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Temperature dependence of the energy gaps in semi-conductors Cuden, Ciril Bernard 1969

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THE TEMPERATURE DEPENDENCE OF THE ENERGY GAPS IN SEMICONDUCTORS by CIHIL BERNARD CUDEN D i p l . ing., Univerza v L j u b l j a n i , 1 9 6 6 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1 9 6 9 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agr e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s thes,is f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f S I C S The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada CONTENTS Page ABSTRACT i ACKNOWLEDGMENT i l l CHAPTER 1 INTRODUCTION 1 CHAPTER 2 STUDY OF ENERGY GAPS IN INTRINSIC SEMICONDUCTORS 5 Experimental Results 5 (a) Optical absorption 5 (b) H a l l constant and conductivity 6 (c) Optical and thermal gaps 7 Theory of Temperature-Dependent Gaps 8 (a) Thermal l a t t i c e expansion 8 (b) Thermal broadening of le v e l s 9 (c) Thermal s h i f t i n g of le v e l s 10 (e) V i o l a t i o n of momentum sel e c t i o n rules . . 11 (f) Creation of new states 11 Interpretation of Physical Significance of Temperature-Dependent Gaps. 13 CHAPTER 3 TEMPERATURE DEPENDENT ENERGY LEVELS l6 CHAPTER b TEMPERATURE DEPENDENT THERMAL GAP 29 (a) Hall constant 29 (b) System Hamiltonian 33 (c) Thermal averages for the number of electrons i n the conduction band and holes i n the valence band 35 (d) The c a l c u l a t i o n ofTfE) ^2 c CHAPTER 5 TEMPERATURE DEPENDENT OPTICAL GAP ^8 (a) The o p t i c a l absorption c o e f f i c i e n t . . . . ^9 (b) Discussion of the res u l t s with respect to the previous work 60 (c) Green's functions - nondegenerate case. . 6l CHAPTER 6 SUMMARY AND CONCLUSIONS 73' BIBLIOGRAPHY 77 Supervisor; Professor Robert Barrie i ABSTRACT The question to what extent the poles of the temperature dependent Green's functions have physical significance as temperature dependent energy l e v e l s Is considered. Our contention Is that such l e v e l s should not be thought of i n terms of i n t e r n a l energy or free energy. They have no physical s i g n i f i c a n c e other than perhaps to say how the s t a t i s t i c a l mechanical averaging was divided into i t s two steps. I t Is shown that the knowledge of the energy and l i f e time of quasi p a r t i c l e s i s not very h e l p f u l i n studying the H a l l Constant. To study the H a l l Constant one would have to use the appropriate two p a r t i c l e Green's Function and a number of approximations have to be made before one has a simple r e l a t i o n "between the H a l l Constant and the average occupation numbers fo r the q u a s l - p a r t l c l e s . (We have discussed the H a l l Constant since t h i s Is usually the technique discussed i n the l i t e r a t u r e . ) In the l i t e r a t u r e on the energy gap i n semiconductors, the single p a r t i c l e e x c i t a t i o n energies (mechanical quantities) were found. They were made temperature dependent by replacing phonon occupation numbers by t h e i r thermal averages, and I t was assumed then that the gap could be simply written down as a difference of two of these energies. I t was not c l e a r whether t h i s should be Interpreted as the o p t i c a l or the thermal gap. We have shown that the question, many times raised i n the l i t e r a t u r e , whether the two gaps are the same or not, i s not the relevant question at a l l . Due to the completely d i f f e r e n t physical nature of the Hall and absorption phenomena, the i i d e f i n i t i o n s of the gaps In both cases are e n t i r e l y d i f f e r e n t . We have shown that c e r t a i n approximations involved i n our c a l c u l a t i o n of the H a l l constant and o p t i c a l absorption co-e f f i c i e n t lead to the thermal and o p t i c a l gaps being the same. The o p t i c a l gap i s the same as the thermal one because our c a l c u l a t i o n of o p t i c a l absorption involved only the zero-phonon process, neglect of the vertex part of the corresponding two p a r t i c l e Green's Function, neglect of the damping and neglect of the frequency dependence of the l e v e l s h i f t . In summary, i t appears that the picture i n which one thinks of temperature-dependent energy l e v e l s and t r a n s i t i o n s between these i s often too misleading and that i t Is safer to carry out the c a l c u l a t i o n of macroscopic behaviour i n the correct fashion. One may by the l a t t e r method, obtain the r e s u l t one would have obtained using one's physical I n t u i t i o n , but the approximations f o r t h i s are now spelled out. i i i ACKNOWLEDGMENT I would l i k e to thank my advisor Dr. Robert Barrie for his constant assistance and patience he has shown through the entire period of my research. I am also indebted to the University of B r i t i s h Columbia for the University of B r i t i s h Columbia Fellowships. Part of thi s work was supported by NRC grant A - 7 1 3 . CHAPTER 1 INTRODUCTION The primary purpose of this thesis i s an attempt to cla r i f y the statement that the poles of certain temperature-dependent Green's functions are to be interpreted as temperature-dependent energy levels. We shall use as an example the problem of the temperature dependence of the energy gap in an intrinsic semi-conductor. This problem w i l l not be solved in complete detail, but a solution w i l l be found that illustrates the points we wish to make. Our study of this problem w i l l also enable us to correct erroneous statements which appear In the literature. Green's functions have been used frequently to study the dynamics of many-body systems. This problem i s purely mechan-i c a l in nature; the concept of temperature does not enter. For this mechanical problem one can show that the poles of the Green's function are related to the excitation energies of the many-particle system. We shall give here only the minimum necessary to appreciate this point; details may be found in ft) Schultz A Green's function is defined for t)>0 , as e(p.t) =-K€ l v t ' c > l € > where is the exact ground state of the N-particle system and i s the operator which, acting on an N-particle state, adds to that state a particle with momentum f . c T("t) i s the corresponding annihilation operator in the Helsenberg picture. The above may be written as z 6 ( 1 * ) - - I I 1^ ),/. e , P [ A(C - )t] («) where e.o Is the ground state energy of the N-particle system and the energy of the stationary state u of the ( N + l ) -p a r t i c l e system. If we write C-C = ( C ' - 0 + ( C - C ) - ( C- e . H) + C ' (») we see that In the exponential we have the exact e x c i t a t i o n energy £^ +' of the (N+l) - p a r t i c l e system. For a system of non-correlated p a r t i c l e s applied to<p 0 creates a state with addi t i o n a l energy , c denoting a single p a r t i c l e e x c i t a t i o n energy. For a system of correlated p a r t i c l e s , creates a l i n e a r combination of stationary states with a spread i n energy and we can talk about the probab-i l l t y £ (f/£)d£ that the state <P0 has a ce r t a i n energy l n the range £ » £+<=l£ . Making a Fourier transform on the time variable one can show that + other term a r i s i n g u» £ - ( £ N + 1 _ £ N + S) + ^  from case i n which 1'*' t<0 and the pole of i n the lower half-plane, considering £ complex, gives the energy and damping of the elementary 3 e x c i t a t i o n of momentum f . For metals £ N + I - £ N « s . £ N - but th i s i s not so for o o o semiconductors. For non-zero temperatures one can define a Green's function l i k e (I.) , but with the average rather t h a n ^ l n g ^ o v e r the ' - —-mechanical ground state being a statistical-mechanical average over a grand canonical ensemble. The poles of the Green's function now become temperature-dependent and i t Is assumed that these s t i l l have the property of being e x c i t a t i o n energies and are referred to as the temperature-dependent quasi-particle e x c i t a t i o n energies. These poles being temperature-dependent are i n t h i s sense statistical-mechanical quantities, but they r e f e r to a single p a r t i c l e and i t i s not clear that these micro-scopic quantities have a clear-cut physical significance In the in t e r p r e t a t i o n of a macroscopic measurement. Although temperature-dependent, they are not thermodynamic quantities. (2) Bonch-Bruevich and Tyablikov claim that Green's functions have led to the clearest understanding of the concept of temperature-dependent energy l e v e l s . We intend to show, r e l y i n g (3) heavily on the work of Elcock and Landsberg that Green's functions have not r e a l l y added to our understanding of th i s concept. To do so we s h a l l use the temperature-dependence of the energy gap i n i n t r i n s i c semiconductors as an i l l u s t r a t i v e example. In Chapter 2 we s h a l l present an outline of the history of the study of temperature-dependent energy gaps i n semiconductors as It i s relevant to our study. Chapter 3 contains a si m i l a r treatment of the concept of temperature-dependent energy l e v e l s . There i s l i t t l e that i s o r i g i n a l In these two chapters; the s i g n i f i c a n t o r i g i n a l contribution i s to point out misinterpret-ations that have appeared i n the l i t e r a t u r e . These misinter-pretations are concerned with the c a l c u l a t i n g of thermal and o p t i c a l gaps of semiconductors and with the thermodynamic significance of these gaps. (Chapter 2 & 3) The relevant one- and two-particle Green's functions f o r our problems are calculated i n Chapters k and 5 respectively. These are o r i g i n a l c a l c u l a t i o n s . Chapter 6 also contains mostly o r i g i n a l workj i t contains the conclusions we have drawn as regards the in t e r p r e t a t i o n of the poles of Green's functions and also a comparison of our thermal and o p t i c a l gaps f o r semi-conductors. 5 CHAPTER 2 STUDY OF ENERGY GAPS IN INTRINSIC SEMICONDUCTORS Experimental Results (a) Optical absorption In many semiconductors i t was found that the long wave-length l i m i t of the o p t i c a l absorption band s h i f t s toward shorter wavelength with decreasing temperature. The steeply r i s i n g edge of the absorption band s h i f t s without changing i t s shape. See F i g . ( l ) . co F i g . ( l ) Schematic representation of the absorption c o e f f i c i e n t versus photon frequency at three d i f f e r e n t temperatures. Near the threshold frequency the absorption c o e f f i c i e n t behaves l i k e where co(T) i s defined as the temperature-dependent energy gap. Sab The theory f o r a r i g i d l a t t i c e gave <T(to) oc (co - £° where £j i s the temperature-independent energy gap parameter, and was extended to {TC<oj oc (co — £ T simply by replacement of by a temperature-dependent energy gap £^  i n order to reach agreement with experiment. CE^ -Z-I ) • Different calculations to obtain £j were made and usually they lead to the l i n e a r approximation £ T = £° + oof (2.1) (b) Hall constant and conductivity The equilibrium concentrations of electrons n e and holes fv were determined from Hall and conductivity measurements. Schematic plots f o r U| RHl!«>^  ^  and lo^ej^) versus ^ are given i n F i g . (2),,(3) and (k). 11RH Fig.(2) Diagram for the l og of the Hall const, versus j/j . Fig.(3) Diagram for Fig.(^) Diagram for the l og of the cond- the log of the prod-u c t i v i t y versus l/r. uct of the electron and hole concen-trations versus It was found that the product of the concentrations f i t t e d a well known r e l a t i o n CO " . - T V - r C 2 T W k B T / ^ 3 . ( ^ ) % e - t f (2-3) i n many cases, but the values for the numerical factors preceding the exponential were much larger than those found by other methods. This discrepancy was explained as r e s u l t i n g from a change i n the energy gap with temperature. Namely the empirical value f o r \ fv could be brought into agreement with the the o r e t i c a l one by replacement of the temperature-independent energy gap parameter £^  with a temperature-dependent energy gap £ j . Experiments indicated that tj should be roughly of the form W i t h oO = JVIOCJOG' where <C} accounts f o r the proper numerical factor, (c) Optical and thermal gaps Methods based on the Hall constant and conductivity gave values for the energy gap which were smaller than those deduced from the o p t i c a l data. This smaller gap from e l e c t r i c a l measurements was taken to be due to the fac t that i n t h i s case the electrons stay In thermal equilibrium with the phonons whereas i n the o p t i c a l absorption technique the f i n a l state had a l a t t i c e " d i s t o r t i o n " . This " d i s t o r t i o n " arises from the fac t that the electron absorbs the r a d i a t i o n i n a much shorter time than the time of relaxation f o r electrons i n t e r a c t i n g with the phonons. The photon absorbed has then not only to excite the electron but also to provide the energy of thi s d i s t o r t i o n . The two gaps obtained by the above experimental-analytical techniques are referred to as the thermal gap (the e l e c t r i c a l measurements) and the o p t i c a l gap. 8 Theory of Temperature-Dependent Gaps In the l a s t two decades many measurements of the o p t i c a l absorption, H a l l constant and conductivity i n both I n t r i n s i c and e x t r i n s i c semiconductors were made. Several attempts to explain the temperature v a r i a t i o n of these quantities appeared i n the l i t e r a t u r e . I t has become Increasingly apparent i n recent years that the observed absorption spectra, H a l l constant and con-d u c t i v i t y are a l l intimately connected with the i n t e r a c t i o n between electrons and l a t t i c e v i b r a t i o n s . A l l explanations seem to have one common point; the Idea that the temperature-dependent energy gap, defined i n one way or another, Is responsible f o r the observed temperature dependence. Moreover, the same temperature dependence of the energy gap was supposed to be the cause f o r the temperature v a r i a t i o n of the above mentioned d i f f e r e n t physical phenomena. In the following we s h a l l outline b r i e f l y d i f f e r e n t interpretations of the temperature dependent energy gap appearing i n the l i t e r a t u r e which were supposed to explain both o p t i c a l and c a r r i e r concentration data at the same time. (a) Thermal l a t t i c e expansion (S) Mogllch and Rompe pointed out the e f f e c t of the thermal l a t t i c e expansion on the energy gap. Later t h i s approach was followed by Bardeen and Shockley. Temperature dependence of the gap was obtained i n terms of a deformation potential theory, under the assumption that the temperature v a r i a t i o n i s e n t i r e l y due to thermal expansion. There seems to be no known case where such a treatment would give adequate description of the experimental data. The e f f e c t i s f a r too small. (b) Thermal broadening of l e v e l s (7) Radkowsky has considered the e f f e c t of the broadening of each one-electron l e v e l caused due to the Uncertainty P r i n c i p l by the c o l l i s i o n s of the electron with the l a t t i c e v i b r a t i o n s . In polar c r y s t a l s where the 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 i s large, the scattering time, hence the l i f e t i m e of the states, i s small and th i s e f f e c t may be quite s i g n i f i c a n t . In order t get temperature dependence the quantum numbers for given v i b r a t i o n a l states i n the expression f o r the l i n e width were replaced by thermal average quantum numbers f o r the correspond ing states of the decoupled c r y s t a l . A schematic description of Radkowsky*s approach i s given i n F i g . (5.) T = 0 T ^ O F i g . (5) Schematic procedure of Radkowsky's treatment of the temperature dependent energy gap. £j?- Is then defined as the temperature-dependent energy gap. f indicates that the l i f e t i m e e f f e c t due to the electron-l a t t i c e i n t e r a c t i o n i s taken into account. This treatment turned out to be compatible with experiment to an order of magnitude f o r polar c r y s t a l s . For nonpolar c r y s t a l s t h i s 10 approach f a i l e d . The e f f e c t was found to be n e g l i g i b l e , (c) Thermal s h i f t i n g of l e v e l s (8) Fan has l a t e r suggested that the e f f e c t of the l a t t i c e vibrations on the energy gap should be treated on the basis of s h i f t i n g rather than broadening the energy l e v e l s . A v i b r a t i n g l a t t i c e not only scatters electrons, but also changes the electron energy, the energy change depending on the state of the electron. Thus, when an electron changes i t s state i n a t r a n s i t i o n from the top of the valence band to the bottom of the conduction band, the energy of the t r a n s i t i o n i s modified by the difference i n the int e r a c t i o n energies associated with the two states. Of course i t i s assumed that the l i f e times of the two states separated by the energy gap are long compared to the l a t t i c e v i brations. The temperature-dependent energy l e v e l s were again introducedtKr^replacement of quantum numbers f o r given v i b r a t i o n a l modes of the decoupled crystal,, by the mean occupation quantum numbers i n the shi f t e d energy l e v e l s . A schematic representation of Fan's treatment i s given i n Fi g . (6). F i g . (6) Schematic procedure of Fan's treatment of the temperature dependent energy gap. II i s temperature-dependent energy gap. s Indicates that the s h i f t of the energy l e v e l s due to the e l e c t r o n - l a t t i c e i n t e r -action i s taken into account. Fan's approach gave reasonable agreement for both polar and nonpolar c r y s t a l s . (d) Others Thereawere other e f f e c t s considered i n which the temperature dependence was introduced i n a si m i l a r way as before. For our discussion these e f f e c t s are not es s e n t i a l , however we s h a l l mention two of them j (e) V i o l a t i o n of momentum se l e c t i o n rules If the minimum of the conduction band occurs at a d i f f e r e n t value of momentum from that of the maximum of the valence band, a breakdown of the momentum sel e c t i o n rule allows low-energy t r a n s i t i o n s not otherwise possible. These non-v e r t i c a l t r a n s i t i o n s have been c a l l e d i n d i r e c t t r a n s i t i o n s . A treatment has been given of the e f f e c t of l a t t i c e vibrations on these t r a n s i t i o n s , i n which i t i s shown that absorption or emission of a phonon of suitable momentum enables these non-v e r t i c a l t r a n s i t i o n s to occur. Temperature dependence was obtained as before. This theory with s l i g h t modifications has been applied with at lea s t q u a l i t a t i v e success l n some of the semiconductors. (f) Creation of new states The e f f e c t s mentioned above were a l l computed using the elect r o n i c states which would exi s t i n the periodic non-vib r a t i n g l a t t i c e , emphasis being variously placed on the s h i f t i n g or broadening of the l e v e l s , or on the breakdown of sele c t i o n r u l e s . Thus each e f f e c t was described i n terms of a (9),<|o) p e r t u r b a t i o n treatment. Dexter pointed out the e f f e c t which i s concerned w i t h the new e l e c t r o n i c s t a t e s which must be taken i n t o account when one considers the n o n p e r i o d i c l t y of the l a t t i c e p o t e n t i a l i n a v i b r a t i n g c r y s t a l . Interpretation of Physical Significance of Temperature-Dependent Gaps 00 In his work James claims to have shown that the temperature dependent energy gap has the nature of free energy. His source of the temperature dependence i s again the electron-phonon i n t e r a c t i o n . He assumes that the change i n the average energy of the c r y s t a l due to electron-phonon coupling when the electronic system i s In the i - t h state, i s proportional to the number of electrons and of holes present i n the i - t h state, and that i t i s independent of just what electron and hole states are occupied. Namely, AE.(T) = V A E(T) (n x i s the number of electrons i n the conduction band and of holes i n the valence band i n the i - t h state of f j , e electronic system, and not the occupation quantum number which can take only two values, 0 or 1.) If t h i s assumption i s accepted, then from 0.) ,0t.), (n},(|£.)(13.;) and (19) of ;,his paper i t follows that V A E < T = 1 , > >' , E. — n I, I . e - ( ^ H ; ) / K , T * Lt-%? and from (22) and (31) of his paper that L e Vk,T i 14 where E. - the energy of >tkelectronic system i n the i - t h electronic state <>HKedecoupled c r y s t a l . • Uj - the energy of Ike v i b r a t i n g l a t t i c e i n the j - t h v i b r a t i o n a l normal mode o\.\\^ decoupled c r y s t a l . \.: - the change i n the energy of the state of the c r y s t a l due to the electron-vibration coupling. ri^ - the number of electrons i n the conduction band and of holes i n the valence band i n the i - t h state of the electronic system of decoupled c r y s t a l . AE^CT) - the change i n the average energy of the c r y s t a l due to electron-vibration coupling when the electronic system i s i n the i - t h state. A£(T) i s defined by AR (T) = \-Al^CO , w h e r e AF^ CT) i s related to AE^(T) by the Gibbs-Helmholtz equation James then shows from the above that i n the expression (2-3) the quantity that appears i n the exponential i s — l ^ / i ^ T where FCCO = ^ + AFg(T) i e . , the temperature dependent part of the gap has the nature of a free energy. (2#) and (2..8) c e r t a i n l y cannot be s a t i s f i e d i n the case when the electron-phonon inte r a c t i o n i s present, and hence the statement that the gap has the nature of free energy i s incorrect. He i d e n t i f i e d Fan's r e s u l t with AfI(T) . He claims that Fan's r e s u l t for high enough temperatures i s of the form \s At(T) = ^ ( ^ + /3T) where <C and/3 are constants. Fan's energy-level structure does not s a t i s f y (hi), and so there i s no j u s t i f i c a t i o n to interpret his r e s u l t as Ai^ (T) Moreover, his r e s u l t for high enough temperatures i s not of the form ^(oo-t-pT) • CHAPTER 3 TEMPERATURE-DEPENDENT ENERGY LEVELS In the case of op t i c a l absorption, conductivity and Hall constant we are dealing with thermodynamic systems. In fa c t , a thermodynamic system i s any macroscopic system. A macrostate of the system i s spec i f i e d by giving the values, of a certa i n set of compatible observables the number of which i s much • less than the number of observables i n the complete set. When we act u a l l y measure an observable i n the laboratory we measure not i t s instantaneous value but a time average. In other words, i n the experiment one can follow only the time evolution of macrostates and not mlcrostates. In the case of macroscopic systems i n equilibrium the number of degrees of freedom i s large, and instead of time averaging we perform rather easier ensemble averaging. With t h i s step i s concerned ergodic theory, and we s h a l l not discuss i t here. Via s t a t i s t i c a l mechanics we can predict the macroscopic behaviour of the system ( i . e . evolution of macrostates) given an i n i t i a l macrostate and macroscopic physical model of the system. The usual procedure i s given schematically i n the following picture, F i g . (7). macro state physicdil model micro stale d y n a m i c s of micro stales ftafhsVical wechoiTiics evolution of macro slates F i g . (7) Schematic representation of the procedure to be followed i n order to obtain the desired macroscopic behaviour of the thermodynamic system. * The time i n t e r v a l should possibly be short compared to the resolving time of the measuring apparatus but long compared to scattering times or periods of molecular motion. 17 Previously discussed work i n Chapter 2 obviously does not follow the above scheme. A lack of appreciation of th i s has led to i n s u f f i c i e n t understanding of the o p t i c a l , Hall and conductivity phenomena i n semiconductors. The concept of temperature dependent energy l e v e l s The usual s t a t i s t i c a l mechanical treatment of a system i s based on purely mechanical energy l e v e l s . However i n some problems the s t a t i s t i c a l mechanical averaging i s divided into two steps. (Indicated by the dashed l i n e i n F i g . (7).) In the process of p r i o r p a r t i a l averaging over some states of a system of the ensemble under consideration one may encounter so-called temperature-dependent energy l e v e l s . The remaining part of the averaging must of course be done properly to ensure the same answer as would be obtained by the usual s t a t i s t i c a l mechanics techniques. For a given macroscopic system d i f f e r e n t kinds of p r i o r averaging are possible, depending on the nature a f t t ' e information available or, a l t e r n a t i v e l y , on the nature <£:4Ke i n f ormati on i t i s desired to obtain. In fa c t there i s an i n f i n i t e number of ways of defining temperature-dependent energy l e v e l s . And i t i s exactly because of t h i s freedom of choice that they have no physical s i g n i f i c a n c e . Such l e v e l s are usually introduced for reasons of expediency and convenience, and t h e i r introduction should not imply any lack of knowledge about the complete set of mechanical energy l e v e l s of a system of the ensemble. In the following we s h a l l construct the equilibrium s t a t i s t i c a l mechanical formulae f o r thermodynamic functions i n terms of such temperature dependent energy l e v e l s . We s h a l l * The adequacy of the basic postulates l n the quantum s t a t i s t i c a l mechanics w i l l be assumed. consider now, to be s p e c i f i c , an electron-phonon system. This rather special case should not be regarded as a convenient method f o r c a l c u l a t i o n but rather as a simple i l l u s t r a t i v e (3) example of the general treatment by Elcock and Landsberg. I t w i l l give us some further insight into the physical concept of temperature-dependent energy l e v e l s . Our l i m i t a t i o n to the electron-phonon system w i l l enable us to discuss s p e c i f i c r e s u l t s i n the l i t e r a t u r e . Let us suppose that the system of int e r a c t i n g electrons and phonons can be eventually decomposed into noninteracting electrons, noninteracting phonons and electron-phonon i n t e r a c t i o n . In the f i r s t approximation, neglecting the inte r a c t i o n , the state of the free electron system can be spe c i f i e d s o l e l y by the group of electron quantum occupation numbers {n^ , and the state of the system of noninteracting phonons solely by the group of phonon quantum occupation numbers fv^j . It i s supposed that to a good approximation, the i n t e r a c t i o n has the e f f e c t that the state of a complete system may s t i l l be characterized by the two sets {Tik} , {^} ( i . e . weak interaction) (If the in t e r a c t i o n removes any degeneracy a further l a b e l i would be necessary). The energy of the complete system i n such a state can be written as Let us define the pr o b a b i l i t y PCi**}) , that at temperature T the system i s i n a state characterized by a given group of quantum numbers {^K} , b u t w i t h any s e t o f w i t h 2 s ^ 2 s I -/3 E ( W ( ( v f } ) e where f a r e so c a l l e d t e m p e r a t u r e dependent energy l e v e l s . E x p l i c i t t e m p e r a t u r e dependence i s due t o the p r i o r a v e r a g i n g o v e r the s t a t e s o f t h e ensemble under the c o n s i d e r a t i o n . We assume t h a t ^(x) i s a s t r i c t l y d e c r e a s i n g f u n c t i o n o f X . T h i s r e s t r i c t i o n on ^ (x) i s n o t n e c e s s a r y but e n s u r e s t h a t P({nk}j d e c r e a s e w i t h an i n c r e a s e i n (^^ K}) f o r g i v e n s e t {VlK} , thus p r e s e r v i n g t h i s u s u a l b e h a v i o u r between energy and p r o b a b i l i t y . S i m i l a r l y we d e f i n e P « V ) = <4> where and Z = 2- e From(3-z),(3-3), and(3-4) i t follows that the £ 5 can be constructed i n an i n f i n i t y of ways. The d i f f e r e n t sets of t's are given by d i f f e r e n t choices of g. Differentiating(3-2.)with respect to the temperature and summing over a l l values of (*lk) , i t can be shown that and s i m i l a r l y It i s understood here that i n a l l p a r t i a l derivatives the volume and the number of electrons i s kept constant. The convention i s made that, a f t e r £({*«}) have been defined, Ij[/2£(tv*^ ] s h a l l be defined by the l e v e l s £({vf}) being then again defined by(3-2),fc3)andG-H). Combining(3-*U3,,J) and(3-io), one obtains easily f o r the inter n a l energy U *. 2.1 By using the r e l a t i o n together with(3-'°) and(3«ll) one gets e a s i l y f o r the entropy To get the expression f o r the free energy f , i t i s s u f f i c i e n t to note that From f3-K);sand (3-13) one gets then F = - k B T l o ? I dtpHi**})] - k s Tlo-£ gf[/3£(f^))] { s ) From now on i t w i l l be said that a thermodynamic function i s i n i t s standard form when expressed i n terms of and Ps . From (315T) i t i s easy to see that with the choice = (3-16) the standard form for the free energy becomes the usual one, namely With th i s choice i t follows from(3-2.)and(3-l0 that It i s seen, i n c i d e n t a l l y , that here Eil**}) has the "nature" of a free energy. Similar conclusions are v a l i d f o r £-(l\Y) As before one gets From(3ii) ,(3-13) and(3-ifc) one gets f o r the int e r n a l energy and entropy U = L P f t ^ f e f f V ) - T J p £ffi.,J)J + I P(£v,})f £(tvt}) - T | f £({v,})j (3.20) Both f u n c t i o n s a n d ( 3 - i i ) i n t h e i r standard form, d i f f e r from the usual expressions by the appearance of the l a s t terms. The omission of these terms from either or both of the thermodynamic functions i s incorrect and leads to v i o l a t i o n of standard thermodynamic r e l a t i o n s . If I t i s desired to have a usual standard form f o r U i t follows from (3-ii) that gK*) must be chosen so that T where X i s a fixed temperature. With t h i s choice i t follows f rom(3-2),(3-t) and (3-H) that £n*> *V /L P(K},K}) S i m i l a r l y v fCfv^J - J_ = i:P(K} ({v;.EaM,{v?})/ - u 7 {^K} Here £'s have the "nature" of an in t e r n a l energy. From(3^3)and (3,|s) one gets f o r the entropy and free energy S = - KI P f tMj f lo^PrM-/3£(w; - j - ^ 1 J T j + T - Kl Ptft»{>f P ^ ) - f - ^ J T F = - kBT Io With the choice(3-22) for ^  both functions, i n t h e i r standard forms, d i f f e r from the usual expressions. Neither of the choices(3-lO or(3-22-) of the function g puts the expression f o r the entropy i n a usual standard form IH This i s independent of the choice of <=j because the PCC71*}) and PftV,2}) are. Therefore, there i s no function 3 which w i l l achieve f o r the entropy what(3-i£) and(3-22)achieve f o r the free energy and i n t e r n a l energy respectively. There i s c l e a r l y an i n f i n i t y of choices of the function g f o r which none of the thermodynamic functions i s i n a usual standard form. Let us now have a closer look at Fan's r e s u l t s i n the l i g h t of the above introduced temperature dependent energy l e v e l s . The equations(3-2),(3-3) and(3-<f) may be written as the single equation Moreover, l e t us put and suppose that to y = j(x) corresponds a unique r e c i p r o c a l r e l a t i o n and conversely. Then, a f t e r f i x i n g ^ , £(£*KW and £({%}) are well defined. From(3-8), (3-9) and (3-28) one gets and (3'3<>)may be rewritten i n the following form Af t e r expanding the f i r s t exponential one finds Neglecting the fluctuations i n the phonon part system, and with the choice (3-iOfor q; In the Bloch one-electron approximation the electronic energy l e v e l s are characterized by the wave vector k . The quantum mechanical energy l e v e l structure^!), can be then s i m p l i f i e d For t h i s special case(3-3H)becomes (8) This i s nothing but Fan's r e s u l t . (12.) Using F r o l i c h ' s r e s u l t as a s t a r t i n g point Fan i s dealing at the beginning with the energy l e v e l structure of the form given by 0-1 ). However, the electron quantum occupation numbers are not written out e x p l i c i t l y . The exclusion p r i n c i p l e Is taken into account by simple p r e s c r i p t i o n f o r the summation over the unoccupied states. Because i n a semiconductor only a few states i n the conduction band are occupied and the valence band i s p r a c t i c a l l y f i l l e d , the correction due to the Pauli p r i n c i p l e i s l a t e r neglected. If Fan would r e t a i n the Pauli exclusion p r i n c i p l e , then the energy l e v e l s and thus the gap would depend on the d i s t r i b u t i o n of the electrons l n the conduction and valence band, i . e . on {>,,} Neglecting a l l electron-electron correlations, i . e . electron-electron i n t e r a c t i o n and exclusion p r i n c i p l e , he i s avoiding such dependence by making Bloch one electron approx-imation. Namely, he i s dealing with the energy l e v e l structure given by(3-35j. In the next step temperature dependent energy l e v e l s of the form<3'3*) are introduced f o r two p a r t i c u l a r electronic states (\j&t) • According to Fan, the difference between the two •temperature dependent energy l e v e l s defines the temperature dependent energy gap. It i s apparent from our disucssion i n th i s chapter that there are no grounds for assuming that such defined temperature dependent energy l e v e l s and gap have any physical significance whatsoever. If one would l i k e to evaluate a macroscopic quantity, l e t us say the free energy f o r our electron phonon system, one would have a long way to go yet. It would be necessary to calculate the temperature dependent energy l e v e l s for the phonon part system of the form (3-31) and then the corresponding expression f o r the free energy(3-\5). Having committed oneself to a cert a i n p r i o r averaging one has to be sure that the remaining part (thermodynamics using the correct*] -function) i s done c o r r e c t l y . In our case, t h i s way of c a l c u l a t i n g the free energy would c e r t a i n l y have no advantage over the more f a m i l i a r procedure. Moreover, when one i s dealing with nonequilibrlum i r r e v e r s i b l e processes l i k e o p t i c a l absorption or Hall e f f e c t , the Introduction of such temperature dependent l e v e l s would make i t extremely d i f f i c u l t to carry out the program f o r c a l c u l a t i n g a macroobservable i n question discussed at the beginning of th i s chapter. We s h a l l not discuss i n d e t a i l the work done by the other authors mentioned previously. Although they are treating d i f f e r e n t microscopic mechanisms the concept of the temperature dependent gap i s b a s i c a l l y the same i n a l l cases. As i n the above example, the gap i s defined as the difference 11 between the two s p e c i f i c temperature dependent energy l e v e l s obtained by p a r t i a l s t a t i s t i c a l mechanical averaging of mechanical states over some states of the ensemble. Moreover, such a gap was supposed to explain conductivity and absorption data at the same time. Looking f o r a ce r t a i n macroobservable, i t i s unnatural to stop calculations at a given stage before the desired s t a t i s t i c a l mechanical averaging i s completed over a l l the states of the system under the consideration. There are no grounds to assume that the byproduct of such incomplete averaging over the mechanical energy l e v e l structures — temperature dependent energy l e v e l s — can explain by i t s e l f the temperature behaviour of any macroscopic system. S p e c i f i c a l l y , there i s no ju s t i f i c a t i o n - . to assume that the temperature independent gap parameter i n semiconductors £| should be replaced by the "temperature dependent energy gap" obtained by Fan and others i n the expression for the o p t i c a l absorption or Hall constant l n the e x i s t i n g theories. With regard to the work by James, we would l i k e to point out that from our discussion of temperature dependent energy l e v e l s i n chapter 3 i t i s apparent that i t i s neither necessary nor desirable to emphasize whether such quantities have the "nature" of Internal energy or free energy. The defining equations(i'2.),(3-3)(3-H)allow'. i n f i n i t e choices f o r a , as and when convenient, for which the temperature dependent energy l e v e l s have neither the "nature" of an i n t e r n a l energy nor a free energy. CHAPTER b TEMPERATURE-DEPENDENT THERMAL GAP In the case of the Ha l l e f f e c t the thermal temperature-dependent energy gap may be defined v i a an exponent as +73 E (T) R H(T) - const.- e S*P (H One can of course force any expression f o r the Hall constant into exponential form. To talk about the gap does not make much sense then. -A more natural procedure', i f possible, would be to get such exponential form automatically through the process of c a l c u l -a t i o n a f t e r imposing c e r t a i n reasonable approximations on the "exact" expression f o r the Hall constant. In fa c t , we s h a l l see <13) l a t e r that for the Hall constant defined i n terms of the Kubo type conductivity tensors, the desired mathematical form, and thus the thermal temperature-dependent energy gap can be obtained for the in t e r a c t i n g electron-phonon system i n i n t r i n s i c semiconductors i n a rather natural way within c e r t a i n reasonable approximations and assumptions. However, i f one would l i k e to perform better (more exact) c a l c u l a t i o n without making such approximations and assumptions, the concept of the thermal temperature dependent energy gap would lose i t s context, (a) The.Hall Constant In the following the emphasis w i l l be put on the l o g i c a l structure of the theory that w i l l give us a kind of a c t i v a t i o n energy, the so c a l l e d thermal temperature-dependent energy gap. 30 Such a gap with the e x p l i c i t temperature dependence, due to the electron-phonon Interaction can be defined v i a the high magnetic f i e l d H a l l constant. (The experimental r e s u l t w i l l of course include temperature-dependence due to other processes such "as thermal expansion.) By choosing the magnetic f i e l d H p a r a l l e l to the 2 axis the general expression f o r the H a l l c o e f f i c i e n t i s given by R = ^ (W) 77 where 6jlv 5-s "the conductivity tensor i n the presence of H # If one neglects the electron-hole interactions the t o t a l conductivity tensor b^ .v can be written as a sum of two parts where the upper indices e, A indicate the conduction electron and valence hole part respectively. According to the theory of (i"0 i r r e v e r s i b l e processes developed by Kubo, the e l e c t r i c DC conductivity tensor 6fa can be expressed i n the following way oo ft 0-4-J J i/ J xO,*(-^)3>> («) (Through t h i s paper we s h a l l be working with the units where * - i ) 1^ i s the electron (hole) current density operator. In the occupation number space i t can be written i n the form v w I I (*) (A) e The Heisenberg operator (t) represents the natural motion of current i n the absence of external e l e c t r i c f i e l d , namely where H i s the Hamiltonian of the system, excluding the external e l e c t r i c f i e l d but including the magnetic f i e l d , and the average denoted by brackets <( • )> lnCt-H) means the equilibrium average with the equilibrium density operator ^ for the grand canonical ensemble, i . e . + ae and a e are the creation and a n n i h i l a t i o n operators f o r a (fo db . \ / charge c a r r i e r i n state I , and V i s the volume of the system. e Further d e t a i l s concerning the matrix elements are (is) U given in;the paper by Luttlnger. The H a l l e f f e c t shows d i f f e r e n t features depending on the difference i n the strength of the magnetic f i e l d and i n the scattering mechanisms involved. The magnetic f i e l d strength reglon :. we s h a l l be interested i n , can be characterized i n terms of the cyclotron angular frequency 60c , the mean free time ( i . e . the relaxation time of the e l e c t r i c current), and representative energy of the charge c a r r i e r s £~ i n the following way 32 e where c. i s the chemical potential r- f o r degenerate electrons (holes) and the thermal energy kgf f o r nondegenerate electrons (holes). In our treatment we s h a l l neglect the eff e c t s of the quantization of electronic motion i n transport phenomena through changes i n the density of states of the charge c a r r i e r s and the scattering mechanisms at strong magnetic f i e l d s . We s h a l l not be interested i n the case when at low temperatures such e f f e c t s make t h e i r appearance, mainly i n the Shubnikov-(16) de Haas o s c i l l a t i o n . For the sake of s i m p l i c i t y we s h a l l assume that i s much smaller than ^ due to the stronger scattering and larger e f f e c t i v e mass/ of holes i n the valence band, and ** 71 xy y x ( it . 7 ) due to the strong magnetic f i e l d . Then one gets f o r the Hall constant the simple expression P = _ J — (13) Furthermore at strong f i e l d s ^ - - V " " — H T ~ ( M 1 and so I R H = ~ ff-e-c OHO) where n c i s the thermal average number of electrons i n the 33 conduction band. Similar derivation and approximations by using(w),C»-3) andO-H) (13) were made by Kubo and others, and the reader i s referred f o r more d e t a i l s to these papers. We s h a l l then take i t that an experimental a n a l y t i c a l treatment of the Ha l l constant w i l l give us a value for TTC and now st a r t on the t h e o r e t i c a l c a l c u l a t i o n of t h i s quantity, (b) System Hamiltonian At f i n i t e temperatures the free charge c a r r i e r s i n a pure semi-conductor consist of electrons i n the conduction band and holes i n the valence band. The l a t t i c e vibrations interact with t h i s system of free charge c a r r i e r s thus r e s u l t i n g i n the change of energy of the system. We s h a l l assume the density of electrons i n the conduction band and holes i n the valence band i s s u f f i c i e n t l y small that the Coulomb in t e r a c t i o n between them can be neglected. The Hamiltonian of such a system of electrons and holes l n a v i b r a t i n g l a t t i c e , i n the second quantized form, i s where Hee i s t h e H a m i l t o n i a n o f f r e e e l e c t r o n s i n t h e v a l e n c e and c o n d u c t i o n h a n d . By we have d e n o t e d t h e e n e r g y o f t h e e l e c t r o n , measured f r o m t h e c h e m i c a l p o t e n t i a l y«? i n t h e s t a t e c h a r a c t e r i z e d by t h e p r o p a g a t i o n wave v e c t o r k and band i n d e x fi" . F o r s m a l l k we s h a l l assume t h e f o l l o w i n g band s t r u c t u r e (Fig.(*)) c T = "A (H-15) T -k V 2.™: e ~F 1 V y C £° = e° 0 F i g . ( 8 ) D i a g r a m f o r t h e e l e c t r o n i c e n e r g y TA as a f u n c t i o n o f t h e wave vector k. + &t and oi. a r e t h e c o r r e s p o n d i n g c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s . H p k i s t h e H a m i l t o n i a n o f t h e l a t t i c e v i b r a t i o n s . The e n e r g y o f t h e v i b r a t i n g l a t t i c e i n t h e v i b r a t i o n a l mode c h a r a c t e r i z e d by t h e wave v e c t o r % i s 60^ , H e r e , t h e c o r r e s p o n d i n g c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s f o r a g i v e n mode o f v i b r a t i o n a r e a n d ^ • The l a s t t e r m , H j * t . r e p r e s e n t s t h e 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 . V ( f f ' ^ j i s t h e m a t r i x e l e m e n t f o r t h e t r a n s i t i o n o f t h e c a r r i e r f r o m a s t a t e £ t o }> , due t o t h e i n t e r a c t i o n , a n d # i s t h e c o u p l i n g p a r a m e t e r w h i c h measures t h e s t r e n g t h o f i n t e r a c t i o n . By a s s u m p t i o n 36<$C I , and \ / ( | \> fj a r e n o n z e r o over a wide range of co% . The operators c x + , a , , and Q> s a t i s f y the usual r e l a t i o n s I f 2 f a a 1 = i . J-(c) Thermal averages f o r the number of electrons i n the conduction band and holes i n the valence band  In order to obtain an expression f o r the temperature-dependent energy gap we s h a l l calculate the thermal average number of the electrons i n the conduction band n c , and holes i n the valence band pv . The method of c a l c u l a t i n g the thermal averages i s based on the Identity V' ..; \ .V <BA> = |im A J *m ^ — where f o r any two operators A and B, the temperature dependent double time Green's function ^A|B^> i s defined everywhere outside the r e a l axis as r 'A <A|B> E = v. L J o l t - e < [ACt) ,B]> j{ L E < 0 - ^ J c t i . e ' E t < t A ( D , B ] > If l « t > 0 with [A,B] = AB — 7 BA = ±1 I 3 " l ? Ct-19) The thermal average i s taken over the Grand canonical ensemble, i . e . - -« H Tr Hence the thermal average number of the electrons i n the conduction band can be written i n terms of the temperature dependent double time Green's functions i n the following manner (t-20) By using "the equation of motion technique" or the one of " p a r t i a l d e r i v a t i v e s " one can obtain the expression for the single p a r t i c l e Green's function <^ a k|^ ^  . The solution i s of the form (Ml) where Z.,(E) i s t n e so c a l l e d mass operator. Xi^) w i l l be calculated i n the next section. Furthermore we Introduce the re a l and imaginary part of the mass operator, fl CE) and & A 37 as follows £-> + 0 4 S f By using (<t-2|) and (H.zz) one can write f£ as ^ = T = = Tl E t ^ 3 ft In the case when the damping i s very small, and f l * ^ ) and Vi(m) are slowly varying functions of 60 , the function under the i n t e g r a l i n (1-23) has a steep maximum at some value <±> — £ k , which i s the solution of the equation for the c. elementary excitations c c c We expand the function F l t C 1 0 ) l n a P° wer series i n to near these values and take into account that c c The r i g h t hand side of O 2 - 5 ) can be further approximated correct to the second order In ^ as (4-2t) 38 Then we get f o r the number of electrons i n the conduction band the expression where Replacing the Lorentzian function c in(H-*7) f or very small values of the damping | ^ ( ^ j ) h y « J L c function,(t*7) becomes (Wo) A f t e r converting the summation in(M-3o) into integration one gets - if2- J L 1 V + % , - E J l e . * + U r d J t (WD where V i s the volume of the system, and we have assumed that the 1 - d i s t r i b u t i o n i s i s o t r o p i c . 39 By using the rela t i o n s (His) and (4^) dk ell j£ A ' dA. (4-32) so we can write i f * . (4-33) or *H J (t-3t) and thus (HIT) £4 i s an even function of k . We expand £ 1 for small values of I up to the quadratic term as follows 2- { AX i-o withOt-vO and MO one gets % J l — — I (H-37) where the renormalized temperature dependent mass fo r the quasi-electrons i n the conduction band i s defined as I _ J _ _ + (Ml) ] , ~ w» ldAV|L-o ( l t' 3 8 ) / Pf* , r' The function ^ C + 1 / decreases strongly as one moves up i n the conduction band. Hence to evaluate the thermal average number of electrons i n the conduction band one i s e s s e n t i a l l y interested i n the density of states near the bottom of the conduction band. We s h a l l assume that £| 't kBT . The term unity i n the denominator of (^ "J) may be neglected to a good approximation. The i n t e g r a l can then be reduced to the type 0 0 k and one obtains e (Wo) Similar calculations can be done fo r the number of holes i n the valence band pv. V within the same approximation as before, and a f t e r neglecting small terms one gets (WO •H where J_ J / <\\ n*0 ml Since we have the conservation of charge, the number of electrons i n the conduction band must be equal to the number of holes i n the valence band \ * f i r ( W ) F i n a l l y we get the expression f o r the thermal average number of electrons In the conduction band or holes i n the valence band As a matter of convenience we s h a l l c a l l the quantity i n the exponent the temperature-dependent thermal gap, i . e . , E Cr) = Z - K This gap corresponds to the system i n thermal equilibrium. It can be determined, as discussed i n section (a) of t h i s chapter, by measuring the Hall constant as a function of temperature. We emphasize again thatOt-HS) i s an approximation only i f f o > ^ J " ] l £ o | y^Yf and i f the damping could be neglected. If these conditions are not f u l f i l l e d the gap i s not well defined any*more. Actually to talk about the thermal temper-Hi-ature dependent energy gap makes sense only i f one can obtain, without forcing, the e x p r e s s i o n ^ ) f o r the Hall constant l n the exponential form. We have seen that t h i s can be done as f a r as the approximations leading to the express!ons(4-io) andC^ s) f o r the Hall c o e f f i c i e n t and the average number of conduction electrons can be tolerated, (d) The c a l c u l a t i o n of In the following we s h a l l examine the e f f e c t of the electron-phonon i n t e r a c t i o n on the r e a l and imaginary part of the Fourier transformed double time temperature dependent single p a r t i c l e Green's function. By using "the equation of motion technique" to obtain the single p a r t i c l e Green's function ^ o ^ l a ^ " ^ one gets the following set of equations for our system ; on the r i g h t hand side of On) , A.£|^>, wd Z and for the higher order Green's functions appearing rr At that stage we discontinue the i n f i n i t e hierarchy of equations by decoupling the higher order Green's functions appearing on the r i g h t hand sides of(4-18)and(4-^ ). The order estimation of decoupling i s s i m i l a r to that used fl7) by Nishikawa and Barrie (1963) for the two p a r t i c l e Green's function. Further bearing i n mind the r e l a t i o n between the moments and cumulants, we decompose a higher order Green's function into a sum of products of lower order Green's functions and averages,as f a r as possible,so the remainder can be regarded as s t a t i s t i c a l l y independent and small i n magnitude. where „-r r ff 4 where and The approximations (t-so) are not v a l i d for certa i n p a r t i c u l a r values of E . However, f o r these p a r t i c u l a r values of E , l i t t l e error i s Introduced to the c a l c u l a t i o n of C q | J q A X ' provided that the phonons of Importance are not confined to a narrow range of co^ . By substituting(H-S'o)into(4-H8)and(,i-'tf)we have a closed set of equations which can be solved f o r C ^ & i ^ X The solution i s of the form \ i i °!» X 2T ' E - \ - -*t\_ (E) fr*2) 45 or *N l ( E )-*-|v (*t.)|-C - r ^N- + t ^ t ) « r r s-Since we are interested i n the energy expression i n the denominator of the Green's function, we write correct to the second order i n K as ( |— l e ^ i d ) ' • Within that approximation we have la \ c^\ = —1 ! r A r where we have denoted 7(E) - * L ( E ) + * ( E - T 4 ) . N ( E ) which i s the so c a l l e d mass operator. By using the asymptotic re l a t i o n s lim ! = P(IT) +• ^  ^ and (H-ST) where P denotes the Cauchy's p r i n c i p a l value, one gets e a s i l y the r e a l and imaginary part of the mass operator, namely e->to r f P with N H - *PIM* ^ ^ - = ^ r - r-'T;:r )+ f J, «•' r' r 4 £ and r tr' * + 0 * + l - \.J^(M-\-t--w%)] + s (Wl) For r e a l systems with i n t e r a c t i o n the Green's functions have a cut along the r e a l axis and have poles only i n f i r s t approxim-ation. The damping i s f i n i t e . If one neglects damping, the approximate Green's functions have poles which are usually labeled as the temperature dependent energy of elementary excitations ( q u a s i - p a r t i c l e s ) . In our case the equation f o r these quasi-electrons (excitations) , which are determined by the r e a l part of the mass operator f l , , i s of the form The imaginary part of the mass operator accounts f o r the r 11 "broadening"due to a f i n i t e " l i f e time of quasi-electron i n the state £ , as a re s u l t of electron-phonon in t e r a c t i o n at nonzero temperatures. Due to the slowly varying nature of we can approximate r f| (d>) by f l . (T) . Furthermore, because of the f a c t that one may ignore the part of f i t (T.) due to the band to band scattering . Within these approximations the thermal temper-ature dependent energy gap becomes E = £ - £ - o Gap c v -(. 2 £1 i + /2 m * In the above we neglected the damping which was the e f f e c t considered by Radkowsky. If we look at our damping (equation ) we f i n d that i t reproduces Radkowsky's r e s u l t for the broadening of a single l e v e l when the n^o^l. In the Green's e. M function method, the s h i f t and broadening .of a single l e v e l r e s u l t from the one c a l c u l a t i o n . CHAPTER 5 TEMPERATURE-DEPENDENT OPTICAL GAP In the case of the o p t i c a l absorption, the o p t i c a l temperature dependent energy gap can be defined by extrapolating the absorption curve near the edge. See Pig.(8) . Here the gap i s not defined l n advance v i a special mathematical form as i n the H a l l constant case. Any theory of the op t i c a l absorption, i f good enough, should give us the proper spectrum shape near the edge, and thus the gap. It w i l l be shown i n t h i s chapter how the op t i c a l temperature dependent energy gap can be obtained v i a the Kubo type formul-ation of the absorption constant f o r the int e r a c t i n g electron-phonon system i n i n t r i n s i c semiconductors. T > T, F i g . (8) Graphical representation of the o p t i c a l temperature dependent energy gap. a) The o p t i c a l absorption c o e f f i c i e n t The absorption constant can be written i n terms the d i e l e c t r i c s u s c e p t i b i l i t y as (^u>) = const • • Im u^(to) / with £(u>) = ~ |im J t - e ^ <[M(i) M l \ where the symbols .have the following meaning* M = the electronic dipole moment operator M(t) - e. - M - 1 i In the above formulae the following assumptions were involved i ) The system i s i s o t r o p i c i i ) The external e l e c t r i c f i e l d F(t) has been applied a d i a b a t i c a l l y s t a r t i n g from the distant past* h-m £(t) =0 i l l ) The system was i n thermal equilibrium at t=-°° . iv) The Hamiltonian associated with the external f i e l d Is of the form SO v) The external f i e l d i s s u f f i c i e n t l y weak that i t s e f f e c t can be treated i n f i r s t - o r d e r perturbation theory. The function X(-°°) , given byte-*) i s clo s e l y related to a temperature dependent double-time Green's function. Such a Green's function <CA^Ji^ £ f o r operators A andj_ i s a two branch an a l y t i c function of E defined everywhere outside the r e a l axis i n a usual way. J e l l - e < [ A ( t ) , B ] > jf U E < 0 i f j d t . / <[&«U]> i { | „ E>0 In terms of the above Green's functions the absorption constant becomes £(co) = c o n s t .-co-U U ^1 ^ ^ v l MvXo-i.e In the case of the weak electron-phonon int e r a c t i o n , i t i s possible to extend the d e f i n i t i o n of the Condon approximation to the case i n which the adiabatlc approximation i s not v a l i d . (17; Within the framework of the generalized Condon approximation the dipole-moment operator can be written i n the second quantized form as where i s the matrix element of the dipole moment rr' operator corresponding to the tr a n s i t i o n s between the two elec t r o n i c states £ and . In t h i s approximation i ^ u 1 does not depend on the phonon creation and a n n i h i l a t i o n operators l>£ , H>2 , as i t would i n the general case and i s a number. Here the upper indeces & and I' stand f o r electronic state wave vectors, and the C and C denote the corresponding bands. Substituting (s-H) into (5'3) one gets $(yS) = cons' U '77' i - " - lim lm( M . w - M , , < o . % J a . a > \ # . By using — u , - — A t and "co + X.£ (5"-7) one can write (s-S) as b v yy We now divide (S"S) into the sum of two terms where ( * « ) The f i r s t term comes from the d i r e c t t r a n s i t i o n s between the two electronic states whereas the interference term comes from the interference of tr a n s i t i o n s among three or four electronic states, i n the presence of the electron-phonon i n t e r a c t i o n . We assume that the electronic states are nondegenerate, and that the main contribution to the absorption c o e f f i c i e n t comes from the f i r s t term. The c a l c u l a t i o n of the approximate two-particle Green's function appearing i n (rio) w i l l be done i n the next section. Here we Just state the r e s u l t , namely A f t e r introducing the r e a l and imaginary part of J~ (E) as U' tt' one finds the following expression for the absorption c o e f f i c i e n t r(<*o = co.st,u>-^ M t 1 » — ^ — c c — - — — jp * [ « - T f + T r n u » ] V r > ) The expression (SMS-) i s v a l i d f o r a l l co to the second order i n * . . Flan'^ 0) a n d , a r e g l v e n e x p l i c i t l y (for the one-PS' phonon processes) i n the next section. In the case when H. &nd [T.» ("•>) are slowly varying functions of W , and damping i s very small, the function under the summation has a steep maximum at some value to = ccf^, which i s the solution of CO Expanding the function H^ '^ ^ i n power series i n near these values and assuming that we may approximate the expression for co close to uT , correct up to the second order i n ~& as r t = const -co- ^ A A' 6-f where we denoted r i A ' /OJ= CO AA, ("-"Is) 1 ' + i , t o as with /-_ - I Afc'-(ST-19) (s--w) The approximation which leads to the modified (modified because of the presence of the factor f I — (^ M'^) _ 1 f t ' i n the denominator of(5"-i8) as well as transformational*) ) Lorentzian curve i n the v i c i n i t y of the peak of the absorption l i n e i n the expression (sis) i s no longer possible i n higher orders, since the co dependence of F^ ' C00) cannot be Ignored i n general orders. Provided that Hy^  ("^  and [yji1^) & r e ff' ff slowly varying functions of co , we can think of f^4< by analogy rr with T-0 case as a " l i f e time of the quasi--electron-hole pair, the state of which i s characterized by AA' . fL/ can be ff 1 » ; interpreted as the average energy gained by the quasi electron-ss l l ' hole pair i n the state i n virtue of i t s correlations with the other p a r t i c l e s i n the system. At that point we emphasize, that the quasi electron-hole pairs are e x p l i c i t l y temperature dependent, and i t i s thus impossible to make a purely mechanical in t e r p r e t a t i o n from the point of view of the energy spectrum of the t o t a l Hamiltonian. Due to the small damping we further replace the Lorentzian function i n (H8) by delta function »' This i s the so-called zero-phonon approximation. i ) Direct allowed t r a n s i t i o n s The summation i n (rz.1) i s a sum over a l l electronic states i n the bands which may be of i n t e r e s t i n the energy range covered. We s h a l l now consider photons with such energies that cause the lowest possible induced t r a n s i t i o n s from the valence to the conduction band. We s h a l l characterize the corresponding i n i t i a l and f i n a l states by the propagation vectors k and 4 respectively. Then we expect the c h a r a c t e r i s t i c photon frequency should be l n the range co T & '— T » • We assume that the medium i s otherwise transparent i n the energy range being considered. The s e l e c t i o n rule f o r a t r a n s i t i o n i n which only a photon i s involved i s t - i ±(J = 0 (5-22) where J i s the photon propagation vector. We consider the case i n which fj i s small, being several orders of magnitude smaller than most values of A and A' Accordingly we set fl? = 0 , leading to the approximate statement that a t r a n s i t i o n i s f o r -bidden when the electron changes i t s momentum. Hence the se l e c t i o n rule f o r such d i r e c t t r a n s i t i o n s i s and we can write the absorption c o e f f i c i e n t as V c At t h i s stage we s h a l l specify a band structure. The simplest model we treat Is the case of spherical energy surfaces, with a minimum i n the conduction band and a maximum i n the valence band f o r zero momentum, at which the energy separation i s £° . (Fig.(?)) F i g . (9 ) Schematic energy band structure i n the case of spherical energy surfaces. A-B represent-; d i r e c t t r a n s i t i o n f o r a p a r t i c u l a r value of the electronic wave vector. \\ 0 1 I t i s important to note that with t h i s energy model, the lowest possible t r a n s i t i o n i s also an allowed t r a n s i t i o n according to the momentum sel e c t i o n r u l e . Thus imperfections which would destroy momentum as a good quantum number would have r e l a t i v e l y l i t t l e influence on the absorption band shape. In the case of symmetry allowed interband d i r e c t t r a n s i t -ions may be considered independent of | and £ , and can be removed outof+hit summation. This i s of course not true except i n a small range of & , when the symmetry of the usual l a t t i c e periodic functions occurring wthe Bloch wave functions i s such that does not vanish anywhere i n the v i c i n i t y of AtC the edge. In t h i s case the v a r i a t i o n of the with A A r c . V i s much slower than the res t of the integrand with £ , and the above approximation becomes s u f f i c i e n t l y : v a l i d . A f t e r changing the summation into integration one gets g(o>S) = const -co. d to /co=co (ft) V C where 1rc We n o t e f r o m t h a t a n d s o Arc 4 I' d5^C« J ( « < where t h e e f f e c t i v e mass m i s d e f i n e d by 3* — — * ' a n d t h a t t h e e x c i t a t i o n e n e r g i e s k^CA) a r e e v e n f u n c t i o n s i n A , i . e . So we c a n expand co (&) f o r s m a l l A , a s s u m i n g t h a t ^ k l t ( \ ~ \ ) i s a r e g u l a r f u n c t i o n i n t h a t r e g i o n * CO (A) = 60 (o) +-5 - T l^ll -£ v c l ^ 2 /._ * T | Wlth(s--26) , (5-i8), and(5--3l) one g e t s where tti*tR i s the renormalized e f f e c t i v e mass f o r the quasi electron-hole p a i r defined by J I , / /l i l t ) ~Zf~ = - — * ~ + Z i J (5-33 In the Hartree approximation for the t o t a l s e l f energy part of the two p a r t i c l e Green's function, quasi electrons and holes "move" independent of each other although they interact with the phonons, and I I 1 -tr<.R V R c R Due to the presence of the J*"-function i n (532.) the integration can be e a s i l y c a r r i e d out, and one has for the absorption c o e f f i c i e n t near the edge Further from the edge the presence of TJA and \ i n (5"32) gives an exponential dependence on co , As f a r as the damping can be neglected we see from (£-3?) that the threshold frequency, and thus the o p t i c a l temperature dependent energy gap i n I n t r i n s i c semiconductors i s determined z. by the pole,correct to the order if- i n electron-phonon i n t e r -action, of the Fourier transformed double time temperature dependent two p a r t i c l e Green's function f o r the d i r e c t J U — * l-o t r a n s i t i o n s . Namely % ( T ) * * L < 0 ) ( M l ) eo In the Hartree approximation 60 (T) = £ - £ (S-37) C V i i ) Direct non-allowed tra n s i t i o n s If the t r a n s i t i o n i s not allowed, one may assume that M ^ l s proportional to the wave vector £ , and the absorption constant becomes, a f t e r s i m i l a r arguments as before, proportional to the P ° w e r °f the Photon frequency r - l3/z 6-(co) oc [ O J - toc(o)J (5.38) which gives the same o p t i c a l gap as was obtained i n the case of allowed d i r e c t t r a n s i t i o n s . b) Discussion of the re s u l t s with respect to the previous work It i s cl e a r from the above discussion that the o p t i c a l temperature dependent energy gap cannot be defined i n terms of temperature dependent energy l e v e l s (Fan), or i n terms of temperature dependent e x c i t a t i o n energies only. It cannot be defined s o l e l y i n terms of "temperature dependent"life time" e f f e c t s either (Radkowsky). In f a c t , i f one would include the l i f e time e f f e c t s of quasi-electrons (holes), then the absorption curve would never approach zero at a l l . For the sound d e f i n i t i o n of the o p t i c a l temperature dependent energy gap, one has to calculate the absorption curve i t s e l f ; w h i c h i , w i t h i n the quasi-particle approximation, neglect of l i f e time e f f e c t s and zero-phonon approximation i s given by 61 the envelope of the quasi electron-hole e x c i t a t i o n energies. See Fig.Oo). F i g . (10) Schematic representation of the o p t i c a l temperature dependent energy gap. The region II indicates the v a l i d i t y of the calculated t h e o r e t i c a l absorption curve £ ( » ) • « C « > - t h the region I, co -dependence, the proper band structure and damping should be taken into account. In the region III the absorption curve depends strongly on the damping and interference terms. Region IV indicates the background (noise). The above t h e o r e t i c a l predictions are i n agreement with the experiment. See Fig.(l). c) Green's functions - nondegenerate case In t h i s section we s h a l l discuss two p a r t i c l e double time temperature dependent Green's function f o r the electron-phonon system given In (z«n - 2.-IH) . We assume that the electronic states are nondegenerate, and that the c h a r a c t e r i s t i c phonon frequency col i s comparable to the gap Z° . The exact two p a r t i c l e Green's function i s usually divided into Hartree, Hartree Fock, and vertex part. The graphical representation i s given l n Fig.(n). X > X C(u>) e H H F V F i g . (11) Graphical representation of the exact two p a r t i c l e Green's function. H, HF and V denotes the Hartree, Hartree Fock, and vertex part of the exact Green's function. The single l i n e s correspond to the exact single p a r t i c l e Green's functions and the shaded square the vertex part. In the following we s h a l l be interested i n the expansion of the poles of the two p a r t i c l e Green's function, rather than i n the expansion of the Green's function i t s e l f , i . e . , we s h a l l be interested i n the contribution to the po3ies of the two p a r t i c l e Green's function from the corresponding Hartree, Hartree Fock, and vertex s e l f energy part. The corresponding graphical representation can be written as F i g . (12) Graphical representation of the exact s e l f energy part of the two p a r t i c l e Green's function divided into. Hartree, Hartree Fock, and the remaining vertex part. By using "the equation of motion method" one gets the following equation f o r the Green's function ^ a& aA-1 appearing in(5-lo), + and f o r the higher order Green's functions on the ri g h t hand side of (ws) , and At t h i s stage we break the chain of equations for the higher order Green's functions by using the following decoupling scheme where V ( « * T * + i)" and In writing down the system Hamiltonian, we assumed that the density of charge c a r r i e r s i s low, and therefore we neglected a l l interactions among them. The charges exhibit long-range coulomb forces, and can be assumed to move i n an average e l e c t r i c f i e l d , the care of which i s taken by the d i e l e c t r i c constant. That i s why we neglected a l l d i r e c t coulomb as well as the exchange e f f e c t s terms i n the above decoupling scheme. Quasi electrons and holes may interact among each other as well v i a exchange of v i r t u a l phonons. This would be important for i n d i r e c t t r a n s i t i o n s . Examples for such vertex parts are given i n Fig.(13) , i ) i i ) Fig . (13) Scattering processes for an excited p a r t i c l e - h o l e p a i r . i ) Direct Coulomb scattering, i i ) Exchange phonon scattering. The above decoupling hence corresponds to the Hartree approxim-ation for the t o t a l s e l f energy part. Hence we have to get and where f~L , (], » fT' and (7 are given i n (<<-4l) f>.6i) . In the quasi particle-model language, fl^ , \ ^ ) represents the "average energy" gained by. the quasi-electron (hole) i n virtu e of i t s correlations with the phonons. S i m i l a r l y ^ l ' (^) i s the? l i f e time of the quasi-electron (hole) i n the state >^ ( r ) . The approximations in^-1*2-) are of the order . ( S t r i c t l y speaking, the above approximations are not v a l i d for certa i n p a r t i c u l a r values of E . However, f o r these p a r t i c u l a r values of E , l i t t l e error Is introduced to the c a l c u l a t i o n of <ah\iail , provided that the phonons of importance are not E confined to a narrow range of <x>z . ) With the above decoupling we get the set of the self-contained equations for The solution i s — C £ ! ££ r r . ft-= £- Ci. 2T ^ v v = v i r ™ where with v z + i - —£L_ CO. (5-19) 68 and E " TH\ + T A + Z + cOz /I (5-50) Putting E = <o± ^ £ and using the asymptotic r e l a t i o n s 60 - X + i £ ini we can write [ i + ( r - x > P ( T 3 = 7 - ) + ^ ( y - x ) . ^ « o - y ) MTn _ where (SSI) if, rr V ^ <*> 69 + v9 + n£t to -V* + TA + W X 4 jfi V + 'A+* + 2 + + r' / — CO •f- A' A^ CO V T i n oj -Ti: + Tt, + CO ~ ) < C « I -_ A a n d 2 1 0 + < c ^ > ( v v ^ - V > ^ ) ] } (5-55) T By taking into account the equations of motion f o r 0, F OL , 4 t , and i> , and the fac t that one gets e a s i l y and (r-57) - v.-n4 CO. By putting co=X i n equation (S'Si) and then Integrating over y one gets no contribution to the "l i n e width" V (~[ ,-X) IX' fc, 4 ; > 0" f 71 from the imaginary part The l i n e width i s therefore given by To calculate "the peak p o s i t i o n " o^.^ , w e s-f note that i f we put co = \> -Ti = X. i n equation (ST-£2) the r e a l part of that equation vanishes. Hence we have no contribution to the peak from R e [ ( « » - T t ; + T r x e ) . N t t , ( t o - « ) ] . Furthermore, because 1 » ! one can neglect the contributions due to the band to band scattering i n the expression f o r L^, (\> -T^  ) • Taking 12 into account the above approximations one gets f o r the o p t i c a l temperature dependent energy gap the following expression C O . • f + /2< V-+ I - t i t co„ V9 + tl s /2 m* + C O , ( * « ) 13 CHAPTER 6 SUMMARY AND CONCLUSIONS One of the questions considered i n t h i s thesis i s to what extent the poles of Green's functions have physical significance as temperature-dependent energy l e v e l s . For a purely mechanical problem the poles of c e r t a i n Green's functions are recognizable quasi-particle e x c i t a t i o n energies as discussed i n Chapter 1. When these Green's functions are made temperature-dependent i t i s tempting to think of the corresponding poles as temperature-dependent quasi-particle e x c i t a t i o n energies. How then do these energies manifest themselves i n the macroscopic behaviour of the system? Is i t s t i l l legitimate to think of adding an extra p a r t i c l e to each member of an ensemble and to have the poles of the Green's functions give information on the "energy" and " l i f e time" of such an excitation? This "energy" i s now of course temperature-dependent and our contention i s that i t i s not to be thought of i n terms of Internal energy or free energy -i t i s s t i l l i n i t s nature microscopic and should be thought of as a temperature-dependent energy i n the sense given by Elcock and Landsberg. It then of course has no d i r e c t physical significance other than perhaps to say how the s t a t i s t i c a l -mechanical averaging process was divided into i t s two steps. We have shown for instance that, i n general, knowledge of the energy and l i f e time of quasi-particles (poles of single-p a r t i c l e Green's functions) i s not very h e l p f u l i n studying the Hall constant. To study the Hall constant, one would have to use the appropriate two-particle Green's function and a number of approximations have to be v a l i d before one has a simple r e l a t i o n between the H a l l constant and the average occupation numbers for the qua s i - p a r t i c l e s . A better example would have been the c a l c u l a t i o n of the s p e c i f i c heat. (We have discussed the H a l l constant since t h i s i s usually the technique discussed i n the l i t e r a t u r e . ) One would calculate the s p e c i f i c heat of the electrons and holes In a semiconductor by f i r s t f i n d i n g the int e r n a l energy using the correct r e l a t i o n between th i s and Green's functions. In an attempt to simplify or p i c t o r i a l l s e t h i s c a l c u l a t i o n one would f i n d the temperature-dependent "energies" of the qua s i - p a r t l c l e s . One would then have to f i n d the correct g-functlon to complete the statistical-mechanical averaging c o r r e c t l y . This second step i s so d i f f i c u l t that i t would be simpler and safer to forget the idea of temperature-dependent energy l e v e l s and to do the c a l c u l a t i o n by the d i r e c t method. Of course one might have a simple g-function under ce r t a i n simplifying assumptions. In studying the energy gap i n semiconductors, Fan found the s i n g l e - p a r t i c l e e x c i t a t i o n energies (mechanical quantities), made them temperature-dependent by replacing phonon occupation numbers by t h e i r thermal averages, then assumed that the gap could be simply written down as a difference of two of these energies. It was not clea r ( James) whether t h i s should have been interpreted as the o p t i c a l gap or the thermal gap. Basic-a l l y Fan ignored the second step i n the statistical-mechanical averaged process and based his r e s u l t on physical i n t u i t i o n . In physics we often avoid mathematical d i f f i c u l t i e s i f possible and we usually want the res u l t s i n a simple form easy to IS survey. In order to achieve that, we introduce cert a i n "reasonable" approximations. Of course, due to the completely d i f f e r e n t physical nature of the H a l l and absorption phenomena, one i s dealing with p h y s i c a l l y d i f f e r e n t assumptions and approx-imations. Following the program f o r c a l c u l a t i n g macroobservables mentioned i n chapter. 3 the approximations and assumptions can be made at d i f f e r e n t sections of the scheme. From what we have said, i t appears clear that the question, many times raised i n the l i t e r a t u r e , whether the two gaps are the same or not, i s not a relevant question. The question to ask would be - Under what "reasonable" assumptions and approximations do the thermal and o p t i c a l gap become the same? Is i t possible at a l l to make such "reasonable" assumptions and approximations? We have shown that the approximations involved i n our c a l c u l a t i o n of the one- and two-particle Green's functions lead to the thermal and o p t i c a l gaps being the same and that our gap agrees with Fan's. In equation (s«63) putting the electron occupation numbers zero f o r the conduction b^nd and unity f o r the valence band gives Fan's r e s u l t . In equation Ct-W) putting i n the same values f o r the electron occupation numbers again gives Fan's r e s u l t . (Using the correct thermal averages f o r these occupation numbers changes the numerical r e s u l t s very l i t t l e f or t y p i c a l semiconductors.) The o p t i c a l gap i s the same as the thermal one because our c a l c u l a t i o n of o p t i c a l absorption involved only the zero-phonon process, i . e . there i s no change i n the v i b r a t i o n a l quantum numbers when a photon i s absorbed. (Our c a l c u l a t i o n i s not based on the adiabatic approximation.) Inclusion of multiphonon processes would have led to a difference between the o p t i c a l gap and the thermal gap - the p o s s i b i l i t y of simultaneous absorption of phonons would have led to a non-zero absorption c o e f f i c i e n t below to^ and to a larger absorp-t i o n c o e f f i c i e n t above To . It would also have le d to our V C o p t i c a l gap being d i f f e r e n t l y defined and being d i f f e r e n t from Fan's gap. Since Fan r e l i e d on physical i n t u i t i o n i t i s not cle a r from his paper under what conditions his r e s u l t can be interpreted as an o p t i c a l gap. We f i n d that i t can be so interpreted under conditions which make the thermal and o p t i c a l gaps the same» zero-phonon processes only? neglect of the vertex part; neglect of damping; neglect of the frequency-dependence of the l e v e l s h i f t . If we look at the damping we f i n d that the Green's function method reproduces Radkowsky's answer f o r t h i s e f f e c t . In summary then I t appears that the picture i n which one thinks of temperature-dependent energy l e v e l s and t r a n s i t i o n s between these i s often too misleading and that i t i s safer to carry out the c a l c u l a t i o n of macroscopic behaviour i n the correct fashion. One may, by the l a t t e r method, obtain the r e s u l t one would have obtained using one's physical i n t u i t i o n , but now the approximations required for t h i s have been spelled out. Our other main r e s u l t i s that the temperature-dependent gap i n i n t r i n s i c semiconductors (see equation (4«M5) ) should not be thought of as free energy as suggested by James. BIBLIOGRAPHY d J s h u l t z , T.D., Quantum F i e l d Theory and The Many-Body Problem ( 1 9 6 ^ ) . Gordon and Breach Science Publishers, N.Y. (^Bonch-Bruevlch, V.L. and S.V. Tyablikov, The Green Function Method i n S t a t i s t i c a l Mechanics ( 1 9 6 2 ) . North Holland Publishing Company, Amsterdam. Intersclence Publishers, I.N.C., New York. ^ E l c o c k , E.W. and P.T. Landsberg, Proc. Phys. Soc. LXX, 2-B ( 1 9 5 6 ) . ( 4 ) Fowler, R.H., S t a t i s t i c a l Mechanics, Cambridge University Press, London ( 1 9 3 6 ) , second ed i t i o n , Chapter I I . ^ M o g l i c h , F. and Z. Rompe, Tech. Physik 119, ^ 7 2 ( 1 9 4 2 ) . ^ B a r d e e n , J . and W. Shockley, Phys. Rev. 80, 7 2 ( 1 9 5 0 ) . ( 7Radkowsky, A., Phys. Rev. 7 3 , 7 ^ 9 (19^8). ( 8 ) F a n , H.Y., Phys. Rev. 82, 9 0 0 ( 1 9 5 1 ) • ^ D e x t e r , D.L., Photoconductivity of Solids by Richard H. Bube, page 1 5 5 , John Wiley & Sons, Inc. ^ 1 0^Dexter, D.L., Photoconductivity Conference (Atlantic City 1 9 5 ^ ) . New York, John Wiley & Sons, Inc. London, Chapman & Ha l l Limited. ( 1 1 ) James, H.M., Photoconductivity Conference (Atlantic City 195*0. New York, John Wiley & Sons, Inc. London, Chapman & Ha l l Limited. ( l 2 ) F r b l l c h , H., Phys. Rev. 7 9 . $ + 5 ( 1 9 5 0 ) (^Kubo, R. and J.M. Satoru, Soli d State Physics, Vol. 17, 2 6 9 , Academic Press ( 1 9 6 5 ) * ( 1^Kubo, R., J . Phys. S o c , Japan, Vol. 1 2 , No. 6 , 5 7 0 (1957) ( l 5 ) L u t t l n g e r , J.M., Phys. Rev. 1 1 2 , 7 3 9 ( 1 9 5 8 ) ( l 6 ) Shubnikov, I. and W.J. de Haas, Leiden Commun. 2 0 7 a, c, ds 2 1 0 a ( 1 9 3 0 ) ^ l 7^Nishikawa, K. and R. Barrie, C.J.P., Vol. 4 l , 1 1 3 5 ( 1 9 6 3 ) others ( l 8^Baym, G., Phys. Rev. 1 2 7 , 1391 ( 1 9 6 2 ) . 18 (^Baym, G. and L.P. Kadanoff, Phys. Rev. 124, 287 ( l 9 6 l ) ^ 2 0^Kubo, R.J., Phys. Soc. Japan, Vol. 17, No. 7 ( 1 9 6 2 ) (21 ) v - L /Abrlkosov, A.A., Gorlkov and Dzyaloshlnskll, Quantum F i e l d Theoretical Methods i n S t a t i s t i c a l Physics, Pergamon Press ( 1 9 6 5 ) • (77) K * ;Zubarev, D.N., Vspekhl F i z . Nauk, 71, 71 ( i 9 6 0 ) ( 2 3)j31och, G. and C . De Dominicis, Nuclear Phys. 7 , ^ 5 9 ( 1 9 5 8 ) ^Lecture notes on Advanced S t a t i s t i c a l Mechanics. The course being given at U.B.C. by Dr. L. De Sobrino ( 1 9 6 6 - 6 7 ) . 

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