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Electron mobility in germanium at high temperatures Eastman, Philip Clifford 1960

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ELECTRON MOBILITY I N GERMANIUM AT HIGH TEMPERATURES  by PHILIP CLIFFORD EASTMAN B.Sc,  M.Sc,  McMASTER UNIVERSITY", 1955 McMASTER UNIVERSITY, 1956  A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  %  DOCTOR OF PHILOSOPHY  i n t h e Department of Physics  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA January  i960  In p r e s e n t i n g  this thesis i n p a r t i a l fulfilment of  the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t t h e L i b r a r y s h a l l make it  freely  a v a i l a b l e f o r r e f e r e n c e and s t u d y .  agree t h a t p e r m i s s i o n f o r e x t e n s i v e  I further  copying of t h i s  thesis<  f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s .  I t i s understood  that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n  Department o f  PHYSICS  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date  Jaoauary 4,  i960  permission.  LIST  OF  PUBLICATIONS  %[}t  1. A t o m i c Masses of N i ^ and N i ^ Eastman, Isenor, Bainbridge, Duckworth Phys. R e v . 103, 145-6, 1956  Pitfaerstig  of ^British  Glolmubra  Faculty of Graduate Studies  2. T h e Effect of R e s i d u a l Gas Pressure upon the Spacing of Mass Spectroscopic Doublets. Isenor, Bainbridge, Eastman, Duckworth C . J . P . 34, 943-5, 1956 3. Some R e c e n t Mass Determinations at M c M a s t e r University Duckworth, Bainbridge, Isenor, Kerr, Eastman " N u c l e a r Masses and their D e t e r m i n a t i o n " Permagon Press, 1957 4. Effect of N e u t r a l Impurities on M o b i l i t y on Nondegenerate Semiconductors. Sodha and Eastman Phys. R e v . 108, 1373-5, 1957 5. Effects of E l e c t r o n - E l e c t r o n Scattering o n H a l l M o b i l i t y of Electrons i n n-Semiconductors. Sodha and Eastman Prog. T h e o r . Phys. 103, 344-5, 1958. 6. M o b i l i t y of Electrons i n Nondegenerate Semiconductors C o n s i d e r ing E l e c t r o n - E l e c t r o n Scattering. Sodha and Eastman Z . f. Phys. 150, 242-6, 1958 7. V a r i a t i o n of H a l l M o b i l i t y of Carriers i n Nondegenerate conductors with E l e c t r i c F i e l d . Sodha and Eastman Phys. R e v . 110, 1314, 1958 8. H a l l M o b i l i t y of Carriers i n Impure Nondegenerate Sodha and Eastman Phys. R e v . 112, 44, 1958  Semi-  FINAL ORAL  EXAMINATION  FOR T H E DEGREE OF  DOCTOR OF PHILOSOPHY of PHILIP  EASTMAN  B. Sc., McMaster University, 1955 M . Sc., McMaster University, 1956 IN R O O M 301 PHYSICS B U I L D I N G M O N D A Y , J A N U A R Y 4th A T 3:00 p. m .  Semiconductors.  9. Drift and H a l l M o b i l i t i e s of Electrons i n Nondegenerate n-Semiconductors. Sodha and Eastman Prog. T h e o r . Phys. 21, 214-5, 1959.  PROGRAMME OF THE  COMMITTEE IN CHARGE D E A N G . M . S H R U M : Chairman  Impure  R. B A R R I E  J.  R. E . B U R G E S S  C. A . M c D O W E L L  NORRIS  K . N . R. T A Y L O R  .T. R. H . D E M P S T E R  J. B. BROWN-  E.  External Examiner:  V. B O H N  Professor H : Y . F A N  Purdue University, Lafayette,  Indiana  ABSTRACT  A study is made of the temperature dependence of the l a t t i c e scattering m o b i l i t y of electrons i n g e r m a n i u m . Previous work on this subject has been restricted to a range ofttemperature f r o m - 1 0 0 ° K to 3 0 0 ° K . In this range it is possible to use specimens i n w h i c h the only scattering the electron suffers is that due to the l a t t i c e vibrations; the lattice m o b i l i t y can then be deduced i n a straightforward manner from measurements of the H a l l constant and c o n d u c t i v i t y of the m a t e r i a l . It was found that over this restricted temperature range the temperature dependence of the lattice m o b i l i t y c o u l d be represented approximately by the format*. e * T - l ° 66, It has, however, been p r e d i c t e d , o n t h e o r e t i c a l grounds, that such a simple power law d e p e n d ence is insufficient, e s p e c i a l l y when the temperature range is greater. T h e present work carries out an extension of the measurements to higher temperatures and studies more c a r e f u l l y the approximation of a simple power law dependence. It is found that if the lattice m o b i l i t y is expressed i n the f o r m ^ ^ ^ T - " , then a has to be considered as increasing from about' 1. 7 to 1. 9'between.200iand 4 0 0 ° K . These results are i n qualitative agreement with the t h e o r e t i c a l predictions. a  " In order to extend the temperature range, strongly h - t y p e specimens of g e r m a n i u m were required. Several basic and p e r m a n ent crystal preparation f a c i l i t i e s , i n c l u d i n g a crystal grower and w i r e saw cutter, were designed and constructed. T h e conductivities and H a l l coefficients of severalispecimens, prepared with different c o n centrations, were measured over the appropriate temperature range. T h e l a t t i c e m o b i l i t y i n these specimens c a n not be d e d u c e d directly from such measurements as the electrons also suffer scattering from the ionized impurities present. A n analysis is g i v e n w h i c h enables the lattice effects to be separated from the i m p u r i t y effects, This analysis is based on; an assumed power, law/dependence of the l a t t i c e r e l a x a t i o n t i m e , o n temperature and of the i m p u r i t y scattering relaxation t i m e , o n temperature and i m p u r i t y concentration. T h e separation of these two scattering effects-is performed i n a way almost independent of-the other-factors: on-which they depend. S o m e information was also obtained on the impurity scattering m o b i l i t y , T h i s slightly favours a-, screened rather than a c u t - o f f C o u l o m b scattering p o t e n t i a l ; -  GRADUATE  Field  of  Study:  Solid  STUDIES  State  Physics  Electromagnetic Theory Quantum Theory of R a d i a t i o n Noise i n Physical Systems Physics of the Solid State Crystal Structure and X - r a y s Servomechanisms A n a l o g u e Computers Structure of Metals C o m p u t a t i o n a l Methods  ii  ABSTRACT  A s t u d y i s made o f the temperature dependence o f the  lattice  s c a t t e r i n g m o b i l i t y o f e l e c t r o n s i n germanium.  P r e v i o u s work on t h i s s u b j e c t has been r e s t r i c t e d t o a range of  temperatures from 100°K t o 300°K  o  I n t h i s range i t i s  p o s s i b l e t o use specimens i n w h i c h t h e o n l y s c a t t e r i n g the e l e c t r o n s u f f e r s i s t h a t due t o the l a t t i c e the  vibrations;  l a t t i c e m o b i l i t y can t h e n be deduced i n a s t r a i g h t f o r w a r d  manner f r o m measurements o f the H a l l c o n s t a n t and conducti v i t y o f the m a t e r i a l *  I t was f o u n d t h a t o v e r t h i s  restrict'  ed temperature range the temperature dependence o f t h e l a t t i c e m o b i l i t y c o u l d be r e p r e s e n t e d a p p r o x i m a t e l y by t h e form  -cc o r  I  I t h a s , however, been p r e d i c t e d , o n  t h e o r e t i c a l grounds, t h a t s u c h a s i m p l e power law dependence i s i n s u f f i c i e n t , e s p e c i a l l y when the t e m p e r a t u r e range i s greater,, The p r e s e n t work c a r r i e s out an e x t e n s i o n o f the measurements t o h i g h e r t e m p e r a t u r e s and s t u d i e s more c a r e f u l l y t h e a p p r o x i m a t i o n o f a s i m p l e power l a w dependence. I t i s found that i f the l a t t i c e m o b i l i t y i s expressed i n the form  °(  about 1.7  I , t h e n a has t o be c o n s i d e r e d as i n c r e a s i n g f r o m t o 1.9  between 200 and l4.00°K.  These r e s u l t s a r e  i n q u a l i t a t i v e agreement w i t h t h e t h e o r e t i c a l p r e d i c t i o n s . I n o r d e r t o e x t e n d t h e temperature r a n g e , s t r o n g l y n-type specimens o f germanium were r e q u i r e d . and permanent c r y s t a l p r e p a r a t i o n f a c i l i t i e s ,  Several basic including •  ill  a c r y s t a l grower and wire-saw c u t t e r , were designed' and constructed. several  The c o n d u c t i v i t i e s and H a l l c o e f f i c i e n t s o f  specimens, p r e p a r e d w i t h d i f f e r e n t c o n c e n t r a t i o n s ,  were measured o v e r the a p p r o p r i a t e temperature r a n g e .  The  l a t t i c e m o b i l i t y I n t h e s e specimens can not be deduced d i r e c t l y from s u c h measurements as the e l e c t r o n s a l s o s c a t t e r i n g from the i o n i z e d i m p u r i t i e s p r e s e n t .  suffer  An  a n a l y s i s i s g i v e n w h i c h e n a b l e s the l a t t i c e e f f e c t s t o be s e p a r a t e d from the i m p u r i t y e f f e c t s .  This analysis i s  based on an assumed power law dependence o f the l a t t i c e r e l a x a t i o n time on temperature and o f the i m p u r i t y s c a t t e r i n g r e l a x a t i o n time on temperature and i m p u r i t y concentration,,  The s e p a r a t i o n o f these two s c a t t e r i n g e f f e c t s i s  p e r f o r m e d i n a way almost independent of the o t h e r f a c t o r s on w h i c h t h e y depend.  Some i n f o r m a t i o n was a l s o o b t a i n e d  on t h e i m p u r i t y s c a t t e r i n g m o b i l i t y .  This  slightly  f a v o u r s a s c r e e n e d r a t h e r than a c u t - o f f Coulomb s c a t t e r i n g potential.  lv  T A B L E  OP  C O N T E N T S  Chapter 1  2  3  Page INTRODUCTION 1.1  Semiconducting C r y s t a l s  1  1.2  Scattering of Carriers  10  1.3  H a l l E f f e c t and H a l l M o b i l i t y  16  METHOD OF ANALYSIS 2.1  Introduction  20.  2.2  Form o f R e l a x a t i o n Time  2!;  2.3  S e p a r a t i o n o f L a t t i c e and Impurity E f f e c t s  28  2.1L  Determination  37  2.5  Summary  o f of and  US  EXPERIMENTAL PROCEDURE 3.1  P r e p a r a t i o n f o r Experimental Measurements  i|6  3ola  Preparation of M a t e r i a l  I4.6  3.1b  P r e p a r a t i o n o f Samples  3«lc  Temperature C o n t r o l  3.Id  Magnetic F i e l d  3»1©  Measurement o f C o n d u c t i v i t y and  51  55  Hall Coefficient  56  3o2  Measurement o f C o n d u c t i v i t y  59  3.3  Measurement o f H a l l C o e f f i c i e n t  62  V  T A B L E  OP  C O N T E N T S  Chapter k  Page ,RESULTS OF THE INVESTIGATION o f of  I4..I  Determination  and <K  ii»2  Use o f the or- if Data  71  I4..3  Lattice Scattering  77  li.ii  Impurity  79  Scattering  Conclusions  69  8l  APPENDIX A C o n s t r u c t i o n o f a C r y s t a l Grower  82  APPENDIX B C o n s t r u c t i o n o f a Wire-Saw f o r Crystal Cutting  85  APPENDIX C Design and C o n s t r u c t i o n of a Small Oven and Temperature C o n t r o l  89  BIBLIOGRAPHY  93  '  LIST OF FIGURES Figure  Title  Page  1  Electron  2  H a l l F a c t o r f o r Mixed L a t t i c e  Energy L e v e l s i n a Semiconducting C r y s t a l  6  and I o n i z e d  Impurity S c a t t e r i n g  18  3  V a l i d Ranges o f N and T  If.2  \±  C r y s t a l Grower  lj.7  •£ 6  I4.8  C r y s t a l Saw Instrument f o r F a b r i c a t i o n  of A l l o y $0  Contacts 7  Schematic Diagram o f Measuring C i r c u i t s  8  Hall Coefficient  and C o n d u c t i v i t y  9  Sources o f E r r o r  in Hall  6lj.  10  D e t e r m i n a t i o n of H a l l C o e f f i c i e n t  11  Plots  12  I l l u s t r a t i o n o f E r r o r Due to Changes in m Values o f Slope and <=< - I n t e r c e p t from oc~y Data Oven Temperature C o n t r o l C i r c u i t  13 lij.  Data  61  Potential  Measurements  of cx-f  f>7  -  6f> 72 7tj. 76 90  vli  ACKNOWLEDGMENTS  The author would l i k e t o express h i s a p p r e c i a t i o n t o Mr. J.B. Gunn f o r s u g g e s t i n g  the t o p i c o f the  i n v e s t i g a t i o n and f o r s u p e r v i s i o n d u r i n g I t s i n i t i a l stages.  H i s a p p r e c i a t i o n i s a l s o extended to Dr. R. B a r r i e  and to other members o f h i s committee  f o r h e l p f u l discuss-  ions during the a n a l y s i s o f the r e s u l t s and p r e p a r a t i o n o f this thesis. F i n a n c i a l a s s i s t a n c e from the N a t i o n a l Research C o u n c i l o f Canada i n the form o f a studentship, i s g r a t e f u l l y acknowledged. The i n v e s t i g a t i o n was c a r r i e d out w i t h  research  f a c i l i t i e s p r o v i d e d by the Defence Research Board o f Canada under grant number 380I-O8.  1  C H A P T E R I d  1 - I N T R O 0 U C T I O N  SEMI CONDUCT! MG  CRYSTALS  A qualitative structure  i n a crystal  h a p p e n s when a g r o u p to occupy  the s i t e s  picture c a n be  o f the e l e c t r o n e n e r g y  band  d e v e l o p e d by c o n s i d e r i n g  what  o f i s o l a t e d atoms i s b r o u g h t t o g e t h e r  of a c r y s t a l  wave f u n c t i o n s o f t h e atoms o v e r l a p , energy l e v e l s This  be c a r r i e d f u r t h e r  the u s u a l development  within  the d i s c r e t e  o f the band  the c r y s t a l .  The  from  electron  electron bands.  (Shockley, 1939)»  theory consists  'the h e l p o f t h e B l o c H t h e o r e m (Dekker, 1 9 5 7 )  of a  atoms o f t h e  crystal.  boundary solutions  o f the l a t t i c e .  conditions  The  by  resulting of quasi-con-  t o t a l momentum o r e n e r g y c a n n o t - change  so  not f r e e  i n t o which  energy be  Hence t h i 3 band  the e l e c t r i c a l  gaps.  they can  they.are e f f e c t i v e l y processes.  a function  i n a completely f i l l e d  b a n d have no n e a r by e n e r g y s t a t e s their  and w i t h  c a n be o b t a i n e d  spectrum i s found to c o n s i s t  the e l e c t r o n s  moving periodic  t i n u o u s a l l o v j e d e n e r g y bands a n d f o r b i d d e n e n e r g y Since  but  to a  w h i c h a r e p l a n e waves m o d u l a t e d  w i t h the p e r i o d i c i t y e l e c t r o n energy  i s subject  the i n d i v i d u a l  the a p p l i c a t i o n o f s u i t a b l e  excited,  electron  o f the S c h r o e d i n g e r e q u a t i o n f o r e l e c t r o n s  'potential arising By  the  form a s e t of q u a s i - c o n t i n u o u s energy  a p p r o a c h may  solution  As  lattice.  to partake i n transport structure  c a n e x p l a i n many o f  p r o p e r t i e s - of c r y s t a l s .  c r y s t a l h a v e enough e l e c t r o n s  to f i l l  I f t h e atoms o f a  t h e l o w e r deep  lying  2  bands and h a l f f i l l  the uppermost o c c u p i e d band, t h e n  are many c l o s e by unoccupied e l e c t r o n s i n t h i s band. has a m e t a l l i c  energy l e v e l s a v a i l a b l e  This s i t u a t i o n occurs i n  A s i m i l a r case can a r i s e i f t h e r e a r e  enough e l e c t r o n s t o j u s t f i l l  an upper band but i t o v e r l a p s  the next h i g h e r band v j i t h no f o r b i d d e n gap. available  to the  Hence they are f r e e and 'the c r y s t a l  conduction nature.  the a l k a l i m e t a l s .  there  Here the  energy l e v e l s o f t h i s next h i g h e r band f r e e the  e l e c t r o n s f r o m the band below.  This s i t u a t i o n arises  the a l k a l i n e e a r t h s * - On the o t h e r hand, t h e r e may enough e l e c t r o n s to f i l l  in  be  just  the uppermost o c c u p i e d band (at  a b s o l u t e zero) w i t h a f o r b i d d e n energy gap betvreen the  top  of t h i s band ( v a l e n c e band) and the bottom o f the next  higher  band ( c o n d u c t i o n b a n d ) . exhibit  I n t h i s case, the c r y s t a l  the p r o p e r t i e s of an I n s u l a t o r or a  may  semiconductor  depending , upon the w i d t h o f the energy gap and the tempera1  ture e The  a l l o w e d c a r r i e r s t a t e s i n the c r y s t a l may  d e s i g n a t e d by the wave number v e c t o r , k. is  be  F o r each band, k  l i m i t e d t o a s e t of allox-jed v a l u e s , and the s m a l l e s t  volume o f k-space c e n t r e d about k a 0 which c o n t a i n s t h i s s e t i s c a l l e d the B r i l l o u i n zone. an energy extrem'um E  Q  A 3imple band x^ould have  a t k : 0 and near t h i s extremum the  energy c o u l d be e x p r e s s e d by the q u a d r a t i c r e l a t i o n E = + w k , where w i s a c o n s t a n t .  Surfaces of constant  E  Q  energy  i n k-space would then be c o n c e n t r i c spheres about the o r i g i n .  3  T h i s s i m p l i f y i n g assumption i s a good guide I n g e n e r a l a n a l ysis. A t a b s o l u t e z e r o , a l l c r y s t a l s w i t h an energy gap between the v a l e n c e and c o n d u c t i o n bands a r e i n s u l a t o r s because they have no f r e e e l e c t r o n s .  At higher  temperatures  some e l e c t r o n s can become t h e r m a l l y e x c i t e d a c r o s s t h e f o r b i d d e n gap and occupy energy l e v e l s i n the c o n d u c t i o n band. The  number o f these c o n d u c t i o n e l e c t r o n s p e r u n i t volume  ( d e n s i t y ) a t temperature  n  * T T  =  T i s g i v e n by  (kj^Lj' ' 7  f f(E) E' JE A  Bo *  (1.1)  1  f o r a s i m p l e band where f ( E ) i s the F e r m i - D i r a c f u n c t i o n g i v i n g the p r o b a b i l i t y o f occupancy o f an energy l e v e l a t energy E , E  C  i o n band, E  w  i s the energy" o f the bottom edge o f the conducti s the Fermi l e v e l a t w h i c h f s -|, k i s  Boltzman's c o n s t a n t , h i s P l a n c k ' s c o n s t a n t , i r ^ i s t h e e f f e c t i v e mass o f an e l e c t r o n i n the c o n d u c t i o n band, T i s t h e a b s o l u t e temperature  A/  C  and  = 2.[nrmh  kTJ  i / 2  i s the e f f e c t i v e d e n s i t y o f energy s t a t e s i n the c o n d u c t i o n band c o n s i d e r e d t o be grouped a t EQ*  The upper l i m i t o f  the I n t e g r a l may be t a k e n as I n f i n i t e hecause o f . t h e r a p i d l y decreasing nature of f ( E ) f o r higher E .  The l a s t  k  approximation h o l d s i f E  F  i s more than a few kT below E Q .  These e l e c t r o n s which are e x c i t e d t o the c o n d u c t i o n band l e a v e vacant energy l e v e l s behind i n the v a l e n c e band.  This  allows the r e m a i n i n g valence e l e c t r o n s a c e r t a i n measure of freedom.  As l o n g as the d e n s i t y o f these v a c a n c i e s i s  r e l a t i v e l y s m a l l , i t i s more convenient t o consider, them as mobile h o l e s w i t h p o s i t i v e  charge and mass than t o c o n s i d e r  the e f f e c t of the remaining valence e l e c t r o n s . convenience  i s not without ample a n a l y t i c a l  This  justification.  The d e n s i t y of these h o l e s i n the v a l e n c e band Is g i v e n by  k  H  where  =  V  2. ( 2 T T hl  '  T  (1.2)  fcj)^  p  i s . the e f f e c t i v e d e n s i t y o f energy l e v e l s  i n the valence  band c o n s i d e r e d t o be grouped a t Ey, the top edge of t h i s band, irip I s the e f f e c t i v e mass o f h o l e s i n the v a l e n c e band, and the approximation holds f o r Ep more than a few kT above Ey. I f a l l the e l e c t r o n s m i s s i n g from the v a l e n c e band a r e In the c o n d u c t i o n band, w i t h no o t h e r s , i t f o l l o w s that  h  «  p  =  * V ~ (Nc^v)  f  e*p(~ ^/KT) E  (1-3)  where n^ Is the i n t r i n s i c c a r r i e r d e n s i t y and E Q = E Q - Ey, the energy gap w i d t h .  Prom ( 1 . 3 )  i t i s apparent  the energy gap i s g r e a t enough (e.g, EQ. ^ lOOkT), i n t r i n s i c c a r r i e r density i s I n s i g n i f i c a n t l y c r y s t a l w i l l remain e s s e n t i a l l y  an i n s u l a t o r .  that i f then the  s m a l l and An  the  example  5  of  t h i s i s diamondo  But i f the energy gap I s s m a l l e r , the  i n t r i n s i c c a r r i e r d e n s i t y can become a p p r e c i a b l e a t r e a s o n a b l e tempera cures and. the c r y s t a l w i l l be an i n t r i n s i c conductor,,  Two  examples of t h i s are s i l i c o n and germanium©  S i m i l a r l y , i f the energy gap  i s n e g l i g i b l y s m a l l , the  c a r r i e r d e n s i t y a t normal temperature m a t e r i a l w i l l be a good conductor,, of  t h i s case,,  semi-  i s q u i t e h i g h and the G r a p h i t e i s an example  O b v i o u s l y these d i s t i n c t i o n s are ones o f  degree, but most cases are r e a d i l y  classifiable  D  T h i s q u a l i t y of n and p- can be d e s t r o y e d by i n t r o d u c i n g u n e q u a l l y h o l e s o r e l e c t r o n s i n some e x t r i n s i c mannero  S m a l l t r a c e s of i m p u r i t y atoms, v a r i a t i o n f r o m  s t o i c h i o m e t r i c c o m p o s i t i o n , or l a t t i c e i m p e r f e c t i o n s can this,,  do  I f the " r e s u l t i s an i n c r e a s e i n the e l e c t r o n d e n s i t y  r e l a t i v e t o the h o l e d e n s i t y , then the c e n t r e . w h i c h i t i s c a l l e d a donor.  S i m i l a r l y , i f the r e s u l t i s an  i n c r e a s e ' i n the h o l e density'-the c e n t r e i s c a l l e d acceptor.  caused  an  These c e n t r e s i n t r o d u c e l o c a l i s e d energy  i n t o the o t h e r w i s e f o r b i d d e n energy gap.  levels  E l e c t r o n s from  the v a l e n c e band can t h e n be e x c i t e d i n t o the vacant  acceptor  l e v e l s , l e a v i n g h o l e s h e h i n d , and e l e c t r o n s from the donor l e v e l s can be e x c i t e d t o the c o n d u c t i o n band.  These  energy  l e v e l s and t h e band s t r u c t u r e are shown s c h e m a t i c a l l y i n F i g u r e 1. densities N  I f b o t h donors and a c c e p t o r s are p r e s e n t w i t h D  and  r e s p e c t i v e l y , e l e c t r o n s from the donor  l e v e l s can drop t o the a c c e p t o r l e a v i n g an e f f e c t i v e donor  6  FIGURE I  ELECTRON ENERGY LEVELS IN A SEMICONDUCTING CRYSTAL  CONDUCTION  BAND  CONDUCTION LEVELS  DONOR  E  C  G  LEVELS  FORBIDDEN ENERGY GAP  ACCEPTOR LEVELS  A VALENCE  BAND  VALENCE LEVELS  d e n s i t y of | N -  or acceptor (1.2)  0  N |.  Equations  A  (1,1)  and  s t i l l govern the e x t r i n s i c case f o r simple bands  and  non-degenerate s t a t i s t i c s so t h a t  T h i s r e l a t i o n may a c t i o n lav/.  be derived-more g e n e r a l l y from the mass  C o n d i t i o n s of l o c a l charge n e u t r a l i t y l e a d t o  a second r e l a t i o n s h i p  n  -  p  K/  =  D  —  N  (i.5)  A  i f a l l donors and acceptors are a c t i v a t e d . conductor  may  t u r e s where  Hence a semi-  e x h i b i t e x t r i n s i c p r o p e r t i e s a t low -%j  «  tempera-  and i n t r i n s i c p r o p e r t i e s at  h i g h e r temperatures where the r e v e r s e I n e q u a l i t y i s t r u e . The work t o be  d e s c r i b e d In t h i s r e p o r t deals w i t h  some p r o p e r t i e s of germanium, a group f o u r element which i n pure s i n g l e c r y s t a l form Is an i n t r i n s i c a zero temperature energy gap value o b t a i n e d by f i t t i n g t u r e s between 100  (1.3)  semiconduetorwith  of 0 . 7 8 5 e l e c t r o n v o l t s , a e x p e r i m e n t a l l y f o r tempera--  and 300° K (Morin and M a l t a , 195U)o  The  room temperature i n t r i n s i c c a r r i e r c o n c e n t r a t i o n i s 13  2o5 x 10  o  per cm- . >  Since the c r y s t a l c o n t a i n s about  22 Ij.o5 x 10  atoms per cm-,  i t Is apparent t h a t v e r y  small  p r o p o r t i o n s of donor or a c c e p t o r c e n t r e s c o u l d a p p r e c i a b l y a l t e r the conduction p r o p e r t i e s of the c r y s t a l at t h i s temperature.  Group three and group f i v e elements such as  g a l l i u m and indium  or a r s e n i c and antimony can e n t e r the  germanium l a t t i c e s u b s t i t u t i o n a l ^ and a c t r e s p e c t i v e l y as  8  a c c e p t o r s and donors.  T h e i r l o c a l i z e d energy l e v e l s l i e i n  the energy gap about 0.01  the v a l e n c e and c o n d u t i o n bands r e s p e c t i v e l y , so t h e y a l l be i o n i z e d a t normal  of  e l e c t r o n v o l t s f r o m the edges  will  temperatures.  More d e t a i l e d t h e o r e t i c a l i n v e s t i g a t i o n s o f s p e c i f i c c r y s t a l s demand a b e t t e r r e p r e s e n t a t i o n o f the r e l a t i o n between the e l e c t r o n energy and wave number v e c t o r t h a n assumed f o r the s i m p l e s p h e r i c a l band.  was  One model w h i c h  has been used s u c c e s s f u l l y f o r the c o n d u c t i o n band i n germanium has  c o n s t a n t energy s u r f a c e s i n k-space which are  e l l i p s o i d s o f r e v o l u t i o n l y i n g a l o n g the [ l l l j w i t h minima at the zone b o u n d a r i e s .  The  v a l l e y s share the c o n d u c t i o n e l e c t r o n s .  directions  r e s u l t i n g four The  surfaces  of  c o n s t a n t energy are i m p o r t a n t because f o r many of the s c a t t e r i n g processes t c be d i s c u s s e d , the e l e c t r o n energy remains p r a c t i c a l l y unchanged so t h a t the wave number v e c t o r must t e r m i n a t e on the same energy s u r f a c e a f t e r the i n g p r o c e s s as b e f o r e . The  *  r e l a t i o n between E and k a l s o determines  e f f e c t i v e mass, .which may  The  scatter-  the  be r e p r e s e n t e d by the t e n s o r .  s p h e r i c a l s u r f a c e a p p r o x i m a t i o n l e a d s t o an  Isotropic  e f f e c t i v e mass v a l u e , but l e s s s i m p l e models i n g e n e r a l make I t a n i s o t r o p i c .  The  e l l i p s o i d a l model l e a d s t o two  e f f e c t i v e mass components, a l o n g i t u d i n a l one, corresponding  m-^,  to k p a r a l l e l to the a x i s o f r o t a t i o n of the  9  e l l i p s o i d and a t r a n s v e r s e this axis.  Cyclotron  one, m^., f o r k p e r p e n d i c u l a r  resonance measurements a t l|°K  ( D r e s s e l h a u s e t a l 1955*  Dexter e t a l 19>6)  have y i e l d e d  v a l u e s f o r m^ and m^ f o r c o n d u c t i o n e l e c t r o n s of l o 6 m  0  of a f r e e  and 0 . 0 8 2 m  o  to  r e s p e c t i v e l y , where m  Q  i n germanium  i s t h e r e s t mass  electron. Most o f t h e f o l l o w i n g t h e o r y i s based on t h e  spherical surface  a p p r o x i m a t i o n , b u t the e f f e c t s o f t h e more  c o r r e c t model a r e i n d i c a t e d where p o s s i b l e .  10  1.2  SCATTERING OF CARRIERS Germanium c r y s t a l l i z e s i n a diamond type  lattice  i n w h i c h e a c h atom l i e s a t the c e n t r e o f a t e t r a h e d r o n w i t h an atom a t each o f t h e f o u r v e r t i c e s .  This i s equivalent  t o two i n t e r - p e n e t r a t i n g f a c e - c e n t r e d c u b i c s u b l a t t i c e s , o r t o one f a c e - c e n t r e d c u b i c l a t t i c e w i t h e a c h l a t t i c e  site  b e i n g o c c u p i e d by a group o f two atoms, one a t (0,0,0) and the o t h e r a t (Ti>4si)o i s f?.62 Angstroms.  The l a t t i c e c o n s t a n t o f t h e l a t t e r The atoms do n o t r e m a i n f i x e d on t h e i r  l a t t i c e s i t e s b u t undergo t h e r m a l v i b r a t i o n s about t h e i r equilibrium positions.  These d i s p l a c e m e n t s  can be e x p r e s s e d  as t h e sum o f a s e t o f r u n n i n g e l a s t i c waves s u p p o r t e d by the c r y s t a l .  These normal modes range' i n w a v e l e n g t h f r o m  t w i c e the atomic dimensions.  s p a c i n g up t o the o r d e r o f t h e c r y s t a l  E x c i t a t i o n and d e - e x c i t a t i o n o f these modes  can be thought o f as e m i s s i o n o r a b s o r p t i o n o f phonons o f energy to'where Vis the f r e q u e n c y  o f t h e mode and h i s  Planck's  constant. Because t h e r e a r e two atoms p e r u n i t c e l l , two d i s t i n c t types o f v i b r a t i o n can a r i s e  ( B r i l l o u i n , 19i+6).  I n a c o u s t i c a l modes, b o t h atoms v i b r a t e almost t o g e t h e r , w i t h o n l y a s m a l l phase d i f f e r e n c e between them.  The f r e q u e n c y ~y  and w a v e l e n g t h /) a r e r e l a t e d b y = c,at l e a s t a t s m a l l ~V, where c I s t h e speed o f sound I n t h e c r y s t a l , about 5> x 10-^cm p e r second i n germanium. ^can  S i n c e J can be q u i t e l a r g e ,  be q u i t e s m a l l and the a c o u s t i c a l phonons can have l o w  11  energies. and  T h i s t y p e o f v i b r a t i o n i s f a m i l i a r as sound waves,  hence t h e name a c o u s t i c a l .  because i n p o l a r  I n o p t i c a l modes, so c a l l e d  c r y s t a l s t h e y can g i v e r i s e t o o p t i c a l  a b s o r p t i o n , t h e two atoms v i b r a t e a s m a l l phase d i f f e r e n c e  i n phase o p p o s i t i o n  between c e l l s .  with  T h i s produces i n  e f f e c t a h i g h f r e q u e n c y v i b r a t i o n o f one s u b - l a t t i c e w i t h r e s p e c t t o t h e o t h e r , modulated by a r u n n i n g wave.  Both  types o f v i b r a t i o n can be l o n g i t u d i n a l o r t r a n s v e r s e , i••"•>• ;  v  Because o f t h e h i g h and r e l a t i v e l y f i x e d f r e q u e n c y o f t h e o p t i c a l modes, about.,,8,3 x 10  p e r second i n germanium,  t h e r e i s a c h a r a c t e r i s t i c temperature a s s o c i a t e d w i t h them. T h i s i s t h e t e m p e r a t u r e f o r w h i c h kT e q u a l s of t h e o p t i c a l phonons.  hy  %  the energy  Various- t h e o r e t i c a l e s t i m a t e s o f  •this c h a r a c t e r i s t i c temperature have boen p u b l i s h e d i n v a l u e f r o m 275°K t o £60°K ( H a r r i s o n ,  ranging  1956), but recent  o b s e r v a t i o n s o f slow n e u t r o n s c a t t e r i n g - have y i e l d e d an e x p e r i m e n t a l v a l u e o f ! ; 0 0 K ( P e l a h e t a l , 1957) ,o  Such l a t t i c e v i b r a t i o n s the  lead t o a deformation o f  c r y s t a l l a t t i c e i ^ h i c h changes t h e l o c a l v a l u e of t h e  energy l e v e l s i n t h e band s t r u c t u r e . the  Such p e r t u r b a t i o n s o f  l o c a l p o t e n t i a l can l e a d i n t u r n t o t r a n s i t i o n s o f  c a r r i e r s between s t a t e s  i n the c r y s t a l .  Other f a c t o r s b e s i d e s these l a t t i c e v i b r a t i o n s c a n cause s c a t t e r i n g o f the c a r r i e r s by p e r t u r b i n g t h e l a t t i c e periodicity.  Some examples a r e l a t t i c e d e f e c t s ,  ions, impurities  dislocat-  and e s p e c i a l l y i o n i z e d i m p u r i t i e s .  For  12  each such s c a t t e r i n g p r o c e s s , an a s s o c i a t e d  r e l a x a t i o n t i m e J"*  may o f t e n be d e f i n e d such t h a t it's r e c i p r o c a l i s the p r o b a b i l i t y p e r u n i t time t h a t a c a r r i e r w i l l be by the p r o c e s s .  I f more than one independent  scattered  scattering  mechanism i s a c t i v e , t h e s e p r o b a b i l i t i e s are u s u a l l y to be a d d i t i v e so t h a t  assumed  the e f f e c t i v e r e l a x a t i o n time i s  obtained, by t a k i n g t h e r e c i p r o c a l o f the sum o f t h e r e c i p r o -  cals  o f the i n d i v i d u a l r e l a x a t i o n t i m e s .  f i e l d i s applied  I f an e l e c t r i c  t o a group o f m o b i l e charge c a r r i e r s i n a  s e m i c o n d u c t i n g c r y s t a l , they w i l l a c q u i r e a net d r i f t  velo-  c i t y l i m i t e d i n v a l u e by the r a n d o m i z i n g e f f e c t o f the s c a t t e r i n g mechanism.  Carrier m o b i l i t y i s  .  t  d e f i n e d by the  r e l a t i o n ^v-^- = -#E where i r i s c a r r i e r v e l o c i t y , ^ /> denotes a v e r a g i n g over the c a r r i e r v e l o c i t y d i s t r i b u t i o n and E i s I n general, M. i s a t e n s o r i n  the a p p l i e d e l e c t r i c f i e l d .  the p r e s e n c e o f a magnetic f i e l d H , b u t becomes a s c a l a r when H E 0. A general theory of c a r r i e r m o b i l i t y  applicable  when s c a t t e r i n g p r o c e s s e s a r e e l a s t i c t o a good a p p r o x i m a t i o n has been developed by C o n w e l l (1952).  3 tn*  \  "  tr  This gives  <*• ~"~  K  /  where C£ and th a r e the charge and e f f e c t i v e mass o f the carriers.  Spherical  energy -surfaces have been assumed.  When M a x w e l l - B o l t zman s t a t i s t i c s a r e a p p l i c a b l e , v e l o c i t y d i s t r i b u t i o n i s g i v e n by  the c a r r i e r  where N(v)  - dv i s the  density  between v and v+-dv and  N  Q  of c a r r i e r s w i t h  i s a normalizing  speeds  constant.  above g e n e r a l e x p r e s s i o n f o r UA. t h e n r e d u c e s t o the  The form  d e r i v e d by S h o c k l e y (1950),  fn*  <  H e r r i n g (1955) has  JkT  z^<>  N  (1.6)  7  shown t h a t t h i s f o r m a l s o h o l d s f o r  e l l i p s o i d a l energy s u r f a c e model w i t h m*  r e p l a c e d by  the  a  g e o m e t r i c a l average of the e f f e c t i v e mass components. S h o c k l e y and Bardeen (1950) have c a r r i e d out  a  t r e a t m e n t of c a r r i e r s c a t t e r i n g by a c o u s t i c a l modes of . l a t t i c e v i b r a t i o n b a s e d on a l i n e a r r e l a t i o n between l a t t i c e d e f o r m a t i o n and p e r t u r b a t i o n s p h e r i c a l model. J  of c a r r i e r p o t e n t i a l u s i n g  They found a r e l a x a t i o n time of the A  o c f T v ) '  the form (1.7)  1  giving a mobility —  °<  -  T  (1.8)  A s i m i l a r t r e a t m e n t i n d i c a t e d no modes, but H e r r i n g (1955) has  s c a t t e r i n g by o p t i c a l  considered c a r r i e r s c a t t e r i n g  u s i n g the e l l i p s o i d a l model w i t h low energy phonons c a u s i n g s c a t t e r i n g o f c a r r i e r s t o s t a t e s i n the same v a l l e y valley scattering)  (intra-  and h i g h energy phonons c a u s i n g s c a t t e r -  i n g to states i n other v a l l e y s  (inter-valley scattering).  Assuming a s i n g l e i n t e r - v a l l e y phonon of energy h " K i n  addition  t o the low energy I n t r a - v a l l e y ones he f o u n d a r e l a x a t i o n time  Ik  o f the form -i - »  (-b&n  (1.9)  where W i s a constant depending upon the r e l a t i v e ' ' importance o f I n t e r - and i n t r a - v a l l e y s c a t t e r i n g and the l a s t term i s zero f o r / c T ^ ^ i n c e i t i s due t o h i g h energy phonon e m i s s i o n .  U s i n g t h i s r e l a x a t i o n time, he computed  the m o b i l i t y as a f u n c t i o n of temperature f o r d i f f e r e n t values o f the mixing constant W and showed t h a t i f the m o b i l i t y were approximated as  (1.10) b y analogy w i t h ( 1 . 8 ) ,  then a would be s l i g h t l y  I n c r e a s i n g from 1.5  dependent,  temperature  at very low temperature t o  a maximum v a l u e a t a temperature below the c h a r a c t e r i s t i c temperature o f the h i g h energy phonon and then d e c r e a s i n g back towards  1.5  at h i g h e r temperature.  The amount o f  change i n a would depend on the value of W,  but a s m a l l -  p r o p o r t i o n o f h i g h energy phonon s c a t t e r i n g was  found  s u f f i c i e n t to e x p l a i n the e x p e r i m e n t a l a v a l u e s o f around 1.66  o b t a i n e d f o r e l e c t r o n s i n germanium between 100 and  300°K Evans,  (e.g. Debye and Conwell, '195U» M o r i n and M a l t a , 195U, 1957)» i  I n c r y s t a l s t h a t have been doped w i t h donor or a c c e p t o r i m p u r i t i e s , s c a t t e r i n g of c a r r i e r s can occur due to the l o c a l i z e d changes  i n p o t e n t i a l a r i s i n g from the  15  ionized impurities.  T h i s problem has  been t r e a t e d u s i n g  v a r i o u s s c a t t e r i n g p o t e n t i a l s (e.g. S c l a r , 1956).  Use  of  the Born approximation l e a d s to a r e l a x a t i o n time of the form  The f u n c t i o n g i s of n e a r l y the same form f o r the scattering potentials. ed to be  different  This function is usually consider-  s l o w l y v a r y i n g compared to the re3t of the  inte-  grand a r i s i n g i n the e v a l u a t i o n of the m o b i l i t y and usually evaluated of t h i s i n t e g r a n d .  Is  at the value of v which maximizes the  rest  T h i s g i v e s *an i o n i z e d i m p u r i t y m o b i l i t y  of the form -sOLx Sclar  A/  o<  (1956) and B l a t t  /  (1.12)  (1957) have a l s o c o n s i d e r e d  i m p u r i t y s c a t t e r i n g at low  temperature and h i g h  where the Born a p p r o x i m a t i o n i s not v a l i d . q u i t e d i f f e r e n t from (1.12)  ionized  impurity  Mobilities  can a r i s e under these  circum-  stances. S c a t t e r i n g by n e u t r a l i m p u r i t i e s may i n h e a v i l y compensated c r y s t a l s .  be  important  Carrier-carrier scatter-  i n g can a f f e c t the m o b i l i t y i f the Boltzman v e l o c i t y  distri-  b u t i o n i s d i s t u r b e d by n o n - e q u i l i b r i u m  Dis-  l o c a t i o n s c a t t e r i n g can be i s low  important  conditions.  i f the c r y s t a l q u a l i t y  and e l e c t r o n - h o l e s c a t t e r i n g can be important  minority c a r r i e r density i s high. important  i n the work to be  further described  here.  None of these  i f the are  d i s c u s s e d , so they w i l l not  be  16  1.3  HALL EFFECT AND  HALL MOBILITY  Charge c a r r i e r s moving i n a r e g i o n of raagnetifc f i e l d experience a f o r c e o r t h o g o n a l to b o t h t h e i r v e l o c i t y and the magnetic f i e l d .  T h i s can produce a g r a d i e n t  of  c a r r i e r d e n s i t y t o e s t a b l i s h an e l e c t r i c f i e l d to balance this force. H  z  I f an e l e c t r i c f i e l d E  x  and a magnetic  are a p p l i e d to a c r y s t a l , then an e l e c t r i c f i e l d Ey i s  developed by t h i s e f f e c t .  The  s u b s c r i p t s x, y and  to d i r e c t i o n s along the three c o - o r d i n a t e  axes.  constant R i s d e f i n e d by the r e l a t i o n R m Rj/tr E CT i s the c o n d u c t i v i t y of the c r y s t a l . may  be  M-n =  d e f i n e d as  JUL  H  =  R cr  3_  a n (  < * V  to* for  field  i  The  i t can be  z refer  The H  x  z  Hall where  Hall mobility shown t h a t  J'>.  < > * T >  the s p h e r i c a l s u r f a c e model and  (1.13)  a l s o f o r the  ellipsoidal  model i f m*- i s r e p l a c e d by a f u n c t i o n of the e f f e c t i v e mass components.  As l o n g as T  i s a f u n c t i o n of v,  (1.13)  y i e l d a v a l u e of m o b i l i t y d i f f e r e n t from ( 1 . 6 ) .  The  will ratio  of these m o b i l i t i e s can be expressed by a dimensionless H a l l factor  I -  ^cc  •  T h i s can be expressed as the  o f a s c a t t e r i n g f a c t o r and a band shape f a c t o r as  The  follows.  f a c t o r D depends upon the c a r r i e r e f f e c t i v e mass a n i s o -  tropy. and  product  D i s u n i t y f o r the s p h e r i c a l energy s u r f a c e model  depends only on the r a t i o m-j/m^ f o r the  model ( H e r r i n g , 1955).  The  ellipsoidal  c y c l o t r o n resonance mass  values  17  mentioned e a r l i e r f o r conduction e l e c t r o n s i n germanium g i v e a value of about 20 f o r t h i s r a t i o , which y i e l d s a D value of  0.78. The  lattice  s c a t t e r i n g f a c t o r i s 1,18  scattering,.  for acoustical  F o r mixed i n t r a - and  inter-valley  s c a t t e r i n g i t i s s l i g h t l y temperature dependent, i n c r e a s i n g from l o l 8  at low temperatures  to a peak below the  c r i t i c a l temperature and then d e c r e a s i n g a g a i n . (1953) r e p o r t s an experimental value of r  =  1,1  Prince for  l a t t i c e s c a t t e r i n g of e l e c t r o n s i n germanium at room temperature.  s c a t t e r i n g g i v e s a value of 1 , 9 3  Impurity  for  r i f the slowly v a r y i n g g f u n c t i o n s are c a n c e l l e d .  How-  ever, a lower value i s o b t a i n e d i f the d i f f e r e n c e i n p o s i t i o n of the maxima of the i n t e g r a n d s a r i s i n g i n the t i o n are not The  evalua-  Ignored, value of r when s c a t t e r i n g by both  lattice  a c o u s t i c a l v i b r a t i o n s and i o n i z e d i m p u r i t i e s i s p r e s e n t has been s t u d i e d by Johnson and L a r k - H o r o v l t z Mansfield  (1956b),  i n t e g r a l s while the other authors  lower  The  TP  d i d not.  a d i f f e r e n c e a t low N or h i g h T v a l u e s .  former treatment,  and  Both used the s p h e r i c a l model, but  M a n s f i e l d r e t a i n e d the g f u n c t i o n from  shown i n F i g u r e 2 .  (195D  inside  the  This leads to The r e s u l t s  are  upper curve r e s u l t s from the  or the l a t t e r f o r low H or h i g h T,  curve i s f o r the l a t t e r treatment  The  at h i g h N or,, low  T,  18  FIGURE  2  HALL FACTOR FOR MIXED LATTICE AND IONIZED IMPURITY SCATTERING  20,  li«6  r D  01 + Oi  19  Prince  (1953) a l s o r e p o r t s experimental v a l u e s of r at room  temperature  f o r e l e c t r o n s which decrease from 1.1  p u r i t y germanium to 0.95 qualitative  at N  =  10  per cmr»  agreement w i t h the i n i t i a l  curves In F i g u r e 2.  f o r high  This i s i n  decrease of the  T h i s a n a l y s i s w i l l be used l a t e r t o  estimate changes i n r , but not absolute v a l u e s .  20  2-MET.HOD  C H A T T E R  2.1  OP.  A N A L Y S I S  INTRODUCTION The  I n t r o d u c t i o n to o a r r i e r m o b i l i t y g i v e n i n  the p r e v i o u s chapter has  i n d i c a t e d t h a t the study of t h i s  m o b i l i t y and i t s temperature dependence I n a semiconducti n g m a t e r i a l can y i e l d i n f o r m a t i o n about the s c a t t e r i n g mechanisms i n v o l v e d .  I t was  have been made f o r l a t t i c e room temperature.  a l s o noted t h a t such s t u d i e s  s c a t t e r i n g i n germanium below  Most of these p r e v i o u s i n v e s t i g a t i o n s  have taken advantage of the f a c t t h a t f o r e x t r i n s i c germanium of reasonably h i g h p u r i t y there i s a l a r g e temperature range over which the i o n i z e d i m p u r i t y s c a t t e r i n g i s n e g l i g i b l e compare d to the l a t t i c e s c a t t e r i n g and the m i n o r i t y c a r r i e r concentration i s also n e g l i g i b l e .  The  majority  c a r r i e r c o n c e n t r a t i o n remains e s s e n t i a l l y constant t h i s temperature range. m o b i l i t y may  Hence the m a j o r i t y C a r r i e r  lattice  be measured d i r e c t l y , and i t s temperature  dependence determined Previous expression  over  AX  L  —  i n v e s t i g a t o r s have f i t t e d the m o b i l i t y  A  r e a s o n a b l y good f i t s t u r e s between 100  B  T  ^  to t h e i r data and have o b t a i n e d  f o r a constant val ue of a at tempera-  and 300° K.  However, t h i s method of  a n a l y s i s would not be too s e n s i t i v e to s m a l l changes w i t h temperature i n the value of a. a t i o n , the experimental  Indeed, on c l o s e examin-  data of P r i n c e (1953)  for electron  m o b i l i t y i n germanium would appear to i n d i c a t e some i n c r e a s e  21 i n a w i t h temperature around room t e m p e r a t u r e , a l t h o u g h the a u t h o r f i t t e d the d a t a w i t h a c o n s t a n t v a l u e o f a.  The  temperature dependences o f m o b i l i t y o b t a i n e d i n t h i s temperature range have been e x t r a p o l a t e d t o much h i g h e r ranges i n t h e c o u r s e of some a n a l y s e s (e.g.' M o r i n and M a l t a , 1951+).  C o n w e l l (1953) has p o i n t e d out t h a t such e x t r a p o l -  a t i o n s s h o u l d not be made as the temperature dependence p r o b a b l y changes a p p r e c i a b l y o u t s i d e t h i s range. I t was  t h e r e f o r e f e l t t h a t a study o f e l e c t r o n  l a t t i c e m o b i l i t y i n germanium e x t e n d i n g to h i g h e r temperature ranges s h o u l d be made.  The p r i m a r y aim of such a s t u d y would  be t o f i n d the temperature dependence of t h i s m o b i l i t y i n an attempt t o l e a r n more about the s c a t t e r i n g mechanisms i n v o l v e d . The r e m a i n i n g p o r t i o n s o f t h i s c h a p t e r p r e s e n t the t h e o r e t i c a l i n v e s t i g a t i o n s which were u n d e r t a k e n i n o r d e r to f i n d  an  e x p e r i m e n t a l method s u i t a b l e f o r such a s t u d y . The p r e v i o u s s t u d i e s of e l e c t r o n m o b i l i t y i n germanium mentioned above have not gone much beyond room temperature because the e f f e c t o f h o l e s on the measured m o b i l i t y c o u l d no l o n g e r be n e g l e c t e d .  The e f f e c t s of these  m i n o r i t y c a r r i e r s c o u l d have been s u p p r e s s e d u n t i l h i g h e r temperatures were a t t a i n e d by the a d d i t i o n of more donor impurities.  However, the a d d i t i o n o f t h e s e donors  have i n c r e a s e d the i m p u r i t y s c a t t e r i n g as w e l l .  would  This i n  t u r n would have caused an i n c r e a s e i n the temperature above w h i c h i m p u r i t y s c a t t e r i n g c o u l d be n e g l e c t e d .  Since- t h i s  22  lower temperature l i m i t Increases much more r a p i d l y than the upper one  f o r i n c r e a s i n g donor c o n c e n t r a t i o n , the a d d i t i o n  of s u f f i c i e n t  donors to r a i s e the upper temperature l i m i t  by  an a p p r e c i a b l e amount would have made I t i n c o r r e c t to n e g l e c t the e f f e c t of i m p u r i t y scattering,, Since  suppression  of the m i n o r i t y c a r r i e r s  was  c o n s i d e r e d e s s e n t i a l f o r the proposed measurements of e l e c t r o n m o b i l i t y , the e f f e c t s of the i m p u r i t y s c a t t e r i n g were c o n s i d e r e d .  As mentioned In the previous  t h e o r e t i c a l expressions been developed. and  chapter,  f o r i o n i z e d i m p u r i t y m o b i l i t y have  Methods f o r s e p a r a t i n g or combining  lattice  i m p u r i t y m o b i l i t i e s when both s c a t t e r i n g mechanisms are  a c t i v e have a l s o been p u b l i s h e d 1950, M a n s f i e l d , 1 9 5 6 a ) .  (Johnson and  Thus a t h e o r e t i c a l value of the  Impurity m o b i l i t y c o u l d be used to o b t a i n the m o b i l i t y from, the e x p e r i m e n t a l l y ever, I t was  felt  should be avoided,  Lark-Hprovitz,  t h a t the use  lattice  determined m o b i l i t y .  of c o r r e c t i o n s of t h i s  because any e r r o r s i n the  Howtype  theoretical  i m p u r i t y m o b i l i t y values x^rould be c a r r i e d over to the l a t t i c e m o b i l i t y under study, p a r t i c u l a r l y when the  two  m o b i l i t i e s were comparable In magnitude. The  f o l l o w i n g a n a l y s i s was  performed i n search  a b e t t e r method f o r s e p a r a t i n g the e f f e c t s of l a t t i c e impurity a study  scattering.  The  r e s u l t was  and  a method which r e q u i r e d  of the m o b i l i t y over a range of i m p u r i t y  i o n as w e l l as temperature.  of  concentrat-  T h i s method, however, had  the  23  decided advantage  of depending almost e n t i r e l y on the  f u n c t i o n a l form of the m o b i l i t i e s their values.  I n t h i s way  i n v o l v e d r a t h e r than on  the effects of assumed v a l u e s  f o r parameters such as e f f e c t i v e mass was  minimized.  0  2k  2.2  FORK OP RELAXATION TIME I n the  strongly  n-type germanium necessary f o r  t h i s i n v e s t i g a t i o n of e l e c t r o n m o b i l i t y , i o n i z e d donor atoms d i s t u r b s the  crystal lattice.  s c a t t e r i n g of the  the presence of  the p e r i o d i c p o t e n t i a l  These p e r t u r b a t i o n s can  electrons  moving w i t h i n  the  within  then l e a d  buted c o n d u c t i o n e l e c t r o n s  s c r e e n i n g due  and  c u t - o f f due  neighbouring ions can modify t h i s form. i n g p o t e n t i a l has time may  be  been determined, the  The  to the to the  c a l c u l a t i n g the r e q u i r e d  the  a i d of the Born (1926)  A f t e r the associated  approximation.  The  which the  theory i s v a l i d .  i n Figure  3.  This  relaxation (I936)  use  (1956)  has  ing potentials,  times were -given by  inhere E i s the constant of the  the  c a r r i e r energy, ?C c r y s t a l and  of  impurity  t h i s problem,  cut-off.  The  and  scatter-  resulting  expression  i s the  N i s the  the  indicated  summarized the r e s u l t s of v a r i o u s both screened and  x-jith  temperature f o r  limitation is  S e v e r a l authors have t r e a t e d  of  scatter-  scattering cross-sections  c o n c e n t r a t i o n and/or a lower l i m i t on the  relaxation  distrieffects  l a t t e r approximation imposes an upper l i m i t on the  Solar  the  c a l c u l a t e d by employing the method o f Mott  and  to  crystal.  b a s i c form of the p o t e n t i a l about such Ions i s g i v e n by simple Coulomb p o t e n t i a l , but  the  dielectric  density  of  ionized  26  scattering centres.  The f u n c t i o n g depends on the e x a c t  f o r m o f the s c a t t e r i n g p o t e n t i a l employed,, cases w i l l be c o n s i d e r e d l a t e r . the v a r i a b l e X  s  Some s p e c i f i c  For l a t e r convenience,  E/kT i s i n t r o d u c e d and the r e l a x a t i o n t i m e  i s r e - w r i t t e n as  As p o i n t e d out i n t h e i n t r o d u c t o r y  chapter,  s c a t t e r i n g o f c a r r i e r s by a c o u s t i c a l modes o f l a t t i c e v i b r a t i o n y i e l d s a r e l a x a t i o n time, o f t h e f o r m  The i n c l u s i o n o f s c a t t e r i n g by h i g h energy phonons was outlined, leading (1.9).  also  t o a much more complex r e l a x a t i o n time  However, H e r r i n g ' s n u m e r i c a l work (1955) showed  that only a small proportion is required  o f h i g h energy phonon s c a t t e r i n g  t o e x p l a i n the experimental r e s u l t s f o r e l e c t r o n  m o b i l i t y i n germanium.  T h i s , combined w i t h the temperature  dependence o f h i s c a l c u l a t e d m o b i l i t y v a l u e s , s u g g e s t s i t might be p e r m i s s i b l e of t h e f o r m  that  t o assume a l a t t i c e r e l a x a t i o n time „  /  a t l e a s t over s h o r t ranges o f t e m p e r a t u r e * These r e l a x a t i o n t i m e s ,  (2.1)  and  (2.2),  w i l l be  used i n t h e f o l l o w i n g a n a l y s i s , and i t w i l l be assumed t h a t no o t h e r s c a t t e r i n g mechanisms a r e i m p o r t a n t . Under t h e u s u a l a s s u m p t i o n t h a t the two s c a t t e r i n g  2?  mechanisms are i n d e p e n d e n t , the s e p a r a t e r e l a x a t i o n t i m e s . g i v e n above may  be combined by a d d i n g t h e i r r e c i p r o c a l s t o  g i v e an e f f e c t i v e r e l a x a t i o n time  & T h i s r e l a x a t i o n time may  J  (2.3)  t h e n be used i n the c a l c u l a t i o n o f  c a r r i e r m o b i l i t y , o r o t h e r t r a n s p o r t phenomena.  28  2.3  SEPARATION 0 ? LATTICE AND IMPURITY  EFFECTS  S i n c e a l l p r o p o s e d measurements were t o be made at low e l e c t r i c f i e l d s and under c o n d i t i o n s o f t h e r m a l e q u i l i b r i u m , t h e assumption o f a M a x w e l l i a n v e l o c i t y t r i b u t i o n of carriers certain  limitations  i n the c r y s t a l was made.  dis-  This placed  on t h e p e r m i s s a b l e v a l u e s o f N and T i n  o r d e r t o keep t h i s d i s t r i b u t i o n f u n c t i o n non-degenerate. Hence the m o b i l i t y as g i v e n by ( 1 . 6 ) i s  3 XT  3 t**s*  J  J  where C i s a c o n s t a n t and I i s t h e i n t e g r a l . S i n c e t h e m o b i l i t y c o u l d be measured over a range o f b o t h N and T v a l u e s , the f o l l o w i n g two o b s e r v a b l e q u a n t i t i e s t-jere d e f i n e d and d i s c u s s e d . cX  =  —  3 Zr\AL  P ^ T  —  <x  —  _L  I  <? I  3i*T  29  V a l u e s o f 0(  may range from about 1»5 f o r l a t t i c e m o b i l i t y  dominant t o about -lo5> f o r i m p u r i t y m o b i l i t y dominanto Similarly  ^ may range from 0 t o 1.  p r e s s i o n s f o r oC and 0<  Prom the above e x -  ^ i t follows that  CL -f-  —  (2.5) w h i c h i n d i c a t e s t h a t a p l o t ' o f of  against  a curve w i t h an e x ' - i n t e r c e p t o f a.  Y would y i e l d  S i n c e the a n a l y s i s  showed t h i s promise o f a u s e f u l r e s u l t , the d i f f e r e n t i a t i o n s i n d i c a t e d i n (2<>5) were performed. between  oC and  ^ t h e n became  a -  cx- =z  a  +c b  Jf^  by  A.  -g-  23. dx  dX (x" ^K) d  X  r *" N 3(X) -  C  b  i s g i v e n by the same e x p r e s s i o n w i t h  ^  (2O6)  +- Jk  C+dL  J  Y  - J", ,.  where  with K=  The r e l a t i o n  59  replaced  I f these J terras were independent o f N and T,  t h e n the oC~c curve would be a s t r a i g h t l i n e whose ^ V - i n t e r c e p t would g i v e the v a l u e o f a and whose s l o p e tarould g i v e some i n f o r m a t i o n about b and c.  I f 0(-  ^  l i n e s were p l o t t e d a t d i f f e r e n t t e m p e r a t u r e s , any temper-  30  a t u r e dependence o f a would be shown by a c h a n g i n g oC~intercept value. I n o r d e r t o e s t i m a t e how c l o s e l y such p l o t s s h o u l d approximate t o s t r a i g h t l i n e s , i t was n e c e s s a r y a t t h i s p o i n t t o consider a s p e c i f i c form of s c a t t e r i n g p o t e n t i a l t o f i x t h e f u n c t i o n g and p e r m i t a c a l c u l a t i o n o f the .J terms I n ( 2 . 6 ) . Brooks,  S e v e r a l authors  195>1* H e r r i n g , u n p u b l i s h e d ,  ( M o t t , 1936*  D i n g l e , 1955, Mans-  f i e l d , 1956a) have c o n s i d e r e d a s c r e e n e d Coulomb p o t e n t i a l o f the f o r m  ct  V,(r) where r  g  ,  -  e  =^-r  «  i s the screening r a d i u s .  The e x p o n e n t i a l s c r e e n -  i n g f a c t o r i s due t o t h e d i s t r i b u t i o n o f o p p o s i t e l y c h a r g e d c a r r i e r s about t h e s c a t t e r i n g i o n and r relation  ^  ^  / ^  T_Y  h  s  i s g i v e n by the  A  (^Tr^/v/  5  f o r t h e e x t r i n s i c case under c o n s i d e r a t i o n i n w h i c h t h e c a r r i e r d e n s i t y equals t h e e f f e c t i v e i m p u r i t y d e n s i t y N. T h i s l e a d s t o a r e l a x a t i o n time g i v e n by (2.1) x-iith t h e f u n c t i o n g g i v e n by  , ^  where _  i.f  x  \o ,lf  T/ty  V a l u e s o f K o 16 and m-"- * 0.20m were used f o r t h i s o  evaluation.  Hence  31  2  3'  =  2  2_2J  and  '<t>,X —_  2_  -f.'M  This s i m p l i f i e d (2.6) s l i g h t l y to  <Y - a. -  a  4 C  i>  - 2 —  TT7\ (2.7)  TTi  wnere  7T  -  ^00  (x^^TW I n o r d e r t o p e r m i t an e s t i m a t i o n o f the v a l u e o f TT over t h e ranges o f N and T.of i n t e r e s t , v a l u e s o f 3 / 2 X  and l / 2 were used r e s p e c t i v e l y f o r the i n d i c e s c and d  0  B o t h i n t e g r a n d s a p p e a r i n g i n TTi then c o n s i s t e d o f a f a c t o r  x e-* ( x ^ ^ f 3  w h i c h was f o u n d t o be s h a r p l y peaked about a v a l u e o f X = X  Q  somewhere between 0 and 3 depending upon the v a l u e o f K, and the f u n c t i o n f ^ ( X ) o r g^(X) .These, were f o u n d t o be slowl y v a r y i n g i n X compared t o t h e above f a c t o r except f o r v e r y small values of  (p,  •  These f u n c t i o n s f  and g  were  t h e r e f o r e c o n s i d e r e d constant, and e v a l u a t e d a t X s X , Q  g i v i n g the approximate r e s u l t  77;  3'pr.)  32  Values  of X  were o b t a i n e d f r o m  0  the c o n d i t i o n f o r m a x i m i z i n g g^ h e l d c o n s t a n t .  the I n t e g r a n d s w i t h f ^ and  The. s u b s c r i p t  h e l d constant i n i t too. and  lined  t h a t g^ i s  The v a l u e s o f K f o r d i f f e r e n t  T  i n t e g r a l h a s b e e n e s t i m a t e d by t h e m e t h o d o u t -  above.  results  Values  of  ML  were o b t a i n e d f r o m 193>U)  (Debye and C o n w e l l ,  n e c e s s a r y , w h i l e M. Herring formula  T  d e r i v e d from  was f i r s t  Q  The  was  found  (p  o  accuracy of this  Hence t h e e f f e c t  w h i c h meant range  The r a t i o  that  Values Q  in K  c  Q  i f n e c e s s a r y when  method  was  o f N a n d T x-rtiich l e d t o also l e d to large  of the assymetry  t h e I n t e g r a n d was  o f X compared t o the* v a l u e X  t h e more  potential  g^(X )/g^(3)  approximate  v a l u e s o f K and h e n c e o f X t  the Brooks-  0  i n c r e a s e d b y two f a c t o r s . small  from  t h e same s c a t t e r i n g  t a k e n as u n i t y b u t was r e v i s e d  a value of X  experimental  e x t r a p o l a t e d where  v a l u e s were c a l c u l a t e d  u s e d i n t h e above a n a l y s i s .  by  on K d e n o t e s  N v a l u e s were e s t i m a t e d b y n o t i n g t h a t  where t h i s  of  the e x p r e s s i o n  important Q  values  o f the i n t e g r a n d , over a  a t t h e p e a k , was  larger offset  s l o w l y v a r y i n g n a t u r e o f f-^ a n d g ^ f o r t h e  33  (p,  larger  values.  Also f o r small values  of (j?,  when  these f u n c t i o n s were l e s s s l o w l y v a r y i n g , b o t h i n t e g r a l s In  TT  were a f f e c t e d i n a s i m i l a r manner so t h a t the e r r o r  i n t r o d u c e d i n t o TJ~, was  by i g n o r i n g t h i s was  e v a l u a t e d by numerical  values of accuracy  reduced.  i n t e g r a t i o n s f o r the extreme  (j>, used i n the approximations as a cheek on o f the approximate method.  The  T7T of from one  was  the  results indicated  t h a t the assumption of s l o w l y v a r y i n g f u n c t i o n s errors i n  77T  to s i x per c e n t .  introduced  This  accuracy  c o n s i d e r e d more than s u f f i c i e n t f o r the estimates  being  made at t h i s stage of the a n a l y s i s . For N s 10"^  per cm-*  and T s 300° K, values  s e n t a t i v e of the ranges l a t e r used i n the measurements, a value of this region  777 s 0.25> was  TTJ' i n c r e a s e d by about $0$  f a c t o r of 10  repre-  experimental  determined.  In  f o r an i n c r e a s e by  i n i m p u r i t y c o n c e n t r a t i o n while  it  decreased  by about 20% as a r e s u l t of d o u b l i n g the temperature. e f f e c t of such v a l u e s of jy of ( 2 . 7 )  and t h e i r changes on the  t  The curve  i s d i s c u s s e d below. 2.7  may cv  he w r i t t e n i n the form =  o_ - rn, cK  where  i -  7  ^  "  a  K  the approximation f o l l o w s from the assumptions t h a t  (2.8)  3k  a  s  c a 3/2,  assumption  b  =  1 and  V, «  I.  A l t h o u g h the  last  i s not a good one i n t h i s i n s t a n c e , i t l e a d s on 777 »  to a reasonable estimate of the dependence of m^ The r e s u l t i n g changes i n m^ a l i n e w i t h a slope  a l t e r the  oc - / r e l a t i o n  t o a curve whose tangent has  from a  slope g i v e n by  - j? - -fa Assuming the  cv - J curve to be l i n e a r i s e q u i /  <  v a l e n t t o n e g l e c t i n g the second term i n t h i s expression© The q u a n t i t a t i v e e f f e c t s of such an assumption  are d i s c u s s e d  i n chapter I4., where the experimental v a l u e s necessary f o r t h e i r e v a l u a t i o n are a v a i l a b l e and are found t o be q u i t e small• I f the c o n c e n t r a t i o n of i o n s i s h i g h enough* the d i s t r i b u t i o n of s c r e e n i n g c a r r i e r s assumed i n the  develop-  ment of the s c a t t e r i n g p o t e n t i a l used above w i l l be by n e i g h b o u r i n g i o n s , and the e f f e c t i v e range of the t e r i n g p o t e n t i a l w i l l a l s o be l i m i t e d by these  affected scat-  neighbours.  The use of a c u t - o f f r a t h e r than a screened Coulomb potent i a l might be b e t t e r i n such c a s e s .  A scattering potential  of the form  VIM = _1_  r4r  Xr  -  o  r  t  yr  c  35  l e a d s t o t h e Coratfell-Weis3kopf (1950) by  (2»1).  given  The f u n c t i o n g i s g i v e n by  y (x) =  +  t  where  d).  Conwell  r e l a x a t i o n time  X  =  and W e i s s k o p f (1950)  k T  & X  )  2  n  chose t h e c u t - o f f r a d i u s r  c  as  one h a l f t h e average i o n i c s e p a r a t i o n , but S c l a r (1955) suggested  that while r  c  could-be assumed p r o p o r t i o n a l t o t h i s  average s e p a r a t i o n , i t s v a l u e s h o u l d be f i x e d by c o n s i d e r i n g the v a l u e o f t h e c u t - o f f d i s t a n c e a t the* c o n c e n t r a t i o n where the a c t i v a t i o n energy o f the donors V a n i s h e d the Bohr r a d i u s o f bound e l e c t r o n s .  t o be e q u a l t o  F o r germanium, t h i s  gave a c u t - o f f r a d i u s o f about one q u a r t e r the average I n t e r i o n i c spacing.  Using t h i s value of r &  =  2-  c  r y N  IO 1  l  /  3  F o r t h i s case, the e x p r e s s i o n ( 2 . 6 ) becomes  cxr =  a  -  a  f  -  c  2 TT1 (2.9)  where 771  gi  w i t h g-, and f - • • 1  v  e  n  D  J the same r a t i o o f i n t e g r a l s as Try  r e p l a c e d by g  f  2  =  2  and f  2  where  -4>,XL-  i 1- <p r t  A p p l i c a t i o n of the approximations  used above .  y i e l d e d a v a l u e o f Jf^_ -,.lj.O w h i c h v a r i e d by o n l y about 5$ ,„when it'.was e v a l u a t e d  over the ranges o f IT and T m e n t i o n e d  36  above.  Also  _ .  the  slope of  i f C  - 2 TTj  b  *~  m  (2.7) ^  _  +-C  b  2  (2.8)  was  applied.  of t h i s c u t - o f f p o t e n t i a l p r e d i c t s not  changes i n  only  Thus  the  smaller  Hz. o v e r the proposed range of measurement, but o C - J ^ c u r v e s l o p e on these changes.  a l s o l e s s dependence o f the The  CL  - 6 " I  when the method used t o o b t a i n use  reduced to  cut-off radius  i s l e s s than the  screening radius  i n d i c a t e s t h a t the former imposes the more s e r i o u s the s c a t t e r i n g p o t e n t i a l and hence the  limit  a n a l y s i s based on  c u t - o f f Coulomb s c a t t e r i n g p o t e n t i a l s h o u l d be However, the two  on the  favoured.  r a d i i are of the same o r d e r of magnitude,,  so b o t h l i m i t i n g f a c t o r s p r o b a b l y a f f e c t the a c t u a l p o t e n t i a l to some degree.  two  scattering  Knowledge of the e x a c t form of  the s c a t t e r i n g p o t e n t i a l i s not by c o n s i d e r i n g  which  essential.  s i m p l e but r a t h e r  T h i s was  shown  a r b i t r a r y screened-  and  s q u a r e - w e l l p o t e n t i a l s i n t r o d u c e d by S c l a r (195>6>)  V (r)  =  3  - _ L ^  i ( - J -  r «  Vi(r) -V,(r)  =  O  w h i c h l e a d t o s i m i l a r e x p r e s s i o n s f o r the  - hi  3  -  o<- ^  g  b  {  6  )  K r>Tc  curve  slope,  37  2.Lj.  DETERMINATION OF c V AND The  analysis  q u a n t i t i e s oC a n d / were i n t r o d u c e d  and defined  (Subscripts  '.  M~h »  i n terms o f the e l e c t r o n m o b i l i t y  n and p are used i n the f o l l o w i n g  differentiate  i n t o the  electron  sections  and h o l e m o b i l i t i e s . )  to  However, t h e  p r o p o s e d e x p e r i m e n t a l measurements were t o y i e l d v a l u e s o f conductivityo~ analysis ships  and H a l l  c o e f f i c i e n t R.  following  was p e r f o r m e d t o d e v e l o p t h e n e c e s s a r y r e l a t i o n -  among t h e s e For  quantities.  strongly  n - t y p e germanium w i t h a d o n o r  o f N and a n e g l i g i b l e a c c e p t o r local  The  charge  density,  n e u t r a l i t y imposes  density  the c o n d i t i o n o f  the c o n d i t i o n  that  n - p * N - n . d  n  and p are the e l e c t r o n and h o l e  density  of electrons  density  of un-ionized  donors.  A/  t, =  where t h e s e c o n d t e r m  Since  (  i s the  this analysis  n o t a t low t e m p e r a t u r e s ,  t o rev/rite t h i s e x p r e s s i o n  :  a n d n ^ i s -the  bound i n donor l e v e l s , t h a t  with e x t r i n s i c material ful  densities  deals  i t i s use-  In the form  ' ~  (t - v»)  i n each bracket  i s 3mall  compared t o  unity.. The by  electrical  the e x p r e s s i o n  ^  conductivity _  ^  {h  M.  h  o f the c r y s t a l i s g i v e n p-^p)  = I * uh ( ' * IT h)  38  where  t> -  -^Tp ^  2  a t room t e m p e r a t u r e .  Use o f the  above e x p r e s s i o n f o r n l e a d s t o  0  from w h i c h  N i f n >/> p and N ^ n^ a r e assumed. din  Hence  T  dJUT  ^JUT  (2.10)  The H a l l c o e f f i c i e n t o f the c r y s t a l i s g i v e n by  R-  where r  =  •  _ _ L P M^Mp-h M M Htn  and r  P  - r  n  has been assumed.  This I s not  t r u e i n ' g e n e r a l , b u t they a r e o f t h e same o r d e r and the f a c t t h a t t h e h o l e d e n s i t y i s s m a l l i n the m a t e r i a l under" c o n s i d e r a t i o n w i l l reduce any e r r o r so I n t r o d u c e d n e g l i g i b l e amount.  Hence  to a  n  39  under the previous assumption t h a t n >> p and N ^> n^, and  T h i s r e l a t i o n s h i p i s used below i n d e v e l o p i n g an e x p r e s s i o n f or  y.  Y ~  -  ?J^N  3 JU  ^JLI^J  PLM  C7ZnA/  (2.11)  The f i r s t terms of the r i g h t hand s i d e s of (2.10) and (2.11) can be e v a l u a t e d from e x p e r i m e n t a l l y determined c o n d u c t i v i t y and H a l l c o e f f i c i e n t v a l u e s .  The remaining  terms were estimated as o u t l i n e d below and found t o be n e g l i g i b l e w i t h i n p r e s c r i b e d ranges of donor d e n s i t y and temperature. S i n c e the product f o r germanium, then  r? p  =  tt;  <V  T  €  . T  ^  T h i s approximation i s s u f f i c i e n t l y accurate f o r the e s t i mations which f o l l o w . s i d e of (2.10) i s  The second terra o f the r i g h t  hand  I|0  /v/ 'e At h i g h e r temperatures  ['  ~f  —r~ \  j  ttfhere the m i n o r i t y c a r r i e r  t i o n , and hence t h i s term, becomes more important,  concentrathe  m o b i l i t y Is due mainly t o the l a t t i c e s c a t t e r i n g so t h a t v a l u e s o f the order of 1 5> 0  are expected*  Thus the n e g l e c t  of t h i s terra i * i l l i n t r o d u c e an e r r o r of about 2% f o r p/n O.OOlo  The  »  e x p o n e n t i a l f a c t o r i n n^ w i l l reduce t h i s s m a l l  e r r o r v e r y r a p i d l y f o r lower  temperatures.  F i g u r e 3 shows  the l i m i t s imposed upon N and T by t h i s c o n s i d e r a t i o n . The  term n ^ / N  may  be e v a l u a t e d by c o n s i d e r i n g the  occupancy of the donor energy l e v e l s .  I f non-degenerate  s t a t i s t i c s are assumed as b e f o r e , t h i s givesc.the value  hL = ze x p  N  r  -(LLZIJL  KT  and the l a s t term In (2.10) becomes  z e.x p The  e x p o n e n t i a l f a c t o r keeps t h i 3 terra v e r y s m a l l  perhaps a t low T and h i g h N v a l u e s when the Fermi r i s e s towards the donor l e v e l .  except level  C a l c u l a t e d Fermi l e v e l s f o r  the h i g h e s t N and lowest T values l a t e r Used i n d i c a t e d t h a t the e r r o r a r i s i n g from n e g l e c t of t h i s term would be and t h a t t h i s s m a l l e r r o r would be q u i c k l y reduced  small  with  N)  ia  i n c r e a s i n g temperature. S i m i l a r l y the l a s t terra i n ( 2 . 1 1 ) i s  w h i c h was a l s o f o u n d t o be n e g l i g i b l e .  The second term  from t h e end I n ( 2 . 1 1 ) i s  a  o  since b  '. 1  i *  7  1  *•  u  w i l l n o t be s t r o n g l y ^-dependent  *  "31/  i n the region  where ( n ^ / H ) becomes a p p r e c i a b l e and l a t t i c e s c a t t e r i n g i s 2  dominant.  The l a t t e r statement i s i n agreement w i t h t h e  d a t a o f P r i n c e (1953) g i v i n g b temperature.  1  as a f u n c t i o n o f N a t room  Hence t h i s term, i s a l s o n e g l i g i b l e under t h e  l i m i t a t i o n on the v a l u e o f p/n imposed  above.  The f a c t o r o u t s i d e t h e square b r a c k e t s i n ( 2 . 1 1 )  w h i c h l i e s between z e r o f o r a l l i m p u r i t y s c a t t e r i n g and unity for a l l l a t t i c e scattering.  Hence t h e l a s t two' terms  f r o m t h e square b r a c k e t s a r e comparable w i t h t h e l a s t two terras o f t h e e q u a t i o n , b u t o f o p p o s i t e a l g e b r a i c s i g n , so t h a t the. e f f e c t o f .these s m a l l terms i s f u r t h e r r e d u c e d . T h i s l e a v e s o n l y t h e term a r i s i n g from changes i n r w i t h Ho  A n e s t i m a t e o f t h e magnitude  o f t h i s term was  made by c o n s i d e r i n g t h e b e h a v i o u r o f r under c o n d i t i o n s o f mixed impurity-.and l a t t i c e s c a t t e r i n g shown i n F i g u r e 2 .  VALID RANGES OF N and T  ro  i+3  F o r v a l u e s o f N a r o u n d 10 t o 5 0 0 ° K, this  p e r crn^ and T i n t h e r a n g e  t h e v a l u e o f r l a y i n t h e b r o a d minimum r e g i o n  c u r v e and t h e t e r m  _x  -i—-"  was  zero.  •values o f N i n c r e a s e d t h e c o n t r i b u t i o n o f t h i s still  200  remained  s m a l l compared t o (2.10)  Thus  and  (2.11)  ^  of  Other  term, but i t  .  reduce t o the  approximate  expressxon  9  cr JU T  d  U\R\  Xnr\  Q  The  r a n g e s o f N and T o v e r w h i c h t h e s e r e s u l t s  shown i n F i g u r e lines  shown.  3 •  The  are v a l i d  P e r m i s s i b l e v a l u e s l i e between the  upper l i n e  i s imposed  by  the l i m i t  are two  on t h e  t ratio  of hole to e l e c t r o n c o n c e n t r a t i o n .  the l i m i t  Imposed by  assuming  s  o f non-degener*acy  were l e s s  severe.  These  C  a  The  density  to y i e l d  some u s e f u l An  liniitations  imposed by  screening  the  radius  indicated that f o r  the p e r m i s s i b l e range  i s not l a r g e , but i s p r o b a b l y s u f f i c i e n t results.  analysis  o f the c o n d u c t i v i t y  t r a n s i t i o n from the e x t r i n s i c r a n g e was  limits  or the l a r g e r  m e a s u r e m e n t s above room t e m p e r a t u r e impurity  lower l i n e i s  o f the B o r n approximation"  the c u t - o f f r a d i u s r  conditions r  t h e use  The  c a r r i e d out.  I t was  curve near  to the i n t r i n s i c thought  that  tween N and nj_ a t t h e minimum p o i n t m i g h t  the  temperature  a relation  permit a  be-  deter-  of  m i n a t i o n o f N independent o f H a l l measurements t o be made. However, i t was f o u n d t h a t the temperature dependences o f ^^-and M-p were n e c e s s a r y f a c t o r s so the method was i n a p p l i c a b l e  i n the r e s u l t i n g  i n t h i s worko  relation  US  2.5  SUnMARY The  that  a n a l y s i s o f the past  a study o f c o n d u c t i v i t y  range of temperature extrinsic functional  and H a l l c o e f f i c i e n t  and i m p u r i t y  concentration  n - t y p e germanium c a n g i v e  shown  over a  i n strongly  information„about t h e  f o r m s o f t h e m o b i l i t i e s a n d s c a t t e r i n g mechanisms  involved. higher  few s e c t i o n s h a s  Since  temperature  this  information  r a n g e s , i t was  was  not a v a i l a b l e i n the  decided  t o p r o c e e d v j i t h an  i n v e s t i g a t i o n b a s e d o n t h e s e methods o f a n a l y s i s . experimental  work a n d i t s a n a l y s i s a r e d e s c r i b e d  remainder o f t h i s  thesis.  The. in'the  C H A P T E R 3.1  3 - E X P E R I M E N T A L  P R O C E D U R E  PREPARATION FOR EXPERIMENTAL MEASUREMENTS To f a c i l i t a t e  the r e q u i r e d measurements o f  c o n d u c t i v i t y and H a l l c o e f f i c i e n t , the f o l l o w i n g apparatus and experimental techniques were developed,, 3.1(a)  PREPARATION OF MATERIAL S i n c e a wide range o f c r y s t a l s would be needed f o r  t h i s and o t h e r experiments planned I n t h i s l a b o r a t o r y , i t was  thought a d v i s a b l e t o design and c o n s t r u c t a c r y s t a l  grower.  The C z o c h r a l s k i  (1918) method was used i n which an  o r i e n t e d seed c r y s t a l i s dipped i n t o a p o o l of molten germaniumv'maintained  a t a temperature  the  The seed i s then r o t a t e d about I t s a x i s  melting point.  a few degrees above  to ensure r a d i a l homogeneity, and s l o w l y withdrawn from the melt.  I f the withdrawal r a t e and melt temperature a r e c a r e -  f u l l y c o n t r o l l e d , a uniform s i n g l e c r y s t a l w i l l grow as an e x t e n s i o n o f the seed c r y s t a l .  I n the f i n i s h e d instrument,  shown i n F i g u r e l|. and d e s c r i b e d more f u l l y i n Appendix A, the  melt i s h e l d i n a g r a p h i t e c r u c i b l e , h e a t e d by a g r a p h i t e  r e s i s t a n c e h e a t e r and s t i r r e d by i n d u c t i o n t o m a i n t a i n homogeneity and a c l e a n s u r f a c e . an atmosphere o f h i g h p u r i t y argon.  The c r y s t a l i s grown i n Several large high  q u a l i t y c r y s t a l s o f germanium w i t h room temperature  resis'i^  t i v i t i e s o f up t o f?0 ohm-cms were grown from zone r e f i n e d germanium t o t e s t the o p e r a t i o n o f the i n s t r u m e n t . proven q u i t e  satisfactory.  I t has  FIGURE  CRYSTAL  k-  GROWER  1*9  Two  were  antimony tion due  n - t y p e germanium c r y s t a l s  gradient to  grown w i t h along  the h i g h  of o v e r l a p p i n g ,  crystals  be  stirring  concentration  their  rates  these  from a b o u t 10  Several perpendicular was  cutting.  This  tungsten The  saw  wire  and  having  checking  sensitive 1959)  the  surfaces  be  > and  exploring with  f o r a Tauc  no  type by  of  crystals A  fine  c a r b o r u n d u m powder.  and  polished  of  evidence  probe  slices  later  light  output  i n this  to  conduct-  A l l of the  H o m o g e n e i t y was  voltage  in  concentration  the v a l u e  with'an intense  (195>>0  wire  a moving  impurity  0.l5ram»  uniform.  a  crystal.  a commercial f o u r p o i n t  of  filaments  Again,  impurity  d e s c r i b e d more f u l l y  method r e c e n t l y d e v e l o p e d o  seed  crystal  of f i n e  amount  d e t e r m i n e d by  for this  the  the  These  3  s l i c e s were l a p p e d  spacing  quite  c h e c k e d by - s c a n n i n g  (Cox,  cut  c h e c k e d by  a probe  cm- .  as  the  p a r a l l e l faces, their  their  were f o u n d t o  while  constructed  After  h o m o g e n e i t y was over  of  a range o f  axes f o r maximum h o m o g e n e i t y .  instrument  and  axis  by v a r y i n g  s l i c e s were c u t f r o m t h e s e  carrying a slurry  .  ensure f l a t  unit  thin  i s shown i n F i g u r e 5  Appendix B  ivity  (110)  X-ray study  to t h e i r  designed  a  per  x o  in  With a small  covered  to 1 0  gradient,  of antimony  1957).  with  concentra-  This  t o some e x t e n t  crystals  were grown a l o n g  impurity  symmetry.  (Pfann,  two  high  coefficient  controlled  Laue b a c k r e f l e c t i o n  saw  axes o f  segregation  germanium, c o u l d g r o w t h and  a purposely  h e a v i l y doped  re-  spot  signal,  a  department  of inhoroogeneity  was  found.  50  51  3.1(b)  PREPARATION OF SAWxLKS. The  samples used f o r H a l l  coefficient  and c o n d u c t -  i v i t y measurement were f i l a m e n t s o f r e c t a n g u l a r • c r o s s s e c t i o n . Sample  samples u s e d slices in  with  i n p r e l i m i n a r y s t u d i e s were the wire  the experimental  and were p r e p a r e d  saw.  However,  from t h i n  slices  and e l e c t r i c a l  were made u s i n g In used i n t h i s 'was h e a t e d  These  wire  was- t h e n  Gold  observed  to the s i d e  ri-type  purity  into  of a l l o y  with  was  desired contact area.  of  arms  crystals  nitrogen  containing  0.6$  and c a r r i e d  i n glass  t >e moment o f u s e . v  o f commercial  contact with  micro-  o f f o u r t o one.  the  heated  p o s i t i o n i n g o f the c o n t a c t  through a b i n o c u l a r microscope.  a small pool  a process  a movement r e d u c t i o n r a t i o  germanium f o r m an a l l o y  the  wire  to p r o t e c t I t u n t i l  germanium s u r f a c e , t h e e x a c t  contacts  a c l e a n f i l a m e n t o f germanium  electrolytically  brought  used  technique.  were m o u n t e d i n a p a i r  having  samples  probes f o r c o n d u c t i v i t y  atmosphere o f h i g h  coil.  cleaned  dispensers  manipulators The  investigation,  designed  the c r y s t a l  carrying contacts to  t h i s method, as a p p l i e d , t o t h e  a s m a l l nichrome  dispensers  the f i n a l  connections  the g o l d a l l o y  i n an i n e r t  a n t i m o n y was  cut from  by u s i n g  Current  t h e ends o f t h e s a m p l e , p o t e n t i a l measurements,  Some  measurements h a d side-arm. H a l l  masking and sand b l a s t i n g .  by  x 10 M j  d i m e n s i o n s were a b o u t 0.3 ..x 0.7  Since  an e u t e c t i c p o i n t  gOld  p o o l was  and  a r o u n d 360°C  f o r m e d w h i c h c o u l d be worked When t h i s  being  slowly  Into cooled,  52  some o f t h e excess germanium "would r e - c r y s t a l i z e b e f o r e t h e r e m a i n i n g a l l o y f r o z e , u s i n g the u n d e r l y i n g s u b s t r a t e as a seed.  . T h i s regrox^th l a y e r was v e r y h e a v i l y doped w i t h  antimony f r o m t h e g o l d "wire, and hence the r e s u l t i n g  contact  was a l i g h t - h e a v y j u n c t i o n w i t h an a t t a c h e d m e t a l l e a d . W i t h two m i c r o m a n i p u l a t o r s ,  i t was p o s s i b l e t o make c l o s e l y  spaced c o n t a c t s s i m u l t a n e o u s l y . p r o t e c t e d from t h e r m a l  ;  Other c o n t a c t s were  damage by the c o n t i n u o u s  flow of the  n i t r o g e n atmosphere w h i c h kept the h e a t i n g l o c a l i z e d . c o n t a c t s were made u s i n g .005 i n c h g o l d w i r e .  Most  The apparatus  used i n making these c o n t a c t s i s shown i n F i g u r e 6 . F i l a m e n t temperature was measured w i t h  chromel-  a l u m e l thermocouples made by d i s c h a r g i n g a l a r g e c a p a c i t o r through  t h e c o n t a c t j u n c t i o n between a p a i r o f .002 i n c h  chromel and a l u m e l w i r e s . " The r e s u l t i n g s m a l l w e l d was trimmed, c l e a n e d and'gold p l a t e d , w h i c h enabled i t t o be wet by tne a l l o y p o o l o f a c o n t a c t d u r i n g p r e p a r a t i o n .  This  gave the thermocouple e x c e l l e n t thermal and m e c h a n i c a l  contact  w i t h the f i l a m e n t , and had no a p p a r e n t e f f e c t on the e l e c t r i c a l p r o p e r t i e s o f the j u n c t i o n . The were s u p p o r t e d mechanical  samples were mounted on ceramic  holders.  They  o n l y by the g o l d x-jire l e a d s t o m i n i m i z e  s t r a i n on t h e sample and c o n t a c t s .  These h o l d e r s  were then f i t t e d i n t o s m a l l t e s t tubes f o r sample p r o t e c t i o n and  positioning. A f t e r t h e technique  o f making m e c h a n i c a l l y  repre-  53  d u c i b l e g o l d a l l o y c o n t a c t s had been m a s t e r e d , s e v e r a l experiments were p e r f o r m e d t o t e s t t h e i r e l e c t r i c a l and mechanical p r o p e r t i e s .  These t e s t s and t h e i r r e s u l t s a r e  described b r i e f l y below  0  Some f i l a m e n t a r y samples were made w i t h a l l o y cont a c t s a t each end and an a l l o y p o t e n t i a l probe near the centre. was  The r e s i s t a n c e between the c e n t r a l and end l e a d s  measured f o r a wide range o f c u r r e n t i n b o t h d i r e c t i o n s .  Ho dependence on c u r r e n t d i r e c t i o n was n o t e d , so t h e c o n t a c t s were judged t o be n o n - r e c t i f y i n g .  Other f i l a m e n t s  were p r e p a r e d w i t h a l l o y end c o n t a c t s and the o v e r a l l r e s i s t a n c e between the g o l d l e a d s was measured as a f u n c t i o n of c u r r e n t d e n s i t y .  T h i s r e s i s t a n c e was found t o v a r y by  l e s s than 0.5/= when the c u r r e n t d e n s i t y was changed by a f a c t o r o f over 5 0 . ohmic b e h a v i o u r .  T h i s was judged t o be s a t i s f a c t o r i l y However, l a t e r measurements were i n no  case dependent upon the ohmic nature  o f t h e c o n t a c t s as  p o t e n t i a l measuring c o n t a c t s never s e r v e d as c u r r e n t c a r r y i n g contacts. An i n v e s t i g a t i o n • o f the p e n e t r a t i o n o f a l l o y c o n t a c t s i n t o the germanium c r y s t a l was a l s o made.  Some  s e c t i o n s c u t t h r o u g h a l l o y c o n t a c t s were p r e p a r e d and m i c r o s c o p i c a l l y s t u d i e d a f t e r v a r i o u s degrees o f p o l i s h i n g and e t c h i n g , but no a l l o y p e n e t r a t i o n c o u l d be observed. second t e s t c o n s i s t e d o f scanning  a f i l a m e n t I n the r e g i o n  of an a l l o y s i d e c o n t a c t w i t h the Tauc d e v i c e mentioned  A  earlier, spot  A l a r g e p o t e n t i a l o u t p u t was  p a s s e d the r e g i o n  when the c o n t a c t the  surface.  was  This  o b s e r v e d as t h e  of the c o n t a c t , b u t  removed by l i g h t l y  was  p e n e t r a t i o n by t h e h e a v i l y doped c o n t a c t was  not  bulk  affect  3.1(c)  the u n d e r l y i n g  a true  disappeared  g r i n d i n g or  i n d i c a t e d that there  appealed t h a t the c o n t a c t  this  no  etching  appreciable  region.  surface  Thus i t  contact  and d i d  material.  TEMPERATURE CONTROL To p r o v i d e  tube f u r n a c e pieces.  was  temperatures  constructed  A second furnace,  above a m b i e n t ,  a  siuall  t o f i t b e t w e e n t h e magnet  better i n s u l a t e d to avoid  constructed  f o r measurements n o t  requiring  field.  oven  the magnetic  The  and c o n t r o l l e d e l e c t r o n i c a l l y ,  themselves  using  sensor.  constant  to a small f r a c t i o n  and p e r f o r m a n c e  i n c l u d e d as A p p e n d i x For  low  the oven  of a degree.  were  windings  Temperatures  t o o v e r 750° & c o u l d be p r o d u c e d  ambient  design  as the t e m p e r a t u r e  temperatures  pole  small  t h e r m a l g r a d i e n t s , was  set  light  from  and m a i n t a i n e d The  circuit  t e s t s of the e l e c t r o n i c  c o n t r o l are  C„  temperature  c o n d u c t i v i t y measurements, the  mounted sample c o u l d be  immersed i n a dewar o f p e t r o l e u m  ether.  o f t h e e t h e r was  The  by p a s s i n g  temperature  a small quantity of l i q u i d nitrogen  of copper t u b i n g was  lowered i n steps  a l s o immersed i n t h e  c i r c u l a t e d by a s m a l l  dewar.  s t i r r i n g motor.  through a The  With  coolant  this  coil  55  a p p a r a t u s j temperatures from ambient  down t o below 1 7 5 °  c o u l d be a c h i e v e d e a s i l y , and no change i n sample  K  temper-  a t u r e c o u l d be n o t e d over p e r i o d s of time much l o n g e r than t h a t r e q u i r e d t o complete the measurements a t a g i v e n temperature o  A s m a l l e r d i a m e t e r dewar was made t o f i t i n the  magnet gap t o p e r m i t H a l l measurements below room  temper-  ature,, As mentioned above, sample temperature  was  measured w i t h c h r o m e l - a l u m e l thermocouples imbedded i n the sample c o n t a c t s .  T h e i r c a l i b r a t i o n was  n i t r o g e n and b o i l i n g water f o r each  3.1(d)  checked i n b o i l i n g  sample.  MAGNETIC FIELD To p r o v i d e the magnetic f i e l d n e c e s s a r y f o r the  H a l l measurements, an e l e c t r o m a g n e t was c o n s t r u c t e d c a p a b l e of p r o d u c i n g a f i e * l d o f up to t e n k i l o g a u s s u n i f o r m o v e r a 1-jj i n c h diameter s e c t i o n I n a 3/k-  i n c h gap.  As a r e s u l t  of c a r e f u l a n n e a l i n g d u r i n g the m a c h i n i n g o f the s o f t  iron  yoke, a n e g i b l e amount o f h y s t e r e s i s c o u l d be d e t e c t e d , and the f i e l d was six kilogausso  a l i n e a r f u n c t i o n of c u r r e n t up t o about  The magnet was  calibrated  using a b a l l i s t i c  galvanometer and f l i p c o i l s t a n d a r d i s e d w i t h a l a r g e p e r manent magnet checked by p r o t o n resonance measurements i n this  department,,  56  3.1(e)  MEASUREMENT OF CONDUCTIVITY AND HALL COEFFICIENT The  e l e c t r i c a l c i r c u i t r y was k e p t q u i t e  simple.  F o r most measurements, the sample c u r r e n t was s u p p l i e d by a lead storage b a t t e r y with a s e r i e s r e s i s t a n c e for  regulation.  C u r r e n t was measured by the p o t e n t i a l drop a c r o s s an a c c u r a t e l y known s m a l l s e r i e s r e s i s t a n c e .  One o f two p o t e n t i o -  meters was used as d i c t a t e d by t h e magnitude p f t h e potent i a l t o be measured.  One, (Rubicon model 2730), capable o f  measuring from 0.000 t o 0.161 v o l t s o r from 0.00 t o 1.61 v o l t s on a second s c a l e , was u s u a l l y used f o r measuring v o l t - ages-along a filament  and thermocouple o u t p u t s .  Another,  (Leeds and N o r t h r o p model 7622) , capable o f m e a s u r i n g from 0.000 t o 9.999 m i l l i v o l t s was u s u a l l y used f o r measuring  voltages  and c u r r e n t v a l u e s .  i n s t r u m e n t s was i d e n t i c a l .  The c a l i b r a t i o n o f t h e two A switching  system was i n c l u d e d  t o p e r m i t r a p i d r e v e r s a l o f p o l a r i t y i n any c u r r e n t t i a l lead.  HallL  o r poten-  The e s s e n t i a l components o f t h e measuring  c u i t r y a r e shown i n F i g u r e 7 .  cir-  The r e v e r s i n g s w i t c h e s have  been o m i t t e d and o n l y one thermocouple has been i n c l u d e d clarity.  for  f  Some p r e l i m i n a r y an a l t e r n a t i n g c u r r e n t .  t e s t measurements were made u s i n g The s i g n a l from a v a r i a b l e low  f r e q u e n c y s i n u s o i d a l g e n e r a t o r was used t o d r i v e a s m a l l power a m p l i f i e r whose o u t p u t f e d a c i r c u i t i d e n t i c a l DC case d e s c r i b e d  t o the  above, w i t h the o m i s s i o n o f t h e r e v e r s i n g  s w i t c h e s and the i n c l u s i o n o f s h i e l d e d l e a d s .  The AC  SCHEMATIC  DIAGRAM OF MEASURING  CIRCUITS  58  v o l t a g e s were 'measured w i t h a wave a n a l y s e r  (General Radio  model 736-A) w i t h a f o u r c y c l e band pass c a p a b l e of measuring v o l t a g e s from 0,000 m i l l i v o l t s t o 300 v o l t s i n 28 r a n g e s . The use of the wave a n a l y s e r as a s e n s i t i v e v o l t m e t e r  ensured  r e j e c t i o n of s p u r i o u s s i g n a l s , and p e r m i t t e d an easy check-f o r s i g n a l d i s t o r t i o n by harmonic a n a l y s i s t o be made.  59  MtiPiSliREriEXT 0? CONDUCTIVITY  3.2  In  a f i l a m e n t of uniform  o f w i d t h w and t h i c k n e s s  rectangular  t cm c a r r y i n g a c u r r e n t  amperes, t h e c o n d u c t i v i t y i s g i v e n crwhere  =  LI  separated  s e p a r a t i o n , between  determine centre  the a l l o y  filaments  several side contacts  Under c o n s t a n t  current  between p a i r s o f s i d e tion  ratios  using  agreement was edges was  side  be t a k e n  unequally  contacts  slope  with  were  The  one s i d e . '  ratios  so t h i s  correctness  separaexact  value of  was  this  o f a sample  a l a r g e r area.  o f the c o n t a c t  just  The  equal  to  edges.  c o n d u c t i v i t y measurements f o r  salable a r e shown i n F i g u r e 8 at the high temperature  contacts  between c l o s e s t  change i n m e a s u r e d r e s i s t a n c e was  of these  end  this  Essentially  separation,  and c o v e r  determine  compared w i o h t h e  l a t e r v e r i f i e d when t h e c o n t a c t s  The r e s u l t s  the  used to  spaced along  when the d i s t a n c e  change i n s e p a r a t i o n  one  To  c o n d i t i o n s , the v o l t a g e  became h o t enough t o f l o w resulting  between  as t h e c e n t r e - t o -  were p r e p a r e d  t a k e n as t h e c o n t a c t  p r o c e d u r e was  measured  contacts  both p o s s i b l e c r i t e r i a .  obtained  1  o f L c>,i m e a s u r e d a l o n g  used i n a l l f u r t h e r c a l c u l a t i o n s .  the  (ohtn-Ctn)'  Some doubt e x i s t e d as t o w h e t h e r  conductivity should  uniform  of I  expression  o r minimum e d g e - t o - e d g e d i s t a n c e , ,  some l o n g and  V)~'  by a d i s t a n c e  l e n g t h o f the fllarnento the  (wt  by the  V i s the p o t e n t i a l d i f f e r e n c e i n v o l t s  contacts  cross-section  „  The  abrupt  -reversal of  e n d i s c a u s e d by t h e  Increase  60  In c a r r i e r intrinsic low  c o n c e n t r a t i o n as temperature  temperature  the m a t e r i a l  region*  o c c u r s because  mechanism becomes d o m i n a n t . with d i f f e r e n t general  impurity  The  scattering  Similar* curves f o r  and  intrinsic  conductivity from below  carriers  temperature  to i n c r e a s i n g reversal correct  c o n c e n t r a t i o n s were o f t h e same  was  m e a s u r e d as a f u n c t i o n  became i m p o r t a n t . limit  o f c u r r e n t was  was  shown by  The  a l o n g the s a m p l e .  due  n e c e s s a r y f o r t h i s were a r o u n d 1%  to  The 0  t h i s was  was  t r a c e d t o an a l l o y  temperature.  f o r sample  to  .  A few v a l u e s after  the  damage o r perman-  a p e r m a n e n t change n o t e d contact  A  slight  largest  a t w i d e l y s p a c e d t e m p e r a t u r e s t>rere r e - m e a s u r e d  O n l y once  this  at e a c h t e m p e r a t u r e  thermo-electric potential  of each run to test  of  temperature.  corrections  change.  of  a change f r o m d e c r e a s i n g  for increasing  performed  effect  gradient  completion  crystal  17$°K up t o t h e t e m p e r a t u r e a t w h i c h  conductivity  f o r any  the v a r i o u s  temperature  ent  samples  form.  temperature  upper  decrease of slope a t  the i m p u r i t y  Samples were p r e p a r e d f r o m slices  approaches i t s  softening  and  at too h i g h a  62  MEASUREMENT 0 ? HALL COEFFICIENT  3.3  If  a long  filamentary  section  carries  subject  t o a magnetic  sample o f r e c t a n g u l a r  a l o n g i t u d i n a l current field  o f I amperes a n d i s  o f H gauss a c t i n g p a r a l l e l  sample t h i c k n e s s , t h e n a p o t e n t i a l o f V v o l t s w i l l oped a c r o s s Hall  t h e w i d t h o f t h e sample n e a r  coefficient will  R  _  i o  V t  8  cm  current  i s distortion  by t h e c u r r e n t  flow,  contacts  than  Hall  ed  sample w i d t h s  This  apparent H a l l with  contacts  who f o u n d no s h u n t i n g but  their  p o t e n t i a l i s meas-  technique.  and a s e r i e s o f t e s t s  were s h u n t i n g  contact  This  sizes  ones u s e d h e r e w h i c h p r o b a b l y  result  i s In disagreement  due t o s o l d e r e d H a l l  accounts  two o b s e r v a t i o n s . .  indicat-  decrease i n the  o f F r i t z c h e and L a r k - H o r o v i t z effect  using  some o f t h e sample  s a m p l e s were more t h a n f i v e  between these  non-axial  were a p p l i e d t o f i l a m e n t s  caused a considerable  the observations  of the H a l l  error i s introduced.  and d i f f e r e n t  coefficient.  '  sample w i d t h s f r o m t h e  c r o s s - s e c t i o n by t h e g o l d a l l o y  that these  current.  Hear t h e f i l a m e n t  and p o s s i b l e  three  contacts  However, i n c o n s i s t e n t r e s u l t s different  i n cm.  sample, n e g l i g i b l e  Initially,  and t h e  3  but i n p r a c t i c e , i f the H a l l  of a uniform  of uniform  the centre  due t o s h o r t - c i r c u i t i n g  u r e d a t a p o s i t i o n no l e s s end  be d e v e l -  coulomb  where t i s t h e sample t h i c k n e s s  potential  to the  be g i v e n b y t h e e x p r e s s i o n  HI  ends, t h e r e  cross-  (195>1;) contacts,  t i m e s as wide as t h e f o rthe difference  Because o f t h i s  shunting  63  effect, favour  this  simple  type  o f the small The  carefully  crystal  Hall  with  are f i e l d  ments was t o p l o t  to locate H a l l  there  i s current  flowing  The u s u a l methods e m p l o y e d t o o v e r -  r e v e r s a l or external b i a s i n g to zero b u t t h e method used i n these  the p o t e n t i a l  this  l i n e a r p l o t was t h e n  Hall  coefficient.  A typical plot  measurement e r r o r s o v e r  f r o m 6>00 g a u s s  direction.  w o u l d be i m m e d i a t e l y or l o n g i t u d i n a l  o f the  i s shown i n F i g u r e i o .  of averaging  several points;  o r any f i e l d  through  The s l o p e o f  used i n the c a l c u l a t i o n  T h i s method h a d the advantage  experi-  d i f f e r e n c e between t h e H a l l  z e r o t o 6500 g a u s s i n t h e r e v e r s e  effect  contacts  Is usually a small  as a f u n c t i o n o f m a g n e t i c f i e l d  Transverse  These a r e d e s c r i b e d  between them when t h e r e  at zero f i e l d ,  efficient  to'the  t h e methods o f e l i m i n a t i n g them, a n d a r e  no m a g n e t i c f i e l d .  resistance  measure-  can c o n t r i b u t e  t h e sample.  one a n o t h e r ,  residual potential  leads  coefficient  I n a d d i t i o n t o the  other e f f e c t s  i tis difficult  opposite  come t h i s  f o rHall  i n Figure 9 o  Since  voltage  arms.  1952),  (Lindberg,  potential,  below, t o g e t h e r Illustrated  was abandoned i n  The p r o b l e m s a s s o c i a t e d w i t h H a l l  p o t e n t i a l measured a c r o s s  but  side  o r i e n t e d In the magnetic f i e l d  ments a r e w e l l know  exactly  contact  samples were m o u n t e d I n t h e s m a l l o v e n a n d  determination.  expected  of H a l l  o u t any random  a l s o a n y magneto-  dependence o f t h e H a l l c o apparent  thermal  as a c h a n g i n g  gradients  slope.  can introduce  64  FIGURE  SOURCES OF ERROR  9  IN HALL POTENTIAL MEASUREMENTS  Hall Contact Conduction Current  >  H= 0  OFFSET CONTACTS  Thermal Current  O  A V  1  H  —  9  AT  H  o  H  NERNST E F F E C T  *  T  O  >  Thermal Current > RIGHI-LEDUC  AT  EFFECT  Conduction Current  >  ETTINGSHAUSEN  EFFECT  65  FIGURE  DETERMINATION  I 5  •  -  .  \  4  -  2  1  10  OF H A L L  •  1 O  MAGNETIC FIELD  COEFFICIENT  '  1 — 2  (Kilogauss)  66  unwanted p o t e n t i a l s .  A transverse gradient introduces a  t h e r m o - e l e c t r i c p o t e n t i a l , b u t c u r r e n t r e v e r s a l removes t h i s error.  A l o n g i t u d i n a l thermal gradient r e s u l t s i n a l o n g i -  t u d i n a l t h e r m a l f l o w w h i c h i n the presence o f the magnetic f i e l d produces a t r a n s v e r s e N e r n s t p o t e n t i a l o r a t r a n s v e r s e Righi-Leduc  temperature d i f f e r e n c e . w h i c h i n t u r n g i v e s a  thermo-electric potential. these e r r o r s .  C u r r e n t r e v e r s a l removes b o t h  N e i t h e r c u r r e n t nor f i e l d r e v e r s a l can remove  the e r r o r i n t r o d u c e d by the E t t i n g s h a u s e n  e f f e c t whereby the  l o n g i t u d i n a l c u r r e n t and the magnetic f i e l d produce a t r a n s v e r s e temperature d i f f e r e n c e l e a d i n g t o an a d d i t i o n a l p o t e n tial.  Due t o the t h e r m a l c a p a c i t y o f t h e f i l a m e n t , t h i s  e f f e c t can be e l i m i n a t e d by AC measurements. ments were p e r f o r m e d a t f r e q u e n c i e s between  Such measure-  £o  aa d $00  cycles  p e r second on a few specimens c o v e r i n g the i m p u r i t y range used and i t was f o u n d t h a t n e g l i g i b l e e r r o r would r e s u l t f r o m u s i n g the DC measurements and i g n o r i n g t h i s l a s t The use o f germanium side-arms as H a l l c o n t a c t s  effect. considerably  reduces those e f f e c t s i n w h i c h the unwanted p o t e n t i a l a r i s e s from a t r a n s v e r s e temperature g r a d i e n t . I n t h i s manner, the H a l l c o e f f i c i e n t v a l u e s were determined over the range of temperature used i n the conducti v i t y measurements.  F i g u r e 8 shows the H a l l  coefficient  v a l u e s f o r one sample i n a d d i t i o n t o the c o n d u c t i v i t y v a l u e s . The o b s e r v e d decrease i n H a l l c o e f f i c i e n t a t h i g h  temperature  r e s u l t s from the i n c r e a s i n g e l e c t r o n c o n c e n t r a t i o n and the  67  e f f e c t of h o l e s .  The  absence of any  dependence o f H a l l  co-  e f f i c i e n t on temperature over most o f the temperature range i n d i c a t e s a constant i o n i z a t i o n of the It e f f i c i e n t due  c a r r i e r concentration  due  t o complete  donors.  i s possible that a small increase i n H a l l  co-  t o some t r a p p i n g o f e l e c t r o n s by donors a t  temperature c o u l d be masked by a c o r r e s p o n d i n g the H a l l f a c t o r r ,  low  decrease i n  However, the p r e v i o u s l y mentioned  a n a l y s i s o f r under c o n d i t i o n s o f m i x e d l a t t i c e and s c a t t e r i n g (Figure 2 ) p r e d i c t s that r w i l l  Impurity  decrease  slightly  ,at the lower temperatures used f o r the more pure samples and I n c r e a s e  s l i g h t l y f o r the more impure ones.  Such changes  would be s m a l l and i t i s improbable t h a t the masking s u g g e s t e d above o c c u r s f o r these samples. a n a l y s i s I n any  event.  It  would not a f f e c t the  T a b l e I l i s t s the s a t u r a t i o n r a n g e  H a l l c o e f f i c i e n t f o r the samples s t u d i e d .  The  donor d e n s i t y  o b t a i n e d under the assumptions of n e g l i g i b l e a c c e p t o r and a v a l u e o f u n i t y f o r r i s a l s o g i v e n . H a l l m o b i l i t y values was  a comparatively  Conductivity  at 300°K are a l s o i n c l u d e d .  and  Sample 1  h i g h p u r i t y sample used o n l y as a check  on the m o b i l i t y d e t e r m i n a t i o n methods. p u b l i s h e d v a l u e s was  density  Agreement w i t h  q u i t e good (Debye and  Conwell,  The g i v e n m o b i l i t y v a l u e f o r t h i s sample i n c l u d e s a c o r r e c t i o n f o r the h o l e  concentration.  19510*  yft>  68  T A B L E  I  SOME DATA -CONCERNING THE SAMPLES STUDIED  Saturation  Range  Values  o  nrple mber  cmcoulomb  -1  , cm •>  3  1  2.27  x 10^  2  2.01L x 1 0  3  1.00  x 10  (ohm-cm)  0.167  a t 300 K c.  9J3 volt-sec  2.75  x 10 "  2  3.06  x 10  1 6  12.14.  2530  2  6.25 x 1 0  1 6  22.7  2270  36.0  2020  1Ll  3900-"-  k  56.2  1.11  x 1CT  5  39.3  1.59  x 10  1 7  1+6.6  1910  6  3U.7  1.80 x 1 0  1 7  50.3  1750  7  19.5  3.21  7U.8  H4.60  *  This  value  x 10  1 7  corrected  1 7  f o r holes  69  C H A P T E R  I4..I  I4. -  R E S U L T S  DETERMI NAT I OH OP ^ The  AND  OP T H E I N V E S T I G A T I O N  .  e x p e r i m e n t a l measurements d e s c r i b e d i n  chapter 3 p r o v i d e d values o f the H a l l c o n d u c t i v i t y o f n-type impurity  c o e f f i c i e n t and  germanium o v e r a r a n g e o f b o t h  c o n c e n t r a t i o n and temperature.  The r a n g e s c o v e r e d  f o r an a p p l i c a t i o n o f the o f - ¥ a n a l y s i s  were s u f f i c i e n t  d e v e l o p e d i n c h a p t e r 2. r e g i o n i n which  In fact  t h e y i n c l u d e d most o f t h e  t h e a s s u m p t i o n s made a n d t h e a p p r o x i m a t i o n s  u s e d were v a l i d .  The q u a n t i t i e s o f a n d ^ w e r e  obtained from  t h i s - e x p e r i m e n t a l d a t a by e v a l u a t i n g  the d i f f e r e n t i a l s  and  ,.  0 For  the purpose o f these e v a l u a t i o n s ,  and g r a p h s o f for  —  £r\ cr  as a f u n c t i o n o f Jim T  e a c h o f t h e samples  studied.  by n u m e r i c a l d i f f e r e n t i a t i o n points,  °C  tabulations  were p r e p a r e d  was t h e n d e t e r m i n e d  at a series  o f fixed, temperature  usi'ng'va method o f d i v i d e d d i f f e r e n c e s .  The s l o w  v a r i a t i o n o f sample c o n d u c t i v i t y w i t h t e m p e r a t u r e and t h e closely  s p a c e d e x p e r i m e n t a l o b s e r v a t i o n s made t h i s method  quite accurate.  Temperatures  r a n g i n g upwards f r o m 200°K i n  i n c r e m e n t s o f 25°K were c h o s e n f o r t h e p o i n t s o f e v a l u a t i o n . The  upper temperature  limit  f o r e a c h sample was i m p o s e d by  70  the l i m i t i n g h o l e d e n s i t y as shown i n F i g u r e 3» s i m i l a r manner,  In a  ¥ was determined f o r each sample from t h e  H a l l c o e f f i c i e n t and c o n d u c t i v i t y d a t a a t the same p o i n t s on t h e temperature a x i s as used  above.  Thus a s e t o f p o i n t s i n the  oc - ¥ p l a n e was  o b t a i n e d over a range o f temperature and I m p u r i t y d e n s i t y values.  These r e s u l t s  are d i s c u s s e d i n t h e next s e c t i o n .  71  !+»2  USE OF THE CV^-^DATA I t was found t h a t a t any f i x e d t e m p e r a t u r e , the  (of,^)  line.  p o i n t s c o u l d be f i t t e d q u i t e w e l l w i t h a s t r a i g h t T h i s was  done, u s i n g the method o f l e a s t s q u a r e s , o  f o r each o f the chosen t e m p e r a t u r e s , up t o q.00 F i g u r e 11 shoxvs two examples o f t h i s d a t a .  K. V a l u e s o f of  were determined a t h i g h e r temperatures t h a n t h i s f o r some of the more impure samples used. . Hoi^ever, the l i m i t e d range o f i m p u r i t y d e n s i t y which c o u l d be used a t t h e s e temperatures made a c c u r a t e d e t e r m i n a t i o n o f impossible. was imposed.  values  Hence the upper temperature l i m i t o f l|fJO°K A few ^ar <f) p o i n t s were determined f o r the  more pure samples a t temperatures above the l i m i t by the h o l e c o n c e n t r a t i o n c r i t e r i o n .  imposed  These p o i n t s were  f o u n d t o l i e below the f i t t e d l i n e s as would be e x p e c t e d to the e r r o r s d i s c u s s e d i n s e c t i o n  due  2.1;.  B e f o r e the s i g n i f i c a n c e • o f these i s o t h e r m a l l i n e s i s d i s c u s s e d , i t s h o u l d be r e c a l l e d t h a t a d i s c u s s i o n of one p o t e n t i a l source of e r r o r i n -the a n a l y s i s d e f e r r e d from s e c t i o n 2.3.  .>TMs was  was  the e r r o r i n t r o d u c e d  by changes i n s l o p e a r i s i n g from c h a n g i n g v a l u e s o f 717 TJ7  f o r d i f f e r e n t temperatures and donor d e n s i t i e s .  was hot d i s c u s s e d q u a n t i t a t i v e l y i n s e c t i o n 2.3 dependence o f ^ on N and T was n o t known? The r e l a t i o n between &C and <K was oc  =  o.  -  hnthl.T)  because  seen t o be <K  or It the  FIGURE  PLOTS  OF  cc-y  II  DATA  73  The  use o f i s o t h e r m a l p l o t s removes the T dependence o f the  slope.  The slope o f a s t r a i g h t l i n e f i t t e d to(of  i n the neighbourhood o f  The  term  points  }  w i l l be  i s absent because T i s constant and a i s i n d e -  pendent of N, and hence o f / .  I f a i s taken as the  I n t e r c e p t o f the f i t t e d l i n e , t h i s  corresponds  to an e x t r a -  p o l a t i o n of the l i n e Of  CL  =r  t o the of-)f  the tangent  -t~ l~n ' M  curve.  y  , of —  However, the c o r r e c t  I n t e r c e p t o f the of~y curve w i l l be g i v e n by an e x t r a p o l a t i o n /  of the l i n e (X  -  CL  —  \  rn(y J 0  s y  as shown i n F i g u r e 12. Thus the e r r o r i n the Of - i n t e r c e p t i n t r o d u c e d i n Y  t h i s manner i s ( gl ¥'y  i  S  n  e  e  shoxtf t h a t  Y  of  d  0  e  d  *  (-jVz) ^  and an estimate o f the term  & - i t of the l i n e s  T l l e  ood  f  can be c o n s i d e r e d constant over the range used i n any one ploto  centre o f t h i s range.  J  P~  Yo  and  itfas taken as the oi tn  can be w r i t t e n as -7  oLlT> j~  to the data  ^  •- •  correspondence  ,  • .  N  was estimated i n chapter 2 f o r the two cases tyi^TTzj •  at hi  fTl  (Tr^  From the work o f t h i s chapter, a one-to-one between v a l u e s o f Y  and N i s a v a i l a b l e f o r  74 FIGURE  12  75  each temperature u s e d i n the  can  ¥ h e n the  m  u s e d , the and  the  has  be  value  made  ^~^T  of  resulting error  i n the  so t h a t  no  contribution  it  appears that  no  important  i n g values of slope The  with  fitted  increase  of  -intercept  b and  are  J  and  both  less  A s i m i l a r estimate  much s m a l l e r  to  tri (TTiJ  to  d i f f e r e n c e between m"  e v a n l e s s a t l4.00°K.  b e e n assumed t h r o u g h o u t t h a t  N,  appropriate  the  v a l u e s o f TT^ g i v e s  the  definite  an  0  r e s u l t s show t h a t  t h a n 2% a t 2 0 0 ° K and b a s e d on  hence  AM.  estimate of  is  c a l c u l a t i o n s and  c are  errors.  It  independent  a r i s e s f r o m them. errors  will  o c c u r due  of  Thus to  chang-  ^'. Of - Y  isothermal  i n both  °C  -intercept  lines displayed and  magnitude  a of  o slope  for increasing  Between 300 slight slope  of  [uOO°K, t h e  d e c r e a s e i-Jhile t h e and  careful has  and  temperature  i n d i c a t e d that <xf - i n t e r c e p t as  slope  are  and  300  values  shown i n F i g u r e  K»  showed a  became l e s s s t e e p .  a p p r o x i m a t i o n s and  changes o f  and  sucho  of  -Intercept  slope  i n t e r c e p t v a l u e s are  consideration  discussed  Of  between 200  13»  These The  possible  errors  t h i s o r d e r i n measured  values  s i g n i f i c a n t and  should  be  VALUES OF S L O P E AND cc - INTERCEPT F R O M o c - y DATA  77  [{..3  L A T T I C E SCATTERING The  b e h a v i o u r o f the  temperature range  Of - i n t e r c e p t  over the  of the o b s e r v a t i o n s i n d i c a t e s  lattice relaxation  that  t i m e i s a p p r o x i m a t e d by t h e s i m p l e power  (2.2)  law o f e q u a t i o n  • •• Tu = A  T^X"  1  then the v a l u e of the i n d e x a i s not c o n s t a n t but with temperature leads  i n a manner shown i n F i g u r e  to an I d e n t i c a l  mobility.  These  temperature  ;  by  100  a t o be  These  independent  changes  in section  valley  He the  1.2,  scattering  scattering  that  This  an a v e r a g e  he h a s by low  of  of Herring  predictions  agreement  (1955).  As  c o n s i d e r e d an a d m i x t u r e  e n e r g y phonons and  then c a r r i e d out n u m e r i c a l c a l c u l a t i o n s  quantitative  obtained:;;fv,  T.  time.  Tt  pointed  of  Intra-  inter-valley  by h i g h e n e r g y p h o n o n s w i t h a s i n g l e  resulting relaxation  a  t o 3 0 0 ° K -would be i n  i n a are In q u a l i t a t i v e  w i t h the t h e o r e t i c a l r e s u l t s out  13.  agreement w i t h p r e v i o u s v a l u e s o f 1.66  assuming  increases  dependence f o r l a t t i c e  values of a indicate  over t h e t e m p e r a t u r e range reasonable  i f the  energy v a l u e .  of mobility  using  i s d i f f i c u l t t o make  from h i s r e s u l t s .  However,  any  our  experimental r e s u l t s  are i n q u a l i t a t i v e agreement w i t h h i s  calculations  he f o u n d t h a t  increasing  i n that  temperatures  a first  Increased with  and t h e n d e c r e a s e d .  The  fact  that  o o u r h i g h e s t v a l u e o f a o c c u r s n e a r 300 K w o u l d H e r r i n g ' s work t h a t  indicate  the c h a r a c t e r i s t i c temperature  from  o f the  78  h i g h e n e r g y phonons i s somevjhat g r e a t e r  than the  a l value  1.2.  o f i4.00°K m e n t i o n e d i n s e c t i o n Harrisson  by  a c o u s t i c a l and  only  intra-valley  values  do  not It  not  be  the  value  (1956) has a n a l y s e d l a t t i c e s c a t t e r i n g  optical  behaviour particular made t h i s  lattice  scattering.  support  i s unfortunate  However, t h e  the  His  dependent v a l u e  t o w a r d s 1.5  limitations  large effect  of  a.  investigation could  temperatures t o see  germanium a t h i g h  impossible.  predicted mobility  that the  of a would decrease  of the  vibrations considering  a temperature  c a r r i e d to higher  Herring.  experiment-  as  whether or predicted  Imposed b y  the  temperatures,  of s m a l l hole  and  not by  intrinsic in  concentrations,  79  l±ok IMPURITY SCATTERING The curve  i n s e c t i o n 2.3  of  the  in  this  ( e q u a t i o n 2.6)  q u a n t i t i e s a, b  and  the  eratures the  c.  used i n the  potential.  oc-  <f c a l c u l a t i o n s ,  t h a t the p r o b a b i l i t y ionized  may  b r e a k down f o r v e r y h i g h  s h o u l d n o t be  A recent  silicon.  relaxation  time Using  2.8  relation  (2.1)  This  centres.  (but not  i t l e d to values  of  the  This  apparent  as at  individuals the  donor  the  9  densities  t o be  general  valid  form of  the  investigation.  o f u n i t y f o r b, the  ions  and M y e r s , (1959),  m o b i l i t y formula  supports  be  assumption  donor d e n s i t i e s xtfhen t h e  used i n t h i s  a value  from  p r o p o r t i o n a l to  i n v e s t i g a t i o n by L o n g  t h e m e a s u r e d s l o p e s and  Equation  one,of?  t h a t an e l e c t r o n w i l l  and-cease to s c a t t e r  shown t h e B r o o k s - H e r r i n g  i n n-type  and  such scattering  t o aggregate  arises  donor i s d i r e c t l y  density of  used here.  use; o f  l e a d to a d e f i n i t e  t h e o r e t i c a l value b = 1  effect  the  c was  estimated  i t s approximate  calculated 777"  values.  form) was  used  c w h i c h r e m a i n e d c l o s e t o 1.55  for  o temperatures  ;  . •  the  this  both v  The  an  of  upon  a t the v a r i o u s , t e m p -  c.  s c a t t e r e d by  using  i n terms  appeared  Since values  and  assumption  has  slope  J termswhich  s l o p e have b e e n d e t e r m i n e d  between b  but  The  s c a t t e r i n g p o t e n t i a l s can  begin  gave t h e  y  e x p r e s s i o n made i t s I n t e r p r e t a t i o n d e p e n d e n t  the a s s u m e d s c a t t e r i n g a and  oc -  g e n e r a l e x p r e s s i o n d e r i v e d f o r the  b e t w e e n 200  and  325  K but  dropped to  around  80  1.35  as t h e  77*  l e d to higher  temperature rose  o f c i s 1.5 the  with  the  s c a t t e r i n g models,  theoretical values  are  values  are u n c e r t a i n .  r a d i u s remains c o n s t a n t .  the more i m p o r t a n t effective the  larger.  s u c h as  screening radius increases with off  one,  t o the  favour  i n magnitude^, These  radii  e f f e c t i v e mass activation  Furthermore,  the  and  energy,  temperature while Since  cut-  undergoes  so  the  the  cut-  smaller l i m i t  is  a transition  the  from  c u t - o f f model f o r i n c r e a s i n g temperature  i n the r e g i o n s t u d i e d .  S u c h an  e f f e c t would keep  value  o f c d e r i v e d by  the  level  over  t e m p e r a t u r e range*.  the  data  value  i t i s p o s s i b l e t h a t the f o r m o f  scattering potential  screened  as  s c r e e n i n g and  comparable  donor c o n c e n t r a t i o n f o r z e r o  their relative  for  theoretical  these  f o r the  screening radius s l i g h t l y  depend upon assumed p a r a m e t e r s critical  of  potential.  g i v e n i n s e c t i o n 2.3  off radii  use  d r o p p e d t o 1.7  Since the  O  Coulomb  The  Similarly,  325°K, b u t  and  t o i|.00 K.  f o r both  screened  K.  c v a l u e s w h i c h s t a y e d a r o u n d 1.9  between 200  temperatures the  t o 2|00  temperature rose  entire  above method a t a more  the  constant  81  CONCLUSIONS The ium  investigation  of electron  mobility  a t h i g h t e m p e r a t u r e w h i c h was u n d e r t a k e n f o r  given i n section  2.1 h a s b e e n c o m p l e t e d ,  e r a t u r e r a n g e was o f n e c e s s i t y originally lattice  and impurity  more l i m i t e d t h a n h a d i p e e n .  applied  t o t h e r e s u l t s . • •• T h i s  analysis  has i n d i c a t e d  mobility  i s expressed i n the form  = F i g u r e 13.  that  i f the l a t t i c e  T  A  of a varies  This  separating  e f f e c t s w h i c h was d e v e l o p e d i n c h a p t e r  2 has been s u c c e s s f u l l y  then the value  reasons  a l t h o u g h t h e temp-  The a n a l y t i c a l m e t h o d o f  expected.  i n german-  scattering  _ F L L  w i t h t e m p e r a t u r e a s shown i n  v a r i a t i o n was f o u n d t o be i n - q u a l i t a t i v e  agreement w i t h H e r r i n g ' s  (1955)  theory f o rl a t t i c e  scatter-  ing. Some i n f o r m a t i o n h a s a l s o impurity rather  scattering.  than a c u t - o f f ,  This  been g a i n e d  about  f a v o u r s t h e use o f a screened,  (Toulomb s c a t t e r i n g p o t e n t i a l , .  32  A F P E  N  D I X  DESIGN AND  A  CONSTRUCTION,OF A CRY3TAL GROWER Before  p a r e d i t was grower.  and  and  a temperature  instrument melt  c r u c i b l e h e a t e d by cooled ing  construct  a  i s contained a graphite  electrodes.  age  power was induce  the  crystal  chosen t o permit  connections  causes the-melt  i s m o n i t o r e d by  radiation pipe. desired value  which g i v e s This  High current  emf  and  growth.  a continuous  u n i t a l s o notes  the  a motor  keeps the  The  pen  three low  voltand  induction armature.  surface  the  If  edges  clear away  c r u c i b l e temperature  which c o l l e c t s  from the  s e t i n t o the  heat-  i n t h e m o l t e n germanium.  c r u c i b l e by means o f The  below.  graphite  graphite heater  p r o p e r l y phased t h i s  a thermopile  bottom o f the  quality  c e n t r i f u g i n g i t to  r e g i o n of c r y s t a l  The  described  h u n d r e d amperes a t  a robust  the melt  f o r e i g n m a t t e r by  from the  the  stir  text.  power i s f e d t o t h i s  t o r o t a t e somewhat l i k e  This motion helps any  are  type  element f e d from water  six legs.  l a r g e eddy c u r r e n t s  heater  heating  to three  through each o f the  and  i n a high  S i x p h a s e AC up  pre-  c o n t r o l l e d melt c r u c i b l e  i s shown i n F i g u r e  element, p a s s i n g  volts  a  design  a s e e d t-Jithdrawal s y s t e m as m e n t i o n e d I n t h e  The  of  to  c o u l d be  general requirements f o r a Czochralski  grower are  finished  to  d e s i r e d germanium c r y s t a l s  necessary  The  crystal  the  '  r a d i a t i o n from  a polished  thermopile  sapphire  i s compared  c o n t r o l s of a commercial  unit  record  to  of the  thermopile  signal.  e r r o r between t h e  thermopile  signal  83  and  the  set  c o n t r o l value  d e p e n d e n t upon the integral able  to  of t h i s  error.  produces  the  time  The  proportions  compensate f o r t h e  control.  This  p h a s e l a g g i n g AC conduction these  value,  and  composite  of  tubes o p e r a t e s  current  feeding  one  d e r i v a t i v e and  response of  s i g n a l which  a set  performed very  a c r u c i b l e temperature 0,1°C, a p e r f o r m a n c e The provided  by  the  d r i v e n by  separate  transmissions. minute and been used  passes  of  outer  the  rates  working area  the  o f 25  o f 0.1  Each  of  to  and  an  atmosphere The  of the-  within growing. s e e d were quill speed  revolutions  inches  per  per  h o u r have  i n a molybdenum rod.  This  i s contained  rod region by  inner Vycor w a l l  c o o l i n g water c i r c u l a t i n g  inner  inhabitable.  250  pull  region  the  maintaining  i n t o a c o n t r o l l e d atmosphere This  a  completes; the  variable  seed i s h e l d  c y l i n d e r c o n s i s t i n g of  the  -  of  controls  spindle  t o 10  a stainless s t e e l  crucible,,  cools  level  satisfactory for crystal  The  Pyrex w a l l with This  i s capable-  head w i t h  adjust-  temperature c o n t r o l "  s y n c h r o n o u s m o t o r s and  through a bellows  double w a l l e d  them.  press  Rotation  end  DC  w i t h d r a w a l m o t i o n s f o r the  successfully.  surrounding  an  quite  time  to c o n t r o l  This  The  w e l l and  withdrawal rates  chuck at the  in turn  heater.  the  s y s t e m .under  o f a r o u n d 1000°C c o n s t a n t  r o t a r y and  a drill  the  through a transformer l e g of  signal  of each are  of s i x t h y r a t r o n s ,  necessary feed-back c o n t r o l loop. s y s t e m has  voltage  s i g n a l c o n t r o l s the  voltage  periods  a DC  and  a and  between  a l s o keeps  a t m o s p h e r e 'around the  the growing  814-  o p e r a t i o n c a n be p u r e been c o m m e r c i a l l y  helium  freshly  supplied purified  t o keep a l l t r a c e s o f m o i s t u r e atmosphere.  boiled  but has u s u a l l y  argon.  I t i s imperative  out o f the c r y s t a l  A l l i n t e r i o r metal  growing  s u r f a c e s have been, r h o d i u m  p l a t e d f o r easy c l e a n i n g . During  the i n i t i a l  p e r i o d o f use, several' working  Improvements have been n e c e s s a r y has  now b e e n u s e d  successfully  germanium, some o f n i g h p u r i t y levels. ive  dislocation  crystal The  T h e s e have  to  grow s e v e r a l c r y s t a l s o f  and o t h e r s doped t o v a r i o u s  been i n v e s t i g a t e d f o r t w i n n i n g ,  and f o u n d  satisfactory  g r o w e r h a s become a r e l i a b l e  modifications  instrument;,-, b u t . i t  excess-  d e n s i t y , u n c o n t r o l l e d i m p u r i t y content and  orientation,  crystal  i n this  silicon  c r y s t a l s might a l s o  i n a l lrespects.  tool  and w i t h  be grown.  some  85  A P P E N D I X DESIGN AND  B  CONSTRUCTION  OF A WIRE-SAW FOR  A r e c i p r o c a t i n g wire e d f o r use i n c r y s t a l In this  i s carried  a fine  s l i g h t l y w i d e r . t h a n . t h e x^ire  instrument was u s e d  wire  longitudinally  over the s u r f a c e of a c r y s t a l .  must be f e d t h r o u g h  CUTTING  designed and c o n s t r u c t -  c u t t i n g f o r s e e d a n d sample p r e p a r a t i o n .  type o f instrument  substance  saw was  CRYSTAL  an a b r a s i v e  under a t r a n s v e r s e l o a d  T h i s c u t s a groove diameter.  the c r y s t a l  i s shown i n f i n a l  carrying  t o show t h e w i r e p a t h )  The w i r e ,  t o complete  form  only of course,  the c u t .  The  i n F i g u r e 6 (a heavy  and d e s c r i b e d i n some  wire detail  below. Since .essential,  pulleys  plate.  The w i r e wound f r o m  turning  freely  rewound on a s e c o n d  and  and turning  on c l o s e l y  seven  i n t h e same  large  An a d j u s t a b l e s p r i n g  drums was  tension. necessary.  i t s cutting  to reduce  bearing  guide  mandrels,  path, and f i n a l l y with the  A l l rotating rotary  inertia  shafts to a  l o a d e d d e v i c e b e t w e e n t h e two  included to permit  Since the wire  of half  o f two i n c h  aligned b a l l  direction.  were  inch  I n c h drum mounted c o a x i a l l y  p u l l e y s vjere o f a l u m i n i u m  minimum.  alignment  a seven  drum, f o l l o w e d a p a t h o v e r a s e r i e s  two o f w h i c h g u i d e d t h e w i r e o v e r  first  and p r e c i s e  a l l m o v i n g p a r t s were m o u n t e d on a c h a s s i s  i n c h aluminium diameter  absolute r i g i d i t y  adjustment  l e n g t h was f i n i t e ,  T h i s was a c c o m p l i s h e d  o f the wire  reversal  by a r e v e r s i n g  was  relay  86  t r i g g e r e d by a m i c r o s w i t c h t r i p p e d i n t u r n by a p i n  fixed  t o a Geneva w h e e l m o u n t e d on t h e b a c k o f one o f t h e l a r g e pulleys. utions  This  reversed  the d r i v e a f t e r each e l e v e n r e v o l -  o f the l a r g e p u l l e y , and thus spread  w i r e o v e r a l e n g t h o f more t h a n  twenty f e e t .  l o a d i n g between t h e d r i v e p u l l e y s a l s o absorber to l i m i t  wire s t r a i n  t h e wear o n t h e  during  The s p r i n g  served  as a shock  reversal.  p u l l e y s were o r i g i n a l l y mounted d i r e c t l y  The d r i v e  on t h e o u t p u t  shaft  of a s m a l l  DC m o t o r a n d g e a r r e d u c t i o n  u n i t , b u t despite the  low  rotary  inertia  unit f a i l e d after a  few  months u s e .  chain unit  o f the system, t h i s  I t was r e p l a c e d b y a b a l l  d r i v e n by a h e a v i e r which drove  the wire  duty  bearing  DC m o t o r a n d g e a r  shaft  reduction  a t s p e e d s o f up t o 125 f e e t p e r  minute and has p r o v e n q u i t e s a t i s f a c t o r y . A  s m a l l m i l l i n g machine  vice incorporating a swivel  b a s e mount a n d m i c r o m e t e r c r o s s - f e e d was u s e d as a h o l d e r the m a t e r i a l  being  cut.  This  w h i c h c o u l d be r a i s e d t o f e e d The  arm and v i c e c o u l d  v i c e was mounted o n a l o n g arm the m a t e r i a l  nique  arc point etched plane.  using  (Edelman, 1956). source  i n t o the wire.  a l s o be swung up t o a s e c o n d p o s i t i o n  where a s u i t a b l y p r e p a r e d sample aligned within limits  for  c o u l d be c r y s t a l o g r a p h i c a l l y  a simple  light  reflection  I n t h i s method, l i g h t  i s r e f l e c t e d from  the s u r f a c e  to b r i n g out the c h a r a c t e r i s t i c When t h e i n c i d e n t l i g h t  desired plane,  the cumulative  etch p i t walls  gives  from a  axis  effect  a characteristic  zirconium  o f a sample  etch pits i s normal  of a desired to t h i s  of reflections pattern  tech-  from t h e  on a s c r e e n  also  8?  normal  to the i n c i d e n t  the r e f l e c t e d p a t t e r n i.e.  about  into  Alignment  symmetry  consists  about  of  bringing  the i n c i d e n t  axis,  a s m a l l h o l e i n t h e c e n t r e o f the s c r e e n .  method was etchant  axis.  t e s t e d on a  (110)  plane using  to develop the e t c h p i t s  and was  The  a hydrogen peroxide f o u n d t o be  quite  satisfactory. Initially, denum w i r e was longitudinal  .003  inch f i f t y  used, but t h i s  splits.  percent  continually  f i f t e e n h o u r s has were t r i e d , hundred was  feet been  carborundum  a wire l i f e  found normal.  By u s i n g a v e r y s l o w r a t e  . 0 0 ^ i n c h e s t h i c k have b e e n  o f t e n to^  Various cutting  o i l or l i g h t  grease  and p i c k e d up by  of feed,  wire  found d i f f i c u l t cut i n a s l i g h t  discs  cut with t h i s  time r e s p e c t i v e l y . i n g an i n d i c a t o r  This  circuit  between t h e c h a s s i s  and  insulating glass plate. was  known t h a t  Another  complete  was  the c r y s t a l ,  overcome by  t h e w i r e was  s o o n as no l o n g e r  w h i c h was  c o n v e n i e n t w o r k i n g improvement was  or  install-  continuity  mounted on  c o n t i n u i t y was cutting  as t h e  be s t o p p e d  used to check e l e c t r i c a l  As  discs, i t  w i t h a wastage o f f i l a m e n t s  difficulty  than  instrument.  Hence t h e c u t w o u l d  e i t h e r t o o soon o r too l a t e  which  o f germanium l e s s  t o j u d g e when t h e c u t was arc.  agents  the w i r e .  'When f i l a m e n t s were b e i n g p r e p a r e d from was  speed of  b e i n g a p a s t e o f number s i x  powder i n h e a v y  s p r e a d on t h e l a r g e p u l l e y s  inch tungsten  With a c u t t i n g  per minute,  t h e most s a t i s f a c t o r y  developed short  made t o . 0 0 2  A change was  wire which has p r o v e n s a t i s f a c t o r y . a r o u n d one h u n d r e d  tungsten-molyb-  the  an  lost, i t  crystal.  the i n s t a l l a t i o n  of  88  a c o n s t a n t p r e s s u r e f e e d system and an a u t o m a t i c s h u t - o f f switcho I n c o n c l u s i o n , . t h e w i r e saw has f u l f i l l e d i t s design requirements  and has become a r e l i a b l e t o o l f o r the  r o u t i n e p r e p a r a t i o n o f c r y s t a l s and specimens.  I t has  a l s o been used f o r c u t t i n g o t h e r m a t e r i a l s , such as g l a s s , c e r a m i c s , and f e r r i t e s w i t h e q u a l s u c c e s s .  89  A P P E N D I X  DESIGN AND  C  CONSTRUCTION OF  A SMALL OVEN AND  TEMPERATURE  CONTROL. As m e n t i o n e d i n t h e was  constructed f o r high  heating  coil  was  t e x t , a simple  temperature measurements.  wound on  a thin walled  number f o r t y Hytempco a l l o y w i r e This  a l l o y was  serve  as b o t h  coefficient  important a heater  of r e s i s t i v i t y ,  because the  and  and  t h e w h o l e a s s e m b l y was  muffle  arm AC  then  s u c h o v e n was  of a r e s i s t i v e - i n d u c t i v e source.  The  The using  sensing  bridge  to  device.  a ceramic  e n c a s e d i n an  paste  asbestos insulating field.  oven w i n d i n g formed f e d from a s i x t y  a d j u s t a b l e r e s i s t a n c e arm  with  latter  w i n d i n g was  tube w i t h  the  glass.  combined  r e q u i r i n g the m a g n e t i c  seen i n F i g u r e l l i ,  oven  fibre  the  encased i n a l a r g e r  f o r measurements n o t As  single  a temperature  w i n d i n g s were c e m e n t e d t o t h e  One  tube  insulated with  The  block.  brass  chosen f o r i t s h i g h r e s i s t i v i t y  h i g h temperature property being  tubular  was  one  cycle  a precision  r e s i s t a n c e box  used' t o s e t t h e r e q u i r e d o p e r a t i n g  temperat-  ure.  i n d u c t i v e arms were made by  modifica-  The  two  slight  t i o n o f t h e f i l a m e n t w i n d i n g s o f a power t r a n s f o r a e r . have a t u r n s r a t i o  o f one  wound i n o p p o s i t i o n .  The  transformer  s e r v e d as  This  output  s i g n a l was  with  a g a i n of over  h u n d r e d t o one  were, o f  high voltage winding of  a pickup  one  and  fed into  f o r the  bridge  then  into  course,  this  output  a phase-corrected  t h o u s a n d and  They  same  signal.  amplifier  a phase  120 Volts AC  DECADE RESISTANCE BOX OVEN I WINDINGS  6  O  PHASE SENSITIVE DETECTOR  O  CONTROL TRIODES  CD c  3) m  O  AMPLIFIER  3000  IsisisiQsmsumy •-^MYWuiruTuir^— 1  too  OVEN TEMPERATURE CONTROL CIRCUIT ID  o  9.  1  sensitive  detector.  d e t e c t o r was triodes  The  f e d to the  the  output voltage  i n p a r a l l e l push-pull  the  bridge  control  operation  across  the  the p r i m a r y o f w h i c h was  circuit.  the  Hence t h e r e  secondary  in series  vjas I n s e r i e s  coil  bridge  signal,  the  the h e a t i n g  and  s y s t e m r e s p o n s e was  t e m p e r a t u r e was  sensing  c o n t r o l l e d , the  o v e n t e m p e r a t u r e f o l l o w the  ure q u i c k l y . recorder  The  the  voltage  p l a c e d i n the temperature  u n i t was  closed  interior thin  ovens.  For  magnetic f i e l d ,  o v e n u s e d f o r the gradients  detected  but  oven w a l l ,  not  i n the  detected  s a m p l e s u s e d o r by  output o f a sample.  The  by  the  temperatwith  a  pen  thermocouple  o r l o n g terra  slight  axial  smaller  temperat-  recorder  sensit-  temperature oven u3ed i n  the  l a r g e r , more h e a v i l y i n s u l a t e d  c o n d u c t i v i t y measurements.  c o u l d be  cylindrical  t e m p e r a t u r e s f r o m room  short  i n the  was  a l l c o n t r i b u t e d t o make  o b s e r v e d w i t h i n the A  same  oven whose  c o n t r o l l e d heater  various  o f l e s s t h a n 0.£°C. c o u l d be  of the  t e s t e d by m o n i t o r i n g  t o a r o u n d U.$0°C, no  the  Although i t  output of a chromel-alumel  v a r i a t i o n s c o u l d be  gradient  u n i t was  excellent.  symmetry and m i n i m a l a x i a l h e a t f l o w  o f the  was  which c o n s t i t u t e d a  the h e a t i n g . c o l l r a t h e r t h a n the  ivity  with  loop. . Since  ure  this  o v e n w i n d i n g s a v a r i a b l e impedance whose v a l u e  c o n t r o l l e d by  the  from  c o n t r o l g r i d s o f a s e t o f f o u r power  o f a power t r a n s f o r m e r , with  DC  the  two  Temperature  t h e r m o c o u p l e s on  some  thermo-electric potential  ovens a l s o a t t a i n e d new  temperat-  92  u r e s r a p i d l y but w i t h no o v e r s h o o t , a l t h o u g h temperature C: decreases were s l o w e r t h a n i n c r e a s e s s i n c e the ovens had t o l o s e t h e excess heat t h r o u g h t h e i r i n s u l a t i n g j a c k e t s . The performance  o f the oven and the temperature  c o n t r o l u n i t have been q u i t e s a t i s f a c t o r y t h r o u g h of hours o f o p e r a t i o n a l use.  hundreds  B I B L I O G R A P H Y  B l a t t , F„ J . , J o u r . Phys. Chem. S o l i d s , 1,  2 6 2 , (1957)  Born, I C , Z e i t s . f . P h y s l k , 3_8, 8 0 3 , (1926) ft  B r i l l o u i n , L., Wave P r o p a g a t i o n i n P e r i o d i c 'Structures, McGraw-Hill,  (l9i|-6)  Brooks, H., Phys.. Rev. 83_, 8 7 9 , (1951) Conwell, E . M., Phys. Rev. 8 8 , 1 3 7 9 , (1952)  Conwell, E . M. P r o c . I n s t . Radio Engrs., l ^ ,  Conwell, E . M. and Weisskopf,  1281, (1958)  V. F., Phys. Rev., 2Z» 3 8 8 ,  (1950)  Cox,  C D . , Ph. D. T h e s i s , U.B.C., (1959)  Debye.P.P., and Conwell S. M., Phys. Rev. 9 2 . , 6 9 3 , (195U)  Dekker, A. J . , S o l i d S t a t e P h y s i c s , P r e n t i c e - H a l l , I n c . , (1957) *  Dexter, R. N„, Z e i g e r , H. J . and Lax, B., Phys. Rev. IOI4., 637,  (1956)  9k  D i n g l e , R. B., P h i l . Mag.,  1^6, 831, (1955)  Dresselhaus, G,, K i p , A. F „ , and K i t t e l , C , 2*>  Phys. Rev.  368, (1955)  Edelman, S. and Hancock, R. D., Rev. S c i . I n s t . , 2£, 1082, (1956)  Evans, D  0  M., P r o c . Phys. 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