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Configuration interaction in the internal acceptor states in silicon Bhatia, Krishan Lal 1970

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CONFIGURATION INTERACTION IN THE INTERNAL ACCEPTOR STATES IN SILICON by KRISHAN LAL BHATIA B.Sc. (Hons. S.), U n i v e r s i t y of Panjab India, 1960 M.Sc. (Hons. S.), U n i v e r s i t y of Panjab India, 1961 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 Ju l y , 1970 In p r e s e n t i n g t h i s t h e s i s in 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 at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r 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 s t u d y . 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 p u r p o s e s 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 . It 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 t h e s i s 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 PHYSICS The U n i v e r s i t y o f B r i t i s h Co lumbia V a n c o u v e r 8, Canada i i ABSTRACT The presence of configuration i n t e r a c t i o n , between the " i n t e r n a l " acceptor states of group I I I impurities in s i l i c o n and the hand Bloch states, has been observed. The e f f e c t of impurity-impurity i n t e r a c t i o n on the l i n e shape of the boron, gallium and indium i n t e r n a l acceptor l i n e s has been studied. The observed c h a r a c t e r i s t i c asymmetric l i n e shape i s explained by in-: troducing the i n t e r a c t i o n of the i n t e r n a l state with the degenerate ?2/2 v a ^ - e n c e hand Bloch states, and the inhomogeneous impurity-impurity i n t e r a c t i o n . The i o n i z a t i o n energy of the s u b s t i t u t i o n a l impurity i n the host l a t t i c e i s found to be a deciding factor in t h i s i n t e r a c t i o n . The i n t e r n a l and external absorption spectra of s i l i c o n doubly-doped with boron and indium acceptors i s studied. The observed broadening of the boron external absorption l i n e s in Si(B,In) i s explained by neutral impurity s c a t t e r i n g . The presence of indium i n Si(B,In) c r y s t a l l a t t i c e , modifies the valence band Bloch states and hence the configuration i n t e r a c t i o n . T h i s modification i s found responsible for the observed features of the boron 2 p ' i n t e r n a l l i n e i n Si(B,In). The coupling between the l a t t i c e and the impurity bound c a r r i e r for deep monovalent acceptors in s i l i c o n , such as gallium and indium, i s found to be stronger than for the shallow boron impurity. This suggests the possible existence of phonon-as s i s t e d t r a n s i t i o n s associated with these deep impurities. Such t r a n s i t i o n s are observed in the absorption spectrum of i n -dium doped s i l i c o n . The phonon-assisted t r a n s i t i o n s are super-imposed on the photoionization continuum t r a n s i t i o n s of the i n -dium acceptors. Interference e f f e c t s between the phonon-assisted t r a n s i t i o n and the t r a n s i t i o n s to the continuum states modify the i i i the position and l ine shape of the transit ions. Using the phonon dispersion curves for s i l i c o n , interpretation of the results is presented. As a supplementary study, the temperature de-pendence of the indium external l ine 2, is investigated. The temperature dependence of halfwidth of indium l ine 2 in Si(In) supports the stronger electron-phonon coupling in Si( In) . i v TABLE OF CONTENTS P a g e Abstract : i i Table of Contents i v L i s t of Tables v i i L i s t of Figures • v i i i Acknowledgements • x Chapter 1 . INTRODUCTION' 1.1 General Introduction 1 1.2 Purpose of This Thesis 2 1 .3 Outline of This Thesis 4 Chapter 2 THEORY 2.1 E f f e c t i v e Mass Theory for the Acceptor States 6 2.2 Quantum Defect Method 8 2 .3 Configuration Interaction 11 i Chapter 3 GENERAL EXPERIMENTAL PROCEDURES 3.1 Apparatus 19 3.2 Samples 20 3.3 Determination of Impurity Concentration 21 3.4 Measurement and C a l c u l a t i o n of Absorption C o e f f i c i e n t 22 Chapter A CONFIGURATION INTERACTION IN THE INTERNAL ACCEPTOR STATES 4.1 Introduction 26 4.2 Experimentally Observed Absorp-tio n Spectrum (a) Boron Internal State 2p'--- 27 V : ,.££!£ (b) Gallium Internal State 2p /- 29 (c) Indium Internal State 2p' -- 29 4.3 Configuration I n t e r a c t i o n (a) Low Impurity Concen-t r a t i o n 29 (b) E f f e c t of Impurity-Impurity I n t e r a c t i o n 31 4.4 Interp r e t a t i o n and Discussion 34 4.5 Deeper Impurities (Gallium and Indium) • 43 Chapter 5 A STUDY OF SILICON DOUBLY-DOPED WITH BORON AND INDIUM ACCEPTORS 5.1 Introduction 46 5.2 Experimentally Observed Spectrum of Si(B,In) - 47 (a) External Absorption Spec-trum of Si(B, In) 47 (b) Internal Absorption Spec-trum of Si(B,In) 50 5.3 Neutral Impurity Scattering 54 5.4 Broadening of Boron Impurity States i n Si(B,In) 55 5.5 Configuration Interaction 60 5.6 P o s s i b i l i t y of Boron-Indium Complex Formation 62 Chapter 6 PHONON-ASSISTED TRANSITIONS IN THE DEEP ACCEPTOR IMPURITIES 6.1 Introduction 63 6.2 Experimental Results 64 6.3 Theory of Phonon-Assisted T r a n s i t i o n s 66 v i ££££ 6.4 I n t e r p r e t a t i o n 69 6 .5 Estimate of the Strength of the Electron-Phonon Interaction 75 Chapter 7 TEMPERATURE DEPENDENCE OF THE 7.1 7.2 7.3 INDIUM EXTERNAL LINE 2 Introduction 77 Experimental Results 77 Interpretation and Discussion 79 Chapter 8 CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK 80 Appendix A LORENTZIAN CORRECTION TO THE INTEGRATED AREA UNDER AN ABSORPTION LINE ---84 Bibliography "86 v i i LIST OF TABLES i Table page 3.1 Boron and Indium Impurity Concentrations i n Si(B,In) 23 4.1 Estimated Values of Parameter "Q" for the 2p' Internal Lines of Group I I I Impurities i n S i l i c o n 44 5.1 Integrated Cross-section for the Boron 2p / Line in Si(B,In) - 53 6.1 Calculated Strength of Electron-Phonon Int e r a c t i o n in Group I I I Acceptors i n S i l i c o n 70 6.2 Phonon-Assisted T r a n s i t i o n s i n Si(In) -- 73 v i i i LIST OF FIGURES Figure page 1.1 The Acceptor States and the Valence Band States i n S i l i c o n 3 2.1 Plot of the Asymmetric Line Shape Function for the D i f f e r e n t P o s i t i v e Values of Q 15 4.1 Observed Line Shape of the 2p XAbsorption peak in Boron-Doped S i l i c o n and i t s Im-p u r i t y Concentration Dependence 28 4.2 Observed Line Shape of the 2p' Absorption peak in Gallium-Doped S i l i c o n 30 4.3 Broadening of the Absorption Lines (a) Homogeneous Broadening 33 (b) Inhomogeneous Broadening 33 4.4 Computed Peak S h i f t i n the Function S(E) as a Function of the Concentration Broadening Halfwidth — 38 4.5 Computed Function S(E) for Values of Q and f 1 Appropriate to Boron-Doped S i l i c o n 39 4.6 E f f e c t of Impurity-Impurity Interaction on the Configuration Interaction 41 5.1 (a) Broadening of the Boron External Line 2-49 (b) Boron External Absorption Spectrum of the Two Si(B, In) Samples 49 J - X (c) Boron E x t e r n a l Abso rp t i on Spectrum o f the Two S i ( B ) Samples 49 5.2 (a) The Boron 2p / I n t e r n a l Peak i n S i ( B , In) - 5 1 (b) The I n t e r r u p t i o n "Broadening of the Boron 2p' Peak i n S i ( B , I n ) - 5 1 6 . 1 Phonon-Ass i s ted T r a n s i t i o n s i n S i ( I n ) 6 5 7 . 1 Temperature Dependence of the Ha l fw id th o f the L i n e 2 i n S i ( I n ) and S i ( B ) 78 . X ACKNOWLEDGEMENTS It gives me great pleasure to thank my super-v i s o r , Dr. J . W. Bichard for h i s continuous i n t e r e s t and h e l p f u l advice throughout these i n v e s t i g a t i o n s and during the thesis preparation. I am indebted to Prof. R. Barrie for many constructive and valuable discussions. The research described i n t h i s thesis was supported by the National Research Council of Canada, Grant NRC A-2204, and the Defence Research Board of Canada, Grant No. 9510-35. The author i s thankful to the Canadian Common-wealth Scholarship and Fellowship Committee for the award of a Commonwealth Scholarship. I am also thankful to my wife for her contin-uous co-operation and patience during the course of t h i s work. CHAPTER I INTRODUCTION 1.1 General Introduction The introduction of an atom from group I I I of the pe r i o d i c table, into the s i l i c o n c r y s t a l l a t t i c e , produces a hole l o o s e l y bound to the impurity s i t e (Kohn 1957). The energy states which t h i s hole can occupy are c a l l e d the "acceptor states". At low temperature, far i n f r a - r e d absorption l i n e s are observed due to photon-induced e x c i t a t i o n of t h i s hole from the acceptor ground state to the acceptor excited states. From the observed posi t i o n s of these absorption l i n e s the acceptor states of group I I I impur-i t i e s i n s i l i c o n have been associated with each of the ^^/2 anc* P^2 valence band maxima r e s p e c t i v e l y . The former, l y i n g i n the forbidden energy gap, are c a l l e d "external acceptor states" and the l a t t e r , being degenerate with the Pg/2 c o n t i n u u m states are c a l l e d " i n t e r n a l states". O p t i c a l l y induced t r a n s i t i o n s from the acceptor ground state to the external states give r i s e to an ex-te r n a l spectrum. Similar t r a n s i t i o n s from the ground state to the i n t e r n a l states produce an i n t e r n a l spectrum. For these acceptor states, only approximate solu-tions of the wave function and energy eigenvalues are possible i n the e f f e c t i v e mass approximation theory (Kohn 1957; Mendelson and Schultz 1969). The agreement with the experimental spectrum i s poor (Onton et a l . 1967). In p a r t i c u l a r , the species depend-ent binding energy of the acceptor impurity i n s i l i c o n i s not very well understood (Morita and Nara 1966). The quantum defect method (Bebb and Chapman 1967, 1969) has been used to determine the ground state wave functions which are dependent upon the bind-ing energy of the impurity. This approach seems to explain the im-p u r i t y i o n i z a t i o n energy dependent behaviour of photoionization and ; photoexcitation spectrum of the group I I I acceptor impurities i n s i l i c o n . Unlike the external acceptor states, the i n t e r n a l states are degenerate with the valence band states (See f i g . 1.1). This puts the i n t e r n a l states in a d i f f e r e n t environment and i t should modify the wavefunction and c h a r a c t e r i s t i c properties of the i n t e r n a l states. The s i t u a t i o n becomes equivalent to a d i s -crete electron state l y i n g i n a continuum of states. The i n t e r -a c t i o n between the d i s c r e t e state and the overlapping continuum i s c a l l e d "Configuration i n t e r a c t i o n " (Fano 1961). Configuration i n t e r a c t i o n has been observed, in atomic spectra (Fano and Cooper 1965), in the study of exciton absorption spectrum in semiconduct-ors and i n s u l a t o r s ( P h i l l i p s 1966) and in the Breit-Wigner scat-t e r i n g c ross-section for nuclear reactions ( B l a t t and Weisskoff 1952). In a l l the previous experimental studies of the i n t e r n a l acceptor states of group I I I impurities in s i l i c o n , t h i s configur-a t i o n i n t e r a c t i o n has not been taken into account (Onton et a l . 1967; Parscns and Bichard 1967; Parsons 1968a). Peterson (1964) t h e o r e t i c a l l y considered the fact that the hole bound to the ac-ceptor impurity, when excited to i n t e r n a l state, eventually pro-pagates to the degenerate P3/2 Bloch states. In h i s concept the i n t e r n a l states are non-stationary and a l i f e t i m e i s associated with t h i s eventual decay into a ^^/2 v a l e n c e band. 1.2 The Purpose Of This Thesis The present thesis has two main purposes. (a) The f i r s t o b j e c t i v e i s to understand how the i n t e r -nal acceptor states of the group III impurities i n s i l i c o n l a t t i c e are a f f e c t e d by the i n c l u s i o n of configuration i n t e r a c t i o n , and then to explore new features i n the experimentally observed i n t e r -n a l absorption spectrum. (b) The second main purpose of t h i s work i s to study the i n t e r n a l and external acceptor spectra of s i l i c o n c r y s t a l s doped with the two acceptor impurities (to be r e f e r r e d as double-3. T 1 r i T \- i n t e r n a l spectrum-i o n i z a t i o n a b s o r p t i o n T r g,' ground s t a t e e x t e r n a l spectrum_ ^. l e v e l \\ / e x t e r n a l n , r\f a c c e p t o r — l e v e l 2V , K j l e v e l s — l e v e l 3J •p / Valence n J'* Bands i n t e r n a l a c c e p t o r l e v e l s P i A> Valence x / * Band 10 a —3} 0 4 8 k // <110> 10 k ( x 10 cm k // <100> F i g . (1.1) The Acceptor States and, the Valence Band States in S i l i c o n . doped s i l i c o n ) , and then to investigate the e f f e c t of including c o n f i g u r a t i o n i n t e r a c t i o n , on the absorption spectra of such a system. In the past, a l l the studies of the i n t e r n a l and external acceptor states of acceptor impurities in s i l i c o n , have been con-centrated on s i l i c o n c r y s t a l s s i n g l y doped with one of the group I I I impurities (Onton et a l . 1967, Parson and Bichard 1967). No attempt has been made, to explore the semiconductors doped with two acceptor impurities, so f a r . A secondary purpose of t h i s thesis i s to explore the p o s s i b i l i t y of observing the optical-phonon a s s i s t e d t r a n s i -tions in the deep acceptor impurities i n s i l i c o n and then to i n -v e s t i g a t e the e f f e c t of any configuration i n t e r a c t i o n , present. So far such o p t i c a l ; phonon-assisted t r a n s i t i o n s i n homopolar c r y s t a l s has not been reported except in Diamond type I l - b (Hardy et a l . 1962) A supplementary objective was to s t -udy the temperature dependence of the external acceptor state-2 i n the indium doped s i l i c o n . Such type of study, in indium doped s i l i c o n , i s not i n the l i t e r a t u r e . 1.3 Outline Of The Thesis Three sets of experiments were performed. ( i ) The i n t e r n a l acceptor l i n e 2p^ was studied in s i l i -con s i n g l y doped with boron, gallium and indium impurities respect-i v e l y . The concentration dependence of the l i n e shape of 2p / ab-sorption l i n e was explored. Fano's theory (1961) was modified to take into account the e f f e c t of impurity-impurity i n t e r a c t i o n on the configuration i n t e r a c t i o n . The observed features of 2p' l i n e were explained by applying t h i s modified theory. This study i s included i n Chapter 4 of t h i s t h e s i s . ( i i ) In the second set of experiments, impurity states i n s i l i c o n doubly-doped with boron and indium impurities (Si(B,In)^ were studied. External and i n t e r n a l spectra of boron acceptors were observed i n the d i f f e r e n t samples, with increasing indium con-c e n t r a t i o n . Line shape, halfwidths and integrated cross sections of the boron absorption l i n e s were determined. This data was used to i n v e s t i g a t e the e f f e c t of the presence of indium acceptors on the (a) absorption spectra of the boron impurity and (b) the con-f i g u r a t i o n i n t e r a c t i o n of the 2p /'-state and the degenerate continuum s t a t e s . Chapter 5 discusses these studies. ( i i i ) In the t h i r d set of experiments t r a n s i t i o n s , with the p a r t i c i p a t i o n of o p t i c a l phonons, were observed i n indium doped s i l i c o n . The presence of these t r a n s i t i o n s , i n indium doped s i l i c o n and t h e i r absence in the boron doped s i l i c o n , was explained on the basis of stronger electron-phonon coupling i n S i ( I n ) . The presence of co n f i g u r a t i o n i n t e r a c t i o n , i n t h i s case, i s also stressed. Chapter 6 contains these r e s u l t s . Experiments were also performed to study the ef-fe c t of t-emperature on the halfwidths of the external absorption l i n e 2 i n indium doped s i l i c o n . The r e s u l t s are given i n Chap-ter 7. Various studies described i n t h i s thesis f a l l into separate chapters (but e a s i l y i n t e r r e l a t e d ) . The general chapters on the introduction, theory, procedure and conclusions ( i . e . Chap-ters 1, 2, 3, and 8) apply to a l l the r e s u l t s . The intermediate chapters are s e l f contained studies (with t h e i r own introduction, r e s u l t s and i n t e r p r e t a t i o n ) . CHAPTER 2 THEORY 2.1 E f f e c t i v e Mass Theory For Acceptor States In a perfect c r y s t a l l a t t i c e each valence electron moves in a p e r i o d i c p o t e n t i a l , Vi~f) , produced by the ion cores and the charge d e n s i t i e s of a l l the other electrons. The Hamil-tonian f ° r each valence electron i s 2^0 (2.1) The one electron wavefunctions which s a t i s f y y\C.£)ave the Bloch functions 1 ' ' Jv? . ™>k (2.2) Where IX has the p e r i o d i c i t y of the l a t t i c e ; yn i s the hand i n -dex, K i s the wave vector, and V i s the volume of the un i t c e l l . In the presence of an acceptor impurity, one con-si d e r s a p o s i t i v e l y charged hole, moving i n the periodic potent-i a l of the pure c r y s t a l plus a perturbing coulomb-like p o t e n t i a l 2 -e /K^Vwhere i s the s t a t i c d i e l e c t r i c constant of the pure cry-s t a l produced by the negatively charged acceptor ion (Kohn 1957)). In the e f f e c t i v e mass approximation the Hamiltonian }^CX) f ° r t n e hole i s given by H e r ) = K 0 c r ) - e 2 , v ( 2 . 3 > In t h i s approximation the following important assumptions are made (Kohn and Luttinger 1955) ( i ) The acceptor p o t e n t i a l i s weak and v a r i e s slowly over several l a t t i c e spacings. ( i i ) The Hamiltonian } H * C £ ) i s replaced by an e f f e c t i v e Hamiltonian i n v o l v i n g an e f f e c t i v e mass tensor m*. Within t h i s approximation, the e f f e c t i v e mass equation for the case of degen-erate valence bands becomes (2.4) The indices rn,yn/ run over oL degenerate valence bands; £ i s an ac-c<(3 ceptor state energy. Replacing the e f f e c t i v e mass parameters n f ^ _ i{x c c \ - W by a s i n g l e e f f e c t i v e mass m* f V — 0/« ) the e f f e c t i v e mass equation becomes In t h i s e f f e c t i v e mass theory the wavefunctions for the acceptor states are taken to have the form d. vn = l (2.6) where lK,Yn^ are the Bloch functions at the top of the valence bands i n the unperturbed c r y s t a l , and the summation i s over <L bands. The functions f" (Jf) are slowly varying hydrogenic functions ( c a l l e d en-Yn velope functions) which modulate the Bloch functions and s a t i s f y equation (2.4). For a simple band s o l u t i o n to equation (2.5), the envelope functions become the modified hydrogenic wavefunctions (Kohn 1957). The ^^/z a n <* ^1/2 v a ^ e n c e hand maxima of s i l i c o n , seen i n F i g . (1.1), are 4-fold and 2-fold degenerate, r e s p e c t i v e l y . The e f f e c t i v e mass /wavefunctions for the external and i n t e r n a l ac-ceptor states are . 1 ' t— _ 3/*™> 3 / i ^7 . / a W 1 'W (2.8) w = - m = -yz . . 8. where IK,>r)> and | k , w \ are the Bloch functions at the P„ 3/2 and v a i e n c e band maxima, r e s p e c t i v e l y . F^y^OC) and hy a Yyi / ^ --^  are the corresponding envelope functions. In Chapter 3 i t w i l l be shown that, i n general, i n t e r n a l acceptor states wave functions should be expressed as in the equation (2.8). 2.2 Quantum Defect Method Simple e f f e c t i v e mass theory, given in the l a s t s ection, p r edicts a unique binding energy for a l l the group I I I acceptor impurities in s i l i c o n that depends only on the host cry-s t a l parameters such as the d i e l e c t r i c constant and the e f f e c t i v e mass. Experimental studies show approximate agreement with the ef-f e c t i v e mass theory in the case of shallow boron acceptors only (Onton et a l . 1967). The observed binding energies of the group II I acceptors are species dependent and theory cannot account for d i f f e r e n c e s , say in the o p t i c a l properties between B, A l , Ga, and In impurities i n s i l i c o n . E f f o r t s to account for the observed chemi-c a l s h i f t have enjoyed l i m i t e d success (Morita and Nara 1966). (The species dependence of the binding energy of impurities i s c a l l e d chemical s h i f t ) . It i s h i g h l y desirable to determine good approxi-mate wave functions which are s e n s i t i v e to the impurity binding energy. The eimplest means i s to assume hydrogenic wave functions A 2 and scale the e f f e c t i v e Bohr radius a* =TK/m*e ^ t o reproduce the observed impurity binding energy £(obs.)= - ~ — ( K o h n 1957). ^K^a^ However t h i s procedure does not change the functional form of the wave functions but only their s p a t i a l extent. This requires as-suming values of the e f f e c t i v e mass, m*,and d i e l e c t r i c constant, K^, d i f f e r e n t from those appropriate to the host c r y s t a l . The method does not y i e l d good agreement with the experimental r e s u l t s for the deep impurities such as indium. .Recently the quantum defect method (QDM) has been 9. applied to the theory of deep impurities i n semiconductors (Bebb and Chapman 1967, 1969). Taking into account the changes i n the ground state binding energies of s u b s t i t u t i o n a l group III acceptor ; impurities, a r i s i n g from deviations of the impurity core p o t e n t i a l from that of a simple point charge, photoionization and photoexcit-ation spectra of the centers have been explained reasonably w e l l . QDM has i t s o r i g i n in atomic spectroscopy. I t was found that the emission l i n e energies of the a l k a l i atoms could be reproduced by assuming =-7^1, < 2- 9 ) Where the parameter, yl , i s c a l l e d the quantum defect; Id i s the 2 p r i n c i p a l quantum number; and ft the Rydberg (e /2a,where a i s the hydrogenic Bohr r a d i u s ) . The parameter i s a measure of the pene-t r a t i o n of the valence electron into the subshells of electrons. The picture formed i s that of a nucleus of charge Ze (Z i s the at-omic number of the atom) c l o s e l y surrounded by a core of (Z-l) e l -ectrons outside of which, at some distance, i s a s i n g l e electron moving i n a hydrogen-like p o t e n t i a l . The e f f e c t of the net core charge, ( Z - l ) e , i s such as to make the system hydrogen-like. But i n non-hydrogenic atoms (where Ze i s greater than in hydrogenic atom) the s state electron spends an appreciable time within the core where the e f f e c t i v e core charge i s greater. This greater pene-t r a t i o n of the ground state o r b i t into the core gives r i s e to the so c a l l e d quantum defect in the quantum number Yl and consequently changes the energy value of the state. In the impurity problem, the QDM r e l i e s on the ob-2 servation that the p o t e n t i a l becomes coulombic,-e /K^f* for largey* ( / ;> Yc) ( i s ~ n e a r e s t neighbour distance). The solutions to the e f f e c -t i v e mass equation in the e x t e r i o r region are therefore also c o u l -ombic (but not n e c e s s a r i l y hydrogenic). The coulombic wave functions must be continuous with the core function, which i s unspecified. It i s through the c o n t i n u i t y requirement that the exterior coulomb 10. function r e f l e c t s the core p o t e n t i a l . Since the core p o t e n t i a l i s unknown, i t s e f f e c t on the e x t e r i o r region i s estimated by looking for a s o l u t i o n of the wave equation (2.5) (in the simple band ap-proximation) with the eigfinenergy replaced by the observed bind-ing energy, £ (obs.). In e f f e c t the absence of knowledge about the p o t e n t i a l in the core region i s replaced by empirical i n f o r -mation about the binding energy which i s s e n s i t i v e to the core. The quantum defect functions are solutions of the equation [ - f y * * — l ' v - « ° B S J \ U > = ° < 2 - 1 0 ) and are v a l i d i n the region of large / . Since the observed ener-§y» 6 ( 0 ^ s * ) ' *-s n o t » i° general, an eigenvalue of the d i f f e r e n t i a l equation, the function w i l l not remain f i n i t e at the o r i g i n . How-ever, divergence of the s o l u t i o n at y* = o does not e f f e c t i t s v a l i d -i t y away from the o r i g i n . The general s o l u t i o n to t h i s equation i s a Whittker function (Seaton 1958). The ground state s-wave-function i s F S ( J T ) - l » < « y . ( < » ' 4 0 ( 2 . n ) where P CJC) i s the r a d i a l part of the wavefunction and given by P 5 ( - 0 = V e * * (2-12) The normalization constant, f4 , i s given by C^®-''^ } l s the angular part of the complete wavefunction and i s c a l l e d the s p h e r i c a l harmonic. The quantity ^ i s r e f e r r e d to as the e f f e c t i v e p r i n c i p a l quantum number, where c L J ^ (2.13) 2 Here, R i s the e f f e c t i v e hydrogenic Rydberg e /2K^a* • The e f f e c t i v e Bohr radius for the host c r y s t a l i s a* = ^ K j j / n ^ e 2 where m* i s the e f f e c t i v e mass in the valence band for the host c r y s t a l . Comparison of equation (2.9) and (2.13) implies the r e l a t i o n (2.14) Thus Ij. takes care of the e f f e c t of penetration of the electron o r b i t into the e l e c t r o n i c core of the atom. From the quantum defect method point of view, the wavefunctions are scaled from the binding energy ini; terms of ^ rather than i n terms of the e f f e c t i v e mass parameter a* or ft*. Thus, neglecting normaliza-t i o n , the hydrogenic . wavefiinction and the quantum defect wave-function, for the ground state, w i l l have fhe form .!?(*> - * . (2.15) V (2.16) where ^ = I for hydrogenic wavefunctions. In (chapter 6 of t h i s t h e s i s , quantum defect wave — function given by the equation (2.12) w i l l be used for the ca l c u -l a t i o n of the strength of electron-phonon i n t e r a c t i o n i n group I I I acceptors i n s i l i c o n . These quantum defect wavefunctions w i l l also be u t i l i z e d in the i n t e r p r e t a t i o n of the experimental r e s u l t s of the double-doped samples. 2.3 Configuration Interaction When a d i s c r e t e e l e c t r o n i c state i s degenerate with a continuum of states and the i n t e r a c t i o n Hamiltonian couples 12. them, then the eigenfunctions and the properties of the two are mixed. This i n t e r a c t i o n w i l l be c a l l e d "Configuration i n t e r a c t i o n " (Fano 1961). Due to t h i s i n t e r a c t i o n , the d i s c r e t e e l e c t r o n i c state i s modified and the l i n e shape of the corresponding e l e c t -r o n i c t r a n s i t i o n has a c h a r a c t e r i s t i c form. The exact c o i n c i -dence of the two i n t e r a c t i n g states makes the ordinary perturbation theory inadequate. The method discussed by Fano (1961) i s adopted. Consider o p t i c a l absorption by a system whose e x c i t -ed states c o n s i s t of a d i s c r e t e unperturbed state (with no i n t r i n -s i c degeneracy) L<\ , and a set of continuum s c a t t e r i n g states \k.^ > i n the zeroth approximation. The elgenfunction of the ground state of the system i s described by J<^\, . The Hamiltonian"}-{, d e s c r i b i n g the system i s H=K 0 + k / (2.17) where l<0 and the are the eigenstates ofT-^. Y~' repre-sents the i n t e r a c t i o n between \«()> and the • That i s K 0 H > = E j ' O H J * > -, e J k > (2.18) (2.19) «•< ' (2.20) where i s the matrix element representing the i n t e r a c t i o n bet-ween the states and I k/> I f lU.y represents the eigenstate of the i n t e r -acting system (in c l u d i n g the d i s c r e t e state ) then |0> = CL\cty + jckEk b ( f j IK> (2.21) where CL and b(E ) are functions of energy. The |t<y>4 are nor-13. malized per unit energy range (Fano 1961). The l i n e shape of the o p t i c a l absorption of the i n t e r a c t i n g system, apart from unimportant f a c t o r s , can be ex-pressed as (Shibatani and Toyozawa 1968) 1 CE) = V W N W > J a £ < 2 - 2 2 > 3 3 where {v\ i s the e l e c t r i c dipole t r a n s i t i o n operator. Following Fano (1961), the state \d^> i s modified by an admixture of states of the continuum, to the form / ! ( ? > = H > + " P P E K - ^ - (2.23) where "p i n d i c a t e s " P r i n c i p a l part of". The t r a n s i t i o n p r o b a b i l i t y to the state of the system i s 4*|rt|*> = ^  <P/M/f>SiyiA-<K|w|g>coSA (2'24) where the phase s h i f t i s introduced i n analogy with nuclear s c a t t e r i n g ( B l a t t and Weisskoff 1952) and i s given by F(E) represents a s h i f t of the resonance p o s i t i o n with respect to £ The phase s h i f t A a r i s e s due to the con f i g u r a t i o n i n t e r a c t i o n of |c<^> with the state |k"> . A changes r a p i d l y by~ f f as E traverses an i n t e r v a l | about the resonance at £ =r f The l i n e shape of the resonance t r a n s i t i o n to the d i s c r e t e state, which overlaps and i n t e r a c t s with the continuum states, i s given by (Fano 1961, Fano and Cooper 1965) <rCE) _ I W W 2 (Q2-1) 2 Q £ 14. Where 0 ^ 1 8 t n e absoprtion cross section for the photon of energy £ corresponding to the t r a n s i t i o n from the ground state, \ c^J to the state lU/> of the system, and CTJ*CE) represents the ab-sorption cross section corresponding to t r a n s i t i o n s to the c o n t i n -uum which i n t e r a c t s with the d i s c r e t e state \<£^ > . The other q u a n t i t i e s are defined as follows; t p 3 E 0 = -f- F i s the s e l f consis-r tent resonant energy of the modified state | P^> ; P = ATT H. and Q = (2.27) ^ i W i r R * P = 2.77" f-Kj,,^ } > an index of the strength of the configuration i n t e r a c t i o n , has the dimensions of an energy since the states |fc"^ > are normalized'per u n i t energy!1 according to Fano (1961). That i s , consequently Q w i l l be dimension less quantity. The terms on the r i g h t hand side of equation (2.26) may be interpreted r e s p e c t i v e l y as, continuum t r a n s i t i o n s , a symmetrical Lorentzian associated with the resonance t r a n s i t i o n and a term which represents the ef-f e c t of interference between resonant and continuum t r a n s i t i o n s . p i s the half-width of the resonant state J £•> and i s a measure of i t s l i f e t i m e . The excess t r a n s i t i o n p r o b a b i l i t y due to the resonant state over the continuum states determines the parameter Q. The r i g h t hand side of the equation (2.26), when plo t t e d as a function of £ for d i f f e r e n t values of Q, gives a fam-i l y of c h a r a c t e r i s t i c asymmetric curves for each value of Q with a dip ( c a l l e d anti-resonance) on the low (high) energy side for p o s i t i v e (negative) values of Q. Each of the curves, shown in F i g . 2 . 1 , i s for a fixed value of Q in the evergy region of i n t e r e s t . The appearance of a c h a r a c t e r i s t i c asymmetric l i n e shape i n the spectrum, has i t s , o r i g i n i n the equation (2.23) F i g . (2.1) Plot of the Asymmetric Line Shape function for d i f f e r e n t p o s i -t i v e Values of Q. The Anti-resonance Dip w i l l be on the High Energy Side for Negative Value of Q. 16. and (2.24), according to which the res u l t a n t i n t e n s i t y of the t r a n s i t i o n s should be obtained by adding the amplitudes of the state \V.y separately, and then taking the square of the i r sum. Equation (2.24) ind i c a t e s that the contributions to matrix e l e -i n t e r f e r e with opposite phase on the two sides of the resonance p o s i t i o n £ , since A i s an even function o f ( E — E , — F ), o <*• whereas C o s A i s an odd function of t h i s v a r i a b l e . At £ = — Q , there i s de s t r u c t i v e interference between the t r a n s i t i o n amplitude corresponding to the resonance absorption and the amplitude for t r a n s i t i o n s to the continuum states. This produces the a n t i -resonance dip in F i g . 2.|. The sign of Q determines on which side of the resonance the dip occurs. The sign of Q i s decided by the nature of the i n t e r a c t i o n (and, therefore by the experimentally observed l i n e shape). Q i s p o s i t i v e when the i n t e r a c t i o n i s r e -pu l s i v e (resonances of neutrons scattered by n u c l e i , corresponding to r e p u l s i v e i n t e r a c t i o n s outside the nuc l e i ) and negative when, the i n t e r a c t i o n i s a t t r a c t i v e (excitons, present case of i n t e r n a l acceptor states) ( P h i l l i p s 1966). The matrix elements i n equation (2.27) w i l l determine the sign of Q. In t h i s work the sign of Q i s determined from the observed absorption l i n e shape. width function p and the cross section xjr^ , can be regarded as independent of £ i n the proximity of the l i n e , the i n t e g r a t i o n of equation (2.26) gives the excess integrated cross section due to t r a n s i t i o n s to the d i s c r e t e state \eCy as t r a n s i t i o n to the modified d i s c r e t e state (f^> and the continuum , by the matrix elements /.PlMl'J) and <(K[ , I f the parameter Q, the l i n e s h i f t function p, the (2.28) (2.29) The theory of configuration i n t e r a c t i o n presented i n t h i s section w i l l be applied to explain, ( i ) the concentration dependent l i n e shape of t r a n s i -tions to the i n t e r n a l acceptor states i n s i l i c o n , s i n g l y doped with one of the group I I I impurities, ( i i ) the changes produced i n the boron i n t e r n a l spec-trum i n the s i l i c o n doubly-doped with boron and indium acceptors, due to the presence of indium impurity, and ( i i i ) the p o s i t i o n and structure of the phonon-assisted t r a n s i t i o n s corresponding to the external excited states i n S i ( I n ) . . The theory considered so far i s appli c a b l e to a si n g l e i s o l a t e d resonant state /<<^> embedded i n a continuum of states ( fc^> • When more than one resonance i s present and th e i r halfwidths are comparable to th e i r energy separation, these reson-ances can i n t e r a c t among themselves v i a the common continuum. The phonon-assisted t r a n s i t i o n s i n Si(Tn) e x h i b i t such properties. The theory discussed below w i l l be used i n the i n t e r p r e t a t i o n of p o s i -t i o n and structure of these phonon-assisted t r a n s i t i o n s i n S i ( I n ) . i Consider two close l y i n g resonances o( and & , whose eigenvectors_and •• eigenvalues are denoted by |rf^ > and [6)> , and E ^ and E ^ , r e s p e c t i v e l y . These resonant states are em-bedded i n a common continuum of states |K^ > and they can i n t e r a c t v i a t h i s common continuum. For such a system, the expression for the phase . A y y , , equivalent to the equation (2.25) i s ( P h i l l i p s 1966a, Fano 1961) " i a - n A = : — -r — : — - — 1 ™ E - E < - 5 E - £ * - £ . (2.30) Using t h i s value of A w i n equation (2.24), the l i n e shape f o r -mula becomes <^|M|J> = <k|M| 2 ) Cos A j L Q . t a n A . - l ] ( 2 > 3 1 ) or 18. Equation (2.32) takes care of the e f f e c t of i n t e r a c t i o n between the two close l y i n g resonances, v i a a common continuum. The be-haviour of equation (2.31) should be compared with equation (2.24) as E i s varied about the resonance p o s i t i o n . CoS "tan £>J; plays the same r o l e i n equation (2.31) as S'VI & does in equation (2.24), because both remain f i n i t e at the resonance point := U- ( i - e - £ = Ei ) and C o s vanishes at t h i s point. Consequently i n equation (2.32) both resonances, o( and & w i l l e x h i b i t t h e i r usual asymmetric l i n e shapes. Equation (2,31) has an important d i s i m i l a r i t y with equation (2.24). I t i s to be no-t i c e d that £. % to.v> £ • = T_ ^ H l H ^ l /(E-£i) takes on a l l values from — o o " t o + ^ i n the i n t e r v a l between the two reso-nances and thus causes a rapid v a r i a t i o n of <£MJ M . i n par-t i c u l a r ^JX\ Ml^y vanishes once, i n the i n t e r v a l betx^een the two resonances. This vanishing of ^ U | M | $ ^ > produces a sharp 'V' shaped dip between the two resonance peaks. The l i n e shape fun-c t i o n (2.32) w i l l e x h i b i t t h i s property. I f more than one continuum i s present, the s i t u a t i o n i s further complicated ( P h i l l i p s 1966). In actual p r a c t i c e c^UIMl^may not go to zero due to the presence of d i s s i p a t i v e e f f e c t s ( P h i l l i p s 1966) but a shallower dip w i l l be present. In chapter 6 i t w i l l be seen that the phonon-assis-ted peaks do show such behaviour. CHAPTER 3 GENERAL EXPERIMENTAL PROCEDURES 3.1 Appara tus Three d i f f e r e n t monochromators were used to cover the various desired spectral regions. ( i ) A Perkin-Elmer (Model 83) monochromator, modified to use a Bausch and Lomb grating with 30 grooves per mm., and blazed v. at i n the f i r s t order, was used i n the s p e c t r a l region 30 meV. to 44 meV. (263 C^o' to 370 CTT/ ). NaF r e s t s t r a h l e n plates were used, in the entrance and e x i t o p t i c s , to produce the r a d i a t i o n i n the desired s p e c t r a l region with minimum stray r a d i -a t i o n . The spectrometer was c a l i b r a t e d using atmospheric water va-pour absorption l i n e s (Blaine et a l . 1962). The s p e c t r a l s l i t width was, 0.16 mety". ( i i ) A Perkin-Elmer (Model 98G) monochromater, equipped with Bausch and Lomb gratings, was used in the sp e c t r a l regions 80 meV. to 86 mey., 140 to 154 meV., and 193 to 216 meV. The gratings used have the following c h a r a c t e r i s t i c s (a) a grating with 60 grooves/mm. and blazed at 16y^ in the f i r s t order (b) a grating with 75 grooves/mm. and blazed at 12/ty in the f i r s t order. Perkin-Elmer interference f i l t e r s #221-1790 & 221-1789 were used, with the grating blazed at 16// , to pro-duce the desired s p e c t r a l regions, i n the f i r s t order of the grat-ing, and i n the second order of the grating, r e s p e c t i v e l y . A Perkin-Elmer interference f i l t e r #221-200719 was used, with grat-ing blazed at \2yU , to produce the desired s p e c t r a l region i n the second order of the grating. The spectrometer was c a l i b r a t e d using atmospheric water vapour absorption l i n e s i n the 206 mev. region, Polystyrene f i l m absorption l i n e s i n the 150 meV. region and CO2 absorption band i n the 78 mev. region (Tables of ..wavenumber s 1961). The s p e c t r a l s l i t widths in the 78 and 150 mev. regions were 0.2 and 0.23 meV. r e s p e c t i v e l y . ( i i i ) A Perkin-Elmer (Model E-l) monochromator, equipped with a Perkin-Elmer grating blazed at 11/^ and with 57.6 lines/mm. was used to cover the 108 meV to 115 meV. spectral region. A s u i t -able interference f i l t e r was used to reduce unwanted r a d i a t i o n . The s p e c t r a l s l i t width was oz 0.12 meV. In a l l the three spectrometers, l i g h t from a globar source was chopped at 13 cycles/sec. near the entrance s l i t s and the transmitted r a d i a t i o n detected with C s l window vacuum thermocouple. The amplified s i g n a l was displayed on a chart recorder. During the experiments the spectrometers were continuously flushed with n i t r o -gen gas which removed atmospheric CO^ and water vapour absorption in the appropriate s p e c t r a l region. The metal l i q u i d dewar was positioned i n the spect-rometer such that the cooled p a r a l l e l sided s i l i c o n samples were at a focus point of the i n f r a - r e d beam. The sample holder in the l i q u i d helium dewar was connected to the helium bath v i a a t h i n -walled s t a i n l e s s s t e e l tube. A s t r a i n free sample mounting was used (White 1967a) and a small amount of grease, mixed with f i n e l y d i -vided s i l v e r powder, was applied to improve the thermal contact. Depending upon the type of study to be made, measurements were made at three temperatures corresponding to the b o i l i n g point of Helium (4.2°K), the b o i l i n g point of l i q u i d nitrogen (77°K)and nitrogen at approximately 2mm.. Hg pressure (T 55°K) . 3.2 Samples A l l samples were cut from commercially obtained, 21. si n g l e c r y s t a l Ingots grown along (III) c r y s t a l axis by the cz o c h r a l s k i method. A l l samples were uncompensated (no donor impurities were added to the c r y s t a l ) . Single c r y s t a l Ingots of ( i ) Si(Ga), ( i i ) S i ( I n ) , ( i i i ) Si(BJn) and (iv) Si(B,Ga),were studied. The sample thicknesses were chosen to give optimum mea-surements of absorption c o e f f i c i e n t , which was a compromise bet-ween the transmission, at the absorption l i n e maximum and at i t s base. Samples of proper dimensions were ground with #600 mesh car-borundum on glass and then polished with #600 mesh carborundum on Astromat c l o t h to a fi n e f i n i s h . In the case of the double-doped samples, Si(B,In) and Si(B»Ga),it was found d i f f i c u l t to get si n g l e c r y s t a l s with con-centration of In and Ga impurities greater than 2 x 1 0 ^ and 1 x lO^cm r e s p e c t i v e l y . 3.3 Determination Of Impurity Concentration Impurity concentrations i n s i n g l y doped samples, were determined by room temperature r e s i s t i v i t y measurements using a v a i l a b l e curves (Irvins 1962) with the proper c o r r e c t i o n , for the i o n i z a -t i o n energy of the impurityjapplied. To determine the concentrations of the two i n d i v i -dual acceptor impurities i n the double-doped samples, a s p e c i a l method was adopted. The method for Si(BIn) samples i s outlined below. Due to small binding energy ( ^ 4 4 meV.) of boron i n s i l i c o n , a l l the boron can be considered ionized at room temp-erature. In iridium, the ground state binding energy (155 meV.) i s about ten times larger than that for the f i r s t excited state. Thus excited states of indium, w i l l have n e g l i g i b l e population i n thermal equilibrium at the temperature of i n t e r e s t (Blakemore 1968) . Following Blakemore (1962) for a nondegenerate p-type si n g l y doped semiconductor P = o (3.1) N„- -3 where N g i s the acceptor impurity concentration (atoms cm )> P Q i s the number of free holes at temperature T°K, the i o n i z a t i o n energy of acceptor impurity, N v the e f f e c t i v e density of states i n the valence band at temperature T°K,-^t the Boltzman constant and f3 the degeneracy of impurity l e v e l . As a f i r s t approximation, the number of free holes, p, present in the c r y s t a l Si(B jIn) at temperature T°K i s the sum of free holes, p , coming from boron and p , holes coming from indium B I impurities. That i s P = P B + Pj (3.2) The hole density, p, was determined from room temperature r e s i s t i v i t y measurements and compared with I r v i n ' s (1962) curve and equation (3.1). These values of p for various samples were i n agreement, with i n experimental error, with the values obtained from room temperature H a l l e f f e c t measurements (Lamont 1968). (The free hole concentration due to the indium impurities was determined s p e c t r o s c o p i c a l l y following the method used by White (1967a) for S i ( B ) . Using these values of p and p^ . i n the equation (3.2), v a l -ues ofp were determined for d i f f e r e n t samples, Table (3.1) B shows values of N (indium impurity concentration and N (boron I B impurity concentration) for d i f f e r e n t Si(BIn) samples used. 3.4 Measurement And C a l c u l a t i o n Of Absorption C o e f f i c i e n t In a l l the experiments the absorption c o e f f i c i e n t 23. TABLE (3.1) BORON AND INDIUM IMPURITY CONCENTRATIONS IN S i ( B , L n ) SAMPLES Sample No. p ohm-cm -3 p cm N D cm n + 307o XT " 3 Nj cm + 15% 1 .5 + .03 3.3 x 1 0 1 6 1.9 x 1 0 1 6 4.9 x 1 0 1 6 2 .45 + .03 • 3.5 . x 1 0 1 6 1.9 x 1 0 1 6 5.6 x 1 0 1 6 3 .37 + .03 16 4.6 x 10 2.5 x 1 0 1 6 8.6 x L O1 6 4 .32 + .03 16 5.5 x 10 2.6 x 1 01 6 1.8 x 1 0 1 7 i s the measured room temperature r e s i s t i v i t y o f the sampl i s the f ree ho le c o n c e n t r a t i o n , at room temperature i n the samples. i s the boron c o n c e n t r a t i o n i n the samples i s the indium c o n c e n t r a t i o n i n the samples N B N I 24. o(.Clj) » o f t n e e x t r i n s i c sample under i n v e s t i g a t i o n was desired. An i n t r i n s i c sample was always mounted along with an e x t r i n s i c one. E x t r i n s i c transmission, "T^and i n t r i n s i c transmission, TQi were measured in each experiment. From the quotient transmission the e x t r i n s i c absorption c o e f f i c i e n t o((pJ was ca l c u l a t e d using the formula o-R 2)^e^s) -% "7IR2 C'***s) ( 3 - 3 ) where i s the e x t r i n s i c sample thickness, and R i s the r e f l e c t i -v i t y of s i l i c o n . The value of R chosen was obtained from McCarthy (1963) for the appropriate s p e c t r a l region being studied, c^(ij) as calculated from the equation (3.3), i s not s e n s i t i v e to the un-c e r t a i n t y i n R. Within experimental error, the r e f l e c t i v i t y for both the i n t r i n s i c and e x t r i n s i c s i l i c o n was taken constant over each s p e c t r a l region of i n t e r e s t (Parsons 1968b). Care was taken to prepare samples with uniform thickness. In the case of thin samples l i t t l e e r r or(:s ly£ ) was introduced by using a s l i g h t l y n o n - p a r a l l e l surface. Except i n the sp e c t r a l region of 16^4 , a l l the other regions studied did not show any s i g n i f i c a n t s i l i c o n l a t t i c e absorption. (Parsons 1968b), and Parsons and Bichard (1967) found that ( i ) i n the photon energy region 82 to 86 meV-, the l a t t i c e absorption c o e f f i c i e n t was approximately l i n e a r and constant, and ( i i ) the absorption c o e f f i c i e n t due to impurities and due to the l a t t i c e absorption may be taken as ad d i t i v e q u a n t i t i e s . Therefore, i n the study of the boron i n t e r n a l l i n e s (which appear in the region £r 82. to 86. raev.) the absorption c o e f f i c i e n t was determined from equation (3.3) and the known l a t t i c e absorption was subtracted from i t . 25. The i n t e r n a l acceptor l i n e s are superimposed on the external photoionization continuum. Two experimental quan-t i t i e s desired for these i n t e r n a l l i n e s are, ( i ) f u l l width of the l i n e at h a l f of i t s i n t e n s i t y preferred as halfwidth H) and ( i i ) the integrated cross section, vTj~ » of the l i n e . The separation of the absorption peak from the photoionization continuum i s c a r r i e d out by i n t e r p o l a t i n g v i s u a l l y the continuum away from the peak p o s i t i o n . This separation i s only approximate because photoionization continuum may behave in a d i f f e r e n t manner near the peak. (This has been discussed i n section (2.3)). The area under the absorption l i n e so obtained i s determined by cut-t i n g i t along i t s wings at a distance I £ from i t s peak p o s i t i o n . The p o s i t i o n of depends upon the measurability of the area under the peak p r o f i l e near the wings. In order to determine ap-proximately the area, l e f t out due to the termination of the ab-sorption l i n e at Vc , the Lorentzian function was integrated from to oO (see Appendix A). 26. CHAPTER 4 CONFIGURATION INTERACTION IN THE INTERNAL ACCEPTOR STATES 4.1 Introduction The external acceptor states in s i l i c o n , s i n g l y doped with group III impurities, l i e in the band gap of the host cry-s t a l . On the other hand, the i n t e r n a l acceptor states l i e i n the valence band i t s e l f and are degenerate with P^/2 • B^ o c^* s t a t e s ' The f a c t that the i n t e r n a l states are degenerate with the P^/2 continuum states, should modify t h e i r properties as compared to the external st a t e s . In the past these features have not been taken into account. The interference between a sharp e l e c t r o n i c t r a n s i -t i o n of an atom or molecule and an overlapping continuum of i o n i z i n g t r a n s i t i o n s , i s a known phenomenon (Fano 1961). A s i m i l a r effect, has been observed i n the exciton spectra i n semiconductors and i n s u l a t o r s ( P h i l l i p s 1966), i n f r a - r e d and Raman spectra of some perovskites (Rousseau and Porto 1968). The Ligand f i e l d spectra of transition-metal and other ions i n s o l i d (Sturge 1969) also show t h i s e f f e c t . In t h i s case, the interference i s between sharp purely e l e c t r o n i c t r a n s i t i o n s and broad v i b r o n i c bands. Due to t h i s i n t e r a c t i o n between a d i s c r e t e state and the continuum a c h a r a c t e r i s t i c asymmetric l i n e shape i s observed (see section 2.3). No experimental evidence of t h i s e f f e c t for impuri-t i e s i n homopolar semiconductors, such as s i l i c o n , has been report-ed i n the past. In the present study such an e f f e c t has been i d -e n t i f i e d i n group I I I acceptors in s i l i c o n . I t i s found that, ( i ) t h i s e f f e c t i s modified by impurity-impurity i n t e r -a ction and ( i i ) the i o n i z a t i o n energy of the impurity i n the host 27. c r y s t a l i s a c o n t r i b u t i n g factor to t h i s i n t e r a c t i o n . The concentration dependence of i n t e r n a l acceptor states i n boron-doped s i l i c o n (see F i g 4.1) has been studied pre-v i o u s l y (Parsons 1968a). The concentration dependent behaviour of the i n t e r n a l acceptor 2p' absorption l i n e was interpreted as due to the possible presence of a 2 s / i i k e state very near the 2p' state. I t w i l l be shown i n t h i s chapter that the observed l i n e shape i s , very l i k e l y due to the interference e f f e c t s between the 2p' state and the Regenerate continuum states. Internal ab-sorption l i n e s , due to gallium and indium acceptors in s i l i c o n , were also studied. Asymmetric l i n e shapes, s i m i l a r to boron i n t e r -nal l i n e s , were also abserved in the case of these i m p u r i t i e s . Quantitative estimates of the strength of configuration i n t e r a c t i o n and the non-stationary l i f e t i m e of the i n t e r n a l acceptor state 2p' has also been made. 4.2 Experimentally Observed Absorption Spectrum Internal absorption sptecta of Si(Ga) and Si(In) s i n g l e c r y s t a l samples were studied using standard experimental procedures (chapter 3). The gallium impurity concentrations in the samples studied were, 1.4 x 10*"* and 2.5 x 10*^ cm? Due to the low s o l -u b i l i t y of gallium in s i l i c o n , s i n g l e c r u s t a l samples with much 17 -3 greater concentration ( ^ 10 cm. were not commercially a v a i l a b l e . The impurity concentrations of the indium doped samples studied 16 17 3 ^  were 3.x 10 and 2 x 10 cm.) For the boron doped s i l i c o n some of the experimental r e s u l t s of Parsons (1968a) w i l l be used. (a) Boron Internal State 2p / In F i g . (4.1) some of the p l o t s , of absorption c o e f f i c i e n t vs. photon energy for the 2p^ i n t e r n a l l i n e , has been taken from Parsons (1968a). D i s t i n c t features of the spectrum are 81.0 81.2 81.4 81.6 81.8 82.0 82.2 82.4 82.6 82.8 83.0 83.2 83.4 83.6 83.8 PHOTON ENERGY (meV) ' ' • . Fig. (4.1) Observed Line Shape of the 2p^ Absorption Peak i n Boron-Doped S i l i c o n and i t s Impurity Concentration Dependence. The cross points represent the computed Fano function (with s u i t a b l e values of Q,r and H corresponding to the impurity concentration of = 4 . 3 x l 0 1 6 atoms/cm3) of equation (4.3). • 29. ( i ) a low energy t a i l i n the absorption peak with strong asymmetry about the peak maximum, and . ( i i ) a peak p o s i t i o n s h i f t towards the low-energy side, with increasing concentration, as well as an enhancement of the asym-metry. (b) Gallium Internal State 2y' Similar r e s u l t s for the gallium 2p' "line are shown in F i g (4.2). Here the asymmetric peak shape i s more evident. However, due to the r e l a t i v e l y low gallium impurity concentration, the features, such as observed for the boron i n t e r n a l -2p / peak 1 7 - 3 at a concentration of 4.5 x 10 cm., does not appear. The boron and gallium absorption l i n e p r o f i l e s along the wings, on the two sides of the peak p o s i t i o n , should be compared for nearly the same concentration i n F i g . (4.1) and F i g . (4.2). (c) Indium Internal State 2p"^  Results for the indium i n t e r n a l absorption l i n e 2p' also show s i m i l a r o v e r a l l features. Due to a very small photo-, e x c i t a t i o n cross section, large .photoionization absorption and the presence of atmospheric water vapour absorption peaks i n the region of the wings of the absorption l i n e , quantitative estimates were not attempted. Nevertheless, the c h a r a c t e r i s t i c asymmetric l i n e shape i s well established. 4.3 Configuration Interaction The theory of configuration i n t e r a c t i o n , discussed in section (2.3) w i l l be applied to the i n t e r n a l acceptor states i n s i l i c o n . (a) Low impurity concentration F i r s t a single acceptor impurity, introduced sub-PHOTON ENERGY (meV) F i g . (4.2) Observed Line Shape of the 2p X Absorption Peak in Gallium-Doped S i l i c o n and i t s Impurity Concentration Dependence.Error Flags Represent the Mean Deviation f o r Several Scans. 31. s t i t u t i o n a l l y in the host l a t t i c e of s i l i c o n , i s considered. The i n t e r n a l acceptor states, so produced, are degenerate with continuum Bloch states. This gives r i s e to two possible e x c i t a t i o n channels leading to the same f i n a l continuum state. One of these channels i s d i r e c t , photoexcitation to the continuum state, and the other i s e x c i t a t i o n to the resonant state ( i n t e r n a l state) with subsequent decay to the continuum state. The interference between these two channels leads to a modified absorption l i n e , depending upon the strength of the (interference) i n t e r a c t i o n . The presence of only one i n t e r n a l state (2p x) i s considered, where there i s no i n t r i n s i c degeneracy i n either 2p^ state or i n the con-tinuum states. Following the procedure of section (2.3), the modi-f i e d l i n e shape of the 2p / absorption l i n e w i l l be given by the equation (2.26). Shibatani and Toyozawa (1968) have reformulated Fano's (1961) theory i n order to apply i t to a s u b s t i t u t i o n a l i m -p u r i t y i n a host l a t t i c e taking into account the pe r i o d i c l a t t i c e s t ructure. They have also taken into account the energy dependence of P , Q and F . But the general form of the l i n e shape remains unchanged and i s the same as in equation (2.26). There new para-i( V r-meters p , Q and p a r e re l a t e d to y , Q and p i n a simple manner. In order to bring out the importance of th i s i n t e r a c t i o n i n the group I II acceptors in s i l i c o n , equation (2.26) w i l l be used for the i n t e r a c t i o n between an i n t r i n s i c a l l y non-degenerate state Jo^> and the overlapping continuum states | \C^> (b) E f f e c t of impurity - impurity i n t e r a c t i o n Next, the e f f e c t of impurity-impurity i n t e r a c t i o n (Baltensberger 1953) on the l i n e shape of the t r a n s i t i o n to the di s c r e t e state i s studied. At higher impurity concentrations, the impurity absorption l i n e s are considered to be broadened due to the impurity-impurity i n t e r a c t i o n . The concentration broaden-32. ing mechanism i s inhomogeneous i n nature and can lv- v i s u a l i z e d as f o l l o w s . An i m p u r i t y c e n t e r , i n the absence of any p e r t u r b a t i o n , g i v e s a c o n t r i b u t i o n to the a b s o r p t i o n l i n e at energy E of the l i n e . When the presence of p e r t u r b a t i o n due to the i m p u r i t y - i m p u r i t y i n -t e r a c t i o n i s taken i n t o account then, dvie to the random d i s t r i b u -t i o n of the i m p u r i t i e s , d i f f e r e n t i m p u r i t y centers w i l l experience v a r y i n g p e r t u r b a t i o n e f f e c t s . Consequently, the c o n t r i b u t i o n of a p a r t i c u l a r i m p u r i t y center to the a b s o r p t i o n l i n e w i l l he at s l i g h t l y d i f f e r e n t energy (E + /J^E). S u p e r p o s i t i o n of a l l such c o n t r i b u t i o n s coming from v a r i o u s randomly d i s t r i b u t e d i m p u r i t y cen-,~;.H.,„..ters g i v e s the observed c o n c e n t r a t i o n broadened l i n e p r o f i l e . F i g . (4.3) compares homogeneously and inhomogeneously broadened l i n e s . In c o n c l u s i o n , such an inhomogeneously broadened s t a t e should be con s i d e r e d as a band of c l o s e l y l y i n g .sharp s t a t e s . The process should produce a c o n c e n t r a t i o n broadened i n t e r n a l acceptor s t a t e i n the absence o f any i n t e r a c t i o n between the sharp i n t e r n a l s t a t e and the P^/2 c o n t i n u u r a s t a t e s . When i n t e r a c t i o n between the i n t e r n a l s t a t e and the continuum s t a t e s i s present, the c o n t r i b u t i o n of the s t a t i s t i c a l l y d i s t r i b u t e d i m p u r i t i e s to the l i n e , should be dressed w i t h the resonant l i n e shape f u n c t i o n given by equation (2.26). M a t h e m a t i c a l l y the new l i n e shape f u n c t i o n (apart from other broadening mechanisms (Parsons 1968b)) can be worked out as f o l l o w s : . - ... S t a r t i n g w i t h a c o n c e n t r a t i o n broadened l i n e ( w i t h -out c o n f i g u r a t i o n i n t e r a c t i o n between the sharp s t a t e and the con-tinuum)of L o r e n t z i a n shape and of e x p e r i m e n t a l l y observed co n c e n t r a -t i o n broadening h a l f w i d t h H (Parsons 1968b), the L o r e n t z i a n f u n c t i o n can be w r i t t e n as LCE> = - -~r E i - - • <4-'> (4.3) Broadening of the Absorption Lines (a) Homogeneous Broadening (b) Inhomogeneous Broadening The Fano l i n e shape function of equation (2.26) can be wr i t t e n as RE) = or FCE; - a (0 -0 . J L ^ + ^ TTP r (4.2) The resultant l i n e shape function i s obtained by con-v o l u t i n g the Lorentzian of-equation (4.1) with the function (4.2). The convoluted l i n e shape function i s (4.3) where p - p + H The peak p o s i t i o n of the function S(E) i s (4.4) (Deta i l s of t h i s c a l c u l a t i o n are given i n Appendix B). 4.4 In t e r p r e t a t i o n And Discussion The above arguements w i l l be used to understand the experimentally observed l i n e shape of the i n t e r n a l absorption l i n e . I t has been shown (Peterson 1964) that a hole bound to an accep-tor impurity, when excited to an i n t e r n a l state, spends some time near the impurity s i t e but may eventually propagate away i n the degenerate Bloch s t a t e s . This propagation of the hole can take place due to two perturbations; 35. ( i ) the impurity p o t e n t i a l i t s e l f which causes the non-sta t i o n a r y l i f e t i m e , of the i n t e r n a l state, and ( i i ) electron-phonon i n t e r a c t i o n which produces a phonon l i f e t i m e broadening of the observed absorption l i n e . A t h e o r e t i -c a l estimate (Parsons 1968 b; Peterson 1964) of the non-stationary l i f e t i m e of the i n t e r n a l states in boron-doped s i l i c o n gave the value /v 1 0 ^ sees. Thus, the state i s comparatively long l i v e d due to the fact that the coulomb p o t e n t i a l can not e a s i l y supply the required momentum to d e l o c a l i z e the hole. Experimental studies showed (Parsons 1968b; Parsons and Bichard 1967) that the non-sta t i o n a r y l i f e t i m e of the boron i n t e r n a l states can be neglected -12 i n comparison to th e i r electron-phonon l i f e t i m e ( r^r 10 sees.). Hence the energy eigen values of the i n t e r n a l states are not s i g -n i f i c a n t l y affected by the presence of the continuum. These con-clus i o n s were drawn from the study of samples of low impurity con-15 -3 centration ( 2 x 10 era.) where concentration broadening i s not s i g n i f i c a n t . I t w i l l be shown that even t h i s large non-st a t i o n a r y l i f e t i m e of ^ 10 sees, plays a very important part i n the i n t e r n a l spectrum. The presence of large phohon broadening introduces an incoherent or d i s s i p a t i v e character in the resonance l i n e and i t always tends to suppress the antiresonance (discussed in section ( 2 . 3 ) ( P h i l l i p s 1966a). Coherent broadening of the resonance l i n e due to the configuration i n t e r a c t i o n i s represented by p , the strength of the i n t e r a c t i o n . The parameter y / of the incoherent broadening can be introduced by convoluting the resonant l i n e shape given by equation (2.26) with a Lorentzian or Gaussian shape of width y . For V ^> P , the antiresonance disap-pears but the o v e r a l l asymmetry about the resonance p o s i t i o n £ o s t i l l remains. Any inhomogeneous e f f e c t on the resonant l i n e w i l l broaden i t incoherently. In order to demonstrate the importance of the con-36. f i g u r a t i o n i n t e r a c t i o n and i t s consequences for the boron i n t e r n a l state 2p"' i n s i l i c o n , the r a t i o ^L.* (see equation (2.29) has been determined from the experimental data (since the modified eigenfunctibn |P>^ >is not known e x p l i c i t l y ) . For the boron i n t e r -nal state 2p / the r a t i o obtained i s Assuming that V ^ .0066 mev., corresponding to the non-stationary l i f e t i m e of 10 ^ sees., one obtains \ Q \ % \b . An important as-sumption made here i s that the l o c a l i z e d impurity state and the i n -t e r a c t i n g continuum states are a l l i n t r i n s i c a l l y non-degenerate. For any i n t r i n s i c degeneracy of the two sets of states Fano's theor might have to be modified. • For such a small Value of p and large phonon broadening ( ~ 0.25 meV.) (Parsons and Bichard 1967), af t e r convoluting the shape function (2.26) with a Gaussian or Lorentzian of halfwidth Oc 0.25 meV., i t was found that a n t i -resonance dip i s almost suppressed. Only the c h a r a c t e r i s t i c a--symmetry about the resonance p o s i t i o n remains. In F i g . (4.1), thet 15 -3 curve for a boron impurity concentration of 2 x 10 cm. (where concentration broadening i s ne g l e g i b l e ) , shows only a small asym-metry around the peak p o s i t i o n . Even for a larger value of P and corresponding smaller value of Q obtained from equation (4.5), the convolution of the shape function (2.26) with the phonon broad-ening p r o f i l e , obscures the antiresonance i n the re s u l t a n t l i n e shape. Next i t i s necessary to explain the experimentally observed e f f e c t of impurity-impurity i n t e r a c t i o n (shown i n F i g . (4.1)) on the l i n e shape of the boron 2 p x l i n e . An important obser ved feature of the spectrum i s the s h i f t i n the peak p o s i t i o n of the absorption l i n e with increasing impurity concentration. This B h i f t , R(obs.), w i l l be taken as an experimentally observed para-meter. This s h i f t , R (obs.), w i l l be compared with the computed *In the experimental determination of O"Q, no d i s t i n c t i o n was made between the heavy and l i g h t hole valence bands i n s i l i c o n . 37. s h i f t , R ( c a l c ) , obtained by computation of the peak p o s i t i o n E x'. i n equation (4 ,4) for d i f f e r e n t values of concentration broadening halfwidth H- The function S(E) was computed (according to the pro-cedure given i n Section (4.3b) for d i f f e r e n t values of V and the corresponding values of Q (as given by equation (4.5)) where the concentration broadening halfwidth . H , was given the values 0.25, 0.5, L0 and 2.0 meV. F i g . (4.4) shows the computed r e s u l t s for R(calc.) as a function of the concentration broadening, H . Neg-2 a t i v e values of Q were taken because the experimental l i n e shape i n F i g (2.2) supports t h i s sign. Taking the experimentally ob-served peak s h i f t , R(obs.), i t appears, from F i g ( 4 . 4 ) , that for the boron acceptor impurities i n s i l i c o n the best f i t i s obtained for P 5^ 0.13 meV., and Q » -2.5 (see p l o t (c) i n F i g . ( 4 . 4 ) ) . This corresponds to a non-stationary l i f e t i m e of 10 ^ sees, for the boron 2p' state. But one should also note that even a small value of P ( 0.0066 meV.) gives a s i g n i f i c a n t peak s h i f t R ( c a l c ) , at higher concentration broadened halfwidths (see curve (a) i n F i g . ( 4 . 4 ) ) . At low values of H ( l e s s than 0.2 meV.), the s h i f t , R ( c a l c ) , i s n e g l i g i b l e . For the boron itf state, H ^ 0-2 16- -3 mev. corresponds to an impurity concentration 4=. 4 x 10 cm. (see Parsons 1968b). Thus, for the boron 2p' state, the e f f e c t discussed here w i l l not show a measurable s h i f t , R(obs.), at im-16- ' - 3 p u r i t y concentration £ h x 10 cm. Curves of F i g . (4.1) support t h i s contention. Comparison of R(calc.) and R(obs.) i n F i g . (4.4) for the boron 2p / state suggests that reasonable agreement i s obtained for ^ = .13 raetf. and Q-=--2.5. F i g . (4.5) shows the computed l i n e shape, S(E), a f t e r being convoluted with -phonon and d i s l o c a t i o n broadening-(Parsons 1968b) for these values of p and Q taking values of H'= 0.25, 1.0 and 2.0 meV. These 16 17 correspond to boron impurity concentrations of 4.3 x 10 ; 1 x 10 1 7 - 3 and 4.5 x 10 cm. r e s p e c t i v e l y _ * A c a r e f u l comparison of the pro-A computed curve for the l i n e corresponding to N - 4 . 3 x l 0 1 6 atoms/ci has-also been plotted i n F i g . (4.1) for comparison with the corres-ponding experimental curve. 0.6 - T 0.4 ^ 0.2 < LU QL 0 0 F i g . (4.4) T 1—^~ 1 r COMPUTED PEAK SHIFT ' h COMPUTED EXPERIMENTAL ( c ) - 9 \ (b) _ o (Q = - 2 . 5 , r = O.I3) . _ , roe* Si (B) 0", 77- = I.I wev, (Q = - 7 p r = 0.0066) Si (Ga) 07 = 0.5 wc/. (o - ie , r=o .oo66) Si(B) = 1.1 TYieV, 0.25 0.50 0.75 1.0 C O N C E N T R A T I O N BROADENING 'hi (meV) 1.25 Computed Peak S h i f t i n the Func t ion S ( E ) , as a Func t i on o f the C o n c e n t r a t i o n Broadening H a l f w i d t h , f o r D i f f e r e n t Combinat ions of Q and p Approp r ia te to Boron and G a l l i u m I m p u r i t i e s . The R a t i o T / ^ -i s f o r the I n t e r n a l Abso rp t i on L i n e Corresponding to Boron or G a l l i u m I m p u r i t i e s . Expe r imen ta l " P o i n t s (w i th Exper imen ta l E r r o r F l ags ) are fo r the 2p^ Boron I n t e r n a l L i n e . The E r r o r F l a g s i n the 00 Computed P o i n t s Represent the E r r o r In t roduced i n Computat ion due to the Expe r imen ta l E r r o r i n 07 yc-c F i g . (4.5) Computed Function S(E) ( i n Equation (4.4)) af-ter being Convoluted with Phonon and D i s l o c a -t i o n Broadening ( ~ -3 meV.), for Values of Q and P appropriate to Boron-Doped S i l i c o n , for D i f f e r e n t Concentration Broadening Hal f -, widths H. The Error Bars on the Computed Curve Represent the Error Introduced i n Computation due to the Experimental Value of °T/oS used i n Computa-t i o n . _ J • 1 _JL 1 I L _ 81.0 8 2 . 0 83 .0 8 4 . 0 •PHOTON E N E R G Y (meV) so. f i l e s - of F i g . (4.5) with the experimentally observed p r o f i l e s i n F i g . (4.1), confirms the importance of the i n t e r a c t i o n between a d i s c r e t e state and overlapping continuum in the case under study. Configuration i n t e r a c t i o n becomes more prominent with increasing concentration broadening. The l i n e becomes very asym-metric with a s h i f t i n i t s peak p o s i t i o n towards low energy ( t h i s s h i f t i s produced by concentration broadening). It i s shown i n Appendix B that, 6 the parameter of asymmetry i n the l i n e shape, i s dependent upon the concentration broadening halfwidth of the •'line/ Due to the weakness of configuration i n t e r a c t i o n (Fano 1961) i n the present case, i t s e f f e c t w i l l not be very prominent at low impurity concentration. I t i s the r e s u l t of im-purity-impurity i n t e r a c t i o n , which enhances the e f f e c t i v e i n t e r a c t i o n between an impurity band of sharp states ( c o n s t i t u t i n g the 2 p / state at higher impurity concentration) and the overlapping continuum states. This e f f e c t i v e enhancement can be v i s u a l i z e d as follows. Each of the sharp states in the impurity band w i l l i n t e r -act with a group of unperturbed continuum states in an energy range f about the energy of the sharp state. The energy value of each sharp state i s s l i g h t l y displaced from the p o s i t i o n of the 2p^ state without any impurity banding. The band width, f , of the continuum states, with which the s l i g h t l y displaced sharp state w i l l i n t e r a c t , w i l l be of a s l i g h t l y d i f f e r e n t energy than the band width which i n t e r a c t s with the 2p^ state without any im-p u r i t y banding. Thus on the whole, an e f f e c t i v e l y greater number of continuum states w i l l take part i n the i n t e r a c t i o n and so an e f f e c t i v e i n c r e a s e - i n the i n t e r a c t i o n r e s u l t s . F i g . (4.6) shows t h i s e f f e c t i v e increase in the i n t e r a c t i o n . The e f f e c t of phonon broadening on the p r o f i l e s i n F i g . (4.5) w i l l be only to broaden the l i n e as a whole without suppressing the c h a r a c t e r i s t i c asym-metric features of the p r o f i l e . One of the possible explainations of the p r o f i l e s i n F i g . (4.1) given by Parsons (1968a), was based on the presence of a 2s'' state very near to the 2p' state. As the impurity Ik> I a > A 3t Ik > ig (a) (b) F i g (4.6) The E f f e c t of Impurity-Impurity Interaction on the Configuration Interaction, (a) Low Impurity'Concentration (b) Higher Impurity Concentration. concentration i s increased the 2s and 2p states form impurity bands which couple together. As a r e s u l t of th i s coupling the over a l l l i n e shape should be as observed experimentally. But i n view of the theory presented here i t appears that configuration i n t e r -action can also produce the experimentally observed absorption l i n e shape for the boron 2p"^ absorption l i n e . From F i g . (4.1), i t i s c l e a r that there i s an observable c h a r a c t e r i s t i c ^symmetric feature i n the spectrum even at the low impurity concentration of 15 -3 2 x 10 cm. At t h i s low impurity concentration (where concen-t r a t i o n broadening i s n e g l i g i b l e ) the e f f e c t of coupling between the 2 s / state and the 2p' state should be small. Similar asym-metric behaviour, of 2p / absorption l i n e , also appears i n the case of the Ga-and I n — 2p' i n t e r n a l absorption l i n e s (see section 4.2). Therefore i t i s l i k e l y that i t i s the configuration i n t e r a c t i o n between the 2 p / state and the Pg/2 continuum states, which i s re s -ponsible for the observed absorption l i n e shape. The non-stationary l i f e t i m e of the boron 2p' state, as estimated from the present study, i s ^ 10 ^ sees. P h i l l i p s (1966a) has indicated that the phonon broadening can produce two co n t r i b u t i o n s ; ( i ) i t could broaden the resonant l i n e i.incoherently (which would reduce the magnitude of the antiresonance), and, ( i i ) i t could act to broaden the resonance coherently (by increasing the value of P ) and p a r t i c i p a t e i n interference e f f e c t s . The second c o n t r i b u t i o n of phonon broadening might be quite small as compared to the f i r s t one. Consequently the value of f ^ ,13 meV, as determined i n t h i s section, might not be completely due to the non-stationary l i f e t i m e of the 2p^ state. I t might contain some coherent c o n t r i b u t i o n of the phonon broad-ening. The non-stationary l i f e t i m e may be neglected compared to the electron-phonon coupling l i f e t i m e of the 2p / state, when one i s studying a low impurity concentration sample (in which the im-purity-impurity i n t e r a c t i o n e f f e c t i s n e g l i g i b l e ) . But when higher concentration samples are studied, c o n f i g u r a t i o n i n t e r a c t i o n can not be neglected. 43. 4.5 Deeper Impurities (Gallium And Indium) The preceding arguments may be applied to the i n -terpretation, of the spectra of the deeper impurities, gallium and indium, which e x h i b i t s i m i l a r behaviour but with d i f f e r e n t strengths of i n t e r a c t i o n between the 2p' i n t e r n a l state and the con-tinuum states. Recently the quantum defect method has been applied to the theory of deep impurity centers (Bebb and Chapman 1967, 1969) (see section (2.2)). I t i s found that the presence of deeper centers i n the host l a t t i c e modifies both the ground state and the continuum state wavefunctions. The t r a n s i t i o n p r o b a b i l i t y from the ground state to the 2p"" l i k e state decreases more r a p i d l y as one goes from the shallow to deeper centers, than expected from the scaled hydrogenic model (Bebb and Chapman 1969) . The photoionization absorption shows a corresponding increase i n i t s o s c i l l a t o r strength and the d i s t r i b u t i o n of absorption in the i o n i z a t i o n continuum depends upon the i o n i z a t i o n energy of the im-p u r i t y in the host l a t t i c e . These features of photoexcitation are also exhibited by the 2p / i n t e r n a l s tate. The integrated absorption cross sections for the 2p^ state, associated with boron, gallium and indium impurities i n s i l i c o n are compared in Table (4.1). I t i s the r a t i o fT_ , i n equation (4.5), which determines the parameter Q. Any r e d i s -t r i b u t i o n between & <^ -, and the p o s i t i o n of the 2p / absorption l i n e on the photoionization cross section spectrum, w i l l be the deciding factors for the value of Q. Thus, the l i n e shape of the 2p^ absorption w i l l be correspondingly modified. Such a modifi-c a t i o n should be r e l a t e d to the modification in both the ground state and the continuum states eigenfunctions, |$^ > and of the deeper centers. Using the experimentally mea-sured, \rp and eg- » values of Q (taking P ~ 0.0066 mev.) are given i n Table (4.1) for gallium and indium centers. A value of Q Ziz -7 for the gallium center as compared to Q = -16 for the boron centers, stresses greater m o d i f i c a t i o n of the i n t e r n a l TABLE (4.1) ESTIMATED VALUES OF PARAMETER "Q" FOR THE 2p''-INTERNAL LINES OF GROUP III IMPURITIES IN SILICON 3 Impurity E Q QJ-E I ^ (Tngy.) 1.9 1.1 -16 Ga 1.54 0.5 In 1.25 0.1 -3 a) The values of Q are for T = .0066 meV. E i s i o n i z a t i o n energy of the impurity. 1 / E i s 2p -peak p o s i t i o n , o 4 5 . s t a t e |oT> of the g a l l i u m c e n t e r s . In F i g . (4.2) the curve for 16 -3 g a l l i u m (2.5 x 10 atoms cm. ) has a greater asymmetry than the 16 *"3 curve i n F i g . (4.1) for boron (4 x 10 atoms cm. ). (Using the v a l u e of P =.13 meV., as i n the boron case, the corresponding value of Q obtained f o r g a l l i u m i s approximately -1.9). Greater s h i f t i s expected, f o r the g a l l i u m 2 p / t r a n s i t i o n than f o r the boron 2p^ t r a n s i t i o n , f o r the same value of H- The reason f o r t h i s i s the lower value of Q ( i f f remains the same i n g a l l i u m as i n boron) which i s due to the f a c t that 2p t r a n s i t i o n occurs at the maximum of the p h o t o i o n i z a t i o n c r o s s - s e c t i o n of the g a l l i u m i m p u r i t y . The higher value of Q i n boron i s because the boron 2p / / t r a n s i t i o n l i e s on the decreasing t a i l of the p h o t o i o n i z a t i o n ab-s o r p t i o n . The magnitude of 0~~ i s an important q u a n t i t y i n d e t e r -mining the v a l u e of Q. The t r a n s i t i o n energy of the indium 2p^ t r a n s i t i o n occurs on the r a p i d l y i n c r e a s i n g part of the p h o t o i o n i z a t i o n ab-s o r p t i o n cross s e c t i o n . The low value of 2E_ oc 0.1 mel/.gives Q ^ -3. f o r P ~ 0.0066 mev. Due to the very small photo-i o n i z a t i o n c r oss s e c t i o n and the presence of atmospheric water va-pour peaks on e i t h e r s i d e of 2p X t r a n s i t i o n , q u a n t i t a t i v e study has not been attempted. Moreover, phonon-broadening h a l f w i d t h and c o n c e n t r a t i o n broadening h a l f w i d t h (Parsons 1968b) determination i s not p o s s i b l e f o r the indium 2p"' t r a n s i t i o n . However the over-a l l behaviour of the 2p^ s t a t e of indium, shows the same c h a r a c t e r -i s t i c asymmetric fe a t u r e s i n i t s l i n e shape, as f o r boron and g a l l i u m . CHAPTER 5 A STUDY OF SILICON DOUBLY-DOPED WITH BORON AND INDIUM ACCEPTORS 5.1 Introduction The acceptor impurities i n a s i l i c o n c r y s t a l l a t -t i c e may be perturbed by l a t t i c e v i b r a t i o n s , c r y s t a l imperfections and other impurities. A l l these perturbations contribute to the halfwidths of the acceptor impurity absorption l i n e s . In the past a l l the studies of group I I I acceptor impurities i n s i l i c o n have been concerned with s i l i c o n doped with a s i n g l e impurity species (Onton et a l . 1967; White 1967; Parsons and Bichard 1967). In the present i n v e s t i g a t i o n two d i f f e r e n t acceptor im-p u r i t i e s , boron and indium, were introduced into the same s i l i -con ingot. Such c r y s t a l s are r e f e r r e d to as double doped s i l i -con ( S i ( B , I n ) ) . The purpose of studying such systems i s to i n -ve s t i g a t e the e f f e c t of perturbation produced by the presence of the one impurity species on the impurity absorption l i n e s of the other species. T r a n s i t i o n s to the external and i n t e r n a l acceptor states associated with both, the boron and indium impurities, were studied i n the Si(B,In) samples with varying indium concentrations. The experiments showed s i g n i f i c a n t e f f e c t s of the presence of i n -dium impurities on the boron acceptor states. I t was found that an increasing indium concentration in Si(B,In) a f f e c t s the boron absorption spectrum i n a d i f f e r e n t way than an increasing boron concentration in S i ( B ) . The boron i n t e r n a l absorption l i n e s were broadened more than the external ones. In the boron i n t e r n a l spectrum of Si(B,In), h a l f -widths and integrated absorption cross sections of the boron 2p' l i n e were determined. A s i g n i f i c a n t decrease i n the integrated absorption cross section of the boron 2p' l i n e was observed, due to the presence of indium impurities in these samples. A peak s h i f t towards the low energy side, in the peak p o s i t i o n of the boron 2p' l i n e , was also observed. The indium external and i n -t e r n a l absorption spectra of Si(B,In) were also studied. The boron concentration of ~ 2 x 10 atoms cm in these samples was not s u f f i c i e n t to a f f e c t appreciably the indium external spectrum. These experimental r e s u l t s are presented in sections 5.2 (a) and 5.2 (b). The theory of neutral impurity s c a t t e r i n g was used to i n t e r p r e t the broadening of the boron external and i n t e r n a l absorption l i n e s in Si(B,In). The presence of indium impurities, produced changes i n the configuration i n t e r a c t i o n between the boron 2p^ state and the v a i e n c e hand Bloch states. The de-crease in the integrated absorption cross section and s h i f t i n the peak p o s i t i o n of the boron 2p^absorption l i n e i n s i ( B , I n ) , were associated with t h i s change i n the configuration i n t e r a c t i o n . Sec-tions 5.3, 5.4 and 5.5 discuss i n t e r p r e t a t i o n of the experimental r e s u l t s . 5.2 Experimentally Observed Spectrum Of Si(B,In) The external and i n t e r n a l absorption spectra of Si(B,In) s i n g l e c r y s t a l samples were studied using standard ex-perimental procedure (chapter 3). The impurity concentrations of boron and indium centers, in the various samples studied, were determined according to the procedure discussed in section (3.3). Table (3.1) summarises the r e s u l t s for the impurity concentrations. (a) The external absorption spectrum of Si(B.In) The external spectrum of boron i n Si(B,In) was o studied at low temperature (less than l o K) for increasing indium concentration in the four samples l i s t e d i n Table (3.1). The over-4 8 . a l l behaviour of the boron external l i n e s in Si(B,In), with i n -creasing indium concentration, was compared with a s i m i l a r study of the boron external l i n e s in S i ( B ) . F i g (5.1) (b) shows the r e s u l t s for the absorption cross-section for the boron l i n e s 2 through 9 (see Colbow 1963 for the notation) in Si(B,In) (Samples #1 and 4). Similar r e s u l t s for the boron l i n e s in the two Si(B) 16 16 samples (boron concentration of % 1.3 x 10 and 4.3 x 10 atoms -3 cm r e s p e c t i v e l y ) are presented in F i g . (5.1) (c) for comparison. In F i g . (5.1) (a) the halfwidth, .(H - H ), of the boron l i n e 2 in the d i f f e r e n t Si(B,In) samples has been plotted, 1/3 1/3 as a function of (N^ .) . (The quantity, (N^ .) has been chosen as abscissa, to compare these r e s u l t s with the corresponding re-s u l t s of White (1967a) for the boron ...line 2 in S i ( B ) ) . i s the experimentally observed t o t a l halfwidth of the l i n e 2 in Si(B,In), H i s the c o n t r i b u t i o n to the halfwidth of l i n e 2 due to the i n -t e r a c t i o n of neighbouring boron impurities in Si(B,In) (Baltens-berger 1953), and N^ . i s the indium impurity concentration in Si(B,In). Similar r e s u l t s for the t o t a l halfwidth, Hg , of the boron l i n e 2, i n S i ( B ) , are plotted for comparison. F i g (5,1)(a) shows that the halfwidth of peak 2 in Si(B,In) samples, increases slowly with the increasing impurity concentration from 4.9 x 1 0 ^ to 1.8 x 1 0 ^ atoms cm ^. The h a l f -width of the boron peak 2 in Si(B) increases at a faster rate with the increasing boron concentration. Peaks 3, in Si(B,In) i s very much broadened with the increasing indium concentration and i t over-laps with peak 4 (see F i g . (5.1) (b)). Such behaviour i s not found i n peak 3 and 4 i n Si(B) with increasing boron concentration ( F i g . (5.1) ( c ) ) . F i g . (5.1) (b) shows that the peaks 5, 6, 7, 8 and 9 become very broad and are washed away by increasing indium concen-t r a t i o n . In Si(B) (at least up to impurity concentration 4.3 x 16 —3 10 atoms cm ) these peaks do not become so broad ( F i g . (5.1) ( c ) ) . This d e s c r i p t i o n of the absorption spectrum in F i g . (5.1) (a), (5.1) (b) and (5.1) (c) shows that the increasing indium con-ce n t r a t i o n i n Si(B,In) a f f e c t s the boron absorption spectrum in a d i f -ferent way than the increasing boron concentration in S i ( B ) . 49. F i g . (5.1) (a) The Broadening of the Boron External Line 2 i n Si(B,In). H i s the T o t a l Ha Ifwidth of the Boron External Line 2 in Si(B,In) Corrected for Instrumental Broadening, and 11 i s the Concentration Broadening of the Boron External Line 2 Due to the Presence of Boron Impurities i n S i (B,In). The Concentration Broadening Halfwidth, H , (up to (N )?** = 2.3 x 10 5 cm" ) for the Boron External Line 2 in Si(B) i s from White (1967). N and N are the Indium and the Boron Concentration Respectively. T « 6 K. The Error Flags Indicate the Mean Devi-ation for Several Scans (b) The Boron External Absorption Spectrum of Si(B,In). Spectral S l i t w i d t h as 0.16 meV. T « 6 K. The Peak P o s i t i o n of Various Absorption Peaks i s shown by a V e r t i c a l Line. (c) The Boron External Absorption Spectrum of Si ( B ) . 1 < Si(B) £ Sl(B,In) (a) Boron Line 2 (N T or N „ ) 1 / 3 ( x 10 5 cm 1 B T T 30 15 ( b ) A Boron External Spectrum Si(B.In) , 7 l i N =2.6 x 10 w ,N =1.8 x 10 atoms/cm N =1.9 x 10 ,N =4.9 x 10 B atoms/cm 30 15 33 37 41 Photon Energy (meV.) 45 50. In order to see the e f f e c t of the boron impurities on the external absorption spectrum of the indium impurities i n Si(B,In), the indium external spectrum of Si(B,In) samples was also studied. The d i f f e r e n t absorption l i n e s of indium were not 16 affected, by the presence of boron impurities ( 2 x 10 atoms -3 cm ) i n various samples, l i k e those of boron in Si(B,In). A study of Si(B,In) samples, with much greater boron concentration, would be needed to observe any s i g n i f i c a n t e f f e c t on the indium absorp-tio n l i n e s , (b) Internal Absorption Spectrum of Si(B,In) The i n t e r n a l absorption spectra of the samples, with the same impurity concentrations as i n those discussed in section (5.2) (a), were studied at low temperature. F i g . 5.2 (a) shows the plots of absorption c o e f f i c i e n t (including the back-ground photoionization continuum) vs. photon energy for the boron 2p' i n t e r n a l peak i n the two Si(B,In) samples; A s i m i l a r plot for the boron 2p' i n t e r n a l peak in Si(B) (having boron concentration 16 -3 2J 1.2 x 10 atoms cm ) i s also given for comparison. F i g . 5.2 (a) shows that the boron 2p / peak i s strongly affected by the presence of indium impurities in Si(B,In). The peak height de-creases and the absorption l i n e becomes broadened with increasing indium concentration. The peak p o s i t i o n of the boron 2p / l i n e i n Si(B,In), with respect to i t s p o s i t i o n in S i ( B ) , s h i f t s towards lower energy with increasing indium concentration. There appears to be a c h a r a c t e r i s t i c asymmetry about the peak p o s i t i o n i n the l i n e shape of the boron 2p / peak i n Si(B,In) samples. The halfwidths, H^, of the 2p' l i n e i n the d i f f e r e n t Si(B,In) samples were measured and corrected for instrumental broadening (Parsons 1968c). The presence of boron impurities i n Si(B,In) samples produced the concentration broadening of the 2p' l i n e due to i n t e r a c t i o n of neighbouring boron impurities (Baltensberger 1953). This c o n t r i b u t i o n , H , was subtracted from 51. F i g . (5.2) (a) The Behaviour of the Boron 2p Peak i n Si(B,In) (Sample #1 N ft; 1.8 x 10 1 7 , N » 2.6 x 10 atoms cm ; #4 N K.4.9 xlO 1 0 , Ng = 1.9 x 10^ Qatoms cm ) i s Plotted for Com-parison. T 6 K and Spectral S l i t w i d t h 0.2 meV. The Error Bars Represent Mean Dev-i a t i o n for Several Scans. (b) The Interruption Broadening of the Boron 2p' Peak i n Si(B,In). H i s the Tot a l H a l f -width of the Boron 2p' Peak in Si(B,In), and H c i s the Concentration Broadening Halfwidth of the Boron 2p' Line Due to the Presence of Boron Impurities i n Si(B,In). The Concentra-ti o n Broadening Halfwidth, H , for the Boron 2p'Line in Si(B) i s from Parlons (1968). N and N are the Indium and the Boron Concentra-R o tions Respectively. T % 6 K. The Error Bars Represent the Mean Deviation for Several Scans. HALFWIDTH ( H T - H C ) OR H B (meV.) A B S O R P T I O N C O E F F I C I E N T ( c m " 1 ) ro O J ^ 52. the experimentally observed halfwidth, H^, of the boron 2p'/ peak in Si(B,In). The remaining halfwidth, (H -H ), i s plotted i n F i g . 5.2 (b) curve ( i ) as a function of . (N^ i s the indium im-p u r i t y concentration in Si(B,In). Similar r e s u l t s for the t o t a l halfwidth, H , of the boron 2p' peak in the various samples B 15 17 of Si(B) (having a boron concentration of 2.6 x 10 to 4.5 x 10 -3 atoms cm ) are plotted in F i g 5.2 (b) ?curve ( i i ) for comparison (Parsons 1968b). The curves ( i ) and ( i i ) i n F i g . 5.2 (b), when extrapolated to zero impurity concentration, give an intercept on the ordinate. This intercept i s the phonon l i f e t i m e broadening and the d i s l o c a t i o n broadening for the boron 2p^ peak (Parsons and Bichard 1967). The plots ( i ) and ( i i ) of F i g . 5.2 (b) appear, to follow the same behaviour up to the indium and boron concentra-tions of 4.9 x 1 0 ^ and 4.3 x 1 0 ^ atoms cm^ r e s p e c t i v e l y . For the higher impurity concentrations the two plots deviate s i g n i f i -c a n t l y from each other. This d i s s i m i l a r behaviour could be as-sociated with the d i f f e r e n t nature of boron-indium i n t e r a c t i o n and the boron-boron i n t e r a c t i o n . The integrated absorption cross-section for the boron 2p^ peak was also measured in the various samples of Si(B,In) In determining the integrated absorption cross-sections of the various 2p^ peaks a Lorentzian c o r r e c t i o n (see Appendix A) was taken into account. In Table (5.1) the integrated absorption cross sections for the boron 2p^ peak in d i f f e r e n t samples of Si(B,In) are tabulated. A s i g n i f i c a n t decrease i n the absorption cross section was found due to the presence of indium impurities, as compared to the sin g l y doped boron sample. Within the experi-mental error ( ^ 407o) no change was found i n the photoionization absorption cross section i n the v i c i n i t y of boron 2p^ l i n e . The 2p indium i n t e r n a l peak in Si(B,In) was also observed in samples # 1 and 4. The integrated i n t e n s i t y appeared to be greater in the Si(B,In) samples, compared to i t s i n t e n s i t y i n the S i (In) samples of approximately the same indium concentra-TABLE (5.1) INTEGRATED CROSS-SECTION FOR B-2p' LINE IN Si(B,In) Sample # N^ cm + 15% N cm B + 30% 2 meV-cm + 40% 1 END-1 1.8 x 1 0 1 7 2.6 x 1 0 1 6 3.5 x 1 0 1 6 2 MIDD-lb 8.6 x 1 0 1 6 2.5 x 1 0 1 6 3.7 x 1 0 1 6 ' 3 MIDD-la 5.6 x 1 0 1 6 " 16 1.9 x 10 5.1 x l 5 1 6 4 MIDD-2a 4.9 x 1 0 1 6 1.9 x 1 0 1 6 5.0 x 1 0 1 6 Si(B) o 1.2 x 1 0 1 6 l l x l O 1 6 i s the indium impurity concentration i s the boron impurity concentration i s the integrated absorption cross s e c t i o n . N„ t i o n . Due to the very small absorption cross section and the pre-sence of atmospheric water vapour absorption l i n e s , q u antitative estimates of the absorption cross section becomes uncertain. The study of the indium 2p'" l i n e i n Si(B,In) samples, having a boron concentration much greater than was a v a i l a b l e (boron concentration 16 3 i n the samples a v a i l a b l e was 2 x 10 atoms cm ), i s needed to make conclusive decision about the increase of absorption cross section i n the indium 2p' l i n e . 5.3 Neutral Impurity Scattering In this section the theory of s c a t t e r i n g of a free c a r r i e r , by neutral impurities i n the host c r y s t a l , i s b r i e f l y discussed. I t w i l l be used i n the i n t e r p r e t a t i o n of the external and i n t e r n a l absorption spectra of Si(B,In) samples. The s c a t t e r i n g of a free c a r r i e r by randomly d i s t r i -buted neutral impurities i n the host c r y s t a l l a t t i c e i s treated as e l a s t i c s c a t t e r i n g of slow electrons by hydrogen l i k e atoms (Massey and Moiseiwitsch 1951; Erginsoy 1950). The c a r r i e r i s free and i s not bound to any p a r t i c u l a r impurity. I t undergoes an e l a s t i c s c a t t e r i n g event such that the c a r r i e r makes a t r a n s i -t i o n from an i n i t i a l state corresponding to the wave vector « , to a state with wave vector . N e g l i g i b l e energy i s l o s t by the c a r r i e r during an e l a s t i c c o l l i s i o n with the impurity atom. Therefore, the wave vector terminates on the same constant energy surface as the wave vector . For a simple band s t r u c -ture, the t o t a l e l a s t i c s c a t t e r i n g p r o b a b i l i t y for such e l a s t i c c o l l i s i o n s i s r e l a t e d to the d i f f e r e n t i a l s c a t t e r i n g cross section by the expression (Smith 1963) = fv* v- Ja.Tr o-(e) sine (i-cose) de ( 5 > 1 ) where -£ i s the re l a x a t i o n time, \)- i s the v e l o c i t y of c a r r i e r , 9 i s s c a t t e r i n g angle, cr(©) i s the d i f f e r e n t i a l s c a t t e r i n g cross section, and f\J t e r i n g centers. i s the concentration of neutral impurity seat-To determine the s c a t t e r i n g cross section, $-{Q) » Erginsoy's (1950) approach i s adopted. For a small incident car-r i e r v e l o c i t y , the s c a t t e r i n g i s s p h e r i c a l l y symmetric and only the zeroth order phase s h i f t need be considered. The c a r r i e r ex-change e f f e c t s and the e f f e c t of p o l a r i z a t i o n of the atom by the incident charged c a r r i e r are taken into account. The d i f f e r e n t i a l s c a t t e r i n g cross section, under these conditions, can be written as 2 0 \ <rte) = 4 r rw !> / (5.2) where O and ync are the v e l o c i t y and e f f e c t i v e mass of the scat-tered c a r r i e r r e s p e c t i v e l y . The parameter a*(=iZK^/m*e^)is the e f f e c t i v e Bohr radius of the impurity atom (Kohn 1957) s c a t t e r i n g the c a r r i e r . i s the d i e l e c t r i c constant of the host l a t t i c e and m* i s the e f f e c t i v e mass of the c a r r i e r bound to the s c a t t e r i n g center . Combining equation (5.1) and (5.2), one obtains J _ 20 f a*N •C ml (5.3) Equation (5.3) w i l l be used in explaining the broad-ening of the external and i n t e r n a l absorption peaks of boron im-p u r i t i e s i n Si(B,In). 5.4 Broadening Of Boron Impurity States In Si(B,In) In the Si(B,In) samples, when the boron i n t e r n a l and external l i n e s are photoexcited, the photon energies are i n -s u f f i c i e n t to excite the indium acceptor impurities. The i o n i z a -t i o n energy of the indium impurity ( 155 meV.) i s much greater than that of the boron impurity ( 44 meV.) in s i l i c o n . Thus, the indium impurities remain neutral during the e x c i t a t i o n of the 56. boron e x t e r n a l or i n t e r n a l absorption l i n e s . In the hydrogenic approximation, the hole bound to the acceptor i m p u r i t y could be thought of as moving i n a Bohr o r b i t w i t h an e f f e c t i v e Bohr r a d i u s a*. When t h i s h o l e i s e x c i t e d to the v a r i o u s e x c i t e d acceptor s t a t e s ( i n t e r n a l or e x t e r n a l ) i t moves i n the o r b i t s of greater r a d i i , but s t i l l remaining l o o s e l y bound to the acceptor i o n . When the hole i s e x c i t e d to the continuum s t a t e s , i t moves i n the Bloch s t a t e s extending throughout the c r y s t a l u n t i l i t i s f i n a l l y captured by an i o n i z e d acceptor and r e t u r n s to the ground s t a t e . The boron h o l e , i n an acceptor s t a t e , can be t r e a t e d as moving i n -x the valence band but w i t h an e f f e c t i v e mass, w»c and remaining i n the r e g i o n near that boron i m p u r i t y . T h i s e f f e c t i v e mass de-pends upon the b i n d i n g energy of that acceptor s t a t e . The h o l e i n the higher e x c i t e d s t a t e s w i l l have an e f f e c t i v e mass c l o s e r to that of a f r e e hole i n the valence band. The n e u t r a l indium atoms i n the S i ( B , I n ) c r y s t a l perturb the holes bound to the boron i m p u r i t i e s . T h i s p e r t u r b a t i o n broadens the acceptor s t a t e s . This e f f e c t can be thought of as a r i s i n g from the s c a t t e r i n g of the boron hole (considered to be moving i n a Bohr o r b i t , w i t h i n the e f f e c t i v e mass approximation) i by the n e u t r a l indium i m p u r i t i e s l y i n g i n the v i c i n i t y of i t s o r -b i t . In order to determine the r e l a x a t i o n time for t h i s s c a t t e r i n g process, the theory of n e u t r a l impurity s c a t t e r i n g , discussed i n s e c t i o n (5.3), can be used. From the u n c e r t a i n t y p r i n c i p l e the f u l l w idth at h a l f maximum of the absorption l i n e , due to the t r a n s i t i o n between the broadened acceptor s t a t e s , i s given by Av= A=- (5-4) ^ 2.TT Z where ~c i s the mean l i f e t i m e of the hole (and i s given by equa-t i o n ^ . 3 ) ) . T h i s c o n t r i b u t i o n to the width of the absorption l i n e w i l l be c a l l e d the " I n t e r r u p t i o n broadening h a l f w i d t h " (Breene 1957). Equation (5.3) shows that the r e l a x a t i o n time, -^ , w i l l depend upon the indium i m p u r i t y c o n c e n t r a t i o n i n the S i ( B , I n ) c r y s t a l . The parameter y-c i s i n v e r s e l y proportional to the ef-f e c t i v e mass of the boron hole. This means that 1-r , for the boron hole i n the higher excited states, w i l l have a larger value because the e f f e c t i v e mass of the boron hole i n the higher ex-c i t e d states becomes smaller and approaches the e f f e c t i v e mass of a free hole i n the valence band. Consequently the c o n t r i b u t i o n of i n t e r r u p t i o n broadening to the halfwidth of the absorption l i n e w i l l be greater for higher energy absorption l i n e s . In order to make quan t i t a t i v e estimates of the two contributions to the r e l a x a t i o n time T discussed above, equation ( 5 . 3 ) w i l l be used. I t i s assumed that the hole bound to boron, and i n an excited state (with the p r i n c i p a l quantum number n), be-haves as i f i t were moving i n the valence band with an e f f e c t i v e mass tt?c . I t i s further assumed that the boron hole does not r e -main bound to a sing l e impurity but can move through the c r y s t a l . (In the actual s i t u a t i o n the boron hole remains i n the region near one impurity atom). Within the e f f e c t i v e mass approximation the e f f e c t i v e mass of the c a r r i e r and the binding energy of the accep-tor state are r e l a t e d by t, % 3, ™C- = - - ^ - 7 7 — • £ 6 ( 5 . 4 a) e7 The neutral indium atom i s taken as a sc a t t e r i n g center for the scattered boron hole. Within the e f f e c t i v e mass approximation, the hole bound to indium i s also treated as moving around the indium ion in i t s ground state o r b i t . Since indium i s a comparatively deep acceptor impurity, a refinement to the e f f e c t -ive mass approximation approach i s made. The usual scaled hydro-genic model (see section 2 . 2 ) i s not used i n the determination of the e f f e c t i v e radius, Y£ , for the ground state o r b i t of the hole bound to the indium atom. The quantum defect ground state wave-function i s used to make a better estimate of Yj, (see section ( 2 . 2 ) ) . The p r o b a b i l i t y of f i n d i n g the hole between a distance 58. and Y*WY from the indium nucleus i s proportional to where i s the r a d i a l part of the ground state wave-function of the impurity given by the quantum defect method (see equation (2.12)). For indium, ^ = 0.5, The ground state wave-function of the hole i s taken to be s p h e r i c a l l y symmetric about the o r i g i n . The most probable value, Yj, , i s obtained from the equation OS J (5.6) which gives Y*^  ^ T j - a * , where a* i s the e f f e c t i v e Bohr ra-dius for the hydrogenic impurity i n s i l i c o n . This value , gives an estimate of the radius of the hole o r b i t around the indium ion. For a hydrogenic impurity such as boron(1= = /J, Vjj = a* ~ 14*^ . For indium, where ^  = 0.5 , Vj, = 0.5 a* 7°fj . Using equation 1 7 - 3 (5.3) for the Si(B,In) sample #1 (N oc 2 x 10 atoms cm ) and taking yYlc = .36 times the free electron mass for the boron hole moving i n the external state 2 the expected i n t e r r u p t i o n broad-ening i n the boron external l i n e 2 i s HB = **V - 4 oco.5- meV. . This r e s u l t i s true for a free c a r r i e r moving i n the valence band with an e f f e c t i v e mass VWC . (The c a r r i e r i s not bound to any p a r t i c u l a r impurity). In the case of the Si(B,In) c r y s t a l , the hole which i s heing scattered by the neu-t r a l impurity, remains bound to i t s boron ion always. Therefore the value of Hg = .5 meV. should be taken as an upper i i m i t of the i n t e r r u p t i o n broadening. When the hole bound to the boron im-p u r i t y i s excited to a higher excited state (5, 6, 7, 8, 9) then due to i t s much more extended wavefunction, i t w i l l behave almost l i k e a free c a r r i e r . Consequently higher boron excited states are expected to be more broadened i n t h i s approximation. These ar-guments are consistent with the experimental r e s u l t s of F i g . (5.1) (a) and (5.1) (b) . The halfwidth of the boron l i n e 2 i n sample #1 i s 0.2 + 0.05 meV., a f t e r removing a l l - the other contributions (White 1967a). This i s the magnitude of the i n t e r r u p t i o n broad-ening for the l i n e 2. Much greater broadening of higher states 5, 6, 7, 8, 9 i s understood to be due to the greater s p a t i a l exten-sion of t h e i r wavefunctions. The boron peak 3 i n Si(B,In) behaves d i f f e r e n t l y than i n S i ( B ) . This d i f f e r e n c e i s not completely understood. A possible explanation i s that the peak 3 and 4 l i e quite close to each other and when broadened, w i l l overlap with each other. The re s u l t a n t peak shape could look l i k e that i n F i g . (5.1) (a). In order to make a quan t i t a t i v e estimate of the broadening produced i n the 2p' i n t e r n a l absorption l i n e , equation (5.3) can be used again. The binding energies of boron i n t e r n a l states are l e s s than that of the external states (Onton et a l 1967). Consequently, the i n t e r r u p t i o n broadening of the boron i n -te r n a l l i n e s i s expected to be greater than that of the external l i n e s . (Within the e f f e c t i v e mass approximation, the Bohr radius of the boron hole i n the 2p i n t e r n a l state and the external state 2 i s c= 26 % and 15$ r e s p e c t i v e l y ) . Using the value of -f^_ -X .24 (obtained from equation (5.4a)) for the boron hole moving i n the 2p' i n t e r n a l state, the i n t e r r u p t i o n broadening of the 2p' i n t e r -nal l i n e i s H % 0.8 meV. The experimentally observed value (a f t e r removing other broadening contributions) i s % (0.48 + 0.07) meV. These estimates show how the i n t e r n a l acceptor state 2p of the boron impurity i n Si(B,In) should have greater i n t e r r u p t i o n broadening than the external states. I t should be noted that the ca l c u l a t e d i n t e r r u p t i o n broadening represents only the upper lim-i t of the in t e r r u p t i o n broadening c o n t r i b u t i o n . The p r o b a b i l i t y of thermal i o n i z a t i o n of indium impurities i n s i l i c o n i s very small up to 77°K. Thus according to the equation (5.3), neutral impurity s c a t t e r i n g i s independent of temperature. Study of the temperature dependent halfwidth of the boron 2p / l i n e i n Si(B,In) samples showed a behaviour si m i -l a r to that i n Si(3) samples within the experimental error. This i s consistent with equation (5.3). The observed l i n e a r dependence of the halfwidth of boron 2p' l i n e i n Si(B,In), as a function of the indium concentration, i s also supported by equation (5.3). In order to understand more about the i n t e r r u p t i o n broadening, samples of s i l i c o n doubly doped with boron and gallium impurities were also studied. A maximum gallium concentration of 16 -3 1 x 10 atoms cm was a v a i l a b l e i n the double doped samples. In these samples no appreciable e f f e c t on the boron l i n e s was ob-servable. According to the equation (5.3) there should be no. ap-16 - 3 prec i a b l e broadening at 1 x 10 gallium atoms cm 5.5 Configuration Interaction The modification of the Bloch states i n the valence band, due to the presence of indium impurities, s i g n i f i c a n t l y af-fect s the configuration i n t e r a c t i o n between the i n t e r n a l state and the Pg/2 v a l e n c e band Bloch states. The observed s h i f t i n the peak p o s i t i o n of the boron 2p' l i n e i n Si(B,In) and the decrease i n i t s integrated cross section are r e l a t e d with this modification due to conf i g u r a t i o n i n t e r a c t i o n . In order to make an estimate of t h i s change the theory presented i n section (2.3) i s used. Equation (2.28) may be rewritten as £HL - ' tr 2 ^ - ^ T ( Q -» ) < (5.7) where OJ^ ; i s the modified integrated cross section of the boron 2p / l i n e ; <TnT0 i s the modified continuum absorption cross sec-t i o n at the peak p o s i t i o n E 0 . Experimentally i t was found that there i s no appreciable change, within the experimental error ('k.L+oy )> in the continuum absorption cross section away from the f 0 peak p o s i t i o n . I t i s f e l t that the change produced in CTf^ , , due to the presence of indium, i s spread throughout the broad photo-i o n i z a t i o n absorption cross section. The observed decrease in UJ^ j" i s a t t r i b u t e d to the modification produced i n the i n t e r n a l state wave function due to configuration i n t e r a c t i o n . This would change the value of Q. Introducing the concept of inhomogeneous broadening due to impurity-impurity i n t e r a c t i o n (see section (4.3 b ) ) , a s h i f t i n the peak p o s i t i o n towards the low energy side (for negative value of Q) i s obtained. In order to obtain a quan-t i t a t i v e estimate of these e f f e c t s , the experimental value of for the boron 2p / peak i s used. For the sample #1, ^ / ^ _ % 0 . 6 + 3G7o meV. Tak-ing a value o f f 0 . 1 3 meV. and using the experimentally obser-ved peak s h i f t , R(obs.) 0.125 + 0.05 meV., leads to a value of I Q/ of approximately 2.0. The computation method of section 4.3 (b) gives the concentration broadening £s 0.38 meV. for Q = -2 and P = 0 . 1 3 meV. This concentration broadening i s the i n -t e r r u p t i o n broadening due to the boron-indium i n t e r a c t i o n . Theo-r e t i c a l l i n e shapes can be generated exactly as in section 4.3 (b). In a l l the samples of Si(B,In) studied, the boron and indium concentrations were in the r a t i o or 1:5 to 1:10 with the boron concentration almost constant. The observed apparent increase in the i n t e n s i t y of the indium 2 p ' i n t e r n a l l i n e sug-62. gests that the indium centers can also f e e l the presence of bor-on impurity, and there seems to be an increase i n the configuration i n t e r a c t i o n parameter Q for the indium 2p' l i n e . In order to make a d e f i n i t e d e c i s i o n about t h i s behaviour, samples with greater boron concentration than that of indium, i n Si(B,In), should be studied. 5.6 P o s s i b i l i t y of Boron-Indium Complex Formation One could speculate the formation of a boron-in-dium complex due to the i r mutual i n t e r a c t i o n . The hole bound to a boron atom, when photoexcited to the i n t e r n a l acceptor state, . could be scattered into one of the 2^/2 ^ o c n s t a t e s - This hole could now spend some time i n the v i c i n i t y of an indium atom r e -s u l t i n g in the formation of an I n + ion. The d i r e c t experimental observation of such a state may be d i f f i c u l t . CHAPTER 6 PHONON ASSISTED TRANSITIONS IN THE DEEP ACCEPTOR IMPURITIES 6.1 Introduction When an impurity electron or hole i s o p t i c a l l y excited from the ground state to some e l e c t r o n i c state of higher energy, the c r y s t a l may or may not remain i n the same i n i t i a l v i b r a t i o n a l s t a t e . I f the c r y s t a l remains i n the i n i t i a l v i b r a t i o n a l state the t r a n s i t i o n i s r e f e r r e d to as a zero-phonon t r a n s i t i o n , and the t r a n s i t i o n energy i s the d i f f e r e n c e i n energy of the i n i t i a l and f i n a l e l e c t r o n i c states. However, multiphonon processes, i n which the e l e c t r o n i c t r a n s i t i o n i s accompanied hy the emission or absorption of one or more phonons, may also be possi b l e . Such t r a n s i t i o n s y i e l d absorption peaks on either side of the zero-phonon peak at energies appropriate to the phonon taking part. The r e l a t i v e i n t e n s i t i e s of the zero-phonon and multiphonon tran-s i t i o n s i s determined by the strength of the electron-phonon i n -t e r a c t i o n (Toyozawa 1966). Both, o p t i c a l and acoustic, phonons can .take part i n t h i s process depending upon the strength of the electron-phonon i n t e r a c t i o n . When the t r a n s i t i o n s between the two impurity states take place with the emission or absorption of one or more o p t i c a l phonons, these t r a n s i t i o n s are c a l l e d phonon-ass i s t e d t r a n s i t i o n s . Kane (1960) found that for the shallow hydrogenic impurity centers in s i l i c o n , electron-phonon coupling i s weak and the dom-inant o p t i c a l l y induced t r a n s i t i o n s between impurity bound states are zero-phonon t r a n s i t i o n s . I f phonons do take part they are Longitudinal acoustic. Due to the weakness of electron-phonon i n t e r a c t i o n l o n g i t u d i n a l acoustic modes are important. Using quantum defect ..wavefunctions for the impurity ground state, Bebb and Chapmann (1969) f i n d that for deeper acceptor impurities (Ga, 64 In) the strength of the carrier-phonon i n t e r a c t i o n increases both for the acoustic and o p t i c a l modes, and p a r t i c i p a t i o n of o p t i c a l phonons becomes important. This suggests that phonon-assisted t r a n s i t i o n s ( e l e c t r o n i c transitions- with the emission or absorption of o p t i c a l phonons), such as those observed in group II - VI and I V compounds (Hopfield 1959, Williams 1967), and Diamond - l i b (Hardy et a l . 1962), should also be present in deep impurities such as indium acceptors in s i l i c o n . Moreover, due to the small dis p e r s i o n i n the o p t i c a l phonon branch of the s i l i c o n l a t t i c e v i b r a t i o n spectrum, the one phonon peak w i l l be r e l a t i v e l y sharp and should be observable. In the present i n v e s t i g a t i o n such phonon-assisted t r a n s i -tions have been observed i n indium-doped s i l i c o n . These t r a n s i -tions are superimposed upon the indium photoionization continuum. The i n t e r a c t i o n between the phonon-assisted t r a n s i t i o n and the overlapping photoionization t r a n s i t i o n s (see section (2.3); Hop-f i e l d et a l . 1968), w i l l modify the l i n e shape and structure. Using the phonon dispersion curves for the s i l i c o n l a t t i c e (Johnson 1965 ), the experimentally observed spectrum i s i n t e r -preted in section (6.3). In section (6.4) an estimate of the strength of the electron-phonon i n t e r a c t i o n i n Si(In) i s made from the experimental data. 6.2 Experimental Results The absorption spectrum of S i ( In), in the s p e c t r a l region 193 - 216 meV., was studied at l i q u i d Helium temperature using standard experimental procedures (see shapter 3). The indium con-16 17 centrations in the samples studied were 3 x 10 , 2.9 x 10 and 17 -3 5 x 10 atom cm . The r e s u l t s for a sample with impurity concen-17 -3 t r a t i o n of 5 x 10 atoms cm i s presented in F i g . (6.1). The i n t e n s i t i e s of the observed l i n e s were found to be impurity con-centration dependent, as expected. The peaks became very weak in the case of the sample with an impurity concentration of 3 x 1 0 ^ -3 atoms cm and i t was d i f f i c u l t to observe them. 202 208 214 220 226 PHOTON ENERGY (meV) F i g . (6.1) Phonon-Assisted Tr a n s i t i o n s i n S i ( I n ) . Indium Impurity Concentration ~ 5 x 10 1 7 atoms era"3 Sample Thickness 2^  •SZO.U cm. Spectral S l i t w i d t h 0.25 meV. Error Flags Represent the R e p r o d u c i b i l i t y of Several Scans. 66. The spectrum shows two prominent peaks at approxi-mately 206.5 and 216,3 meV. ( +0.5 meV.), designated P(\x and P^ , r e s p e c t i v e l y . Two weaker peaks occur at approximately 202. and 203.5 meV. ( + 0.5 meV.) designated P#, and pfl' r e s p e c t i v e l y . Be-cause of the proximity of the strong peak^ ( PA^) these weaker peaks cannot be completely resolved. A strong "V" shape dip i s evident between the two stronger peaks. Another comparatively shallower dip also appears between the peak at 206.5 meV. and an-other peak at 197.6 meV. In the next section the observed peaks w i l l be associated with phonon assi s t e d t r a n s i t i o n s with simultaneous emission of o p t i c a l phonons of appropriate energies. 6.3 Theory Of Phonon-Assisted T r a n s i t i o n s An impurity i n a host c r y s t a l l a t t i c e i s considered. The t o t a l wave function describing the e l e c t r o n i c state of the e l e c t r o n (hole) and the v i b r a t i o n a l state of the l a t t i c e can be written, within the Boon-Oppenheimer approximation, as W> * l*>T£j \t i> ~• ( 6 . i , where |ft^> i s the impurity eigenfunction (see section (2.1)) and the |^ are harmonic o s c i l l a t o r states of the c r y s t a l approp-r i a t e to the th mode of wave vector ^ . For a given t r a n s i t i o n between states a and b, the electron-phonon i n t e r a c t i o n couples the state \%^) and J and the o p t i c a l absorption c o e f f i c i e n t i s proportional to where * s t n e matrix element for o p t i c a l t r a n s i t i o n s between the e l e c t r o n i c state (A&}> and //Jfa^ . Following Hardy et a l . (1962), Hardy (1962), and Kane (I960), the t r a n s i t i o n p r o b a b i l i t y X between the two states with either o p t i c a l or a c o u s t i c a l phonon p a r t i c i p a t i o n i s proportional to " (6-3) II m where V? i s the number of phonons emitted or absorbed. At low temperature the number of thermally excited phonons for o p t i c a l modes may be set equal to zero. Thus, processes i n which phonon emission takes place w i l l be of importance. The coupling strength, y , i s rela t e d to the appropriate electron-phonon coupling constant o( through a sum over the l a t t i c e momentum by the expression where ^ represents the mean number of phonons in a normal mode with wave vector . U)q^ i s the angular frequency of the emitted o p t i c a l phonon, in a normal mode, with wave vector ^ The r a t i o . - o f the i n t e n s i t y of the t r a n s i t i o n with V? o p t i c a l phonons to that with no phonon i s (Hardy 1962) fly, _ • J L Here, i s the difference i n energy between the two bound e l e c -tron states. fly, and f\0 are the integrated absorption cross sections for the phonon-assisted t r a n s i t i o n and zero-phonon t r a n s i t i o n , r e s p e c t i v e l y . The energy of o p t i c a l phonon- emitted i n a phonon-assisted t r a n s i t i o n i s ^f\U>o (neglecting the di s p e r s i o n i n the o p t i c a l branch of l a t t i c e absorption spectrum of the host c r y s t a l l a t t i c e . The phonon emission peaks w i l l occur at energies yl"^W 0 (where V\ =1,2...) above that of any zero phonon— t r a n s i t i o n of energy . The experimental observation of phonon-assisted t r a n s i t i o n s (hereafter r e f e r r e d to as Pfl. t r a n s i t i o n s ) depends upon two f a c t o r s ; ( i ) the magnitude of the coupling strength y (The magnitude of Y- determines the r e l a t i v e i n t e n s i t y of zero-phonon t r a n s i t i o n to the pfl. t r a n s i t i o n ) and, ( i i ) dispersion in the l a t t i c e v i b r a t i o n spectrum of the host l a t t i c e . In the l a t t i c e spectrum of various semiconductors, the acoustic phonon branch has quite high dispersion and the op-t i c a l phonon branch has only a l i t t l e d i s p e r s i o n (Johnson 1965). Consequently the t r a n s i t i o n s with the emission or absorption of a-c o u s t i c phonons w i l l give a broad structure in the spectrum (when the electron-phonon coupling i s strong). The t r a n s i t i o n s with the p a r t i c i p a t i o n of o p t i c a l phonons w i l l produce a r e l a t i v e l y sharp l i n e s tructure. As the number of phonons taking part i n a t r a n s i t i o n increases, the l i n e becomes broader. So the one phonon t r a n s i t i o n w i l l be the sharpest among the observable transions. The magnitude of the coupling strength parameter, ^ , as a function of the impurity binding energy can be e s t i -mated for impurities in a semiconductor using the quantum defect method. The ground state r a d i a l wave function of a s u b s t i t u t i o n a l impurity i n a semiconductor i s given as (Bebb and Chapman 1969) i s the normalization constant, and ^ i s r e l a t e d to the observed binding energy ^ (obs.) of the impurity by equation (2.13). Using these quantum defect wave functions for the impurity ground state, the coupling strength parameter $1 has been c a l c u l a t e d by Bebb and Chapman (1969) for d i f f e r e n t values of & In if the subscript S r e f e r s to the coupling (with o p t i c a l or acoustic phonon) ; £ - 1 for deformation coupling for acbustic phonons, and ;5=2 for deformation coupling for o p t i c a l phonons. Obviously J i s dependent upon the binding energy 69 of the impurity. Table (6.1) summarises the c a l c u l a t e d strengths of electron-phonon i n t e r a c t i o n for the group III acceptors in s i l i c o n . For ^ — 1 equation (6.6) reduces to a hydrogenic ground state r a d i a l wave function. Values of the coupling strength parameters, for a hydrogenic impurity are Y' = . 15 and ^ = .03 (Hardy et a l . 1962). I t i s clear that due to the weakness of the coupling, Pf\. t r a n s i t i o n s should not be observable in t h i s case. For the deeper acceptor impurities (gallium and indium) i n s i l i c o n the strength of coupling with the o p t i c a l phonons i s not n e g l i g i b l e (see Table 6.1). In p a r t i c u l a r , for the indium acceptors ^ i s quite large as compared to Y( for a boron impurity. Consequen-t l y , PA. t r a n s i t i o n s i n the deep impurities in s i l i c o n should be more e a s i l y observable. The experimental observation of such tran-s i t i o n s i n Si(In) supports these arguments. The presence of Pft-t r a n s i t i o n s in Diamond type l i b i s also consistent with t h i s theory At low temperature, only PA- t r a n s i t i o n s with emission of phonons w i l l be observable. T r a n s i t i o n s with simultaneous absorption of an o p t i c a l phonon w i l l not be observable due to the low phonon density at low temperature. 6.4 I n t e r p r e t a t i o n According to the discussion i n the previous sec-t i o n , phonon-assisted t r a n s i t i o n s may be expected to occur in indium-doped s i l i c o n . These phonon-assisted t r a n s i t i o n s are superimposed on the P , and P valence band continuum t r a n s i -3/2 1/2 tions in indium-doped s i l i c o n . The theory of configuration i n t e r -a c t i o n , between a d i s c r e t e state and an overlapping continuum (Fano 1961; Hopfield 1968; section (2.3)) should be applicable here. Within the context of t h i s theory, the spectrum shown in F i g . (6.1) may be at l e a s t q u a l i t a t i v e l y explained. TABLE (6.1) CALCULATED STRENGTH OF ELECTRON-PHONON INTERACTION IN GROUP-III ACCEPTORS IN SILICON IMPURITY %<.) £(obs.)(b) meV. BORON 1. 45. l 1 0.15 0.03 GALLIUM 0.7 72. 4. 9. 0.6 0.27 INDIUM 0.5 155. 10.5 16. 1.5 0.48 ACCEPTORS IN DIAMOND TYPE II b 0.35". « 3 7 0 - - 0.2 0.2 00 E . r m (b) £.(obs.) i s the i o n i z a t i o n energy of the acceptor impurity. 71. The l o n g i t u d i n a l and t r a n s v e r s e o p t i c a l phonon ( h e r e a f t e r r e f e r r e d as Lo ; TO ) bands o f s i l i c o n show l i t t l e d i s p e r s i o n (Johnson 1965). I f C r e p r e s e n t s an e x c i t e d s t a t e o f the a c c e p t o r i m p u r i t y bound h o l e , t h e n p h o t o e x c i t a t i o n t o t h e i m p u r i t y s t a t e Cft-O^To) r e p r e s e n t s the e x c i t a t i o n o f the im-p u r i t y bound h o l e t o the s t a t e C w i t h the s i m u l t a n e o u s e m i s s i o n o f l o n g i t u d i n a l ( t r a n s v e r s e ) o p t i c a l phonons. ( T h i s s t a t e can be a p p r o x i m a t e l y c h a r a c t e r i s e d as h a v i n g s i n g l e e n e r g y . ) . The e m i t -te d phonon may escape to a d i s t a n t p a r t o f the c r y s t a l and the system w i l l be l e f t i n the f i n a l s t a t e C +" (/To). Due t o the p r e s e n c e o f the o v e r l a p p i n g p h o t o i o n i z a t i o n c o n t i n u u m , the i n c i -d e nt photon can a l s o i o n i z e the i m p u r i t y g e n e r a t i n g a f r e e h o l e i n the v a l e n c e band. These two p r o c e s s e s a r e r e p r e s e n t e d as p h o t on — > C-t- L-0 QTo) photon — > F r e e h o l e i n the v a l e n c e band. When i n t e r f e r e n c e between t h e s e two p r o c e s s e s i s t a k e n i n t o a c c o u n t , e x c i t a t i o n o f the s t a t e C + near the i m p u r i t y t a k e s p l a c e . By r e a b s o r p t i o n o f a Lo(To) phonon a f r e e h o l e i s p r oduced i n the v a l e n c e band. (For t h i s p r o c e s s , the s t a t e C-t~l~o(_To) i s s i m p l y a r e s o n a n t i n t e r m e d i a t e s t a t e and the f i n a l s t a t e i s i n 'the v a l e n c e band c o n t i n u u m ) . photon •> C-f LO C~To) ^ i n t e r m e d i a t e ^ > F r e e h o l e i n v a l e n c e band. (6.8) The a b s o r p t i o n t o the f i n a l s t a t e C-f L°(To) w i 11 have the g e n e r a l shape o f an o r d i n a r y a b s o r p t i o n l i n e . The o b s e r v e d l i n e shape f o r the p r o c e s s (6.8) w i l l be g i v e n by one o f the asymmetric c u r v e s shown i n F i g . (2.1) ( H o p f i e l d e t a l . 1968). Due t o the p r e s e n c e o f two c o n t i n u a (^2/2 a n C * ^1/2 v a ^ e n c e bands) a s u p e r -p o s i t i o n o f s e v e r a l forms l i k e t h o s e o f F i g . (2.1) w i l l r e s u l t . 72, In addition to the above discussed interference ef-f e c t , i n t e r a c t i o n between the two close l y i n g resonances (the two neighbouring phonon-assisted t r a n s i t i o n s ) v i a the common continuum states can also take place ( P h i l l i p s 1966; section (2.3)). This i n t e r a c t i o n w i l l give a "V" shape dip between the two p a r t i c i p a t i n g resonances and w i l l also produce a r e l a t i v e s h i f t in energy bet-ween the i n t e r a c t i n g resonances (see equation 2.32). The dip shown in F i g . (6.1) between neighbouring peaks Pfl^ and , i s at t r i b u t e d to t h i s e f f e c t . In - the external acceptor spectrum of indium-doped s i l i c o n , there are seven observed zero-phonon l i n e s (Onton et a l . 1967) of which l i n e 2 and 4 are the most intense and l i n e 1 i s of lesser i n t e n s i t y . According to equation (6.5), the i n t e n s i t y of the phonon-assisted t r a n s i t i o n i s dependent upon the i n t e n s i t y of the zero-phonon l i n e . Therefore, experimentally, only phonon-as s i s t e d t r a n s i t i o n s corresponding to l i n e s 2 and 4 should be d i s -t i n c t l y observable. In Table (6.2) the expected energy values for tr a n s i t i o n s corresponding to l i n e 1, 2, 4, in the Si(In) external spectrum are given. Transverse and l o n g i t u d i n a l o p t i c a l phonon energies for the symmetry points L, )( and W were taken (Johnson 1965). It should be noted that in t h i s process, c o n t r i b u t i o n from the f u l l phonon branch i s possible (Hardy et a l . 1962) since momentum i s conserved by r e c o i l of the acceptor im-p u r i t y . On comparing the expected energy values of Table (6.2) with the experimental l i n e structure of F i g . (6.1), the following i n t e r p r e t a t i o n can be made. Peak PAJL i s due to a PA- t r a n s i t i o n , associated with the zero-phonon l i n e 2, with the possible p a r t i c i p a t i o n of ~fO phonons. Peak Pflj. has a halfwidth of approximately 3 meV. This suggests that phonons from the whole To branch are taking part (Johnson 1965). Peak PAu i s associated with the zero-phonon l i n e 4 and i s again due to the p a r t i c i p a t i o n of TO phonons from the whole branch ending at the symmetry points f and \j>J i n TABLE ( 6 . 2 ) PHONON-ASSISTED TRANSITIONS IN Si(In) Symmetry Point Phonon Energy of Pho-non meV. PA- t r a n s i t i o n corresponding to l i n e 4 ( 1 5 2 . 8 meV.) Vfl. t r a n s i t i o n corresponding to l i n e 2 ( 1 4 5 . 8 ) meV. PP- t r a n s i t i o n corresponding to l i n e 1 ( 1 4 2 . meV.) r LO& T O 6 4 . 4 2 1 5 . 2 2 1 0 . 2 2 0 6 . 4 L To 6 0 . 9 2 1 1 . 7 2 0 6 . 7 2 0 2 . 9 L LO 5 1 . 8 2 0 2 . 6 1 9 7 . 6 1 9 3 . 8 X TO 5 6 . 9 2 0 7 . 7 2 0 2 . 7 1 9 8 . 9 w TO 6 0 . 5 1 2 1 1 . 3 2 0 6 . 3 2 0 2 . 5 P , L j X , W are the symmetry points i n the B r i l l o u i n zone of the l a t t i c e . the B r i l l o u i n zone. The halfwidth of peak Pfl^is also approxi-mately 3 meV. PR^ and Pft^ are s l i g h t l y s h i f t e d i n energy from the expected value and there i s a dip in between these two l i n e s . This behaviour i s a t t r i b u t e d to the interference e f f e c t between pf l^and P^if v i a the common photoionization continuum (section (2.3) ; P h i l l i p s 1966). The pfij t r a n s i t i o n , associated with the zero-phonon l i n e 1 appears on the low energy side of Pfi^ a t approximately 202. meV. The two bumps on low energy side of Pfi^ could be due to the superposition of several weak P/h t r a n s i t i o n s . The p a r t i -c i p a t i o n of To (L) and To(W) phonons i n Pflt t r a n s i t i o n could pro-duce weak peaks near Pfl^peak. - Pfi. t r a n s i t i o n s due to the p a r t i c i p a t i o n of Tc (X) phonons associated with zero-phonon l i n e 2 could also appear near Pfl^- A weak Pfi. t r a n s i t i o n , due to the p a r t i c i p a t i o n of Lo^LJ associated with, the zero-phonon l i n e 4 w i l l also have i t s energy near Pfl^ • 1° conclusion the peaks P,3y and Pft^ a r e °* u e t o t n e p a r t i c i p a t i o n of phonons at d i f f e r e n t symmetry points and are associated with the zero-phonon l i n e s 1 and 2. From table (6.2) i t can be seen that a Pfi. t r a n s i -tion associated with the zero phonon l i n e 2 and with the p a r t i c i -pation of aLo(L) phonon, i s expected at about 197.6 meV. The peak, P/) , of F i g . (6.1) i s a t t r i b u t e d to t h i s t r a n s i t i o n . The dip between PA and ( P/)Z J Pfi^^Pfi,) i s associated with the inter -ference e f f e c t between the adjoining phonon-assisted t r a n s i t i o n s v i a the common continuum ( P h i l l i p s 1966 ). It appears that the above assignments of various absorption l i n e s , i n the spectra of indium-doped s i l i c o n , are in reasonable agreement with the c a l -culated energy values of phonon-assisted t r a n s i t i o n s . ; S i m i l a r t r a n s i t i o n s could also be present i n the gallium-doped s i l i c o n . But they w i l l be much weaker in i n t e n s i t y and may not be observable. 75. 6.6 Estimate of the Strength of Electron Phonon Int e r a c t i o n Using equation . (6.5) one can estimate the coupling strength, ")f , from the observed phonon-assisted l i n e s (Hardy 1962a). One requires the integrated cross section, $j , for the one phonon t r a n s i t i o n , (n = 1) and fl0" , for zero-phonon l i n e . Peak Pftx , which corresponds to the zero-phonon l i n e 2, was chosen for i n v e s t i g a t i o n . f\Q was determined, for the zero-phonon l i n e 2, from the external absorption spectrum of S i ( I n ) . In order to determine /)j , i t was assumed that the peaks Pflj and PA^  are weak as compared to peak Pfi^ . The separation of the peak Pfi^ from the photoionization background i s not wel l defined due to the presence of configuration i n t e r a c t i o n . An estimate of the boundary l i n e between peak P^and the back-ground continuum, was made. The value of f\i , so determined i s taken as a rough estimate. S u b s t i t u t i n g values of the measured quan t i t i e s into equation (6.5), one obtains ^,2. approximately equal to 0.2. This gived an estimate of the electron-phonon coupling strength for o p t i c a l phonons i n indium-doped s i l i c o n . The values of lC i shown in Table (6.1) were ob-tained by taking values of the l a t t i c e momentum ^- • , up to i n -f i n i t y (Bebb and Chapman 1969). This over estimates the coupling. Previous i n v e s t i g a t o r s (Hardy 1962a; Kane 1960; Lax and Burstein 1955) also made the same approximation of extending to i n f i n i t y for i n t e g r a t i o n . Thus a l l the values i n Table (6.1) for / are overestimates. I f i n t e g r a t i o n i n equation (fr/+)is c a r r i e d out for values of <?, between zero and (where a* i s the Bohr — -%a* radius of the c a r r i e r i n the l a t t i c e ) , then for <cf-=:'. i t i s found that the values of 2^ <. are reduced by ~ 0.87 (for g = 1), and 0.713 (for s = 2) r e s p e c t i v e l y (Bebb/ and Chapman 1969). For £j O » v a l u e s a r e expected to be reduced by about the same proportions as for "£ =./ • Hence the experimental estimate of ^^,2. O'X for Si(In) becomes closer to the t h e o r e t i c a l l y predicted value. CHAPTER 7 TEMPERATURE DEPENDENCE OF THE INDIUM  EXTERNAL LINE 2 7.1 Introduction The temperature dependence of the boron external l i n e s i n Si(B) have been studied extensively by White (1967) . Such d e t a i l e d study of the indium external l i n e s i s not in the l i t e r a t u r e . Newman (1955) studied only the gross features of the temperature dependence of the indium external l i n e s i n S i ( I n ) . The purpose of studying the temperature dependence of the indium external l i n e s i s to determine i f the spectra ob-tained are species dependent. The temperature dependence of the halfwidth of the indium external l i n e s should also throw l i g h t on the strength of electron-phonon i n t e r a c t i o n i n Si(In) (see chapter 6). In the present i n v e s t i g a t i o n , the temperature dependence of the halfwidth of the indium external l i n e 2 i s determined. (The other indium external l i n e s start, overlapping with each other on increasing the temperature of Si(In) sample. In F i g . (7.1), the r e s u l t s are compared with the corresponding r e s u l t s of the boron external l i n e 2. The ob-served rapid r i s e i n the halfv:idth of the indium l i n e 2 in Si(In) with temperature i s at t r i b u t e d to the stronger electron-phonon coupling i n S i ( I n ) . 7.2 Experimental Results The temperature dependence of the halfwidth of l i n e 2 in Si(In) samples (of impurity concentration 7c 3 x 1 0 ^ -3 atoms cm ) was studied at the b o i l i n g point of l i q u i d Helium F i g . (7.1) Temperature Dependence of Halfwidth of the Line 2 i n Si(In) and S i ( B ) . The Error Flags Represent the Mean Deviation for Several Scans. 79. o o (4.2 K) and at pumped Nitrogen temperature ( ?6 55 K). At l i q -uid Nitrogen temperature (77°K) the l i n e becomes very broad and one cannot obtain r e l i a b l e data on the halfwidth. In F i g . (7.1) the t o t a l measured halfwidth of the indium l i n e 2 has been p l o t -ted, as a function of temperature. Similar r e s u l t s , for the boron l i n e 2 are also plotted for comparison. The halfwidth of the indium l i n e 2 increases at a much faster rate with temperature, than the corresponding boron l i n e . At 55°K, an apparent s h i f t towards higher energy in the peak p o s i t i o n of indium l i n e 2 r e -l a t i v e to the 4.2°K p o s i t i o n , was also noticed ( s h i f t <ss 0.187 + .07 meV.). 7.3 Interp r e t a t i o n and Discussion F i g (7.1) shows that the indium external l i n e 2 broadens with temperature at a much faster rate than the cor-responding boron l i n e 2. The temperature dependent behaviour of the h a l f -width of the boron external l i n e 2 was explained by applying B a r r i e and Nishikawa (1963) theory of phonon l i f e t i m e broadening (White 1967). The Barrie and Nishikawa (1963) theory gives the l i f e t i m e broadening of the excited states due to weak el e c t r o n -phonon coupling with the acoustic phonons only. In S i ( I n ) , the s i t u a t i o n i s d i f f e r e n t . I t was shown in chapter 6 that for the deep impurities, such as indium, in s i l i c o n the electron-phonon coupling becomes stronger for both the acoustic and o p t i c a l phonons. Consequently, the Ba r r i e and Nishikawa (1963) theory should not be applicable in the case of S i ( I n ) . 80. CHAPTER 8 CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK Three types of studies were undertaken in t h i s thesis ; (i ) the e f f e c t of configuration i n t e r a c t i o n on the pro-p e r t i e s of the i n t e r n a l states of group I II impurities in s i l i c o n , ( i i ) the i n v e s t i g a t i o n of the external and i n t e r n a l ab-sorption spectra of s i l i c o n doubly doped with the boron and indium acceptors, and ( i i i ) the strength of the electron phonon i n t e r a c t i o n producing phonon as s i s t e d t r a n s i t i o n s in the indium doped s i l i c o n . The presence of configuration i n t e r a c t i o n in the i n t e r n a l acceptor states of the group III impurities i n s i l i c o n has been recognized for the f i r s t time.. The experimentally observed l i n e shape of the 2p i n t e r n a l l i n e in gallium- and indium-doped s i l i c o n showed an asymmetric l i n e shape si m i l a r to that of the 2-p i n t e r n a l l i n e i n boron-doped s i l i c o n (Parsons 1968a). The ob-served impurity concentration dependent features of the 2p^ ^ i n t e r -nal l i n e in gallium-doped s i l i c o n were s i m i l a r to those of the 2p/ i n t e r n a l l i n e in boron-doped s i l i c o n . (The e f f e c t , of increasing the impurity concentrations, on the l i n e shape of the 2p/ l i n e was to enhance the asymmetry and to produce a s h i f t in the peak p o s i t i o n ) . The l i n e p r o f i l e was also influenced by the impurity i o n i z a t i o n energy. Fano's (1961) theory of the configuration i n t e r -action was invoked to explain these new features. This theory was modified to take into account the e f f e c t of concentration broadening on the configuration i n t e r a c t i o n and the l i n e shape. In order to s i m p l i f y the a p p l i c a t i o n of th i s theory to the experi-mental data, c e r t a i n approximations were made. It was assumed that 81. both, the i n t e r n a l state and the degenerate Bloch states are i n -t r i n s i c a l l y non-degenerate and that the l i n e shape parameters Q, r and the peak s h i f t , F , was independent of energy in the energy range of i n t e r e s t . In the actual case of a c r y s t a l l a t t i c e Q , V and F vary in a complicated manner (Shibatani and Toyo-zawa 1968). Within these approximations t h i s modified theory was used to compute the t h e o r e t i c a l l i n e shape and the r e s u l t s were f a i r l y consistent with the observed features of the boron 2p l i n e . The l i n e shape parameter Q and the peak s h i f t , R(calc.) were also computed for the 2p' l i n e in the gallium- and indium-doped s i l i -con samples. The non-stationary l i f e t i m e of the 2p' i n t e r n a l state in boron-doped s i l i c o n , estimated from t h i s theory, i s ap-proximately 10 ^ sees. These investigations suggest that the l i n e shape of the boron 2p l i n e can be explained quite well with-out postulating the presence of a 25 l i k e state near the 2p state. The experimental study of the i n t e r n a l and exter-nal acceptor spectra of s i l i c o n doubly-doped with boron and i n -dium acceptors, represents the f i r s t undertaking of t h i s type. The presence of indium impurity in s i l i c o n a f f e c t s s i g n i f i c a n t l y the i n t e r n a l and external boron spectrum. The i n t e r n a l l i n e s are more broadened than the external. The i n t e r n a l boron l i n e , 2p showed a decrease in i t s integrated absorption cross-section and i t s peak p o s i t i o n s h i f t e d towards the low energy side, with increasing indium content of the samples. Broadening of the absorption l i n e s was understood on the basis of i n t e r r u p t i o n broadening due to the hole impurity s c a t t e r i n g . The s h i f t in the peak p o s i t i o n and the decrease i n the integrated cross section of the i n t e r n a l l i n e , were a t t r i b u t e d to the modification produced in the configuration i n t e r a c t i o n by the presence of the indium impurities. The a p p l i c a t i o n of the theory of i n t e r r u p t i o n broadening due to the neutral impurity scat-t e r i n g , i s only an approximation to the actual s i t u a t i o n . In order to draw d e f i n i t e conclusions about the apparent increase i n the i n t e n s i t y of indium 2p/ l i n e in Si(B,In) samples, study of the Si(B,In) samples with boron concentration greater than that of indium i s needed. This study might also throw more l i g h t on the a p p l i c a b i l i t y of the in t e r r u p t i o n broadening to the spectra of double-doped s i l i c o n . The importance of much stronger electron-phonon coupling i n deep impurities i n s i l i c o n , has been discussed. As a consequence of t h i s i n t e r a c t i o n , phonon-assisted t r a n s i t i o n s ( s i m i l a r to those observed i n Diamond type II b) are expected i n S i ( I n ) . During experimental studies of Si(In) c r y s t a l s , such phonon-assisted t r a n s i t i o n s were observed. Using the phonon d i s -persion curves for the s i l i c o n l a t t i c e , the observed structure i s a t t r i b u t e d to phonon-assisted t r a n s i t i o n s with the p a r t i c i p a -tion of transverse o p t i c a l phonons associated with the en t i r e d i s p e r s i o n curvesJalong various allowed nymmetry d i r e c t i o n s in the l a t t i c e . These phonon-assisted t r a n s i t i o n s are superimposed on the 3 n c* ^3/2 v a ^ e n c e band continua of indium acceptors i n S i ( I n ) . Thus, the interference e f f e c t s between these phonon-as-s i s t e d t r a n s i t i o n s and the background continua, modify the expected t r a n s i t i o n l i n e shape and,energy values of these t r a n s i t i o n s . Tak-ing into account t h i s interference e f f e c t , the shape and structure of the observed spectrum, has been interpreted. Using the observed l i n e structure, an estimate of the strength of electron-phonon i n -te r a c t i o n i n S i ( I n ) , has also been made. This estimate i s f a i r l y c lose to the t h e o r e t i c a l l y expected value. Due to the very small value of the coupling constant, ^  , in S i ( B ) , such phonon-assisted t r a n s i t i o n s w i l l not be observable. However, such t r a n s i t i o n s might be observable in Si(Ga) although their i n t e n s i t i e s are ex-pected to be quite low. Further proof of the stronger electron-phonon i n -te r a c t i o n i n S i ( I n ) , was found in the study of the temperature dependence of the halfwidth of indium external l i n e 2 in S i ( I n ) . I t was found that the broadening, of the indium external l i n e 2, with temperature was much greater than that of the corresponding boron l i n e . APPENDIX A LORENTZIAN CORRECTION TO THE INTEGRATED AREA UNDER AN ABSORPTION LINE In the determination of the integrated absorption cross section of an absorption l i n e , the area under the absorp-t i o n l i n e i s needed. The impurity l i n e s i n s i l i c o n are found to be approximately Lorentzian with very extended wings, and the area i s not a good measurable parameter for such l i n e s . When a l i n e i s superimposed on an i o n i z a t i o n background, the problem "be-comes more d i f f i c u l t . The method adopted in determination of the area under an absorption l i n e i s as follows. It i s assumed that the observed l i n e i s Lorentzian i n shape. The l i n e i s cut at a distance 1/c ( ^ i s in cm^) from the center of the l i n e . The point i s chosen so that the area under t h i s l i n e , with the cut o f f at yi , can be e a s i l y measured. The f r a c t i o n of the t o t a l area thus excluded, assuming that the ^ a i l s continue to be Lorentzian, i s JL. [ Ja = icorY^-) TT H J H~/^f • * ( A , 1 ) where H i s the f u l l width at h a l f maximum of the l i n e . For 2£ 2>> H , the excluded f r a c t i o n of area i s ~i^TT ' This excluded f r a c t i o n of the t o t a l area of a l i n e i s determined and added to the measured area. asymmetr i c To determine the Lorentzian c o r r e c t i o n for the i n t e r n a l l i n e , the following method i s adopted. 85. The i n t e r n a l absorption l i n e has approximately Lorentzian p r o f i l e on the higher energy side of i t s peak p o s i t i o n . This higher energy part of the l i n e i s folded about i t s peak p o s i -t i o n g i v i n g a symmetrical l i n e . Equation (A.l) i s used to determine the Lorentzian c o r r e c t i o n for t h i s symmetrical l i n e . This assump-ti o n of symmetrical l i n e shape w i l l have only l i t t l e e f f e c t , i n the present case, on the estimated value of the Lorentzian cor-r e c t i o n . 86. APPENDIX B (B.I) Convolution of Lorentz i a n with Fano Function The L o r e n t z i a n f u n c t i o n of h a l f w i d t h H| at the h a l f maximum i s , L(E) = l _ I .(B. I ) TTH, I + E 2 H 2 where H = 2 H( The fano f u n c t i o n of equation (4.2) can be w r i t t e n as F (E,p, Q) = j _ . (Q2 - I) + 2 Q / p\ (B.2) IT r 1 +_£ ir P.; i + E 2 where F? = P H ' • •' ^ 2 According t o the con v o l u t i o n theorem (see f o r example the Mathematics of Physics and Chemistry by Margenan and Murphy, Van Nostrand Co., 1956, page 262) the i n t e g r a l x S(E) = f L(x) F (E-x) dx = L ( E ) * F (E) (B.3) i i s c a l l e d the co n v o l u t i o n of L(E) and F(E). The F o u r i e r transform of the c o n v o l u t i o n i s the product of the F o u r i e r transforms of L(E) and F(E). sT (E) =T (E) * (E) (B.4) where A represents the F o u r i e r transform of A. The convoluted f u n c t i o n . S(E) i s the inverse F o u r i e r transform of S(E). The transforms are evaluated as f o l l o w s : L(E) = I j t c E ) e E w dE = e ~ H l *W' 2 TT _^ a n d F(E) = ( Q 2 - l ) - »7 VW| + 2_Q_ d_ e «7<wi e i n I - I f ' l - H . M - -.WE i f ° r ^ 2 w — f? lw/ O A - p |w| 1 -H.fwJ - iw E = I ( Q - I ) e i ~2Q e V sgnw e 1' ' a 2 F J L Y J — 06 - r n 2 i > 2 p ™ f - IJ I w| - iwE = (Q -I ) . li - 2Q sgnw e e r,'2. ,-2 i J " . • dw 87. where V = H, + H •' Hence S ( E ) = ( Q 2 - l ) 2 f? - 2Q H . T . I Ie ft 1 W 1 e 1 w E dw *rr n ' 2 + E 2 where H . T . r e p r e s e n t s t h e H i l b e r t t r a n s f o r m . S(E) = _ l _ ( Q 2 - l ) 2 I? 4Q . 1 ( Q 2 - l ) + 2Q . E / f / • = F ( E , Q , f •) (B.5) ' p'2 ' f ? 2 T h i s i s a new Fano f u n c t i o n wi t h H . +n = (B.2) Some P r o p e r t i e s o f t h e Fano F u n c t i o n F ( E , Q , fj )., ( i ) F o r t h e maxima and m in ima i n F ( E , Q , p ) , t a k e d ¥{£,0tfi ) = 0 dE T h i s g i v e s t h e p o s i t i o n o f t h e maximum and minimum r e s p e c t i v e l y , as where (B.6) Q • r,Q (B .7) - Q 2 (B .8) ( i l ) D i s p l a c e m e n t o f t h e peak f r om t h e mean-o f h a l f power p o i n t s , The V a l u e o f Fano f u n c t i o n a t t h e h a l f power p o i n t s i s i Fornax., Q , f J ) = Q 2 = ( Q 2 - l ) . I + 2Q_ E/ft r,2 r,2 w h i c h g i v e s t h e v a l u e s o f £ a t t h e h a l f power p o i n t s as f? 2 1/2 '/ 1 1 0 1 E + — Now /2 [ ( = ), + / E > H i s t h e mid energy between t h e ha I f, power p o i n t s (•I ) mid =2 o r E mid. = 2 f d F The d i s p l a c e m e n t , o f t h e e n e r g y a t t h e peak p o s i t i o n ( = o ) f r o m t h i s mid e n e r g y i s = /7 ( 1-2) = -Pi = - E — — m a x . Q In t h e c o n v o l u t e d Fano f u n c t i o n F ( E , Q , l~J ) -( V + H . ) 8 * — -Q T h i s shows t h a t <£ i s d e p e n d e n t upon H j . £ i s a measure o f t h e asymmet ry i n t h e l i n e shape o f Fano f u n c t i o n F ( E , Q , fj ). BIBLIOGRAPHY Baltensberger, W. (1953). P h i l . Mag., 44, 1355. B a r r i e , R. and Nishikawa, K. (1963). Can, J . Phys., 41, 1823. Bebb, H. Barrie and Chapman, R. A. (1969). Proceedings of the I I I International Conference on Photoconductivity (to be published). Bebb, H, Barry and Chapman, R. A. (1967). J . Phys. Chem. 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