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A study of the condensed phase in phosphorus doped silicon Halliwell, Robin Ernest 1973

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A STUDY OF THE CONDENSED PHASE IN PHOSPHORUS DOPED SILICON by ROBIN ERNEST HALLIWELL B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1967 M.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In the Department of Physics We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department The University of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT Photoluminescent studies give evidence that a condensed phase of electrons and holes can be formed i n s i l i c o n containing phos-phorus donor concentrations from 9.0 x 1 0 1 5 cm""3 to 4.3 x l O 1 ^ cm - 3. When the impurity concentration i s l e s s than that required f o r d e r e a l -i z a t i o n of the donor el e c t r o n s , the condensed phase takes the form of *! ft 3 electron-hole drops with c a r r i e r concentrations of 3.0 x 10 ° cm - . For Impurity concentrations greater than that required for d e r e a l i z a -t i o n of the donor electrons the experimental observations are i n t e r -preted as evidence for a condensed phase i n v o l v i n g only holes. Heat treatment of s i l i c o n doped with phosphorus i s found to produce marked changes i n both the photoluminescence and the e l e c t r o n paramagnetic resonance for samples containing impurity concentrations 2.2 x 1 0 1 8 cm-3^ N D < 1.3 x 1 0 1 9 cm"3. These changes are interpreted i n terms of the production of n e u t r a l traps by r e s i d u a l impurities such as carbon and oxygen. The e f f e c t of t h i s heat treatment on the photo-luminescence lends support to the hypothesis of a condensed hole phase for high impurity concentrations. TABLE OF CONTENTS Page Abstract i Table of Contents . . i l List of Tables v List of Figures v i Acknowledgements . v i i i Chapter 1 INTRODUCTION 1.1 General Introduction 1 1.2 Purpose and Outline of this Thesis 2 Chapter 2 PHOTOLUMINESCENCE OF INTRINSIC SILICON 'AND GERMANIUM 2.1 Introduction 3 2.2 Free Excitons 4 2.3 Excitonic Molecule 8 2.4 Bose-Einstein Condensation 10 2.5 Electron-Hole Drops (EHD) 10 2.6 Experimental Evidence for Biexcitdns and Electron-Hole Drops 16 Chapter 3 EXPERIMENTAL 3.1 Sample Preparation 19 3.2 Photoluminescent Spectrometer 20 3.3 Electron Paramagnetic Resonance Spectrometer 22 3.4 Temperature Measurements 22 i i i Page Chapter 4 CONCENTRATION DEPENDENCE OF THE PHOTO-LUMINESCENCE OF Si:P 4.1 Introduction 29 4.2 Concentration Dependence of the Photoluminescent Spectra of Si:P at 2K 30 4.3 Low Concentration (9 x 10 1 5 cm - 3 1 N D < 3.6 x 10 1 7 cm"3) 30 4.4 T r a n s i t i o n Range (1.1 x 10 1 8 cm - 3 1 N D < 5.5 x 10 1 8 cm"3) 34 4.5 High Concentration (N^ > I O 2 9 cm - 3) 36 Chapter 5 TEMPERATURE DEPENDENCE OF THE PHOTO- , LUMINESCENCE OF Si:P 5.1 Introduction 37 5.2 Temperature Dependence of the Photoluminescence of Si:P 38 5.3 Low Concentration 1) 9 x 10 1 5 cm"3 38 2) 3.6 x 10 1 7 cm"3 41 ! i 5.4 T r a n s i t i o n Range (1.1 x 10 1 8 cm"3 < N D < 5.5 x 10 1 9 cm - 3) 41 5.5 High Concentration 1) 1.3 x 10 1 9 cm - 3 43 2) 4.3 x 10 1 9 cm - 3 43 Chapter 6 DISCUSSION OF RESULTS 6.1 Screening of Excitons \ 44 6.2 In t e r p r e t a t i o n of Spectra 48 i v Page 6.3 Low Concentration (9 x 10 1 5 cm - 3 < N D < 1.1 x 10 1 8 cm - 3) 49 6.4 T r a n s i t i o n Range (2.2 x 10 1 8 cm"3 < N D < 5.5 x 10 1 8 cm - 3) 54 6.5 High Concentration (N^ > 1.3 x 10 1 9 cm - 3) ....... 56 6.6 I n t e n s i t y of Photoluminescence 58 6.7 Temperature Dependence of Photoluminescence 60 Chapter 7 EFFECT OF HEAT TREATMENT ON Si;P 7.1 Introduction 63 7.2 Photoluminescence from Heat Treated Si:P 64 7.3 EPR of Heat Treated Si:P 67 7.4 Discussion of Results ...; 72 Chapter 8 CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY 8.1 Conclusions 78 8.2 Suggestions f o r Further Study 79 Appendix A Correction f o r Instrumental Broadening 81 Appendix B C a l c u l a t i o n of Lineshape f o r Electron-Hole Drops 84 Bibliography 88 V LIST OF TABLES Table Page 2.1 I n t r i n s i c Recombination Radiation Bands of S i l i c o n at 26K 7 2.2 Band Parameters for S i l i c o n 12 2.3 T h e o r e t i c a l E q u i l i b r i u m Density and Ground State Energy of EHD 14 2.4 Experimental Values of Equ i l i b r i u m Density and Ground State Energy of EHD v i LIST OF FIGURES Figure Page 2.1 Band Structure of I n t r i n s i c S i l i c o n at 2K 6 2.2 Ground State of EHD i n S i l i c o n 14 3.1 O p t i c a l Configuration of Photoluminescent Spectrometer . . . 21 3.2 D e t a i l s of Temperature Control System 24 3.3 Temperature C a l i b r a t i o n Procedure 26 3.4 Temperature Difference Between Sample and Sample Mount . . . 28 4.1 Photoluminescence for Si:P with N D = 9 x 101 5 cm-3 at 2K . 31 4.2 Photoluminescence for Si:P with = 3.6 x 1 01 7 cm- 3 at 2K 32 4.3 Concentration Dependence of the Photoluminescence of Si:P at 2K 35 5.1 Temperature Dependence of the Photoluminescence f o r i N D = 9 x 10 1 5 cm-3 ' 39 5.2 Temperature Dependence of the Photoluminescence f o r N D = 3.6 x 10 1 7 cm-3 40 5.3 Temperature Dependence of the Photoluminescene f o r N D - 1.3 x 10 1 9 cm - 3 42 6.1 Concentration Dependence of the Threshold Energy E 50 v i i Figure Page 6.2 Half-Width of TO Assisted Peak vs. Impurity Concentration 51 6.3 Comparison of Calculated and Experimental Line Shapes at 2K 52 6.4 I n t e n s i t y of TO Assisted Peak vs. Impurity Concentration 59. 7.1 Photoluminescence vs. Impurity Concentration f o r Heat-Treated Si:P at 13K 65 7.2 Photoluminescence vs. Temperature for Heat-Treated Si:P with N D = 3.0 x 1 0 1 8 cm - 3 • 66 7.3 EPR Spectra f o r = 2.0 x 1 0 1 8 cm - 3 69 7 .4 EPR Spectra f o r N = 6.2 x 1 0 1 8 cm-3 70 v i i i ACKNOWLEDGEMENTS I wish to express my thanks to my supervisor Dr. R.R. Parsons f o r h i s support, guidance and friendship during these studies. To Dr. R. B a r r i e , I am g r a t e f u l f o r h i s c r i t i c i s m , advice, and many invaluable discussions. I would l i k e to thank Dr. J . Marko for providing .the o r i g i n a l impetus f o r t h i s work and guiding i t through the ea r l y stages, also Dr. J . Quirt f o r advice and suggestions on some of the experimental aspects. A s p e c i a l thanks must go to Dr. C.F. Schwerdtfeger f o r h i s support and encouragement during my ea r l y years as a graduate student. G r a t e f u l acknowledgement i s given to the National Research Council and the H.R. MacMillan family f o r f i n a n c i a l support. This research was supported by grants to Dr. C.F. Schwerdtfeger CNRC 67-2228) and Dr. R.R. Parsons (NRC 67-6714). F i n a l l y I wish to thank my wife Janet, without whose, help t h i s t h e s i s may never have been completed. - 1 -CHAPTER 1  INTRODUCTION 1.1 General Introduction A s u f f i c i e n t l y high density of photocreated electrons and holes i n a semiconductor can condense to form what i s known as an (1 2) "electron-hole d r o p l e t " ' As a r e s u l t of t h i s condensation some regions of the c r y s t a l contain two plasmas, one due to the electrons moving i n a uniform p o s i t i v e background and one, to the holes moving i n a uniform negative background. These two plasmas are e l e c t r i c a l l y n e u t r a l and can be treated as being non-interacting. These regions are macroscopic and i n quasi-thermal equilibrium with the l a t t i c e . Only semiconductors having an i n d i r e c t band gap, such as s i l i c o n and germanium, are expected to have free c a r r i e r l i f e t i m e s long enough f o r such a condensation to occur. Consequently there has been considerable experimental and t h e o r e t i c a l work done on these two m a t e r i a l s , both to e s t a b l i s h the existence of such a condensate, and (3) to study i t s p r o p e r t i e s . Some work has been done on the d i r e c t band (4 5) gap semiconductor GaAs but the r e s u l t s are inconclusive ' . To date, work i n t h i s f i e l d has been r e s t r i c t e d to i n t r i n s i c materials or to mat e r i a l s with impurity concentrations low enough that impurity—impurity i n t e r a c t i o n s may be neglected. One i n t e r e s t i n g feature that has come from these studies i s that the e q u i l i b r i u m density of the condensed phase i s approximately the same as the impurity concentration required for the d e r e a l i z a t i o n of the impurity s t a t e s . This immediately r a i s e s the question of what - 2 -happens to the condensed phase when the impurity concentration Is close to or above t h i s c r i t i c a l concentration f o r d e r e a l i z a t i o n . 1.2 Purpose and Outline of t h i s Thesis The main purpose of t h i s thesis i s to in v e s t i g a t e the photo-luminescence of s i l i c o n heavily doped with phosphorus. In p a r t i c u l a r the nature of the condensed phase i s studied as the material undergoes the t r a n s i t i o n from "semiconducting" to " m e t a l l i c " behavior. This i s co r r e l a t e d with a subsidiary study of the e f f e c t s of heat-treatment on the photoluminescence and el e c t r o n paramagnetic resonance. The thesis i s divided into seven chapters. The t h e o r e t i c a l and experimental i n v e s t i g a t i o n s of excitons and various condensed phases i n i n t r i n s i c s i l i c o n and germanium are discussed i n Chapter 2. Chapter 3 o u t l i n e s the experimental procedures and precautions. The photolumin-escent data as functions of impurity concentration and temperature are presented i n Chapters 4 and 5, these r e s u l t s are then analysed and d i s -cussed i n Chapter 6. A study of the e f f e c t s of heat-treatment on the photoluminescence and electron paramagnetic resonance i s presented i n Chapter 7 and an attempt i s made to c o r r e l a t e t h i s with the r e s u l t s of the previous chapters. A f i n a l chapter provides a summary of the r e s u l t s and conclusions. - 3 -CHAPTER 2 PHOTOLUMINESCENCE OF INTRINSIC SILICON AND GERMANIUM 2.1 Introduction In recent years photoluminescence has developed into a power-f u l t o o l f o r the i n v e s t i g a t i o n of semiconductors. When a c r y s t a l absorbs a photon with energy greater than the band gap energy an e l e c -tron i s ejected i n t o the conduction band, leaving a hole i n the valence band. This photo-created electron and hole r a p i d l y lose energy and f a l l to the bottom of the conduction band and the top of the valence band r e s p e c t i v e l y , where they recombine with the emission of a photon. Di r e c t Band Gap Indirect Band Gap - 4 -This recombination may be d i r e c t band to band or i t may involve an electron-hole complex, such as an exciton or the more r e c e n t l y observed electron-hole d r o p l e t . Before i t i s p o s s i b l e to understand the e f f e c t s of s p e c i f i c i m p u r i t i e s on the properties of a material i t i s f i r s t necessary to understand i t s i n t r i n s i c p r o p e r t i e s . Because s i l i c o n i s r e l a t i v e l y easy to prepare i n a pure state, and to dope by prescribed amounts, i t i s a nearly i d e a l m a t e r i a l with which to work. This chapter w i l l present the two t h e o r e t i c a l models f o r free excitons and the various exciton condensate models that have been pro-posed to explain the recent photoluminescent data from i n t r i n s i c s i l i c o n and germanium. 2.2 Free Excitons Excitons were f i r s t described by Frenkel i n 1931^^ as a r e s u l t of i n v e s t i g a t i o n s into the absorption processes that take place i n i n s u l a t i n g s o l i d s . He proposed that t r i a l excited states of the c r y s t a l could be constructed from the ground and excited states of i s o -l a t e d atoms. In t h i s model, the ground state of the c r y s t a l i s w r i t t e n ! as an antisymmetrized product of non-overlapping orthonormal one-e l e c t r o n wave functions. In an analogous manner an excited state can be constructed by assuming that an atom at one s i t e i s i n an excited s t a t e , and s u b s t i t u t i n g the appropriate one-electron excited state wave function i n t o the ground s t a t e wave function. The true stationary excited states of the c r y s t a l can be expressed as l i n e a r combinations of these l o c a l i z e d e x c i t a t i o n s , each having a wave vector K i n the r e c i p r o c a l space. - 5 -The Frenkel model, which i s a c t u a l l y a t i g h t binding model, (7 8) .has been used extensively i n the study of molecular c r y s t a l s ' where e x c i t a t i o n only involves a rearrangement of atomic o r b i t a l s w i t h i n the molecule and does not appreciably increase the overlap between mole-cu l e s . The model has been found to have only l i m i t e d usefulness i n rare (9) gas s o l i d s despite the r e l a t i v e l y t i g h t binding i n t h e i r ground stat e s , and i n general i s not a p p l i c a b l e to most other c r y s t a l l i n e i n s u l a t o r s . Of more i n t e r e s t i n the present work on s i l i c o n i s the approach adopted by Wannier He viewed the exciton as a conduction band e l e c -tr o n bound to a valence band hole, although p o s s i b l y with considerable separation. The p a i r would move through the l a t t i c e , t h e i r centre of mass having t o t a l wave vector K. The Frenkel and Wannier models are r e l a t e d , j u s t representing the two extremes of exciton radius r : f o r the Frenkel model, r < a, the interatomic distance; f o r the Wannier model, r » a. The Hamiltonian f o r the Wannier exciton i s that f o r a two p a r t i c l e system, i n t e r a c t i n g v i a the coulomb i n t e r a c t i o n modified by the d i e l e c t r i c constant of the medium: "&2V2 T i 2 V 2 2 K = £. k - £ 2 (2.2-11 * * icr 2me 2^ This has hydrogenic energy l e v e l s which can be written as: E (1) m-Ji£— f K 2 \ (2.2-2) n 2n2K2n2 2(m* + n^) * * where and m^ are the e f f e c t i v e masses of the el e c t r o n and hole - 6 -E i • C o n d u c t i o n band i E n = 1.1698 eV 1 \ r Va lence bands > (0,0,0) ! (1,0,0) FIGURE 2 . 1 Band Structure of Intrinsic Silicon at 2 K . Table 2.1 I n t r i n s i c Recombination Radiation Bands of S i l i c o n at 26K (12,14) Threshold Phonon Phonon Rel a t i v e Energy (eV) Energy (meV) Assignment Intensity 1.1550 ^0 No phonon ^0.004 1.1365 18.3 TA 0.035 1.0998 55.3 LO -1.0978 57.3 TO 1.00 1.074 80.8 TO + I V a 0.016 1.051 103.8 TO + IV b M3.008 1.0315 122.3 TO + 0 F 0.07 1.013 142 TO + 0 r + I V 3 -vO.0025 0.968 187 TO + 20 F ^0.01 TA: Transverse a c o u s t i c a l momentum conserving TO: Transverse o p t i c a l momentum conserving i v a 'k; Phonons selected f o r i n t e r v a l l e y s c a t t e r i n g 0 F: Centre of reduced zone - zero wave vector - 8 -r e s p e c t i v e l y , K i s the d i e l e c t r i c constant, and y i s the reduced mass of the system. Since the wave vector K can range over the f i r s t b r i l l o u i n zone there are exciton "bands" rather than s t a t e s . The energy required to d i s s o c i a t e the e l e c t r o n and hole when the exciton i s i n i t s ground st a t e , more c o r r e c t l y , when the c r y s t a l i s i n i t s lowest excited s t a t e , i s c a l l e d the exciton "rydberg" and i s given by: G= yel* *" (2.2-3) 2 h 2 K 2 Excitons were f i r s t observed i n s i l i c o n by Macfarlane et a l . while doing absorption studies at the band edge. They observed several knees i n the absorption c o e f f i c i e n t of the band edge which were e x p l a i n -ed, i n terms of Wannier excitons with an exciton rydberg of 10 meV. More (12) r e c e n t l y Shaklee and Nahory , using wave length d e r i v a t i v e techniques, have r e f i n e d t h i s value to G = 14.7 meV by d i r e c t observation of the n - 2 exciton excited state and the onset of the exciton continuum. The photoluminescence produced by the recombination of free excitons with the simultaneous emission of a photon and one or more pho-(13) nons was f i r s t observed i n s i l i c o n by Haynes et a l . i n 1960 . Because s i l i c o n has an i n d i r e c t band gap (Figure 2.1) the recombination must be phonon a s s i s t e d , with at l e a s t one phonon required i n order that momentum be conserved. Table 2.1 l i s t s the energies of the i n t r i n s i c recombina-t i o n r a d i a t i o n bands and the phonons involved with each. 2.3 E x c i t o n i c Molecule Lampert^ 1" 5^, i n 1958,'first suggested that i t should be p o s s i ~ b l e f o r two excitons to bind together to form an e x c i t o n i c molecule. His - 9 -contention was based upon the e a r l i e r t h e o r e t i c a l p r e d i c t i o n of W h e e l e r o f the existence of positronium and a corresponding p o s i -tronium molecule. Conservation considerations require that a phonon be emitted when two excitons bind to form an e x c i t o n i c molecule or what i s usually r e f e r r e d to as a b i e x c i t o n . For two excitons at r e s t , energy conserva-t i o n r equires: AE = E b - Tuai (2.3-1) where i s the d i s s o c i a t i o n energy of the b i e x c i t o n , Tiw^ i s the energy of the emitted phonon, and AE i s the k i n e t i c energy of the b i e x c i t o n acquired by the emission of the phonon. Biexcitons have been extensive-l y studied by Asnin et a l . who show that there should be a quasi-e q u i l i b r i u m between f r e e excitons, b i e x c i t o n s , and the emitted phonons. This can be expressed by: nL C nf + 1 ) = nnfe -AE/kT (2.3-2) where Y = ex f > \ 3/z m 2 urn kT ex ex * h 2 and where g ^ and g^ are the degeneracy f a c t o r s of the exciton and b i -exciton states; n ^ , n^, and n^ r e f e r to the number of excitons, b i -excitons, and phonons i n a given state r e s p e c t i v e l y . For low e x c i t a t i o n l e v e l s at low temperatures the number of phonons i s given by the e q u i l i -brium number: - 10 --nto /kT n f » n -x. e * C2.3-3) o and s u b s t i t u t i n g t h i s i n t o equation (2.3-2) gives n 2 ^ a n^, the number of b i e x c i t o n s v a r i e s as the square of the number of f r e e excitons. At high e x c i t a t i o n l e v e l s the number of phonons created by the formation of b i e x c i t o n s exceeds the e q u i l i b r i u m v a l u e , therefore n^ ^ n^ and s u b s t i t u t i o n i n t o equation (2.3-2) gives n g x a n^, the number of b i -excitons v a r i e s l i n e a r l y as the number of free excitons. On the basis of experimental data r e l a t i n g to bound excitons, Haynes estimates that E^ 0.1 G, where G i s the exciton rydberg. This would give - 1.5 meV as the binding energy of a b i e x c i t o n i n s i l i c o n . 2.4 Bose-Einstein Condensation (19) I t was postulated by Moskalenko i n 1961 that there could e x i s t a condensed phase i n a semiconductor c o n s i s t i n g of excitons bound by Van der waals a t t r a c t i o n . As the exciton has i n t e g r a l s p i n , such a condensation would have the c h a r a c t e r i s t i c s of a Bose-Einstein condensa-t i o n , and l i k e He I I possess s u p e r f l u i d i t y . 2.5 Electron-Hole Drops (EHD) The existence of a condensed phase c o n s i s t i n g of a degenerate electron-hole plasma was f i r s t suggested by K e l d y s h ^ on the basis of photoconductivity studies made by R o g a c h e v ^ ^ . High m o b i l i t i e s were found i n germanium at low temperatures and high photo e x c i t a t i o n l e v e l s which were q u a l i t a t i v e l y explained i n terms of degenerate electron-hole plasmas. - 11 -The problem of mobile c a r r i e r s of one sign moving i n a homo-geneous background formed by c a r r i e r s of the other sign was solved by (21) Pines . The problem of two mobile c a r r i e r s i s approached by summing the s o l u t i o n s f o r an el e c t r o n plasma and a hole plasma. Such a s o l u t i o n has the obvious defect of not including any electron-hole c o r r e l a t i o n s . At present there i s no method f o r c a l c u l a t i n g t h i s term, however i t i s believed that the electron-hole c o r r e l a t i o n i s of the same magnitude as (3) the e l e c t r o n - e l e c t r o n and hole-hole c o r r e l a t i o n s . In a degenerate system the average energy per electron-hole (21) p a i r can be written as : E = 3/5 (F + F J + E e S . + E h h . + E 6 6 + E™1 (2.5-1) e h exch exch corr corr ee F g and F^ are the Fermi energies of the electrons and holes, ^ e x c i 1 » E 6 6 , and E*1*1 E*1*1 are the exchange and c o r r e l a t i o n energies of the corr exch' corr electrons and holes, r e s p e c t i v e l y . As noted above, no attempt has been made to include electron-hole i n t e r a c t i o n s . S i l i c o n i s characterized by s i x parabolic conduction band minima which are located along the (100) d i r e c t i o n s , about two-thirds of the way to the zone edge, and a two-fold degenerate valence band with i t s maximum at K = 0. The ele c t r o n and hole Fermi energies, i n u n i t s of the e x c i t o n rydberg, may be wri t t e n as: 3/5 F - ^ 1 e 2 r z s , 2 / 3 6 m, de (2.5-2) - 12 -3/5 F. = 2.21 1 + 3/2 (2.5-3) where r g i s the mean spacing between electrons or holes i n un i t s of the exciton Bohr radius, a e* x u m m , K oe oh (2.5-4) a = exciton Bohr radius x m oh 1 2 1 + 1 m lh 'V (2.5-5) m oe 1 3 2mt ^ (2.5-6) _ 2/3 1/3 m, = m ' m. / de t 1 (2.5-7) The appropriate values for the d i e l e c t r i c constant K and the e f f e c t i v e masses are given i n Table 2.2. Table 2.2 Band Parameters f o r S i l i c o n Conduction Band Valence Band ni = 0.1905 m t o m. = 0.9163 m 1 o m, = 0.32 m de o m = 0.26 m oe o m,,. H 0.154 m In o = 0.523 mQ m . = 0.237 m oh o p = 0.14 m,. K » 11.4 - 13 -The exchange energies have been c a l c u l a t e d by Brinkman and (4) R i c e using the d e r i v a t i o n of the exchange energy i n an e l l i p s o i d a l (22) band by Combescot and Nozieres . T h e i r r e s u l t can be w r i t t e n as: Ee e . + Eh h u = - 1^51 (2.5-8) exch exch r The c o r r e l a t i o n energies do not len d themselves to s t r a i g h t -(23) forward c a l c u l a t i o n s . Brinkman et a l . have used a g e n e r a l i z a t i o n of a scheme suggested by Hubbard based on the random-phase approxima-t i o n . Although t h i s method giv e s q u i t e accurate r e s u l t s f o r a simple band s t r u c t u r e i t cannot be e a s i l y extended to i n c l u d e complicated band s t r u c t u r e s as are found i n s i l i c o n and germanium. S i m i l a r r e s u l t s (22) were obtained by Combescot and Nozieres u s i n g an i n t e r p o l a t i o n method which i m p l i c i t l y i n c l u d e s the d e t a i l s of the band s t r u c t u r e . These terms may be added and the energy per e l e c t r o n - h o l e p a i r p l o t t e d as a f u n c t i o n of r as i s done i n Figure 2.2. Here the s energy i s expressed i n terms of the e x c i t o n rydberg G. The ground s t a t e energy of the condensed phase i s obtained by i m i n i m i z i n g the energy per e l e c t r o n - h o l e p a i r w i t h respect t o r . The s corresponding value of r g i s then r e l a t e d to the e q u i l i b r i u m d e n s i t y n Q by — = | 7rr 3a 3 C2.5-9] n 3 s x o These r e s u l t s are summarized i n Table 2.3. - 14 --2 .0 (4) FIGURE 2.2 Ground State of EHD in Silicon. From Brinkman and Rice Table 2.3 Theoretical Equilibrium Density and Ground State Energy of EHD Equilibrium Concentration Ground State Energy o' Ge (a) 1.8 x 10 1 7 cm"3 5.3 meV Cb) 2.0 x 10 1 7 cm"3 6.1 meV S i (a) ' 3.4 x 10 1 8 cm"3 20.4 meV (b) 3.1 x 10 1 8 cm-3 21.0 meV Ca) ref. 4 (b) ref. 22 - 15 -(2) P o k r o v s k i i and Svistunova have used a simple model to o b t a i n the dependence of the recombination r a d i a t i o n from the EHD on e x c i t a t i o n l e v e l and temperature. They considered the condensed phase as a cloud of s p h e r i c a l e l e c t r o n - h o l e drops of r a d i u s r occupying a s m a l l f r a c t i o n of the t o t a l volume of the sample. T h i s assumption was necessary so tha t generation of c a r r i e r s w i t h i n the drops could be i g n o r e d . Under steady s t a t e c o n d i t i o n s the f l u x of e x c i t o n s i n t o the drop must be equal to the sum of the recombination r a t e i n s i d e the drop and the l o s s of c a r r i e r s out of the drop due t o thermal e m i s s i o n . When t h i s c o n d i t i o n i s combined w i t h the steady s t a t e g e neration r a t e of c a r r i e r s , g, i t i s found that a necessary c o n d i t i o n f o r the e x i s t e n c e of the EHD i s t h a t : 8 - >0 ' (2.5-10) where <}> i s the work f u n c t i o n f o r emission of c a r r i e r s from the EHD, T i s the e x c i t o n l i f e t i m e , v i s the thermal v e l o c i t y and A i s the Richardson constant (A = 7.5 x 1 02 0 cm- 2 s e c- 1 d e g- 2) . This immediately shows that t h ere i s a c r i t i c a l e x c i t a t i o n l e v e l , gcr> and a c r i t i c a l temperature, T c r» which are r e l a t e d through: -<|>/kT 4 AT 2 g m - (2.5-11) p c r vx On the b a s i s of t h i s model i t i s a l s o p o s s i b l e to show that f o r low e x c i t a t i o n l e v e l s , but gr e a t e r than g c r> the i n t e n s i t y of the recombina-t i o n r a d i a t i o n from w i t h i n the EHD, I d , i s p r o p o r t i o n a l to g3. At h i g h - 16 -e x c i t a t i o n l e v e l s 1^ i s l i n e a r l y p r o p o r t i o n a l to g. These c a l c u l a t i o n s (3) have been extended somewhat by Pokrovskii to show that the i n t e n s i t y of the EHD recombination i s r e l a t e d to that of the free exciton, I , ' ex' through: I , a I 3 (2.5-12) d ex This i s independent of temperature and e x c i t a t i o n l e v e l i f the concen-t r a t i o n of drops i s c o n t r o l l e d by nucleation centres. The t h e o r e t i c a l l i n e shape f o r recombination r a d i a t i o n w i t h i n an EHD can be obtained i n a straightforward manner. I t i s necessary only to assume that the matrix element f o r recombination i s independent of the e l e c t r o n and hole energies, g i v i n g : I d ( h v ) a n(E ) n ( E j f (E ) f ( E j 6(hv-E -E -E ) dE dE^ (2.5-13) e h e h g e h e h 4 4 o o where hv i s the energy of the emitted photon; and n ( E e ) , n(E^), f ( E e ) , and f ( E ^ ) are r e s p e c t i v e l y , the density of states i n the conduction and valence bands, and the Fermi d i s t r i b u t i o n functions f o r electrons and holes. 2.6 Experimental Evidence for Biexcitons and Electron-Hole Drops At present there i s strong experimental evidence that a con-densation of excitons can occur i n i n t r i n s i c germanium and s i l i c o n . The k i n e t i c s of i t s formation however, have not yet been agreed upon. - 17 -(18) In 1966 Haynes observed a new ser i e s of peaks i n the photoluminescent spectrum of i n t r i n s i c s i l i c o n at high e x c i t a t i o n l e v e l s . On the basis of p o s i t i o n , width, and e x c i t a t i o n l e v e l depen-dence these l i n e s were a t t r i b u t e d to r a d i a t i v e a n n i h i l a t i o n of a b i e x c i t o n . The analogous s e r i e s of l i n e s were also observed i n german-(24) ium These spectra, p a r t i c u l a r l y that of germanium, have been (25—28 17) extensively studied by a number of workers. Asnin et a l . ' have done a number of experiments studying the photoluminescence, l i f e t i m e s , and photoconductivity which suggest that the spectra are associated with b i e x c i t o n s at low e x c i t a t ion l e v e l s and with EHD at high e x c i t a t i o n l e v e l s . On the b a s i s of s i m i l a r experiments other workers, notably, (2) (29) P o k r o v s k i i and Svistunova and Benoit a l a Guillaume , have con-cluded that the b i e x c i t o n phase does not e x i s t at any e x c i t a t i o n l e v e l , but that the EHD phase does e x i s t f or e x c i t a t i o n above a c r i t i c a l value. Both these conclusions are based upon the d e t a i l e d dependence of the luminescent i n t e n s i t y and exciton concentration on the e x c i t a t i o n l e v e l c l o s e to the c r i t i c a l value. I t i s u n l i k e l y , however, that the a v a i l a b l e i data i s r e l i a b l e enough to favour one point of view or the other. There i s ample evidence from photoconductivity, d i f f u s i o n , and l i g h t s c a t t e r i n g (3) -studies (see f o r example Pokrovskii ) which c l e a r l y demonstrates the existence of macroscopic regions of the c r y s t a l having properties compat-i b l e with the assumption of EHD. I t i s only the k i n e t i c s of formation which are not c l e a r , whether the EHD i s i n equilibrium with an exciton gas or a b i e x c i t o n gas, or perhaps both. There i s as yet no evidence f o r the existence of a Bose-Einstein condensation as suggested by Moskalenko. - 18 -Because the l i f e t i m e and quantum e f f i c i e n c y of the condensed (2) phase are much l a r g e r i n germanium, T A a 20 ys and n = 0.8 , than i n s i l i c o n , T 0 = 0.18 ys and n = 5 x 10 - 1 +^^\ i t i s possible to create a much higher average density of c a r r i e r s and to obtain much more r e l i a b l e ^ data. Consequently there i s considerably more experimental data a v a i l -able f o r germanium than s i l i c o n . Experimental values of the concentra-t i o n of electrons and holes within the EHD and i t s ground state energy are given i n Table 2.4. These appear to be i n good agreement with the p r e d i c t e d values i n Table 2.3. '** Table 2.4 Experimental Values of E q u i l i b r i u m Concentration and Ground State Energy of EHD E q u i l i b r i u m Concentration Ground State Energy Ge (a) 1:95 X 10 1 7. _3 cm 0 6.0 meV (b) 2.6 X 10 1 7 cm - 3 6.3 meV Cc) 1.0 X 10*6 cm - 3 (d) 2.0 X IO*7 cm - 3 S i Ce) 3.7 X IO1*3 cm - 3 21.5 meV (a) r e f . 29 (b) r e f . 2 (c) r e f . 27 (d) r e f . 31 (e) r e f . 32 - 19 -CHAPTER 3  EXPERIMENTAL 3.1 Sample Preparation The samples were cut from single crystal ingots of phosphorus doped silicon purchased from Ventron and General Diode Corporations. The impurity concentration was determined by resistivity measurements using a four-point probe on the face of the crystal before and after cutting the slice. The face was polished and etched with hot concen-trated KOH before measurements were made. A series of ten or more measurements were made at different positions on the crystal face. These were averaged to give the resistivity of the face. The final value was taken to be the average of the resistivities obtained before and after cutting the slice. The impurity concentrations were deter-(33) mined from the measured resistivities using the Irvin chart and were estimated to have an accuracy of ± 5%, subject to the validity of the Irvin chart. Samples to be used in photoluminescent studies were polished with 40 um abrasive on a napped cloth and then etched for 10 seconds in a mixture of HNO^  and HF (20:1) before being mounted in the spectro-meter. For the heat treatment studies the samples were heated to 1150 C in a helium atmosphere for 30 minutes. At the end of this time the samples were quenched in acetone and then etched. If i t was not possible to perform the measurements immediately the samples were stored in liquid nitrogen. The samples could be stored at this temperature for several weeks with no noticeable annealing. - 20 -To avoid the d i f f i c u l t i e s encountered when studying the EPR spectra of samples larger than the microwave sk i n depth i t was necessary that at l e a s t one dimension of the sample be small compared to the sk i n depth. To achieve t h i s the samples were ground to a f i n e powder with a mortar and p e s t l e . At each concentration studied two samples were pre-pared, one was heat treated at 1150 C i n helium f o r 30 minutes before being ground, the other was not heat treated and was used as a standard f o r the purpose of comparison. A f t e r grinding, the powder was etched i n a mixture of HNO^ and HF (20:1). A f l o a t a t i o n separation procedure was then used to grade the powder i n t o three l o t s with average p a r t i c l e s i z e 1 - 5 um, 5 - 1 5 um, and greater than 15 ym. The s i z e s were determined by examination with an o p t i c a l microscope. The samples were then weighed, sealed i n l u c i t e capsules and stored i n l i q u i d nitrogen u n t i l used. 3.2 Photoluminescent Spectrometer A Spectra Physics 165 Argon l a s e r with a continuous maximum o a v a i l a b l e power of 1.8 watts at 5150 A was used as an e x c i t a t i o n l i g h t source. The photoluminescence was analysed with a Perkin Elmer 98G grat-ing spectrometer f i t t e d with a Bausch and Lomb grating blazed at 1.0 ym i n the f i r s t order and having 600 lines/mm. A PbS detector cooled with dry i c e and acetone, together with a Princeton Applied Research (PAR) 113 p r e a m p l i f i e r and a PAR JB-5 l o c k - i n a m p l i f i e r operating at ^ 80 Hz gave a minimum detectable i n t e n s i t y of 1 0 - 1 3 watts. The o p t i c a l c o n f i g -u r a t i o n used i s shown i n Figure 3.1. Corning f i l t e r s 4-96 and 9-92 were used on the l a s e r output to eliminate i n f r a r e d l a s e r emission, and f i l t e r s 7-56 and 2-64 were used before the spectrometer entrance s l i t s A laser filters 4-96,4-92 chart recorder lock-in amplifier 1 10 sample in cryostat chopper filters 7-56,2-64 FIGURE 3.1 Optical Configuration of Photoluminescent Spectrometer. - 22 -to prevent any laser light from appearing in the second order. Care had to be exercised in both the selection and order of f i l t e r s , as many of the dyes used i n their manufacture are photoluminescent. 3.3 Electron Paramagnetic Resonance Spectrometer (34) A standard X-band homodyne EPR spectrometer was used to obtain the EPR results. Through the use of a microwave bias system and automatic frequency controller which "locked" the microwave frequency to that of the resonant cavity, the spectrometer was tuned to measure the absorption due to the sample. A pair of Helmholtz coils mounted on the magnet pole faces were used to provide magnetic f i e l d modulation at 500 Hz. The resultant audio frequency signal was then detected and amplified by a PAR HR-8 lock-in amplifier. The spectrometer was fitted with a double-sample modulation-switched cavity so that control samples which had not been heat-treated could be compared directly with the heat-treated samples. Details of this cavity design and i t s operation (35) are presented i n the thesis of J.D. Quirt . 3.4 Temperature Measurements Electron paramagnetic resonance studies were made at three fixed temperatures, 77K, 4.2K, and 1.1K; Because of high dielectric losses and excessive bubbling i t was not practical to obtain 77K by immersion of the microwave cavity i n liquid nitrogen. Instead, a helium exchange gas system employing an outer dewar f i l l e d with liquid nitrogen and an inner dewar f i l l e d with helium gas was used. It was found that 1 1/2 hours was sufficient for such a system to equilibrate and that the resultant temperature was within 1 degree of liquid nitrogen temperature. - 23 -A temperature of 4.2K was obtained by immersion of the sample and c a v i t y i n l i q u i d helium, and 1.1K was obtained by pumping on the helium to reduce i t s vapour pressure. The microwave power input was s u f f i -c i e n t l y low that there was no heating of the sample. Because of the need for high e x c i t a t i o n power l e v e l s some d i f f i c u l t y was experienced i n obtaining sample temperatures for the photoluminescent studies. A nominal 2K temperature was obtained by immersing the sample i n a l i q u i d helium bath kept at 1.9K. Superfluid helium ( i . e . helium below the A point) does not provide a good heat sink i f the sample must d i s s i p a t e large amounts of heat. To obtain an estimate of the sample temperature the p o s i t i o n and width of the photo-luminescent peaks were studied as a function of the e x c i t a t i o n power. I t was found that f o r e x c i t a t i o n l e v e l s greater than 0.2 watts the peaks broadened and s h i f t e d to higher energies i n d i c a t i n g that the sample was being heated. Thus a l e v e l of 0.15 watts was used to ensure that as l i t t l e heating as po s s i b l e occurred without reducing the power l e v e l too ser i o u s -l y . Therefore i t i s believed that T = 2K i s not s e r i o u s l y i n e r r o r , a l -though the temperature could be as high as 3K. For higher temperatures a d i f f e r e n t system employing helium gas was used to cool the sample. The experimental d e t a i l s are shown i n Figure 3.2. A PAR model 152 tempera-ture c o n t r o l l e r with a GaAs heat sensor was used to c o n t r o l the temper-ature of the sample mounting block to within ± .5 degrees. The sample was mounted with a screw clamp at one end with indium pads used to increase the thermal contact. Mounted i n t h i s way i t was found that the - 24 -l i q u i d helium excitation light molecular s i e v e sample heater GaAs heat sensor helium gas flow FIGURE 3.2 D e t a i l s of Temperature C o n t r o l System. - 25 -i n t e n s i t y of the EHD photoluminescent peak, which i s quite temperature dependent i n the case of i n t r i n s i c s i l i c o n , correlated very c l o s e l y with the heat sensor output with a time constant of much l e s s than 1 second. This was taken as an i n d i c a t i o n that the sample mount tempera-ture as measured by the GaAs diode was c l o s e l y coupled to the tempera-ture of the sample and responded quickly to any change i n sample temperature. Temperature c a l i b r a t i o n was c a r r i e d out i n two steps. F i r s t the GaAs diode was c a l i b r a t e d by mounting a 33 ohm Allen-Bradley carbon r e s i s t o r on the sample mount as shown i n Figure 3.3 ( a ) . G.C. E l e c t r o n -i c DC-Z9 s i l i c o n e compound was used to ensure good thermal contact. The re s i s t a n c e of the carbon r e s i s t o r was c a l i b r a t e d at four f i x e d p o i n t s ; l i q u i d helium, l i q u i d n i t r o g e n , d r y - i c e acetone, and room temperature. Intermediate temperatures were obtained by f i t t i n g the re s i s t a n c e to the semi-empirical e q u a t i o n ^ ^ : l o 81 0R + i s f ^ R = 2 b + f (3.4-1) where R i s the resistance i n ohms and K, b and a are determined by f i t t i n g at known temperatures. This gave the temperature to with i n 10% over the range 4.2K - 200K. Secondly the sample temperature was measured with the carbon r e s i s t o r as shown i n Figure 3.3 (b) with the l a s e r oper-ating a t 1.8 watts, and t h i s temperature compared to that of the sample mount. The d i f f e r e n c e between the sample temperature and that of the sample mount i s shown i n Figure 3.4. This i n d i c a t e s that although the - 26 -laser li g h t resistor FIGURE 3 . 3 Temperature C a l i b r a t i o n Procedure. (a) C a l i b r a t i o n of GaAs Heat Sensor. (b) C a l i b r a t i o n of Sample Temperature. - 27 -sample i s s i g n i f i c a n t l y heated f o r T < 20K, at higher temperatures the system was able to d i s s i p a t e the heat quite w e l l . R e p r o d u c i b i l i t y of linewxdth and peak p o s i t i o n i n d i c a t e that the temperature was repro-d u c i b l e to ^ IK. - 28 -10 20 30 40 50 60 70 80 90 TEMPERATURE OF SAMPLE MOUNT ( K) FIGURE 3.4 Temperature Difference between Sample and Sample Mount - 2 9 -CHAPTER 4 CONCENTRATION DEPENDENCE OF THE PHOTOLUMINESCENCE OF Si;P • I 4.1 In t r o duction Lampert^^^ predicted i n 1958 that i t should be pos s i b l e f o r an e x c i t o n to be bound to an impurity i n a semiconductor. This was (37) confirmed by Haynes , who i d e n t i f i e d a bound exciton l i n e i n the photoluminescent spectra of s i l i c o n doped with donor im p u r i t i e s . The observed dependence of the binding energy on the donor i o n i z a t i o n energy was found to be inco n s i s t e n t with the p i c t u r e of an exciton bound to a n e u t r a l donor, but r e a d i l y explained i f the ne u t r a l donor f i r s t cap-tured a second e l e c t r o n and then bound a hole. More r e c e n t l y , Dean et C4) a l . have done extensiye studies on the-recombination processes ass o c i a t e d with excitons bound to impurities i n s i l i c o n and germanium. (38) Alekseev et a l . have studied germanium doped with donor i m p u r i t i e s using high photo e x c i t a t i o n l e v e l s . They observed the same photoluminescent spectrum a t t r i b u t e d to recombination w i t h i n an EHD as seen i n i n t r i n s i c germanium. For donor concentrations N^ < 5 x 1 0 1 5 cm - 3 no change was observed i n the p o s i t i o n of the photoluminescent peaks, hence i t was concluded that the EHD observed i n i n t r i n s i c c r y s t a l s di d not i n v o l v e r e s i d u a l i m p u r i t i e s . Recombination r a d i a t i o n at the energy a s s o c i a t e d with the EHD i n i n t r i n s i c germanium was not observed fo r donor concentrations of = 2 x 1 0 1 6 cm - 3. Radiation was observed at lower energy and was a t t r i b u t e d to t r a n s i t i o n s of excess electrons i n impurity l e v e l s i n t o the valence band. - 3 0 -Gbbel et al . have done photoluminescence studies on silicon doped with phosphorus which show a similar trend, with no change in the position of the spectra associated with the EHD for donor concentrations < 6.5 x 10 1 7 cm-3. For unspecified higher concentrations they observed very broad emission bands at lower energy, which were attributed to band tailing. The results of extrinsic photoluminescence studies of silicon doped with phosphorus are presented in this chapter. These samples have impurity concentrations in the range 9 x 10 1 5 cm-3^ i 4.3 x 10 1 9 cm-3. 4.2 Concentration Dependence of the Photoluminescent  Spectra of Si;P at 2K There are two reasons why i t is'convenient to consider the photoluminescent spectra at low temperatures. First, at sufficiently low temperatures the broadening of the lines due to thermal excitation of electrons and holes is reduced. Secondly, the condensed phase, in which this work is primarily interested, is only observed below 20K in intrinsic silicon. Thus i f the EHD is to be considered, i t is logical to start with temperatures below 20K. To make comparisons more meaning-ful a l l samples were excited with the same laser power of 0.15 watt. This i s a factor of 5 greater than that used by Gobel et al. 4.3 Low Concentration Samples with phosphorus concentrations of 9 x 10 1 5 cm-3 and 3.6 x 1 0 1 7 cm-3 were studied to confirm the previous work of GObel et a l . ^ . The photoluminescent spectrum observed from the sample • FIGURE 4.1 Photoluminescene for Si:P with N D «* 9.0 x 1 0 1 5 cm - 3at 2K. The superscript r e f e r to the phonon a s s i s t i n g the recombination; TO-transverse o p t i c a l , TA-transverse a c o u s t i c a l , NP-no phonon. The peaks labeled BE are bound exciton peaks. N D =3.6 X 10 17 PHOTON ENERGY ( eV) FIGURE 4.2 Photoluminescence for Si:P with » 3.6 x 10 1 7 cm~3 at 2K. The peak labeled ND is due to the capture of a free hole by a neutral donor. - 33 -containing 9 x 1 0 1 5 cm - 3 phosphorus impurities exhibited four l i n e s at 2K as shown i n Figure 4.1. Two of these l i n e s have been previously i d e n t i f i e d by Dean et a l . . The l i n e at hv = 1.092 eV i s a t t r i b u t e d to the recombination of a bound exciton with the simultaneous emission of a transverse o p t i c a l (TO) momentum conserving phonon. The l i n e at hv SB 1.151 eV i s a t t r i b u t e d to the phononless recombination of a bound exc i t o n . There i s no phonon involved i n t h i s t r a n s i t i o n ; the impurity atom takes up the extra momentum. The two remaining l i n e s at hv = 1.082 eV and hv = 1.12 eV are i d e n t i c a l i n p o s i t i o n , width and r e l a t i v e (39) amplitude to the l i n e s observed i n pure s i l i c o n and a t t r i b u t e d to TO and TA (transverse a c o u s t i c a l ) phonon a s s i s t e d recombination w i t h i n (12) the EHD r e s p e c t i v e l y . I t should be noted that Shaklee and Nahory have shown that the l i n e s a t t r i b u t e d to TO phonon r e p l i c a s are a c t u a l l y unresolved doublets due to TO and LO ( l o n g i t u d i n a l o p t i c a l ) phonon r e p l i c a s separated by 1.8 meV. For the purpose of t h i s t h e s i s such l i n e s w i l l be r e f e r r e d to as TO phonon r e p l i c a s . At the higher concentration (N = 3.6 x 1 0 1 7 cm - 3) the photo-luminescent spectrum i s comprised of four l i n e s (Figure 4.2). Three of these l i n e s , observed at hv = 1.08 eV, 1.12 eV, and 1.14 eV, are i d e n t i -f i e d with TO phonon, TA phonon, and no phonon r e p l i c a s of the recombina-t i o n r a d i a t i o n associated with the EHD. These were f i r s t observed and i d e n t i f i e d by Gobel et a l . f o r t h i s impurity concentration. A fo u r t h l i n e at 1.065 eV was not observed by Gb*bel et a l . , but has been observed by other workers at lower concentrations and i s i d e n t i f i e d with the r a d i a t i v e capture of free holes at n e u t r a l donors . - 34 -The observed spectra are consistent with the conclusions of Gobel et a l . that recombination radiation from the EHD is observable for impurity concentrations up to 3.5 x 10 1 7 cm-3. 4.4 Transition Range (1.1 x 10 1 8 cm-3 i Np < 5.5 x 10 1 8 cm-3) As the concentration of phosphorus impurities i s increased above = 3 x 10 1 7 cm 3 the wave functions of the excited impurity states begin to overlap. This results in a merging of the higher excited states with the conduction band, producing a conduction band t a i l . When the impurity concentration exceeds = 3 x 10 1 8 cm - 3 the impurity atoms are sufficiently close that the ground state wave functions overlap and the donor electron can no longer be considered to (41) be localized at a particular l a t t i c e site . For a l l higher impurity concentrations the material i s characterized by metallic behavior, e.g. there i s no activation energy for conductivity. The photoluminescent spectra for a number of different impurity concentrations i n the region of this semiconductor-metal tran-s i t i o n are shown in Figure 4.3. This figure illustrates the most important feature of the photoluminescent spectra i n this range; that the l i n e width and shape, except for some t a i l i n g at low energy, remains essentially unchanged with only a monotonic shift to lower energy. As the recombination radiation intensity i s dependent upon the surface of the sample i n a manner which i s not f u l l y understood, i t i s not possible to quantitatively monitor the intensity as a function of impurity concentration. It i s possible to say, at least qualitatively, that the intensity of the luminescence decreases with increasing concen-tration. - 35 -4.3 x IO 1 9 c m - 3 I . 3 x l 0 l 9 c m " 3 5.5x |0 ' ° cm '8 r m - 3 •8 rm-3 3 x | 0 cm '8 r m - 3 1.03 1.06 1.09 1.12 PHOTON ENERGY (eV) FIGURE 4.3 Concentration Dependence of the Photoluminescence of Si:P at 2K. The dashed portion of the spectrum for = 4.3 x 10 l a cm - 3 was inferred from the spectra at higher temperatures. The parallel bars give the spectrometer s l i t width. - 36 -4.5 High Concentration The spectra observed for samples with phosphorus concentra-tions of N Q = 1.3 x 10 1 9 cm-3and Np = 4.3 x 10 1 9 cm"3 are shown i n Figure 4.3. The lines are found to be very much broadened on the low energy side with the width increasing with concentration. There i s a continuing shift of the peak position to lower energy as the impurity concentration increases. CHAPTER 5 TEMPERATURE DEPENDENCE OF THE PHOTOLUMINESCENCE OF Si:P • I 5.1 Introduction An investigation of the temperature dependence of the photo-luminescence is of interest for several reasons. In particular, the cri t i c a l temperature above which a peak is not observed, and the temper-ature variations of the linewidth are of considerable aid in identify-ing the recombination mechanism responsible for the observed spectra. In the case of silicon such studies may make i t possible to determine whether biexcitons are formed, as suggested by Asnin et al.^\ Other reasons for such studies include the understanding of the effects of impurities on the cr i t i c a l temperature and excitation level, and per-haps further insight into the semiconductor-metal transition itself. (c\ (38) Recent work by Gobel et al. and Alekseev et al. indicates at least for low impurity concentrations, that the critic a l temperature is raised and the cri t i c a l excitation level lowered as the impurity concen-tration i s increased. This chapter will present the results of an investigation of the photoluminescent spectra of phosphorus doped silicon in the tempera-ture range 2K < T < 140K. Some of the important features will be discussed briefly, although any detailed discussion will be left to a later chapter. - 38 -5.2 Temperature Dependence of the Photoluminescence of Si:P As was done when studying the concentration dependence of the photoluminescence, a l l experiments were performed with the same incident power. For these experiments the maximum power level, 1.8 watts, was used in order to obtain the largest possible signal. Unfor-tunately, this power level could not be used to study samples at 2K as the poor thermal contact with the superfluid helium resulted in signifi-cant sample heating. The poor signal obtained at low incident power levels also ruled out any meaningful attempt to determine the excitation threshold levels or the variation of photoluminescent intensity with excitation level. 5-3 Low Concentration 1) 9 x 10 1 5 cm"3 This was the only concentration studied for which recombina-tion radiation from free excitons and bound excitons could be clearly identified. As shown in Figure 5.1 the free exciton peak at hv = 1.10 eV was not observed at very low temperatures, T = 2K, but increased in relative magnitude such that for T £ 30K, i t was the dominant peak. The two peaks associated with the bound exciton, TO phonon assisted at hv = 1.092 eV and no phonon at hv = 1.15 eV were only observed for (14) T £ 25K. Other workers have considered samples with phosphorus concentrations of = 8 x 10 1 6 cm-3 and found that with the higher impurity concentration i t was possible to observe the hound exciton peaks up to I = 80K. The peaks at hv - 1.082 eV and hv = 1.12 eV, iden-tified with recombination within an EHD, were observed only for T < 20K. - 39 -32 K 25 K 21 K 13.5 K 2 K — i 1 '. i i i 1.03 1.06 1.09 1.12 1.15 PHOTON ENERGY (eV) FIGURE 5.1 Temperature Dependence of the Photoluminescence f o r N = 9 x 1015 cm - 3. - 4 0 -- 41 -The disappearance of these lines, and the simultaneous increase i n strength of the f r e e exciton l i n e was a t t r i b u t e d to the evaporation of (38) the EHD i n t o an exciton gas . , 2) 3.6 x 1 0 1 7 cm - 3 As shown i n Figure 5.2 the photoluminescent spectrum f o r - 3.6 x 1 0 1 7 cm - 3 was quite d i f f e r e n t from that observed for N D = 9 x 1 0 1 5 cm - 3. There were no l i n e s which could be a t t r i b u t e d to f r e e or bound ex c i t o n s , and there was a new l i n e at hv - 1.065 eV. T h i s l i n e was i d e n t i f i e d i n Section 4.3 as a r i s i n g from the recombina-t i o n of" a f r e e hole with a n e u t r a l donor. The remaining peaks, a t t r i -buted to the EHD, x>irere found to s h i f t monotonically to higher energies and broaden as the temperature increases; behavior which i s i n accord with the p i c t u r e of recombination between-two bands. For T > 45K the TO a s s i s t e d peak developed a shoulder on the high photon energy sid e . Unfortunately, due to the width of the l i n e s at these temperatures i t i s not p o s s i b l e to determine whether there are two photoluminescent peaks present. I t i s p o s s i b l e to set a lower bound of T - 44K for any threshold temperature to be associated with the EHD for t h i s impurity concentration. .5.4 T r a n s i t i o n Range (1.1 x 1 0 1 8 cm - 3 < < 5.5 x 1 0 1 8 cm - 3) The temperature dependence of the photoluminescent spectra i n the concentration range 1.1 x 1 0 1 8 cm - 3 <, <; 5.5 x 1 0 1 8 cm - 3 was found to be very s i m i l a r to that observed f o r = 3.6 x l O 1 7 cm"3. The l i n e due to recombination of a f r e e hole with a n e u t r a l donor was not observe although i t may contribute to the t a i l observed on the low energy side o - 42 -— « 1 1 i 1-03 1.06 1.09 1.12 1.15 PHOTON ENERGY (eV) FIGURE 5.3 Temperature Dependence of the Photoluminescence for N D = 1.3 x 1019 cm"3. - 43 -of the TO assisted recombination peak. A line identified as due to phononless recombination was present i n this concentration range, with i t s relative intensity increasing with concentration. For a l l samples studied i n this concentration range a broad shoulder developed on the high photon energy side of the TO assisted peak for T £ 40K. There appears to be an increase i n the temperature at which the shoulder appears, however due to the uncertainty i n the line shapes, i t i s impossible to make any quantitative verification of this trend. 5.5' High Concentration 1) 1.3 x 10 1 9 phosphorus/cm3 The photoluminescent spectrum, shown i n Figure 5.3, i s com-prised of two broad lines whose energy separation suggests that they are TO phonon and no phonon replicas of the same recombination mechanism. The lines broaden and shift to higher energy with increasing temperature, with the broadening taking place primarily on the high energy side. 2) 4.3 x 10 1 9 phosphorus/cm3 The spectrum i s composed of two broad bands at hv = 1.06 eV i and hv = 1.11 eV which broaden with increasing temperature u n t i l by T = 125K they can no longer be resolved. The peak positions appear to be independent of temperature, although the broadening and overlap make i t impossible to determine this with any degree of r e l i a b i l i t y . - 44 -CHAPTER 6  DISCUSSION OF RESULTS 6.1 Screening of Excitons As a f i r s t step towards identifying the recombination process responsible for the observed photoluminescence, i t i s necessary to con-sider what role, i f any, is played by excitons. In this section i t w i l l be demonstrated that free and bound excitons do not contribute s i g n i f i -cantly to the photoluminescence for N^ > 3.6 x 10 1 7 cm-3. (42) Asnin and Rogachev noted that the energy of the free exciton peak in the absorption spectrum of germanium was independent of impurity concentration, although the peak was found to broaden with increasing impurity, concentration. The maximum width attained was ^ 5 meV. It was found that the excitons disappeared when the decrease in the band gap was equal to the exciton ionization energy. For T = 77K, the impurity concentration at which the exciton line disappeared agreed well with that calculated using the simple Debye expression: ' \l/2 KkT 4ire 2N D (6.1-1) The impurity concentration at which there are sufficient free carriers to screen the coulomb interaction between an electron and hole, and prevent the formation of excitons can be estimated in several ways. Using the criterion of equation (6.1-1) suggested by Asnin et a l . , or a (43) slightly more sophisticated approach adopted by Albers to calculate - 45 -the exciton binding energy, i t is found that there can be no exciton states in silicon for free carrier concentrations greater than n = 2.2 x 10 1 7 cm 3. Both approaches suffer from only being applicable at high temperature, however since screening increases with decreasing temperature, the results will at least constitute an upper bound for (44) the existence of excitons. Mott , using the Thomas-Fermi approxima-tion arrived at the criterion l/3 n a > 0.25 (6.1-2) x For silicon a x -' 4.29 x 10 - 7 cm, which gives a value of n = 2 x 10 1 7 cm-3 as the concentration of free carriers required to prevent the for-mation of a bound state. This approach is applicable at low temperature and its close agreement with the Debye model indicates that i t is reasonable to accept n ^ 2 x 10 1 7 cm"3 free carriers as the maximum allowed for the existence of free excitons. The impurity concentration required to give 2 x 10 1 7 cm""3 free carriers at 2K can be calculated quite readily. If the Fermi level is at least 4 kT below the bottom of the conduction band then the occupation probability of the conduction band can be described by Boltzmann statistics, and the number of electrons in the conduction band given by N n - N — (6.1-3) • . N1 n - 46 -where N' = 6 c m, kT-de 3/2 exp 2TTR2 ~ kT (6.1-4) The activation energy, E^, of phosphorus donors in silicon has been shown by Kuwahara (45) E A = E O - < A > N D to obey the empirical expression l/3 (6.1-5) where E Q = 45 meV and <a> = 3.27 x IO - 5 meV-cm. From equation (6.1-3) i t is found that to have 2 x 10 1 7 cm-3 free carriers at 2K i t is necessary to have = 2.5 x IO 1 8 cm"3. This calculation can be extended to bound excitons in an (43) approximate manner. Albers has shown that the free carrier concen-tration required to screen excitons is proportional to the square of the reduced mass. n g a y 2 (6.1-6) The exciton binding energy is proportional to the reduced mass, 1 G a y (6.1-7) and therefore h a G2 (6.1-8) The binding energy of a bound exciton is ^  1/2 G (.0065 eV) which means that the free carrier concentration required to screen bound excitons is ^ 5 x 10 l e cm-3. In terms of impurity concentration, a phosphorus - 47 -concentration of = 1.0 x 1Q 1 8 cm""3 i s required to ensure that excitons do not bind to impurity atoms at 2K. These r e s u l t s show that i t i s reasonable to assume that there w i l l be no recombination emission due to free or bound excitons for > 2.5 x 1 0 1 8 cm - 3 and 1.0 x 1 0 i 8 cm"3 r e s p e c t i v e l y . The binding energy f o r a b i e x c i t o n i s ^ 0.1 G. This reduces by two orders of magnitude the number of free c a r r i e r s required f o r screening. This e f f e c t i v e l y r u l e s out the existence of the b i e x c i t o n i n the impurity concentration range > 1 0 1 6 cm - 3. I t i s not necessary to consider screening of excitons to e x p l a i n the photoluminescent spectra f o r N Q = 9 x 10^ 5 cm"3, since at t h i s impurity concentration there are i n s u f f i c i e n t free c a r r i e r s . As shown i n Figure 5.1 the TO phonon a s s i s t e d free exciton l i n e at hv =1.098 eV i s observed for T > 13.5K. This l i n e i s not present at T = 2K because the time required f o r the capture of an exciton by the condensed phase i s short compared to the exciton l i f e t i m e at t h i s tem-perature. The p r i n c i p a l l i n e due to the recombination of bound excitons i s that at hv = 1.092 eV and i s observed at T < 32K. Because of the low impurity concentration t h i s l i n e i s weak r e l a t i v e to the f r e e exciton l i n e and d i f f i c u l t to i d e n t i f y at T = 32K, however i t should be present f o r T ^ 80K when the thermal energy i s i n s u f f i c i e n t to cause d i s s o c i a t i o n Although screening arguments alone i n d i c a t e that bound excitons could e x i s t f o r T 1 50K with = 3.6 x 1 0 1 7 cm"3 i m p u r i t i e s , the photo-luminescent spectrum for t h i s impurity concentration does not contain a l i n e at hv = 1.092 eV due to bound excitons. Were i t present t h i s l i n e - 48 -could be r e a d i l y r e s o l v e d . Free excitons are possible f o r T < 100K at t h i s concentration, and t h e i r possible c o n t r i b u t i o n to the spectrum w i l l be discussed i n a l a t e r s e c t i o n . There i s no emission peak observed at hv = 1.098 eV or hv = 1.092. eV i n the concentration range 1.1 x 101 8 cm- 3 £ < 4.3 x 101 9cm- 3 f o r any T i 2K. This observation i s i n accord with the preceeding c a l c u l a t i o n s which show that bound excitons should not e x i s t i n t h i s range and that free excitons should not e x i s t except for T < 50K when = 1.1 x 1 01 8 cm- 3. I t i s concluded that f o r the impurity oncentration range 3.6 x 1 01 7 cm- 3 < < 4.3 x I O1 9 cm- 3 there i s no c o n t r i b u t i o n due to the recombination of excitons to the observed photoluminescence except p o s s i b l y that due to free excitons at = 3.6 x 10 cm . 6.2 I n t e r p r e t a t i o n of Spectra In the previous section' i t was shown that the experimental r e s u l t s cannot be explained i n terms of the recombination of free or bound e x c i t o n s . This leaves two other p o s s i b l e mechanisms. The Obser-ved spectra could be due to simple band-to-band recombination of e l e c t r o n s i n the conduction band with holes i n the valence band without the formation of any form of condensate, or i t could be due to recombin-a t i o n w i t h i n some form of condensed phase. I f the photoluminescence i s due to band-to-band recombination, the observed peak must have photon energies greater than - hv^, where E i s the band gap energy and hv i s the energy of the phonon a s s i s t i n g S P the recombination. At low temperature the width of such a recombination - 49 -l i n e depends on the density of photocreated c a r r i e r s . For the maximum e x c i t a t i o n power used i n these experiments, 1.8 watts, there would be *v> 5 x 10 cm - J c a r r i e r s . This concentration of c a r r i e r s would r e s u l t i n a narrow l i n e (^2.0 meV) at T = 2K whose width would be l i m i t e d by the e f f e c t i v e s l i t width of the spectrometer (y 3 meV). A necessary, but not s u f f i c i e n t , condition for the existence of a condensed phase i s that the binding energy given by E = E - ( E ) + hv (6.2-1) o a g opt p v ' be l e s s than zero. In t h i s equation ( E g ) Q p t I s t n e o p t i c a l band gap and E i s the threshold energy. The threshold energy i s obtained by 3. e x t r a p o l a t i n g the high photon energy side of the photoluminescent peak to the base l i n e . The width of the peak would depend on the equilibrium concentration of c a r r i e r s i n the condensed phase and not on the e x c i t a -t i o n l e v e l . In the following a n a l y s i s i t w i l l be shown that the observed photoluminescence can be explained i n terms of both of the above two mechanisms. For convenience only the TO a s s i s t e d recombination l i n e w i l l be discussed, the others being j u s t phonon r e p l i c a s . .6.3 Low Concentration C9 x 1 0 2 5 cm""3 .< N D .< 1.1 x 1 0 1 8 cm"3) The threshold energy and width at h a l f maximum at 2K f o r the TO a s s i s t e d peak are shown i n Figures 6.1 and 6.2 r e s p e c t i v e l y . The experimental data shown i n these f i g u r e s have been corrected f o r instrumental broadening i n the manner described i n Appendix A. The values used f o r the band gap and o p t i c a l band gap were obtained from l.l 1.10-> >-e> cr. in 2 UJ 1.09 1.08 O g 1.07 CL 1.06 1.05 10 16 (Eg) o p -h I /p Threshold Energy J J L I I M i l l ^7 j I I I I I 11 1 I ( E g ) 0 p - h v p J8 ' ' i i i i 11 ' ' i i *i 111 10" 10'° 10 IMPURITY CONCENTRATION ( P / C m 3 ) 19 10 20 o FIGURE 6.1 Concentration Dependence of the Threshold Energy, Ea. For < 1.0 x 1 0 1 8 cm - 3 Eg * (Eg) ^ and was obtained from Kuwahara^ 5\ For N_, > 6.0 x 1 0 1 8 cm - 3 Eg and (Eg) 6 6 opt ( 4 6 ) D e 6 / o p t were obtained from Balkanski et a l . . FIGURE 6.2 Half-width of TO Assisted Peak vs. Impurity Concentration. Solid curves are calculated assuming parabolic bands and constant effective masses. 9 X 1 0 1 5 3.6 X 1 0 1 7 1.1 X 10 ,18 FIGURE 6.3 Comparison of Calculated and Experimental Line Shapes at 2K, The dashed curves represent the profile for EHD with n » 3.0 x 10 1 8 cm"3. - 53 -Kuwahara^ 4 5^ for < 1.1 x 1 0 1 8 cm"3 and from Balkanski et a l . ^ 6 ^ f o r N D > 6 x 1 0 1 8 cm - 3. It i s clear that f o r 9 x 1 0 1 5 cm - 3 £ < 1.1 x 1 0 1 8 cm - 3 the c r i t e r i o n of equation (6.2-1) f o r the existence of a condensed phase i s s a t i s f i e d . The requirement of band-to-band recom-• b i n a t i o n , that the emitted photons have energy greater than the band gap, i s not s a t i s f i e d and the l i n e width i s found to be independent of e x c i t a -t i o n power. These find i n g s i n d i c a t e that the mechanism responsible f o r t h i s peak i s not simple band-to-band recombination, but must involve a condensed phase. I t i s found that the threshold energy, E , and the halfwidth of the l i n e i n t h i s concentration range are e s s e n t i a l l y the same as those observed i n i n t r i n s i c s i l i c o n f o r the l i n e a t t r i b u t e d to TO phonon a s s i s t e d recombination within an EHD. Figure 6.3 shows the c l o s e agreement (except i n the wings) found between the experimentally observed l i n e shapes for = 9 x- 1 0 1 5 cm - 3, 3.6 x 1 0 1 7 cm - 3 and 1.1 x 1 0 1 8 cm"3 and the c a l c u l a t e d l i n e shape f o r an EHD with n = 3.0 x 1 0 1 8 cm - 3. The d e t a i l s of t h i s c a l c u l a t i o n can be found i n Appendix B. This value for the eq u i l i b r i u m density i s somewhat d i f f e r -ent from the value of 3.8 x 1 0 1 8 cm - 3 found by Pokrovskii et a l . ^ " ^ i n i n t r i n s i c s i l i c o n , however the discrepancy i s due l a r g e l y to the choice of values for the e f f e c t i v e masses. On the basis of the above evidence i t i s c l e a r that the photo-luminescence observed for 9.0 x I O 1 5 cm"3 H 1 1.1 x 1 0 1 8 cm - 3 i s due to recombination wi t h i n an EHD a s s i s t e d by appropriate momentum conserv-i n g phonons. - 54 -6.4 Transition Range (2.2 x 1018 . cnT3.s.N <.5.5 x 1Q18 cm"3) Unfortunately there are no data available on either the band gap or the optical band gap in the concentration range 2.2 x 1018 cm-3 < < 5.5 x 10 8 cm-3. It is possible, though, to make a reasonable estimate of E - hv in this range. From Figure 6.1 i t appears that § P E - hv can be extrapolated over this range by connecting the data of Kuwahara with that of Balkanski et al. with a smooth curve. If this is done, then for a l l samples studied in this concentration range, the criterion of equation (6.2-1) is satisfied, indicating that the recombin-ation may involve a condensed phase. In the next section i t is suggested -that the data of Balkanski et a l . is misleading and this would mean that the above estimate of E^ - hv^ is not a good one. Since the donor ground state is 45 meV below the intrinsic conduction band edge this constitutes 'a lower limit for the band edge when ND ^ 3,0 x l Q i 8 cm"3, the concentra*-tion at which the ground state be.comes delocalized. For this extreme case, i t is s t i l l found that for = 3.0 x 1018 cm"3 an appreciable fraction of the observed photoluminescent line has photon energies less than this value of E - hv . On the basis of these estimates i t is g P | unlikely that the observed photoluminescence is due to simple band-to-band recombination. The linewidth data shown in Figure 6.2 is more informative. If the data is to be analysed in terms of a condensed phase, the simple picture of an EHD must be modified somewhat. When the impurity concen-tration exceeds the equilibrium density (3.0 x 1Q18 cm"3) i t is unreason-able to consider a condensation of electrons. Maintenance of an electron - 55 -density of n = 3.0 x 1 0 1 8 cnf^ would co n s t i t u t e a r a r e f a c t i o n of ele c t r o n s which i s p h y s i c a l l y u n r e a l i s t i c . The modified EHD would have an e l e c t r o n density of n = 3.0 x I O 1 8 cm"3 when 1 3.0 x 1 0 1 8 cm"3, and an e l e c t r o n density of n = when £ 3.0 x 1 0 1 8 cm _ 3 +. I f the problem of a hole plasma moving i n a homogeneous negative back-ground i s solved, i . e . by considering only the terms i n v o l v i n g holes i n the equation (2.5-1), then i t i s found that i n s i l i c o n the lowest energy c o n f i g u r a t i o n i s a hole plasma with density n^ ^  3.0 x 1 0 1 8 cm"3. Thus i t i s reasonable to consider as a f i r s t approximation that the holes w i l l continue to condense to t h i s density i r r e s p e c t i v e of the e l e c t r o n concentration. This i s i n accord with the model f o r the EHD which contains no electron-hole i n t e r a c t i o n s but considers the e l e c -trons and holes to be independent. The linewidth expected of a con-densed phase with e l e c t r o n density NQ and hole density n = 3.0 x 1 0 1 8 cm i s given by curve ( i ) of Figure 6.2. This c a l c u l a t i o n assumes p a r a b o l i c bands and constant e f f e c t i v e masses. The linewidths obtained In t h i s manner are l a r g e r than the measured linewidths i n the impurity concentration range 2.2 x 1 0 1 8 cm"3 < < 5.5 x 1 0 1 8 cm"3. Using a density of states f o r the conduction band of the form suggested by (47) Mott i s expected to give even l a r g e r c a l c u l a t e d linewidths rather than improve the agreement. I f , on the other hand, the data i s analysed i n terms of band-to-band recombination without a condensed phase then The a d d i t i o n of 5.0 x 1 0 1 6 cm photocreated electrons i s not expected to cause any appreciable r e d i s t r i b u t i o n of donor electrons when n » N - 56 -the l i n e w i d t h would be given by curve ( i i ) of Figure 6.2. This c a l -c u l a t i o n a l s o assumes para b o l i c bands and constant e f f e c t i v e masses. The li n e w i d t h s c a l c u l a t e d with t h i s model are smaller than the observed l i n e w i d t h s , but the use of a more s u i t a b l e density of states function may improve the agreement. The concept of a constant hole density used i n the model f o r recombination i n v o l v i n g a condensed phase may be a l i t t l e naive. The e l e c t r o n d e n sity of n > 3.0 x 1 0 1 8 cm"3 i s s u f f i c i e n t to screen any bound impurity s t a t e s , hence the condensed phase w i l l contain p o s i -t i v e i o n cores. Although the d e t a i l e d nature of the e f f e c t s of these p o s i t i v e cores i s not known, i t i s reasonable to expect that t h e i r p o s i t i v e f i e l d w i l l reduce the hole density. Experimentally i t i s observed that i n the concentration range 2.2 x 1 0 1 8 cm"3 < N D < 5.5 x 1 0 1 8 cm"3 the linewidth l i e s between those p r e d i c t e d f o r a condensed phase with constant hole density (curve ( i ) ) , and f o r no condensed phase (curve ( i i ) ) , and a c t u a l l y decreases r e l a t i v e to the l i n e w i d t h of the EHD recombination l i n e i n I n t r i n s i c s i l i c o n . This i s not expected of a condensate with a constant hole density but i s compatible with the concept of a condensate with a decreasing hole d e n s i t y . However, without knowledge about the band gap, the p o s s i b i l i t y of band—to-band recombination cannot be d e f i n i t e l y ruled out. 6.5 High Concentration '.N > .1.0 .x .101 9 . cm"3 In p r i n c i p l e i t should be p o s s i b l e to draw d e f i n i t e conclusions about the existence of a condensed phase for N > 6'x 1 0 1 8 cm - 3. The band gap and o p t i c a l band gap have been determined f o r 6.0 x 1 0 1 8 cm"3 - 57 -< N D < 4 . 9 x 1020 cm""3 at 35K by Balkanski et al, 1 using i n f r a r e d absorption techniques. Together with the photoluminescence e x p e r i -ments t h i s should be s u f f i c i e n t to determine whether or not there i s a condensed phase. Unfortunately the r e s u l t s of Balkanski et a l . must be viewed with some care. To obtain values for the band gap two assumptions were made: f i r s t that the f r e e absorption c o e f f i c i e n t had the c l a s s i c a l A 2 dependence, and secondly that the remaining absorption c o e f f i c i e n t a f t e r s u b t r a c t i o n of the f r e e c a r r i e r concentration could be f i t t e d to a t h e o r e t i c a l expression assuming a parabolic density of s t a t e s . The assumption of X 2 dependence f o r the free c a r r i e r absorp-t i o n may be reasonable, since i t does give good agreement at long wave-lengths. The assumption of a p a r a b o l i c density of states i s c e r t a i n l y not v a l i d f o r impurity concentrations j u s t above the semiconductor-metal t r a n s i t i o n . At these concentrations there i s s i g n i f i c a n t t a i l i n g of the conduction band, a feature w e l l documented by other workers. Further evidence of the existence of such a t a i l on the conduction band i s pro-vided i n Figure 4.3. Here long t a i l s are observed on the low energy side o f the TO a s s i s t e d peaks, p a r t i c u l a r l y f o r < 3.0 x 1 0 1 8 cm""3. Balkanski's value of E = 1.15 would suggest that the conduction band has only been lowered by 20 meV, even though f o r t h i s concentration i t i s known that the donor ground s t a t e , which i s 45 meV below the i n t r i n s i c band edge, i s completely d e l o c a l i z e d . With the proviso that the data of Balkanski et a l . may not be c o r r e c t , Figure 6.1 shows that the c r i t e r i o n of equation (6.2-1) f o r the existence of a condensed phase i s s a t i s f i e d i n the concentration range 1.3 x 1 0 1 9 cm - 3 & N 4 4.3 x 1 0 1 9 cm - 3. - 58 -The measured linewidths are l e s s conclusive. Although there appears to be some agreement between curve ( i ) of Figure 6.2 and the measured va l u e s , corrections for non p a r a b o l i c bands may r e s u l t i n a b e t t e r f i t with curve ( i i ) which represents the width expected i f there i s no condensed phase. I t should be noted that the assumption of a p a r a b o l i c band i s based on a one e l e c t r o n p i c t u r e which ignores c o r r e l a t i o n s between e l e c t r o n s . The i n c l u s i o n of c o r r e l a t i o n e f f e c t s i s expected to produce a further broadening of the peak, e s p e c i a l l y f o r N D -v 1 0 1 9 cm" 3^ 8^. Thus the linewidth data by i t s e l f i s not s u f f i c i e n t to draw any conclusions with regard to the existence of a condensed phase. 6.6 I n t e n s i t y o f Photoluminescence The r e l a t i v e integrated i n t e n s i t y of the TO a s s i s t e d peak i s shown as a f u n c t i o n of impurity concentration at 2K i n Figure 6.4. Although the i n t e n s i t i e s are shown with an uncertainty of 10% the data should be considered only to show a q u a l i t a t i v e trend. The e f f i c i e n c y of photoluminescence depends i n an unknown manner upon the d e t a i l e d c o n d i t i o n of the sample surface. Care was taken to maintain uniformity i n the p r e p a r a t i o n of the samples, but i t was found that changes i n i n t e n s i t y of t» 10% occurred i f d i f f e r e n t samples of the same concentra-t i o n were compared. Small v a r i a t i o n s i n the o p t i c a l alignment of the spectrometer between successive experiments can also cause s i g n i f i c a n t v a r i a t i o n s i n the s i g n a l i n t e n s i t y . Thus the quoted uncertainty of 10% i n F igure 6.4 may be o p t i m i s t i c . However, there i s an order of magnitude change i n the integrated i n t e n s i t y of the TO-assisted peak over the - 5 9 -IMPURITY CONCENTRATION (P/cm3) FIGURE 6.4 Intensity of TO Assisted Peak vs. Impurity Concentration. - 60 -concentration range studied which must be considered r e a l . Such a change i s p l a u s i b l e as the increased number of s c a t t e r i n g centres would enhance non r a d i a t i v e recombination processes. (38) The f i n d i n g of Aleeksev et a l . that the c r i t i c a l e x c i t a -t i o n l e v e l i s lowered as the concentration i s increased would appear to suggest that an increase i n photoluminescence i n t e n s i t y might be expected. Such an increase may occur, however i t could be e a s i l y o f f -set i f the l i f e t i m e w i t h i n the EHD decreased as the impurity concentra-t i o n increased. 6.7 Temperature Dependence of Photoluminescence The temperature dependence of the photoluminescence of (39) i n t r i n s i c s i l i c o n has been studied by Kaminskii et a l . . I t was observed that at low temperature (T i 2K) only recombination r a d i a t i o n from the EHD was present. As the temperature was r a i s e d , peaks due to the recombination of free excitons appeared i n the photoluminescent spectra. The i n t e n s i t y of t h i s recombination r a d i a t i o n , r e l a t i v e to that from the EHD, increased with increasing temperature u n t i l by T > 20K there was no r a d i a t i o n observed from the EHD. The c r i t i c a l temperature f o r t h i s "evaporation" of the EHD was taken to be that at which the r e l a t i v e i n t e n s i t i e s of the two l i n e s were equal. The c r i t i -c a l temperature, T » i s dependent upon the e x c i t a t i o n l e v e l i n the manner shown by equation (2.5-11), and f o r a l l r e a l i s t i c power l e v e l s i s about 15K. As Figure 5.1 shows, a s i m i l a r behavior i s observed f o r samples with N = 9 x 1 0 1 5 cm""3, except that i n t h i s case there are also - 61 -bound excitons present. Although the exact c r i t e r i o n that should be employed f o r determining i s no longer c l e a r , i t i s apparent from Figure 5.1 that 13.5 < T < 21K. This i n d i c a t e s that the EHD-exciton cr e q u i l i b r i u m i s not s i g n i f i c a n t l y affected by the presence of = 9 x 1 0 1 5 cm"3 phosphorus i m p u r i t i e s . I t i s c l e a r from Figure 5.2 that the e f f e c t of the i m p u r i t i e s can no longer be ignored f o r = 3.6 x 1 0 1 7 cm""3. At t h i s concentration the lowest temperature f o r which there i s any evidence of f r e e exciton r a d i a t i o n i s T ^ 45K. At t h i s temperature a shoulder has begun to develop on the high photon energy side of the TO a s s i s t e d peak. This shoulder increases i n r e l a t i v e s i z e u n t i l by T 100K there i s a d e f i n i t e s h i f t of the peak p o s i t i o n to hv = 1.098 eV, the energy expected for recombination of a TO a s s i s t e d free exciton. Unfortunately the thermal broadening of the l i n e s i s such that at no temperature i s i t po s s i b l e to resolve two l i n e s . Thus the p o s s i b i l i t y that the observed peak i s due to the same recombination process at a l l temperatures i s not r u l e d out. Calc u l a t i o n s of the e f f e c t s of screen-ing i n d i c a t e that f r e e excitons can e x i s t up to at l e a s t T ^ 100K f o r samples of t h i s impurity concentration. Thus the i d e n t i f i c a t i o n of the peak at T = 100K with free excitons i s conceptually v a l i d . The temperature dependences of samples with 1.1 x 10"18 cm"3 5, 5 1.3 x 1 0 1 9 cm - 3 impurities are much the same as that of the sample with = 3.6 x 1 0 1 7 cm - 3. In t h i s concentration range i t i s p o s s i b l e to r u l e out any recombination r a d i a t i o n due to excitons. The (43) I R c a l c u l a t i o n s of Albers can be used to show that f o r = 1.0 x 10 cm"*3 f r e e excitons cannot e x i s t at T > 30K, and for N„ > 3 x 1 0 1 8 cm - 3 - 62 -they cannot e x i s t at a l l . On t h i s b a s i s , i f a condensed phase does exist,, the work f u n c t i o n f o r an electron-hole p a i r would not be measured r e l a t i v e to the f r e e exciton, but r e l a t i v e to free electrons and holes •I i n the conduction and valence bands r e s p e c t i v e l y . Again, because of broad : l i n e s and l a c k of information on which to p r e d i c t the l i n e shapes, i t i s not p o s s i b l e to say with any c e r t a i n t y i f there i s any change i n the recombination mechanism responsible f o r the TO a s s i s t e d l i n e as the temperature i s r a i s e d . The peak does appear to develop a shoulder on the h i g h energy si d e f o r T >• 45K f o r a l l samples i n t h i s concentration range. T h i s may be construed as the onset of evaporation of a condensed phase. The temperature dependence of the = 4.3 x 1 0 1 9 cm - 3 sample i s s t r a i g h t f o r w a r d . Within the experimental e r r o r , the two observed peaks broaden with temperature but do not s h i f t appreciably i n energy. Based on the data of Balkanski et a l . these peaks l i e e n t i r e l y above t h e i r r e s p e c t i v e values of E - hv which suggests that the photolumin-i 6 P escence i s due t o recombination of donor electrons and f r e e holes with-out the presence of a condensed phase. The temperature dependence of the peak p o s i t i o n would depend p r i m a r i l y on the e l e c t r o n density and at t h i s donor concentration would be i n s e n s i t i v e to temperature although i t would be subject to broadening. - 63 -CHAPTER 7 ''EFFECT OF HEAT-TREATMENT ON S i : P • ! 7.1 I n t r o d u c t i o n I t i s known that commercially produced s i l i c o n c r y s t a l s may c o n t a i n high concentrations of oxygen and carbon unless s p e c i a l pre-cautions are taken during manufacture. The number of oxygen impur-i t i e s present depends strongly on the method used to grow the c r y s t a l , with C z o c h r a l s k i grown c r y s t a l s having concentrations as high as i q ^(49) 1.5 x 10 3 cm - 3 and vacuum f l o a t zoned c r y s t a l s u s u a l l y having le s s 17 ^ (50) than 1.0 x 10 1' cm . The concentration of carbon impurities has been found to be ^ 5 x 1 0 1 8 cm - 3 r e l a t i v e l y independent of the manner of production . Both carbon and oxygen enter the s i l i c o n l a t t i c e as n e u t r a l i m p u r i t i e s , and as such have no appreciable e f f e c t on the e l e c t r o n i c p r o p e r t i e s of s i l i c o n . Their presence has been detected by (52 53) studies of the f a r i n f r a r e d absorption due to l a t t i c e v i b r a t i o n s ' (54) A report by F u l l e r et a l . on the formation of more than 2 x 1 0 1 6 donors-cm""3 i n i n t r i n s i c s i l i c o n a f t e r heat treatment at 300 - 500C sparked considerable i n t e r e s t i n the e f f e c t s of heat t r e a t -ment on s i l i c o n . K aiser found that prolonged heat treatment (greater than 10 hours) at 1000C r e s u l t e d i n the formation of p r e c i p i -tates w i t h i n the c r y s t a l . A considerable amount of work has been done on i d e n t i f y i n g these p r e c i p i t a t e s and the k i n e t i c s of t h e i r formation (see f o r example, Bullough and Newman ). The p r e c i p i t a t e s are now known to be of two types, l a r g e ones (10 - 40 um) c o n s i s t i n g of a Si02 - 64 -core surrounded by a layer of SiC, and smaller ones (5 - 10 ym) con-sisting only of S I C ^ 5 7 \ It i s also known that heat treatment for periods as short as 15 minutes at 1000C can produce considerable structural changes within the crystal, corresponding to the early stages of p r e c i p i t a t i o n O n e of these early stages is l i k e l y to be the ejection of carbon from i t s usual substitutional sites into i n t e r s t i t i a l sites. In this configuration carbon i s highly mobile and (Kg) can diffuse rapidly, even at 300K '. In this chapter the effects of heat treatment at 1150C for 30 minutes on the photoluminescence and the electron paramagnetic resonance are presented for the phosphorus impurity range 9 x 10 1 5 cm-3 < N D < 4.3 x 10 1 9 cm-3. 7.2 Photoluminescence from Heat Treated Si:P Heat treatment at 1150C for 30 minutes was carried out on samples of s i l i c o n with phosphorus impurity concentration in the range 9 x 10 1 5 cm-3 c < 4.3 x 10 1 9 cm"3. The photoluminescence of heat treated samples was found to differ markedly from that of samples which had not undergone heat treatment in the impurity concentration range 2.2 x 10 1 8 cm"3 < N D < 1.3 x 10 1 9 cm"3. Samples with impurity concen-tration outside this range were found to be essentially unaffected by heat treatment. Within this concentration range heat treatment produced two major changes in the photoluminescent spectra. F i r s t , a broad line developed at hv ^ 1.1 eV. This line was about 200 meV wide and appeared to be independent of both concentration and temperature. - 65 -13 K —J 1 ' 1 : L _ 1.03 1.06 1.09 1.12 1.15 PHOTON ENERGY ( eV) FIGURE 7.1 Photoluminescence v s . Impurity Concentration at 13K f o r Heat-Treated S i : P . - 66 -UJ ( J z UJ u </> Ul 2 2 r> _j O r -o I CL U. o >-(/) z UJ I-N D = 3 X 10 18 103 1.06 1.09 1.12 PHOTON ENERGY (eV) 1.15 125 K 80 K 58 K 36 K 28 K 19 K •13 K FIGURE 7.2 Photoluminescence vs. Temperature for Heat-Treated Si:P with N D - 3.0 x 10 1 8 cm-3. - 67 -Secondly, a new l i n e developed at hv = 1.05 eV which, was only observed f o r T < 20K and whose p o s i t i o n was only s l i g h t l y concentration dependent. This: emission l i n e i s somewhat broader than that observed due to recom-b i n a t i o n w i t h i n the EHD. The photoluminescence at 13K i s shown as a f u n c t i o n of concentration i n Figure 7.1 and the photoluminescence of = 3»0 x 1 0 1 8 cm 3 i s shown as a function of temperature i n Figure 7.2. Allowing the samples to anneal at 300K f o r several days was s u f f i c i e n t f o r the photoluminescent spectra to revert to that observed before heat-treatment . 7.3 EPR of Heat-Treated Si:P E l e c t r o n paramagnetic resonance (EPR) i s the study of magnetic d i p o l e t r a n s i t i o n s between spin states of a system. For phosphorus donors i n s i l i c o n only the ordinary Zeeman i n t e r a c t i o n need be considered f o r impurity concentrations > 2 x 1 0 1 8 cm - 3. This gives a s p l i t t i n g of the donor ground s t a t e AE = gu gH (7.3-1) where g i s the e f f e c t i v e g-value, u i s the Bohr magneton and H i s the p magnitude of the applied magnetic f i e l d . A l l other terms of the spin Hamiltonian, i n c l u d i n g the hyperfine i n t e r a c t i o n with P 3 1 n u c l e i , average to zero due to the d e r e a l i z a t i o n of the donor electrons. S i l i c o n samples from the same impurity concentration range as that used f o r the photoluminescent studies were heat treated at 1150C f o r 30 minutes and t h e i r EPR spectra observed at 77K, 4.2K and 1.1K. As with the photoluminescent studies, i t was observed that the heat t r e a t -- 68 -ment had no e f f e c t upon the spectrum for impurity concentrations N D < 1.1 x 10 1 8 cm"3, w i t h i n the experimental accuracy. The spectrum f o r s i l i c o n containing 2.0 x 1 0 1 8 cm - 3 phos-phorus c o n s i s t s of a s i n g l e symmetric l o r e n t z i a n l i n e as a r e s u l t of the p a r t i a l d e r e a l i z a t i o n of the electrons at t h i s concentration. A f t e r heat treatment, the spectrum (Figure 7.3) observed at 77K i s s l i g h t l y broadened (y 10%), while at 4.2K and 1.1K the spectrum con-s i s t s of a s i n g l e g r e a t l y broadened l i n e s h i f t e d to lower magnetic f i e l d (higher g-value). Figure 7.4 shows the EPR spectrum observed a f t e r heat t r e a t -ment of samples containing 6 x 1 0 1 8 cm - 3 phosphorus i m p u r i t i e s . The changes i n the spectra of t h i s sample at 4.2K and 1.1K are character-i s t i c of the changes observed i n the impurity concentration range 3.0 x 10 l a cm - 3 s $ 1.3 x 1 0 1 9 cm - 3. The spectrum i s a composite peak at g 'v 2.0 which may be separated into two components. One of these components has the same width and g-value as the s i n g l e l i n e due to conduction e l e c t r o n s observed i n samples that have not been heat t r e a t e d . A f t e r s u b t r a c t i o n of t h i s component there remains a broad asymmetric l i n e with a higher g-value. The width, asymmetry and g-value of t h i s l i n e a l l increase with decreasing temperature. Although lack of knowledge as to the shape of t h i s l i n e makes proper r e s o l u t i o n of the two l i n e s impossible, one may say that the g-value i s r e l a t i v e l y insen-s i t i v e to impurity concentration, whereas the linewidth increases r a p i d l y with concentration. A l l of the samples i n t h i s concentration range showed a small broadening of ^ 10% of the l i n e at 77K. FIGURE 7.3 EPR Spectra f o r Np = 2.0 x 1 0 1 8 cm-3. The dashed curves represent the spectra before heat-treatment, the s o l i d curves, a f t e r treatment. - 70 -Nr»= 6.2 X 101 \/ "> MAGNETIC FIELD FIGURE 7.4 EPR Spectra for = 6.2 x 1 0 1 8 cm - 3. The dashed curves represent the spectra before heat-treatment; the s o l i d curves, a f t e r treatment. - 71 -The number of electrons responsible f o r an EPR l i n e i s given by N a e x"(H)dH (7.4-1) where x" Is the complex s u s c e p t i b i l i t y . Due to the modulation technique 9 Y " used, the observed l i n e shape i s pro p o r t i o n a l to — , hence i t i s on. necessary to inte g r a t e the observed l i n e twice to obtain the area under the absorption curve. A comparison technique using a double-sample (35) modulation-switched sample c a v i t y developed by Quirt was used to determine i f Nfi was changed by the heat treatment. Although t h i s method i s accurate to ^ 3% i f the l i n e shapes are known, the uncertainty i n the cor r e c t l i n e shape due to the presence of long absorption t a i l s made i t impossible to determine the area under the curve to better than 15%. To t h i s accuracy and assuming no change i n spin s u s c e p t i b i l i t y , there was no change i n the number of electrons responsible f o r the observed EPR spectra before and a f t e r heat treatment. As with the photoluminescent work, the changes due to heat treatment disappear and the EPR spectra revert to that observed before heat treatment i f the sample i s allowed to anneal at 300K f o r several days. To avoid problems associated with the microwave skin depth i n t h i c k samples, the samples used f o r the EPR studies were ground to a powder with average s i z e < 50 ym. I t was found that i f the sample was ground before heat treatment there were no changes observed i n the - 72 -s p e c t r a . S i m i l a r l y i f the heat treatment was c a r r i e d out f o r times as long as 24 hours there was no change i n the spectra from the untreated. The EPR l i n e f o r heat treated samples with impurity concen-t r a t i o n s - 4.3 x 1 0 1 9 cm"3 was observed to be broadened by 10% at a l l temperatures. This broadening would be compatible with the presence of a second l i n e , however attempts to resolve t h i s were hampered by the width of the observed l i n e . 7.4 Discussion of Results Any attempt to develop a model which can explain the e f f e c t s of heat treatment must also deal with the question of why the e f f e c t anneals out at 300K and why powdered samples are- unaffected. I f the k i n e t i c s of the heat treatment are as outlined i n s e c t i o n 7.1, then these two aspects can be explained quite r e a d i l y . When the sample i s heated f o r only short periods of time and then quenched, carbon impur-i t i e s can be trapped at i n t e r s t i t i a l s i t e s . I f allowed to anneal at 300K these carbon i n t e r s t i t i a l s w i l l d i f f u s e to and be trapped by vacancies, or as i n the case of powdered samples, at the surface. The d i f f u s i o n length of carbon at 1150C i n s i l i c o n i s such that the carbon (59) i m p u r i t i e s would migrate to the surface of the powdered samples during the period of heat treatment where they would be trapped and l a t e r removed by etching. S i m i l a r l y , heat treatment for prolonged times r e s u l t s i n the formation of SiC p r e c i p i t a t e s which e f f e c t i v e l y remove the carbon impurity from the c r y s t a l . Thus the heat treatment and annealing p r o p e r t i e s could be explained i f the change's to the photolumin-escence and EPR were i n some way r e l a t e d to the presence of i n t e r s t i t i a l carbon i m p u r i t i e s . - 73 -The powder samples used for the EPR studies were graded according to p a r t i c l e s i z e , and the e f f e c t of heat treatment on the observed spectra was found to be a function of t h i s s i z e . This cannot be due to the e f f e c t of heat treatment on the surface of the p a r t i c l e s , s i n c e the sample i s ground a f t e r treatment and then etched. Nor i s there any such s i z e dependence i n the EPR of untreated Si:P. There i s an EPR l i n e due to the s u r f a c e b u t t h i s has g = 2.06 and i s quite w e l l resolved from those of i n t e r e s t . Although the problems involved i n separating the two l i n e s at g ^ 2.00 and determining the r e l a t i v e areas under the two absorption curves make i t d i f f i c u l t to make a d e f i n i t i v e statement, there i s not s u f f i c i e n t evidence for b e l i e v i n g that the sur-face plays a major r o l e i n the observed EPR spectra. More l i k e l y , the heat treatment produces l o c a l d i s t o r t i o n s whose d i s t r i b u t i o n w i l l vary throughout the c r y s t a l and which w i l l i nfluence the manner i n which the c r y s t a l f r a c t u r e s when ground to a powder. This would lead to a d i s t r i -bution i n p a r t i c l e s i z e which i s r e l a t e d to the density of these d i s -t o r t i o n s i n the heat treated samples. There are a number of complexes that i n t e r s t i t i a l carbon can (52) form with oxygen and s i l i c o n . These would produce a v a r i e t y of d i s t o r t i o n s and accompanying s t r a i n f i e l d s i n the l a t t i c e . The work of (57) Shul'pina et a l . suggests that such d i s t o r t i o n s would be short range, p o s s i b l y extending only several l a t t i c e spacings. I t i s u n l i k e l y that both the EPR and photoluminescence would show changes i n the same con-c e n t r a t i o n and temperature ranges, and that these changes would have the same annealing c h a r a c t e r i s t i c unless both are associated with the same c r y s t a l d e fect. On t h i s basis any model put forward must explain both sets of r e s u l t s . - 74 -The properties that these complexes must possess can be enumerated. They must be able to trap an e l e c t r o n and form a para-magnetic species. The time required f o r the capture of the ele c t r o n must be short compared to any experimental times so that the system i s i n e q u i l i b r i u m . It must, however, be longer than the time required f o r capture of an ele c t r o n by an ionized donor or for recombination of elec t r o n s and h o l e s . I f t h i s were not true than the e f f e c t s of the complexes would be observed at a l l impurity concentrations. The captured e l e c t r o n must be capable of tun n e l l i n g out of the "trap" by 20K, since the e f f e c t s of the heat treatment are only observed below t h i s temperature. The paramagnetic centre i n v o l v i n g a complex and a trapped e l e c t r o n w i l l contribute i n some way to the photoluminescence. This follows from the observation that there i s a new l i n e i n the photo-luminescent spectra only when the new paramagnetic s p e c i e s , as measured by EPR, i s present. In the v i c i n i t y of t h i s paramagnetic species electron-hole recomination occurs with the emission of a photon with energy hv ='1.05 eV independent of impurity concentration. Any model should also give some explanation f o r the broad background photolumines-cence observed. L The question i s , can these v a r i e d properties be r e c o n c i l e d to any one mechanism or are the experimental r e s u l t s being interpreted i n too r e s t r i c t i v e a fashion? One such mechanism that does appear to s a t i s f y a l l the r e q u i r e -ments i s the formation of n e u t r a l e l e c t r o n traps having a bound state j u s t s l i g h t l y below the impurity ground s t a t e , i . e . about 47 meV below - 75 -the; bottom of the i n t r i n s i c conduction band. The process responsible for. the photoluminescent l i n e at hv = 1.05 eV would be the recombina-tion, of a trapped e l e c t r o n with a photocreated hole. The di s c u s s i o n of the previous chapter i n d i c a t e s that t h i s hole i s probably i n a condensed phase. The only assumption necessary to explain the observed p r o p e r t i e s i s that the time required to trap an e l e c t r o n be long compared to the l i f e t i m e of photocreated c a r r i e r s . The time required for am e l e c t r o n to i n t e r a c t and recombine with a hole i s ^  2 x 1 0 - 6 s e c o n d s ^ a Q ( j m a y ^ e u s e < j a s a i o w e r l i m i t f o r the capture time f o r the t r a p s , s i n c e the capture of electrons by ionized donors i s much f a s t e r C— 1 0 - 9 seconds). The upper l i m i t i s given by the shortest c h a r a c t e r i s t i c time of the experiments and i s c e r t a i n l y l e s s than 1 second-As long as the impurity states aire l o c a l i z e d , the p r o b a b i l i t y of an e l e c t r o n being captured by a ne u t r a l trap i s low, since both capture by i o n i z e d donor and recombination wi t h i n the EHD are much f a s t e r mechanisms. I t i s only when there are s u f f i c i e n t i m purities that there are d e l o c a l i z e d electrons present f o r T < 20K that the proba-b i l i t y of e l e c t r o n capture by a trap w i l l become appreciable. The de-l o c a l i z a t i o n which occurs f o r £ 2.2 x 10-18 cm - 3 r e s u l t s i n an excess of f r e e e l e c t r o n s with i n f i n i t e l i f e t i m e which can now be captured. The trap becomes n e g a t i v e l y charged and repels other electrons, thus the trap w i l l not be screened by e i t h e r the e x i s t i n g donor electrons or by any e l e c t r o n condensate which may form. This negatively charged trap w i l l be paramagnetic and contribute to the EPR. Since there w i l l be a - 76 -range of trap energies there w i l l be a variety of contributions, resulting in an asymmetric resonance li n e . Being negatively charged, the trapped electron w i l l attract holes and recombine resulting in radiative emission. There are two types of holes with which recombin-ation can occur, free holes and holes in a condensed phase. Both processes are l i k e l y to occur, giving photoluminescence at hv = 1.065 eV and hv = 1.05 eV respectively, the condensed holes having a binding energy of ^ 15 meV.^  The width of.the.observed line would be controlled by the variation i n trap energies. Experimentally there i s a photoluminescent peak at hv =1.05 eV indicating that the recombination occurs via a condensed phase. This i s not surprising, since i t i s known that the time for capture by the con-densed phase i s short compared to the free carrier lifetime. This i s the reason recombination from free excitons i s not observed at 2K i n (3) i n t r i n s i c s i l i c o n . The electron would only remain trapped for T < 20K, since above this temperature the probability i s high that i t w i l l tunnel to the donor ground state, which i s only a few meV higher in energy. When the impurity concentration i s high enough,i.e. for > 4.3 x 10 1 9 cm - 3, the conduction band w i l l be lower than the trap level so that there w i l l be no bound state. The broad background luminescence i s not explained by this model, but could be attributed to a low density of traps with a wide range of energies. Recombination with free hole at hv *=> Eg *- hv ~ 47 meV ~ 1,065 eV. o p Recombination with hole i n condensed phase at hv ~ Eg - hv - 47 meV r - o p - 15 meV =1.05 eV. Eg Q = 1.170 eV in t r i n s i c band gap, hv p = 57.3 meV for TO phonon. - l i -lt Is p o s s i b l e to obtain an estimate of the number of such traps in the c r y s t a l . The photoluminescence spectra at 13K and the EPR sp e c t r a at 1.1K i n d i c a t e that the majority of the electrons are asso-c i a t e d with traps when = 2.2 x 1 0 1 8 cm - 3. There i s not a s i g n i f i c a n t c o n t r i b u t i o n to the EPR from conduction electrons, and the i n t e n s i t y of the photoluminescent l i n e a t t r i b u t e d to recombination of trapped e l e c -trons I s much greater than that a t t r i b u t e d to free Cor condensed) e l e c -t r ons. For higher impurity concentrations there are c l e a r l y two c o n t r i -butions to the EPR and photoluminescence, therefore the concentration of traps Is taken to be ^ 2 x 10 cm J which i s i n accord with the concen-t r a t i o n o f carbon expected to be present. This concentration i s expected to vary from sample to sample. The a c t u a l p h y s i c a l s i z e of the l a t t i c e d i s t o r t i o n g i v i n g r i s e to these traps may be several l a t t i c e spacings even though the e f f e c t i v e capture cross-section as determined by the capture p r o b a b i l i t y may be quite small. One outcome of t h i s model i s the i d e n t i f i c a t i o n of a condensed phase of holes f o r impurity concentrations < 1.3 x 1 0 i 9 cm - 3. This condensed phase may have a lower density than the EHD observed i n i n t r i n -s i c s i l i c o n , i n f a c t the r e s u l t s of the previous chapter suggest that i t must. In a simple p i c t u r e , the binding energy of a hole condensate should decrease i f the density changes s i g n i f i c a n t l y from n, = 3 x 1 0 1 8 cm~^ n. however t h i s p i c t u r e does not take i n t o account the e f f e c t of Ionized donors. These may r e s u l t i n a s h i f t of the equilibrium to lower density while maintaining the binding energy. - 78 -CHAPTER 8 CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY • I 8.1 Conclusions The photoluminescence of phosphorus doped s i l i c o n has been studied f o r 2K < T < 125K over the concentration range 9 x 1 0 1 5 cm - 3 -N„ < 4.3 x 1 0 1 9 cm . Si m i l a r studies were made of the photolumines-cence of the samples a f t e r heat treatment at 1150C and corr e l a t e d with studies on the e f f e c t of t h i s heat treatment on the el e c t r o n paramagnetic resonance of Si:P i n t h i s concentration range. On the basi s of t h i s work i t i s concluded that e l e c t r o n hole (39) d r o p l e t s , as observed i n i n t r i n s i c s i l i c o n , can be created i n phosphorus doped s i l i c o n with impurity concentration N^ < 2.2 x 1 0 1 8 cm - 3. Also the EHD r e t a i n s the same equilibrium density of n = 3.0 x 1 0 1 8 cm - 3 e l e c t r o n s and holes throughout t h i s range. The threshold energy and d e t a i l e d l i n e shape of the photoluminescent peaks associated with the EHD are modified by changes i n the conduction band density of states as the m a t e r i a l approaches the semiconductor-metal t r a n s i t i o n at - 3.0 x 1 0 1 8 cm" 3. When the impurity concentration i s greater than 2.2 x 1 0 1 8 cm"3 i t i s p h y s i c a l l y u n r e a l i s t i c f o r an el e c t r o n condensate to occur. There i s evidence, however, that a hole condensate can be formed. The e q u i l i b r i u m concentration of t h i s hole condensate decreases with increas-ing impurity concentration such that f o r N^ > 4.3 x 1 0 1 9 cm - 3 i t no longer makes a s i g n i f i c a n t c o n t r i b u t i o n to the luminescence. - 79 -The c r i t i c a l temperature f o r "evaporation" of the condensed phase i s found to increase to 1 « 45K f o r > 3.6 x 101 7 cm"3, as compared to T = 15K i n i n t r i n s i c s i l i c o n . This increase i s interpreted as being a r e s u l t of the screening of excitons and the replacement of the EHD-exciton eq u i l i b r i u m with an EHD-free el e c t r o n - f r e e hole e q u i l i b -rium. Any conclusions drawn f o r > 2.2 x 101 8 cm-3 must be con-sidered as t e n t a t i v e pending more accurate knowledge -of the concentra-t i o n dependence of the conduction band density of states i n t h i s concen-t r a t i o n range. Heat treatment of phosphorus doped s i l i c o n at 1150C f o r 30 minutes i s explained i n terms of neutral electron traps with a range of energies close to 47 meV below the i n t r i n s i c conduction band. These traps are associated with i n t e r s t i t i a l carbon being ejected from i t s s u b s t i t u t i o n a l s i t e s . The carbon i s present as a n e u t r a l impurity i n s i l i c o n c r y s t a l s unless s p e c i a l precautions are taken to remove i t . The i n t e r s t i t i a l carbon i s quite mobile at 300K, d i f f u s i n g r e a d i l y to vacan-c i e s on the c r y s t a l surface w i t h i n several days. 8.2 Suggestions f o r Further Work Although d e t a i l e d information on the concentration dependence of the s i l i c o n conduction band would be very u s e f u l , there are several experiments which may help c l a r i f y the present work without t h i s i n f o r -mation. Light s c a t t e r i n g experiments, s i m i l a r to those done i n i n t r i n s i c (3) germanium , would confirm or re f u t e the existence of macroscopic EHD f o r - 80 -<. 1.1 x 1Q1B cm"3. The proposal of the existence of a condensed hole phase could also be tested By this technique. A pulsed laser could be used to measure the lifetime of the condensed phase as a function of both concentration and temperature. This may prove to be the only way to settle the question of evaporation of the condensed phase, since i t should be possible to distinguish between different recombination mechanisms. Two other experiments which may be interesting to consider are the replacement of donor impurities with acceptor impurities to check the possible formation of an electron condensate, and replacement of phosphorus with a donor of different ionization energy. If the donor ground state i s different from that of phosphorus then the photolumines-cence and electron paramagnetic resonance lines produced by heat treat-ment should have a different temperature dependence. This information could then be used to gain more insight into the nature of the traps produced. - 81 -APPENDIX A The output of any spectrometer which uses a monochromator as an i n t e g r a l part i s of necessity broadened by the e f f e c t i v e band width of the instrument. In order to study lineshapes or widths i t i s necessary that t h i s broadening be removed. The observed spectra are a c t u a l l y a convolution of the true spectra with a s l i t f u nction. The s l i t function f o r the monochromators used f o r t h i s work are gaussian and given by: F CE - E ) = - i - exp s o r- a * ( E - E o where 2flnl or i s the width at h a l f maximum of the s l i t p r o f i l e . This i s the observed l i n e shape when i t i s known that the true l i n e i s much narrower than the observed l i n e . The observed p r o f i l e L (hv) i s given by: Lp(hv) F s ( E - hv) L T ( E ) d E where hv i s the photon energy and L^CE) i s the true l i n e p r o f i l e . A f a i r l y simple technique f o r obtaining L^,(E) i s given by S a v i t z k y ^ ^ . This technique i s a p p l i c a b l e i n cases where the l i n e width i s more than twice the s p e c t r a l width cr, and employs the i t e r a t i v e procedure - 82 -L (hv) « L (hv) + L (hv) -n n«~l c F (E - hv)L ,(E)dE s n-l LpChv) is used as the first t r i a l function L Q(hv). The following is a sample computer program written in FORTRAN G which converts the spectrometer wave drive readings to photon energies, smooths the data with a nine point smoothing routine, and extracts the true line profile. The inputs required are: WD - the starting value of the wave drive STPSZ - the wavedrive interval between successive data points N - the number of data points SLTW - the effective s l i t width given in eV LF(I) - the data to be analysed. The constants A, B, C, and D are obtained from a calibration of the particular grating used. On output the true profile is contained in OLDOM(I) and the corresponding photon energies in HV(I). - 83 -REAL LF(300),HV CiOO) jNOMEGA(30 0),OLDOMC500},F(300,300),SPECTR(300) 7 READ (S,l,F .NP=i?) WD,STPSZ,N 1 1 FORMA T (2F10,0,13) '• ! Re*n—«T2->-3tT-* • 2 FORMAT (F10.0) READ (5,3) (LF(I),I=l,N) 3 FORMAT M6F5,1 ) — C SMOOTH OAT A M=N-b po io I = O , H — — : — : : — LFCI)=(-36,*(LF (I - 5)*LKI+ 5 ) )+9, * (IF (I •»«)+LF (I •<!) ) , * U F ( I ^ 3) t*LFCI*3)W69,*(LF(I»2)*LF(1+2))+ 8 4 ,*(LF(I-I)+LF(1*1))*B9,* 2LF(I>)/429 t-10 CONTINUE C CONVtRTSPfcCTRUM TO PHOTON ENfcHGIES P s 10,/b%  A = 0,23620 S 3 0.783S2E-02 e — Q r , - u±ja i E - Q b 00 20 1=1,N THETA s A+B*W0*C*W0*N0 HV(l)8lg39 ,a5ai / t.',yO/3IN(THCTA)/l ,0E»03 mo = wO*STPSZ OLOOM(I) = LF(I) 59 CONTINUE '• < C CALCULATE SLIT FUNCTION F SIGMAsSLTw/l,665 : 00 30 1»1,N 00 30 J=1,N FCIrJ) = ,b6il2/SIGMA*EXP(-(CHV(J)-HV(I))/SIGMA)**2) -JO— CONT I NUfc ; C PERFORM CONVOLUTION AND ITERATE NOMEGA(1 ) = LF(1) 0LD1FF a ^ E + gb :  00 40 M=l,10 DIFF s o , DO so—I »2 rN SUMso, OLOVAL = F(I,1)»0LD0M(1) 00 60 J=2,N VAL = F(I,J)»OLDOHtJi : '• : — ~ SUM=SUM*,5*(VAL*0LDVAL)*(HV(J)-.HV(Jwl)) OLOVAL = VAL 60 CONTINUE NOMEGA(I)=OLOOM(I)+LF(I)*SUH DIFF=DIFF*(SUM-.LF (I ) )*(SUM-PLF (I) ) -SO CONTIfWE : : IF (OLOIFF ,LT, OIFF) GO TO 11 ' •_, PRINT 8, M, DIFF —9 FORMAT—(-•—AT- ITER A T ION-«+rI 3 r« SOU ARE—OF—DIFFERENCES - S U M M E D — — 1 F15.3) OLOIFF a OIFF DO 70 1 = 1,N 70 OLDOM(I)sNOMEGAU) 40 CONTINUE — « CONTINUE-- 84 -APPENDIX K The line p r o f i l e expected for radiative recombination between two bands can be calculated in a straightforward manner. The intensity of photoluminescence I (hv) i s given by: i-i ILChv) N (E e) N(E h)f(E e)f(E h) 5(hv-E -E e-E h)dE edE h 0 0 where A i s a constant depending on the matrix elements connecting the states and is assumed to be independent of energy. N(E g) and NCE^) a r e the density of states for the conduction band and valence band respectively and for parabolic bands are given by: N(E) = 2m * V 3 / 2 •1/2 f ( E e ) and f(E^) are the electron and hole probability distribution functions respectively. The Fermi energy for a parabolic band is : readily obtained from N(E): 2m substituting i n the appropriate values for s i l i c o n , allowing for the six conduction band minima and doubly degenerate valence band one obtains - 85 -I* - 1.14 x I0~ lh (|) 2/3 eV E** = 0.697 x l O - 1 4 ( n ) 2 / 3 eV Jc f o r electrons and holes, where n i s the concentration of electrons and holes. At f i n i t e temperatures the e l e c t r o n and hole p r o b a b i l i t y d i s t r i b u t i o n functions are given by: fCE) -exp (E - ?)'/kT + 1 where cfT) - E p - j- ( k T ) 2 9 i 3E l n r \ N(E) E = E, f o r p a r a b o l i c bands N(E) a E 1/2 • r m - F — C K T ) 2 For an electron-hole drop with binding energy EA the i n t e n s i t y of photoluminescence i s given by OO QO 0 f Id(hv) E1/2E1/2 0 0 exp ^ e ' ^ / k T +1 exp ° W / k T +1 x S(hv-E -E -EL+EA+E^+E^+tUo)dE dE g e h F F ' e n - 86 -where hco i s the energy of the phonon a s s i s t i n g the recombination. This I n t e g r a l can be solved quite r e a d i l y -using numerical techniques. The f o l l o w i n g i s a computer program written i n FORTRAN G to solve t h i s i n t e g r a l and p l o t the l i n e p r o f i l e . In t h i s program a phonon energy hu = .0578 eV i s used. The program input required i s T - temperature f o r which the l i n e shape i s required EGAP - band gap ESTP - the number of e l e c t r o n and hole energies to be used N - the number of points i n the output p r o f i l e EA - the binding energy i n eV CON - e q u i l i b r i u m concentration of EHD. The l i n e p r o f i l e i s scaled to 5 inches and the e l e c t r o n and hole Fermi energies are p r i n t e d on output. I - 87 -IMEGEW ESTP REAL K , K T » t E L ( 2 0 0 ) » F D E ( 2 0 0 ) » H V ( 5 0 0 ) # I N T ( S O O ) READCS.l ) T i EGAP RE-AO-fS ^ - K W T N :  9 READ(5,2,EN0=8) EA.COM 1 FO*MAT(2F10.0) — g FOHMAT(F10,0,E7,1) — : 3 FORHATC2I3) C CALCULATE FERMI ENERGIES E-F-E-s-l-r-Hli-4*4C-0N At>-r-)-«*42« / 3 « ) • H F E c 0 . 6 O 7 E - l O * C 0 N * * ( 2 , / 3 . ) PRINT U,EFE, MFE —H FQk'MAT ( i ' ,T10 . IEFE • , T 2 5 , >HFE'V/ (2F15.5)J C CALCULATE CHEMICAL POTENTIALS K=8.62E-05 K^f-K*4 :— FAC = V.8h96>/12.E-0«*KT*KT CPE = EFE-FAC*ALOG( CEFE + 1 . OE-OU)/EFE) CPM = HFfc-FAC*ALOG( (HFt*l ,0E-0tt)/HFE) PRINT b, CHE,CPH 5 FORMAT( 1 I » T 1 0 i , C P E , # T 2 5 » ' C P H ' / / ( 2 F 1 5 . 5 ) ) fcU4^=EFE*S.«KT HUL=HFE*S.**T ESS=EUL/(ESTP-1) : — E E L U ) = 0.0 : : — 00 10 I=2»ESTP EELCI)=EELCI-n+ESS W CONTINUE — C CALCULATE FERMI OIRAC FUNCTION DO 20 1 = 1 , tSTP _ E00M=(EELCI )-CPE)/KT. : '. FDECI)=1.0/CEXP(tDOMj*I .O) 20 CONTINUE PBOBsFDE (ESTP) PRINT 6, EULfPKOB 6 F 0 R M A T ( ' • , T 1 0 , ' E U L ' , T 2 5 , , P R O R ' / / ( 2 F 1 5,5n C _eALCULATE -HHaiOLU.MlNESCE^CE- IHTENSI - IY HV(1 )=EGAP-0.0b7ci-EA-EFE-HFE D E L T A = ( E U L * H U L)/(N - U o o 30 i=2v» : HV (I )=HV (I- l)+DELTA 30 CONTINUE : , D o-jw-j=I,N : : i__ INT(J)=0.0 OLDVAL=0,0 00 UP I=2,ESTP : EH=HV(J) -HV( l)-EELCn IF CEH.LT ,0 .0 ) fcNsO.O J . 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