *W"lO"'*\«> ") Similarly, the only contributions from terms bilinear in 6^ and arise from those involving ^ -^u,^  or ^ - 1 1 , 0 5 a n d the like and these given combinations are nonzero only i f M,- , ^ = °i . The asso-ciated anharmonic term VOtp- - k p \ i s however zero for Bravais lattice or for nonprimitive crystals with inversion symmetry (Maradudin and Fein, 1962) , and one does not expect the presence of an impurity ion or vacancy to change this drastically. (Local or resonant modes are not considered here.) One is thus le f t with terms cubic in . B ^ and 8 ^ in H', which, in the notation of the main, body of.text is (4 .28). 86. APPENDIX B : JUSTIFICATION OF THE DECOUPLING SCHEME In this .appendix the decoupling scheme leading to (4.32) is shown to be valid with the neglect of terms of 0(N 2) or lower, where N is the number of phonon states. The proof is similar to that of Nishikawa and Barrie (1963) and makes use of the relation (Zubarev, 1960) (B.i) «A IB» = 4 l \ ^ U*e'E <BA(*)> One need only consider the f i r s t Green's function ^wA^^J^^ of (4.32), the other cases being completely analogous.. The purpose of decoupling is to express ^ U ^ x ! ^ ^ i n terms of lower order Green's functions such as «BwX|6X;^ • 0 n e begins with CB.2) < i v \ M ; , > -<Tr\ e * B ; M ; i e e M*8t\ J where One now uses the relations for expansion of exponential operator functions 87. (B.5) *PVP" = Z torU,dut...(l*f Ht^ H'w-H"^ H \}\ (B.7) t ^ e " * * = Z C " ' / j^ . ^ . . . a u r H C - f y - - H"(-'H0R(-iH,) = where (B.8) RV) = ew** H"e"*^ and where the symbols \,^ \ , and £ designate that each \ \ is a sum of products of creation and annihilation operators 8* or B• attached with a factor V-^  and hence of the order of N 2. (Situations involving localized modes are not considered here.) Substituting (B.5) - (B.7) into the curly bracket in (B.2) yields One next applies the contraction theorem to (B.9) to obtain (B.io) < £ ' T . L c T ^ f J J J 7 I B X 1 3 J B ^ ^ since any other contraction not involving 6;^  would contain at least one 88. operator from , {^ } or ^ J and hence is of the order of or lower. Setting (B.10) into (B.2) and then into (B.l) leads' immediately to This is the decoupling scheme adopted in arriving at (4.32). Q.E.D. 89. APPENDIX C : DERIVATION OF (6.13) In this appendix the expression (6.13) for the one-phonon part of the contribution to the absorption spectrum is derived. This result is obtained by taking the Fourier transform of that expansion term of (6.1) which is proportional to K1" , and which may be written as ( c . l ) A ^ W J « I t ^ where (C.2) «i = Uo.;£- / - J Z E -^T and (C3) ^ = > -Consider f i r s t the integral in (C.2) and rewrite i t as -CBrO One may use formulae (6.10) to calculate each of the terms in (C.4). When these terms are calculated exp l i c i t l y , the integral (C.4) takes the form: 90. t f i u . I I ^ I 4- ... which upon substitution into (C.2) and regrouping leads to (C.5) where the definition for thn zero-order Laguerre polynomial L n(x) has been used. The integral (C.3) consists of two parts. The f i r s t part i s similar to (6.8) for A'" l1*) • In f a c t i t m a v b e obtained from (6.8) by changing ^ to w-uy and multiplying the result by U^i . Thus from (6.12) i t is The second part of (C.3) is similar to (C.2) except for a change of sign in one of Cc^ , hence i t i s , from (C.5) 91. Combining these two results one obtains for (C.3) (c.6) i = V e ~ ^ \ ! ^ ( ^ L ; ^ - ) j ^ ^ - y - > ^ V r t V Upon substitution of (C.5) and (C.6) into (C.l), one obtains (6.13). 92. APPENDIX D : THE TIME-DEPENDENCE OF THE DIPOLE MOMENT OPERATOR The purpose of this appendix is to show, via a study of the time-dependence of the dipole moment operator, that quadratic electron-phonon interaction is responsible for transitions involving multiple phonons of varying frequencies and hence is responsible for the asymmetric broadening effect. For this purpose i t is sufficient to consider a simplified hamiltonian without the presence of linear electron-phonon interaction: (D.l) H = IT ,«X + + UtUy^ + K*^)^ To study the time-dependence of the dipole moment operator one expresses the operator in» the Heisenberg representation (D.2) tilt) - e Me = I £ e Jk'flfoe. where T is the renormalized electronic energy, (D.3) f K = The expression (D.2) may be further simplified by making use of the formula 93. and the property that for n commutator (D.5) I ' - W i / a - ^ } = [ £ tv^K^f i f A =X(^^^iaW)^k ' 6 = a X' > a n d * V ( * + * M » i l . In arriving * r at ( D.5), the restriction that one deals with a one-electron system has been used. By substituting (D.4) and (D.5) into (D.2), the time-dependent dipole moment operator becomes It i s already clear from (D.6) that the permissible transitions, for the simplified system, are not only those involving electronic levels (sayA->)( ) but also those involving phonon number-conserving process (e.g. >iw^ -» nw^ < ). Owing to the energy conservation, the former processes require a set of discrete photon energy values (these are the differences between the , . electronic levels.) This requirement is relaxed, however, because the latter processes admit, a wide range of energy, with the overall consequence that the discrete spectral lines become broadened. It should be pointed out that the present calculation rests on the use of the simplified hamiltonian (D.l). If the linear electron-phonon interaction is to be added to (D.l), then i t is not possible to obtain a closed form for M(t), such as (D.6), so that the method is not useful in calculating the line-shape function in general. i 94. APPENDIX- E : ON THE DIAGONALIZATION OF THE HAMILTONIAN AS AN INITIAL STEP IN CALCULATING THE LINE-SHAPE FUNCTION The purpose of t h i s appendix i s to demonstrate that the c a l c u l a t i o n of o p t i c a l absorption and emission i s not n e c e s s a r i l y s i m p l i f i e d i f the hamilto n i a n of the system can be d i a g o n a l i z e d . The present d i s c u s s i o n i s e s s e n t i a l l y the same as may be found i n B a r r i e , Sharpe and Jones (1972). Consider the hamilt o n i a n I t i s p o s s i b l e to d i a g o n a l i z e ( E . l ) , i n the sense to be s p e c i f i e d l a t e r , -R w i t h the use of the t r a n s f o r m a t i o n e , where (E.2) R- = *U ( j ^ - ^ ^ V w i t h ( E . 3 ) = ^ + a25*i Under the transformation' e" R, the operators 4>. and ^  assume the form (E.4) ah = e \ e R = a * f [ ^ ^ j t - ^ - t ^ w . ; ' : ^ 95. and hence the transformed hamiltonian is (E.6) H = e * | u R ' wh ere the terms in fifi^fi. are neglected, as may be ju s t i f i e d within the one-electron approximation. Although (E.6) is not manifestly separated into dressed electron and phonon parts, i t i s diagonal in the representa-tion |l^l-V"> > where 11*) is the electronic'eigenstate "\ with eigenvalue ^Z^U'I'^'YHA ' a n d ^ i s t h e eigenstate of £ \ with eigenvalue % K and eigen-energy n^(Sf +)' . The dipole moment operator, on the other hand, is now much more complicated: ~ -<? ft (E.7) M = e It is the complication in (E.7) that makes this representation unsuitable for the calculation of optical absorption and emission. 96. APPENDIX F : ON THE SIGNIFICANCE OF THE NEGLECTED QUADRATIC ELECTRON-PHONON INTERACTION TERMS The discussion of Chapter III and Chapter VII was based on the hamiltonian which includes a selected set of quadratic electron-phonon interaction terms, the significance of the neglected terms w i l l be examined in this appendix. For simplicity, one begins with a simplified hamiltonian such as i.e. the coupling between electron and lattice vibrations is taken to be that part of the neglected quadratic terms that are diagonal in the phonon modes. Some comments w i l l be made on the general case of the f u l l hamiltonian later. Following the general formalism of Chapter II, one calculates the line-shape function (F.l) where (F.3) The calculation is fa c i l i t a t e d i f the operator-dependent part of (F.3) 97. is diagonalized f i r s t . This may be accomplished by the introduction of the Bogolyubov uv-transformation (Bogolyubov, 1967): where K-v, (real) and ^ K (complex) are parameters to be determined as follows. F i r s t , the condition that the transformation (F.4) be canoni-cal is imposed, i.e. (F.5) CVj* 1 = *j which leads to (F.6) = 1 Second, the transformed ^ is required to be diagonal; for that one f i r s needs the transformation inverse to (F.4). The latter may be obtained from (F.4) and (F.6) and is given by (F.7) The use of (F.7) and the diagonalization requirement lead to 98. (F.9) - W ^ v J + < 4 VA l i = o By solving (F.6), (F.8) and (F.9) simultaneously, one gets (F.10) %- h = cosk.?.^ (F.ll) IT. = M . iUCp l£* = & L <f>v where '(F.10) - (F.12) sp e c i f y the parameters needed i n the transformations (F.7) The transformed now reads (F.13) H> = T; + £ Z + I ^B+B wherein (F.i4) L ^ - ^ ^ n ^ A f t e r some straightforward algebra one f i n d s , s i m i l a r l y , 99. (F.15) where Ir I € £ € J Tr I e" Pi S'&h f and (F.17) y , ( ^ ) = " ^ V ^ + V ^ V ^ > It w i l l be observed that (F.15) i s mathematically akin to i t s counterpart (3.9). In fact, i f the steps similar to (3.10) - (3.20) are taken, (F.15) may be converted to (F.18) C Here (F.19) (F.20) and (F.21) = 2 ta 11=0 • e> o 100. wherein . It i s easy to show that the average \^ ><*''^ ,>^ )'"' i s i d e n t i -c a l l y zero i f n i s odd. For n even, one uses the Gaussian approximation as discussed i n Sec. 3 , Chapter IV. Thus, to a good approximation, (F.23) <?«-)> = e« i-iffi*,^ <L^ ,)LK)> j " Y o o T and, by some simple c a l c u l a t i o n , Without any further c a l c u l a t i o n s i t i s poss i b l e to deduce the physi-cal consequences of (F.18) and (F.24) by r e s o r t i n g to analogy with the case studied i n Chapters III and VII. B r i e f l y , the quadratic e l e c t r o n -phonon i n t e r a c t i o n s terms considered here give r i s e to a much more complicated modification of the phonon frequencies, which i n turn leads to an asymmetric broadening e f f e c t , a circumstance assured by the presence . ;tl(5.-^))B'B Av /fit o s of such factors as se- ' / and xO^^/ i n the line-shape function. The only a d d i t i o n a l e f f e c t that emerges here i s indic a t e d by the 1 0 1 . presence of factors e A' , which generate two-phonon processes in single absorption or emission acts. Their effects on the spectra are further discussed on pp. 69-70. The procedure outlined in this appendix can be used to study the f u l l hamiltonian that consists of the basic hamiltonian H of (2.6) and o v a l l the terms quadratic in electron-phonon interaction. The step which corresponds to diagonalizing (F.3) in this appendix, or (3.3) in Chapter III, would in the most general case necessitate the use of a transformation matrix, of which the transformation (F.4), or (3.4), is a special case (Bogolyubov, 1967). Such a calculation w i l l be prohibi-tively cumbersome and is not expected to incorporate any more effects not already covered so far, and hence i t w i l l not be pursued further. 102. APPENDIX G : COMPARISON WITH SILSBEE'S RESULTS The purpose of this appendix is to compare the results of the present study with Silsbee's (1962). As discussed in Sec.3, Chapter VI contact between the two works can only be made i f one neglects terms linear in electron-phonon interaction in the present results. Thus, from (3.46), one has the case of a single phonon mode, 1 ' r> = 0 / »-« where (G.2) T x = T , + U,. is the renormalized electronic energy and where the phonon mode index is suppressed. Hence the absorption line-shape is (G.3) U lw) - i e 1 &w+VTL+" w*-" wP/2 ^ ' Instead of the line-shape function, Silsbee calculated the f i r s t four moments. The moments taken with respect to * s easily t \\ obtained, the m moment being 1 0 3 . (G.4) ( 4t0 Iwtf^-f f M = R - " ^ * Here In p a r t i c u l a r , (G.6) <->N = n = ^ i t ~ 7 " The f i r s t f our moments, according to (G.4), are then Were the moments taken w i t h respect t o the bare energy d i f f e r e n c e ^ - 7 ^ ) the r e s u l t s of S i l s b e e (his equation (15)) are reproduced. 00 (G.5) 104. BIBLIOGRAPHY Barrie, R. and Cheung, C. Y. 1966. Can. J. Phys. 44, 2231. Barrie, R. and Sharpe, I. W. 1972. Can. J. Phys. 50, 222. Barrie, R., Sharpe, I. W. and Jones, B. L. 1972. Can. J. Phys. 50, 231. Bogolyubov, N. N. 1967. Lectures on Quantum St a t i s t i c s . (Gordon and Breach. New York.) Cowley, E. R. and Cowley, R. A. 1965. Proc. Roy. Soc. (London) A287, 259. Dexter, D. L. 1958. Solid State Physics Vol. 6_, edited by F. Seitz and D. Turnbull (Academic Press, New York). Fetter, A. L. and Walecka, J. D. 1971. Quantum Theory of Many-Particle  Systems. (McGraw-Hill, New York). Feynman, R. P. 1951. Phys. Rev. 8_4. 108. Hizhnyakov, V. and Tehver, 1. 1967. Phys. Stat. Sol. 21_, 755. Koehler, T. R. and Nesbet, P. R. 1964. Phys. Rev. 135, A638. Krivoglaz, M. A. 1964. Soviet Physics, Solid State, 6, 1340 Kubo, R. 1957. J. Phys. Soc. Japan, 1_2, 570. Lax, M. 1952. J. Chem. Phys. 2£ , 1752. Maradudin, A. A. and Fein, A. E. 1962. Phys. Rev. 128, 2589. Martin, P. C. and Schwinger, J. 1959. Phys. Rev. 115, 1342. Nishikawa, K. and Barrie, R. 1963. Can. J. Phys. 41, 1135. Pryce, M. H. L. 1966. Phonons in Perfect Lattices and in Lattices with  Point Imperfections, edited by R. W. H. Stevenson (Oliver and Boyd. London). Schiff, L. I. 1968. Quantum Mechanics, third edition (McGraw-Hill, New York) Silsbee, R. H. 1962. Phys. Rev. 128, 1726. Silsbee, R. H. 1963. Phys. Rev. 129, 2835. 105. Toyozawa, Y. 1967. Dynamic Processes in Solid State Optics, edited by R. Kubo and H. Kamimura (W. A. Benjamin, New York) Ziman, J. M. 1960. Electrons and Phonons. (Oxford University Press, Oxford). Zubarev, D. N. 1960. Usp. Fiz. Nauk. 71_, 71. (English transl. Sov. Phys. Usp. 3_, 320) 

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