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Impurity band photoconductivity in Boron-doped silicon Scott, Myrsyl Walter 1966

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IMPURITY BAND PHOTOCONDUCTIVITY IN BORON-DOPED SILICON by M. WALTER SCOTT ., University of B r i t i s h Columbia, Vancouver, B.C., 1962. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1966 In present ing th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r ee l y ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th i s thes i s for s cho la r l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i c a t i on of t h i s thes i s for f i nanc i a l gain sha l l not be allowed without my wr i t ten permission The Un ivers i t y of B r i t i s h Columbia Vancouver 8, Canada Department of The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of MYRSYL WALTER SCOTT B , S c , University of B r i t i s h Columbia,, 1962 WEDNESDAY, MAY 11, 1966 AT 3:30 P.M. IN ROOM 301, HENNINGS BUILDING COMMITTEE IN CHARGE Chairman: B. M.. Moyls J. W. Bichard F. W. Dalby Ro Barrie K . B , Harvey R. C. Williams E. Teghtsoonian External Examiner: E. Burstein Department of Physics Uni v e r s i t y of Pennsylvania Ph i l a d e l p h i a , Pa. IMPURITY BAND PHOTOCONDUCTIVITY IN BORON-DOPED SILICON ABSTRACT The e f f e c t of impurity concentration on the photo-conductive spectrum of boron-doped s i l i c o n , at helium temperatures was investigated. Photoconductivity was observed for e x c i t a t i o n s of the bound hole into the impurity excited states. The photoconductivity i n t h i s region depends strongly on impurity concentration and was interpreted as being conduction through excited state impurity bands. Two bands were observed to form s with excited states 3 and 4 i n "the ber on spectrum forming one band and state 2 forming the other. A q u a l i t a t i v e d e s c r i p t i o n of the mob i l i t y i n the impurity bands was obtained using Baltensperger's theory and the add i t i o n a l assumption that the holes are scattered by 'randomness' in the impurity array, Photoconductivity of holes i n the valence band was also measured i n order to obtain l i f e t i m e s and capture cross sections. Assuming the mobility i n th i s region to be determined by neutral impurity s c a t t e r i n g , the hole l i f e t i m e was estimated to be 10"^ sec and the capture cross section of ionized boron 3 x 10'^cm^. Assuming s i m i l a r l i f e t i m e s for the holes i n impurity bands, the peak mobility i n band 3 and 4 was found to be *— 150 cm^/volt-sec. The d.c. c h a r a c t e r i s t i c s of the various samples s while at low temperature and exposed to room temperature r a d i a t i o n , were measured during the course of t h i s i n v e s t i g a t i o n . A l l samples were observed to have a non-linear dependence between the current and applied f i e l d , terminated by a non-destructive low f i e l d break-down „ GRADUATE STUDIES F i e l d of Study: S o l i d State Physics Quantum Mechanics Waves Electromagnetic Theory,/ Quantum Theory of Solids Noise i n Physical Systems Spectroscopy Special R e l a t i v i t y S t a t i s t i c a l Mechanics Low Temperature Physics W„ Opechowski J . G. Savage G. M Volkoff R, B a r r i e R. E. Burgess A. M. Crooker A. J o Barnard P . R a s t a l l R. Barrie J . B o Brown Related Topics i Applied Electromagnetic Theory G. Bo Walker PUBLICATIONS M, Wo Scott and J. J, White Degeneracy of Impurity States i n Boron-Doped S i l i c o n . Can.J.Phys. 43, 1388 (July, 1965)= Chairman: Professor J.W. Bichard - i i -ABSTRACT The e f f e c t of impurity concentration: on the photoconductive spectrum of boron-doped s i l i c o n at helium temperatures was investigated. Photo-conductivity was observed for e x c i t a t i o n s of the bound hole into the im-purity excited states. The photoconductivity i n this region depends strongly on impurity concentration and was interpreted as being conduction through excited state impurity bands. Two bands were observed to form, with excited states 3 and 4 i n the boron spectrum forming one band and state 2 forming the other. A q u a l i t a t i v e d e s c r i p t i o n of the mobility i n the impurity bands was obtained using Baltensperger 1s theory and the a d d i t i o n a l assumption that the holes are scattered by "randomness" i n the impurity array. Photoconductivity of holes i n the valence band was also measured i n order to obtain l i f e t i m e s and capture cross sections. Assuming the mobility i n this region to be determined by neutral impurity scattering, the hole l i f e t i m e was estimated to be ^ 10"^ sec and the capture cross -10 2 section of ionized boron ^ 3 x 10 cm . Assuming s i m i l a r l i f e t i m e s f or the holes i n impurity bands, the peak mobility i i i band 3 and 4 was found 2 to be ^ 150 cm /volt-sec. The d.c. c h a r a c t e r i s t i c s of the various samples,while at low temperature and exposed to room temperature radiation,were measured during the course of this i n v e s t i g a t i o n . A l l samples were observed to have a non-linear dependence between the current and applied f i e l d , terminated by a non-destructive low f i e l d breakdown. - i i i -TABLE OF CONTENTS Page A b s t r a c t i i T a b l e o f C o n t e n t s i i i L i s t o f T a b l e s v L i s t o f F i g u r e s V i Acknowledgements i x C h a p t e r I - I n t r o d u c t i o n 1 C h a p t e r I I - E x p e r i m e n t a l P r o c e d u r e and A p p a r a t u s A. Sample P r e p a r a t i o n 5 C o n t a c t s 6 B. The S p e c t r o m e t e r 8 C. The Dewar and Sample M o u n t i n g 9 D. D e t e c t i o n System 11 C h a p t e r I I I - R e s u l t s and A n a l y s i s A. S i g n a l f r o m a Non-Ohmic P h o t o c o n d u c t o r . . . 15 B. The P h o t o c o n d u c t o r as a C i r c u i t Element . . 17 C. D.C. C h a r a c t e r i s t i c s 19 D. S p e c t r a l Response 23 E. P h o t o c o n d u c t i v i t y Measurements . 29 C h a p t e r IV - Theory and D i s c u s s i o n A. I m p u r i t y P h o t o c o n d u c t i v i t y 42 - i v Page B. Hole M o b i l i t y and Lifetime 1. M o b i l i t y , 55 2. Lifetime 59 C. Impurity Conduction 1. Conduction Mechanisms 69 2. Conduction i n Impurity Bands 71 Chapter V - Conclusions 84 Appendix A - Performance of Boron-Doped S i l i c o n as a Far Infrared Detector 87 Bibliography 91 LIST OF TABLES Page Table I D.C. C h a r a c t e r i s t i c s of the Samples 22 Table II Valence Band M o b i l i t i e s of the Free Holes 57 Table III Comparison of Detecting A b i l i t i e s of Various Types of Infrared Detectors 88 - V I LIST OF FIGURES Page FIG. I . Energy l e v e l scheme of boron i n s i l i c o n 3 FIG. 1. Shape o f specimens 6 FIG. 2. Method o f sample mounting 10 FIG. 3. B i a s i n g network 11 FIG. 4. P r e a m p l i f i e r C i r c u i t 12 FIG. 5. B l o c k d i a g r a m of complete d e t e c t i o n system . . . 14 FIG. 6. A.C. e q u i v a l e n t o f the p h o t o c o n d u c t o r 18 FIG. 7. C u r r e n t - v o l t a g e c h a r a c t e r i s t i c s o f samples of c o n c e n t r a t i o n 1.2 x 1 0 1 6 / c m 3 and 1.5 x 1 0 1 8 / c m 3 . 20 FIG. 8. C u r r e n t - v o l t a g e c h a r a c t e r i s t i c s o f samples o f c o n c e n t r a t i o n 4.5 x 1 0 ^ / c m 3 and 4.5 x 1 0 ^ / c m 3 . 21 FIG . 9. S p e c t r a l r e s p o n s e f o r v a r i o u s i m p u r i t y c o n c e n t r a t i o n s (a) Samples o f c o n c e n t r a t i o n 1.2 x 10 /cm and 4.5 x l O ^ / c m 3 25 (b) Samples o f c o n c e n t r a t i o n 4.5 x 10^^/cm 3 and 1.5 x 1 0 l 8 / c m 3 26 FIG. 10. P h o t o c u r r e n t i n e x c i t e d s t a t e i m p u r i t y bands a t v a r i o u s i m p u r i t y c o n c e n t r a t i o n s 27 FIG. 11. Measured s i g n a l and n o i s e as a f u n c t i o n o f b i a s i n g c u r r e n t t h r o u g h the samples ( e x c i t a t i o n by 75 meV. photons) (a) 1.2 x 1 0 1 6 / c m 3 30 (b) 4.5 x 1 0 1 6 / c m 3 31 ( c ) 4.5 x 1 0 l 7 / c m 3 32 (d) 1.5 x 1 0 1 8 / c m 3 33 - v i i -Page FIG. 12. P h o t o s i g n a l g e n e r a t e d a t the p h o t o c o n d u c t o r by 75 meV. photons v s . a p p l i e d e l e c t r i c f i e l d . . . 35 (a) 1.2 x 1 0 1 6 / c m 3 35 (b) 4.5 x 1 0 1 6 / c m 3 . 36 (c) 4.5 x 1 0 1 7 / c m 3 37 (d) 1.5 x 1 0 1 8 / c m 3 38 FIG. 13. R e s i s t a n c e due t o c o n t a c t s v s . c u r r e n t t h r o u g h the samples 40 FIG. 14. N o i s e s p e c t r a , f o r 1.2 x l O ^ / c m 3 sample f o r o p p o s i t e d i r e c t i o n s o f c u r r e n t f l o w 41 FIG. 15. Conductance change produced by f l u x o f 75 meV. photons v s . a p p l i e d e l e c t r i c f i e l d 50 (a) 1.2 x 1 0 1 6 / c m 3 50 (b) 4.5 x 1 0 1 6 / c m 3 51 ( c ) 4.5 x 1 0 1 7 / c m 3 52 (d) 1.5 x 1 0 1 8 / c m 3 . ' 53 FIG. 16. Conductance change a t d i f f e r e n t photon e n e r g i e s 16 3 v s . a p p l i e d e l e c t r i c f i e l d f o r the 4.5 x 10 /cm sample 54 FIG. 17. Dependence o f co n d u c t a n c e change on i m p u r i t y c o n c e n t r a t i o n f o r v a r i o u s photon e n e r g i e s . . . . 58 FIG. 18. F r a c t i o n a l change i n c o n d u c t a n c e produced by 75 meV. photons v s . a p p l i e d f i e l d 62 FIG. 19. P h o t o c o n d u c t i v e l i f e t i m e t"(A) and d.c l i f e t i m e v s . a p p l i e d e l e c t r i c f i e l d (a) 1.2 x 1 0 1 6 / c m 3 63 (b) 4.5 x 1 0 1 6 / c m 3 64 - v i i i -Page (c) 4.5 x 10 1 7/cm 3 65 (d) 1.5 x 10 1 8/cm 3 66 FIG. 20. Formation of the Is and 2p impurity bands from hydrogenic wavefunctions (after Baltensperger) . . 72 FIG. 21. (a.) Ratio of e f f e c t i v e mass i n valence band to e f f e c t i v e mass i n 2p impurity band vs. impurity concentrations 77 (b) Theoretical mobility i n the 2p impurity band i n terms of the dimensionless constant vs. impurity concentrations 77 FIG. 22. Measured conductance change produced by e x c i t a t i o n of holes into two d i f f e r e n t impurity bands at various impurity concentrations . 79 - i x -ACKNOWLEDGEMENTS I would l i k e to thank Dr. J.W. Bichard for invaluable assistance and advice throughout this i n v e s t i g a t i o n . I would also l i k e to thank Dr. R. Barrie for advice and c r i t i c i s m i n the preparation of th i s thesis. I would also l i k e to express my gratitude to the National Research Council for the award of a studentship. The research for this thesis was supported by the National Research Council, grant number A-2204. CHAPTER I INTRODUCTION I f a group I I I i m p u r i t y , such as b o r o n , i s i n t r o d u c e d i n t o a p e r f e c t s i l i c o n l a t t i c e an i n f i n i t e s e r i e s o f s t a t i o n a r y s t a t e s l o c a l i z e d a t the i m p u r i t y w i l l be formed (Kohn 1957). As the i m p u r i t y c o n c e n t r a t i o n i s i n c r e a s e d , the average s p a c i n g between i m p u r i t i e s w i l l d e c r e a s e and o v e r -l a p w i l l o c c u r between w a v e f u n c t i o n s a s s o c i a t e d w i t h p a r t i c u l a r i m p u r i t i e s . T h i s o v e r l a p between i m p u r i t y w a v e f u n c t i o n s w i l l b roaden the l e v e l s , as has been e x p e r i m e n t a l l y o b s e r v e d (Newman 1956, Colbow 1963). A t s u f f i c i e n t l y h i g h c o n c e n t r a t i o n s the e x c i t e d s t a t e w a v e f u n c t i o n s o f n e i g h b o u r i n g i m p u r i t i e s w i l l o v e r l a p t o such an e x t e n t t h a t the l e v e l s b e g i n b a n d i n g . By t h i s we mean t h a t the e x c i t e d s t a t e w a v e f u n c t i o n s can no l o n g e r be c o n s i d e r e d as b e i n g l o c a l i z e d a t a p a r t i c u l a r i m p u r i t y , b u t ex t e n d t h r o u g h o u t the c r y s t a l . These bands o f n o n - l o c a l i z e d s t a t e s a r e c a l l e d the i m p u r i t y bands. As the c o n c e n t r a t i o n i n c r e a s e s , t h e s e bands w i l l b e g i n o v e r l a p p i n g w i t h each o t h e r and the v a l e n c e band, e f f e c t i v e l y r e d u c i n g the i o n i z a t i o n e n e r g y o f the i m p u r i t y . I n o r d e r t o e x p l a i n the anomalous b e h a v i o r o f the r e s i s t i v i t y and H a l l c o n s t a n t i n s e m i c o n d u c t o r s a t low t e m p e r a t u r e s (see l i s t o f r e f e r e n c e s i n Ray and Fan 1960) E r g i n s o y (1950) s u g g e s t e d t h a t c o n d u c t i o n c o u l d o c c u r t h r o u g h t h e s e e x c i t e d s t a t e i m p u r i t y bands. The a n o m a l i e s o b s e r v e d can be summarized as f o l l o w s . As the t e m p e r a t u r e i s d e c r e a s e d the r e s i s t i v i t y and H a l l c o n s t a n t i n c r e a s e e x p o n e n t i a l l y , as e x p e c t e d . A t a c e r t a i n t e m p e r a t u r e (which depends on i m p u r i t y c o n c e n t r a t i o n ) the r e s i s t i v i t y b e g i n s i n c r e a s i n g a t a. d i f f e r e n t r a t e w i t h t e m p e r a t u r e , c o r r e s p o n d i n g t o a d i f f e r e n t a c t i v a t i o n e nergy o f the h o l e s . T h i s change 2 i n a c t i v a t i o n energy i s believed to correspond to the e x c i t a t i o n of holes into the excited state impurity bands. Baltensperger (1953) used a simple model i n which the impurities form a regular s u b - l a t t i c e to investigate the formation of impurity bands. The e f f e c t of varying the impurity concentration on both the broadening of the levels and the mobility in these bands was discussed using hydrogenic wavefunctions to describe the impurity states. The e f f e c t s of disorder on the formation of impurity bands was investigated by James and Ginzbarg (1953) f o r a one dimensional array of impurities. In t h i s randomly d i s t r i b u t e d array the impurity band no longer has d e f i n i t e edges as calculated by Baltensperger, and, i n f a c t , the only remaining i n d i c a t i o n of the o r i g i n a l impurity l e v e l i s a maximum in the density of states at the o r i g i n a l energy. In addition, the density of states t a i l s o f f on both the low and high energy sides of the impurity l e v e l . The previous experimental investigations of impurity band conduction have always used thermal generation as the means of e x c i t i n g c a r r i e r s into the impurity bands. The d i f f i c u l t y with this method i s that the holes assume an energy d i s t r i b u t i o n which w i l l extend over the valence band as w e l l as the impurity bands. The e f f e c t s of the impurity bands w i l l be masked by conduction i n the valence band. A more d i r e c t method of e x c i t i n g the c a r r i e r s w i l l be used i n this experiment. By photo-exciting the holes with monochromatic l i g h t only into the impurity excited states and measuring the resultant conductivity changes the existence of the impurity bands could be revealed without being masked by conduction i n the valence band. The photoconductive process i n semiconductors can be described quite 3 simply i n i t s most general sense. Incident photons excite current c a r r i e r s (holes) from one state into an excited state i n which the mobility i s d i f f e r e n t from the o r i g i n a l state. The c a r r i e r w i l l l i v e f o r an average length of time *£" i n this excited state before returning to i t s o r i g i n a l state, either d i r e c t l y or through some intermediate states. When in the excited state, the change i n mobility of the c a r r i e r w i l l change the conductivity of the material. This change i n conductivity can be detected as a change i n current through the sample. The energy levels of the boron impurity i n r e l a t i o n to the valence band structure of s i l i c o n i s shown in f i g . I. The hole i s shown i n the impurity ground state, where i£ w i l l normally be unless excited e i t h e r thermally or o p t i c a l l y . Incident photons of energy less than (wavelength longer than 25 JLL ) w i l l excite the holes into the impurity excited states whereas photons with energy greater than Ej w i l l excite the. holes into either the P3/2 o r 1^/2 v a - l e n c e bands. Ground state E x c i s e d states FIG. I Energy l e v e l scheme of boron i n s i l i c o n . E-j- = 46 meV., A = 44 meV. The e x c i t a t i o n s into impurity excited states i s of p a r t i c u l a r i n t e r e s t since no conductivity change w i l l r e s u l t from these e x c i t a t i o n s u n t i l such time as the excited states have formed non-localized impurity bands with a. non-zero mobility. In addition to depending on the change i n mobility of the hole between the i n i t i a l and f i n a l state, the observed change in conductivity 4 w i l l depend c r i t i c a l l y on the l i f e t i m e s of the excited hole. This l i f e -time i s determined ei t h e r by the rate of recombination of c a r r i e r s d i r e c t l y with impurity ground states, or by the rate with which they become trapped by l o c a l i z e d excited states of the impurities. These trapping centers w i l l be boron impurities which have been ionized e i t h e r by compensating centers or by incident r a d i a t i o n . The mechanism of cap-ture of free holes by these centers has been discussed by Lax (1960) i n his theory of giant traps. In this mechanism the capture p r o b a b i l i t y decreases r a p i d l y as the energy of the recombining holes i s increased, so that the l i f e t i m e could be increased by e x c i t i n g the holes to a. high energy or by increasing t h e i r average energy with an applied f i e l d . In this experiment the photoconductivity was measured over a range in wavelengths covering e x c i t a t i o n s into the excited states (impurity bands) to e x c i t a t i o n s into the valence band. These measurements were ca r r i e d out at a. fixed b i a s i n g current through each sample to obtain the r e l a t i v e s p e c t r a l response curves for the d i f f e r e n t impurity concen-tr a t i o n s . To investigate the e f f e c t s of the applied f i e l d on the l i f e - , time of the excited holes, the photoconductivity was measured as a function of the bias current through the sample for fixed wavelengths of l i g h t . The i n t e r p r e t a t i o n of the data from these measurements was complicated by a. non-ohmic behavior of the sample resistance. This "non-ohmicity" of the samples increases with the applied f i e l d and i s associated with a heating of the holes by the applied f i e l d . At s u f f i c i e n t l y strong f i e l d s the average energy of the holes i s increased to such an extent that impact-ionization of neutral impurities produces a non-destructive breakdown of the sample. 5 CHAPTER II EXPERIMENTAL PROCEDURE AND'APPARATUS The experimental problems involved i n observing photoconductivity and the e f f e c t of band formation on the photoresponse are as follows: (a) F a b r i c a t i o n of e l e c t r i c a l contacts on the s i l i c o n which would not generate e l e c t r i c a l noise at low temperatures. (b) I r r a d i a t i o n of the sample with long wavelength infra-red r a d i a t i o n while the sample i s kept at l i q u i d helium temperature. (c) Detection of the resistance change produced by this r a d i a t i o n . A. SAMPLE PREPARATION Wafers of the desired material were cut from s i n g l e - c r y s t a l ingots of s i l i c o n using a diamond abrasive wheel. These wafers were ground f l a t with #600 SiC on a glass plate. The impurity concentration was then determined from a four-point probe measurement of the room-temperature r e s i s t i v i t y . The r e s i s t i v i t i e s used were 1.2-fL-cm^, 1 1 2 0.47 XI-cm , 0 . 1 1 f l - c m , and 0.047 XI-cm corresponding to impurity concentrations of 1.2 x 10 1 6/cm 3, 4.5 x 10 1 6/cm 3, 4.5 x 10 1 7/cm 3, 18 3 1.5 x 10 /cm r e s p e c t i v e l y . In order to measure the f i e l d , the sample was cut with an u l t r a - s o n i c impact grinder as shown i n f i g . 1. The specimen was epoxied to a. 3/4" diameter synthetic sapphire window Supplied by Dow-Corning Supplied by Semi-Elements Inc. 6 0.040" thick to provide support for the weak side arms and also to provide e l e c t r i c a l i s o l a t i o n and thermal contact f o r the samples. prepared low v i s c o s i t y epoxy suitable The epoxy used was a s p e c i a l l y for forming thin layers which did not FIG. 1 Shape of specimens. ( about 2x actual size ) crack when cooled (Glass 1964). The samples were ground down to the desired thickness, where possible, as mentioned above and then etched i n 16 hot NaOH. The thicknesses used were 2.26 mm (1.2 x 10 ) s 0.74 mm i c 17 18 (4.5 x 1 0 i 0 ) , 0.23 mm (4.5 x 10 ) and 0.23 mm (1.5 x 10 ), with a possible deviation of £ 0.02 mm on a l l samples. The samples were then ready f o r the contacts to be put on. Various methods of producing acceptable e l e c t r i c a l contacts to the s i l i c o n were t r i e d . These contacts were then judged according to the following requirements: temperatures without change i n t h e i r c h a r a c t e r i s t i c s (c) S u f f i c i e n t physical strength i n order to minimize the necessity of s p e c i a l handling techniques (d) Ease of f a b r i c a t i o n . The following methods were t r i e d i n order to produce the contacts, but unfortunately did not prove too successful. It. i s worthwhile to indicate the d i f f i c u l t i e s encountered with these methods and why they CONTACTS (a) Low noise (b) A b i l i t y to cycle from room temperatures to helium 7 were not as successful as the method used. Pure indium was applied to the etched s i l i c o n surface with an u l t r a -sonic soldering iron . This procedure greatly increased the breakage rate of the samples and the time of f a b r i c a t i o n , but did not improve the noise c h a r a c t e r i s t i c s of the contacts. Pure gallium was applied to the f r e s h l y etched surface. This contact was easy to produce and had lower noise than the indium but suffered from a serious drawback. The contact cracked a f t e r c y c l i n g between 4.2 °K and room temperature, i n v a r i a b l y breaking o f f part of the sample. Gold was plated on the surface and alloyed with the s i l i c o n . This contact was poor i n that i t had a high contact resistance and produced a high l e v e l of e l e c t r i c a l noise. These contacts also had the p r a c t i c a l disadvantage of being quite d i f f i c u l t to apply. The simplest method found to produce the most consistent low-noise contacts was to apply 100% indium solder to the s i l i c o n . The indium was melted on the t i p of a. small soldering iron and then rubbed against the contact area. By rubbing the s i l i c o n with the molten indium, the outer oxide layer on the indium bead could be broken and the s i l i c o n "wet" with fresh indium. Gold leads were then attached to the indium. Care had to be taken with the thin samples since an uneven heating caused them to crack. This prevented the use of an optimum thickness for the two heavier doped samples. The contacts were tested by measuring the sample resistance, with a Wheatstone bridge for the two d i r e c t i o n s of current flow through the sample. If the resistances in the two d i r e c t i o n s were not within one percent of each other the contact could generally be improved by sparking through the contact with a Tesla. c o i l . This broke down r e c t i f y i n g 8 b a r r i e r s which were sometimes present (Ray and Fan 1961). If the resistances could not be matched within the s p e c i f i e d tolerance by sparking, the indium was removed and the surface of the s i l i c o n l i g h t l y ground and re-etched. A new contact was then made and tested. This procedure was repeated u n t i l the contacts were s a t i s f a c t o r y . B. THE SPECTROMETER The spectrometer consisted of a Model 12B Perkin-Elmer infrared monochromator with e x i t optics modified to accommodate a l i q u i d helium 3 cryostat. Over the range 25 to 35 microns (35 to 50 meV. ) a. plane d i f f r a c t i o n grating, with. 30.lines per mm., blazed at 30 microns was used as the dispersing element1. Two NaF r e s i d u a l ray plates were also used in this range to reduce unwanted shorter wavelength r a d i a t i o n ( s h o r t e r than '~15//<l). This was kept to less than one percent of the desired r a d i a t i o n . The grating provided f a i r l y high r e s o l u t i o n (0.25 meV. at a maximum s l i t width of 2.0 mm.) but could not cover a l l the desired wave-lengths . To cover the shorter wavelengths the grating was replaced with a. KBr prism. This covered the range 15 to 25 microns (50 to 100 meV.) but had a lower r e s o l u t i o n than the grating (0.5 meV. at 25 microns to 1.3 meV. at 15 microns). Two L i F r e s i d u a l ray plates limited the stray r a d i a t i o n to less than one percent of the desired r a d i a t i o n in this region. A Perkin-Elmer thermocouple with a. C s l window was used to detect the l i g h t which was supplied by a globar source operating at about 1300 ° C . The responsivity of the thermocouple was taken to be 4 microvolts per The notation meV. w i l l be used to denote 10" J electron v o l t s . 9 m i c r o w a t t , a c c o r d i n g t o m a n u f a c t u r e r s s p e c i f i c a t i o n s , and the s i g n a l produced f r o m i t was used to c a l c u l a t e the photon f l u x on the samples. The l i g h t beam was chopped a t the s o u r c e a t 870 cps w i t h a s e c t o r e d w h e e l d r i v e n by a synchronous motor. The chopper assembly had t o be mounted on gromets t o p r e v e n t v i b r a t i o n s f r o m the w h e e l r e a c h i n g the c r y o s t a t . The s p e c t r o m e t e r was c a l i b r a t e d w i t h w a t e r vapour a b s o r p t i o n l i n e s p r e s e n t a t t h e s e w a v e l e n g t h s ( B l a i r e t a l . 1 9 6 2 ) . I n o r d e r t o make photo-c o n d u c t i v e measurements, however, t h e s e w a t e r vapour a b s o r p t i o n l i n e s had to be e l i m i n a t e d as much as p o s s i b l e . T h i s was done by f l u s h i n g the s p e c t r o m e t e r c o n t i n o u s l y w i t h d r y n i t r o g e n o b t a i n e d by b o i l i n g l i q u i d n i t r o g e n . A l l b u t the s t r o n g e s t o f t h e s e w a t e r vapour a b s o r p t i o n l i n e s c o u l d thus be e l i m i n a t e d . C. THE DEWAR AND SAMPLE MOUNTING A m e t a l dewar was used t o c o o l the sample. I t c o n s i s t e d o f an o u t e r l i q u i d n i t r o g e n j a c k e t and an i n n e r h e l i u m c o n t a i n e r t o w h i c h the sample was a t t a c h e d . L i g h t r e a c h e d the sample t h r o u g h a C s l window w h i c h was removable t o a l l o w sample mounting. S a p p h i r e s p a c e r s were used t o e l e c t r i c a l l y i n s u l a t e the sample from the dewar, y e t p r o v i d e d s u f f i c i e n t h e a t c o n d u c t i o n t o keep the sample a p p r o x i m a t e l y a t h e l i u m t e m p e r a t u r e s . The sample was h e l d i n p l a c e w i t h a copper r e t a i n i n g p l a t e screwed snug a g a i n s t the s a p p h i r e b a c k i n g . T h i s would a l s o p r o v i d e a d d i t i o n a l c o o l i n g f o r the sample. The method o f mounting i s i l l u s t r a t e d i n f i g . 2 . E f f e c t s o f s t r a i n due t o d i f f e r e n t i a l c o n t r a c t i o n f r o m b o t h f r o m the s a p p h i r e b a c k i n g and f r o m the dewar i s n e g l i g i b l e s i n c e the e x p a n s i o n c o e f f i c i e n t f o r s a p p h i r e i s o n l y s l i g h t l y d i f f e r e n t t h a n f o r s i l i c o n . Any e f f e c t on the b o r o n s p e c t r u m would be n e g l i g i b l e s i n c e , a t the h i g h c o n c e n t r a t i o n s used h e r e , the l i n e s a r e 10 Copper r e t a i n i n g plate FIG. 2 : Method of sample mounting. 11 e x c e e d i n g l y b r o a d t o b e g i n w i t h . I n a d d i t i o n , the s t r a i n would be l o c a l i z e d t o the back f a c e o f the specimen whereas the b u l k o f the a b s o r p -t i o n o c c u r s i n the f r o n t p o r t i o n . The sample was mounted on the s i d e o p p o s i t e the window i n o r d e r t o r e s t r i c t i t s v i e w o f room t e m p e r a t u r e r a d i a t i o n f r o m the i n s i d e o f the s p e c t r o m e t e r . The d e t e r m i n i n g a p e r t u r e was the o p e n i n g i n the n i t r o g e n s h i e l d s u r r o u n d i n g the sample h o l d e r . From g e o m e t r i c c o n s i d e r a t i o n s , the p h o t o c o n d u c t o r was e s t i m a t e d t o subtend a s o l i d a n g l e o f a p p r o x i m a t e l y 0.14 s t e r a d i a n s . Assuming the s p e c t r o m e t e r box r a d i a t e s as a b l a c k body a t 300 °K, the t o t a l number o f background photons i n c i d e n t on the sample was e s t i m a t e d t o be 1.5 x 10^ photons per second ( S m i t h , Jones and Chasmar 1958). D. DETECTION SYSTEM I n o r d e r f o r d e t e c t i b l e s i g n a l t o be o b t a i n e d f r o m the p h o t o c o n d u c t o r , i t had t o be b i a s e d f r o m a f a i r l y s t a b l e s o u r c e . A s a t i s f a c t o r y method i s to use a r e g u l a t e d v o l t a g e s u p p l y i n s e r i e s w i t h an R-C f i l t e r . The f i l t e r e l i m i n a t e d any r i p p l e and s h o r t term i n s t a b i l i t i e s w h i c h would a f f e c t the measurements. The b i a s i n g network i s shown i n f i g . 3. The s i g n a l f r o m the photo-c o n d u c t o r was a p p l i e d t o the i n p u t o f 680 K i l 330 KJQ. 1 0 0 K - A — v W - i — A A ^ - r ^ V V — I a. h i g h impedance p r e a m p l i f i e r ( v - ^ 5 0 M i l a t 870 c p s ) , k e p t as c l o s e t o dewar as p o s s i b l e i n o r d e r t o m i n i m i z e e l e c t r i c a l p i c k - u p and c a b l e c a p a c i -t a n c e . N e v e r t h e l e s s , w i t h the f a i r l y h i g h impedance samples and l o a d r e s i s - FIG. 3 B i a s i n g Network. t o r s u s ed, c o n s i d e r a b l e 60 cps s i g n a l 12 was p r e s e n t a t the i n p u t . A CK 5886 e l e c t r o m e t e r tube was used as the a m p l i f y i n g element i n the p r e a m p l i f i e r . I t was o p e r a t e d i n a s t a n d a r d p l a t e - f o l l o w e r c i r c u i t w i t h a. g a i n of a p p r o x i m a t e l y 1.5 o v e r a wide f r e q u e n c y band. E l e c t r o m e t e r tubes a r e d e s i g n e d t o o p e r a t e a t low v o l t a g e s and w i t h v e r y s m a l l g r i d -13 c u r r e n t s (2 x 10 amp.) i n o r d e r t o p r o v i d e h i g h i n p u t impedances. To p r o v i d e a l e a k a g e p a t h f r o m the g r i d , a 10**.ft. r e s i s t o r was used. F i g . 4 shows a s c h e m a t i c d i a g r a m of the p r e a m p l i f i e r c i r c u i t used. 0.0001 /Af 600 V ' II £ 0.1/xf .- _ \ CK5886 56 Kfl =_ 24 V FIG. 4 P r e a m p l i f i e r C i r c u i t . F u r t h e r a m p l i f i c a t i o n was o b t a i n e d w i t h a T e k t r o n i x M odel 122 low n o i s e p r e a m p l i f i e r w i t h a bandpass of 80 cps t o 1 Kc. The r e m a i n d e r of the 60 cps s i g n a l (and h a r m o n i c s ) were f i l t e r e d out w i t h a. wide band f i l t e r . T h i s f i l t e r was n e c e s s a r y t o p r e v e n t the i n p u t s t a g e s of the f o l l o w i n g tuned a m p l i f i e r (40 cps b a n d w i d t h a t 870 c p s ) , from b e i n g o v e r d r i v e n . The tuned a m p l i f i e r a l s o d e t e r m i n e d the n o i s e b a n d w i d t h o f the system. The s i g n a l was t h e n r e c t i f i e d and d i s p l a y e d on a Brown s t r i p - c h a r t r e c o r d e r . 13 The d.c. c h a r a c t e r i s t i c s of the samples were measured using a. standard four-point probe measuring technique. The current was measured with a Model 425-A Hewlett-Packard D.C. micro-volt ammeter and the f i e l d with a. Fluke, Model 801, d i f f e r e n t i a l voltmeter connected across two of the side arms. The t o t a l voltage drop across the sample was measured to determine the contact resistance. Noise spectra f or the samples were measured with a. General Radio Model 736-A waveanalyzer. A block diagram of the complete detection system i s shown schematically i n f i g . 5. R-C FILTER LOAD RESISTOR VARIABLE VOLTAGE SUPPLY FLUKE NULL-METER PRE-AMPLIFIER TEKTRONIX AMPLIFIER SAMPLE 0HEWLETT-PACKARD KROHN-HITE BAND-PASS FILTER 40 CPS TUNED AMPLIFIER BROWN STRIP-CHART RECORDER FIG. 5 : Block diagram of complete detection system. 15 CHAPTER III : RESULTS AND ANALYSIS A. SIGNAL FROM A NON-OHMIC PHOTOCONDUCTOR The problem i s to calculate the photocurrent produced by r a d i a t i o n incident on the sample with an a d d i t i o n a l complication of a. non-linear dependence of detector current on applied f i e l d (Pultey 1964). The photoconductor i s connected i n a c i r c u i t as shown i n the accompanying f i g u r e . The voltage V„ applied to the load r e s i s t o r R^ and the bulk resistance of the photo-conductor R w i l l e s t a b l i s h the operating point VR V B R i = B , v = B (1) R L + R R L + R t V RT R J The sample resistance, R = R(Q,V), appearing i n equation (1) depends on both the voltage V and the incident r a d i a t i o n Q. For the purposes of th i s discussion we w i l l assume that the amplif i e r connected across the sample has an i n f i n i t e input impedance so no current i s drawn from the photodetector. The e f f e c t s of a f i n i t e input impedance w i l l be discussed i n the following section. When inf r a r e d r a d i a t i o n illuminates the sample, the sample r e s i s -tance w i l l change and, as a r e s u l t , the voltage appearing across the sample w i l l change. The quantity of i n t e r e s t , therefore, i s the change in voltage, dV, produced by the incremental change i n the photon f l u x , dQ. From equation (1), 16 dV = dQ V R L + R dR dQ V BR (R L + R) dR dQ or dV = dQ IR T R L + R dR dQ (2) Since R = R(Q,V), dR = + ( SR\ dQ V * Q / v \*VJ dV dQ (3) Equation (2) can now be written as I R L dV _ 1_ dQ " R L + R 2>jl\ + /^_R \ dV The quantity] 3 (4) can e a s i l y be evaluated from the current-l y Q voltage c h a r a c t e r i s t i c s of the sample. At any point on this curve the resistance i s defined as R(Q,V) = V / l . At a p a r t i c u l a r value of the f l u x Q, the resistance along the c h a r a c t e r i s t i c w i l l vary as 1_R 3 V /Q V 3 7> (5) Defining the dynamic (or a.c.) resistance as R = / 2> 1) gives 1R 2> v l i l -R R ac Equation (4) w i l l then be w h i c h c a n b e s o l v e d f o r t h e c h a n g e i n v o l t a g e — d Q I R d V dQ R a c R S Q V R a c i + ~R7 I n m o s t p r a c t i c a l a p p l i c a t i o n s t h e c i r c u i t i s d e s i g n e d s o t h a t R ^ ) > ^ > R a c . F o r t h i s c a s e t h e c h a n g e i n v o l t a g e a p p e a r i n g a c r o s s t h e p h o t o c o n d u c t o r i s I R d V = a c dR \ dQ (6) R d Q D e n o t i n g t h e q u a n t i t y [-^-^ 1 d Q b y t n e s i g n a l v o l t a g e i s \ OQ. / V d V = I R . •ac R (7) T h i s c a n b e p u t i n t e r m s o f t h e g e n e r a t e d p h o t o c u r r e n t a n d s a m p l e c o n d u c t a n c e a s i = I s A G (8) B . T H E PHOTOCONDUCTOR A S A C I R C U I T E L E M E N T I f we c o n s i d e r t h e p h o t o c o n d u c t o r a s a v o l t a g e s o u r c e , i t i s c l e a r t h a t t h e b e s t o p e r a t i n g p r o c e d u r e i s t o m a k e t h e i n p u t i m p e d a n c e o f t h e d e t e c t o r c i r c u i t r y l a r g e i n c o m p a r i s o n w i t h t h e r e s i s t a n c e o f t h e p h o t o -c o n d u c t o r . T h i s c o n d i t i o n c o u l d n o t b e f u l f i l l e d , h o w e v e r , s i n c e t h e 18 l e a d - i n w i r e s to the h e l i u m c o n t a i n e r loaded the p r e a m p l i f i e r i n p u t w i t h 113 p f . c a p a c i t a n c e , r e d u c i n g the i n p u t impedance t o 1.6 M f l . C o r r e c t i o n s had to be made to the measured s i g n a l i n o r d e r to d e t e r m i n e the s i g n a l a p p e a r i n g a c r o s s the p h o t o c o n d u c t o r . I n a l l samples the m e t a l - s e m i c o n d u c t o r c o n t a c t r e s i s t a n c e was comparable t o the sample r e s i s t a n c e . T h i s f u r t h e r reduced the measured s i g n a l . The sample behaved as a. v o l t a g e s o u r c e w i t h an i n t e r n a l r e s i s -t ance made up of the b u l k r e s i s t a n c e o f the m a t e r i a l i n s e r i e s w i t h the c o n t a c t r e s i s t a n c e . The a.c. e q u i v a l e n t o f the d e t e c t o r i s as f o l l o w s : FIG. 6 : A.C. e q u i v a l e n t o f the p h o t o c o n d u c t o r . R i s the a.c. r e s i s t a n c e a t the c o n t a c t s d e t e r m i n e d ca. from the t o t a l v o l t a g e drop a c r o s s the sample R i s the a.c. r e s i s t a n c e o f the b u l k sample sa. r e g i s the v o l t a g e g e n e r a t e d by the sample R i s the l o a d r e s i s t o r i n p a r a l l e l w i t h the i n p u t r e s i s - t a n c e o f the p r e a m p l i f i e r C i s the s t r a y c a p a c i t a n c e a c r o s s the sample e i s the v o l t a g e a p p e a r i n g a t the i n p u t o f the o a m p l i f i e r . From e l e m e n t a r y c i r c u i t t h e o r y the l o a d impedance Z 0 i s g i v e n by J X c R L , where X (A)C 19 For a l l cases X <C<C RT » s o Z S j X o c The s i g n a l applied to the a m p l i f i e r i s Z + R a + R o ca s a e s In terms of measurable quantities, the measured s i g n a l i s 1 + R c a + R s a . 2 - 1 1 Because of the capacitive loading of the sample, the measured voltage i s only a f r a c t i o n of the generated voltage. This did not a f f e c t the measurements i n general since the l i m i t i n g noise was also generated i n the sample. As a r e s u l t , the noise was decreased by the same fa c t o r , keeping the s i g n a l to noise r a t i o constant. The use of a. lower modulation frequency would reduce the e f f e c t o the stray capacitance and increase the measured s i g n a l . However, the s i g n a l to noise r a t i o decreased at lower frequencies because of increasing noise generated by the sample. This, therefore, made i t advantageous to use the high modulation frequency. C. D.C. CHARACTERISTICS The current-voltage c h a r a c t e r i s t i c s f or the various impurity concentrations are shown i n f i g s . 7 and 8. In a l l cases the samples were exposed to the same amount of background r a d i a t i o n and were approximately at the same temperature, except for the sample with 18 3 1.5 x 10 impurities/cm which was maintained at 2.4 K. A l l samples FIG. 7 : Current - voltage c h a r a c t e r i s t i c s of two samples at helium temperatures. Both samples are exposed to the same room temperature rad i a t i o n . FIG. 8 : Current - voltage c h a r a c t e r i s t i c s of two samples at helium temperatures. Both samples are exposed to the same room temperature radiation.' exhibited a non-linear dependence of current on applied f i e l d even at the lowest f i e l d s measurable. This n o n - l i n e a r i t y increased with f i e l d u n t i l breakdown occured. The breakdown i s a non-destructive "run-away" of the current associated with c a r r i e r m u l t i p l i c a t i o n as neutral impur-i t i e s are impact-ionized by energetic holes (Sclar and Burstein 1957, Koenig and Gunther-Mohr). No apparent damage i s done to the material since the r e s u l t s can be reproduced many times during a run. The break-down f i e l d was found to vary approximately as N "3/4 over the impurity j\ 16 3 17 ^ range 10 impurities/cm to 5 x 10 impurities/cm . The observed values are given i n table I. The log I vs log £, curves were linear at low f i e l d strengths with a slope which decreased as the impurity concentration increased. The value for the slopes are given i n table I. TABLE I D. C. C h a r a c t e r i s t i c s Impurity Concentration Breakdown F i e l d I n i t i a l Slope atoms/cm3 volts/cm 1.2 x 1 0 1 6 45 + 3 1.35 + 0.05 4.5 x 1 0 1 6 130 + 10 1.27 + 0.10 4.5 x 1 0 1 7 700 + 30 1.10 + 0.05 1.5 x 1 0 1 8 2.2 + 0.1 1.03 + 0.02 The value of slope quoted corresponds to an average obtained from a number of separate runs along with the spread i n the measured values. The differences i n slope measured for each sample were attributed to s l i g h t temperature v a r i a t i o n s during the consecutive runs. This was a. r e s u l t of the mounting method and could not be eliminated. The c h a r a c t e r i s t i c s were checked at a lower temperature 23 (approximately 2.4 °K) for the three lower concentration samples and i t was found that the breakdown f i e l d did not change with temperature but the i n i t i a l slope decreased with temperature. The resistance of the samples also increased, i n d i c a t i n g that the recombination rate ( or lif e t i m e ) of the free holes generated by background r a d i a t i o n i s s l i g h t l y temperature dependent. This e f f e c t of temperature on the recombination rate w i l l be discussed l a t e r . Using the c h a r a c t e r i s t i c curves and assuming a. mobility given by neutral impurity s c a t t e r i n g (Erginsoy 1950, Sclar 1956), the equilibrium number of free holes was calculated to be about 10 7 for the concen-trations 1.2 x 10 1 6, 4.5 x 1 0 1 6 and 4.5 x 10 1 7/cm3 at low applied f i e l d s . For the most heavily doped sample there were about lO-^ free holes at 2.4 °K. 18 3 The sample with 1.5 x 10 Boron/cm had a d.c. c h a r a c t e r i s t i c which did not conform with that of the other samples. I t had a. much lower resistance and lower breakdown f i e l d than expected (2.2 volts/cm). At t h i s impurity concentration the material i s nearly degenerate, so thermal generation of holes w i l l be important even at these temperatures. This i s evident i n the estimate of the number of holes present i n the sample. In addition, the energy needed to ionize n e utral impurities i s much less than the normal i o n i z a t i o n energy of the boron, r e s u l t i n g i n a low breakdown f i e l d . In order to observe photoconductivity i n this sample the temperature had to be reduced to 2.4 °K. D. SPECTRAL RESPONSE The s p e c t r a l response, or generated photocurrent per incident photon, was measured f or the four impurity concentrations and the 24 r e s u l t s are shown i n f i g s . 9 and 10. The curves shown are the generated photosignal normalized to a constant photon f l u x of 5 x 10^2 photons/sec incident on the samples. A l l samples were exposed to the same f l u x of background r a d i a t i o n and a l l operating at 0.5 microamp. bias, except for 18 the 1.5 x 10 sample which was operated at 40 microamperes. F i g . 9 shows the region extending over wavelengths shorter than the wavelength required for i o n i z a t i o n . The large dips at 64 meV. and 82 meV. are due to l a t t i c e absorption ( C o l l i n s and Fan 1954). Each sample appears to have a maximum response at photon energies between 90 to 100 meV., with the peak response decreasing with increasing impurity concentration. The increase i n response i n t h i s region i s believed to be p a r t l y due to the e f f e c t of the Pl/2 valence band. Holes excited into this band have a smaller e f f e c t i v e mass than those in the other bands (Zwerdling et a l . 1959) so we would observe a larger photo-current. Previous measurements of impurity photoconductivity at lower impurity concentrations (Burstein et a l . 1954) show a d e f i n i t e threshold energy, corresponding to the impurity i o n i z a t i o n energy, below which there was no response. A f a i r l y d e f i n i t e threshold was observed at the concentration 1.2 x 10 /cm but had almost completely disappeared at 18 3 1.5 x 10 atoms/cm . Considerable f l a t t e n i n g o f f of the response had occured f o r this sample for photon energies up to 20 meV. below the top of the valence band. A more detailed picture of the response i n the region of the excited states i s given i n f i g . 10. Here i t i s evident that the excited states are d e f i n i t e l y a f f e c t i n g the photo response at low photon energies, line f i r s t kink i s present even at the lowest concentration 15 3 (1.2 x 10 atoms/cm ) and i s at the same energy as the group of 9Z FIGURE 10 Photo-current i n the region of the impurity excited states. The photo-current i s normalized to a uniform f l u x of 12 5 x 10 photons/sec at a l l energies. The arrows marked 2 to 9 indicate the p o s i t i o n of impurity excited states. V e r t i c a l scales used f o r the various concentrations are as follows: + - 1.2 X 10 1 5/cm 3 - a r b i t r a r y scale i s used, curve gives r e l a t i v e response only. x - 1.2 X i n 1 6 / 3 10 /cm as indicated. ( U- 1»c*W ) o - 4.5 X 10 1 6/cm 3 - multiply l e f t hand scale by 1/2. A - 4.5 X 10 1 7/cm 3 - multiply l e f t hand scale by 1/5. • - 1.5 X 10 1 8/cm 3 - use r i g h t hand scale. 48 --46 44 42 40 38 36 34 Photon Energy i n meV. 28 excited states 5 to 9, indicated by the arrows (Colbow 1963, Burstein et a l . 1956). The response moves out to the states 3 and 4 at a concen-t r a t i o n of 1.2 x 10^/cm 3 and f o r state 2 some s i g n a l was measured at a 16 3 concentration of 4.5 x 10 /cm . I t appears that the response does not increase s i g n i f i c a n t l y i n the region of state 2 with another 10 f o l d 17 3 increase i n concentration (to 4.5 x 10 impurities/cm ). At th i s concen-t r a t i o n i t does not appear that much response w i l l be obtained from state 1 although this could not be checked. Increasing the concentration by 18 ^ another f a c t o r of 3 (to 1.5 x 10 /cm ) produced a large change i n the response, probably extending i t to energies w e l l below the f i r s t excited state. For the two lowest concentration samples the response had a d e f i n i t e cut-off as indicated i n f i g . 10, with no response observed below the respective c u t - o f f s . For the 10 f o l d increase i n concentration from 1.2 x 1 0 ^ to 1.2 x 1016 atoms/cm3 t h i s cut-off s h i f t e d 4 meV. to lower energies. For a l l higher concentrations used no d e f i n i t e cut-off was observed, only a gradual decrease i n response out to the lowest attainable energies. The response curves f o r a l l the samples have an i n f l e c t i o n point around 46 meV., the quoted low concentration i o n i z a t i o n energy of boron (Burstein et a l . 1956). This "shoulder" i s most evident for the lowest concentration sample and could possibly be due to other excited states of the boron l y i n g between 44 and 46 meV. The rapid r i s e at higher energies i s due to t r a n s i t i o n s d i r e c t l y into the valence band. The very sharp shoulder at 46.5 meV. observed f o r the 1.2 x 10l6/ c m3 sample i s l i k e l y due to an uncertainty i n the determination of the number of incident photons. A strong water vapour absorption peak occurs at this energy and complete c a n c e l l a t i o n of the e f f e c t s of these peaks proved 29 troublesome. F a i r l y strong water vapour absorption peaks also occured at 40.5 and 43 meV., with the r e s u l t that the kinks at these points may not be quite as we l l defined as i s shown. However, the maximum error at these two points w i l l not exceed + 20% so that changes i n response are d e f i n i t e l y occurring i n these regions. 15 3 The curve f o r the concentrations 1.2 x 10 /cm has been included f o r a quantitative comparison only. The response shown i s only a r e l a t i v e -12 response with the value at 48 meV. a r b i t r a r i l y taken at 100 x 10 amp. The reason quantitative measurements could not be made on this sample was pa r t l y because a sample thick enough to absorb a l l the incident r a d i a t i o n could not be mounted i n the cryostat. In addition, the d.c. c h a r a c t e r i s t i c s for t h i s sample could not be reproduced from run to run even a f t e r several attempts had been made at improving contacts to the material. During each run the r e l a t i v e photo response was measured and was reproducible to within 5 % from run to run. Because of th i s r e p r o d u c i b i l i t y i t i s believed the re s u l t s are due to the boron and not extraneous contact e f f e c t s . E. PHOTOCONDUCTIVITY MEASUREMENTS The quantities which are of p r a c t i c a l importance i n photoconductive measurements are the s i g n a l and noise at the input of the amplifier, and how these quantities vary with bias current. These quantities are shown in f i g s . 11 for d i f f e r e n t impurity concentrations. Since the signals have not been corrected f o r the detector impedance, a l l samples appear to have a d i f f e r e n t behavior with bias current. The e f f e c t of the decreasing detector resistance with increasing current i s very noticeable i n the 17 3 sig n a l measured from the 4.5 x 10 /cm sample. At large currents the measured s i g n a l begins increasing r a p i d l y because of the decreasing sample resistance. 30 FIG. 11(a) : Measured s i g n a l and ijgise as a function of bi a s i n g current through the 1.2 x 10 /cm3 sample. 31 0.1 1.0 10 Bias current, I (yU.a ) FIG. 11(b) : Measured s i g n a l and Ogise as a function of biasing current through the 4.5 x 10 /cm3 sample. 32 X Bias current, I ( ILa ) FIG. 11(c) : Measured s i g n a l and noise as a function of biasing current through the 4 . 5 x 10^ 7/cm 3 sample. 33 8.2 x 10xz photons/sec @ 75 meV. FIG. 11(d) : Measured s i g n a l and ngise^as a function of b i a s i n g current through the 1.5 x 10 /cm sample. 34 18 3 The s i g n a l from the 1.5 x 10 /cm sample has a completely d i f f e r e n t current dependence at high currents because of the i n i t i a l low resistance of the sample. The low resistance of t h i s sample meant that a l l the s i g n a l generated at the sample was applied to the input of the a m p l i f i e r without loss. If the e f f e c t of the sample resistance i s taken into account, the photosignal has a f i e l d dependence as shown i n f i g s . 12. For a l l samples the generated photosignal increases with f i e l d up to the region where the c h a r a c t e r i s t i c s becomes quite non-linear. For further increases i n the f i e l d the photosignal saturates or else decreases. This behavior i s understandable i f one considers equation (7) derived e a r l i e r , e = I R s ac A G where e g i s the generated photosignal. The s i g n a l which i s generated depends on the f i e l d dependent dynamic resistance, R , which decreases ac r a p i d l y with applied f i e l d near breakdown, thereby decreasing the photo-s i g n a l . A comparison with the c h a r a c t e r i s t i c s ( f i g s . 7 and 8) shows that the saturation e f f e c t does occur i n the region of increasing non-l i n e a r i t y . The measured noise voltage, also shown i n f i g s . 11, generally increases with increasing current through the samples. A l l the samples had peaks i n the noise curves which were quite reproducible although could be changed by tampering with the contacts. Because of these large v a r i a t i o n s i n noise output a f t e r a modification i n the contacts the noise i s assumed to be associated with current flow through p o t e n t i a l b a r r i e r s at the contact. In addition to the noise, however, the contacts i n t r o -duce an e f f e c t i v e resistance i n series with the photoconductor which 35 0.1 1.0 10 E l e c t r i c f i e l d , £ (volts/cm) 16 3 FIG. 12(a) : Photo-signal appearing across the 1.2 x 10 /cm sample as a function of applied e l e c t r i c f i e l d . 36 5 E l e c t r i c f i e l d , cS (volts/cm) FIG. 12(b) : Photo-signal appearing across the 4.5 x 10 /cm sample as a function of applied e l e c t r i c f i e l d . 37 E l e c t r i c f i e l d , o (volts/cm) 17 3 FIG. 12(c) : Photo-signal appearing across the 4.5 x 10 /cm sample as a function of applied e l e c t r i c f i e l d . 38 E l e c t r i c f i e l d , £, (volts/cm) FIG. 12(d) : Photo-signal appearing across the 1.5 x 10 /cm sample as a function of applied e l e c t r i c f i e l d . 39 contributes to the loss i n s i g n a l (see section B). The measured contact resistance i s shown i n f i g . 13 f o r d i f f e r e n t currents through the samples. The current dependence observed here i s s i m i l a r to that found i n boron-doped germanium photoconductors with s i m i l a r contacts (Wallis and Shenker 1964). The noise c h a r a c t e r i s t i c s of the samples were generally quite d i f f e r e n t f o r the two d i f f e r e n t d i r e c t i o n s of current flow through the samples. For each sample the d i r e c t i o n of minimum noise was determined i n order to obtain the largest s i g n a l to noise r a t i o when making photo-conductive measurements. The curve of rms noise voltage vs frequency i s shown i n f i g . 14 for the two d i r e c t i o n s of current flow through the 1.2 x 10 /cm sample. This sample had the greatest difference i n noise c h a r a c t e r i s t i c s f or the two current d i r e c t i o n s i n addition to the lowest s i g n a l to noise r a t i o of a l l the samples. For both current d i r e c t i o n s , however, there was a d e f i n i t e s i m i l a r i t y i n the behavior of the noise with frequency. The observed decrease i n the noise at higher frequencies prompted the use of 870 cps as the modulating frequency rather than a lower frequency. The s i g n a l to noise r a t i o goes through a maximum before decreasing near breakdown f o r each of the samples. The maximum observed S/N r a t i o decreased with decreasing impurity concentrations. I t i s not known i f the higher concentration samples are inherently less noisy or i f i t i s just a property of these p a r t i c u l a r contacts. FIG. 13 : Resistance due to contacts vs. bias current through samples. 41 42 CHAPTER IV: THEORY AND DISCUSSION A. IMPURITY PHOTOCONDUCTIVITY The problem of impurity photoconductivity can be treated i n a. simple yet i n s t r u c t i v e manner since d i f f u s i o n and space charge e f f e c t s can be neglected (Burstein 1954, Bube 1950, Putley 1964, Rittner 1954). In addition, we w i l l assume that the e f f e c t of traps present i n the material can be neglected, so that the c a r r i e r i s free to p a r t i c i p a t e i n conduction during the time between generation and recombination. Consider a p-type semiconductor at low temperatures containing o 3 N A acceptors/cm - 3 and N donors/cm with N_ ^  N.. For shallow impurities D D A (group III) i n s i l i c o n l i q u i d helium temperatures are required to keep the number of thermally ionized acceptors, and hence the number of free holes, small. The "dark" conductivity of the sample w i l l then be determined by the f l u x of background photons incident on i t . Because of an exponential decay of the photon f l u x through the sample thickness, the den s i t i e s of generated c a r r i e r s w i l l vary throughout the sample volume. Consider a layer of the material of thickness dx, lying a distance x below the surface. Let n,(x), n and n. be the densities of free holes, f v '' u 1 ' unionized acceptors and ionized acceptors i n this layer. We w i l l assume that the following conditions hold: n f ( x ) « n u, n f ( x ) « n±, i i . & N D , n u = N A - N D - n f(x) N A - N Q Because N n and N. are uniform throughout the sample, n. and n w i l l not D A l u depend on the depth below the surface. 43 The e q u i l i b r i u m number of holes w i l l be determined, by the balance between the t o t a l r a t e of generation and the t o t a l r a t e of recombination (Koenig 1962). d n f ( x ) dt = A t ( N A - N N ) + A B ( x ) ( N A - N D ) - B T N D n f (x) + - N D ) n f ( x ) 2 - .B n f ( x ) N D = 0 (10) The terms A (N. — N ) and. A (x)(N. — N,,) are the r a t e s of generation X A D B A D of holes by thermal e x c i t a t i o n and by background r a d i a t i o n . At l i q u i d helium temperature the thermal generation term. A^,(N • — N^) w i l l be n e g l i g a b l e i n comparison to A^(x) (N^ — N^). The term A^n^(x)(N^ — N n) i s the r a t e of i o n i z a t i o n of the n e u t r a l i m p u r i t i e s caused by the c o l l i s i o n of e n e rgetic f r e e holes w i t h .the i m p u r i t i e s . The terms c o n t a i n i n g Brj, and Bj. w i l l be the corresponding recombination r a t e s of the f r e e h oles. BrpNp i s the r a t e of recombination of the f r e e holes v i a a d i r e c t recombination w i t h the i o n i z e d i m p u r i t i e s w i t h the excess energy of the holes being c a r r i e d o f f by phonons. The g i a n t trap mechanism (Lax 1960) i s assumed to be the method of capture of the holes by the i o n i z e d i m p u r i t i e s . In t h i s process the hole i s f i r s t captured i n t o one of the h i g h l y e x c i t e d s t a t e s of the i m p u r i t y , w i t h the excess energy and momentum being c a r r i e d away by a c o u s t i c phonons. The. hole then decays i n steps to the ground s t a t e w i t h the emission of phonons or photons at each step. These f i n a l steps w i l l undoubtedly l i m i t the r a t e of decay i n t o the ground s t a t e , but the capture r a t e w i l l be determined by the i n i t i a l capture i n t o an e x c i t e d s t a t e . In order f o r a s t a t e to be e f f e c t i v e as a. t r a p , i t s binding, energy 44 w i l l have to be greater than kT to prevent the hole from being thermally re-excited into the band. Lowering the temperature permits contributions from states of increasing radius r e s u l t i n g i n increased capture cross sections. This explains the increase i n sample resistance on lowering the temperature from 4.2 °K to 2.4 °K as was mentioned i n chapter I I I . 2 Bj.n^ (x)Np, on the other hand, represents recombination v i a the Auger process i n which the excess energy and momentum of the recombining hole i s ca r r i e d off by another free hole. This term w i l l be negligable for the low f i e l d measurements when n^ i s small, but may become important near breakdown. Since we are interested: i n the lower f i e l d regions, we w i l l neglect t h i s term. The equilibrium concentrations of holes, within the l i m i t s of these approximations, i s A B ( x ) ( N A _ N D ) n f(x) = 1 « _ (11) B T N D - V N A - V The rate of generation of free holes by the background at x can be written as rK V X ) ( N A ~ V> (1 - R) N ( A) (*( A ) e - ~ (X ) X dX (12) where: © < ( A ) i s the absorption constant and R the r e f l e c t i o n c o e f f i c i e n t at wavelength A , A t i s the long wavelength cut-off of photons capable of i o n i z i n g the impurities, N Q ( A ) i s the number of photons/sec with wavelengths between A and A + dA incident on the surface. 45 This number i s calculated assuming that the background r a d i a t i o n i s emitted by a 300 °K black-body. The t o t a l rate of generation i n the sample i s obtained from equation (12) by an integ r a t i o n over the sample thickness, d, with the r e s u l t V Y ~ V = j ( l - . R ) N 0 ( X ) (1 - e _ 0 < < A ) d ) dX . -'o Since a l l the samples were made thick enough to absorb a l l the ra d i a t i o n , we have 1 _ e - o , ( A ) d ^ l i The above equation can then be written as A (N - N ) = Q , (13) B A D g where Q = (1 —1 R)Q ' i s the number of photons/sec absorbed by the sample B B and X Q ' = B r (1 - R) N ( X ) dX J i s the t o t a l number of photons/sec incident on the sample. We w i l l use R = 0.31 as the r e f l e c t i o n c o e f f i c i e n t of s i l i c o n (Bichard and Giles 1962). Using equation (13), the t o t a l number of free holes i n the sample i s QB N f = ; (14) B r * D - A ( N A _ N D ) I t i s convenient to introduce a l i f e t i m e , ^ . def ined as 1 B T N d - A I ( N A _ N N ) Z- = , (15) which represents the time during which holes are free to contribute to conduction. When impact i o n i z a t i o n i s unimportant t h i s time i s the usual 46 recombination time, l / B r r N ^ , of the free holes with ionized acceptors. Using this d e f i n i t i o n , equation (14) can be written N f = Q B t. _ (16) When photoionization r a d i a t i o n i s incident on the sample the number of free holes r i s e s above this e q uilibrium value and the sample conduc-tance increases. If there are Q^ ' ( A ) photons per second of wavelength X to A + d}\ incident on a sample of r e f l e c t i v i t y R, the rate of gener-ation at a distance x below the surface w i l l be :(x) = Q,( A) C* ( A ) e V where Q ( A) = Q. ' ( l — R) i s the number of photons/sec being absorbed i i by the sample. The new equilibrium concentration of free holes w i l l be determined by dn f'(x) A B ( x ) n u ' - B Tn.'n f'(x) + A ^ ' n ' (x) + f ( x ) dt = 0 (17) where n^'(x), n 1 and n V are the new concentration of free holes, unionized acceptors and ionized acceptors. For the case of small signals n N D and n ^ ~ N A - N Q Thus equation (16) becomes dn '(x) - = A g ( x ) ( N A - N D ) _ B TN Dn f'(x) + A^(N^ _ N D ) n f 1 (x) + f(x) = 0 (18) The change i n free hole concentration i n this layer i s e a s i l y shown to be f(x) n ' ( x ) - n ( x ) = / i n (x) = , (19) B ^ . A I ( N A _ N D ) or n f(x) = f(x) r • Because of the v a r i a t i o n i n free c a r r i e r density with depth i n the sample, the s i g n i f i c a n t parameter w i l l be the t o t a l number of generated c a r r i e r s , A N f, obtained from equation (19) by an int e g r a t i o n over the thickness d; A N f = Q. Z . (20) These equations are p a r t i c u l a r l y u seful i n explaining the non-ohmic behavior of the samples at low f i e l d s and also the phenomena, of breakdown (Koenig 1958,Sclar and Burstein 1957, Koenig and Gunther-Mohr 1957). To do so, however, we have to consider the f a c t that the c o e f f i c i e n t s B^ and Aj w i l l be functions of the l a t t i c e temperature and the hole d i s t r i -bution function f ( £ ) . The d i s t r i b u t i o n w i l l i t s e l f be dependent on the applied f i e l d ct and the l a t t i c e temperature. At low f i e l d strengths impact i o n i z a t i o n i s n e g l i g i b l e so the number of holes i s determined by the recombination rate, which decreases as the energy of the hole increases (Lax 1960). This has a. d i r e c t consequence on the d.c. charac-t e r i s t i c s , with the n o n - l i n e a r i t y at low f i e l d s being due to the changing d i s t r i b u t i o n function and the resultant modification i n recombination rate (Picus 1962). The change i n the d i s t r i b u t i o n w i l l occur i n order to e s t a b l i s h a new equilibrium between the rate of absorption of energy from the f i e l d and rate of d i s s i p a t i o n of energy to the l a t t i c e . The increase i n the average energy corresponds to an increase i n temperature of the system of holes. At s u f f i c i e n t l y high f i e l d s the average energy w i l l become of the order of the impurity i o n i z a t i o n energy and impact ion-i z a t i o n w i l l occur. 48 The net e f f e c t of an increasing f i e l d i s a decrease i n the recom-bination rate and an increase i n the impact-generation rate with the r e s u l t that the l i f e t i m e begins increasing r a p i d l y . The d.c. character-i s t i c s of each sample display t h i s behavior: a r a p i d l y increasing conduc-tance for f i e l d s preceding breakdown followed by a much more rapid increase at breakdown. Using equations (16) and (20), we can r e l a t e the measured photo-current (equation (8)) to more fundamental parameters c h a r a c t e r i z i n g the holes. A material with free holes (given by equation (16)), with mobility JUL w i l l have a conductance N f e JJL Q B t &M G = X = - _ , (21) 12 2 1 where T i s written as the l i f e t i m e of these holes, and 1 the distance between contacts. The change i n conductance produced by the photoion-i z a t i o n r a d i a t i o n of wavelength X i s c l e a r l y Q i ( A ) eyU(X ) f ( A ) A G ( A ) = (22) I 2 where JLA. ( A ) i s the mobility and t ( A ) i s the l i f e t i m e of the holes excited by r a d i a t i o n of this wavelength. Because both parameters (y<X (^ ) and t ( A ) ) may depend on the f i n a l energy of the excited hole, we w i l l have to d i s t i n g u i s h between the l i f e t i m e s and m o b i l i t i e s appearing i n equations (21) and (22). The dependence of these two parameters on the energy of the excited holes w i l l be discussed in the following sections. For constant i l l u m i n a t i o n the photocurrent produced (equation (8)) w i l l depend d i r e c t l y on the product yU- ( \ ) ?: (A) for the excited holes. F i r s t introduced by Gudden and Pohl (1921), the (A) f (A) product (or schubweg) gives the distance a free hole w i l l t r a v e l i n the d i r e c t i o n 49 of the applied e l e c t r i c f i e l d i n a time t ( A ) per unit applied f i e l d . The longer this distance can be made, the greater the change in conduc-tance for a fixed i l l u m i n a t i o n . An obvious method f o r increasing the distance i s through the l i f e t i m e . For low f i e l d strengths t ( A ) i s inversely proportional to the concentration of compensating centers, 13 3 estimated to be 1 x 10 atoms/cm , so i t would appear that i t could be made as long as desired by a further reduction i n N . In practice, D however, much lower compensation i s v i r t u a l l y impossible with the present methods of c r y s t a l preparation. In addition, the e f f e c t of other c r y s t a l defects and recombination mechanisms would place an upper l i m i t on t (A). The conductance change produced by i r r a d i a t i n g the samples with 75 meV. photons i s shown i n f i g . 15 f o r d i f f e r e n t values of applied e l e c t r i c f i e l d . The factor Q^(^)e/1 appearing i n equation (22) i s a constant at this wavelength ( i . e . f i e l d independent) so the f i e l d depen-dence of A G( f\ ) w i l l be due to changes i n / 4 (A ) ZT (A) with f i e l d . For each sample there i s a region at low f i e l d strengths for which yCC(X) t ( A ) increases with f i e l d before reaching a plateau. The d r i f t distance per u n i t f i e l d of the holes then remains r e l a t i v e l y constant as the f i e l d i s further increased. Near breakdown f i e l d s , however, each sample shows a r a p i d l y i n c r e a s i n g ) due to a rapid increase in l i f e t i m e . This v a r i a t i o n with e l e c t r i c f i e l d was measured at three other photon energies and the r e s u l t s are shown i n f i g . 16 for the 4.5 x 10^/cm^ sample. Two of the photon energies used were s l i g h t l y lower than the low concentration i o n i z a t i o n energy (46 meV.) and one was s l i g h t l y higher. xl(T 1 1 1 I I | I I 1 | 1 I I I I i 1 — r 10' photons/sec @ 75 meV. 10 11 0.2 1 10 60 E l e c t r i c f i e l d i n volts/cm. FIG. 15(a) : Conductance change A G( X) i n the 1.2 x 10^/cm 3 sample produced by 75 meV. photons vs. applied e l e c t r i c f i e l d . t—1 E l e c t r i c f i e l d i n volts/cm. FIG. 15(b) : Conductance change A G( A ) i n the 4. 5 x 10 1 6/cm 3 sample produced by 75 meV. photons vs. applied e l e c t r i c f i e l d . 0.2 8.2 x 1 0 1 2 photons/sec @ 75 meV. J 1 1 I I I I I I ! I I I 1 I I I I I I I ?- 10 10 2 i o 3 E l e c t r i c f i e l d i n volts/cm. FIG. 15(c) : Conductance change AG(X) i n the 4.5 x 10 1 7/cm 3 sample produced by 75 meV. photons vs. applied e l e c t r i c f i e l d . 1 1 1 1 — . M M 1 — f 10+-o 0-4 4J ho 0.02 I—5—$• 12 8.2 x 10 photons/sec @ 75 meV. -L I I I I I M 0.1 Electric field In volts/cm. FIG. 15(d) : Conductance change ^ G(X) l n C a e x 10^/cm^ sample produced by 75 meV. photons vs. applied electric field. 54 55 The conductance changes at these d i f f e r e n t wavelengths have e s s e n t i a l l y the same f i e l d dependence as the changes measured for this sample at 75 meV. (shown i n f i g . 15). This s i m i l a r i t y at the d i f f e r e n t 16 3 17 ^ wavelengths was also observed for the 1.2 x 10 /cm and 4.5 x 10 /cm samples. Since t h e / A ( \ ) ' Z " ( A ) product has the same f i e l d dependence at a l l the wavelengths measured, the r e l a t i v e shape of the s p e c t r a l response curves for a p a r t i c u l a r sample ( f i g s . 9 and 10) w i l l not depend on the value of f i e l d used when making the measurement. The exact reasons for the p a r t i c u l a r f i e l d dependence of AG( A) shown i n f i g . 15 i s d i f f i c u l t to determine because neithery^*-( A ) nor t ( A ) could be measured separately i n t h i s experiment. A quantitative estimate of e i t h e r one of these parameters w i l l involve estimates about the other, so w i l l r e l y on a c e r t a i n number of assumptions being made. The method used to separate the two factors i s discussed i n the following sections. B. HOLE MOBILITY AND LIFETIME 1. MOBILITY As we have seen i n the previous section, the observed conductance changes depend on both the mobility and l i f e t i m e of the photoexcited holes. In this section we s h a l l show how the mobility may be estimated i n order to obtain the l i f e t i m e of the holes. The m o b i l i t i e s obtained i n this section w i l l apply to holes i n the valence band and are not assumed to apply to holes i n impurity bands. The mobility of holes i n these bands w i l l be discussed i n a l a t e r section. Since the energies of e x c i t a t i o n are never very large, the holes w i l l remain i n the v i c i n i t y of the valence band maximum. The regular 56 valence bands of s i l i c o n w i l l , therefore, be replaced by a single para-b o l i c band with a constant e f f e c t i v e mass. Because of this we can write the mobility as /A . p <T> m where <0~^  i s an i n t e g r a l involving the momentum relaxation time, T* ( c a p i t a l tau), of the holes.^ i n general T w i l l depend on the energy of the holes, so changing the d i s t r i b u t i o n e i t h e r by applying an e l e c t r i c f i e l d or by photo-excitation of the holes with photons of d i f f e r e n t energy w i l l change the mobility. However, f or impurity concentration 16 3 greater than 10 /cm the dominant sc a t t e r i n g mechanism i s assumed to be e l a s t i c s c a t t e r i n g from the neutral impurities (Sclar and Burstein 1957, Sclar 1956, Erginsoy 1950, Yamashita 1960). In this case T i s independent of the hole energy so we could write = e _2L m* (23) for the ent i r e band. The mobility was calculated by Erginsoy (1950) to be 1.43 x 102*- ^  2 AA. = cm /volt-sec where K. i s the d i e l e c t r i c constant, ^ = m*/me , and i s the density of neutral impurities. Using a. density of states e f f e c t i v e mass, m = 0 . 6 me, the mobility of holes i n the valence band w i l l be taken as 7.15 x 1 0 2 0 /t>C = c m /volt-sec NN 4 see for example R.A. Smith 1963 "Wave Mechanics of C r y s t a l l i n e S o l i d s " (Chapman and H a l l Ltd., London) 2nd E d i t i o n , p. 322. Since N M i ^ N A — N n and N _ « N , this can be written as 7.15 x 10 20 cm /volt-sec (24) N Table II gives the valence band m o b i l i t i e s of the four d i f f e r e n t samples used i n subsequent l i f e t i m e c a l c u l a t i o n s . Table II Theoretical M o b i l i t y of Holes i n the Valence Band at 4.2 °K (Erginsoy 1950) Concentration atoms/cm Mo b i l i t y cm /volt-sec 1.2 x 1 0 1 6 6.0 x 10 4 4.5 x 1 0 1 6 1.6 x id* 4.5 x 1 0 1 7 1.6 x 10 3 1.5 x 1 0 1 8 4.8 x 10 2 The assumption of e l a s t i c s c a t t e r i n g i s questionable at high elec-t r i c f i e l d s where the average energy of the holes i s of the order of the i o n i z a t i o n energy (Yamashita 1961). The possible i n t e r a c t i o n s with impurities i n this case would include e i t h e r (a) impacjt i o n i z a t i o n of the impurity, (b) e x c i t a t i o n of the bound hole to an excited state, or (c) i n e l a s t i c s c a t t e r i n g from the impurity accompanied by emission or absorption of a. phonon. The e f f e c t of these a d d i t i o n a l loss mechanisms on the mobility of holes i s not known at this time. Because of the 1/N^ dependence of mobility inferred by neutral impurity scattering, the measured change i n conductance Z^G ( A ) , given by equation (22), w i l l also vary as 1/N^. The e f f e c t of concentration on A G()i ) i s shown i n f i g . 17 for holes excited by photons of four d i f f e r e n t energies. These energies range from e x c i t a t i o n well into the 58 FIG. 17 : Dependence of conductance change A G ( A ) on impurity concentration for various photon energies. 59 valence band (75 meV.) to e x c i t a t i o n into the excited state bands (39 meV.). At a l l wavelengths considered A G(A ) has e s s e n t i a l l y the same concentra-ti o n dependence, with A G(A ) varying approximately as N " 3 / 2 with impurity concentration. As we have mentioned previously, the values of A G( A ) measured here depend on both the mobility and l i f e t i m e of the holes. This deviation of A G(A) from a 1/N^ dependence i s therefore a t t r i b u t e d to a. v a r i a t i o n i n l i f e t i m e f o r the d i f f e r e n t samples, due probably to s l i g h t l y greater levels of compensation as the impurity concentration i s increased. The important point regarding f i g . 17 i s the f a c t that there i s a si m i l a r concentration dependence at photon energies corresponding to e x c i t a t i o n i n to the excited state impurity bands as w e l l as into the valence band. This would imply that the impurities act as hole scatterers  even i n these excited state bands. The implication of this w i l l be d i s -cussed i n the section dealing with impurity band conduction. 2. LIFETIME Using the values of mobility obtained i n the previous section we can now determine the l i f e t i m e of holes excited into the valence band. The l i f e t i m e of c a r r i e r s (holes) i s of i n t e r e s t both f o r p r a c t i c a l and theoret-i c a l reasons. The l i f e t i m e determines the si g n a l obtained from the photo-conductor, r e s t r i c t i n g i t s "detecting a b i l i t y " and frequency response. Through the l i f e t i m e , information can be obtained regarding capture mechanisms of free c a r r i e r s and capture cross sections of the capturing centers. In actual photoconductivity measurements, the measured l i f e t i m e i s an average over holes d i s t r i b u t e d i n energy with a d i s t r i b u t i o n function f ( 6 ) . This l i f e t i m e i s determined by a t o t a l recombination p r o b a b i l i t y , B T, which 60 can be related to the capture of i n d i v i d u a l holes by the ionized impurities i n the following manner. If B ( £ ) i s the capture p r o b a b i l i t y f or a free hole of energy £ by an . ionized impurity (Lax 1960), then the capture cross section of the impurity, 0-(€), i s defined as B ( 6 ) = <T(6 ) v ( £ ) where v ( £ ) i s the speed a hole of energy £ would have. This defines the recombination time (or l i f e t i m e ) f or holes of this energy, 1 r ( € ) = NDB(e) where i s the density of capture centers (Ionized i m p u r i t i e s ) . For the d i s t r i b u t i o n f ( £ ) , the average capture p r o b a b i l i t y , B,p, i s defined as B T = < T T < V > where 0~(6) v(e) f (€.) d fe T <v> i s the t o t a l capture cross section and < v > -J measured l i f e t i m e i s Tr = r v ( e ) f ( €) d e o i s the average v e l o c i t y . 6^. i s the maximum energy the holes can have. The B T N D If we assume there i s no i n t e r a c t i o n between the free holes, the d i s t r i b u t i o n can be separated into two parts according to the o r i g i n of the holes. The background r a d i a t i o n , by the nature of i t s s p e c t r a l d i s t r i -61 bution, w i l l excite c a r r i e r s over a large energy spread. The averages over this d i s t r i b u t i o n , denoted as 2T , B T and ^ , are instrumental i n deter-mining the d.c. c h a r a c t e r i s t i c s of the samples. ..On the other hand, holes photoexcited by the infr a r e d r a d i a t i o n w i l l have a. spread i n energy, dfe , determined by the spectrometer s p e c t r a l s l i t width. The average over this d i s t r i b u t i o n determines the measured photoconductive l i f e t i m e £ ( A ) , which in turn determines the measured conductance change A G ( A ) ( f i g s . 15). Using equations (21) and (22), the f r a c t i o n a l change i n conductance produced by ra d i a t i o n of wavelength A w i l l be A G ( A ) Q i(A)'7J ( A )yU{X) Q If we r e s t r i c t the following discussion to e x c i t a t i o n into the valence band, then we can use the assumption of a constant mobility over the whole band. In t h i s caseyU-( \ ) = ^ JL > so the f r a c t i o n a l change i n conductance reduces to A G ( X ) Q i ( A ) £ ( A ) G QB Z The f r a c t i o n a l change i n conductance produced by 75 meV. r a d i a t i o n i s shown i n f i g . 18 for the four samples used. For each sample AG(A ) /G i s s l i g h t l y f i e l d dependent, i n d i c a t i n g that the l i f e t i m e of holes excited by 75 meV. photons ( Z ( A ) ) has a f i e l d dependence d i f f e r e n t from the l i f e -time of holes excited by background r a d i a t i o n ( £ ). Using the assumption that the mobility i s constant throughout the valence band and determined by equation (24), we ;can c a l c u l a t e "2T and'?J(A) from equations (21) and (22). The d.c. c h a r a c t e r i s t i c s ( f i g s . 7 and 8) determine the conductance G, from which we can obtain t , whereas the measured conductance changes, A G ( X ) , at 75 meV. ( f i g s . 15) were used to 0.02 0.1 1 1—T~\ I I I 1.0 3xl0 5, J ± I 1 1 1 I I I | E l e c t r i c f i e l d i n volts/cm. 10 . _ _ ^ 16 / 3 o - 4.5 x 10 /cm x - 1.2 x 10 1 6/cm 3 A - 4.5 x 10 1 7/cm 3 18 3 Q - 1.5 x 10 /cm ( Top scale ) i I I I I I I 1 rnr TTT J I L E l e c t r i c f i e l d i n volts/cm. FIG. 18 : F r a c t i o n a l change i n conductance A.G(A)/G produced by 75 meV. photons vs. applied e l e c t r i c f i e l d . 1 10 10 Z E l e c t r i c f i e l d i n volts/cm. FIG. 19(c) : Photoconductive l i f e t i m e 7: ( ^  ) at 75 meV. and d.c. l i f e t i m e vs. applied e l e c t r i c f i e l d f o r the 4.5 x 10 /cm3 sample. 40 1 I 1 1 I | I • u CD CO o o co 3 C CD E l e c t r i c f i e l d i n volts/cm. FIG. 19(d) : Photoconducti^e l i f e t i m e f ( \ ) at 75 meV. vs. applied e l e c t r i c f i e l d for the 1.5 x lO^/cm sample. obtain £ ( A ) . These l i f e t i m e s are shown i n f i g . 19 for d i f f e r e n t applied f i e l d s on the samples. In f i g . 19 (d) only 2T(A ) has been plotted f or the 18 3 7Z 1.5 x 10 /cm sample since C could not be determined. In this sample the "dark" conductance ( G ) was determined by thermal generation of free holes and not photo-generation by the background as i n the other samples. Since this thermal generation rate i s unknown, no estimate of ^ could be obtained. For a l l samples ^ has a f i e l d dependence s i m i l a r to that obtained from l i f e t i m e measurements made on germanium (Koenig et a l . 1962, Shenker et a l . 1964). The photoconductive l i f e t i m e , however, appears to be considerably d i f f e r e n t . At low f i e l d s f ( A ) increases more r a p i d l y with f i e l d than t , but then reaches a plateau p r i o r to breakdown. The reason for this plateau i s not known, but may be the r e s u l t of assuming a f i e l d independent mobility i n the c a l c u l a t i o n s . The f a c t that ^ l i e s below & ( A ) for three samples should not be taken as being s i g n i f i c a n t because of the uncertainties involved i n deter-mining the incident photon fluxes. Q R i s obtained from the t o t a l photon f l u x emitted by a black body while Q^(A) r e l i e s on an estimate of the thermocouple responsivity obtained from manufacturer's s p e c i f i c a t i o n s . Both estimates have an uncertainty of at least a factor of two, so the exact p o s i t i o n of t r e l a t i v e to t ( A ) i s uncertain. The dependence of (A ) on the incident photon energy i s assumed to be the cause of the energy dependence i n the measured photocurrent ( f i g . f 9 ) . At the higher photon energies considerable time i s spent by the energetic holes i n thermalizing with the l a t t i c e before recombination occurs (Hoenig 1960, L e v i t t and Hoenig 1961, Koenig 1958). This increased l i f e t i m e pro-duces the increase i n the measured s i g n a l at higher photon energies. This increase i n photosignal cannot be e n t i r e l y a ttributed to l i f e t i m e e f f e c t s , 68 however, since there i s the p o s s i b i l i t y of conduction i n the Pl/2 band at these energies, as has been previously mentioned. An estimate of the t o t a l recombination p r o b a b i l i t y for the holes, B T , and the capture cross section of the ionized impurities, G^ . , can be obtained from t . The rms average v e l o c i t y of holes with a density of states e f f e c t i v e mass, m = 0.6me, described by a Maxwell-Boltzmann d i s t r i b u t i o n at the temperature of the l a t t i c e i s 1.6 x 10^cm/sec. Using this v e l o c i t y and the low f i e l d l i f e t i m e of 2 x 10"*^ sec obtained from f i g s . 19, the recombination p r o b a b i l i t y B^ i s _ 1 B™ = ^ 5 x 10"^cm3/sec , * N D and the capture cross section i s ^"T = 3 x 1 0 - 1 ° c m 2 . Using Lax's theory (1960), the capture cross section for a Boltzmann -9 2 d i s t r i b u t i o n was calculated to be ^ 5 x 10 cm , which i s larger than the observed cross section. The d i f f e r e n c e between the two r e s u l t s may be due to the r e l a t i v e l y large impurity concentrations used here. The theory includes contributions from the very highly excited states of the impurity. However, these states have become non-localized because of overlap, so a hole i s no longer trapped at any p a r t i c u l a r impurity when i n these states. This decrease i n the number of a v a i l a b l e trapping states would make the capture cross section smaller, as observed. This could also explain the unusually large l i f e t i m e measured for the 18 3 1.5 x 10 /cm sample ( f i g . 19 (d) ). Since t h i s sample i s nearly degen-erate, a l l the excited states are i n e f f e c t i v e i n the capture process. The only means of decay, therefore, i s by a t r a n s i t i o n d i r e c t l y from the 69 e x c i t e d s t a t e bands t o the ground s t a t e band. From f i g . 19 ( d ) , the l i f e -time o f h o l e s i n t h i s sample i s Z (*) » 3 x 1 0 " 5 sec so the c o r r e s p o n d i n g r e c o m b i n a t i o n p r o b a b i l i t y c o n s t a n t and c r o s s s e c t i o n f o r t h i s sample a r e -9 ~\ B — 1 3 x 10 cm /sec and T S- tf 2 x 1 0 - 1 5 cm 2 . T h i s v a l u e o f B i s the same o r d e r o f magnitude as the p h o n o n - r e c o m b i n a t i o n T p r o b a b i l i t y c o n s t a n t f o r a d i r e c t r e c o m b i n a t i o n v i a phonon e m i s s i o n d e r i v e d by Gummel and Lax (1955). C. IMPURITY CONDUCTION 1. CONDUCTION MECHANISMS — — — _ _ _ _ _ _ _ _ _ _ _ ^ Because o f the v a r i e t y o f e f f e c t s w h i c h a r e l o o s e l y c l a s s e d as impur-i t y c o n d u c t i o n o r i m p u r i t y band c o n d u c t i o n , i t i s n e c e s s a r y to c l a r i f y the s e p a r a t e c o n d u c t i o n mechanisms. The e f f e c t s o f i m p u r i t i e s can g e n e r a l l y be c l a s s i f i e d a c c o r d i n g t o t h e i r c o n c e n t r a t i o n i n the m a t e r i a l i n the f o l l o w i n g manner. a) A t low i m p u r i t y c o n c e n t r a t i o n s the average s p a t i a l s e p a r a t i o n o f i m p u r i t i e s i s l a r g e and hence the en e r g y s p e c t r u m o f each i m p u r i t y i s u n a f f e c t e d by the p r e s e n c e o f o t h e r i m p u r i t i e s . A t low t e m p e r a t u r e a l l the ground s t a t e s o f the i m p u r i t i e s w i l l be o c c u p i e d e x c e p t f o r a s m a l l f r a c t i o n w h i c h remain i o n i z e d because o f co m p e n s a t i o n . C o n d u c t i o n i s p o s s i b l e i n t h i s system t h r o u g h a p r o c e s s o f p h o n o n - a s s i s t e d t r a n s i t i o n s between the n e u t r a l and i o n i z e d c e n t e r s . T h i s c o n d u c t i o n v i a a. " h o p p i n g " p r o c e d u r e i s n o r m a l l y r e f e r r e d t o as i m p u r i t y c o n d u c t i o n ( M i l l e r and 70 Abrahams 1960, Kasuya 1958, Pollack 1965, Mott and Twose 1960). Conduction vi a this process depends c r i t i c a l l y on the impurity concentration and the degree of compensation. Since conduction occurs by a process of discon-tinuous jumps, the usual d e s c r i p t i o n i n terms of mobility and mean free paths of the c a r r i e r i s no longer appropriate. b) At very large impurity concentrations considerable overlap between ground state impurity wave functions broadens these levels to such an extent that the levels merge with the valence band. At these concen-trations the impurities should no longer be treated as a. perturbation on the normal energy spectrum of the l a t t i c e . Instead, the problem becomes one of f i n d i n g the e f f e c t s of f l u c t u a t i o n s i n the periodic p o t e n t i a l caused by the impurities on the band structure of the material. The e f f e c t s of random d i s t r i b u t i o n s of impurities on the band structure have been d i s -cussed by Kane (1963), Bonch-Breuvich (1962) and Matsubara and Toyozawa (1961). In this case the material i s described as being degenerate with conduction being e s s e n t i a l l y a. m e t a l l i c form of conduction. Once the impurity states have merged with the band, the impurities w i l l act as s c a t t e r i n g centers for the c a r r i e r s . A further increase i n impurity concentration r e s u l t s i n a reduced mobility due to increased scatterings. c) Between the extremes of low concentration, and high concentration, a. v a r i e t y of e f f e c t s occur which are classed as "impurity band" conduction. At f a i r l y high concentrations, but lower than for (b), interactions be-tween impurity ground states w i l l produce a band extending throughout the c r y s t a l which i s separated from the valence band. This band i s generally referred to as "the impurity band". The width of this band increased with impurity concentration u n t i l i t overlaps the valence band. The main factors determining the mobility i n this impurity band are 71 the c o r r e l a t i o n s between holes associated with d i f f e r e n t impurity centers and the extent of overlap between impurity wavefunctions. When corr e l a t i o n s are dominant (at lower concentrations), then i t i s a good approximation to assume that the holes w i l l be l o c a l i z e d at the impurity s i t e s . Compensating centers are again necessary for conduction. When overlap energy i s dom-inant, however, conduction i s possible without compensation because the hole i s no longer l o c a l i z e d at a p a r t i c u l a r impurity. The t r a n s l a t i o n a l energy of a hole i n the band w i l l depend on the amount of overlap, so that an increase i n overlap would increase i t s e f f e c t i v e mobility. A s i m i l a r type of band formation was postulated to occur for the excited states of the impurities (Erginsoy 1950). Because of the larger s p a t i a l extent of these excited states, this banding w i l l occur at concen-trations considerably smaller than those required for ground state banding. Conduction i n these excited state impurity bands was suggested as a. possible explanation of the anomalous behavior of the r e s i s t i v i t y and H a l l constant for p-type germanium at intermediate concentrations (Fritzsche 1955). 2. CONDUCTION IN IMPURITY BANDS The p o s s i b i l i t y of conduction i n the energy bands of an impurity was examined by Baltensperger (1953) f o r the case of a periodic l a t t i c e of impurities. The semiconductor i s characterized by an e f f e c t i v e mass m of the hole i n the valence band, and by a d i e l e c t r i c constant K . A c e l l u l a r method was used i n order to compute the energies of the excited state impurity band edges as shown i n f i g . 20. The Wigner-Seitz polyhedra surrounding each impurity are approximated by spheres of radius r s 0 2 4 6 8 10 12 14 16 18 FIG. 20 : Formation of the Is and 2p impurity bands from hydrogenic wavefunctions (after Baltensperger ). Broadening of the levels vs. distance between impurities i n terms of the e f f e c t i v e Bohr radius a". 73 where i s the density of impurities. Within each c e l l the hole was assumed to s a t i s f y an e f f e c t i v e mass Schrb'dinger equation 2m ,2 U r ' (26) e' where i s the p o t e n t i a l acting on the hole, and €. i s the energy of K r the hole i n the state V . The general solutions of (26) are the regular hydrogenic wave-functions The energy of the hole i s given by e2 (27) n ~ 2 K a V where * 2 m e* (28) and where n i s to be determined by appropriate boundary conditions obtained from the requirement that the wavefunctions have the form of Bloch functions. The excited states i n this way are given "band" prop-e r t i e s , allowing the holes to move throughout the material when excited into one of these "bands". An e f f e c t i v e mass m+ can be introduced to characterize the dynamic property of a hole whose energy l i e s i n an impurity band. This e f f e c t i v e mass i s defined i n terms of the bandwidth, A& , of the impurity band by -ft2 k 2 Ae = 1 1 (29) 2m+ where k i s defined by 74 47T 3 k 3 = N (30) 3 A Combining the previous equations, m+ can be written i n terms of the e f f e c t i v e mass m ; ^ - 0 . 1 7 1 (_!f_\ - ^ i — . (31) m .+ \ a* / e 2/2a* Writing the mobility i n the valence band as /U= -±2L (32) m and the mobility i n the impurity band as - -i<i>' , / m+ then JU+ = _m* / 7XT m+ /U m+ T (33) (34) I t should be noted here that we are able to write the mobility i n the impurity bands as e<jr>+ (33) because we have already assumed that m+ i s a constant over the e n t i r e impurity band. This assumption i s contained i m p l i c i t l y i n equation (29) where we have written the width of the impurity band as h 2 k 2 A€ = . (29) 2m+ This assumption implies that there i s a parabolic dependence of energy on the wavenumber vector f o r the e n t i r e B r i l l o u i n zone. Even though this 75 assumption i s not e n t i r e l y r e a l i s t i c , i t w i l l allow us to make some comparison between theory and experiment. In equation (33), < ^ T ^ i s the average s c a t t e r i n g time or momentum rela x a t i o n time of holes i n the impurity band and T i s the scattering time f o r holes i n the valence band. From the v a r i a t i o n of G(A ) with concentration (shown i n f i g . 17), we have concluded that once there i s s u f f i c i e n t overlap of the wavefunctions so conduction i n an impurity band i s possible, the neutral impurities hinder conduction by sca t t e r i n g the holes. In the case of the perfect impurity, l a t t i c e , however, we have taken the e f f e c t of the impurities into account by assigning an e f f e c t i v e mass m+ to the holes. The holes would then be able to move f r e e l y through-out the c r y s t a l without s c a t t e r i n g from these impurities. This s c a t t e r i n g of holes by the impurities could be accounted f o r i f we make a modification i n the model of a perfect impurity l a t t i c e . In an actual semiconductor the impurities do not form a. periodic array but are randomly d i s t r i b u t e d throughout the material. The r e s u l t i n g impurity bands cannot be expected to have w e l l defined edges as have been calculated, nor can they have such w e l l defined band properties. For example, the requirement that the non-localized wavefunctions have the form of Bloch functions could not be s a t i f i e d since there i s no longer t r a n s l a t i o n a l p e r i o d i t y associated with the impurity l a t t i c e . The con-cept of band formation w i l l have to be interpreted as the t r a n s i t i o n from l o c a l i z e d to extended wavefunctions with a decrease i n the importance of c o r r e l a t i o n between holes. By introducing this randomness we are able to have the holes i n t e r -act with the impurities again. Now, however, the holes do not scatter from the impurities i n the usual way, but scatter from the "randomness" i n the d i s t r i b u t i o n . The v a r i a t i o n of the hole mobility with concen-76 t r a t i o n i n these bands w i l l then be determined by the v a r i a t i o n s of m+ and the sc a t t e r i n g time, + . F i g 21(a) shows the change i n m /m with the concentration of impurities f o r the 2p band as calculated by Baltensperger (1953). Since m i s a property of the host l a t t i c e and can be considered a constant, f i g . 21(a) i n e f f e c t shows the v a r i a t i o n / o f m+ with concentration. There i s an i n i t i a l rapid decrease i n m+ as the band begins forming, followed by a much less rapid decrease as the concentration further increases. The s c a t t e r i n g time, on the other hand, would be expected to decrease as the t o t a l number of impurities which deviate from the regular array increases at higher impurity concentrations. In f a c t , i f we assume the sca t t e r i n g time ^TX> t o D e inversely proportional to the impurity concentration, then we could write the impurity band mobility as m* (32) where the p r o p o r t i o n a l i t y constant ^ = ^T^+/ T i s independent of the impurity concentration. This assumption regarding ^TV" i - s equivalent to s t a t i n g that the number of s c a t t e r i n g centers, which i s the number of impurities out of place i n a periodic array, i s d i r e c t l y proportional to the number of impurities present. I t does not assume that the sc a t t e r i n g times and T* are the same, only that they have the same dependence on concen-t r a t i o n . For this reason we can only f i n d a r e l a t i v e dependence between //^ / (- + andyl/- i n terms of the factor ^ . Using the t h e o r e t i c a l * + values of m /m shown i n f i g . 21(a) and valence band m o b i l i t i e s given i n table I I , we can f i n d the mobility of holes i n the 2p band at d i f f -erent concentrations. The mobility i n the band i s shown i n f i g . 21(b) m / nr 0.1 FIG. 21(a) : Concentration (cm - 3) Ratio of e f f e c t i v e mass i n the valence band m* to e f f e c t i v e mass i n the 2p impurity band vs. impurity concentration ( Baltensperger 1953 ). Concentration (cm" ) FIG. 21(b) : Theoretical m o b i l i ty i n the 2p impurity band i n terms of the dimensionless parameter |g ysl impurity concentration. 78 i n terms of the dimensionless constant . Here we can see the e f f e c t s of two competing processes on yU. , namely the rapid increase as the-band begins to form, followed by the decrease when scatterings become dominant. Comparing the values of /jQ given i n f i g . 21(b) with the m o b i l i t i e s l i s t e d i n table II, we see that ^ 6C+/y_> i s considerably larger than the valence band mobility for a l l concentrations greater 16 3 than 1 / 1 4 x 10 /cm . Since the impurity band mobility i s not expected to exceed the mobility i n the valence band, we would require that ^ 6 ^ 1 . The consequence of assuming p a r t i c u l a r values of ^ w i l l be discussed l a t e r . The formation of the impurity bands and the subsequent changes i n mobility i n these bands with concentration have been experimentally observed f o r two separate impurity bands. F i g . 22 shows the changes i n conductance observed i n the d i f f e r e n t samples when holes are excited into the impurity bands. The top curve i s f o r e x c i t a t i o n into the band com-posed of states 3 and 4, while e x c i t a t i o n into the band at state 2 pro-duces the conductance changes shown i n the bottom curve (states l a b e l l e d as i n Colbow 1963). For both of these bands we see the i n i t i a l , f a i r l y rapid, r i s e i n mobility as the bands begin forming followed by the decrease when scatterings dominate. This t r a n s i t i o n to non-localized wavefunctions occurs at a lower concentration f o r states 3 and 4 than for state 2. This i s to be expected since the higher excited states (3 and 4) have a larger s p a t i a l extent, so overlap becomes important at lower impurity concentrations. The exact behaviour of A G( X ) during the i n i t i a l increase i s d i f f i c u l t to determine, but i t i s f e l t that the curves shown are reason-ably j u s t i f i a b l e . For both bands we have concentrations for which the 79 Impurity concentration (cm ) FIG. 2 2 : Measured conductance change A G ( \ ) produced by excitation of holes into two different impurity bands vs. impurity concentration. 80 measured A G( X ) i s zero or at least A G(A ) ^ 0.001 x 10 /ohm, so that we know conduction does not occur i n these impurity bands f or a l l concentrations. In the case of states 3 and 4, the photocurrent i s zero 15 3 for the 1.2 x 10 /cm sample at this energy, but i s non-zero f o r the 1.2 x 10*^/cm3 sample. A l l that can be said about this band, therefore, i s that the t r a n s i t i o n to non-localized wavefunctions occurs at a range of concentrations intermediate to 1.2 x 10 ^  and 1.2 x 10*^ impurities/cm^. S i m i l a r l y , the increase i n A G(X) for state 2 indicates that these 16 16 states become non-localized between 1.2 x 10 and 4.5 x 10 impurities/cm . In order to compare the measured v a r i a t i o n s i n AG(A) with the re s u l t s of Baltensperger's theory, we have had to make use of an add-i t i o n a l f a c t which should be c l a r i f i e d . Since AG(A) depends on both the l i f e t i m e of the holes as well as t h e i r mobility, what i s a c t u a l l y shown i n f i g . 22 i s the v a r i a t i o n i n Z£(A) /2T(X) with concentration i n the two bands. This i s the same d i f f i c u l t y that was encountered i n deviations i n these curves from a 1/N dependence was att r i b u t e d to an A increase i n l i f e t i m e at the higher concentrations. The decrease i n l i f e t i m e required to produce t h i s deviation from a 1/N^ concentration dependence i s only a factor of about 4 over the concentration range 10*^/cm3 to 5 x 10*7/cm^. The decrease i n A G(X) i n the impurity bands over the same range i n concentration ( f i g . 22) i s much greater than t h i s , being, i n f a c t , almost a factor of 100. Therefore, we can conclude that the concentration dependence observed i n A G(\) ( f i g . 22) i s due almost e n t i r e l y to v a r i a t i o n s i n mobility i n these bands. f i g . 17 where we have yU-(r\ ) t (X ) f o r high ier energy photons. The 81 The e x a c t c o n c e n t r a t i o n dependence of the m o b i l i t y i n these bands c o u l d n o t be d e t e r m i n e d s i n c e the d e c r e a s e i n l i f e t i m e w i t h i n c r e a s e d c o n c e n t r a t i o n i s known o n l y a p p r o x i m a t e l y . The q u a l i t a t i v e p i c t u r e we have o b t a i n e d o f the m o b i l i t y i n these bands i s as f o l l o w s . The m o b i l i t y i n i t i a l l y i n c r e a s e s r a p i d l y as the s t a t e s become n o n - l o c a l i z e d , r e a c h e s a. peak,and th e n i s re d u c e d by f u r t h e r i n c r e a s e s i n i m p u r i t y c o n c e n t r a t i o n . W i t h i n the l i m i t s o f the a s s u m p t i o n s used, B a l t e n s p e r g e r ' s s i m p l e t h e o r y appears to g i v e a r e a s o n a b l y good q u a l i t a t i v e e x p l a n a t i o n of the e x p e r i m e n t a l r e s u l t s . As has been the problem t h r o u g h o u t the e x p e r i m e n t , q u a n t i t a t i v e e s t i m a t e s o f e i t h e r the m o b i l i t y o r l i f e t i m e i n t h e s e i m p u r i t y bands w i l l i n v o l v e f u r t h e r a s s u m p t i o n s . From the t h e o r y we have the m o b i l i t y i n the 2p band i n terms of the d i m e n s i o n l e s s parameter jS I f we assume t h a t the l i f e t i m e s i n each of these bands i s the same as i n the v a l e n c e band, then we w i l l be a b l e t o e s t i m a t e + and from the measured c o n d u c t a n c e changes ( f i g . 2 2 ) . I f we t a k e ( X ) t o be ^ 10"^ sec (see f i g . 1 9 ) , then the peak m o b i l i t y f o r the band a t s t a t e s 3 and 4 i s + 2 . 150 cm / v o l t - s e c F o r s t a t e 2, u s i n g the same l i f e t i m e the peak m o b i l i t y i s + 2 y/U- ^y3 c m / v o l t - s e c T h i s means t h a t the d i m e n s i o n l e s s q u a n t i t y ^ w i l l have t o be 4 / 1 3 x 10" f o r s t a t e s 3 and 4 and s l i g h t l y l e s s f o r s t a t e 2. The low v a l u e s o f peak m o b i l i t i e s f o r these bands was n o t un e x p e c t e d s i n c e c o n d u c t i o n i s o c c u r i n g t h r o u g h a f a i r l y narrow band, w h i c h u s u a l l y i m p l i e s a lower m o b i l i t y f o r the c a r r i e r s . 82 On the other hand, suppose that the decrease i n s i g n a l at these low energies i s due to l i f e t i m e e f f e c t s and that the mobility i n these states i s approximately the same as i n the valence band. In this case P & 0.1 and the l i f e t i m e s are; 2T (A ) ^ 10 sec for holes i n states 3 and 4, 2" (A ) ^ 4 x 10" sec for holes i n state 2. Even though the l i f e t i m e s obtained with the second assumptions are not unreasonable, i t i s f e l t that the f i r s t case i s more representative of what i s occuring. This i s that the l i f e t i m e s i n these states i s _ q 10 sec, but that the m o b i l i t i e s i n these impurity bands are much lower than i n the valence bands. Further modifications in the r e s u l t s of the simple model i n t r o -duced by the randomness are as follows. The e f f e c t s of a. random d i s -t r i b u t i o n on the impurity levels has been investigated for a. one dimensional l a t t i c e by James and Ginzbarg (1953) and for an actual semiconductor by Matsubara and Toyozawa (1961). For. both of these cases the density of states has a maximum at the o r i g i n a l energy but t a i l s o f f on both the high and low energy sides, with the high energy t a i l merging with the valence band. It has been shown (Bonch-Breuvich 1962) that the impurities w i l l also produce a d i f f u s e edge on the valence band. This "band t a i l " w i l l extend well into the forbidden region, but with a density of states which decreases r a p i d l y away from the band edge. This smearing out of both the impurity bands and the valence band would account for the lack of more d e f i n i t e peaks i n the photoresponse ( f i g . 10) and for the lack of a d e f i n i t e cut-off at low energies. In 83 addition to increasing the broadening, i t has been shown (Aigrain 1954) that the i n t e r a c t i o n between the impurities would increase for a random array. This would mean that band formation should occur at concen-trations lower than those indicated by Baltensperger. The degree to which this w i l l a f f e c t the estimates i s not known however, but could The concept of impurity band formation i n the excited states has been questioned by Fritzsche(1955). A d i f f i c u l t y a r i ses i n the model of Erginsoy and Baltensperger because of the use of ordinary impurity wavefunctions i n the formation of the impurity bands. An excited state associated with a p a r t i c u l a r impurity can be occupied only i f a l l the other excited states and the ground state of this impurity are unoccupied. This would l i m i t the number of av a i l a b l e excited states which are able to p a r t i c i p a t e i n the conduction to the number of ionized impurities present. At low temperatures the only ionized impurities present w i l l r e s u l t from compensating centers. The number of a v a i l a b l e states, therefore, would be limited to such an extent that conduction through these states would be exceedingly small. However, once there i s s u f f i c i e n t i n t e r a c t i o n between the excited states to form the band we cannot consider holes i n these states to be associated with any one p a r t i c u l a r impurity, but to the ent i r e system of impurity atoms. In this way conduction through these excited states would proceed as i n a regular band. 21 (b)) toward lower concentrations. 84 CHAPTER 5  CONCLUSIONS From the r e s u l t s of this experiment we may conclude that the excited states of the impurities form bands of non-localized states extending throughout the c r y s t a l . It appears as i f there i s a f a i r l y sharp t r a n s i t i o n region over which the impurity states go from being l o c a l i z e d at the impurities to being band-like states. Over this region of impurity concentrations the mobility of holes in these states increases r a p i d l y as was predicted by Baltensperger. Once these states become non-localized however, i t appears as i f further increases i n impurity concentration act to reduce the mobility i n these bands. This can be accounted for by considering a more r e a l i s t i c semiconductor i n which the impurities are d i s t r i b u t e d randomly throughout the material. For this case we can consider the holes as being scattered by the random f l u c t u a t i o n s i n p o t e n t i a l at d i f f e r e n t positions throughout the c r y s t a l caused by this random arrangement. These scatterings quickly become dominant over further increases i n mobility due to increased overlap, so the mobility decreases with the addition of more impurities. From the nature of photoconductivity measurements, the r e s u l t s are obtained i n terms of the p r o d u c t ) T ( A ) for.the excited holes. This prevented the m o b i l i t i e s i n the i n d i v i d u a l impurity bands from being determined d i r e c t l y . This v a r i a t i o n of the l i f e t i m e - m o b i l i t y product with photon energies was displayed i n the photo-response measurements shown i n f i g s . 9. Here again, the v a r i a t i o n s i n photo-response could not be unambiguously at t r i b u t e d to e i t h e r v a r i a t i o n s i n the hole mobility or l i f e t i m e at these higher energies. To remove 85 this ambiguity b e t w e e n ^ (X ) and t ( X ) a separate measurement would have to be made of eit h e r one. An attempt was made to separate the two factors by assuming the mobility i n the valence band to be governed by e l a s t i c s c a t t e r i n g from neutral impurities. Using the m o b i l i t i e s obtained with this assumption, the l i f e t i m e of holes excited into the valence band (by 75 meV. photons) i s found to be approximately 10"^ sec. This indicates that the ionized -10 2 boron impurities have capture cross sections of about 3 x 10 cm , which i s i n reasonable agreement with the r e s u l t s of Lax's theory of giant traps (1960). A more exact .determination of the capture cross section i s d i f f -i c u l t because of the uncertainty i n the density of compensating centers. In excited state bands i t i s most l i k e l y that the l i f e t i m e of holes i s approximately the same as i n the valence band. This leads to impurity 2 band m o b i l i t i e s less than 150 cm /volt-sec, which i s a. reasonable mobility for conduction through a narrow band. The e f f e c t of impurity banding was also observed on the capture cross section of the ionized impurities. At high impurity concentrations the excited states are no longer capable of trapping the holes so the cross section i s greatly reduced. Because of t h i s , exceedingly long l i f e t i m e s (>—3 x 10"^ sec) were obtained f o r the nearly degenerate sample of concen-18 3 t r a t i o n 1.5 x 10 /cm . A decrease i n the recombination p r o b a b i l i t y with increased hole energy, as indicated by Lax (1960), provides a q u a l i t a t i v e explanation of the non-ohmic nature of the d.c. c h a r a c t e r i s t i c s at low f i e l d s . A l l samples exhibited this non-ohmic behaviour, and also a l l samples were characterized by a c r i t i c a l breakdown f i e l d . This breakdown i s caused by the heating of the holes at high f i e l d s to energies of the order of the impurity i o n i z a t i o n energy. These energetic holes are then capable of impact-ionizing 86 \ the impurities producing the run-away in hole multiplication. The non-ohmic behavior of the sample has d i r e c t repercussions on the photosignal produced by the incident r a d i a t i o n . For a f i x e d i l l u m i n a t i o n on the sample the photo-voltage appearing across the sample reaches a saturation value as the applied f i e l d i s increased. Further increases i n f i e l d decrease the observed photo-voltage as the sample resistance decreases. A more d i r e c t determination of the m o b i l i t y or l i f e t i m e of holes i n the impurity bands would necessitate further measurements. The mobility could be obtained from a photo-Hall experiment while time dependent photo-conductivity experiments would y i e l d l i f e t i m e s . However, the high resistance of the samples at low temperatures combined with the low l e v e l s of i l l u m i n a t i o n a v a i l a b l e i n the f a r i n f r a r e d would make the photo-Hall experiments d i f f i c u l t . Likewise, the short time scales involved in l i f e t i m e determinations would hamper time dependent photoconductive measurements. 87 APPENDIX A: PERFORMANCE OF BORON-DOPED SILICON AS A FAR INFRARED DETECTOR The performance of these p a r t i c u l a r samples as r a d i a t i o n detectors w i l l be rated i n terms of the usual c r i t e r i a , used to measure detecting a b i l i t y (see for example Jones 1959, Jamieson et a l 1963). F i r s t , however, the various ratings used i n measuring the detecting a b i l i t y w i l l be presented. RESPONSIVITY (R): The responsivity of a detector i s defined as the output voltage divided by the power output i n watts. The responsivity i s useful i n determining s i g n a l strengths to be expected from the detector but does not indicate the minimum d e t e c t i b l e power. NOISE EQUIVALENT POWER (N.E.P.): This gives the incident power required to produce an output equal to the noise voltage. The N.E.P. i s obtained by d i v i d i n g the rms noise voltage by the responsivity and i s there-fore given i n units of watts. This r a t i n g of a. detector w i l l determine the minimum power which can u s e f u l l y be detected. DETECTIVITY (D): The r e c i p r o c a l of the N.E.P. i s c a l l e d the detec-t i v i t y . This gives the s i g n a l to noise r a t i o of the detector per unit incident power. D : The D-star or s p e c i f i c d e t e c t i v i t y i s , i n e f f e c t , the d e t e c t i v i t y measured with a bandwidth of one cycle per second and reduced to a responsive area of one cm , and usually given i n units of cm cps 2/watt. I t i s related to the d e t e c t i v i t y by D* = (A _ f ) * D, where A i s the area of detector and A. f i s the bandwidth of the system. Because the d e t e c t i v i t y often varies with d i f f e r e n t operating conditions, the quoted D i s usually given with the parameters associated with the Table III Comparison of Detecting A b i l i t y of Various Types of Infrared Detectors Cut-Off R (Volts/Watt) N.E.P. (Watt) D cm cps%/wa.tt Frequency Thermocouple 5 2-20 10-10 25 cps Go lay C e l l 5 6 x 10 2 x 10 9 25 cps Lead S u l f i d e C e l l 6 l O " 1 * 1 0 U 1 KC Ge ( B ) 7 l O " 1 1 2 x 1 0 U ^~ 10 8 cps S i (B) 1.2 x 16 3 10 /cm 3 2 x 10 l o " 8 5 x 10 8 4.5 x 10 1 6/cm 3 10 3 1.2 x 10" 9 4 9 x 10 8 4.5 x 10 1 7/cm 3 2 3 x 10 8 x l O ' 1 0 5.5 9 x 10 - 10 cps 1.5 x 10 1 8/cm 3 2.5 x 10 2 l O " 9 4.5 x 10 9 Data for Perkin-Elmer Thermocouple and Golay C e l l obtained from Jamieson et a l . 1963. Kodak Ekton Detectors. Shenker et a l . 1964. 89 measurement. For example, the measurements made here w i l l provide D (E» f , A f ) w i t h E = the photon energy (75 meV.), f = 870 cps and A f = 40 cps. A comparison of these d i f f e r e n t ratings i s given i n Table III for various types of infrared detectors and the samples used i n this experiment. As can be seen from Table I I I , the s i l i c o n samples make rather poor detectors i n spite of t h e i r large responsivity . In addition to t h e i r small detecting a b i l i t y , other d i f f i c u l t i e s would be encountered which are not included i n the table. In order f o r these samples to detect r a d i a t i o n they must be maintained at l i q u i d helium temperatures and be well shielded from background r a d i a t i o n . This immediately l i m i t s the p r a c t i c a l usefullness of these materials. In addition, there i s not a uniform responsivity over the range of wavelengths used. This would be p a r t i c u l a r l y troublesome i n the thicker samples i n the region of the l a t t i c e absorption peaks. For an i d e a l detector the l i m i t i n g noise sources which are generally inescapable w i l l be the following ( P e t r i t z 1959, Moore and Shenker 1965, Smith et a l . 1958). (a) Nyquist noise i n the load r e s i s t o r and detector. (b) Amplifier noise. (c) Generation - recombination noise. This l a s t noise source i s the r e s u l t of fl u c t u a t i o n s i n the f l u x of incident photons which cause f l u c t u a t i o n s i n the rate of generation of c a r r i e r s combined with f l u c t u a t i o n s i n the recombination rates of the c a r r i e r s . This process sets the ultimate l i m i t to the detecting a b i l i t y of the material and detectors operated under this l i m i t a t i o n are said to be background limited (van V l i e t 1958). The noise from these sources i s considerably less than the noise encountered, so the detectors are not operating even near th e i r ultimate 90 l i m i t . The excess noise i s att r i b u t e d to the contacts used here. Although the methods we have used to produce contacts are f a i r l y successful f o r germanium detectors (Shenker et a l . 1964), they do not allow the ultimate detecting a b i l i t y to be achieved f o r s i l i c o n photodetectors. 91 BIBLIOGRAPHY Baltensperger, W. 1953. P h i l . Mag. 44, 1355. Bichard, J.W. and Gi l e s , J.C. 1962. Can. J . Phys. 40, 1480. B l a i r , L.R., Plyer, E.K., and Benedict, W.S. 1962. J. Research N.B.S. - A Phys. and Chem. 66, 223. Bonch-Breuvich, V.L. 1962. F i z . Tverd. Tela. 4, 2660. (Translated i n Sov. Phys. - So l i d State 4, 1953 (1963)). Bube, R.H. 1960. Photoconductivity of Soli d s . (John Wiley and Sons, Inc., New York). Burstein, E., Picus, G., and Sclar, N. 1954. Photoconductivity Conference held at A t l a n t i c C i t y (John Wiley and Sons, Inc., New York), p. 353. Burstein, E., Picus, G., Henvis, B., and Wal l i s , R. 1956. J. Phys. Chem. Solid s . 1_, 65. 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So l i d State Physics, edited by F. Seitz and D. Turnbull, Vol. 5. (Academic Press, Inc., New York). Lax, M. 1960. Phys. Rev. 119, 1502. L e v i t t , R. and Hoenig, A. 1961. J. Phys. Chem. So l i d s , 22, 269. Matsubara, T. and Toyozawa, Y. 1961. Prog. Theor. Phys. 26_, 739. M i l l e r , A. and Abrahams, E. 1960. Phys. Rev. 120, 745. Mott, N.F. and Twose, W.D. 1961. Adv. i n Physics 10, 107. Newman, R. 1956. Phys. Rev. 103, 103. Picus, G.S. 1962. J. Phys. Chem. Solids, 23, 1753. Pollack, M. 1965. Phys. Rev. 138, A 1822. Putley, E.H. 1964. Phys. Stat. S o l i d i 6, 571. Ray, R.K. and Fan, H.Y. 1961. Phys. Rev. 121, 768. Rittner, E.S. 1954. Photoconductivity Conference held at A t l a n t i c C i t y (John Wiley and Sons), p. 215. Sclar, N. 1956. Phys. Rev. 104, 1559. Sclar, N., and Burstein, E. 1957. J. Phys. Chem. Solids, 2, 1. Shenker, H., Moore, W.J., Swiggard, E.M. 1964. J. Appl. Phys. 35, 2965. Smith, R.A., Jones, F.E., and Chasmar, R.P. 1958. Detection and Measurement of Infrared Radiation (Oxford University Press), Chapter 2. Stockman, F. 1954. Photoconductivity Conference held at A t l a n t i c C i t y ( J . Wiley and Sons, Inc.) p. 269. Wal l i s , R.F. and Shenker, H. 1963. U.S. Naval Research Laboratory Report, 5996. Wa l l i s , R.F. and Shenker, H. 1964. Naval Research Laboratory Memorandum Report, 1493. van V l i e t , K.M. 1958. Proc. I.R.E. 46, 1004. Yamashita, J. 1961. J. Phys. Soc. Japan, 16_, 720. Zwerdling, S., Button, K.J. Lax, B., and Roth, L.M. 1959. Phys. Rev. Letters, 4, 173. 

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