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Photoluminescence studies of the electron-hole droplet and the impurity band in Si(P) Rostworowski, Juan Adalberto 1977

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PHOTOLUMINESCENCE STUDIES OF THE ELECTRON-HOLE DROPLET AND THE IMPURITY BAND IN S i (P) by JUAN ADALBERTO ROSTWOROWSKI . Sc . , Unfversidad Nacional de Ingen ier fa , Peru, 1 M . S c , The Un i ve r s i t y o f B r i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES C D e p a r t m e n t o f Physics) We accept th i s thes i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA March, 1977 <£> Juan Adalberto Ros two rows k i , 1977 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of PhySJCS  The University of Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 i i ABSTRACT • . , The photoluminescence spectrum of phosphorus-doped silicon at 17 _ 3 1 9 - 3 dopant concentrations ranging from 1.2 x 10 cm to 4.0 x 10 cm is studied as a function of excitation intensity. The spectra are interpreted tn terms of two types of racomMnation events', one' attributed to the recom-bination of oppositely charged carriers fns'ide an electron-hole droplet and the other outside due to the recombination -of free holes with electrons in the impurity band. The latter type of event gives rise to a new photoluminescence peak observed for the f irst time. The line shape of this peak compares very well with a f irst principle calculation.of the impurity band density of states within the Hubbard model. Existing theories for the ground state energy of an electron-hole droplet in n-type heavily doped silicon are reviewed and new numerical results are presented. However, within the present model droplets are not theoretically understood at this time in heavily-doped silicon. i i i TABLE OF CONTENTS Page Abst ract . i i Table of Contents i i i L i s t of Tables y L i s t of Figures v i Acknowledgements x i v Chapter 1: INTRODUCTION . . 1 1.1 General Introduct ion . 1 1.2 Purpose and Out l ine of th i s Thesis 2 Chapter 2: EXPERIMENTAL DETAILS . . . 5 2.1 Sample Preparat ion 5 2.2 Photoluminescence Spectrometer 6 2.3 Signal Averaging 10 Chapter 3: EXPERIMENTAL RESULTS AND ANALYSIS 13 3.1 Introduct ion 13 3.2 The Photoluminescence of S i (P) 13 Chapter 4: IMPURITY BAND 41 4.1 Introduct ion . . . . . 41 4.2 The Impurit ies in a Super !a t t i ce 44 Chapter 5: THE EHD IN HEAVILY DOPED SILICON 54 5.1 Introduct ion 54 5.2 The Or i g ina l Model 54 5.3 The Modif ied Model 63 5.4 Droplets? 67 Chapter 6: SUMMARY AND CONCLUSIONS . 71 i v Page Appendix A: HEAT TREATMENT EFFECTS IN S i (P) 74 A . l Experimental Results 74 A.2 Discuss ion of Results 78 Appendix B: IMPURITY. BAND DENSITY OF STATES - IMPURITY; PAIR MODELS , 82 Appendix C: DATA ANALYSIS PROGRAMME 85 A . l The Subroutines 86 A.2 Examples of Main Programmes 106 A.2.1 Main #1 106 A.2.2 Main #2 107 Appendix D: THE EHD COMPUTER PROGRAMMES . 108 D.l Co r re l a t i on and Impurity Energies 108 D . l . l Valence Band Contr ibut ion 1 ° 9 D.l.2 Conduction Band Contr ibut ion 112 D.2 K i n e t i c and Exchange Energies H 5 D.3 Computer Programmes 115 D.3.1 Programme One 116 D.3.2 Programme Two 119 D.3.3 Programme Three 120 B ib l iography 134 V LIST OF TABLES Table Page 3.1 Phonon Energies 14 3.2 E f f e c t i v e Masses 14 3.3 Best F i t Parameters 18 vi LIST OF FIGURES Figure Page 2.1 Experimental op t i c a l con f i gura t ion 9 2.2 Block diagram of the d i g i t a l equipment and p e r i p h e r i a l devices used f o r s ignal averaging and data ana l y s i s . . . 11 3.1 a) Photoluminescence spectrum of s i l i c o n conta in ing 1 . 2 x l 0 1 7 phosphorus cm" 3 at T = 4.2K and 120 Wcm"2 e x c i t a t i o n l e v e l . The strong broad peak i s a t t r i -buted to the e l ec t ron -ho le drop (EHD), the weaker to the bound exc i ton (BE), b) S o l i d c i r c l e s show the experimental EHD l i n e shape obtained by subt rac t ing the BE l i n e shape from the spectrum shewn in (a ) . The er ror s in subtract ion are shown. The s o l i d curve i s the theo re t i c a l f i t to the EHD l i n e shape 15 3.2 The photoluminescence spectrum of s i l i c o n conta in ing 5.7 x 1 0 1 7 phosphorus cm" 3 at T = 4.2K and 160 Wcm"2 e x c i t a t i o n leve l i s given by.so l i d c i r c l e s . The s o l i d curve shows the t h e o r e t i c a l f i t to the EHD l i n e shape 19 3.4 The photoluminescence spectrum of s i l i c o n conta in ing 17 -3 -2 5.7x10 phosphorus cm at 4.2K and 20 Wcm e x c i t a t i o n l eve l represented by s o l i d c i r c l e s i s compared to that o f a compensated sample conta in ing , both, phosphorus ( 1 0 1 7 cm" 3 ) and boron ( 1 0 1 6 cm" 3) at 4.2K and 8 Wcm"2 exc i t a t i on leve l represented by f l ags (two standard de-v ia t ions from 15 scans) . The peak at 1.045 eV i s vi1 Figure Page ' attributed to donor-acceptor recombination . . . 21 T o 3.4 Photoluminescence spectra of silicon containing 1.8x10 _3 phosphorus cm at 4.2K, a) Solid circles show the spectrum at high excitation level (200 Wcm ). The peak is attributed to the EHD. The solid curve shows the theoretical f i t to the EHD line shape. b) The flags (two standard deviations from 6 scans) show the spectrum at intermediate excitation level -2 (20 Wcm ) and the solid dots (50 scans) the spectrum at low level (.1 Wcm ). The peak at high energies is attributed to the EHD, the other to the impurity band. The spectra have been scaled for comparison 23 3.5 Experimental photoluminescence line shapes for an electron in the impurity band and a free hole of phos-18 -3 phorus-doped silicon containing 1.8 x 18 cm . Im-_2 purity band line shapes at excitation levels of 5 Wcm (long flags) and .1 Wcm (short flags) are shown. The flags represent two standard deviations due to signal averaging and to the subtraction process referred to in the text 24 18 3.6 Photoluminescence spectra of silicon containing 2.45x10 -3 phosphorus cm at 4.2K a) Solid circles show the spectrum at high excitation level (200 Wcm ). The peak is attributed to the EHD. The solid curve shows the theoretical f i t to the EHD l i n e shape. b) The f l a g s (two standard dev ia t ions from 6 scans) show the spectrum at intermediate l eve l (20 Wcm ) and the s o l i d dots (40 scans) the spectrum at low l eve l (.1 Wcm ). The peak at high energies i s a t t r i b u t e d to the EHD, the other to the IB. The spectra have been sca led f o r comparison IB experimental photoluminescence l i n e shape of phos-18 -3 phorus-doped s i l i c o n conta in ing 2.45 x 10 cm" . -2 The e x c i t a t i o n i n t e n s i t y i s approximately .1 Wcm . The f l ags represent two standard dev iat ions due to s i gna l averaging and to the subt rac t ing process r e f e r r e d to in the text . . . 18 a) Photoluminescence spectra of s i l i c o n 3.9 x 10 phosphorus cm at T = 4.2K are shown at two e x c i t a t i o n l e v e l s . At high e x c i t a t i o n l eve l (200 -2 Wcm" , 1 scan) both the impurity band (IB) and the EHD peaks are observed and at low leve l (.2 Wcm- , 60 scans) the IB peak s t rong ly dominates. b) The s o l i d c i r c l e s give the EHD l i n e shape obtained by subt rac t ing the two spectra in f i gure (a) . The s o l i d curve shows the theo re t i c a l f i t to the EHD l i n e shape i x Figure Page 3.9 IB experimental photoluminescence l i n e shapes of 1P _3 phosphorus-doped s i l i c o n conta in ing 3.9 x 10 ° cm . Impurity band l i n e shapes at e x c i t a t i o n l eve l s of 20 -2 2 Wcm (short f l ags ) and 200 Wcm (long f l ags ) are shown. The f l ags represent two standard dev iat ions due to s ignal averaging and to the subtract ing p ro -cess r e f e r r e d to in the text 30 3.10 Photoluminescence spectra of s i l i c o n conta in ing 1.1 19 -3 x 10 phosphorus cm at T = 4.2K are shown at three e x c i t a t i o n l e v e l s . a) At high e x c i t a t i o n l eve l (150 Wcm~ , 5 scans) the EHD peak dominates the spectrum. The s o l i d curve shows the theo re t i c a l f i t to the EHD l i n e shape. - b) At intermediate leve l (20 Wcm , 15 scans) both the IB and EHD peaks are observed. c) At low leve l (2 Wcm , 35 scans) the IB peak domi-nates the spectrum 31 3.11 Photoluminescence spectra of s i l i c o n conta in ing 19 -3 4 x 10 phosphorus cm at T = 4.2K are shown at two e x c i t a t i o n l e v e l s . The s o l i d points show a high e x c i t a t i o n leve l (150 Wcm , 10 scans) spectrum. _2 The f lags correspond to low leve l (5 Wcm , 110 scans) . . 33 3.12 Concentration dependence of the photoluminescence o f phosphorus-doped s i l i c o n at 4.2K using high e x c i t a -t ion i n t e n s i t i e s 34 X Figure Page 3.13 Concentration dependence of the threshold energy E . Open c irc les show data po.ints of Hall iwell pair r " and Parsons^. 36 3.14 Width of half maximum of the TO-assisted peak as a function of phosphorus concentration at 4.2K. Open c i rc les show data points of Halliwell and Parsons^ at 2K 37 3.15 Concentration dependence of the ratio of the relat ive integrated intensity of the sum of the TA and NP replicas to the TO phonon replica 38 3.16 Concentration dependence of the experimental IB photoluminescence line shape of phosphorus-doped s i l i con at 4.2K 39 4.1 The experiemental photoluminescence line shape for an electron in the impurity band and a free hole of s i l i con •jo _3 containing 1.8 x 10 phosphorus cm , represented by f lags, is compared to the theoretical IB l ine shapes calculated in the (—} Heitier-London and ( ) H 2 models. The theoretical bands are shifted in energy and scaled for comparison with experiment 43 4 . 2 The experimental photoluminescence line shapes for the impurity band in-Si(P) at donor concentrations 1.8 x 1 0 1 8 cm" 3 , 2.45 x 1 0 1 3 cm"3 and 3.9 x 1 0 1 8 cm are represented by flags. The solid-dotted curves represent the theoretical impurity band density of state's obtained in the Hubbard Riodel 49 xi Figure Page 4.3 The experimental photoluminescence l i n e shape f o r the Impurity band in S i (P) at donor concentrat ions 1.8 x 18 -3 10 cm" are represented by f l a g s . The s o l i d and chained curves represent the t h e o r e t i c a l impuri ty band dens i ty o f s tates obta ined using the low and high density cumulants, r e s p e c t i v e l y 53 5.1 The c a l c u l a t e d average energy per p a i r as a funct ion of hole dens i ty f o r the i nd i ca ted phosphorus impurity con-cen t r a t i on s . The points on the o rd ina te -ax i s are the c a l c u l a t e d values o f the chemical po tent i a l o f a p a i r in the l i m i t o f zero p a i r dens i ty 60 5.2 The c a l c u l a t e d average energy per p a i r as a funct ion of hole dens i ty f o r the i nd i ca ted phosphorus impurity con-cen t ra t i on s . The impurity energy cont r ibut ion (Equa-t i o n 5.1) i s neg lected. The points on the o rd ina te -ax i s are the ca l cu l a ted values of the chemical po tent ia l o f a p a i r i n the l i m i t of zero p a i r dens i ty 61 5.3 The c a l c u l a t e d average energy per p a i r as a funct ion of hole dens i t y f o r the i nd i ca ted phosphorus impurity con-cen t r a t i on s . The i o n i z a t i o n energy of an i s o l a t e d phos-phorus donor i s used as the energy per e lec t ron outs ide the d r o p l e t . The po int on the ord inate -ax i s i s the c a l -cu la ted value of the chemical po tent i a l o f a p a i r in the 17 -3 l i m i t zero p a i r density f o r n^ = 2 x 10 cm : the c a l -1 7 - 3 cu la ted value for n. = 5 x 10 cm i s o f f sca le 62 n x i i Figure 5.4 Chemical potent ia l of a p a i r as a funct ion of hole density in the fo l lowing cases: a) Droplets are in thermodynamic equ i l i b r i um with a gas phase. b) The gas phase contains no holes in contact with droplets which are e n e r g e t i c a l l y favoured. There i s mechanical equ i l i b r i um 5.5 The ca l cu l a ted chemical po ten t i a l o f a p a i r as a funct ion of hole dens i ty f o r the i nd i ca ted phosphorus impurity concentrat ions. The points on the o rd ina te -axis are the ca l cu la ted values in the l i m i t o f zero pa i r dens i ty . The experimental points (see Chapter 3) are shown f o r comparison 5.6 The chemical potent ia l o f a p a i r as a funct ion o f hole dens i ty . The s o l i d curve i s f o r the dens i ty of i on ized donors ( n ^ ) equal to n^; the dashed curve i s f o r n^. = n^i ( 0 ) ^ ; the chained curve represents a f ree hand i n te rpo l a t i on f o r (0 )<n d l . ( r^H"^ w n e n n d i ^ n h ^ v a n e s s lowly • A . l The e f f e c t s o f heat treatment on the photoluminescence spectra of Si(P) conta in ing impurity concentrat ions in the range 1.8 x 1 0 1 8 cm" 3 to 1.1 x 1 0 1 9 cm" 3 . The dashed l i nes give the spectra before treatment; the s o l i d l i n e s , a f t e r treatment. The spectra have been a r b i t r a r i l y scaled to make comparison of l i n e shapes e a s i e r . . . . . . . . . . x i i i F igure Page A.2 The e f f e c t s of heat treatment on the EPR of S i (P) 18 conta in ing impurity concentrat ions : 2.0 x 10 cm 18 -3 and 6.2 x 10 cm" . Magnetic f i e l d modulation was used and the output s ignal i s proport iona l to the d e r i v a t i v e dX 'VdH where x " i s the imaginary part o f the s u s c e p t i b i l i t y . The dashed l i ne s give the spectra before treatment; the s o l i d l i n e s , a f t e r treatment. The spectra have been a r b i t r a r i l y sca led to make comparison of Tine shapes e a s i e r . The spectra are taken from Reference(23) 77 xiv ACKNOWLEDGEMENTS I wish to thank my supervisor Dr. R. R. Parsons for his friendship, encouragement and helpful guidance throughout this investiga-tion. In the course of this work I have benefited from daily stimula-ting discussions for which special thanks are due to Dr. R. Barrie, Dr. B. Bergersen, Dr. M. Eswaran, Dr. P. Jena, Dr. G. Kirczenow and Mr. M. L. W. Thewalt. I am also grateful to Dr. M. Gush for the loan of a Ge detector. A heartfelt thanks to Tere and Juan Enrique for their sacrifices, patience and understanding. The research of this thesis was supported by National Research Council of Canada grants to R. R. Parsons and B. Bergersen. 1, CHAPTER 1 INTRODUCTION 1.1 General Introduction A s i l i c o n c r y s t a l i s an i n d i r e c t band gap semiconductor. The ground state of the c ry s ta l corresponds to a l l s tates in the valence bands being f u l l and a l l in the conduction band being empty. I f an e lec t ron i s exc i ted across the energy gap, a hole i s l e f t in the valence band and an e lec t ron -ho le p a i r i s c reated. In th i s i nves t i ga t i on only photo -exc i ta t i on is used to create these p a i r s . The oppos i te ly charged c a r r i e r s a t t r a c t each other and at low temperatures can bind to form an exc i ton J Due to the i n d i r e c t band gap of s i l i c o n the r a d i a t i v e a n n i h i l a t i o n o f an exciton requires the creat ion or ann ih i l a t i on of a c r y s t a l momentum-2 conserving phonon. At low temperatures the creat ion o f a phonon i s s t rong ly favoured as a momentum conserving process. Because the p r o b a b i l i t y of s imu l -taneous ann ih i l a t i on of an exc i ton and the creat ion of a phonon i s low, excitons have a long l i f e t i m e , t y p i c a l l y 60 u sec , and therefore a large range of exc i ton dens i t ie s i s experimental ly a t t a i nab le . 3 Keldysh in 1968 pred ic ted that excitons under increas ing concentra-t ion w i l l behave j u s t as a gas behaves under increas ing pressure; at some c r i t i c a l concentrat ion there w i l l be a condensation i n to a " l i q u i d . " A vast 4 amount of experimental work (photoluminescence, photoconduct iv i ty , . l ight-s c a t t e r i n g , e t c . ) has v e r i f i e d the existence of th i s condensed phase which has become known as the " e l e c t r o n - h o l e - d r o p l e t " (EHD). For i n t r i n s i c s i l i c o n , as well as germanium, the ground state propert ies o f the EHD have been 5 6 7 10 extens ive ly s tudied and good agreement between experiment ' and theory has been found. 2. Controversy has. ar i sen concerning the exis tence o f the condensate i n h e a v i l y - d o p e d s i l i c o n . In recent photoluminescence studies H a l l i w e l l a n d Par sons^ w e r e able to i n f e r from photo!imiinescence studies that high d e n s i t i e s of n o n - equ i l i b r i um c a r r i e r s condense into droplets even at impurity concentrat ions where screening e f f ec t s p roh ib i t exci ton format ion. Fur ther -m o r e , t h e y hypothesize that the EHD ex i s t s in samples with donor concentra-18 -' 3 t ions above the c r i t i c a l concentrat ion, n . = 3 x 10 phosphorus cm cnt v r 12 13 for the semiconductor-metal t r a n s i t i o n ' 14 Mart in and Sauer , on the other hand, argue that there i s no E H D i n samples with phosphorus concentrations close t o and above n c n - t . 1.2 Purpose and Outl ine o f th is Thesis The i n i t i a l main purpose of th is thes i s was to f i n d out whether the condensate ex i s t s in m e t a l l i c s i l i c o n . This could best be done by repeat ing t h e photoluminescence experiments o f Ha l l iwe l l and P a r s o n s ^ in heavi ly -doped s i l i c o n with extensive improvement in instrumentat ion. With be t te r s i g n a l -t o - no i se r a t i o a de ta i l ed comparison could be made between the experimental 5 15 a n d theore t i ca l l ine, shapes ' f o r the EHD. Two orders of magnitude improve-ment i n d e t e c t i v i t y allowed the study of the photoluminescence spectrum of S i (P ) as a funct ion of exc i t a t i on in tens i ty and the spectra are found to con-t a i n too components, the f i r s t of these is i n terpreted in terms of the recom-binat ion of an e lec t ron -ho le pair in s ide the drop; the second in terms of the recombination outs ide the drop, in the gas phase. At low e x c i t a t i o n i n t e n s i t y the droplets are few and f a r apart and most recombination events occur in the gas phase. As the e x c i t a t i o n i n tens i t y is increased the EHD l i n e grows. An important aspect of this thesis is to show that by studying the spectra at various exc i t a t i on intensities the recombination emiss ion of e l ec t ron -ho le . 3. pa i r s in the two coex i s t ing phases can be d isentang led. It w i l l be shown that the EHD l i n e shape i s very well descr ibed in terms of the recombination emission i n a degenerate e l e c t r o n - h o l e plasma with f i x e d c a r r i e r dens i ty . The gas phase l i n e shape a r i se s from the recom-b ina t ion o f a f ree hole with e lec t rons in the "Impurity Band"^ 6 ( IB), and i t w i l l be argued that th i s l i n e shape descr ibes very c l o s e l y the impurity band dens i ty o f s t a t e s . Despite the wide va r i e t y o f experimental techniques used in the past ( e l e c t r i c a l c o n d u c t i v i t y , Ha l l m o b i l i t y , magneto-res istance, magnetic 12 s u s c e p t i b i l i t y , NMR and ESR proper t ie s ) the impurity band was not wel l understood f o r donor concentrat ions near n . . . The photoluminescence c n t r detect ion o f the impur i ty band dens i ty o f s tates in samples conta in ing im-p u r i t y concentrat ions above and below n c r j t i s of cap i t a l importance to te s t our understanding of the semiconductor-metal t r a n s i t i o n . In Chapter 2 a desc r ip t i on of the experimental apparatus i s p re -sented. The experimental re su l t s and the ana lys i s of the data are presented in Chapter 3 and i n Appendix A the experimental r e su l t s i n hea t - t rea ted samples are desc r ibed . In Chapter 4 the desc r ip t i on of the Hubbard m o d e l ^ of the impurity band app l i c ab le f o r donor dens i t i e s below n c n -^ . w i l l be presented. The r e s u l t s o f le s s success fu l approaches appearing in the l i t e r a t u r e are a l so shown. The t h e o r e t i c a l background to these approaches i s given in Appendix B. In Chapter 5 the theo re t i c a l model o f Bergersen et a l 1 8 ' 1 9 f o r the EHD in heav i l y doped materia l i s reviewed and corrected numerical r e su l t s are presented. It w i l l be shown that wi th in th i s model, contrary to previous re su l t s 1 * * ' 1 9 the EHD i s not t h e o r e t i c a l l y p red i c ted . A general d i scuss ion and conclusion w i l l be presented in Chapter 6. Appendices C and D are concerned with the numerical work done by the author f o r the d i f f e r e n t aspects o f th i s t he s i s . The main resu l t s o f th i s i nve s t i g a t i on have a lready been pu-b l i s h e d . 1 9 " 2 1 5. CHAPTER 2 EXPERIMENTAL DETAILS 2.1 Sample Preparat ion S ing le c r y s t a l s o f vacuum f l o a t zone phosphorus-doped s i l i c o n purchased from General Diode Corp. and Ventron E l e c t r o n i c Corp. were used. The impur i ty concentrat ion was determined by room temperature r e s i s t i v i t y measurements using a f our -po in t probe on the face of the ingot before and a f t e r cu t t i ng the s l i c e . The c ry s ta l face was po l i shed and etched f o r 30 seconds i n a mixture o f HN03 and HF (5:1) before measurements were made. The r e s i s t i v i t y was determined by averaging at l e a s t ten measurements made at var ious pos i t i ons on the c ry s ta l face. The standard dev ia t ions of the mean were found to be less than two percent. I f the d i f f e r e n c e in the mean r e s i s t i v i t y before and a f te r cut t ing the s l i c e was more than two percent the sample was r e j e c t e d . The concentrat ion of phosphorus impur i t ies was determined from the 22 measured r e s i s t i v i t i e s using the I rv in chart . The accuracy of the donor concentrat ion i s est imated to be with in ±8 percent f o r a l l samples used in t h i s i n v e s t i g a t i o n and mostly r e f l e c t s the author ' s conf idence l i m i t in the 22 I r v in char t . 3 The s l i c e s o f s i l i c o n c rys ta l were cut t y p i c a l l y to 3 x 5 x 20 mm and etched as descr ibed above 'to remove surface damage. No change in the photoluminescence s igna l was found i f the sample was p o l i s h e d ; t h e r e f o r e , sample p o l i s h i n g was discont inued at an ea r l y stage o f t h i s i n v e s t i g a t i o n . The heat treatment studies were done on samples cut from the same s l i c e s mentioned above. The samples were then heated to 1150° C in a 6. helium atmosphere for 30 minutes and immediately quenched to room temperature 1n acetone. The samples were then etched as previously described. 23 Halliwell has pointed out that the effects of the heat treatment will dis-appear i f the samples are left at room temperature for several days. There-fore, when measurments were not performed immediately after heat treatment, the samples were stored in liquid nitrogen. All photoluminescence experiments were done with the sample immersed in liquid helium. The optical cryostat held three litres of liquid helium and provided an average running time of about 15 hours. The tempera-ture could be varied by changing the vapor pressure of the helium liquid. To check whether the sample was in thermal equilibrium with the bath the photoluminescence spectrum of intrinsic silicon was obtained under the same experimental conditions used in the measurement of the doped crystals. The 25 line shape of the recombination emission* due to free excitons , which is highly temperature dependent, did not change for all optical excitation levels reported here. 2.2 Photoluminescence Spectrometer A Spectra Physics Model 120 continuous He-Ne laser with 5mW power output at 6328 A* wavelength was used for low excitation intensities; a Spectra Physics Model 165 continuous argon-ion laser with a maximum power output of o 2W at 5145 A wavelength was used as a source for high excitation intensities. The irradiated area was a spot of 1 to 3 mm in diameter. *The photoluminescence line shape of free excitons was not observed to agree with previously reported work2^, 2 5 and triggered the inter-est of Thewalt and Parsons in our laboratory, who subsequently did very interesting work26 j n this area. 7. A modification of the optical tai l piece of the inner helium can of the cryostat was necessary. The tail piece was originally made of good qua-l i ty pyrex tubing, and this arrangement worked well when the photolumines-cence signals were large as in the case of those reported in this investiga-18 -3 tion for samples containing less than 10 phosphorus cm" . For more heavily-doped samples, however, the photoluminescence signal is very weak and background radiation originating in the pyrex tubing dominates the observed spectrum. Therefore, the lower part of the pyrex tubing was re-placed with a fluorescence cell made of spectrosil, a synthetic s i l i ca, manufactured by Thermal Syndicate Ltd. Although the manufacturer does not mention any low temperature properties, the low temperature fusion technique used in the production of this cell withstands the thermal shock when cooled to liquid helium temperatures. The recombination radiation was collected from the excited surface in order to minimize the effects of free carrier absorption in the heavily-doped samples. The photoluminescence spectra did not need correction for absorption in the sample because the absorption coefficient is known to be 27 slowly varing in the energy region of interest. The optimization of the collection of recombination radiation is not a trivial problem because of silicon's high refractive index n ~ 4. The radiation leaving the sample is concentrated in a small solid angle. An optimum arrangement was found in matching the f/number of the collection optics to that of the spectrometer with an off-axis el l iptical mirror with approximately three fold magnifica-tion. The recombination radiation was analyzed with a homemade 56 cm f/3.5 monochromator of Czerny-Turner design fitted with a Bausch & Lomb Inc. grating blazed at 1.6 u with 600 groves/mm. The grating was used in the second order. The optical configuration is shown in Fig. 2 .1 . To eliminate •infrared emission from the laser a Corning f i l ter CS 4-96 was placed at the output of the argon-ion laser; a Corning f i l te r CS 1-57 was used for the He-Ne laser. A Corning f i l ter CS 7-57 was placed at the exit s l its of the spectrometer to prevent any laser light from reaching the detector. For signal detection a R.C.A. (67-07-B) germanium photodiode detector-preampli-f ier system operated at liquid nitrogen temperature with detectivity D*(1.268u, 91 Hz, 1 Hz) = 4.234 x 1 0 1 3 cm-Hz^ /RMS Watt and noise-equivalent-power NEP (1.268u, 91 Hz, 1 Hz) = 1.056 x 1 0 " 1 4 RMS Watt/Hz*5 was used. -13 iV This value should be compared to 4 x 10 RMS Watt/Hz2 for the dry ice 22 cooled PbS detector used by Halliwell. Further signal amplification was obtained with a low noise pre-amplifier (PAR-11 3, Princeton Applied Research Corp.). The excitation light was chopped at approximately 90 Hz and the detector signal was phase-sensitive detected (PAR-121 Lock-in, Princeton Appried Research Corp.) and finally integrated over times of typically three seconds. The resulting analog signal was electronically converted to a digital one and stored in the memory of a mini-computer (Nova 2 , Data General Corp.). The signal-to-noise of the weaker spectra obtained in this investigation were further improved by signal averaging. To achieve auto-28 matic signal averaging the mini-computer was interfaced with the mono-'chromator. The data were finally punched on paper tape at the end of the experiment for future analysis. It should be pointed out that the improved detectivity used in this 2 Ge detector and p reamp l i f i e r system in l i q u i d U9 3 F i l t e r s 4 Grat ing Figure 2.1: Experimental op t i ca l con f i gura t ion . 10. i n v e s t i g a t i o n and the automatic averaging f a c i l i t y o f the present set-up al lowed spec t ra l l i n e shape ana lys i s on s igna ls over 200 times weaker than those reported by H a l l i w e l l and Pa r son s .^ 2.3 S ignal Averaging A block diagram of the d i g i t a l equipment and pe r i phe r i a l devices used f o r s igna l averaging and data ana lys i s (descr ibed in Chapter 3 and Appendix C) i s shown in F i g . 2.2. The i n t e r f a c e was designed and b u i l t by M.L.W. 28 Thewalt and a f u l l de s c r i p t i on of i t i s found in his thes i s . In th i s sec t i on a general overview only w i l l be g iven. An important pa r t o f th i s i n te r face i s the device c a l l e d the "spectrometer c o n t r o l l e r " . 2 8 j n simple terms, th i s device t rans l a tes a b inary coded number from the computer i n to a t r a i n of pulses. Each pulse increments the monochromators s tepping motor and the ro ta t iona l d i r e c t i o n i s determined by the s ign o f the b inary number (two's complement used fo r negative numbers). In add i t ion to the spectrometer cont ro l l e r i t i s convenient to use a pro-grammable i n t e r v a l t imer (PIT) which allows one.tb set an in tegra t ion time f o r each wavelength p o s i t i o n s . The i n te rac t i on of the mini-computer with the above mentioned dev ices occurs only in g iv ing the devices the parameters o f the task to be performed and in rece i v ing confirmation that the task has been completed. The computer i s consequently f ree f o r many other tasks. In the s i gna l averaging mode the computer i s programmed to reset the i n teg ra to r by means o f a d i g i t a l output device immediately a f t e r the PIT i s s t a r t e d . The i n t e g r a t o r w i l l continue in operation un t i l i t i s reset . While the i n teg ra t i on o f the detected s ignal i s being performed the computer i s used to d i sp l ay on the scope data stored i n i t s memory. The v e r t i c a l 1 1 . Teletype Minicomputer x - y P l o t t e r Scope ~7f Control Unit Paper Tape Reader Interface In tegrator Lock-In \ Amp l i f i e r Spectrometer Figure 2.2: Block diagram of the d i g i t a l equipment and p e r i p h e r i a l devices used fo r s ignal averaging and data a n a l y s i s . 12. analog voltages d i sp layed on the scope are obtained by e l e c t r o n i c a l l y convert ing the d i g i t a l informat ion in a memory locat ion and the hor izonta l voltages are obtained with a d i g i t a l to analog conversion of a number pro -por t iona l to the memory l o c a t i o n . The scope d i sp lay allows the experimen-t a l i s t to observe the spectrum as i t i s measured. At the end of each memory sweep the computer i s programmed to te s t the status o f the PIT. I f the i n -tegrat ion time has exp i red , the computer terminates the i n teg ra t i on o f the s i gna l by t r a n s f e r r i n g cont ro l to the ana log - to -d i g i t a l converter (ADC) whi le conversion of the analog s ignal i s performed the computer s t a r t s the spectrometer c o n t r o l l e r with a new set o f parameters. During the time that the spectrometer c o n t r o l l e r i s performing i t s task the computer w i l l t e s t whether or not the a n a l o g - t o - d i g i t a l conversion has been completed and then compute a new average and standard dev ia t ion f o r the s ignal f o r the l a s t spectrometer p o s i t i o n . The cyc le descr ibed above continues un t i l the l a s t po int o f the spectrum has been measured and i f s igna l average i s to be used the spec t ro -meter i s returned to the f i r s t wavedrive pos i t i on . The exper imenta l i s t may decide a f t e r inspect ing the spectrum displayed on the scope that the s i gna l - t o -no i se r a t i o i s adequate and i n te r rup t the s igna l averaging. The data stored in memory can then be p l o t ted on an x-y recorder. The analog voltages required are obtained using d i g i t a l to analog converters (DAC) as i n the case of the scope d i sp lay . The hor izonta l axis may be c a l i b r a t e d in uni ts o f wavelength or energy. For future use the b inary coded spectrum s tored in memory i s t r ans fe r red to paper tape using the high speed paper tape punch (HSPTP). 13. CHAPTER 3 EXPERIMENTAL RESULTS AND ANALYSIS 3.1 In t roduct ion As po inted out in Chapter 1 rad ia t i ve recombination of an e l e c t r o n -hole p a i r in i n t r i n s i c s i l i c o n is accompanied with the simultaneous emission o f a ssamentum conserving phonon. In doped material the i n t e r a c t i o n o f the p a i r wi th an impur i ty re laxes the momentum conservation and an e x t r i n s i c I, 2 II, 25 27 11 5 component i s observed in both the absorption and photoluminescence. ' Th i s component i s commonly re fer red to as the "no-phonon (NP)" process. The NP process does not change the energy of the t r a n s i t i o n , un l ike the phonon a s s i s t ed recombination which resu l t s in photoluminescence peaks s h i f t e d down by the energy expended in the creat ion of the phonon. Table 3.1 l i s t s the energies o f the phonons a s s i s t i n g the recombi-25 29 na t i on as measured in photoluminescence studies by other i n ve s t i g a to r s . ' The i d t e n t i f i c a t i o n o f the phonon i s done by comparison to the energies mea-sured by i n e l a s t i c neutron s c a t t e r i n g . ^ Since the strongest recombination emiss ion in phosphorus-doped s i l i c o n i s a s s i s ted by c reat ing a transverse 11 14 25 29 o p t i c a l (TO) phonon ' ' ' a l l the l i n e shape ana lys i s has been per-formed-on the TO phonon as s i s ted port ions of the spectrum*. 3.2 The Photoluminescence of Si (P) Figure 3.1a. shows the spectrum of a sample conta in ing 1.2 x 10^ 7 *In r e a l i t y , the TO phonon ass i s ted recombination overlaps with the l ong i tud ina l o p t i c a l (LO) phonon as s i s ted port ion of the spectrum but cannot be reso lved. We w i l l j u s t i f y l a ter on in this Chapter the neglect of the LO phonon r e p l i c a in the analys is o f the spectra. TABLE 3.1 PH.QN0N ENERGIES TRANSVERSE ACOUSTICAL (TA) 18-7 meV LONGITUDINAL ACOUSTICAL (LA) undetected TRANSVERSE OPTICAL (TO) 58.0 meV LONGITUDINAL OPTICAL (LO) 56.1 meV TABLE 3.2 EFFECTIVE MASSES m, .9163 m„ 1 o m t .1905 mQ m* = (m.! m 2 . ) V 3 .3216 mQ m t . .154 ni Ih o 2/3 < - « m l h 3 / 2 + \ i 1 - 5 7 7 3 <"o 15. i i i i r 0> o o to t if) LU 2 o LU O CO LU S' i i i i r -*]*- ;'. Si (P) I.2x!0 I 7 crrT 3 a V J 1 L 1 1 L i.06 1.08 1.10 1.06 1.08 1.10 PHOTON ENERGY (eV) 17 F igure 3.1: a l Photoluminescence spectrum of s i l i c o n conta in ing 1.2xlQ phosphorus c n r 3 at T = 4.2 K and 120 W.cnr2 e x c i t a t i o n l e v e l . The strong broad peak i s a t t r i bu ted to the e l e c t r o n -hole drop (EHD), the weaker to the bound exc i ton (BE). b) S o l i d c i r c l e s show the experimental EHD l i n e shape obtained by subtract ing the BE l i n e shape from the spectrum shown in (a ) . The errors in subtract ion are shown. The s o l i d curve is the theore t i ca l f i t to the EHD l i n e shape. 16. -3 -2 phosphorus cm . The exc i t a t i on i n ten s i t y Is approximately 120 Wcm . The spectrum shows two overlapping peaks: a Broad one at low energies a t t r i -b u t e d ^ ' ^ to the EHD and a sharper one associated with an exc i ton bound to 11 14 25 29 a neutra l phosphorus impurity ' . Since the r e l a t i v e i n t e n s i t y o f these peaks depends on the e x c i t a t i o n leve l the two over lapping peaks can be separated. The bound exciton (BE) peak s t rong ly dominates the spectrum 2 a t very low e x c i t a t i o n leve l (.1 Wcm" ) and i s used, proper ly s c a l e d , to subt rac t the BE peak from the spectra obtained at e x c i t a t i o n i n t e n s i t i e s in the range of 10 to 200 Wcm . In th i s manner the EHD l i n e shape shown in F igure 3.1b. i s obta ined. To r e i n f o r c e the i d e n t i f i c a t i o n of th is p e a k ^ 5 ^ with the EHD 5 15 i t w i l l be shown tha t , as in the case o f i n t r i n s i c mater ia l ' , the l i n e shape of the EHD peak i s well understood in terms o f the recombination r a d i a -t i on of an e l e c t r o n - h o l e pa i r in a degenerate e l ec t ron -ho le plasma of f i xed dens i ty . The s o l i d l i n e in Figure 3.1b. shows an EHD t h e o r e t i c a l l i n e shape 5 15 obtained by a convolut ion integra l of dens i t ies of s ta te ' . . ' T O 0 ™ ' " (3.1) \l\"o N t E e > N ( E h ' f ( E e ) f ' E h ) 6 ( h u - E p a i / E h + E e - E e - E h + h u T 0 ' d E e d E h • where E g and E^ are respect i ve ly the e lect ron and hole energies in the con-duct ion and valence bands; N(E g ) and ^(E^) are the respec t i ve dens i t i e s of s t a t e ; f ( F e ) and f(E^) are the Fermi-Dirac d i s t r i b u t i o n funct ions taken at the helium bath temperature; and E^ the Fermi energies f o r the e lect rons and ho le s ; E p a i r i s the energy required to add one more e l ec t ron -ho le p a i r to the EHD and i s determined, at very low temperatures, by the high energy 31 threshold o f the luminescence peak . A theoret i ca l f i t has been performed 17. by assuming the EHD to be charge iiseutral, that i s , the to ta l e lec t ron density ( n c ) i s equal to the density o f p o s i t i v e l y charged donor ions (n d ) plus the dens i t y o f photocreated holes (n^t. I t w i l l be assumed that the c a r r i e r s e f f e c t i v e l y screen any bound impurity s t a te s , . thus , n^ i s determined by the impur i ty concentra t ion . The bands are assessed to be pa rabo l i c and the e f f e c t i v e masses which descr ibe than were assumed to be independent of doping. The band parameters used in th i s ca l cu la t i on are l i s t e d in Table 3.2. The TO phonon energy i s given in Table 3.1 and as discussed i n the previous Chapter the sample temperature i s that o f the helium bath. The f i t i s per -formed by vary ing two parameters: £ .„ which f i xes the energy po s i t i on and pa i r n^-which changes the l i n e shape and. width fo r a given impurity dens i ty (n^). The parameters g i v ing the best f i t * are l i s t e d in Table 3.3 as a funct ion of impur i ty concentrat ion and the uncerta int ies quoted r e f l e c t the amount by which they have to be var ied so that a c l e a r l y bad f i t i s obta ined. The spectrum shown i n Figure 3.2 was obtained from a sample conta in -17 -3 ing 5.7 x 10 phosphorus cm . Ihe exc i t a t i on i n tens i t y i s approximately _? 160 Wcm . The l i n e shape o f the EHD peak centered at 1.0835 eV i s indepen--2 -2 dent o f e x c i t a t i o n i n t e n s i t y in the range 1 Wcm to 200 Wcm used in th i s experiment. The s o l i d curve in Figure 3.2 shows the t h e o r e t i c a l f i t . The BE peak (1.09 eV) i s not observed. The peak appearing a t low energy (1.061 eV) i s a t t r i b u t e d to the recombination o f an e lectron in the impur i ty states with a f ree ho le . Study o f this peak at low exc i t a t i on leve l i s obscured by the *The t h e o r e t i c a l EHD l i n e shape ca lcu la ted by assuming that the r e -combination i s a s s i s t ed by the creat ion of TO, as well as , LO phonons (10 to 1 r a t i o and 1.8 meV a p a r t 0 2 } i s the same as the one c a l c u l a t e d assuming no LO phonon ass i s tance except f a r in the wings. The systematic e r r o r introduced in the f i t t i n g pa r a s t e r s by neglect ing the 1 0 phonon r e p l i c a are judged to be wel l below the uncer ta in t ie s quoted in Table 3 . 3 . 18. TABLE 3.3  BEST FIT PARAMETERS* n d C x l 0 1 8 cm" 3) E t r CeY) n h C x l 0 1 8 cm" 3 ) .12 1.1471C5) 3.0(1) .57 1.1469(5) 1.9(1) 1.8 1.1394(5) 1.3(1) 2.45 1.1374(5) 1-3(1) 3.0 1.1352(5) 1.3(1) 3.9 1.132 (1) 1.2(1) 5.0 1.131 (1) 1.15(10) n.o 1.130 (1) 1.6(2) The parameters obtained f o r n . = .12 x 10 cm~^ are in agreement with t h e o r e t i c a l c a l cu l a t i on s of Bergersen e t a l l 9 . As w i l l be shown in Chapter 5, there i s to date no r e l i a b l e t h e o r e t i c a l c a l c u l a -t i o n o f th i s parameters f o r higher impurity concentrat ions . 19. i 1 1 r • I ! ! ! f 1.05 1.07 1.09 'PHOTON ENERGY.(eV) Figure 3.2: The photoluminescence spectrum of silicon containing 5.7 x 10 1 7 phosphorus cnr 3 at T = 4.2 K. and 160 Wcm'"2 excitation level is given by solid circles. The solid curve shows the theoretical f i t to the EHD line shape. 20. appearance of a broad peak at 1.045 eV which dominates the spectrum. The spectrum represented by s o l i d c i r c l e s in Figure 3.3 i s obtained from the same sample using an e x c i t a t i o n i n t e n s i t y of approximately 20 Wcm . The spectrum represented by f l ags was obtained from a sample conta in ing both 17 16 phosphorus and boron with concentrat ions in the order of 10 and 10 _3 atoms cm re spec t i ve l y . The e x c i t a t i o n i n t e n s i t y was approximately 8 -2 Wcm . Because of the general agreement o f these two spectra and the ex-pectat ion of observing a broader peak at lower energy f o r the recombination 33 of an e lec t ron in the impur i ty 'band with a hole bound to an acceptor - ion , i t i s reasonable to a t t r i b u t e the broad peak a t 1.045 eV to donor-33 acceptor recombination. Samples with 3.1 x 10^ 7 and 3.7 x 10^ 7 phosphorus c m - 3 , interme-d iate to the impurity concentrat ions of samples discussed above, have a l so been s tud ied . The photoluminescence study of these samples as a funct ion of exc i t a t i on leve l i s observed to be in c lose agreement with tha t of 14 17 -3 Martin and Sauer f o r a sample conta in ing 1.8 x 10 phosphorus cm . A s ing le recombination band i s observed at high e x c i t a t i o n i n t e n s i t y . With decreasing e x c i t a t i o n l eve l th i s band changes l i n e shape and exh ib i t s evidence of s t ruc ture at low e x c i t a t i o n i n t e n s i t y . Martin and S a u e r ^ argued that these changes in l i n e shape were i n d i c a t i v e of a profound change in the e l e c t r o n i c s t a te s . An a l t e r n a t i v e explanat ion i s that the BE peak i s very broad and overlaps the EHD peak to form a s i n g l e broad band. The changes in th is band with e x c i t a t i o n l eve l can be a t t r i b u t e d to the change in the r e l a t i v e i n t e n s i t i e s o f the EHD and BE emissions with e x c i t a t i o n l e v e l . As pointed out in Chapter 1, a great emphasis in th i s work has been 21. CO ;z UJ o LU 1.00 1-02 1-04 1-06 PHOTON ENERGY (eV) Figure 3.3: The photoluminescence spectrum of silicon containing 5.7x10. phosphorus cm-3 at 4.2K. and 20 W.cm-2 excitation leyel repre-sented by solid circles is compared to that of a compensated sample containing, both, phosphorus ( l O ^ cm" 3) and boron (10*6 cm-3) at 4.2K and 8 Wcm-2 excitation level represented by flags (two standard deviations from 15 scans). The peak at 1.045 eV is attributed to donor-acceptor recombination. 17 22. 18 -3 given to the data and ana l y s i s of samples conta in ing 1.8 _x 10 .cm to 18 3 3.9 x 10 cm because the donor concentrat ion range covered by these samples goes from s l i g h t l y below to s l i g h t l y above n c n . t f o r the meta l -semiconductor t r a n s i t i o n . I o Figure 3.4 shows the spectra of a sample conta in ing 1.8 x 10 -3 phosphorus cm . Figure 3.4a. shows the spectrum at high e x c i t a t i o n l eve l (200 Wcm" ) and the observed peak i s a t t r i b u t e d to the recombination wi th in the EHD. This hypothesis i s supported by the good theore t i c a l f i t o f the EHD l i n e shape showed by the s o l i d curve. Figure 3.4b. shows the spectra 2 2 at intermediate (20 Wcm ) and low (.1 Wcm" ) e x c i t a t i o n l e v e l . The spectra show c l e a r l y two peaks and the peak at high energy i s again a t t r i -buted to the EHD. The other peak i s observed f o r the f i r s t time and i s a t t r i b u t e d to the recombination o f an e lec t ron in the impurity band (IB) with a f ree ho le . The IB peak i s observed at approximately 25 meV below 34 the p o s i t i o n p red ic ted f o r an i s o l a t e d impurity given by: hv = E „ , „ - E„. - h v x n , (3.2) gap imp TO v ' where the band gap energy i s 1.1698 eV at 4.2K and the i o n i z a t i o n energy o f 35 phosphorus in s i l i c o n i s 45.3 meV . This s h i f t to lower energies at n ^ l . S x l O cm" may be expla ined by the conduction and valence band deve-lop ing t a i l s ' i n to the forbidden gap or by a lowering of the ground s t a te energy of the bound e lect rons because of the overlap of the donor e l e c t r o n wave funct ions The spect ra in Figure 3.4b. have been sca led so that the low energy t a i l s o f the IB peaks are superimposed. The l i n e shape of the EHD peak ob-ta ined by sub t rac t ing these two spectra i s the same as the one obtained at high e x c i t a t i o n i n t e n s i t i e s shown in Figure 3.4a. The l i n e shape o f the IB peak i s obtained by subt rac t ing the experimental EHD l i n e shape (Figure 23. a o o t CO z: LU h-2 o z: ijj o CO 1x1 n i r i i SKP) n d = L 8 x ! 0 1 8 c m " 3 a 4 — 1 — I — 1 — 1 — 1 || l b I! i ! i I ©o I 0 ! • * ° ol { 1 1 1 L J L 1.02 1.04 1.06 1.08 PHOTON ENERGY (eV) ,18 ph.cs Figure 3.4: Photoluminescence spectra of silicon containing 1.8x10.' phorus cm - 3 a t 4.2 tC. a) Solid circles show the spectrum at high excitation level (200 Wcm"2}. The peak is attributed to the EHD. The soli d curve shows the theoretical f i t to the EHD line shape. b} The flags (two standard deviations from 6 scans} show the •spectrum at intermediate excitation level C20 \>lcm~c\ and the solid dots (50 scans) the spectrum at low level C-l Wcm"2). The peak at high energies is attributed to the EHD, the other to the impurity band. The spectra have been scaled for comparison. I 24. >-H CO LU LU o LU O to LU 3 1.01 1 . 0 3 1 . 0 5 1 . 0 7 PHOTON ENERGY (eV) Figure 3.5: Experimental photoluminescence l i n e shapes f o r an e lec t ron in the impurity band and a f ree hole o f phosphorus-doped s i l i c o n conta in ing 1 . 8 x l 0 1 8 c m - 3 . Impurity band l i n e shapes at exc i t a t i on l eve l s of 5 Wcm-2 (long f lags ) and .1 W.cm-2 (short f lags ) are shown. The f lags represent two standard dev iat ions due to s igna l averaging and to the sub t rac t i on process r e fe r red to in the tex t . 25. 3.4a. l from the low exc i t a t i on leve l (.1 Wcm" ) spectrum. As shown in Figure 3.5 the same IB l i n e shape ar i ses from an intermediate e x c i t a t i o n (5 Wcm ) spectrum. The recombination emission o f e lec t rons in the impurity band and f ree holes in the valence band i s proport iona l to the convolut ion of the dens i t ie s o f s tate of the two bands. Since only those states o f the valence band with in approximately kT of the band maximum are unoccupied and at 4.2 K th i s energy i s n e g l i g i b l e compared to the width of the observed IB peak the experimental IB l i n e shape gives d i r e c t l y the density o f s tates in the impu-r i t y band. In Chapter 4 the ca l cu l a ted impurity band dens i ty of s tates w i th -in severa l models are compared with th i s experimental l i n e shape. 18 Figure 3.6 shows the spectra o f a sample contain ing 2.45x10 -3 phosphorus cm ... The s o l i d dots in Figure 3.6a. show the spectrum at high 2 e x c i t a t i o n i n t e n s i t y (200 '.vcm ). The s o l i d curve i s a t h e o r e t i c a l f i t to the EHD l i n e shape. The f lags in Figure 3.6b. show the spectrum at interme-2 diate i n t e n s i t y (20 Wcm ) whi le the s o l i d c i r c l e s i nd i ca te the spectrum at low i n t e n s i t y (.1 Wcm }. A second peak i s c l e a r l y v i s i b l e at low e x c i t a t i o n leve l which, as in the prev ious ly discussed sample, i s a t t r i b u t e d to IB r e -combination. There i s no d i s ce rn i b l e d i f fe rence between the EHD l i n e shape observed by subt rac t ing the two spectra shown in Figure 3.6b. from the EHD spectrum shown in Figure 3.6a. Figure 3.7 shows the IB experimental l i n e shape obtained by subtract ing the EHD l i n e shape given by the spectrum shown _2 in Figure 3.6a. from the low exc i t a t i on leve l spectrum (.1 Wcm" ) shown in Figure 3.6b. 18 - 3 The spectra fo r a sample contain ing 3.0x10 phosphorus cm" are shown l a t e r in th i s Chapter. The spect ra l analys i s of these data fol lows very c l o se l y the one descr ibed above and i s not inc luded here. The resu l t s are l i s t e d together with those of other samples in Table 3.3. Figure 3.8a. shows two spectra of a sample conta in ing 3.9x10.''8 26. 1.0! 1.03 1.05 1.07 JOJ a o o b to z L i i LU O z: Id O CD lxl Si (P) n , » 2 . 4 5 x ! 0 1 8 c m ~ 3 a b / \ " ' i . n ' " 1.0! 1.03 1.05 .1.07 PHOTON ENERGY (eV) J8 Figure 3.6: Phol-olunrinescence spectra of silicon containing 2.45x10 phosphorus cnr^ at 4.2 K. a) Solid circles show the spectrum at high excitation level (200 Wcm" 2). The peak is attributed to the EHD. The solid curve shows the theoretical f i t to the EHD line shape. b) . The flags (two standard deviations from 6 scans) show the spectrum at intermediate level (20 Wcnr2) and the solid dots (40 scans) the spectrum at low level (.1 Wcnr2). The peak at high energies is attributed to the EHD. the other to the IB. The spectra haye been scaled for comparison. 27. o u t o o OJ c in UJ UJ o UJ o in UJ 1.06 PHOTON ENERGY (eV) Figure 3.7: IB experimental photoluminescence l i n e shape of phosphorus doped s i l i c o n conta in ing 2.45 x 1 0 1 s c m - 0 . The e x c i t a t i o n i n t e n s i t y i s approximately .1 Wcm - 2 . The f lags represent two standard dev iat ions due to s ignal averaging and to the subt rac t ing process r e fe r red to in the text . 28. CD O O V) CO LU 2 LU O LU O CO LU Figure 3.8: a) S i ( P ) n d = 3 . 9 x I 0 , 8 c m " 3 a e V ~ : : > ' ! -1—I—l-H—I—f—+ b 1 L02 1.04 1.06 1.08 PHOTON ENERGY (eV) ,18 Photoluminescence spectra of s i l i c o n 3 .9x10 , u phosphorus c m - 3 at T = 4.2 IC are shown at two e x c i t a t i o n l e v e l s . At high exc i t a t i on leve l 200 Wcm - 2 , 1 scan) both the im-p u r i t y band (IB) and the EHD peaks are observed and a t low leve l (.2 Wcm - 2 , 60 scans) the IB peak s t rong ly do-minates. b) The s o l i d c i r c l e s give the EHD l i n e shape obtained by subtract ing the two spectra in f i gu re Ca)- The s o l i d curve shows the theore t i ca l f i t to the EHD l i n e shape. . 29. -3 2 phosphorus cm . The high e x c i t a t i o n leve l spectrum (200 Wcm ) shows both the EHD and the IB peaks. In the low leve l spectrum (.2 Wcm ) the IB peak s t rong ly dominates the spectrum. The l i n e shape o f the EHD peak obtained by subt rac t ing these two spectra i s shown in Figure 3.8b. In s i m i l a r fashion the EHD l i n e shape has been obtained as a funct ion of e x c i t a t i o n l e ve l s in the range o f 10 to 200 Wcm and th i s l i n e shape i s not observed to change. The s o l i d curve i n Figure 3.8b. shows the theo re t i c a l f i t to the EHD peak. The two superimposed IB peaks shown in Figure 3.9 were obtained by subt rac t ing _2 the EHD l i n e shape (Figure 3.8b.) from intermediate (20 Wcm ) and high e x c i -_o t a t i on leve l (200 Wcm ) spectra . The l i n e shape i s not observed to change i n t h i s range o f e x c i t a t i o n i n t e n s i t i e s and i s very near ly that observed in the low leve l e x c i t a t i o n spectrum. 18 -3 The spect ra o f a sample conta in ing 5 x 10 cm" w i l l be shown l a t e r i n t h i s chapter. The spec t ra l analys i s i s not given here s ince i t may be i n f e r r e d from those o f the prev ious ly discussed samples conta in ing 18 -3 19 -3 3.9 x 10 cm and those of the 1.1 x 10 cm to be discussed below. The numerical values r e s u l t i n g from the ana lys i s are l i s t e d in Table 3.3. 19 -3 Figure 3.10 shows the spectra o f a sample conta in ing 1.1 x 10 cm . 2 The e x c i t a t i o n i n t e n s i t i e s are : a) high (150 Wcm" ), b) intermediate 2 2 (20 Wcm" ) and c) low (2 Wcm ). The EHD peak very s t rong ly dominates the high e x c i t a t i o n l e v e l spectrum. The IB peak very s t rong ly dominates the low leve l one. The photoluminescence i n t e n s i t y f o r samples with impurity con-centrat ions above n c r ^ decreases s t rong ly with increas ing concentrat ion and in add i t ion the r e l a t i v e i n t e n s i t y o f the EHD and IB peaks becomes more s t rong ly dependent on e x c i t a t i o n l e v e l . As shown in Figure 3 . 1 0 c , only the IB peak is observed a t low e x c i t a t i o n l e v e l . The s o l i d curve in Figure 3.10a. 30. 1.02 1.04 1.06 1.08 r~ i i i i i i r -Figure 3.9: IB experimental photoluminescence l i n e shapes of phosphorus-doped s i l i c o n containing 3.9 x 10 1 8 cm"3. Impurity band li n e shapes at excitation levels of 20 Wcm-2 (short flags) and 200 Wcnr2 (long flags) are shown. The flags represent two standard deviations due to signal averaging and to the sub-tracting process referred to in the text. 31. .. 1.02 1.04 1.06 1.08 PHOTON ENERGY (eV) 19 Figure 3.10: Photo!uminescehce spectra of s i l i c o n conta in ing 1.1x10 phosphorus c m - 3 at T = 4.2 K are shown at three e x c i t a t i o n l e v e l s . a) At high exc i t a t i on l eve l (150 Wcm - 2 , 5 scans) the EHD peak dominates the spectrum. The s o l i d curve shows the theo re t i c a l f i t to the EHD l i n e shape. b) At intermediate leve l (20 Wcnr 2 , 15 scans) both the IB and EHD peaks are observed. c) At low leve l (2 Wcnr 2 , 35 scans) the IB peak dominates the spectrum. 32. shows the theoretical f i t to the EHD peak. The EHD line shape is independent -2 of excitation level in the range 80 to 200 Wcm . The spectrum at interme-diate excitation intensity (Figure 3.10b.) can be reproduced by adding the high excitation level spectrum (Figure 3.10a.) to the low level one (Figure 3.10c), properly scaled. 12 For phosphorus-doped silicon a second characteristic concentration, 19 n.cb~2 x 10 cm " a is evidenced in the measurement of the Knight shift of 29 the NMR absorpiton peak for Si as a function of impurity concentration. 12 Alexander and Hoi comb argue that the Fermi level is above the conduction band edge for impurity concentrations greater than n^. 19 -3 Figure 3.11 shows two spectra of a sample containing 4.0 x 10 cm . -2 -2 High (150 Wcm ) and low (5 Won ) excitation intensities have been used. The line shape of the observed peak depends on the excitation intensity with-- 2 - 2 - -in the range 5 Wcm to 200 Wcm . The line shape of the peak shows a de-crease in the slope of the low and high energy side with increasing excita-tion level. These changes in line shape with excitation level could be interpreted in terms of unresolved broad EHD and IB peaks; however, one cannot make firm conclusions because of the absence of structure in the 19 -3 photoluminescence spectrum for 4.0 x 10 cm . A summary of the photoluminescence studies in heavily phosphorus-doped silicon as a function of donor concentration at high excitation levels is shown in Figure 3.12. At this excitation level, the IB peak is only clearly observable at impurity concentrations close to the metal-semiconduc-tor transition. Since Halliwell and Parsons^ used higher excitations than those reported here i t is not surprising that they were unable to detect the IB peak. The monotonic shift of the high energy threshold of the EHD peak to 33. D O O >-to LU 2 UJ o LU O CO LU ID 1.02 1.04 1.06 1.08 PHOTON E N E R G Y (eV) Figure 3.11: Photoluminescence spectra of s i l i c o n conta in ing 4x10 -phosphorus c n r 3 at T = 4.2 K. are shown at two e x c i t a t i o n l e v e l s . The s o l i d points show a high e x c i t a t i o n leve l (150 Wcm - 2 , 10 scans) spectrum. The f l ags correspond to low l eve l (5 Wcm - 2 , 110 scans].. 34. 0 <Ti U if) L_ (TJ <b C b : u l z UJ i — L L I u L L J o -CO U J z j x 1 0 1 8 / | S i C P ) 4 Q / ( \ 11. / / \ / 3 . 9 ^ /k 3 D JI Ir71 1.8 A .12 >/ i i i i i i i i ! > 1 I 1 1 1 < 1 Figure 3.12: Concentrat ion dependence of the photoluminescence o f phosphorus-doped s i -l i c o n at 4.2 K using high e x c i t a t i o n i n t e n s i -t i e s . I.oo LOS I.Jo 115 e n e r g y (eV) 35. lower energies wi th i nc rea s ing doping concentrat ion fo l lows c l o s e l y the de sc r i p t i on o f H a l l i w e l l and Parsons^ 1 as can be seen in Figure 3.13. A comparison of the width at h a l f i n t e n s i t y o f the EHD peak measured in th i s 1 1 i n ve s t i g a t i on wi th those reported by H a l l i w e l l and Parsons ' i s shown i n F igure 3.14. F igure 3.15 shows a p l o t o f the r a t i o of the r e l a t i v e in teg ra ted i n t e n s i t y o f the sum o f the TA and NP rep l i ca s to the r e l a t i v e i n t e n s i t y o f the TO phonon r e p l i c a as a funct ion o f impurity concentrat ion. The sum of the TA and NP r e p l i c a s i s used because they cannot be reso lved over the whole range of impur i ty concentrat ion s tud ied here (see Figure 3.12). The f a c t that the NP plus TA r e p l i c a grows with inc reas ing n^, with respect to the TO phonon r e p l i c a , may be understood because the average in ter -donor d i s tance becomes o f the o n ' e r of the average i n t e r - c a r r i e r d istance in the d rop le t and consequently the p r o b a b i l i t y of a recombination of an e l e c t r o n -hole in the v i c i n i t y o f an impurity increases. It i s present ly not under-1 q o stood why th i s i n t e n s i t y r a t i o should l eve l o f f for'. n^>10 cm to a value approximately equal to the NP plus TA to TO r a t i o f o r the BE recombination r a d i a t i o n . The photoluminescence measurements o f the IB density of s tates in heav i l y phosphorus-doped s i l i c o n are summarized in Figure 3.16. The IB l i n e shapes o f the samples conta in ing 1.8 x 10 cm" , 2.45 x 10 cm"" and 18 -3 3. x IC cm are not s i g n i f i c a n t l y d i f f e r e n t from each other. These samples w i l l be r e f e r r e d to as the lower group. The IB peak, wi th in th i s group, shows a s l i g h t s h i f t to higher energy and an increase in band width with inc reas ing concent ra t ion . The IB l i n e shapes o f the samples conta in ing 3.9 x 1 0 1 8 c m " 3 , 5. x 10 1 8 cm" ' 3 and 1.1 x 1 0 1 9 cm""* are a lso not s i g n i f i c a n t l y 36. 1.15-(0 a 1 1 1 •this work j oHailiwel! & Parsons ? IttLL- J — i I i m i l 1 —» • » " M I , , , • ,,,,] . , lo' \ou w i a , 0 I 9 IMPURITY CONCENTRATION (cm"3) Figure 3.13: Concentration dependence of trie threshold energy F. . . Open c i r c l e s show data points of Ha l l w e l l and p a i r P a r s o n s ' ' . 37. 20h £ J5 X J— Q I 1 0 L L < X ©this w o r k o Ha Iii well & Parsons 11 5 L L _ 1 1—L -X- t I t I 10 is r r ? —1 — ' ' ' 1 1 ' ! [ , Q — J — ' i u x L . — i to' 10' JO 1 9 IMPURIlY CONCENTRATION (cm - 3 ) Figure 3.14: Width, at h a l f maximum of tlie TO-aas isted peak. as. a funct ion of phosphorus concentrat ion at 4.2 K.. Open c i r c l e s show data po ints of H a l l i w e l l and Par sons 1 1 at 2 fC. 38. T A ^ N P TO . 4 0 3 0 H . 2 0 10 (7 10 I i 1 i n f 1 t t t t 111F i i i i i 1 1 1 1 10' .19 J L_1L IMPURIlY CONCENTRATION (cm - 3 ) Figure 3.15: Concentrat ion dependence of the r a t i o o f the r e l a t i v e i n t e -grated i n t e n s i t y of the sum of the TA and NP reoli.cas to the TO phonon r e p l i c a . 39. 1.0! 1.03 1.05 L07 u in © LU H LU O LU O CO LU ZD i 1 1 1 1 r I ' I x10' 08 I il SUP) » I I . I 1. —fy— I 1.8 I .1,111 II 1' 1 .. • „ .', •••••mill 1» II 245 ,»...«•«•«•••* -HIIII« 1 1 '», i I ii i i i V • V , .I1 • •"I"'.1. 1 •'. 3 0 I'II'IIII'1 ; • i l l ! 3.9 .»ni i •in' • I " ' ' / " l l , ! ! , 1 ! 1 | l ' III, | 5 0 11. I Ml, IM'HI 1' 1 1 i i n i 1 i l I ! L LOI 1.03 1.05 1.07 PHOTON ENERGY (eV) Figure 3.16: Concentration dependence of the experimental IB photolumines-cence l i ne shape of phosphorus-doped s i l i c o n at 4.2 IC. 40. different from each other. These samples are referred to as the upper group. Within this group, there is only a slight change in the high energy side of the line shape. The change in line shape between the two groups of samples, is remarkable. The IB peak of the-upper group samples is. shifted & meV to higher energy with respect to the peak of the lower group samples. The long tail of the line shape at high energies characteristic of the lower group samples is no longer observed in the line shape of the upper group; in fact, an edge is observed. It is clear that a change in the nature of the electronic states has occurred by increasing the impurity 18 -3 18 -3 concentration from 3 x 10 cm" to 3.9 x 10 cm" and is attributed to the semiconductor-metal transition. In Chapter 4 the Hubbard model^ for the impurity band is presented. The IB line shapes for samples with phosphorus concentration below n c r j t are successfully described within this model and a plausible explanation for the changes observed at the semiconductor-metal transition is discussed. 41. CHAPTER 4 IMPURITY BAND 4.1 Introduction As mentioned in Chapter 3 the photoluminescence spectrum of heavily phosphorus-doped silicon at low excitation intensities is dominated by a peak associated with the recombination of an electron in the impurity band and a free hole. It has also been said that since the experiments are performed at near zero temperature the IB line shape gives directly the density of occupied states in the impurity band. In the present in-vestigation the comparison of the experimental IB line shape with theore-tical models of the impurity band is restricted to low impurity concentra-18 3 tions, n<n .. = 3 x 10 phosphorus cm"'. v i I t To calculate the density of states in the impurity band two different approaches are taken in the literature. The f i r s t approach is to locate the impurities in a superlattice and then perform a standard band calculation. Such a treatment for phosphorus-doped silicon is pre-sented in the next Section. The second approach is to calculate the energy of a donor electron in all possible impurity configurations. The calculated energies fall in a range of energies referred to as the impurity band and the probability of finding the impurity configuration giving an energy E reflects the density of states of energy E. Lukes et a l ^ have calculated the density of states of the im-purity band in the low impurity concentration limit where they reduced al l possible impurity configurations by considering only impurity pairs and the distance, R, between the impurities was taken to follow the 42. 39 Chandrasekhar d i s t r i b u t i o n . They use a simple hydrogen molecular ion (H^) model of i n t e r a c t i o n between the impur i t ies in a p a i r and the d e t a i l s o f t h i s c a l c u l a t i o n are given in Appendix B. The r e s u l t i n g 18 3 impuri ty band dens i ty of s tates for n d = 1.8 x 10 cm i s compared to the experimental IB l i n e shape in Figure 4.1. The width of the c a l -cu la ted dens i ty of s tates wi th in th is model i s too broad and i s a con-sequence of i gnor ing a l together e l e c t r o n - e l e c t r o n i n t e r a c t i o n s . A hydrogen molecule model of i n t e r a c t i o n , suggested by Macek^, i s used to extend the previous c a l c u l a t i o n and i s solved using the Hei t i e r - L o n d o n ^ method. The d e t a i l s of t h i s c a l c u l a t i o n are a lso given in Appendix B. Figure 4.1 a l so compares with experiment the ca l cu l a ted IB photo!umines-18 3 cence l i n e shape f o r the donor concentrat ion n^ = 1.8 x 10 cm" . The width obtained in the Heit ler -London model i s too narrow which i s a con-sequence of the i s o l a t e d p a i r approximation.* The theory o f the impurity band when the e f f e c t o f randomness in the impur i ty d i s t r i b u t i o n i s neglected i s reviewed in Sect ion 4.2. From the mathematical point of view the treatment i s the same as that o f Eswaran and Bergersen which appears in an a r t i c l e publ i shed in c o l l a b o r a -19 t i on with the author and others . In th i s thes i s a much greater emphasis i s given to the j u s t i f i c a t i o n of the assumptions and approximations of the mathematical treatment. *The rad ius of a sphere of obtained by arranging the impur i t ie s i s c lose to the mean distance of the equal volume to a Wigner-Sei tz c e l l in a regu lar c lose-packed l a t t i c e Chandrasekhar39 d i s t r i b u t i o n . LOI 1.03 1.05 1.07 ~ i i i i i i r~ i i i i i i i i L LOI 1.03 1.05 1.07 PHOTON ENERGY (eV) Figure 4.1: The experimental photoluminescence l i ne shape f o r an e lec t ron in the impurity band and a f ree hole of s i l i c o n conta in ing 1.8 x 1 0 ' 8 phosphorus cm"" 3, represented by f l a g s , i s compared to the theo re t i c a l IB l i n e shapes ca l cu la ted in the ( ) He i t ier -London and ( ) H 2 models. The theoret i ca l bands are s h i f t e d in energy and sca led f o r comparison with experiment. 44. 4.2 The Impurities in a Super l a t t i c e Let us begin by neg lect ing a l l e l e c t r on -e l e c t ron i n t e r a c t i o n s . I f the impur i t ies are placed in a regular l a t t i c e the Hamiltonian i s : H = I t . . a* a . + T E n. (4.1) where a | a ( a ^ a ) creates (destroys) an e lec t ron with sp in o in a Wannier s tate a t s i t e i , n - a = a. a a ^ Q the corresponding number operator , t.. . i s the overlap i n teg ra l of the l a t t i c e Hamiltonian between Wannier funct ions in the nth energy band from s i t e s d i s t an t R. i (=jR. - R-|) apart and i s a lso known as the hopping i n teg ra l and T = I e. , c. being the energies in the k nth band. For impurity concentrat ions below the metal-semiconductor t r a n -s i t i o n i t i s reasonable to describe the impurity band i n the t i gh t binding approximation. The Hamiltonian desc r i b ing the impurity e l ec t ron i s given by H = V 2 + v 0 ( " ) +Z U(r - R.) (4.2) where V ("*"") i s the p e r i o d i c p o t e n t i a l a r i s i n g from the host atoms and U(r - R )^ i s the perturbing po ten t i a l due to the impurity ion (bare po tent i a l ) at s i t e i : U C r - R J ) . ( 4 - 3 ) e j r - R.| where e i s the d i e l e c t r i c constant o f the host. The Hamiltonian in equa-t ion 4.2 may be expressed in terms o f the i s o l a ted donor Hamiltonian • h = "Ik" v" + v° (?) * u(? " ^  (4-4) 45. so that H = H + Z U(r - i g . (4.5) 42 So finally t.- = < * (r - R.)| Z U(7 -R m ) |< {> ("r - R . ) > . (4.6) where the wavefunction (ji("r-Rj) has been taken to be the ground state atomic orbital at site i and is the eigenfunction of the isolated donor Hamiltonian H.. (equation 4.4). We restrict ourselves, to the case where i and j are neighbouring.impurities and the t-y contains one 2-centre integral and five '3-centre integrals. If we know the functions o>(r"-K.) the problem is solved 42 and we obtain a typical tight binding band centered at L with width A = 2z I t . j l (4.7) where the coordination ntimber z will be assumed hereafter tb be 6 applica-ble to a simple cubic lattice. As for all one-electron approximations one runs into the tradi-tional problem of describing an insulator when one has only one s-electron per unit cell. One obtains, due to spin degeneracy, a half f i l l e d band for 42 any inter-donor distance . It has been known for a long time that in a many-electron system this difficulty is remedied by the electron-electron.: 43 interactions . A model that takes into account electron correlations to some degree has been presented by Hubbard^. He adds to the Hamiltonian given by equation 4.1 a term H 1 = h U Z n. n. n . (4.8) The significance of this term is that the Coulomb repulsion, U» between electrons is taken into account only i f the electrons are on the same site. •46. If U is larger than the unperturbed (tight binding) band width one has an insulator since now i t is necessary to ionize at least one donor, and to put the electron back on a distant occupied site for current to flow. The work necessary for this is the ionization energy less the electron 44 affinity , thus approximately U, In the large U limit, the Hubbard model implies that one electron is localized to each impurity site and double occupancy is forbidden. This clearly describes an insulator but when attempting to calculate the band, a question arises on the meaning of the hopping integral. To resolve this problem let us look upon the IB photoluminescence in terms of a transition from an i n i t i a l state given by a hole in the top of the valence band to a final state of a hole in the impurity band, thus we are interested in the hole density of states in the impurity band. In the large U l i m i t , one has an electron in each impurity site except, one and in this picture the hopping integral has again a natural meaning: a hole in site i hops to Site j ; equivalent to the electron in site j hopping to site i . The hopping integral becomes equal to the leading term of the one electron tight binding one since now the bare potentials at nearest neighbour sites m f i , j are screened by the electrons which are localized in those sites and to a good approximation can be neglected. Hence t,j - < $ (r ^ R.)!U(7- RjJI* ( 7 - If.) >. (4.9) which is the sans as that obtained in the mathematical treatment' referred 45 above. In the treatment of Berggren t• • is different - the unperturbed bandwidth defined by him in terms of his integral T is inconsistent with 47. 42 46 AS t r a d i t i o n a l t i g h t binding * ™ and in e r r o r . The in teg ra l L of Berggren corresponds and i s equal to Equation 4.9. Thus, from the d iscuss ion above, i t i s c l e a r that the dens i ty o f s tates in the impurity band in the large U l i m i t i s going to resemble a t i gh t b inding band - sca led to h and f u l l -which f o r narrow bands i s parabo l i c to f i r s t approx imat ion^ . The wave funct ion <{>(r) fo r the ground state o f an i s o l a t e d impu-47 r i t y i s obta ined i n the ef fect ive-mass approximation. Fol lowing Kohn <|>(F) i s wr i t ten as a wave packet cons i s t ing of Bloch funct ions H> r ("r), at 1 the s i x conduction band minima (k^, I = 1, . . .6) o f the s i l i c o n host: <j> (7) = ! a M ^ H F (r) » ( 4 J ° ) £=1 * * Ka where the c o e f f i c i e n t s f o r the ground state are (6 ) "^ f o r a l l £ 4 7 . The are hydrogen - l i ke envelope funct ions which are approximately . 48 ' given by F (^7) = ( i r a 2 b ) ' % exp{ - { (x 2 + y 2 ) / a 2 + z 2 / b 2 } % } (4.11) with the z -ax i s or iented along the long i tud ina l axis of the Ith v a l l e y . The constants a and b are the transverse and l ong i tud ina l Boh r - l i ke r a d i i o f the o r b i t and we choose t h e i r va lues, not in a v a r i a t i o n a l procedure 47 of the e f fec t i ve -mass equations but by requ i r i ng that the eigenvalue 48 be E Q > the observed i o n i z a t i o n energy o f an i s o l a t e d donor : a = ( 2 m t E 0 ) ^ ; b = aim^m^ (4.12) with the transverse and long i tud ina l masses given in Table 3.2. This choice of a and b gives the cor rec t asymptotic behavior c f the envelope functions which i s o f importance in the so lu t i on o f the two centre i n teg ra l 48 t . . . This i n t e g r a l has been solved by M i l l e r and Abrahams and the 48. analytic solution, considering only intra-valley terms, is: t 2 6 -ik -R.. -(R»/a) (A N \ where Upon squaring, keeping only intra-valley terms and finally spherically averaging over the orientations R--^ , one obtains: IV2-<i!s><VW- (4'15) where n ( 1 2 n / 2 ? A = (R) I' dx (1 + ax^) exp (-2(R/a)(lW) } ^ V ) Q (4.16) with a = (a/b) - 1 and R = |R.-{, the inter-donor distance. (The co r re s -45 ponding expressions of Berggren contain algebraic errors.) Figure 4.2 compares the experimental IB line shapes for T O O I O o 1 ft "3 n d = 1.8 x 1 0 ' ° cm , 2.45 x 10 cm and 3.9 x 10 1 cm" with theoretical ones calculated by Bergersen et a l 1 9 following the treatment o f Hubbard17 for finite U. The calculated density of states is not significantly different from a parabolic form: n(E) - ( A / 2 ) " 1 J 1-[(E - E 0 ) / { A / 2 ) ] 2 | * i f |E-£j< A/2 = 0 otherwise. (4.17) The theoretical line shapes have been energy s h i f t e d and sca led f o r compa-rison. As can be seen there is good agreement between the calculated and ob-served density o f s ta tes and the theory p red ic t s s u c c e s s f u l l y the observed broadening of the band with impurity concentrat ion. The observed t a i l s o f the IB photoluminescence are due to the randomness of tfee impurity d i s t r i ! . . 45, 49-51 t ion bu-49. f.Ol t.05 1.05 1.07 i i i i i i i — S i ( P ) A . nd=l.3x|0,8cm~3 . ^ HK" o o to o c >-CO UJ LiJ O LU O CO LU • l r ii : '• I .1 ''Vii, I I t ' l l 1 • 1_ n . = 2.45x|0 l o c m ~ r i .. a • i ' • • i s : ' " i n : u « l ! — HUM'1- « • • • • l « H I 1 L • 1 • 1 • • • w • • • • 1 • • 1 • • l > * • • J.,11 • 1 ! 1 I 3 1.05 1.07 1.0! i PHOTON ENERGY (eV) Figure 4.2 The experimental photoluminescence l i n e sjiapes for the impurity band in S i (P) at donor concentrat ions 1.8 x 1 0 1 8 c m " 0 , 2.45 x 1 0 ' 8 cm" J and 3 . 9 x 10^ 8 cm"^ are represented by f l a g s . The s o l i d - d o t t e d curves represent the theore t i ca l impurity band density of s tates obtained in the Hubbard model. 50. To exp la in the lack of a high energy t a i l of the IB l i n e shape o f 18 - 3 the sample conta in ing 3.9 x 10 phosphorus cm" i t i s necessary to look a t Hubbard's model 1 ^ in a less r e s t r i c t i v e way. Hubbard^ 7 has shown that i f A/U<1.15 one obtains two d i s t i n c t sub^bands separated by an energy gap. The lower sub-band c l o se l y resembles the band, ca lcu la ted above, for the large U l i s r i t . Thus in th is model the semiconductor - metal t r a n s i t i o n occurs at a donor concentrat ion at which the two Hubbard sub-bands begin 17 45 1^ to overlap ' ' ~ . In r e a l i t y , the t r an s i t i on i s more complicated-than tha t . The experimental resu l t s show that the occupied lower band has t a i l s and i t i s reasonable to assume that the upper sub-band has developed t a i l s as w e l l . The t a i l s are made up of l o c a l i z e d states in the Anderson 49-51 sense . In r e a l i t y , the t a i l s of the lower and upper sub-bands w i l l overlap at a lower concentrat ion than in the case of a regular super-l a t t i c e * arid the semiconductor-metal t r a n s i t i o n is now be l ieved to occur not when the bands s t a r t to over lap, but when the Fermi l e ve l l i e s in a 51 52 region of d e l o c a l i z e d states ' . In th is p i c tu re - the Mott-Hubbard-52 Anderson model - i t i s c l e a r that i n the m e t a l l i c region the density of occupied s t a te s w i l l not show a high energy t a i l but ra ther a Fermi edge. Furthermore, the s h i f t to higher energy of the IB peak when n^ goes from s l i g h t l y below n ^ + to s l i g h t l y above i s an i nd i ca t i on that every free e lec t ron helps loosen the remainder' . I t was descr ibed in' Chapter 3 that the r e l a t i v e i n t e n s i t y of the IB and EHD peaks i s exc i t a t i on i n ten s i t y dependent. This experimental *Us ing Hubbard's criterion A/II = 1.15 1 7 to define the semiconductoi metal transition with A as calculated he remand U as given by Berogreni|5 one obtains a c r i t i c a l density of 6 x 10 1 8 cm" 0 . 51. result is an indication that droplets coexist with the lower density phase. In thermodynamic equilibrium the chemical potentials of a pair in each phase are equal. Epa-jr is the chemical potential of a pair in the drop. The minimum energy to create a pair outside the drop assuming the 51 Hubbard model i s , intuitively, an energy close to the mobility edge in the upper sub-band. Thus the high energy edge of the EHD shows approxi-mately where the mobility edge is with respect to the lower sub-band so that for impurity concentrations below the semiconductor-metal transition one expects the I B peak to be at a lower energy than the EHD peak and as the impurity concentration increases one expects the two peaks to overlap -this is indeed observed. The conclusion to the above argument is an important one - E - r is equal to the optical gap. To end this Chapter an outline of the calculation of Eswaran et 21 al of.the density of states in the infinite U limit of the Hubbard model when the impurities are randomly distributed is given. The density of states for a single hole is again of interest. The formalism follows 53 Cyrot-Lackmann and Gaspard who have calculated the density of states for the uncorrected case (U = 0). The Green's function averaged over all configurations is expanded in terms of average moments: the moment u^ , of the density of states being the sum of the hopping contribution of all 54 55 walks of k steps ' which return the hole to its home site. The hopping 53 paths when averaged can be decoupled into irreducible paths or diagrams and the sum of all irreducible diagrams of n steps defines the cumulant of 53 nth order . The importance of the cumulant approach is that i t allows to judge the approximations to be made in different impurity density regimes by neglecting certain diagrams and that those which are kept contribute to 52. the moments o f the dens i ty of s tates to a l l orders. In the high dens i ty regime the important contr ibut ions to the cumulant come from s e l f - a v o i d i n g 21 53 54 diagrams ' ' ; i n the low density regime from those which the hole 21 hops to and from between two impurity s i t e s . 21 F igure 4.3 compares the ca l cu l a ted density o f s tates in the impurity band in the high and low density regimes with the IB experimental l i n e shape f o r two d i f f e r e n t impurity concentrat ions. The widths at h a l f maximum o f the t h e o r e t i c a l bands are in very good agreement with experiment. However, those c a l c u l a t i o n s do not expla in the sign of the s l i g h t skewness of the experimental l i n e shape. The reason f o r th i s discrepancy is not understood. 53. 1.00 1.02 1.04 1.06 1.08 ~i 1 i 1 j 1 — i — > — r Si(P) - I — J 1 i I i L___j_ L_ 1.00 . 1.02 1.04 1.06 1.08 PHOTON ENERGY ( e V ) Figure 4.3: The experimental photoluminescence line shape for the impu-r i t y band in Si(P) at donor concentrations 1.8 x ICH8 cm"^  are represented by-flags. The solid and chained curves represent the theoretical'impurity band density of states obtained using the low and high density cumulants, respectively. 54. CHAPTER 5 THE EHD IN HEAVILY DOPED SILICON 5.1 Introduct ion Th i s Chapter i s concerned with the theory of the ground s tate of the EHD in heav i l y doped s i l i c o n and begins by showing that the p rev ious l y publ i shed t h e o r e t i c a l treatment by Bergersen et a l ^ 8 ' ^ 9 i s in e r r o r . The previous p r e d i c t i o n of the existence of the EHD is e s tab l i shed to be a consequence of sub t le computer programming e r r o r s . I t should be s t ressed that fo r donor concentrat ions above the semiconductor-metal t r a n s i t i o n 56-58 the system under cons iderat ion by Mahler and Birman i s exact ly the same as the one considered by Bergersen et a l 1 8 ' Droplet s t a b i l i t y p red ic ted by Mahler and B i r m a n ^ " ^ is be l ieved to be a consequence of unphysical assumptions. In Sect ion 5.3, the theory of Bergersen et al 1 8 j 1 9 i s presented with a mod i f i c a t i on based on the experimental r e su l t that the EHD coexists with a lower dens i ty e l ec t ron -ho le plasma. This modif ied model i s equa l ly unsuccessful i n p r e d i c t i n g drop let s t a b i l i t y . Speculat ive arguments f o r d rop le t formation in m e t a l l i c s i l i c o n assuming that not a l l donors, are i on i zed are presented in Sect ion 5.4. 5.2 The O r i g i n a l Model The s i l i c o n c r y s t a l i s considered to be at absolute zero tempera-ture and has a volume ft. Let us assume that there exists a drop let of 55. volume V d and i n a d d i t i o n , that a l l photo-created ca r r i e r s are wi th in the d r o p l e t ! thus N(= n^ V^) w i th , N, the tota l number of pa irs kept cons-tant by o p t i c a l pumping. The tota l energy per unit volume e ( n c , n^) i s the sum o f the k i n e t i c , exchange, co r re l a t i on and impurity energies per un i t volume assoc ia ted with the indicated dens i t i e s . e < n C n h } = e k i n ( V n h } + e e x c ( V n h } i+ e c o r r ( V n h ) + e i m P ( V n h ) - ( 5 J ) 18 Bergersen e t a l have worked out these contr ibut ions in the Random Phase Approximation (RPA). The i r treatment of the f i r s t three terms fol lows that of Combescot and Nozieres^and i s general ized to the case when the e lec t ron dens i ty d i f f e r s from the hole dens i ty. The l a s t term i s added to take i n to account the in terac t ion of the impurity ions with charge 18 19 c a r r i e r s . In t h e i r second paper they extend the treatment to inc lude centra l c e l l co r rec t ions - the s i m p l i f i e d a n a l y t i c form of the d i e l e c t r i c 59 funct ion o f the host suggested by Nara and Morita i s used in place of the s t a t i c d i e l e c t r i c constant, which screens the Coulomb i n t e r a c t i o n between charges. The d e t a i l s o f each of the d i f f e r e n t energy contr ibut ions are 18 19 treated ex tens i ve l y by Bergersen et a l ' and are not repeated here. The r e s u l t i n g a n a l y t i c expressions fo r each energy cont r ibut ion are l i s t e d in Appendix D f o r computational purposes. Here, the d i scuss ion of th i s model i s r e s t r i c t e d to the energetics of EHD formation. Fo l lowing Bergersen et a 'P 8 , 1 9 , the tota l energy in the c ry s ta l 15 Crystal = ( f i " V e<V °> + V d e < n d + V nh> ' (5"2> 56. where we have assumed, as in Chapter 3, that the drop is neutral (n £ = n d + n^) and, as we shall assume throughout this chapter, n d fixed. Using N = Vd n^ Equation 5.2 is : Cry s t a l = Q e ( V 0 ) + N {n" h C e^d + V nh> " e< nd» °>]>-(5.3) The free parameter, n^, is determined by the requirement that E c ry S-j- ai 1 S minimum which leads to the condition: F(nd+nh,nh) =1 [e(nd+nh,nh) - e(n d, 0)] = minimum. (5.4) h It is instructive to derive Equation 5.4 from a different starting point, namely, by the requirement of mechanical equilibrium: the pressure inside and outside the droplet is equal. Hence d N out N=o (5.5) and changing variables (N=Vd nh) Equation 5.5 leads to: -^e(n d +n h,n h ) U = ^ W"f\>%) ~ e(nd,0)] . (5.6) 18 By using the identity + n " ^ h {k[e{nd+Vh) " e ( n d $ ° ) ] } ' ( 5 ' 7 ) i t follows that n£ is also determined by Equation 5.4. (If n^, when 57. n R * 0 is not a solution to Equation 5.4 then droplet formation is energetically favorable.) In addition, i t can be easily shown that the chemical potential of an electron-hole pair in a plasma, at zero temperature, i s : u(T=0, n c, nh) = g|- e ( V nh) , (5.8) hence at quasi-equilibrium Epair 5 ^ T = 0 ' V nh> =^nc> nh) > (5'9> which, as pointed out in Chapter 3, is experimentally measurable. Mahler and Birman^ -^ have chosen to formulate the energetics of droplet formation in terms of average energies per carrier. This choice complicates the problem unnecessarily. To solve for the quasi-equilibrium pair density (equal to n^ ) they calculate the pressure inside the droplet in terms of partial pressures: one due to the photo-created pairs, the other due to donor electrons. Since from the onset they assumed all donors within the drop to be ionized, that i s , photo-created and donor electrons are indistinguishable, hence their partial pressure procedure leads to the violation of the Pauli Exclusion Principle. The treatment of Bergersen et a l 1 ^ ' ^ leading to the condition of quasi-equilibrium (Equation 5.4) does not have the difficulties described above. Furthermore, i t is rigorous and therefore will be followed. Before starting with the actual calculation one should determine in which density regime the RPA is valid. Past experience with this high 58. density approximation shows that, in the present context, i t is reasonable to assume that i t works well for such densities where the inter-carrier distance is less than the Bohr radius of an isolated impurity, namely 19 -3 for n >3 x 10 cm . It is then necessary to justify the use of RPA for carrier concentrations which are an order of magnitude less. For the EHD in intrinsic material, where such carrier densities are also encoun-tered, Bhattacharyya et a l ^ argue that one can expect corrections to the correlation energy that go beyond RPA to be about 20 percent. On the 19 r \ other hand, following Bergersen et al , consider the two terms e(n c, n^ } and e(n d, 0) when calculating E(n c > n^): for impurity densities of the * 18 order of magnitude of the quasi equilibrium hole density (n^ ~ 1-3 x 10 cm obtained in Chapter 3), e(n c, nh) and e(n d, 0) are of the same order and their absolute errors could well be about the same and therefore cancel to a considerable degree. The same is true - more so - for the chemical potential. For lower impurity densities the calculation of e(n d, 0) is totally unreliable and was replaced, from physical arguments, by e(n d, 0) = n d E Q (5.10) with E Q the ionization energy of an isolated donor. For these impurity concentrations the cancelling of errors described for higher densities 18 - 3 may not take place and theoretical results for n^ <. 3 x 10 cm should be viewed with' caution. The arguments given here for the correlation energy carry over to the impurity energy contribution so that i t is believed that the RPA should work reasonably well for metallic silicon. 59. Figure 5.1 shows the result of calculating (T (n c, nh) as a 18 3 18 function of n^ for the donor concentrations 3.1 x 10 cm" , 6.2 x 10 -3 19 -3 cm and 1.24 x 10 cm . For all three impurity concentrations i t is quite clear that the condition of quasi equilibrium is found when n h •» 0, therefore, droplet formation is not predicted for metallic silicon within the model of Bergersen et a l 1 8 , 1 9 . ce cq To show that the results of Mahler and Birman " are not a con-sequence of ignoring the impurity energy contribution in Equation 5 .1 , Figure 5.2 shows the corresponding results. Again i t is quite clear that E*(nc, n^) has a minimum when n^ •* 0. By comparing Figures 5.1 and 5.2 one observes that the effect of including the impurity energy contribution is to lower the energy by an amount which is almost independent of n^. The experimental results in this thesis point to the most serious problem in this calculation: the rigid band approximation. For high hole densities this assumption is justifiable by the excellent f i t of the EHD line shape but in the gas phase the experiment shows that the parabolic band assumption is nonsense. Therefore, the calculation of e ( n <j» 0) is suspected. In fact, for example for n^ = 6.2 x 10 cm" , i f one increases efn^, 0) by 2 percent one obtains a minimum in E~ vs. n h with n*, approximately equal to 1. x 1 0 1 8 cm"3 - but E p a i r turns out to be larger than uCn^-* 0) so that droplet formation is energetically un-favorable. Figure 5.3 shows the results of calculating E (n c, n^ ) as a function of n h for the impurity concentrations of 2 x i o " ' 7 cm-3 and 60. f CT) L_ (D X i T J >> cr X 0) c - 4 - h 0 T 1 1 1 I I | D T 1 — i I | i I i T 1—1 » I ' l l J 1- JL 1 t. f 1 7 10 _J 3 t i l l ! • L _! i 1 I t ' l l 18 10 n . f e m 3 ) h Figure 5.1: The calculated average energy per pair as a function of hole density for the indicated phosphorus impurity con-centrations. The points on the ordinate-axis are the calculated values of the chemical potential of a pair in the limit of zero pair density. 61. Figure 5.2: The c a l cu l a ted average energy per p a i r as a funct ion of hole dens i ty f o r the i nd i c a ted phosphorus impuri ty con-cent ra t ions . The impurity energy cont r ibut ion (Equation 5.1) i s neg lected. The points on the o rd ina te -ax i s are the ca l cu l a ted values of the chemical po ten t i a l o f a pair in the l i m i t of .zero p a i r dens i ty . 62. Figure 5.3: The calculated average energy per pair as a function of hole density for the indicated phosphorus impurity con-centrations. The ionization energy of an isolated phos-phorus donor is used as the energy per electron outside the droplet. The point on the ordinate-axis is the cal-culated value of the chemical potential of a pair in the limit zero pair density for nsj_= 2 x 10ly' cm-3: the cal-culated value for n n = 5 x 10 , ; cm"3 is off scale. 63. 1 7 - 3 5 x 10 cm using Equation 5.10 to ca lcu la te e ( n d , 0 ) . In these cases £ ( n c > n^) has a c l e a r minimum. E p a i - r i s within three percent o f the experimental value and i s not exces s i ve ly dependent on va r i a t i on s to e ( n d , 0) . On the other hand, va r i a t i on s to e ( n d , 0) produce very large changes in the qua s i - equ i l i b r i um hole dens i ty . The c o n d i t i o n , -> 0) > E p a - j r » f o r d rop le t to be energe t i c a l l y favorab le i s only 17 -3 met f o r n^ < 3 x 10 cm . Spec ia l s i gn i f i c ance should not be given to th i s re su l t s ince u(n^ 0) cou ld be very wrong f o r th i s low impurity dens i t i e s . 5.3 The Modif ied Model As pointed out in previous Chapters there i s reason to be l ieve that at 4.2K. the EHD and the gas phase coex i s t . Furthermore, s ince N,> the number of photo-created p a i r s , i s kept constant by o p t i c a l pumping, i t w i l l be assumed that the two phases are in quas i - thermodynamic e q u i l i -brium. Hence T g a s = T l i q * <5.11) P g a s = P l i q ' ( 5 ' 1 2 )  a n d y gas = y l i q * ( 5 J 3 ) In the present s i t u a t i o n i t i s more convenient to show the chemical poten-t i a l vs n^ isotherms than the usual pV diagrams f o r l i qu id - ga s phases. A t yp i ca l u vs n^ which descr ibes two phases in equ i l i b r ium*^ i s shown in Figure 5.4.a. The region with dp/dn^ < 0 cannot sus ta in a s tab le phase. The Maxwell construct ion produces the two phases. When we decrease the 64. Figure 5.4: Chemical potent ia l o f a p a i r as a funct ion of hole dens i ty in the fo l lowing cases: a) Droplets are in thermodynamic equ i l i b i rum with a gas phase. b) The gas phase contains no holes in contact with droplets which are e n e r g e t i c a l l y favored. There i s mechanical e q u i l i b r i u m . 65: temperature to zero two things may happen: the hole density in the gas phase may remain larger than zero or may become zero. In the former case the u vs n^ isotherm will be as in Figure 5.4.a. If the stable gas phase has no holes in contact with the liquid the ordinary Maxwell construction does not work but is easily generalized* and as shown in Figure 5.4.b. u > v,. , (5.14) gas liq and there is no thermodynamic equilibrium but formation of droplets is energetically favorable. This quasi-equilibrium situation is the model of Bergersen et a l 1 8 , 1 9 and Mahler and Birman 5 6" 5 8 discussed in the pre-vious Section. In both cases, i f droplets exist, the u vs n^ at T = 0 diagram shows a local maximum and a local minimum, hence i t is unnecessary to de-termine a priori what the density of holes should be in the gas phase. The drawback of this model is that one encounters again the problem that the density of states is far from being parabolic even in metallic s i l i c o n at low hole densities so that we have no hope of calculating realistically u vs n^  for all n^. A brute force calculation, assuming the r i g i d band approximation to be valid for al l n^ in metallic silicon, was performed hoping that a loca l minimum would show up at a high hole density where the' calculation could be be l ievab le - the minimum was not found. For complete-ness the results of this calculation for irapurity concentrations of 3.1 x 10 1 8 cm'3, 6.2 x IO 1 8 cm"3 and 1.24 x 10 1 9 cm"3 are shown in Figure 5.5. * I thank Dr. G. Kirczencw for pointing this out. 66. Figure 5.5: The ca l cu l a ted chemical po ten t i a l o f a p a i r as a funct ion of hole dens i ty f o r the i nd i ca ted phosphorus impuri ty concentra-t i on s . The points on the o rd ina te -ax i s are the c a l c u l a t e d values in the l i m i t of zero p a i r dens i ty . The experimental po ints (see Chapter 3) are shown f o r comparison. 67. 5.4 Drop lets? In Sec t ion 5.2 d rop le t formation was dismissed even though the e q u i l i b r i u m cond i t i on given by Equations 5.4 and 5.9 were met by changing 18 e (6.2 x 10 , 0 ) by two percent because y(n, + 0) < E . . In Sect ion 5.3 n p a i r drop let formation i s again dismissed because u vs n^ does not show a "k ink " which would have had to appear at low hole dens i t i e s . The quest ion a r i s e s : should one conclude that there are no droplets based on the c a l c u -l a t i o n o f the chemical po ten t i a l o f a p a i r f o r low hole dens i t ie s f o r which the r i g i d band approximation i s shown experimental ly to be nonsense? C l e a r l y , the t h e o r e t i c a l c a l cu l a t i on s are inconclus ive one way or the other. The need of more t h e o r e t i c a l work i s ev ident. A promis ing route f o r f u r t h e r theore t i ca l development i s to con-s i d e r l o c a l i z e d s t a t e s . Qu i r t and Marko ' have extens ive ly s tud ied the r a t i o o f d e l o c a l i z e d to l o c a l i z e d e lectrons in Si(P) f o r the same im-purity, concentrat ions used here. They have used sp in s u s c e p t i b i l i t y s tudies to determine that at impar i ty concentrat ions two times above n c r j t 10 to 20 percent of the e lec t rons are l o c a l i z e d , i n f a c t even f o r 19 -3 n d ~ 2 x 10 cm they c la im two percent l o c a l i z e d e lec t rons . I f th i s i s i n f ac t the s i t u a t i o n when no photo-created ca r r i e r s are present then i t i s reasonable to assume-, to f i r s t approximation* that the density o f l o c a -l i z e d e lec t rons i s equal to the dens i ty of neutral donors. Furthermore, l e t us assume that the dens i ty o f i on ized donors, n ^ , and the density o f neutra l donors, n^, are funct ions of the hole density n^. The s implest approach i s to c a l c u l a t e the t o t a l energy per ua i t volume in the plasma by summing two t e r n s : to c a l cu l a te the f i r s t term we ignore a l l the neutra l donors, thus t h i s term is c a l cu l a ted exact ly as i n previous sect ions 68. except that the impurity density i s assumed to be n^.; the second term ca l cu l a te s energy cont r ibut ion of the neutra l donors assuming that average energy of the e lect rons i s E n below the conduction band minimum. The t o t a l energy per un i t volume i s now e ( n c , n h ) = e ( n d l ( n h ) + n h > n h ) + E n n d n (n h ) , (5.15) with n d = n d i ( n h } + n d n ( n h } ' ( 5 ' 1 6 ) f o r a l l n^ and the chemical po ten t i a l at zero temperature i s now: h n d i ( n h ) r ^ ^ " d i ( n h> + V V> n - E J x f n - ; n d i ^ \ h ' h (5.17) which w i l l depend on how the density o f i on i zed impur i t ies changes with the photo-created p a i r dens i ty . The problem could be so lved by using the mechanical e q u i l i b r i u m condit ion to obtain an equation that would then be so lved s e l f - c o n s i s t e n t l y with Equations 5.16 and 5.17 f o r n^(gas) and n^( l iq) with only one parameter, e i t h e r n^(0) or E n > This c a l c u l a t i o n i s beyond the scope of th is thes i s and I w i l l r e s t r i c t myself to specu lat ing on the chemical po tent i a l as given in Equation 5.17. Let us imagine that n d - j ( n n ) var ies l i n e a r l y over a very large range of n^ so that the second term in Equation 5.17 i s neg lected. We ca l cu l a te the f i r s t term fo r the two extremes of n d-j(. n h)> namely fo r n d and n d l - (0 ) . The s o l i d curve in Figure 5.6 shows the chemical po ten t i a l as a funct ion of n^ f o r " d l - ( n ^ . ) = n d : the dashed curve f o r n d j C n ( l ) = n ( j 1 - ( 0 ) < n d . We can envisage now that i f ^ ( n ^ ) i s va'ring slowly that the f i r s t term in 69. Figure 5.6: The chemical po tent ia l of a p a i r as. a funct ion of hole dens i ty . The s o l i d curve is. fo r the dens i ty o f i on ized donors L^AII equal to nd; the dashed curve i s fo r n d l- = n,.(Q)<n • the chained curve represents a f ree hand i n t e r p o l a t i o n f o r nd t(0) < n d i ^ n h ^ < n d w ' i e n n d i^ - n h^ v a n e s s lowly. 70. Equation 5.17 w i l l r e su l t in values which f a l l on a l i n e which resu l t s in i n te rpo l a t i ng the two curves re fe r red ' to above and i s represented by a cha curve in Figure 5.6. I f n^-fn^) var ies in this manner then droplets are formed and the s tab le gas phase contains no holes. Let us examine now the second term in the chemical po tent i a l (Equation 5.17). Test c a l cu l a t i on s o f ( f - . e ( n d + n h , n h ) ) (5.18) d l n h show this term to be always l a r ger ( l e s s negative) than any reasonable value of E n obtained from the IB photoluminescence. Therefore one may assume that the second term in the chemical potent ia l i s always p o s i t i v e and i f "^-(n^) var ies mainly over a narrow range of n^ then u vs n^ w i l l show a pos i t i ve spike in that range o f n^. CHAPTER 6 71. SttMARY AND CONCLUSIONS The work reported in this thesis has involved the measurement of the photoluminescence in heavily-doped Si(P) at liquid helium temperatures. The spectra were studied as a function of excitation intensity and found to contain two components dwe to two distinct recombination phenomena. Care-ful line shape analysis of the spectral component which dominates at high excitation intensity confirms the hypothesis of Halliwell and Parsons11 on the existence of EHD's in metallic silicon. The present study is incon-clusive about the existence of droplets in silicon containing 19 -3 nd > ncb ~ 2 x 1 0 phosphorus cm , the density when the Fermi level is in the conduction band. The second component of the spectra is observed for the fi r s t time. This component dominates the spectra at low excitation intensities and is attributed to the recombination of an electron in the impurity band and a free hole. The line shape of the IB peak is found to be well described by the density of states of the impurity band within the Hubbard model 1 7 for n , < n„ ... The effects of the semiconductor-metal d cn t transition on the experimental IB line shape are well understood in terms 5? of the so-called " Hott-Kubbard-Anderson transition model. The change in the relative intensity of the EHD and IB peaks is indicative of the coexistence of two phases. This hypothesis is further strenghtened because i t predicts the observed relative positions, of the EHD and IB peaks with donor concentration. 72. The theory of the EHD ground state in the model of Bergersen et a l ^ ' 1 9 and Mahler and Birman^" 5 8 v/as reviewed and found in error. New numerical results based on this model were presented which show that the droplet is energetically unfavorable in metallic silicon. On the basis of the experimental results obtained in this work i t is found that the use of the rigid band approximation for low photo-created carrier densities is at fault. It is shown in this work, by hand-waving arguments, that droplets may be energetically favorable i f for low photo-created densities not all donors are ionized in metallic silicon as the work of Quirt and Marko^' 63 strongly suggests. The speculative discussion in Chapter 5 suggests that the co-existence curve for the EHD and IB phases in heavily doped silicon may be very complicated. In fact, depending how we visualize the density of ionized donors as a function of photo-created carriers to change with temperature, several critical temperatures are possible: there could be a range of temperature, in between them, where droplets are energetically unfavorable and possibly a temperature region close to zero temperature where droplets are formed but are not in thermodynamic equilibrium with the gas. Clearly the photoluminescence study of these samples as a function of temperature 64 could yield surprising results. The work of Parsons and Thewalt , re s -tricted only to = 2 x 10 phosphorus cm" , shows a critical temperature of approximately 51K which i s three times higher than the critical tempe-rature for intrinsic s i l i c o n 5 5 . Of additional interest i s that the results 54 of Parsons and Thewalt seem to indicate that between V3 and 20K the chemical potential as well as the equilibrium hole density in the drop show a minimum^. 73. The near-infrared absorption (or reflection) experiments oh these samples at low temperatures are also of great interest to determine whether the coexistence hypothesis is valid since the optical gap measures the chemical potential of a pair in the gas phase. The absorption expe-27 riments at approximately 35IC reported by Balkanski et al indicate an optical gap larger than E . measured in this work. Nevertheless, these r pair results are inconclusive since as pointed out above, at least for a sample 18 -3 containing 2 x 10 phosphorus cm , the chemical potential of a pair seems to increase above 20K with temperature. 74. APPENDIX A HEAT TREATMENT EFFECTS IN S i (P) A . l Experimental Results Figure A . l shows the e f f ec t s on the photoluminescence spectra of the heat treatment descr ibed in Chapter 2. The photoluminescence measurements were done at 4.2 K and the exc i t a t i on i n t e n s i t y i s low, about 2 10 watts/cm . In Figure A . l the dashed l i n e gives the spectrum before treatment; the s o l i d l i n e , a f t e r treatment. No_ e f f e c t s of the heat t r e a t -1 o _3 ment are observed f o r impurity concentrations below = 3.0 x 10 cm . The luminescence peak observed in the range 1.045 eV - 1.055 eV f o r the 1.8 x 1 0 1 8 cm" 3 - 1.1 x 1 0 1 9 cm" 3 spectra in Figure A . l i s the IB peak and the peak observed at about 1.07 eV is assoc iated with the e l e c t r o n -hole d rop le t . Both peaks have been f u l l y s tudied in the main body of th i s 18 ? t h e s i s . As depicted by the 3.0 x 10 cm spec t ra , i t i s more d i f f i c u l t to form the d rop le t a f t e r heat treatment, i . e . h igher e x c i t a t i o n l eve l s are requ i red to obtain the droplet peaks a f t e r heat-treatment. As shown by the 3.9 x 10 cm and 5.0 x 10 cm" spectra a peak i s observed at •j o _ q =s 1.088 eV in the 5.0 x 10 cm" spect ra . At other concentrat ions th i s peak was weak and d i f f i c u l t to separate from the background luminescence. No e f f e c t o f the heat treatment was observed in a sample conta in ing 4.3 x 1 0 1 9 c m " 3 . I f a heat - t rea ted sample is l e f t at room temperature f o r a few days, the photoluminescence peaks at 1.028 and 1.088 eV are reduced in i n t e n s i t y r e l a t i v e to the I B ^ peak. A f t e r about one week at th i s tempe-75. 0 .96 1.00 1.04 1.08 PHOTON ENERGY CeV) Figure A . l : The e f f e c t s o f heat treatment on the photoluminescence spectra o f S i (P ) conta in ing impurity concentrations in the range 1.8 x I O 1 8 cn-j-3 to 1.1 x 10'9 cm~3. The dashed l ines give the spectra before treatment; the s o l i d l i n e s , a f t e r t r e a t -ment. The spectra have been a r b i t r a r i l y sca led to make com-par i son of l i n e shapes ea s i e r . 76. rature the photoluminescence spectrum completely reverts to i t s form before treatment, but the 1.028 eV peak (and probably the 1.088 eV peak) does not disappear upon fu r the r room temperature anneal ing. 23 H a l l i w e l l has s tud ied the e lec t ron paramagnetic resonance 23 (EPR) of s i m i l a r samples under the same heat treatment. H a l l i w e l l uses ft 7 a standard x-band homodyne EPR spectrometer . The spectrometer was f i t t e d with a double-sample modulation-switched cav i ty designed by Qu i r t (see CO also Qu i r t and Marko ) which al lows d i r e c t comparison of untreated and heat - t reated samples. As in the photoluminescence case, the e f f e c t s of the heat-treatment on the EPR spectrum are observed only f o r phosphorus concentra-t ions greater than a ce r t a i n va lue. No changes at a l l were observed f o r T O O n D < 2.0 x 10 c m . As shown i n Figure A.2, taken from Reference (23), f o r sample temperature 1.1 K marked changes were observed with the t r e a t -18 -3 19 -3 ment f o r higher concentrat ions n n < 2.0 x 10 cm . At 4.3 x 10 cm a very small e f f e c t was observed as a 10% broadening o f the phosphorus l i n e . In accord with the photoluminescence resu l t s the EPR spectra show s i g n i f i c a n t annealing e f f e c t s . A f t e r several days at room temperature the spectra completely rever t to t h e i r form before treatment. The EPR comparison technique ' mentioned above was used by 23 Ha l l iwe l l to determine the number n of e lect rons respons ib le f o r the EPR. Although th is method i s claimed to be accurate to about 3 percent i f the l i n e shapes are known and there i s no change i n sp in s u s c e p t i -62 63 b i l i t y ' , the uncer ta in t ie s i n the co r rec t l i n e shape f o r these studies 77. • — M A G N E T I C F I E L D Figure A. 2: The e f f e c t s o f heat treatment on the EPR of S i (P) conta in ing impur i ty concentrat ions : 2.0 x 1018 cm"^ and 6.2 x I O 1 8 cm-3. Magnetic f i e l d modulation was used and the output s ignal is p ropor t i ona l to the der i va t i ve dx"/dH where x " i s the imaginary par t of the s u s c e p t i b i l i t y . The dashed l ines give the spectra before treatment; the s o l i d l i n e s , a f t e r t r e a t -ment. The spectra have been a r b i t r a r i l y sca led to make com-par i son of l i n e shapes ea s i e r . The spectra are taken from Reference(23). 78. 23 made i t impossible to determine n to b e t t e r than 15 percent . To th i s accuracy there was no change in the t o t a l number o f e lec t rons respons ib le f o r the observed EPR spectra before and a f t e r heat reatment. I f the sample was ground to a powder before heat treatment, no 23 heat- induced e f f e c t s could be observed in the EPR spectra . Photo-luminescence spectra could not be obtained on the powdered samples because the surface recombination d r a s t i c a l l y reduced the quantum y i e l d beyond the l i m i t s of our. d e t e c t i v i t y . In the op t i c a l experiments the sample d i -3 mensions were t y p i c a l l y 2 x 5 x 10 mm . The grain s i z e of the powdered samples was about 5 microns diameter. A.2 Discuss ion of Results The changes produced by the heat treatment are near ly conf ined to the same concentrat ion range in the case of both the EPR and the photo-luminescence range in the case o f both the EPR and the photoluminescence p roper t i e s o f S i ( P ) . In add i t ion the anneal ing behaviour observed f o r the two sets o f measurements i s s i m i l a r . I t i s reasonable to assume, there -f o r e , that these changes are re l a ted to a common c ry s t a l de fec t . The con-18 -3 cen t r a t i on thresho ld 2-3 x 10 cm" f o r the induced e f f e c t s i s very I o near ly equal to the c r i t i c a l concentrat ion n c n -| . - 3.0 x 10 phosphorus -3 12 cm f o r the semiconductor-metal t r a n s i t i o n . This f a c t combined with the constant e l e c t ron - sp i n dens i ty r e s u l t in the comparison measurement suggest that the induced defect becomes paramagnetic with the capture of a d e l o c a l i z e d donor e lec t ron and that the induced luminescence i s due to recombination of th i s captured e lec t ron with a ho le , which we assume to be f r e e . In th i s argument the defect trap could s a t i s f y the fo l lowing 79. conditions when the P concentration is less than ^ r i- t: (1) its electron-capture cross-section must be less than that of an ionized P impurity, and (2) the probability of transfer of electrons from the P states to the trap states must be negligible at liquid helium temperatures. We have not been able to show that i t is possible to have a trap with these properties. Therefore, our present discussion of the results in speculative. We do not have an alternate explanation. The energy separation of the 1.028 and 1.088 eV peaks in the 18 3 5.0 x 10 cm spectrum in Figure A.l suggests that the peaks are asso-ciated with the emission of a TO-phonon CO.058 eV) and a no-phonon process respectively. Comparing the energy positions of the former peak with the 1.06 eV of the IB T^ peak in the above spectrum, we conclude that.the electron states of the induced trap form a distribution centered at about 0.032 eV below the center of the donor impurity band. At 4.2 K, there-fore, an electron in a defect state would have, l i t t l e probability for re-excitation to the impurity band. Consistent with the interpretation of the photoluminescence data, the EPR results on the heat-treated samples could be interpreted in terms of two unresolved line shapes: one due to electrons loosely bound to phos-phorus sites; the other due to electrons trapped by the induced defects. The EPR spectrum is located at magnetic field corresponding to g - 2.0 and can be separated into two components. One of these components has the same width and g-value as the single line due to donor electrons observed in samples that have not been heat treated Xdashed lines in Figure A.2). After subtraction of this component there remains a broad asymmetric line with a higher g-value. Although lack of knowledge as to the shape of this 80. line makes proper resolution of the two lines impossible, one can say that the g-value is relatively insensitive to impurity concentration, whereas the linewidth increases rapidly with concentration. The fact that the EPR spectra are unchanged when the samples were crushed before heat-treatment shows that the results are not associated with a surface effect. Presumably, the heat-induced defects in the powdered samples are able to quickly diffuse to the surface where they are removed during the etching procedure. / / /In order to observe the large changes shown in Figure A.2 for ' 1 8 - 3 the ~6 x TO cm EPR spectra the concentration of the heat-induced de-18 fects must be comparable to that of the phosphorus impurities > 10 defects cm-3. This concentration is at least 10 times greater than the typical concentrations of residual impurities, e.g. oxygen, carbon, metals fi8 in vacuum float-zoned samples of the type used here . Consequently, we do not think that the heat-induced defect is due to a chemical impurity. The fact that we do not observe heat-induced, effects for n . < 2-3 x 10 1 8 d _3 cm and that we do not measure a change in the total spin density suggests that the heat-induced defect is not associated with the phosphorus dopant. The origin of the heat-induced trap, therefore, seems to be due to intrinsic lattice point defects: i.e. vacancies or interstitial Si atoms. On the other hand indirect evidence coming from diffusion studies and high voltage transmission electron microscopy69 as well as theoretical calculations^ 0 seem to indicate that dominant high temperature defects are self-intersti-tials and not vacancies, in contrast to metals, thus only this defect will be considered. 81. A self-interstitial in silicon is highly mobile at room tempera-t u r e 6 9 , ^° and, therefore, could not explain the heat-treatment effects observed here which persist, in the case of the luminescence studies, after prolonged room temperature annealing. However, interstitials are 69 known to form aggregates which are stable at room temperature. It is possible that these aggregates possess a distribution of electron traps and could explain our data. The effect of the heat treatment could be under-stood i f large aggregates break up into small clusters of interstitials on high temperature treatment and thereby increase the density of trap centers. The concentration of self-interstitials depends crucially on the 69 growth rate of the crystal and therefore is difficult to estimate theore-tically. Using the diffusion study results of Seeger and Chik^, FOlls et a l . ^ has deduced -10^ self-interstitials cm"3at the melting point 1420°C in silicon. This is in agreement with the concentration deduced here. The fact that heat treatment produces very l i t t l e effect on the photoluminescence and EPR spectra for very high phosphorus impurity concen-19 -3 trations n d > 1 x 10 cm can be qualitatively understood. At sufficiently high donor concentrations all states, including those of the traps, will be screened by the free carriers. 82. APPENDIX B IMPURITY BAND DENSITY OF STATES IMPURITY PAIR MODELS 38 + As mentioned in Chapter 4 Lukes et al have used a simple H2 ion model of interaction between impurities in a doped semiconductor to calcu-late the density of states in the impurity band. The donors are taken to be randomly distributed and the distance, R, between nearest neighbours forming the H^-like ion is assumed to follow the Chandrasekhar distribu-39 tion . Using the Green's function formalism they show that the density of states my be written as follows: /.„ r>2 _ r n3 n t t ) = -ir " J \ \ <ir r exp [-4ir RJ n d/3 ] [ y * - - o i - + y,- (r) E^E++ie E-E +ie dr dR , (B.l) where the antibonding band given by the second term will be ignored. The wave function for the ground state is the same as for an isolated impurity as calculated in the effective-mass approximation and is the same as used 47 in Chapter 4, Equation 4.10. Here, we will follow Kohn by taking the en-velope function to be spherical symmetric with the effective-mass equa-tion4''. Since the wave function is normalized Equation B.l reduces to a trivial one-dimensional integral and one obtains , ' -1 i 2 , 3 dE, n [E + (R )] = -tr 4TT(R ) n d exp {-4n(R ) / | ^ t | (B.2) hence to obtain the density of states we need only to calculate the ground state energy and i f inter-valley terms are ignored one obtains the analytic 83: expression for the energy which is identical to the well known solution 4 1 of the bonding state energy of the hydrogen molecule ion, H^ , with the appropriate Bohr radius. The calculated density of states was compared with the experimental IB line shape with the result already described in Chapter 4. + A natural extension of the H2 ion model of interaction between impurities is the molecule model so that electron-electron interactions are taken into account. Equation B.l also expresses the density of states in this model. The Heitier-London method41 is used to set up the molecular wave functions in terms of the "atomic orbitals", that i s , the isolated impunty wave functions used in the ion model. Since the Heitler-London wave functions are. not normalized the integral over F in Equation B.l is i y + (r) d r = 1 + S (R) 2 (B.3) where S is the atomic orbital overlap integral which will depend on the inter-impurity distance. In this case one writes -HL 4ir(R')2nd exp [-4TT(R') 3 nd/3] [ 1 + S ( R ' ) 2 J / n [ E f <R')] = dE H L dR dx H-nyf n d exp [ > 4 i r X J n/3j [l+S(x)2] (B.4) and again the problem reduces to calculating the ground state energy and i f inter-valley terms are again ignored one obtains the analytic expression for the energy which in this case is identical to that given by the Heitier-London model for the H 2 molecule with the appropriate Bohr radius; 84. the analytic solution of the two center integrals listed by Slater 4 1 was used. In the Heitler-London model, the photoluminescence line shape does not correspond to the calculated density of states because i t is un-reasonable to assume that both electrons associated with the impurity pair will recombine simultaneously. To calculate the photoluminescence line shape i t has been assumed that the recombination of a donor electron with a free hole is a Franck-Condon process that leaves an ion-like behind in its ground state and gives I-[EJJl(R') - E + ( f i ' ) ] a n f e J V ) ] 0,5) where E+(R') is the ground state energy of the ion. When evaluating the line shape care should be exercised since the argument of the L.rl.S. of (B.5) is not a single value function of R* . The calculated line shape was compared to the experimental one with the result already described in Chapter 4. 85. APPENDIX C DATA ANALYSIS PROGRAMME This section describes a computer programme which has been extensively used to analyze the photoluminescence spectra obtained for this work. The programme is written in the BASIC language which is fully des-cribed in "An Introduction to BASIC"?3 An extension to the manufacturers compiler, programmed by TRIUMF at U.B.C. is used. This compiler allows to call Machine Language Subroutines .(MLS) with the instruction CALL. The MLS 2 called by this programme are explained and listed in M.L.W. Thewalt's thesis. For the purpose of this programme a spectrum is a set of four dimensional prints: one dimension for the energy of the measured photon; a second for the intensity; a third for the standard deviation due to signal averaging; and a fourth which was reserved for digitally smoothed data which is no longer used. This implies that i f a spectrum of "n" points is to be stored in memory, given the starting location in the memory Buffer, the pro-gramme will use "4n" consecutive memory locations the f i r s t being the starting location given. Care must be then taken so that the data will not overlap with other information stored in the memory Buffer. The Buffer consists of 2000 memory locations. The programme listed below requires that the memory locations 1850 to 2000 be reserved for the EHD •theoretical line shape calculations. Locations 1600 to 1800 are reserved for the IB theoretical line shape calculations. As pointed out in Chapter 3, the difference between two experimen-tal spectra are taken to disentangle the IB and EHD contributions and to do so the following convention is followed: the second spectra read into memory ' 86. i s subtracted from the f i r s t ; s c a l i n g and base l i n e changes are only done on the second; the r e s u l t i n g d i f f e rence w i l l be stored in the f i r s t "4n" locat ions o f the B u f f e r , where " n " i s the number of points of the second spectrum. Most spectra analyzed in th i s work cons is ted o f less than 125 po in t s , f o r such spectra i t i s convenient to choose memory l oca t i on 500 as s t a r t i n g l oca t ion fo r the f i r s t experimental spectrum read into the Buf fer and loca t ion 1000 fo r the second spectrum. The fo l lowing programme has been wr i t ten in modular form. The subroutines w i l l be l i s t e d f i r s t and two main programmes f o r s l i g h t l y d i f f e r e n t purposes which use these subroutines are l i s t e d at the end. A . l "The Subroutines 9.1.00 R E M J S ; * * * ^ * * * * * * * * * * * ^ * * * * ! * : * * * * * * * * ^ * * * * ^ * 9 1 0 1 R E M L O A D 9103 PR I NT " L O A D T A P E * I NPUT S T A R T I N G P O I N T S OF # * S A N D " 9104 P R I N T " . . • 1 * F O R S T A N D A R D D E V I A T I O N S " 9 1 0 5 1 NPUT. A l * A 2 * C 8 REM This subroutine w i l l load the contents of a paper tape in to bu f fe r s t a r t i n g at l oca t i on A l . The spectrum is represented by A2 po in t s . The " 1 " f o r ' s t andard dev i a t i on s , i s a contro l number and should be set to one i f the paper tape a lso contains the standard dev iat ions o f the data po int s . 9106 C A L L 1 6 i A l * A 2 9107 I F C8<>! S O T O 9110 9.1.08 C A L L !6AA! * 3 * A 2 # A 2 REM The MLS #16 t rans fers data on paper tape to memory Bu f fe r . 9109 G O T O 9 1 1 6 ' . 91.10 PRINT - I N P U T S T . D E V » T O S 1 6 . R A T I 0* *0 ' I F UNKNOWN* 91.1 ! ! N P U f C 9 . 91.12 LET C 9 c C 9 * 1 9 0 0 -87. REM The m u l t i p l i c a t i v e factor 1900 i s required for scaling our data for output due to the requirement of the D/A converters used in this system as t t w i l l appear many times throughout the programme. 9113 FOR 1= 0 TO A2-1 9114 CALL 7,AI+3*A2+!#C9 91.15 NEXT 1 REM The MLS #7 stores i n memory location given by the f i r s t expression the number given by the second. 91.16 FOR l~ 0 TO A2-1 9.118 CALL 8*A1 *l * F t I 3 9120 NEXT | REM The MLS #8 r e c a l l s the contents of memory location given through the f i r s t expression and i d e n t i f i e s i t with the variable l i s t e d next. 9122 REM SCALES DATA 91.24 LET Ml =-20000 9126 LET M2=2<3000 REM The maximum and minimum possible values are ±20000 and are a characteristic of the A/D converters used in this system. This values w i l l appear several times in the programme. 9126 FOR 1=1. TO A2-1 . 9130 IF f l ! >FE I 3 GOTO 9! 34 9132 LET M1=FU 3 ..... . 91.34 IF M2<FC I 3 GOTO 9138 9136 LET M2=FCI3 91.38 NEXT i 91.86 FOR I = 0 TO A2-! 9168 LET FC ! 3 = (FC ! 3-P12) * J 900/CM I -M2) 9190 CALL 7*A1 + UFCT3 9191 IF C G o l GOTO 9195 9192 CALL 3*&S.+3*A2 + i*C 9193. LET C=C*i90S3/€Ml -M2) 9194 CALL 7*A1+3SA2+1Ve 9195 NEXT I 9196 RETURN REM For the spectrometer described in Chapter 2 i t was necessary to recalibrate periodically the wavedrive. The following subroutine calculates the energy axis of a given spectrum. It requires that the calibration parameters used when the spectrum was taken be given. 88. 9400 REM * $ « : ] ( * « i 9 ^ « * $ # $ 4 ^ $ # < ^ * $ # $ ^ $ * $ $ $ # 4 c $ $ « * 9401 REM CALCULATE ENERGY AXIS 9402 REM $##$*jt^i}^************$*$*****#****** 9403 PRINT "NEW CAL.TYPE 1*0 OTHER oSR»l FOR DI SP*THEN 0 TO CONTI" 9404 INPUT Z5 " 9485 IF Z5<>1 GOTO 9410 9406 PR8 NT "INPUT CALIB.PARo" 940? PRINT " . . . 9408 INPUT Z0*Z1 *Z2*Z3.»Z4 9409 DEF FNW(X>=Z9+Z1 *X+Z2*X*X+Z3*X*X«:X4-Z4*X*X*X*X 9410 PRINT "SToWD*STEP SIZE" 9411 PR I NT _ 94.12 INPUT W0*W1 9413 FOR 1 = 0 TO A2-1 9414 LET W2 = FNW(W0) 941.5 LET W2=2*479?I/W2 9416 LET W2=€$2-lo0S5>*2!000 9417 CALL 7J>A! *A'2 + I «W2 9418 LET W0«U0«vr 9419 NEXT i 9426 CALL 9*-l*C REM The MLS #9 reads se lec ted b i t s o f the Switch Reg is ter (SR) by performing a l o g i c a l AND with the MASK given by the f i r s t para-meter and the r e s u l t i s i d e n t i f i e d with the var i ab le l i s t e d next. Thi.s subrout ine i s heav i l y used to i n te r ac t with the computer and in the present case, i f SR=1 i t w i l l d i sp lay on the scope the l a s t spectrum read into the Bu f fe r . The d i sp lay w i l l continue u n t i l the SR i s se t to cero. 942? IF C<>1 GOTO 9432 9423 CALL 2 5 * A 1 + A 2 » A 1 , A 2 9429 CALL 26 . REM The MLS #25 points to the locat ion in the Bu f fe r where the energy values are s t o r e d , by the f i r s t parameter. The second parameter po ints to the loca t i on i n Buf fer where the i n t e n s i t i e s correspon-ding to the previous energy values are located and the t h i r d para-meter g ives the number o f points to be disp layed on the scope. The MLS #26 a c t u a l l y performs the d isp lay on the scope. 9430 CALL 9*-l*C 943! IF C°\ GOTO 9428 9432 RETURN . . . 9500 REM i j t * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 9501 REM DIFFERENCE FOR TWO EXPERIMENTAL 9502 REM *^^^j^*4'^**«t^**'8!******- ,j:»?^^**^>st*»*'$** 9510. PRINT "SCOPE POSIT! ON* I NPUT A tf" 95.11 I NPUT C 951.2 CALL 9^-1 *C 89. REM The contents o f the Sr are read and depending on the number (octa l ) read the programme w i l l do the fo l l ow ing : - SR=1 Returns to main. - SR=2 Wi l l s h i f t the.energy axis of the second spectrum. - SR=4 Wi l l sca le the i n t e n s i t y of the second spectrum. - SR=10 Wi l l change the base l i n e and s lope. - SR=20 Displays the d i f f e rence of the two spectra on the scope. - SR=100 Displays the second spectrum on the scope. - SR=140 Displays both the f i r s t and second spectra s imultaneously on the scope. - SR=200 Punches the data o f the d i f fe rence f o r future use. I f none of the octa l numbers l i s t e d above i s set on the switches the programme w i l l loop f o r another t ry . 93.13 IF C«l ' GOTO 9546 9514 IF C<>2 GOTO 9556 9515 GOSUB 9530 9 5 J . 6 IF C<>4 G O T O 9518 9517 GOSUB 9548 9518 IF C<>8 G O T O 9520 9519 GOSUB .9564 9520. IF C<>! 6 GOTO 9522 f$2l GOSUB. 9580 952-2 I F C<>32 G O T O 9524 • 9 8 2 3 GOSUB 9586 9524 IF C<>64 GOTO 9 5 2 6 9525 GOSUB 9594 . 9526 IF C<>96 GOTO 9528 952? GOSUB.. 9 592 . 9528 IF _ C B ! 2 8 GOTO 9597 9529 GOTO 9552 9530 P R I N T "ENERGY SHIFT" 9532 INPUT C 9534 L E T E8sC*23oS • 9536 FOR 1= 0 T0 C 4 - 1 9538 CALL u<?C3*C4 +1 ,s>E0 . . 9540 C A L L 7!PC3+C4 + 1 *E0+E1 9542 NEXT I 9544 GOSUB 9600 9 5 4 6 RETURN . . . 9540 PRINT M I N T E N S ! TY FACTOR" 955§ INPUT C 9552 FOR 1= 0 TO C4 - I 9554 CALL 8 * C 3 * . * E 0 9556 CALL 7 « C 3+UE0*C 9857 C A L L 8 * C 3 + 3 * C 4 + I , E 0 95S8 CALL 7 * C 3 * C 4 * 3 * ! ' * E 9 * C -9§S9 N E X T I 90. 9560 GOSUB 9600 9562 RETURN 9564 PRINT "BASELINE & SLOPE" 9566 INPUT C*C5 9568 FOR 1= 0 TO C4-I 9570 CALL 8jrC3+IaE0 9572 CALL 7*C3*I*E0+C+J*C5 9574 NEXT I 9576 GOSUB 9600 9578 RETURN 9580 CALL 25*C3+C4* 0*C4 9582 CALL 26 9584 RETURN 9SS6 CALL 25*C1+C2*C!*C2 9588 CALL 26 9590 RETURN 9592 CALL 25*CJ +C2*C1*C2 9593 CALL 26 9594 CALL 25*C3*C4*C3,C4 9595 CALL 26 9596 RETURN 9597 CALL 17* 0*C4 9598 CALL !7*03*04*04 9599 GOTO 945© REM The MLS #17 w i l l punch as many numbers as given by the second parameter s t a r t i n g in Buf fer locat ion given by the f i r s t para -meter. 9450 REM THIS IS PART OF SUB 9500 9452 CALL !7*3*C4^C4 9464 GOTO 9S12 REM S ince in general the energy axis o f the two experimental spectra do not co inc ide exac t l y , the f i r s t spectrum i s l i n e a r l y i n t e r p o -l a t e d to obtain the luminescence i n t e n s i t i e s that correspond to the energies o f the second. For th i s purpose the fo l l owing sub-rou t ine i s c a l l e d . 9600 REM 96® 1 REM INTERPOLATE FIRST SUBTRACT SECOND 966'2 REM .. 96© 3 LET 11=0 9604 FOR I= @ TO C4-1 9605 CALL 7,3*C4+T* 0 9606 NEXT I 9607 FOR J= 0 TO C4-5 CALL 8 J.C3*C4+J*Jl 9610 LET I2"ii . 9612 FOR f M 2 TO C2-! 9614 CALL 8*Gl + C2"+I »J2 9616 IF .J2<J! GOTO 9622 91. 9 6 1 8 L E T 1 9620 N E X T I . . . 9622 C A L L 8 » C . l * C 2 * i l » < J 3 9 6 2 4 C A L L 8»C1 +C2+1 + I 1 #J2 9 6 2 6 C A L L 8'*Cl*ltir#'J4" 9 6 2 8 C A L L 8"*CI '•! I * J 5 9 6 3 0 L E T J 6 = J 5 + ( J l - J 3 > * < J 4 - J 5 ) / < J 2 - J 3 ) 9 6 3 2 C A L L 8*C3-+J « J7 9 6 3 4 C A L L 7 js »J* J 6 "?«J7 9636 C A L L 8»C3+3$C4*J.eX0. . 9638 C A L L 8J.CI +3*C2+! 1 *X1 9640 C A L L 7^3*C4+J*X0*X1 9 6 4 2 NEXT J 9 6 4 4 FOR ! = 0 T O C 4 - 1 9646 " CALL 8*C3+C4+I*X0 9648 C A L L 7*C4+I*X0' 9649 NEXT i 9 6 5 0 RETURM S2Q9 REM a£ ij: <t ifc # ^ <t s?8 $ sjs ifc # & $ j»t # jjs * # jjt $ s?< j?. :5s $ # $ Jjs $ # # sj: # 820 I REM WEWTON-COTES S NTEGRATI ON 8 2 0 2 REM : . . REM A Newton-Cotes formula of fourth degree with error calculation is used. On input X7 = X minimum, X8 = X maximum, N = number of points It also requires, that the integrand be evaluated in the subroutine 8250 and its value returned as FI and error as E4. On output the value of the integral is given in the variable S and the error in El. 8 2 0 4 L E T S ? 0 8 2 © 5 L E T F.l= 0 8 2 0 6 LET D4= € X a-X7)/N 8207 LET Dl=D4/4 8 2 0 8 LET X9=X7 6 2 0 9 LET D 9 = 0 G2W GOSUB S 2 5 9 821.1 LET FC53--FI 6 2 1 2 LET E 2 = E 3 8213 F O R 1=1 T O N. 8 2 1 4 LET FE13=FC53 82.15 FOR J~2 TO 5 .. 8 2 1 6 LET X9*X9+D1 821.7 GOSUB 8 2 5 0 8 2 1 . 3 L E T E2*E2*E3 6219 L E T F€*J3=Fl 8 2 2 8 NEXT J 8 2 2 I L E T V 9 = F nJ+4*FC2}*2*F£3 3*4£FC4 3*FC53 92. 8222 LET U9 ° < F C1 3 *4 *F C 3 3 *F C 5 3 ) *2 8223 LET D2° ( U 9 - V 9 ) /T 5 8224 LET S=S+V9-D2 8225 LET D3<* ABS (D2> 8 2 2 6 LET EI=E1*D3*2*E2 8 2 2 7 IF D3<D9 GOTO 8229 8 2 2 8 LET D9s>D3 8 2 2 9 NEXT I 8230. LET S°S£DI/3 8 2 3 1 LET £i=El*Di/3 8232 RETURN 8 2 5 0 REM i S i ^ ^ ^ ^ ^ ^ ^ ^ ^ i S i ^ ^ ^ ^ ^ j p J t i J S e j j t ^ ^ J S r i j ! ^ ^ * * * ^ * ^ ^ * * * * * * * * 8 2 5 2 REM FUNCTION 8 2 5 4 REM j e t * * * ' * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 8 2 5 6 LET F i ~ FWFCX9) 8 2 5 8 LET EA~ & 826® RETURN 8 4 0 0 REM ^ * * * * ^ * * * * « 3 ! : * $ * * * * * * * > i ; * * * * ^ « ^ « * ^ « < : * j S i 5 § t « 8 4 0 1 REM INITIALIZE EHD THEOR* DATA 8402. . REM i j i * * * * * * * * * * * * * * * * * * * * * * * ^ * * ^ * * * * * * * * * REJ^ J When c a l c u l a t i n g Equation 3.2 in Chapter 3 th i s should be the f i r s t subroutine to be c a l l e d . There are four parameters th i s subroutine w i l l ask to be g iven. - ECT) = EHD energy gap - phonon energy. - E(2) = Density o f e lec t rons . - E(3) = Density of holes. - E(4) = Temperature. Note that E(2) = E ( l ) + impur i ty dens i ty . 8 4 0 8 REM 8 4 0 9 DIM EE53 8 4 1 0 PRINT "SELECT PARAMETERS YOU WANT CHANGED ON S R " 8411 PR IOT "TYPE 1 WHEN YOU ARE READY" 8 4 1 2 - IMPUT C REM On f i r s t c a l l o f th i s subrout ine the SR should read 17 ( o c t a l ) . On subsequent c a l l s the programme w i l l do the fo l lowing depending on the number (oc ta l ) read in the SR: - SR = 0 Wi l l ask f o r a new se lec t i on of parameters. - SR = 15 Wi l l change the EHD energy gap. - SR = 14 Wi l l change the e lec t ron density. - SR = 13 Wi l l change the hole dens i ty. - SR = 12 Wi l l change the temperature. 8 4 1 3 CALL 9 * 1 5 * 0 8 4 1 4 I F C = 0..GOTO 841® 8415 CALL 9*1*C . 8416 I F C<>t GOTO 8420 841? PRi NT "TRIAL EHD GAP - PHONON ENERGY" 93. 8418 PR JOT 8419 INPUT EC!3 8420. CALL 9 #2*0 . 8421 iF C<>2 GOTO 8425 8422 PR5 NT "TRIAL ELECTRON DENSITY" 8423 PR1 NT . 8424 INPUT EE23 8425 CALL 9*4*C 8426 IF C«>4 GOTO 8430 842? PRINT "TRIAL HOLE DENSITY" 8428 PRINT " 8429 INPUT ECS3 043® CALL 9*8*C . 643! IF C<>8 GOTO ,8435 8432 PRINT "TRIAL TEMPERATURE" S433 PR!NT " 8434 INPUT..EC4 3 REM C a l c u l a t i o n and p r i n t o f th\e Fermi energies and chemical poten-t i a l s for the e lec t rons and holes. 0435 LET Ul= 0 8436 LET U2=..0 843? LET Ul«l .14/10 *!4*<EC2J/6> »(2/3> 8438 LET U3 J BC3«14159#EC43*8i>6! 746 /10 *5"> *2/12 8439 PR I NT. ."EL •FERMI. E N . S S ^ U t 8440 LET U1=U!~U3/U1 . 6441 PRINT "ELVCMEMcPOT• IS=*%Ui 8442 LET U2 = .6324/10 ?1 4*EC33 ?<2/3> 8443 PRINT "HOLE FERMI EN. JS=",U2 8444 LET U 2 = U 2 - U 3 / U 2 " ' 8445 PRINT "HOLE CMEMoPOT* IS="*U2 8446 PR I NT " J NPUT 0 OF POINTS TO I NTEGRAT£*<?TO Di SPLAY*" REM Each c a l c u l a t e d point i s 1 meV apart. As a general ru l e the number o f points to be in tegra ted i s chosen to exceed by about 20% the i n teger value of the sum of the two Fermi energ ies. I f the number o f points to be d isp layed exceeds the number to be i n -tegra ted the programme w i l l produce a base l i n e . The maximum number o f points to be in tegrated and/or d isplayed in 49. 844? 8448 INPUT K&sKl RETURN .". 94. 81.00 REM 8101 REM CALCULATE LUM« EHD 6102 REM 4 > & # $ « $ $ $ $ # # $ $ $ « « « $ « « # i $ ^ # $ $ $ i 3 t $ $ $ 8103 DIM LC49I 8104 LET X7* 0 8105 FOR K«l TO K0 8106 LET XB-K/IB&0 8107 LET N=K 8108 DEF FNE(X)sl0»(-17)*< EXP ((X8-X-UI)/8.61746*10-t5/EC4 3) +1 ) 8109 DEF FMH<X)sl0t(-i7)*< EXP <<X-U2)/8.61746*10?5/EC43)*S) 8110 DEF FNF<X)=10?<-34>* SQR <X)*"SQR (X8-X)/ FNE(X)/ FNH(X) 8111 GOSUB 8200 -• 8112 LET LCK3--S 8!13 NEXT K REM FNF i s the. integrand. . The f a c t o r s 1Q~'- / and 10 are inc luded to avoid f l o a t i n g po int over f low in th i s small machine. 81.14 LET Lt 0 3= 3 . SI IS FOR K<=K@ TO Kl 8SJ.6 LET LtK3 = 0 811.7 NEXT K e n s GOSUB a m ® 8119 RETURN 8297 REM ^ ^ ^ . & ^ ^ ^ < » ^ ^ K ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ' 8298 REM SCALES & STORES THEORETICAL LUM. 8299 REM $ ^ ^ J S t ^ 4 J $ * ^ * * ^ * * # * * * * $ * * * * ' S ^ * * * 3 S ^ * * * * ^ * , 1 6300 PR I NT -INPUT BASELiNE" 8301 i NPUT L6 8302 LET L@= 0 8303 FOR 0 TO K0 . 8304 LET LCJ3=L6*L£J3 8305 IF L[J3<L0 GOTO 8307 6306 LET LO-LCJ3. 8307 NEXT J. 8308 FOR J~ 0 TO K0 8309 LET Lrd3<*Li:«J3'M90©/L0 8310 NEXT J 831 ! REM X*Y THEORETICAL DATA FROM 1900 TO TOP BUFFER 8312 FOR J - 0 TO Kl .. 8313 LET J l e f E U 3-1 •0.85*J/1000>*2]000 83 5 4 CALL 7«.I.900+J« J l " 83i 5 CALL 7,l950*Ji.LCJ3 8316 NEXT J 8317 RETURN 8650 REM ****$a&$4«*^***^****#^*$^*S:5t$:^**Ji<^^»3t*^4!*>St '86 5 8 REM DISPLAY FITT! KG 8652 REM . *J*******^*J5! 95. REM This subroutine expects to f i n d the experiemental EHD l i n e shape s t a r t i n g in Buf fer locat ion Al cons i s t ing of A2 po ints with the energy values o f each po int s t a r t i n g at buf fer l oca t i on A l + A2. 8653 PRINT "SET PARAMETERS IN 3R*TYPE 1 WHEN READY" 8654 PR!NT "WANT OUT?SET SR=0 FOR ANOTHER TRY*1 RETURNS TO MAIN" 86 55 PR I NT 8656 INPUT C. 8657 CALL 9,16*C 8658 IF C» 0 GOTO 8679 REM To perform a t r i a l and e r r o r f i t of the EHD l i n e shape the pro-gramme allows two d i s t i n c t leve l s of operator i n te rac t i on s which are chosen by the f i f t h b i t of the SR, that i s , by switch #11. I f switch #11 is down a new EHD l i n e shape w i l l be c a l c u l a t e d . I f switch #11 i s up only changes that do not a l t e r the l i n e shape are performed. The programme w i l l tes t the status o f switches #12 to #14 and w i l l do the fo l lowing depending on the octa l number given by th i s 3 b i t s . - SR(12, 13, 14) = <j> M i l l go to d i sp lay mode. - SR(12, 13, 14) = 2 Wi l l sca le the i n ten s i t y o f the theore t i ca l l i n e shape, a new d i f fe rence of the experimental and theore-t i c a l l i n e shape i s performed and f i n a l l y the programme is d i ve r ted to display mode. - SR(12, 13, 14) = 4 Wi l l s h i f t the theore t i ca l peak, new d i f f e r e n c e i s performed and control given to d i sp lay mode. - SR(12, 13, 14) = 6 Wi l l s h i f t the theoret i ca l peak as well as s ca le i t . New d i f fe rence i s performed and contro l given to d i sp l ay mode. - SR(12, 13, 14) = 10 Wi l l change the base l i n e o f the t h e o r e t i -ca l peak, new d i f ference i s performed and contro l given to d i sp l ay mode. 8659 CALL 9*14*C 8660 IF C«> 0 GOTO 8?It 8661 CALL 2S*Af*A2*A1*A2 8662 CALL 26" . . . " 8663 CALL 9*-!*C REM On d i sp lay mode the contents of the SR are read and depending on the number (octa l ) read the programme w i l l do the fo l l ow ing : - SR = 0 Wi l l r e s ta r t the subroutine for another f i t . - SR = 1 Wi l l return to main. - SR = 32 Wi l l d isplay the experimental spectrum as wel l as the d i f fe rence between this spectrum and the theore t i c a l l i n e shape. On any other SR reading the programme w i l l d i sp lay the experimen-ta l and t h e o r e t i c a l spectra . 96. 8664 IF C= 0 GOTO 8655 8665 IF C = l GOTO 8683 8666 IF C=*32 GOTO 8645 8667 GOTO 8640 8669 IF C<»6 GOTO 8673 8670 GOSUB 8685 8671 GOSUB 8700 8672 GOTO 8661 8673 CALL 9*2*0 8674 IF C<>2 GOTO 8677 8675 GOSUB 8700 S676 GOTO 8661 8677 GOSUB 8685 8678 GOTO 8661 8679 60S03 8420 8680 GOSUB SI 00 8682 GOTO 866 J 8683 RETURN REM This subrout ine continues 8640 CALL 25.»1900 * i 9 50 *KI +1 8641 CALL 26" '.. 8642 GOTO 866.1 6645 CALL 25*!900«i8S0«X!*!' 8646 CALL 26 . 8647 GOTO 8661 REM The fo l l ow ing four subroutines are c a l l e d from the previous subrout ines . 8685 REM J S * * * ^ * * * ^ * ! ; ; * * * * * * * * * * * * * * * * * * * * * * * * * ^ 8686 REM SHIFT ENERGY 8687 REM J ^ ! £ « $ ! ^ * ^ * i & $ ^ « s $ * $ < t ^ ^ $ $ * i « : j 3 t « ! 4 t ! S : $ i 5 t© i } c j « : * ^ ! S 4 5 * * * 8688 PRINT "ENERGY SHIFT IN MEV" 8689 8 NPUT C 8690 LET EC! 3 = E £ U * C / 1 0 0 0 869! LET E i - C ^ S l 8692 FOR 1^0 TO 49 8693 CALL 8*1.900 + 1 *E0 8694 CALL 7„t900*1,E0*E1 8695 NEXT I "' 8696 GOSUB 88®B 8697 RETURN 8700 REM >$c<t$#i9c$<c«^«i(i«iic«;««<t#4:«^$<:««:!t:$4E4:«««#4t«]it# 8701 REM INTENSITY CHANGE 8702 REM &#&®&&%&&%%iri#&#0&&&£%##%## + %#tst&&&%%fr& 3703 PR 1 NT "INTENS1TY FACTOR" 8704 INPUT C 8705 FOR 1 " 0 TO 49 G7«6 CALL 8*1.950*1 *E0 8707 'CALL 7* 2 9S0*! *E«*C 8708 NEXT 1 97. 8709 GOSUB 8800 8710 RETURN 871! REM BASELINE CHANGE 8713 IF C<>8 GOTO 8669 8714 GOSUB 8300 8715 GOSUB 8800 8716 GOTO 8661 8800 REM * * * * # $ s 3 t « * * * * ^ * $ * * * * * * * * * * ! ! < « j e t * * 4 t $ ^ ' ! « * * * * * * * * 8801 REM INTERPOLATE EXPERIMENTAL SUBTRACT TH. 8802 REM ... * * ^ * * $ * * * * « * * . ^ ^ * * * * * * * * * * ' & * 5 ! S * * * ^ * * * ^ * ^ J ! S * ^ « ; 8803 LET 11=0. 8804 FOR j= 0 TO K0 8805 CALL 8* I 900-i-J a J l 8806 LET 12=11 8807 FOR ! *!2 TO A2-1 ... 8808 CALL 8*A!<s-2#A2»S~laJ2 8809 IF J2»Jt GOTO "'8812 88J.0 LET 8811 NEXT I' 8812 CALL Si-Ai +2«A2-1 -i.1.»J3 881S CALL 8'» A i *2 * A2 - 2 I i 9 J2 8814 CALL 8«AJ'«A2-2-U«J4 8815 CALL 8*A1+A2H1-11iJ5 8816 LET J6=J5 + (Jl-J3>#<J4-JS)/<J2-J3) 8817 CALL 8*.!950+Ji'J7 881.8 CALL 7*S850+J J>J6-J7+L6*1900/L0 8819 NEXT J . . . 8820 RETURN 5200 REM 5001 REM NEW PLOTS 5002 REM # * * i ? i ! 5 t ^ * * i » * i J t $ « ! i } £ » « ^ ^ : ^ ! # i ? $ $ i 5 [ j f : $ ^ X 5 $ * * * > j ! 5@§3 INPUT C 5004 CALL 9*-5*C REM The input i s j u s t a pause to get the x-y p l o t t e r ready. The SR is. read and depending on the number (octa l ) read the programme w i l l do the fo l l ow ing : Reads SR again. Wi11 return to main. Wi l l p l o t the f i r s t experimental spectrum read in to b u f f e r . Wi l l p l o t the second experimental spectrum. Wi l l p l o t the d i f f e rence of the two experimental spectra . Wi l l p l o t f i xed energy axis from .99 to 1.18 eV (21000 x .095 = 1995<2000 which i s maximum of our D/A conver ter ) . W i l l p l o t IB t heo re t i c a l l i ne shape. Wi l l p l o t EHD theo re t i c a l l i n e shape. W i l l p l o t the d i f f e rence between the experimental and theo re t i c a l EHD l i n e shape. - SR - SR = 1 - SR = 2 - SR = 4 - SR = 10 - SR =20 - SR =40 - SR= :100 - SR= =200 53©S. IF C» & GOTO 500-4 5006 IF C - l GOTO 5099 5ma IF C«*2 GOTO 5016 5013 LET CS=Cf 5812 LET C6=C2 5814 GOTO 5070 52-16 IF C«»4 GOTO 5024 5018 LET C5=C3 5028 LET C6=C4 SBZ2- GOTO 507.0 . 5924 • IF C<>8 GOTO 5032 5S2A LET C5= 0 mZl •• LET C6=C4 Sfl>30 GOTO 507® SS32 IF C<>16 GOTO 5100 5P33 CALL | U 0 5034 CALL 1.2*2 1000*( - .095) *-50 5036 CALL f 1 *1 ' ' S8-38 FOR i» 0 TO ! 9 5040 LET X0»2f000*(- .095+1 /100) S»42 CALL 120X0*~S0 " 5044 CALL l2»X0«-25'-. 5046 CALL 12#X0*-50 5048 NEXT I 5050 FOR I sj 0 TO 19 5® 52 LET X0=*21000*< .095-1/100) 5054 CALL .!2*X0«1950 •* SBS6 CALL 12*X0*1925 5958 CALL 12^X0^1950 S©6'0 NEXT 1 5@62 CALL !2*X0*-50 50i&4 CALL t 1 0 8 " 5066 GOTO .50® 4 5 0 C A L L li*..0.' 5S71 GALL 9 * - l ,C 50?2 IF C<> 0 GOTO 5071 5073 FOR 1= 0 TO C6-! 59*14 CALL 6*C5+C6+l*X3 5S76 CALL 6*C5+1«Y0 S&7.8 CALL 8*C5*3*C6+I,Y1 5080 CALL I2*X0#Y0+Y1 50-82 CALL t!«1 CALL t2«X8*Y0-Yl 5SK6 CALL 1 ti..0 5 0 C A L L 9*-| »C 5ff90 IF C™1 GOTO 5073 5Q92 IF C»2 GOTO 5003 5094 NEXT I .5096 GOTO 5004 5099 RETURN 5100 IF C«>32 GOTO 5130 5502 CALL 11*0 5104 CALL 9 * - l*C 5106 IF C<>'9 GOTO 5104 5108 FOR I = 0 T O 49 51 10 CALL 8*1600+C7*.!00*I«X0 5112 CALL 8*16S0*C7aj00->i*Y0 5114 CALL 12*X0*Y0 5116 CALL 11*1 • 5118 CALL I 1 * 0 SS20 CALL 9 * - l*C SI 2-2 IF C*I GOTO 5808 5124 IF Cs>2 GOTO S@©3 5126 NEXT.1. 5123 GOTO 5004 Sf 30 IF C<>64 GOTO 516© 5131 LET RC493=1 '5132- CALL t l * . . 0 SI 34 CALL 9*'-l sC 5136 IF C<> & GOTO 5134 5138 FOR S« $ TO KS S140 CALL 8 * 1 9 0 0 *I * X 0 5.142 CALL 8*1850+R149 3«100+I 5144 CALL J2*X0*Y0 5146 CALL 11*1 5148 CALL I I * .0 5150 CALL 9*-I*C 5152 IF C=5 GOTO 5S4® 5154 IF C®2 GOTO 5003 5156 NEXT I 5158 GOTO 5004 516® IF C « » S 2 8 GOTO 5004 5162 LET RC493** 0 5164 GOTO 5132 100. REM The fo l lowing subroutines deal with the c a l c u l a t i o n of the IB l i n e shape. In these subroutines we are forced to use an array f o r v a r i ab l e names because at t h i s stage of the programme we have exceeded the storage a l l oca ted by the compiler to var iab les names which has a disastrous e f f e c t on the l e g i b i l i t y of these subrout ines. 5200 REM 5205 REM IMPURITY INHCIALIZE 5202 REM ^ ^ s ^ ^ s ^ i s t ^ ^ j s s ^ i s t ^ i s i s j ^ ^ ^ ^ * * * * * ^ * * ^ * * 5204 LET RC © 3=s3 »14! 59 5206 LET RC! 3 = 1 1 V33 5208 LET RC23=.0906 5209 LET RC33=.57722 5210 PR I NT "INPUT IMPURITY CONC»" 5211 LET Rf.433 = ! 52*2 INPUT RC41 5214 LET RC53 = (3/4/RC 03/RU J> U 1/3>*1E*8 5215 LET R£53=R£53/RCI 3 5216 LET R £6 3--2*266 5218 RETURN S250 REM : « t***^«*!,l5>*« i*Si i J t* . -X:5>!»<t<is ) t j J t*«c^*i ; t )3t**j } t 5252 . .REM ' IMPURITY BAND 5254 REM ' HYDROGEN MOLECULE I ON 5256 REM *iii* $ $ « 4 c«)tt* 4 t * « * * 4 : 4 c « * ' 4 c « * * 4 t « « * * « « « * 5258 GOSUB 52 SS REM The fo l l ow ing d i c t i onary i s he lp fu l R(8) = V,(R) R(9) = V2(R) R(10)= where V-.(R), V 9(R) and A are defined by Lukes et a l . 526$ FOR l» 0 TO 49... 5262 LET S=RC63/RCt 3 526 4 LET R£83=l/S- EXP <-2*S)*(!+1/S) 5266 LET RC93=(!*S>* EXP C-S) 5268 LET RE !03=»< 1+S+S*S/3>* EXP C-S) 52 70 . LET RCII3=-2*RC23*<R£8 3*RC9 3>/<1*R£ 103) 527! LET RC1 1 3»-«0453+RCU 1/2 5272 LET R£123=-1 /S/S*2* EXP ( -2^S) *< 1 +1 /S/S/2-M /S) 5273 LET QC l'3=RC I 1 1 5274 LET R[.13 3»-S'/S* EXP <-S> 5276 LET R U 4 3*-S*S/3»*<.S+1 )* EXP < -S) S278 LET R M 53*-<RC i 2)*RC!3 J>/<1+RT10 3 > S279 LET REI53=RCI5 3*(R£83+R£9 3)/(1+RE10 3) t2*RC!43 5280 LET RE 15 3=RE1 5 3sRE2-1 52 84 LET R£!!3 = CR£71»S .085+REl1 3)*21090 5286 LET RC16 3»S*S/RC5 3 E X P <-<S/R£53)*3> 52S7 LET L£ ! 3=3-*R£16 3/R£IS3 '5288 LET R£6j=RC6 3+2..26.6 5290 CALL 7i, 1600 + i s R i l I 3 -5292 NEXT t 101. 5294 529 5 52 98 5300 5302 5304 5306 5307 5308 5309 5310 53 J 2 5314 5316 5318 5320 5322 5324 5326 532S 5330 5332 5334 5336 5338 5340 5342 5344 5346 5348 5350 5352 5354 GOSUB 5420 GOSUB 5470 RETURN REM IMPURITY BAND REM HEITLER-LONDQN MODEL REM * * * * * j ) : i S e * # X t * * * * < r * i ) : * * * * * * * * $ * * * * * * * * * GOSUB 5200 GOSUB 5800 . REM The following R(23) R(24) R(25) R(26) R(27) R(28)+R(29) R(30) R(31) R(32) R(33) R(34) R(35)+R(36) R(44) where S, S' dictionary is helpful S S' J K J' K' 3S/3R 3S73R 3J/3R 3K/3R 3J'/3R 3K'/3R |3E+/3R K, J ' J , R' K' are defined by Slater. 41 I NT <X/2)-2#X DEF FNTCX) =1 +4* FOR 1=0 TO 49 . • — • -LET S=RC6 3/RU3 LET RES 7 3= 0 LET RC183s 0 LET RtI93= 0 LET RC20 3* 0 IF S>2 GOTO 5354 LET RC!7 3=RE33+ LOG <2*S) LET RC183=RC33+ LOG <4SS> LET RC193 = !/S LET RC20 3 = 1 /S FOR J=I TO 32 LET RC213=1 FOR K=l TO J LET RC213=Rt213*K NEXT K LET RC223=RC213*J LET LET LET LET NEXT J LET RC233* RC173=RC !73+ R C 1 8 3=R118 2+ R U 9 3 = R [ m -R C 2 0 3=RC20 3-r NT(J)*(2*S) tJ/R[223 FNTCJ)*(4*S> *J/R£223 ; FNT<J-1>*C2*S) ?<J~1)/RC213*2 FNT( J - 1 >*(4*S) »< J-I )/RC21 3*4 EXP <-S>*< 1 • S+S#S>'3> 102. 5356 LET RE243« EXP ( S ) * ( 1 -S+S*S/3) 5358 LET RT25]=-1/S* EXP <-2*S>*<1+1/S> 5360 LET RC26]=- EXP <-S ) *< l+S) 5362 LET RC27]»1'/S- EXP ( -2*S) *< 1 /S+! 1 /8+ .75*S*S*S/6 ) 5364 LET RE283»-i/5* EXP <-2*S)#<-25/8+23*S/4*3*S*S*S*3/3> 5365 LET S0=RE233*R£233*<RT33• LOG <S))+RC24 3*R£243*RC183 5366 LET RE293*6/S/S*<S0-2*RC233*R£243*RE173) 536 7 LET S0 ==R£43 3+2*RE26 3*R £23 3-R £23 3 *R E233«-2*Rf 253+RE27 3+R £28 3 5368 LET RC403 = <S0+RE293)/(1+R£233*R£233>*RE2J 5370 LET RC303=-RE233+ EXP < -S> #i t «-2*S/3> 5372 LET RE3I 3s>RE243 + EXP < S > * ( 2 * S / 3 ~ U 5374 LET RE323*1 / S / S - 2 * EXP (-2*S) #< 1 "• 1 /S + l /S/S/2) 5376 LET RE333=-RE261- EXP <-S> 5378 LET R t 3 4 3 s - l / S / S + EXP <-2*S>*C2/S+2+7/6*S+S*S/3+l/S/S) 5380 LET RE353=-2#RE283- EXP (°2#S>*(23/4+6$S+S*S)/5 5382 LET S0=R£233*R£23 3/S+2*RE24 3*RE313*R£183+R1243*RC243*RC203 5383 LET S0=S0+2*RE233*R£303*<R£33* LOG <S>) 5384 LET R£363«-R£293/S+6/5/S*S0 5385 LET S0=-2*(Rf243*R1303*R£ 173+RE233*RE313*Rf173> 5386 LET RE373=6/5/S*CS0-2*R£23 3*RE243$RE193) 5388 LET S0*-RE433*2$Rf233#R£263-RE233*R£233*2*RE253+R£273+RE283 5389 LET S0**'CS0*RE29 3 > #R £23 3#R £30 3/{ 1 *R£23 3*R£23 3) »2 539® LET RE383=-2*S0 5392 LET S0=2*<R£233*R£333*R£263#R£303)-2*R£233*R£303 5393 LET S0=S0*2*R£32 3+R £34 3 *R £ 35 3+R £36'J*R £37 3 5394 LET R£393*S0/£1+RE233SRE233) 5396 LET RC413=<RE383+RE393>*R£23 5397 LET R£41 3«* ABS <RE413> 5399 LET RE423oS*S/RE53 t 3 * EXP (-(S/RC53)?3> 5400 LET LCI 3=6*RE423/RE4U/R£443*( 1+RE23J*RE233) 540 8 GOSUB 5650 §402 CALL 7*1700+1*RE40 3 5404 LET R£63*RC63+2o266 5406 NEXT ! 5408 GOSUB 5420 5409 GOSUB 5450 5410 RETURN 5412 NEXT I 54 S 4 RETURN 5420 REM ^**^**>i<^^i3t**$^*^$^i3t*^$jf:*^$i^$^!j[^!jij}:^^^(ij!$^$ 5422 REM SCALE THEOR. IMPURITY BAND 5424 REM «**i5i**^#$*<:$i3i*«$J8tiSt*i>*>!t*$iS$«!i55****i?**iS**iS>S! 5426 PRINT "BAS E L I N E I B " 5428 INPUT L6 5430 LET L 0 - 0 5432 FOR «J» 0 TO 49 5434 LET LCJ3=LEJ3+L6 5436 I F LCJ3<L0 GOTO 5440 S438 LET L0sL£J3 5440 NEXT J 5442 RETURN 103. 54 S& REM 5452 REM STORE HE1TLER LONDON 54 54 REM 5456 FOR J= 0 TO 49 5458 LET J8 =LCJ3*1900/L@ 5460 CALL 7 *1?50*»W8 5462 NEXT «i 5464 RETURN 5470 REM ljei}s# ij* >% J*i# sjs « # Us $ ^ ^ t t 5 t * $ $ $ #$Jjt & & $ $ $ $ $ $ & 5472 REM STORE MOLECULAR 8 ON 5 4 7 4 REM 5476 FOR J= 0. TO 49.. 5470 LET J i =LCJ3*1900/L® 5480 CALL 7 5482 NEXT J 5 4 8 4 RETURN 5650 REM S 6 5 2 REM STORE KEITLER-LONDON AXIS .5554 REM 56 5 9 LET RE40 3=RC403-QC13 REM The array Q i s where the energies o f the molecular ion model were s tored ( i n s t r u c t i o n #5273). We are subtract ing i t now because we want to equate (see Appendix B). I CEjLCRI)-E+CRJ)) = n(EjLCR1)) 5660 LET .RC40}»(Rt7]-l .085«RC403)«2|000 • 5662 RETURN 5800 REM * « 4 t # * * * * * * ^ * * * * * # * * j J i * * * # * * * * * * * $ i J : J ! t * J 5 t # * 5802 REM CALCULATE NORMALIZING INTEGRAL 5804 REM * * * ^ * * * * # * J | ! * * * # * * 4 « * * * * * * * # * * * * * * * * * * * * 5806 DEF'" PNH<X)?6*X*X/RCS3«3.* EXP <°<X/RC53>f3> 5808 DEF* FNF(X)« FNH(X3*C 5 *( 1 *X*X*X/3> ?2* EXP <-2*X) 58.1.0 LET X7n* @ . . ' 5812 LET X5«20 5 8 1 4 LET N*I00 5SI6 GOSUB 8200 581 8 PR! NT S 5S20 . LET RC443»S 5S22 RETURN 104. 5500 REM 5502 REM DISPLAY FIT IMPURITY BAND 5504 REM ^t**^*!***^******^**^*****"**!**^** 5506 PRINT "INPUT 0 FNR MOLEC*ION*I HEITLER LONDON" 5508 INPUT C7 5510 PRINT 5512 PRINT "SET PARAM* IN SR «0 ANOTHER TRY THEN ! FOR OUT" 5514 PR IWT 5516 CALL 9*-l*C REM The SR i s read and depending on the number (oc ta l ) read the .pro -gramme w i l l do the fo l lowing^ - SR = 0 Wil l read the SR again. - SR = 1 Wi l l return to main. - SR = 2 Wil l d i sp lay the experimental and t h e o r e t i c a l IB l i n e shapes. r e s u l t . - SR = 4 Wil l s h i f t the IB l i n e shape and d i sp lay the - SR = 10 Wi l l sca le the IB l i n e shape and d i sp lay the r e s u l t . - SR = 20 Wi l l change the base l ine of the IB l i n e shape • SSI8 IF C«..0"GOTO 5516 5520 SF C*t GOTO 56 S8 SS22 IF C«>2 GOTO S544 5524 CALL 25*|6@0+C7*f00*1650+07*100*50 5526 CALL 26" 5528 CALL 25*C4* 0*C4 5530 CALL 26 5531 CALL 9*-l*C . 5532 IF C<> 0 GOTO 5524 5534 GOTO 5516 5544 IF C«>4< GOTO SS50 5546 GOSUB 5560 5548 GOTO 5524 5550 IF C<>8 GOTO 5556 5552 GOSUB 5600 5554 GOTO 5524 5556 IF C<>!6. GOTO ' 5524 5558 PRINT "BASELINE" 5560 INPUT L6 5562 FOR I * 0 TO 49 5563 CALL 8*.f6S0*C7 #100* I *E0 5564 CALL 7* 1650+07*100+ i ^ ES*L6 5566 NEXT I ' " 5568 GOTO 5524 5580 REM 5582 REM ENoSHI FT IMPURTY BAND S 5 B 4 R EM ^ * 4' 4: # * * # # vjj ;*t s£ # i}c * s»a $ jj; j»s $ $ $ $ $ $sjc«$$$ 5586 PR I NT "EN-SKI FT" S588 . INPUT.!C ".. 5590 LET E!«C*2I. 105. 5592 FOR I « 0 TO 49 5594 C A L L 8*1600+07*100+1,E0 5596 C A L L 7,1600+C7*100+I,E0+E1 5598 NEXT i 5599 RETURN 5600 • REM * tf * * * * # * * * $ * # # # $ * # * * * * * * * * * * $ * * 5602 REM INTENSITY FACTOR IMP.BAND 5604 REM 5606 PR 1 NT " 11 NTEN .FACTOR" 560.8 INPUT C t 5610 FOR 1 = 0 TO 49 5612 CALL S*.l 650*07*100+ |,E9 5614 CALL 7; 1653+07*100 + 1 j.E0*C 5616 NEXT 1 56S8 RETURN A.2 Examples of Main Programmes 106. A.2.1 Main #1 Let us suppose that we have two experimental spect ra . In the f i r s t one both the EHD and IB peaks are present whi le in the second spectrum the IB peak dominates. We can then use the l i n e shape of the IB peak of the second spectrum to subtract the IB peak from the f i r s t . The r e s u l t i n g spectrum w i l l show the EHD l i n e shape. We w i l l proceed then as fo l l ows : 2 CALL 1 4 D IMf (255) , L(49) , R(49), Q(49), E(5) 6 GOSUB 9100 REM Wi l l load f i r s t spectrum 8 GOSUB 9400 REM Wi l l generate i t s energy ax i s . The LOAD subroutine defines A l , the s t a r t i n g memory loca t ion in b u f f e r f o r the loaded data, as well as A2, the number of data po int s . We assign the values to another var i ab le name before they are l o s t by loading the second spectrum. 10 LET C1=A1 12 LET C2=A2 REM The var iab le names chosen w i l l permit the use of other subroutines without problems. Now we can proceed to load a second spectrum. 14 GOSUB 9100 16 GOSUB 9400 REM Generates i t s energy ax i s . 18 LET C3=A1 20 LET C4=A2 REM The var iab le names chosen w i l l again permit the use of other sub-routines without problems. We now turn control to the subroutine that w i l l take the d i f f e r e n c e . 22 GOSUB 9500 REM At th i s po int "of the main programme we w i l l have the experimental EHD l i n e shape s tored in the bu f fe r s t a r t i n g in l oca t i on cero and having C4 po ints . I f we eventua l ly w i l l l i k e to f i t to th i s expe-rimental l i n e shape a t h e o r e t i c a l l y ca l cu l a ted one we w i l l turn control to the subroutines that w i l l do the c a l c u l a t i o n of the theore t i ca l EHD l i n e shape. 24 GOSUB 8400 26 GOSUB 8100 REM The f i t t i n g rout ine which w i l l be c a l l e d next expects to f i n d the experimental EHD l i n e shape stored in bu f fe r s t a r t i n g at l oca t i on A l and having A2 points so we have to redef ine A l and A2 so that the f i t t i n g is done to the appropriate data. 28 LET Al=<fr 30 LET A2=C4 32 GOSUB 8650 REM If the f i t was successfu l we would l i k e a p l o t . 34 GOSUB 5000 36 STOP 107. A.2.2 Main #2 A v a r i a t i o n of the previous example may occur i f in the second spectrum the EHD peak dominates. We would fo l low the previous main programme to i n s t r u c t i o n 22 i n c l u s i v e . At that stage of the programme we would have the experimental IB l i n e shape s tored in the bu f fe r . At th i s po int we w i l l turn cont ro l to the subrout ine that w i l l c a l cu l a te the theo re t i c a l IB l i n e shape 26 GOSUB 5250 REM I f the He i t ier -London model i s to be ca l cu l a ted use the next i n s t r u c -t i o n , i f not sk ip i t . 28 GOSUB 5300 REM We w i l l now do the f i t . 30 GOSUB 5500 REM And f i n a l l y we would l i k e a p l o t . 32 GOSUB 5000 34 STOP 108. APPENDIX D THE EHD COMPUTER PROGRAMMES D.l Correlation and Impurity Energies The correlation energy per unit volume is related to the pair 1 g correlation function S(q, w) through •i 1 .2 / \ 1 ( d^ q ( du f 4ire2 S (q ,w) + « c o r r ( V n * ) " ^ ^ H ) 3 ) +• In 1 - 2S(q,a)) . -e(q)q (D.l) 59 where following Nara and Morita 1 - A Q 2 , (l-A)q 2 e'^O) Y 2 , (D.2) eXqT M 2 J 2 2 . - 2 2 . 2 V M / q + a q + 3 q + Y with A «• 1.175, a = 0.7572 a.u., $ = 0.3123 a.u., y = 2.044 a.u. and e(0)=11.4. 1 g To claculate D.l Bergersen et al found i t convenient to distort the w - contour along the imaginary axis w = iz. Similarly the energy per unit volume associated with the impurity 19 interaction is approximately given by 2TrZe2nd( d3 W s ^>°> G i r n p (2n)3 )e(q)? [ q ^ i r e V 1 (q) S (q,0)] In the RPA the total pair correlation function S(q, iz) may be written as a sum of contributions from the conduction valley electrons, c h S and light and heavy holes, S . 109. D.la. Valence Band 'Contribution For further computations i t is useful to define the following: ?c = mt/ml *' rh = W » " t 2> 1 / 3 r C = f 3 iA , A i ^ a • t - n = /Q-n-2., / / U v 3 / 2 ^ 3 kp=(3^nc/v) a Q ; = { 3 ^ / ( 1 - ^ ) } a Q  E C . ^ F ' ) 2 . F h _ ^ " F 2m ' F ~ 2 mhh where v is the number of conduction band valleys and a Q = fi/n^e 1" the atomic Bohr madices so that the electron and hole densities are per The hole polarization function is written^ 8 2 4 T r e S h (a i z E h ) = 1 ( d3p I AU (2TI) 3 )p<^ k^h |izE^ +h2(2p.q + q 2 ) / 2 m £ h h h + c c . ^ ^ cm izEj + -fi2(p + q)2/2mhh - •n2p2/2m h^ j e ( k ^ ) 7 1 f d3p I ^h£_ <*^)p<kj! h ( i 2 E F + ^ + ^ 2 % " ^ P 2 / 2 m h h V + c . c . \ 4T re 2 1*p + f,2(p + q) 2/2m h h . tffa^ ' j S ¥ ^ n o . where A££ = Ahh 1 2 3q2(1 - y 2) p 2 + q 2 + 2pqy (D.5) and \l = Ah£ 1 q2(1 - y 2) 2 2 2 p + q + 2pqy (D.6) with u = cos (p\q) (D.7) 1 g Bergersen et al expressed Equation D.4 in momentum and energy hole Fermi units and have performed the angular integration, they found: 4TC ch/ . rh, ~hhT2 S (q' l z EF } a h Qh (q,z) (D.8) where 2m, a hh eirk. (D.9) and Qh (q,z) = | ( pdp e(y^ - P) x 2 Z 2 y g + ( 2pq + q 2 f Z 2 + (2pq + q 2 ) 2  X l n ,2,2 , ,2,2 + £ n 7 7 + (2pq - q')' Z 2 + (2pq - q 2) + Re 16p - ^ - T - (- 12p2q2 + 3(izy h +q2) ) -X .+ D L n •ZYh-+ P izy. + q + 2pq x In ( 2 ~ ) iZY. + q - 2pq (Continued...) 1 1 1 . + (12p 2 q 2 - 3(1z - y ^ P 2 + P 2 + q 2 ) 2 ) x - 1 2 2 2 i z - Y H P + P + q + 2pq v x £n ( V? ? ? - } i z - Yu P + P + q - 2pq J + _ U R e _ J 2 , N C 2 16p - i z + p ( - 1 2 P V • 3 ( i z + , 2 ) 2 ) * + 2e*) i z + q - 2pq ( - 12p 2 q 2 +3( izY h + p 2 - Y h P 2 + q 2 ) ) x 2 2 2 i z Y h + p - Y h P + q + 2pq x £n ( 2 * ^ * ) i Z Y h + p - Y n P + q ~ 2pq I h" e(p - Y h 2 ) x x 4 2 z 2 + (2pq - qV P h 16p^ - i z + pd ( - 1 2 p 2 q 2 + 3 ( i z + q 2 ) 2 ) £n ( i z + \ + 2 M ) i z + q - 2pq -( - 12p 2 q 2 + 3 ( i z Y h + P 2 - Y h P 2 + q 2 ) ) x i z Y h + P 2 - Y h P 2 + q 2 + 2pq n x £ n ( — * * ^ ? ) i Z Y h + p •• Y h P + q ' - 2pq J (D.10) (The typographical errors appearing in the corresponding formulas to 18 Equations D.4 and D.10 in the paper of Bergersen e t al are here corrected) 112. Equation D.10 is calculated numerically with programme one is listed below and is independent of the hole density since q and z are in hole Fermi units and to avoid unnecessary repetitions of this highly time consuming integration, one should define a priori a grid of q and z values (in hole Fermi units) for which the integrands of Equations D.l and D.3 will be calculated. Care must be exercised when choosing the grid: i f the largest values of q and z are too small then, for low hole densities the integrations given by Equation D.l and D.2 will be in error; for high hole densities the values of q and z may become too coarse. It was found 1R - ? that to ensure accuracy in the low hole density region of 10 cm i t was necessary to extend the grid used by Bergersen et a l ^ ' ^ by at least an order of magnitude. The evaluation of Equation D.10 is prohi-bitive in computer time for such a large grid but may be considerably simplified by performing the angular integration considering only the do-minant terms of Equation D.4 when q and/or z are large. The result in this ^ 1 5 h, , 4 <^ / 2 + D(q 2/Y h) (Y? / 2 + D q 2 Q / W ) = % { - S o o-^ + -4 a — > ( D . H ) 3 z + (q /Y n) 2 z 2 + q 4 which is evaluated in programme two listed below. D.lb. Conduction Band Contribution 18 Bergersen et al -indicate that to calculate the electronic polarization function i t is useful to define 113. q~ll = Y c / 3 q | | ; \ *VC\ (D.12) JL _ JL . JL II and q- = 6"j + q^ where q^ is oriented along the longitudinal axis of a conduction band valley and q" is the projection of q~ on the plane perpendicular to that axis. Clearly, for each of the six valleys in silicon we have a set of definitions as in Equation D.12, namely q 2 j 4 = q 2 ( Y 2 / 3 cos 2 9 + Y ^ 1 / 3 sen 2 9) q? c = q 2 ( Y 2 / 3 sin 26 sin2<J> + Y " 1 / 3 (cos26 + sen2e cos2<|))) cf| 6 = cj2 (Y 2 / 3sen 2e cos2<(. + y^/3 (cos29 + sen29 sen2<J>)) (D.13) where the principal axis is along the longitudinal one of the fi r s t valley and cj) is the azimuthal angle. The polarization function due to the conduction valley i can now be written as: 1 s C ( 3 i ' i z E F > - - 7773-f d ' p i. F c + u 2 / 0 , — ; 2 (2TT) U ryC izEp..+ u '°~ - ' -ht(2p.q. + qj)/2m 1 •1ZE£ + h2(2p q. + q 2)/2l } CD.14): 114. Expressing momentum and energy tn electron Fermi, units, the above Integral can be easily shown to y ie ld : •4irf> r r with a = 2 m 1 1 ,2 2 f c + (cfT - 2q\) q i 4 1 K z' (<L + 2a, ) 2 " , 2q. - q 2 , 2q\ + (J2 - f ^ U a n 1 ( — l j M + tan" 1 ( 1 2 (0.17) Where z and q. are now dimensionless quantities measuring energy and mo-18 19 C mentum. Bergersen et al J approximate S by i ts spherical average replacing q. by q for a l l i . This approximation is adequate when c a l c u l a -ting the correlation energy but for the impurity -energy contribution i is elways zero and the logarithmic singularity in £"(2,0) produces excess ive noise which is bothersome when calculating u, therefore the exact S was used here. As pointed out in the previous subsection the electronic po lar i -zation function (Equation D. 15) has to be evaluated fo r a predetermined grid 115. of q and z (in hole Fermt units). Much of the complexity of programme three arises from this required change of variables. D.2 Kinetic and Exchange Energies This energy contribution in Rydbergs per unit volume can be easily expressed in terms of previously defined quantatives: Q -1 2 ekin C n C n h } - 5 A ( k c ) 2 "c ,.11,2 % m hh (D.18) e (n , n. ) = exc v c' h' 3 m " l "o m. e 2TT A kF n c $ ( V + kf nh * ( V * (D.19) where $(Yc) and IKYc) a r e given explicitly by Combescot and Nozi^res7. D.3 Computer Programmes The following computer programmes are revised and corrected versions of those used by Bergersen et a l ^ 8 ' ^ . The programmes are written in Fortran IV and the IBM 370 computer model 168 is used. As mentioned above, programme one calculates the hole polariza-tion function per Equation D.10 for small q and z; programme two calcu-lates the large q and/or z assymtotic expansion of the hole polarization function per Equation D.ll; programme three uses the results of the pre-vious two and after calculating the electronic polarization function evaluates all the energy contributions. 116. D.3a. Programme One MICHIGAN TERMINAL SYST«=H FORTRAN GJ41336) MAIM 0 1 - 0 4 -C C COMPUTATION OF THE HOLE POLARIZATION FUNCTION FOR C SfALL VALUES OF 0 ANO Z C C BY BERGERSEN,JENA AND BERLINSKY C 0001 OTMFNSION ZA(99) ,QA(99),HST(10),NST(10),XST110) 0002 DIMENSION SHST(99) 0003 CO^MCN C.Z,GH 0004 EXTERNAL SH 0005 REAL KL,HT,^HH,MLH C C TABULATION OF DATA C C COURSE MESH 0006 0.4TA H S T / . 1 , . 2 , . 5 , 2 . / C0C7 DATA NS7/11,17 ,21 ,41/ C SILICGN DATA 0008 DATA M L » M T r f ' H H , M L H t D E / . 9 1 t » 1 9 » . 4 8 , . 1 6 , 1 1 . 4 / 0U09 NU=6 0010 PI = 3.14159 C011 GH=MLH/PHH 00 12 T--SQRT(GHJ 0013 T2=T/2. 0014 T3=T+(l.-T)/2. 0015 PAX=NST(4) C C MESH FOR Z AND Q VARIABLES C 0016 N=i 0017 ZZ=-HST(1) C018 CC=ZZ 0019 DC 10 I = l t 4 0020 NN=NST(I) 0021 H = H S T « I ) 0022 CO 11 J=N,NN 0023 ZZ=ZZ+H 0024 OG=OG+H 0025 Z M J ) = ZZ 0026 11 CACJ)=QQ 002 7 10 N=NN+1 C C INTEGRATE FUNCTION SH CVER P C 0023 SHSTU)=0. 0029 DC 12 1= l .HAX 00 30 Z=ZA(I) 0031 WRITE(6,).01)Z 0032 KRITE(7,104)Z 0033 DO 15 J=2,HAX 0034 C=CA(J) 0035 QSQ=Q*Q C THE INTEGRATIONS OVcR P ARE DCNE USING THE C G4USS-LEGEN0RE INTEGRATION FORMULA BY C-ULING THE C U . 3 . C . COMPUTER CENTRA'S FUNCTION FGAUI6(A ,B , F I . 0036 HS = FGAU16(0. ,T2,SH J+FG5U1.6 IT2. T.SH )• «FGAUlfc<7,T3.SH)tFGAU16(73,L.iSH) 0037 8S--HS*QSC 117. MICHIGAN TERMINAL SYSTEM FORTRAN GI41336) MAIN 0O3fl SHST(J) = HS 0039 15 CONTINUE 0U40 WRITE< 7, 103)tSHST< J ) , J = 1,MAXI 0041 12 CONTINUE 0O42 101 FORMAT!1CX,'Z=',F10.4) 0J43 10? F0RMAT(6E13.6> 0344 1C4 F09MAT(10X,«Z=»,F10..4) 0045 STOP 00 46 FNO -OPTIONS IN =FF=CT* 10,FBCDIC,SOURCE,N0LIST,NOCECK,10AC,NO CAP ^OPTIONS IN EFFECT" NAM? = MAIN , L!NFC NT = 60 ^STATISTICS* SCURCE STATEMENTS = 46,PROGRAM SIZE = 2410 • ^STATISTICS* NO DIAGNOSTICS GENERATED MO ERRORS IN f. A IN 118. MICHIGAN TERMINAL SYSTEM FORTRAN G I 4 1 3 3 6 ) SH 01-04-0001 0002 0003 0004 0005 0006 0007 0008 CO 09 00 10 0011 0012 0013 0014 0015 0016 0017 00 18 0019 0020 00 21 CO 22 0023 0024 00 25 0026 0027 00 28 0029 0030 00 31 00 32 0033 0034 CO 35 0036 0037 00 38 0039 0040 0041 0042 0043 00 44 C045 •OPTIONS •OPTIONS •STATIST! •STATISTI NO ERRORS IN FUNCTION SHIPP) REAL *8 P,0,Z,GH,R1,R2,R3,R4,P012,PQP.PQM,0REAL, •DSQRT.DLCG RC-AL *3 PCPSQ,PQPSC,ZSC,PSQ,QSC,S1 ,TPQ CCHMON CC,ZZ,G • C0MPLEX#16 CDL0G,C!Z,C1,C2,C3,C4,C5,C6,C7,C8, •C9,C10,C11,C12,C13.C14.DCMPLX PI=3.141593 GH=G P=PP Q=QQ z=zz CIZ=CCMPLX(0.00,Z) QSQ=Q*Q PSQ=P*P ZSC=Z*Z TPQ=2.D0*P*0 PQP=QSQ+TPQ POM=OSQ-TPQ POPSO=PQP*POP POPSC=FCP*PCM Rl =1 Z*GH) +*2*PQPSQ R2 =(Z*GH)**2+PQMSQ R3 =ZSQ+POPSC R4 =ZSC+POMSO PQ12=1. 201-»FSC*0SQ CI C2 C3 ("A c5 C6 C7 C8 C9 CIO C l l C12 C13 =CIZ *Gh+(l .D0-GH)*PSQ+PQP C14 =CIZ *GH+( l.D0-GH)*PSOPQM IF (P.GE.OSQRT(GH)) GO TO 1 Sl= (GH *DLOG(R1 /R2 )+0LCG(R3 =-CIZ*GH+PSQ = -CT»Z*PSQ =PQ12-3.D0*(C! 7. = PQ12-3.D0'*( CI Z =PC12-3.CO*(CIZ =PQ12-3,D0"(CIZ =CIZ *Ch+PQP =C!Z *GH+PQM =CIZ -U.DO/GH *r,H + OS CI **? -! 1.00/GH-1.D0)*PSQ+0SG)*,!<2 +QSQ)**2 *GH+(1.OO-GH)*PSQ*QSQ)**2 = CIZ =CIZ = CIZ   I z -(l.CO/GF-+ PCP + PCM 0 -I.00-1.DC)*PSQ+PQP l.C0)*PS0+P0M >>/2.oo ? ? ? 7 /R4 ) )/ ( 16.D0*PSQ) I . )/I16.00*PSO) .D0)/P)/2.00 —ORE AL(GH * ( C 3 *CDLCG(C7 /C8 -C4 *CCLCG(C9 /CIO 1I/C1 -CREAL((C5 *CDLOG(CU /C12 -C6 *CDL0GIC13/ C14 ))/C2 GO TO 2 1 Sl= (DL0GIR3 /R4 ) -1 .500*0*(GH-1 ? -DRFALHC5 *CCLCG(Cll /C12 ) 7 -C6 *C0L0G(C13 /C14 ))/C2 I/(16.DO*PSO) 2 SH=2.D0*S1*P /Q**3 RETURN END IN EFFECT* 10,EQCOIC,SOURCE,NOLI ST,NOCECK,LOAO,NOPAP IN EFFECT* NAME = Sh , L INECNT = 60 CS* SOURCE STATEMENTS = 45,PROGRAM SIZE = CS* NO CIAGNOSTICS GENERATEO SH 3366 119. D.3b. Programme Two MICHIGAN TERMINAL SYSTEM FORTRAN GC41336) MAIN 01-04-C C C C C C CALCULATFS THE ASSYMPTCTIC VALUE OF THE HOLE POLARIZATION FUNCTION FOR LARGE VALUES OF Q ANO/OR Z BY JUAN ROSTWQROWSKI DIMENSION Z(99),T199) SILICON HOLE MASS RATIO G=.48Q0/.16000 G2=G*G G3=1.D0/G**«3.D0/2.D0) DATA MAX , M AX C/79,7 9/ READ(5,100)(Z(I),1=1,MAX) DC 20 1=1,MAX ZZ=Z(I)*Z(I) IF (ZU)-44.) 10,10,12 NN = 41 DO 11 J=l,40 T(J)=0. GOTO 13 NN=1 DO 30 J=NN,MAXO QQ=Z<J)*Z(J) T( J) = l . 3333333 3300*00*1G3+1. DO) * (G / < Z Z + OQ*C0>G2) •+1.D0/(ZZ+C0*QG)) Tl1)=0. WRITE(7,2C0) Z(I) WRIT = (7,300) (T( J) ,J = 1-,MAXC) FORMAT<uFlO.O) FGRMATU0X,'Z=»,F7.2) F0RMAT<6E13.6) STOP END IN EFFECT* ID,EBCDIC,SOURCE,NOLI ST,NOCECK,LOAD,NONAP 0001 0002 C003 0004 00 05 0006 0007 oo ce 0009 CO 10 0011 0012 0013 00 14 0015 0016 00 17 0018 0019 0020 00 2 i 00 22 00 23 0024 0025 •OPTIONS . . _ * OPT IONS IN EFFECT* NAME = MAIN , LIN! ^STATISTICS* SOURCE STATEMENTS = ^STATISTICS* ND DIAGNOSTICS GENERATED NO ERRORS IN MAIM 10 11 12 13 30 20 ICO 2 GO 3C0 CNT = 25,PROGRAM 60 SIZE = 1762 120. D.3c. Programme Three MICHIGAN TERMINAL SYSTEM FORTRAN G141336) MAIN 01-05-77 C C EHD PROGRAM FOR DOPED S EM! CONCL'CTORS GE AN C SI C COMPUTATION OF CORRELATION ENERGY IN HOLE OAND UNITS C VERSION 2 - BY JUAN ROST V>0 RGWSKI C 0001 IMPLICIT REAL*8(A-F.G,t-,K-M.O-Z) 0002 REAL*8 FC(99),FZ(99),FIMPNHI99),FI MP(99),F(99 I,FNH(99),FZMH(99) 0003 CIM cNSION ZA(99) ,QA(99 ) ,FPSH(99,991 ,FZZ(99 ) 0004 FXTE5MAL FEO,FF1,FE , " 2 , fc3 , F=4 " C005 COMMON OTZ ,GE,PT ,KFNE, KFND ,KFNH,MHH,MRAR,NU 0006 CCMMCN /TWO/DEC(99) ,DE,A c ,A H,AEO ,J 0007 CC W MCN /FCUR/FSH(99) 00 08 C0MW0M/FTVE/HST(20),NST(20),NSNP 0009 RF4D(5,112)T!TLE 0010 READ!5,113)ML,"T ,VHH,MLH.DE, ZVAL.SHO.NU 00 11 READ (5,100) NAX ,NAXQ,NSMP 0012 READ (5,101){H.STtJ),J=l,NSfP) 0013 READ (5 , 100) (\ST(J>,J=l,NSMP> C NST(J) IS THE NUMBER OF 0 AND Z POINTS SPACED BY HST(J) C IN H CL E UNITS AN! 0 ARE NECESSARY FOR CALCULATING THE C SIMPSON INTEGRATION IN VARIABLES Q ANO Z-0014 R F *D (5,100) I COR, I IMP , IMARA-C ICPR=0 CALCULATES CORR EL i T I ON ENERGY WITH AVERAGE S C ]IMP=0 CALCULATES IMPURITY cMERGY V;ITH AVERAGE S C ICC-R=1 CALCULATES CORRELATION ENERGY WITH EXACT S C !IMP=1 CALCULATES IMPURITY ENERGY WITH EXACT S C ]N AR A = 0 CALCULATES WITHOUT CENTRAL CELL CORRECTION C INARA=l CALCULATES M TH CENTRAL CELL CORRECTION 0015 \r- (ICGR.EQ.l) IIMP=1 0016 N A = NST(NSMP) C WE READ NEXT THE VALUES OF THE HOLE POLARIZATION FUNCTION C C0R EACH PAIR (Q,Z). 0017 DO 31 1=1,NA 0018 READ 17,104) FZZ ( I ) 0019 REAR (7,103) (FSH(J1,J=1,NA) C020 00 32 J=1,NA 0021 32 FFSHII,J )=FSH(J) 0022 31 CONTINUE 0023 FFSH(I, 1)=SHO 0024 DO 11 J=1,NAX 0025 ZA(J)=FZZ(J) 0026 11 QA(J)=ZA(J) C C TABULATION OF DATA C 0027 PI=3.14159265AC0 C028 A0=.S29L77D-8 00 29 M P AR =(M L * M T * M T)**(1.C0/3.D0) 0030' NAU1=(2.DC/MTU.CO/KL)/3.C0+11.CO/fHH+1-CO/MLH1/2.00 0031 RE50 (5,102) CD 0032 90 READ (5 , 102, Ei\'C = 99)CH 0023 CE=CD*CH 0034 WRIT C(6,300) 0035 IF(ICOR.EC.O) GOTO 5 00 36 W°!TF(6,111) 00 37 5 CONTINUE 0030 IF(IIMP.FO.O) GOTO 6 00 39 WRITE (6,116) 121. MICHIGAN TERMINAL SYSTFM FORTRAN G141336) MAIN 01-05-77 10: 0040 6 CONTINUE 0041 IF (INARA.FQ.O) GOTO 9 0042 WRITE(6,123> 0043 9 CONTINUE 0044 ViRITEI6,114ITITLE,r'L,MT,yHHf fL H„ 0= ,NU , SHO 0045 WRITFI6, 115)CH,CE,C0 0046 WRIT C (6,107) NAX,N»XQ,NSMP 0047 V>R T T E (6,140) 0048 KRITE (6,135) (HST(J),J=1,NSMP) 0049 V>R I T E (6,141) C050 WRITE (6,136) INST(J),J=1,NSMP) CO 51 WRITE (6,130) 0052 GH=MLH/PFH 0053 GG=(l.DO+GH**(3.00/2.DC)) 0054 KFNH=(3.CO*PI*PI*C F7 GG)**( 1.0073.00)*AO 0055 KFNE=(3.DO*PI*PI+CE/NU)**(1.CO/3.DO)*AO 0056 AE =2.D0*U.RAR*NU/0E/PI /KFNE 0057 KFND=( 3.00*PI*PI*C0/NU)**« I.CO/3.00)*AO 0058 ACo=2.DO*MBAR*NU/DE/PI/KFNC 0059 AH=2.DO*NHH/PI/DE/KFNH 0060 SC0=2.D0/A0**3*DE*CE*KFNH*+5*MAM1/MHH /(2.DO*PI)**3 0061 IF (IC0R.S0.1) EC0=EC0/2.00/P! 0062 EIO=2.O0*0E**2*MAMl*KFNH*CO/PI 0063. IF (IIMP.EQ.l) EI0=EI0/2.DO/PI 0064 GP=M T/ML 0065 WRITE!6,105)KFNM,KFND,KFNE,AF,AEO, AH,ECO,CIO 0066 EKO=3.D0/5.DC*tfAMI*DE**2 0067 EXO=-1.500/PI*0=*MAM1 C C HLCUL4TE DI AL5CTRIC FUNCTION FOR HCLE GRID C C068 DEC(1)= CE 0069 IF (INARA.EO.l) GOTO 7 00 70 " 00 8 J=2,NAX C071 . 8 0F0(J)=DE 0072 GOTO 10 0073 7 DO 10 J=2,NAX 00 74 ONARA=QA(J)*KFNH 0075 CcO(J)=ENARA(QNARA) 0076 10 CONTINUE C C CORRELATION ENERGY r. 0077 F (1)=-2.C0*4 C *(KFNE/KFNH)**2 - FFSH{1,1)*AH 00 78 F0(1)=-2.O0*AEO*(KFND/KFN'H)**2 0079 FNH(1)=F(1) C080 F( 1) =F ( D-FO! 1) C081 IF (IC0R. C0.1) GOTO 1 C C CORRELATION ENERGY WITH AVERAGE S BEGINS C 0082 KRITEI6,109)F(1) 0083 DC 12 1=1,NAX 0084 Z=ZA(I) C C 0 INTEGRAND FOR FIXEC Z C 0085 h=NAXO 122. MICHIGAN TERMINAL SYSTEM FORTRAN G<41336) MAIN 01-05-77 10: 0006 IF {I.EC.l.ANC.IIMP.EQ.OJ N= NAX 00G7 OC 15 J = 2,N 0080 FSH(J)=FFSH(I,J) 0080 C=CA(J) 0090 CQ=Q«*2 C C INTEGRATE ELECTRONIC S C C THE ANGULAR !VT = GRATIONS A 0E DCNE USING THE GAUSS-LEGENORE C INTEGRATTGN FORMULA BY CALLING THE U.B.C. COMPUTER CENTRE'S C FUNCTION EGSUXXlA,S,F) , W H E ' E XX CAN BE AN EVEN NUMBER -LC.16 C A NO DENOTES THE MJMF*ER OF PC I NTS USED. A ANC P APE THE C LIMITS OF INT = GRATION 4N0 F IS AM EXTERNAL LY DECLARED C FUNCTION WHICH EVALUATES Tt- C INTEGRAND. C091 S =«=GAU08!0.,1.,FE) + A = *DE/CE0( J) 0092 SSO=FGAU08 (0. , 1. , FE0)**E0*DF/DSQU J 0093 SS=S*AH*FSH (J)*DE/OEG(J) 0094 I F ( I .EC. 1) FIMPNH(J)=SS 0095 I F ( I -EC. 1) FIMP(J)=SSO 0096 F(J)=-SS + 0L0G11.DO*SS/CO)* GO 0097 F0(J )=-SSO + DLOG(l.CO + S SO/QC)*QQ 0098 FNH(J)=F(J) 0099 P(J)=E(Jl-FOIJ> 0100 - 15 CCNTINUE C C INTEGRAL OVER 0 VARIABLE C 0101 FZ!I)= SMPSN(F,NAXO) 0102 FZNHi i i = SMPSN t FNH» NA XQ) 01C3 F(1)=0.D0 0104 F0(1)=0.D0 0105 FNH(1)=0.D0 CIC6 12 CONTINUE 0107 WPITt" (6,142) CI 08 WRIT c(6,13 5) (F Z ( J ) , J = 1,NAX) 0109 W=UTE (6,143) 0110 WRITE(6,135)(FZNHtJ),J=1,NAX) C C INTEGRAL OVER Z VARIABLE C 0111 ECQ"=ECO*SMPSN(FZ ,NAX) 0112 ECORNF=ECO*SMPS.N (FZNH, NAX ) C C CORRELATION ENERGY WITH AVERAGE S ENDS C 0113 GOTO 2 0114 1 CONTINUE C C CORRELATION ENERGY WITH EXACT S BEGINS C " 0115 FNH ( I)=FNH(1)*2.C0*PI 0116 F t l ) = F ( l ) * 2 . D 0 * ? I 0117 DO 42 1=1,NAX 0118 Z=ZA(!) 0119 00 45 J==2,NAXQ 0120 FSH( J) = FFSH(I,J) 0121 C=CA(J) 0122 C0=0*0 123. MICHIGAN TERMINAL SYSTEM FORTRAN GJ41336) MAIN 01-05-77 10: 0123 FNH( J) = F GAUOS( 0. #1 •»Fc 2 I 0124 45 F(J) =FCAU08(0.,1.,FF4) 0125 FZ (I)=SMPSN<F ,NAXO) 0126 FZ'JHI T ) = SMPSN(FNH,N«X<3) 0127 F(1)=0.D0 0128 42 FM((1)=0.00 0129 WRITE (6,142) 0130 WRTT C(6,135)(FZ(J),J=1,NAXC) 0131 WRITE (6,143) 0132 WRITE(6,13 5)(FZNH(J),J=1,NAXO) C C INTEGRAL CVER Z C 0133 ECOR=CCO*SMPSN(FZ ,NAX) 0134 ECORNH=ECO*SMP SN(F ZNH,NA X) C C CORRELATION ENERGY WITH EXACT S ENCS C 0135 2 CONTINUE 0136 F TMPNH( 1)= 1.OO/DE 0137 IF (IIMP.EQ.l) FIMPNHI 1) = FIMPNH( 1!*2.D0*PI 0138 FI.MPll)=C.OO 0139 Z=0.00 C C IMPURITY ENERGY C 0140 00 20 J--2.NAX 0141 FSH(J)=FFSH(l,J) 0142 C=CA(J) 0143 Cr=Q**2 0144 I i (IIMP.cO . l ) GOTO 3 C NEXT 3 CARDS IF ELECTRONIC POLARIZATION HAS BEEN AVERAGED 0145 F!MPNH(J)=FIVPMH( J)/OS C(J >/ICQ + FIMPNH(J)) 0146 F IMP ( J ) = FIMP ( J )/DEQ( J )/( OQ+F Iv;P( J) ) 0147 F!MP(J)=FIMPNH(J)-FIMP(J) 0148 GOTO 20 0149 3 CONTINUE C NCXT TWO CARDS IF ELECTRONIC POLARIZATION IS EXACT 0150 FIMP ( J)=FGAU10(0.,1. ,FE3)/c;-"C(J) 01 51 F I MPNH { J ) =F G AU 10 (0 -, 1. , FE 1)/ DE C (J ) 0152 20 CONTINUE C C INTEGRATION CVER 0 C 0153 EIMP=-FI0*SMPSM(FIMP,NAXQ) 0154 E!M|>NH = -EI0*SMPSM FIMPNH.NAXQ) C C EXCHANEGE ENERGY C 0155 PH=PHT(GE) 0156 FEX--EXO* ( CE*KFNE*PH + CH*KFNF* PS I (GH ) ) CI 57 F_EXO = EX0*Cn*KFNO*PH 0158 FEXN»- = EEX 0159 EEX=EEX-EEXO C C KINETIC ENERGY C 0160 EKIN =EK0*(KFNE«*2*CE/HBAR <• K FNH* <• 2*CH/MHH) 124. MICHIGAN TERMINAL SYSTEM FORTRAN GC41336) MAIN 0 1 - 0 5 - 7 7 0161 PKINO= CKC* KFN0**2*CD/MBAR 0162 FK!\'NH = CKIN 0163 EKIN=FKIN-EKINO 0164 EKX=EKIMFFX 01 65 ETnT=EK!N+ rEXfECCR 0166 FTCN =CTCT+EIMP 01 67 EKX\'H = EK I NNH + - EXNH 0168 ETOTNH = EKXMH +- CORNH 0169 ETCNN'H = ETGTNHfE IMPNH 01 70 FKFNNI = EKTN:\H-EKIN 0171 EEXN I = F*: XNH-EE X 0172 ECOR,NI = ECCRNF-EC0R •01 73 E I VPK1 =E T f'PNH-E T MP 01 74 ETCTNI=ETCTNH-ETOT 01 75 CTCNN!=ETCNNH-ETCN 0176 EKXNI=EKXNH-EKX 0177 WRITE ( 6 , 3 0 0 ) C178 WR!TE(6,132) 01 79 WRITS(6,106)EK!N,EEX,EC0R,EIMP 01 80 WRITE16,110)ET0T,ETCN ,EKX 0181 kR!TE(6,131) 0182 WRITP|6,106)EKINNH,EEXNH,!=C0RNH ,EIMPNH 01 R3 V.RITE(6,110)ETCTNH,ETCNNH, EKXNF 0184 W<?ITE<6, 137) 0185 W°ITC(6,106)< :KINNI ,ECXNI .ECORNI .FIMPNI 0186 UR!TC(6,H0)eTCTM .ETCNNI , EKXNI 0187 EK!NNK=EKINNH/CH 01 88 EKINN!=EKINNI/CH 01 89 C t ^ T k i — r. V 1 M /CM 0190 FEXNF=EEXNH/CH CI 91 5PXNI=EEXNI/CH 0192 EEX =ESX /CH 01 93 EC9RNH=ECCRNH/CH 0194 ECORNI=ECCRNI/CH CI 95 ECQR = FCGR /CH 01 96 EINPNH=EIKPNH/CH 0197 PIMPNI = E I.VPNI/CH 0198 CIMP =EIMP /CH 0199 FTO"l"NH = ETCT,\,H/CH 02 00 ETOTNI=ETCTNI/CH 0201 = TCT = ETQT /CH 02 02 ETCNNH = ETC.NNH/CH 0203 FTCNNI=CTCNNI/CH 02 04 ETCN =E TCN /CH 0205 EKX.NH=EKXNH/CH 0206 EKXMI = C K X M / C H 02 07 EKX =EKX /CH 02 08 K»IT C (6,110) 02 09 K » n F ( 6 ,122) 02 !.0 WRIT- (6, 106)EKr\,PEX,ECOR,FIMP 0211 WRITE16,!10)ET0T,=TCN , CKX 0212 WRITE(6,121) 0213 W"!TC(6, IC6')5KINNH,EEXKH,FC0RNH, ,EI PPNH 0214 WR!T5=(6, 110)FT0TMH,ETCNNH, EKXNH 0215 V-"ITE(6,H7) 0216 W I T E ( 6 , 106 ) EX INNI , FEX NT , EC'TPN I i IEIMPNl 0217 WRITE(6,110!FT CT NI» ETCNN1,EKXNI 0213 GOTO 90 125. MICHIGAN TERMINAL SYSTEM FORTRAN G(41336) MAIN 0 1 - 0 5 - 7 7 10 : 02 19 02 20 0221 02 22 02 23 02 24 02 25 0226 02 27 02 20 0229 C230 02 31 C232 02 33 0234 02 35 100 FORMAT (f!I 101 101 FORMAT (5C15.2) 102 FC'MAT ( C I O . 2 ) 103 FORMAT!6E13.6) 104 FORMAT(10X,'Z = ',F7.2) 105 FORMA T' (1X ,'KFNH = ' ,013.6.2X, •KFND=«,013.6,2X, 'KFNF=<,013.6,/,IX, 1"6' :=',013.6,2X,'AEG= ,,013.6 I2X, ,1M = ',D13.6,/,1X, ,ECO = ,,D13.6,2X, 2'EI0=>,013.6/) 10 6 FORMAT{/IX , « E ( K I N E T I C ) = ' , D 1 5 . 9 , 3 X , ' E ( E X C H A N E G E ) = ' , 0 1 5 . 9 , 3 X , 1 • J=(CORRELA" ION ) = • , 015. 9, 3X, ' E( IMPUR ITY ) = » , CI 5. 9/) • 1C7 FORMAT ( I X , • N A X = ' , 1 3 , 1 0 X , 1 N A X C = ' , ! 3 . 1 0 X , « N S M P = • , 1 3 / ) 103 FORMAT (312) 109 F O R M A T ( I X , « F ( 1 ) , FNDP' ,012 .5/) 110 FORMAT!IX,•E(TOTAL> = ' ,C15.9,3X , •ET CN(I MP ) = *, 1 0 1 5 . 9 , 3 X , « E ( K I N t E X ) = « , 0 1 5 . 9 / ) 111 FORMAT (/,10X,'YCU HAVE USED EXACT EL cCTKONIC POLARIZATION TO CAL 1CULATE T H c CORRELATION ENERGY'/) 112 FCPMAT(A4) 113 F0RMAT(7C10.0,!2) 114 F 0 R M A T ( / A i 5 , 3 X , ' M L = « , 0 1 0 . 3 , 3 X , ' M T = • , D 1 0 . 3 , 3 X . « P H H = « , 0 1 0 . 3 , 3 X , 1 • MLH = * , 0 1 0 . 3,3X, '0E=' , 0 1 0 . 3 , 3X , 'N'J = ' , I i , 3X , « SHO= ' , 0 1 0 . 3 / ) 115 F O R M A T ( i O X f « C H = « , Q l l . 3 , l C X , ' C E = , , D 1 1 . 3 , l C X , , C D = ' , C l l - 3 / ) 116 FORMAT (/.lOX.'YCU HAV? USEO EXACT ELECTRONIC POLARIZATION TO CAL 1CULATE THE IMPURITY ENERGY'/) ARE THE CONTRIBUTION TO TOTAL ENERGY, E(ND,0 r n MT o T D I IT T rm CONTRIBUTION TO NO 0236 117 F0RMAT(/5X,'FOLLOWING 1 ) /NH 1,/ ) 0237 118 FORMAT (/IX,• 1XXX",/) ~ - -» •»-»•« <- o r, • • » -r / / C v % r-.n* i r<\.i r \ir> A O C TuC \j£ OO l i l f j ^ i ' i M : 1 / J A I i J L L I . " 4>io iM-1H,NH)/MH',/» 0239 122 FORMAT(/5X,•FOLLOWING ARE THE 1,(E(ND+NH,MH)-E(NO ,0)) /NH',/) 0240 123 FORMAT(/10X , 'CON TR I BUT ION S INCLUDING 0241 130 FORMAT (/*IF NAXCNAXO •ARE NOT ODD THEN THIS 0242 131 FORMATI/5X,'FOLLOWING 1H.NH) ',/) 0243 132 FORMATf/5X,»FOLLOWING ARE THE 1,(E(ND+NH ,NH)-c(ND,0)) «,/) 0244 135 FORMAT!101IX ,012.5)) 0245 136 FORMAT (10( 1 X , U 2 ) ) 0246 137 FORMATj/5X , •F3LLCWING ARE THE 1) ',/) 0247 140 FCRMAT (IX,'LIST OF FST"S'/) 0248 141 FORMAT(/IX, "LIST OF NST"S'/) C249 142 F0RMAT(/1X,'LIST OF FZ"S'/) 0250 143 FORMAT(/IX, 'LIST OF FZNH"S«/) 0251 300 FORMAT('l') 0252 99 STOP 02 53 END *QPTTONS IN EFFECT* ! D , E BCD! f,, SOUP CE , NOL I S T, NO CE CK , LOAC, N O P AP *OPTIONS IN EFFECT* KAMF = MAIN • LINECNT - 60 •STATISTICS* SCURCF STATEMENTS = 253,PROGRAM SIZE = •STATISTICS* NO CIAGNOSTICS GENERATED ERRORS IN MAIN TO T O T A L ENEP.GYs EINO+N DIFFERENCE ENERGIES CENTRAL CELL CORRECT ICNS'/) AR F NOT C.NE OF THE NST"S AND IF THE NST"S OUTPUT IS GARBAGE'/) ARE THE CONTRIBUTION TO TOTAL ENERGY, E(NDfN CONTRIBUTION TO DIFFERENCE ENERGIES CONTRIBUTION TO TOTAL ENERGY, E(ND,0 5 5 2 2 0 126. MICHIGAN TERMINAL SYSTEM FORTRAN G I 4 1 3 3 6 ) 01 -0 5-NO 00 01 f FUNCTION FE IX) L C r THESE FUNCTIONS ARE REQUIRED FOR AVERAGE S 0002 C IMPLICIT RF AL *3 I A-F.G ,H,K-M,0-T,W,Y,Z) 0003 COMMON P, T , GE..P! ,KFNE, KFNO . K FNH, MHH , M BAR , NU 00 04 COMMON /ThO/DEC(99) iDE,AE »A F , A E 0 ,.J 0005 KFNC =KFNE 0006 GCTO 1 0007 ENTRY FEO (X) 0008 KFNC=KFNC 0009 1 CONTINUE 00 10 G3 = GE*M I. DO/3.00) 0011 Q =DSCRTi(G3*X)**2 + (1 .DC-X*X)/G3)+P CO 12 Q=0/KFNC*KFNH C013 CC=0*0 0014 Z=T*{KFNH/KFNC)**2*MBAR/MHH 00 15 zz=z*z 00 5 6 02P=2.nO*Q*QC CO 17 O2M=2-DC*C-C0 0013 C2M2=C2M*C2M CO 19 IFIZ.LT. 1.D-51GGT0 10 00 20 Y=0AT*N{C2M/Z)+0ATAN(02P/Z) 0021 GCTO 21 00 22 10 Y = P.I 00 23 IF (C2M2.GT.1.0-10) GOTO 21 0024 FE =1.0C*<KFNC/KFNH)**2 0025 FE0=1.D0*(KFNC/KFNH)**2 00 76 .RFTIIRM 0027 21 *• = ( l.DO - lCO-ZZ/CC)/4.CO)* CLOG( (ZZ+Q2M21/IZZ+C2P 00 28 S =1.D0-.5C0*(K+Z*Y)/Q C029 S =S *(KFNC/KFNH)**2 00 30 FE =S 00 31 FC0=S 0032 RETURN 00 33 END •PORTIONS IN EFFECT* ID.ERCDICSOURCE.NCL I ST, NO CECK , LO AD, NO V. AP *0°TIONS IN EFFECT* NAM= = FE » LI NEC NT = 60 *STATISTT CS* SCURCE STATEMENTS = 33,PROGRAM SIZE = * ST AT I ST ICS * NO DIAGNOSTICS GENERATED ERRORS IN FE 1178 127. MICHIGAN TERMINAL SYSTEM FORTRAN G(41336) FF.l 01-05-77 10 0001 00 02 0003 00 C4 00 05 00 06 00 07 •00 03 00 09 00 10 0011 C012 0013 0014 0015 00 16 0017 o  ie CO 19 00 20 0021 C C r C C C FUNCTION FE1IX) THESE FUNCTIONS ARE RECUIREO-FCR EXACT S CALLED TO CALCULATE TOTAL EIMP COMMON /THREE/XO,JJ,JJ1 EXTERNAL FEOS.FES JJ1 = C JJ=1 XO = X FE1=4.*FGAU14(0.,1.5708,FES) RFTURN ENTRY FE2 1X! CALLED TO CALCULATE TOTAL ECOR JJ1=1 J J = 1 XC=X FE2=4.*FGAU06<C.,1.5708,FECS) RETURN ENTRY FEi(X) CALLED TO CALCULATE EIMP DIFFERENCE JJ1 = 0 JJ=2 xo=x FF3=4.*FGAU14( 0. , 1.5708,FES) RETURN ENTRY FE4(X) CALLED TO CALCULATE ECOR DIFFERENCE JJ1=1 J J = 2 XC=X FE4=4.*FGAU06<0.,1.5708,FEOS) RETURN END ^OPTIONS IN EFFECT* ID,EECDIC,SOURCE,NOLIST,NODECK,LOAD,.NOMAP * OPT IONS IN F FFEC T* NAME = F E1 , LI NEC NT = 60 *STATISTTCS* SOURCE STATEMENTS = 27,PROGRAM SIZE = • STATISTICS* NO DIAGNOSTICS GcNERATED NO ERRORS IN FE1 00 22 0023 0024 00 25 C026 00 2T 904 128. MICHIGAN TERMINAL SYSTEM FCTRAN 0(41336) FES 01-05-77 10: 0001 FUNCTION FFS(U) C C THESE FUNCTIONS CALCULATF THE INTE GRANOS C FOR EXACT S CALCUL4 TIONS C 0002 IMPLICIT RT-1L *8(A-F,G,H.K-M,0-T , W , Y,Z) G003 CCVnN P,T ,GE, PI ,KFNC, KFNC ,KFNW,MHH,MBAR,NU C004 COMMON /TWO/0-C(59) , HE ,AE. ,Ar-,AE0 , J 0005 CCMMCN / T r P . E E / X , J J , J J 1 00C6 COMMCN /FOUR/^SHfgg) 0007 DIMENSION 0 T13),S(3) ,SES(2),SE0S(2) 0008 GC TC 1 CO09 ENTRY FEOS(U) 0010 1 CU=U 0011 V=DCCS(CU) .0012 G3=GE**( 1.D0/3.00) 0013 V2=V**2 0014 X2=X**2 0015 SX=1.D0-X2 0016 SV=1.D0-V2 C017 G32=G3**2 0018 pp=P**2 0019 0T(1)=0SCRT(G32*X2rSX/G3)*P 0020 Q T(2)=DSCR T(G32-*SX*SW • ( X2 + SX* V2 ) / G 3 >* P 0 0 2 1 Q T ( 3 ) = D S C R T ( G 3 2 * S X * V 2 * ( X 2 T S X * S V ) /G 3 ) * P 0022 DC 3C J 1 = 1 , J J CO 22 DC 15 1 = 1 , 3 0024 KFNC = KF NC 0025 I F ( J ) . F O . l ) KFNC. = KFNF 00 26 C=QT(I) 0027 G=0/KFNC*KFNH 0028 00=0**2 CO 29 Z=T*(KFNH/KFNC)**2*M2AR/MHH C030 ZZ=Z**2 0031 C2P=2.00*0+00 C032 Q2M=2.00*C-CC 0033 02M2 = 02f*«'2 0034 !F(Z.LT-l.n-5)G0T0 10 0035 Y = DATAN(C2M/Z)*0AT AN(02P/Z ) C036 GCTO 21 0037 10 Y=PI 0038 IF ( Q 2 M 2.GT.1.D-10) GOTO 21 0 0 3 9 sm=i.oo 00 40 GOTO 22 0041 21 V=('l.00-(CQ-ZZ/QC)/4.0C)*CLOG( ( Z Z + 02M 2 ) / ( Z Z+02 P*0 2P ) ) 0042 S(!)=1.D0-.5D0*(*+Z*Y)/Q 0043 22 St T)=S( I )*(KF.\'C/KFNH)**2 C044 A = A E C 0045 IF ( J l . E C . l ) A=AF C0'>6 S ( I ) = S ( I ) * A *CE/DEC( J )/NU 00 47 15 CONTINUE 0048 SFFS=2.00*(S ( 1)»S(2)+S«3)I C049 IF ( J l . E C . l ) Sr-ES=SFES + SF kFSH( J)*OE/DEC( J) 0050 SES( J l > = SFr-S/( o : >*SFES) 0051 IF ( J J 1 . E C . 1 ) SES(J1)=-SFES*DLCG(I.C0+SFES/PP)*PP 00 52 30 CONTINUE 00 53 IF ( J J - 1 ) 2,2,3 0054 2 F=S=SES(1) 129. MICHIGAN TERMINAL SYSTEM FORTRAN GI41336) FES 0 1 - 0 5 -0055 IF ( J J 1 . E C . I I FECS=FES 0056 R C TURN 0057 3 FES=S C S (D-SE.S (2 ) 0050 IF U J 1 . E Q . 1 ) FECS=FES 0059 R C TURN 0060 END •OPTIONS IN EFFECT* ID,EBCDIC,SOURCE.NOLI ST,NO DECK,L0AO,NOfAP • OPTIONS IN EFFECT* N A w £ = FES , L INEC NT = 60 • STAT I ST ICS * SOURCE STATEMENTS = 6C,PROGRAM S IZE = 2076 • ST AT I ST IC S* NO CIAGNCSTICS GENERATED NO ERRORS IN FES 130. MICHIGAN TERMINAL SYSTEM FORTRAN G( A1336I PSI 01-05-77 0001 00 0 2 CQ03 0004 0005 0006 0007 • 0008 0009 0010 0011 0012 c c c c C013 0014 •OPTIONS IN *QOTT3NS IN • STATIST ICS* # STAT I ST ICS* NO ERRORS IM PS I REAL FUNCTION PSI*8(GH) EQUIVALENT T0 CN EQUATION 26 IF DIVIOED BY (11G H * * (3/2) )**(1/3) IVPLICIT Pf?AL«8 t A-J.L-Z) G12=0SQRT(GH) G32= C-l 2**3 G2=GH**2 SUM=C.DO 00 1 K=1,99,2 EX=K/?.DO NUMPT=2.00*(l.00-GH**EX) S=NUMRT/K**2 1 SUM=SUV+S PS 1 = 1.00/11.00+G32)*(-3.00/16.DO*(l.DO-GH)**2*DLCG((1.DO+G12)/ 1 (1.D0-G12) ) + (G2*-3.D0*G 32+3 .DC*G12+1.00 )/4.C0-r-3.00/16.00* (l.DO-2*SUM ) RETURN END EFFECT* ID,EBCDIC,SOURCE,NOLI ST,NOCECK.LOAD,NO MAP CFF=CT* N A '* E = PSI , L I NEC NT = 60 SOURCE STATEMENTS = 14,PROGRAM SIZE = 796 NI DIAGNOSTICS GENERATED 131. MICHIGAN TERMINAL SYSTEM FORTRAN G{41336) PHI 01-05 0001 RFAL FUNCTION PHI*S(GE) C C EQUIVALENT TO CN EQUATION 24 C C002 IMPLICIT RE AL* 8 IA-Z) C003 G6=GE**t1.00/6.DO) 0004 IF (GE.LT.1.D0I GO TO 1 C005 IF (GE. CC.1.00) GOTO 2 0006 PHI = CARSIN'(DSCRT(1.00-1.D0/GE) )/DSORT(GE-1.DO)*G6 00C7 RETURN 0008 1 PHI=0ARSIN(DSQRT(1.00-GE))/DSORT(1.DO-GE)*G6 C009 RE TURN C010 2 PHI=1.D0 0011 RETURN CO 12 END • OPTIONS IN EFFECT* ID,EOC01C,SOURCE,NOLIST,NOCECK,LOAD, NOfAP • OPTIONS IN EFFECT* N A M E = PHI , L INECNT = 60 • STATISTICS* SOUR.CE STATEMENTS = 12, PROGRAM SIZE = 680 *STATIST1CS* .NO DIAGNOSTICS GENERATEC NO ERRORS IN PHI 132. MICHIGAN TERMINAL SYSTEM FORTRAN GI41336) SMPSN 01-05-77 10 NO 0001 REAL FUNCTION SMPSN*8(F , M A X) 00C2 IMPLICIT P. E A L * 8 { A-H, K , L , C-Z J 0003 DIMENSION F(99) 0004 CCMMON/FIVF/HST<20),NST{20),NSfP 0005 SUM=0.00 C006 N" = l 00C7 IMAX=0 0008 DO 24 J=1,NSMP 0009 24 IF (NST(J).EC.MAX) IMAX=J C010 DO 26 J=1,IMAX 0011 NN = .NST(J) 0012 NM=NN-2 0013 SM=0.00 0014 DO 28 JJ=NM,MM,2 0015 28 SM=SMfF! J J)+4-00 *F ( J J+1)+F IJ J <• 2) 0016 SM=SM*HST<Jl/3.00 0017 SUM=SUM+SM 0018 26 N" = NN 0019 SMPSN=SUM 0020 RETURN 00 21 ENO * CPT IONS IN EFFECT* 10, EBCD IC , SOURCE, NOL I ST, NODECK., LOAD, NO MAP •(•OPTIONS IN EFFECT* NAME = SMPSN , LI NEC NT = 60 ^STATISTICS* SOURCE STATEMENTS = 21,PROGRAM SIZE = *STATISTICS* m DIAGNOSTICS GENERATED ER.ROPS IN SMPSN 722 133. MICHlG<\N TCRMINAL SYSTEM FORTRAN G141336) ENARA 01-05-77 10 0001 CQ02 REAL FUNCTION EN AR A* B ( Q ) IMPLICIT REAL*OlA-H.C-Z) CIELECTRIC FLINT I CN OF KARA 0003 C ATA AAN, ALN , ABN.AGN, DE/1.. 17500, .757200, . 3123D0, .2C44D + 1 ,11.4D0/ 0004 FNARA = A AN*G**2 / ( ALN**? fQ**21 +( 1. CO-A A N ) -*C*ft 2/ ( ABN **2+Q** 2) 1 «- l.CC/06*AGN**2/( AGN**2 •1-0**21 0005 ENARA=1.CO/FNARA 00C6 RCTURN 0007 END • OPTIONS IN EFFECT* IC,EBCD IC,SOURCE,NCLI ST,NOCECK,LOAO,NOMAP • OPTIONS IN EFFECT* NAME = ENARA , LINECNT = 60 •STATISTICS* SOURCE STATEMENTS = 7,PROGRAM SIZE = 454 •STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN ENARA NO STATEMENTS FLAGGEC IN THE ABOVE COMPILATIONS. NAME NUMBFR OF FRRQRS/WARNINGS SEVERITY MAI N 0 0 FE1 0 0 FFS 0 0 FE 0 0 PSI 0 0 PHI 0 0 SfPSfJ 0 0 ENARA 0 0 EXECUTION TFRMINATED 134. BJBLIOGRAPHX 1. G. Wannier, Phys. Rev. 52, 191 (19371; G. Dresselhaus, Phys. Chem. Solids ]_, 14 0956); G.G. MacFarlane, T.P. McLean, J.E. Quarrington and V. Roberts, Phys. Rev. l_n, 1245 (1958). 2. J.R. Haynes, M. Lax, W.F. Flood, J. Phys. Chem. Solids 8, 392 (1959); "Proc. International Conf. on Semiconductor Physcis, Prague 1960", p.423, Czechocoslavak Acad. Sci., Prague (1961). 3. L.V. Keldysh, "IX International Conference of the Physics of Semicon-ductors. 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