Infrared-Absorption Studies Of Localized Vibrational-Modes And Lattice-Modes In Germanium - Silicon Alloys

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Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 .... 74-17,333 C O S A N D , Albert Edm und, 1942- IN FR A R ED A B S O R P T IO N STUDIES O F LOCALIZED VIBRATIONAL M O D E S A N D LATTICE M O D E S IN G ERM ANIUM -SILICO N ALLO YS. University of Southern California, Ph.D., 1974 Physics, solid state I University Microfilms, A X E R O X Com pany, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. INFRARED ABSORPTION STUDIES O F LOCALIZED VIBRATIONAL MODES AND L A T TIC E MODES IN GERM ANIUM -SILICON ALLOYS by A lb e rt Edm und Cosand A D is s e rta tio n P re s e n te d to the FA CU LTY O F THE GRADUATE SCHOOL UNIVERSITY O F SOUTHERN CALIFORNIA In P a r tia l F u lfillm en t of the R e q u ire m e n ts fo r the D egree DOCTOR O F PHILOSOPHY E le c tric a l E ngineering Ja n u a ry 1974 UNIVERSITY O F SOUTHERN CALIFORNIA THE GRADUATE SCHOOL. UNIVERSITY PARK LOS ANGELES, CALIFORNIA 9 0 0 0 7 This dissertation, written by _ _ _ _ A lb e rt Edm und C osand under the direction of h .D is se rta tio n Com mittee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of requirements of the degree of D O C T O R OF P H IL O S O P H Y DISSERTATION COMMITTEE Chairman ACKNOW LEDGEM ENT The a u th o r is g re a tly in d eb ted to s e v e r a l people fo r th e ir contributions to th is w o rk . I w ish to p a rtic u la rly thank P ro fe s s o r W .G . S p itze r fo r su g g estin g th is p ro je c t, and fo r guidance and en co u rag em en t in the c o u rse of th is w o rk . P r o f e s s o r J . Sm it contributed g re a tly to m y u n d e rsta n d in g in som e a re a s through c ritic a l d isc u ssio n s, and P ro f e s s o r J .M . W helan w as often of i c o n sid erab le help w ith e x p e rim e n ta l p ro b le m s. A ll of th ese a re to be thanked fo r c r itic a l rea d in g of th is m a n u s c rip t, an d fo r the g re a t : am ount of p atience they have show n. I m u st a ls o thank W orth A llred , • w ithout w hose advice and a s s is ta n c e it m ig h t have b een im p o ssib le to i ; grow the c ry s ta ls u se d in th is study. The a u th o r a lso is g ra te fu l fo r fin a n c ia l su p p o rt fro m the ; N ational Science F oundation in the f o rm of a G rad u ate T ra in e e sh ip , I and to the U. S. C. E le c tro n ic S c ien c es D iv isio n fo r a re s e a rc h j a s s is ta n ts h ip . ii Table of C ontents A cknow ledgem ent L is t of Illu s tra tio n s L is t of T ables I. Introduction and B ackground 1 A. Introduction 1 B. B ackground 3 1. Ge S i1 alloys X J L “ X 2. V ibrational m odes of im p e rfe c t c ry s ta ls 3. L attice dynam ics a . P e rfe c t la ttic e b. Im p e rfe c t la ttic e i. Isotopic im p u rity ii. M ore g e n eral m odels 4. P re v io u s o b se rv atio n s of lo c a liz e d v ib r a tional m odes II. A pproxim ate calcu latio n m ethods fo r lo c a liz e d m o d es 32 A. P rev io u s w ork 32 B. A "new" m ethod 38 1. Sim ple lin e a r chain 2. D iso rd e red lin e a r chain 3. E xtension to th re e d im en sio n s a. F o rm u latio n b. E xam ples i. F o rc e constant ch an g es fo r the B lo ca l m ode ii. Iso topic su b stitu tio n a . C e n tra l-fo rc e m o d e l b. In clu sio n of n o n -c e n tra l fo rc e s 4. C om parison of the Ite ra te d M a trix M u ltip li cation M ethod w ith O ther M ethods in C. C ondition fo r the o c c u rre n c e of a lo c a liz e d m ode th e th re sh o ld value fo r the m a s s d efect E x p e rim e n ta l p ro ced u re A. C ry sta l grow th 1. S i-ric h Ge Si. x 1-x 2. G e -ric h Ge Si. x 1 -x 3. G erm anium B. C om pensation by lith iu m diffusion 1. A lloying of L ith iu m 2. Ge S i. alloys X X “ X 3. G erm anium C. R e sistiv ity and type m e a su re m e n t D. D e term in atio n of com position E . P o lish in g F . In fra re d m e a su re m e n ts In fra re d ab so rp tio n of la ttic e m o d es and the silic o n lo c a l m ode in Ge S i, x 1-x A. V ib ratio n al a b so rp tio n of d e fe cts in th e diam ond type la ttic e B.. P re v io u s e x p e rim e n ta l o b se rv a tio n s C. E x p e rim e n ta l re s u lts and d isc u ssio n 1. Ge S i. w here 0 < x < 0. 12 x 1-x — — 2. Ge S i. w here 0 .8 8 < x < 1 x 1-x — — L o c a liz e d v ib ratio n a l m o d es of B -L i p a irs in S i- r ic h G e-S i a llo y s A. B ackground B. E x p e rim e n ta l C. R e su lts D. D isc u ssio n 155 E . M odels 156 F . L in esh ap e calculation 159 G. B -L i p airin g d istan c e 169 H. S um m ary and conclusions 172 VI. L o ca liz ed v ib ratio n a l m odes of p h o sp h o ru s and of Li p a ire d w ith Ga in germ anium 174 VII. S u m m ary 185 A. C alculation of lo calized v ib ra tio n a l m ode fre q u e n c ie s 185 B. In fra re d ab so rp tio n of la ttic e m o d es and the Si lo ca l m ode 185 C. L o calized v ib ratio n al m odes of B -L i p a ir s in S i-ric h Ge Si. alloys 186 X X ” X D. L o calized v ib ratio n a l m odes of P and L i in Ge 187 R E FE R E N C E S 189 A PPE N D IX 195 v LIST OF ILLUSTRATIONS 1. V ib ratio n al sp e c tra of Ge and Si. 6 2. Sim ple o n e-d im en sio n al m odels fo r la ttic e v ib ra tio n a l s p e c tra . 13 3. P ro p e rtie s of a local m ode in silico n fo r a n iso to p ic im p u rity m odel, as calculated by D aw ber an d E llio tt. 21 4. A p proxim ate d istrib u tio n of lo cal m ode fre q u e n c ie s for lo c a l m o d es of an im p urity of m a s s aM in a random chain of m a s s e s M and 2M. 53 5. C alculated lo cal m ode frequency fo r b o ro n in silic o n | v s . re la tiv e fo rce constant cp^ g i ^ S i g i • ^ : 6. A n atom in a diam ond la ttic e w ith its fo u r n e a r e s t n e ig h b o rs, and th e ir rela tio n sh ip to th e cu b ic a x es of th e c ry s ta l. 64 I 7. A pproxim ations to the phonon d en sity of s ta te s fo r a re a l c ry s ta l. 74 I 8. e . v s . (cu - to )/ui . 76 m in m ax m m m ax j j 9. P h a se d iag ra m fo r the G e-Si sy ste m . 81 10. S teady s ta te d iffu sio n -lim ited grow th fro m a m e lt containing an im p u rity . 84 11. R e sista n c e -h e a te d C zo ch ralsk i grow th s y s te m fo r Ge c ry s ta ls . 92 12. Q u artz c ru c ib le a rra n g e m e n t fo r rap id ly q u en ch in g a sa m p le a fte r Li diffusion. 101 vi L 13. O ptical layout of the double b e a m fa r in fra re d sp e c tro m e te r. 108 14. A b so rp tio n coefficient v s. w avenum ber for S i-ric h Ge S i. alloys at room te m p e ra tu re and liquid X X * * x n itro g e n te m p e ra tu re . 123 J 15. A b so rp tio n coefficient v s. w avenum ber for room te m p e ra tu re Si, and fo r Ge 88 a t room te m p e ra tu re and liq u id n itro g en te m p e ra tu re ; com pared to a b so rp tio n p re d ic te d by D aw ber and E llio tt for a heavy im p u rity (As) in Si. 125 16. cp(z) as c alcu late d by B ellom onte and P ry c e , and fo r Ge in Si. 130 17. O ptical a b so rp tio n co efficien ts p red ic te d by D aw ber and 19 -3 E llio tt fo r 10 cm im p u ritie s of v a rio u s e le m e n ts in group III and group V. 132 18. A b so rp tio n co efficien ts v s. w avenum ber fo r n e u tro n ir r a d ia te d Si and Ge gg • 134 19. A b so rp tio n co efficien t v s. w avenum ber for Ge . 88S i. 12 a t liquid He te m p e ra tu re , liquid ^ te m p e ra tu re , and ro o m te m p e ra tu re , and fo r Ge at liq u id te m p e ra tu re . 135 20. A b so rp tio n co efficien t v s. w avenum ber fo r p u re Ge and Ge ggSi a t room te m p e ra tu re . A rro w s in d ic a te d e n s ity -o f-s ta te s peaks fo r G e. 136 21. A b so rp tio n co efficien t v s. w avenum ber at liquid ^ te m p e ra tu re fo r Ge, Ge qgSi Qg , and Ge ggSi ^ » vii I 22. A b so rp tio n coefficient v s. w avenum ber fo r the Si lo c a l m ode in G e -ric h Ge S i. alloys, a. O bserved X 1 " X a b so rp tio n , b. L ocal m ode band w ith background su b tra c te d . 23. C o n cen tratio n dependence of the in te g ra te d a b so rp tio n s tre n g th fo r the Si lo cal m ode band. 24. A b so rp tio n co efficien t v s. w avenum ber fo r b o ro n doped lith iu m com pensated sam p les at 80°K . A: Si, n o rm a l 7 7 b o ro n , L i. B: Be Q^Si , n o rm a l b o ro n , L i. C: G e. 12S i. 88 ’ 1C > B ’ 6 U - 10 7 25. D iffe re n tia l ab so rp tio n betw een Ge ^ S i gg : B , L i and undoped Ge ^ S i ^ a t liquid He te m p e ra tu re . 26. An ato m in a diam ond la ttic e with its four f ir s t n e ig h b o rs, tw elve second n e ig h b o rs, and a te tra h e d ra l in te r s titia l s ite . 27. L o ca liz ed v ib ra tio n a l m ode absorption of b o ro n in B ” - L i+ p a irs in Ge . 12S i. 88 ‘ 28. A b so rp tio n co efficien t v s. w avenum ber a t liquid n itro g e n te m p e ra tu re fo r doped and pure g e rm a n iu m sa m p le s. 139 140 152 154 157 166 178 v iii LIST O F TABLES C o m p a riso n of lo ca l m ode freq u en cies calculated by tw o d iffe re n t m ethods w ith the e x p erim en tal v a lu e s . In itia l tr ia l v e c to r, and n o rm aliz e d v e c to rs re su ltin g fro m ite ra te d m a trix m u ltip licatio n , for an im p u rity m a s s of 0.1 of the h o st m a s s . L o c a l m odes of a lig h t im p u rity of m a ss a M i n a ra n d o m chain of m a s s M and 2M. D e s c rip tio n of s ite s in a diam ond la ttic e . F re q u e n c y and d isp lac em e n ts fo r th e lo ca l m ode of in Si v s. th e B -S i fo rc e constant. F re q u e n c ie s and d isp lacem en ts fo r local m odes in a diam o n d la ttic e fo r a m odel w ith only c e n tra l fo rc e s . F re q u e n c ie s and d isp lac em e n ts fo r local m odes in a d iam o n d la ttic e fo r a m odel w ith both c e n tra l and no n c e n tr a l fo rc e s . R ecom m ended filte rin g fo r the P e rk in -E lm e r m odel 301 f a r in fra re d sp e c tro m e te r. L o c a liz e d m ode fre q u e n c ie s at 80°K for boron and lith iu m in Si and G e-S i allo y s. D ire c tio n of m o tio n of th e n eig h b o rs of an im p u rity fo r arlO cal m ode v ib ra tio n of the im p u rity along each of th e cubic a x e s. ix 37 48 51 56 63 66 69 109 155 181 CH A PTER I INTRODUCTION AND BACKGROUND A . In tro d u ctio n ; The p re s e n c e of defects in a c ry s ta l m ay m odify the v ib r a - j ; tio n a l s p e c tru m a t the c ry s ta l. Some d efects m ay in tro d u c e I i lo c a liz e d v ib ra tio n a l m odes o utside the bands of fre q u e n c ie s allo w ed I in the u n p e rtu rb e d c ry s ta l; d efects m ay a lso p ro d u ce in -b a n d I ' j re so n a n c e s w hich d ra s tic a lly a lte r the fo rm of som e of the band ! m o d es. The study of th e se m odes can be u se d to obtain in fo rm a tio n j i j on the n a tu re of the d e fe c ts, in te ra c tio n s betw een d e fe c ts, and som e ! of th e p ro p e rtie s of the h o st la ttic e . In p a rtic u la r, sin ce the fre q u e n c ie s of lo c a liz e d v ib ra tio n a l m odes of a d efect a r e v e ry se n - I ! sitiv e to the im m e d ia te en v iro n m en t of the d efect, the stu d y of lo c a liz e d m o d es can be u se d to probe s tru c tu re on a n a to m ic sc a le a ro u n d the d efect. I T his d is s e rta tio n p re s e n ts re s u lts of in fra r e d a b so rp tio n stu d ie s of lo c a liz e d v ib ra tio n a l m odes of lig h t im p u ritie r, and stu d ie s of d e fe ct-in d u ce d in f ra r e d a b so rp tio n of in -b a n d m o d e s. T his c h a p te r c o v e rs g e n e ra l background re le v a n t to the e x p e r im e n ta l w o rk p re s e n te d in C h a p te rs IV, V, and VI. F i r s t , so m e of the p re v io u s w o rk done on Ge Si a llo y s is rev iew ed . The n ext X x " ■ X se c tio n rev iew s the th e o ry of v ib ra tio n a l m o d es of a la ttic e w ith d e fe c ts . The fin a l se c tio n is a b rie f rev iew of p rev io u s o b se rv a tio n s of lo c a liz e d m o d es. C h a p te r II: A pproxim ate c a lcu latio n m ethods fo r lo c a liz e d I m o d e s. P re v io u s a p p ro x im ate calcu latio n s of lo c a l m ode fre q u e n - 1 c ie s and e ig e n v e c to rs a re review ed. A new technique th a t is p a rtic u la rly ap p licab le to com plex d efects is p re s e n te d , and s e v e ra l e x am p le s a re given. The la s t se c tio n of th is c h a p te r c o n s id e rs the re la tio n sh ip betw een the la ttic e d e n s ity -o f-s ta te s n e a r u u and ■ m ax i the m inim um m a s s defect re q u ire d to give a lo c a l m ode. C h ap ter III: E x p e rim e n ta l T echniques. T echniques of s e m i- : co n d u cto r c ry s ta l grow th a re d isc u sse d w ith p a r tic u la r e m p h a sis j on allo y c ry s ta l grow th. D etails a re given of m ethods fo r I ; m e a su rin g co m p o sitio n and hom ogeniety of the c r y s ta ls , fo r sam p le j p re p a ra tio n , and fo r in fra re d a b so rp tio n m e a s u re m e n ts . I C h a p te r IV: In fra re d a b so rp tio n of la ttic e m odes and the ! s ilic o n lo cal m ode in Ge Si a llo y s. In -b an d a b so rp tio n due to a X X “X sin g le-p h o n o n p ro c e s s is o b se rv e d in S i-ric h a llo y s ( x < . 12 ). In G e -ric h allo y s ( x > . 88 ) in fra re d a b so rp tio n of the Si lo c a l m ode i j | is o b se rv e d a s w ell a s in -b an d a b so rp tio n . It is not c le a r fro m the te m p e ra tu re dependence w h eth er the in -b an d a b so rp tio n in the G e- ric h a llo y s is a one-phonon p ro c e ss o r an en h an cem en t of n o rm a l tw o-phonon p ro c e s s e s . C h a p te r V: L o c a liz e d v ib ra tio n a l m odes of b o ro n -lith iu m p a ir s in S i-ric h Ge S i. a llo y s. The e ffect of the a llo y c o m p o sitio n X x -X ! _ -f J on the b o ro n lo c a l m odes of B -L i p a irs in Ge S i, is o b se rv e d r x 1 -x in the ran g e of 0 < x < . 12 . A m odel is p ro p o se d to fit the j o b se rv e d s tru c tu re . The re s u lts su g g est ra n d o m d istrib u tio n of the I ! Ge in the la ttic e . New in fo rm atio n on the p a irin g of d istan c e of : L i and B is obtained. ■ I ‘ f C h a p te r VI: L o calized v ib ra tio n a l m o d es of phosphorus and lith iu m in g erm an iu m . The lo ca liz ed v ib ra tio n a l m odes of iso la te d p h o sp h o ru s and of L i p a ire d w ith Ga in Ge a r e re p o rte d . A nother i band, te n ta tiv e ly a ttrib u te d to the lo cal m ode of P p a ire d w ith Ga, | is a ls o o b se rv e d . C h a p te r VII is a su m m ary of the e n tire d is s e rta tio n . B . B ackground i 1. Ge Si A lloys X X - X j , ■ C ry s ta ls of g erm an iu m and silic o n both have the diam ond i ' | s tr u c tu r e , w ith n e a rly the sam e la ttic e p a ra m e te r; the la ttic e j p a ra m e te r of Ge, 5. 658 X, is about 4% l a r g e r th a n th a t of Si, 5.431 X. The two a re m iscib le in a ll p ro p o rtio n s . The allo y s have the sa m e b a sic la ttic e but any site m ay be o c c u p ie d by e ith e r a Ge o r a Si ato m . No long range o rd e rin g of Ge a n d Si h a s b een found in Ge S i, a llo y s. T h ere is little d ire c t e v id e n c e e ith e r fo r o r x 1 -x 1 a g a in st v e ry sh o rt range o rd e rin g . In s e v e r a l in s ta n c e s , e x p e ri m e n ta l d ata involving lattic e v ib ratio n s in so m e m a n n e r h a s b een in te rp re te d a s im plying sh o rt-ra n g e o rd e r, but m o re re c e n t j th e o re tic a l w o rk on v ib ratio n a l s p e c tra of d is o rd e re d allo y s has ; show n th at the re s u lts do not re q u ire any s h o rt-ra n g e o rd e rin g . ; The study of the boron lo ca l m odes in the S i- r ic h ( x < . 12 ) Ge Si a llo y s w as u n d ertak en w ith the o b jec tiv e of applying the X X - X ; in fo rm a tio n on the lo ca l m ode s p e c tra to the p ro b le m of m e a su rin g j the d istrib u tio n of Ge and Si in the alloy. The G e-S i allo y sy ste m w as f ir s t stu d ied by S to h r and K lem m (1939) who show ed that Ge and Si fo rm e d so lid so lu tio n s throughout the ran g e and d e te rm in e d the phase d ia g ra m . T h e ir phase d ia g ra m ! h a s been a c c e p te d u n til rec en tly , w hen S te in in g e r (1970) published a p h ase d ia g ra m c alcu late d fro m th erm o d y n am ic a rg u m e n ts w hich ! su g g e sts th a t the e x p e rim e n ta l d e te rm in a tio n of the so lid u s is not i a c c u ra te . B oth phase d iag ra m s a re show n in F ig . 9, Chap. Ill; the effectiv e se g re g a tio n co efficien ts th a t w ere o b se rv e d w hile | grow ing the Ge Si c ry s ta ls a re m o re n e a rly in a g re e m e n t w ith i X! X * • X j i the c a lc u la te d phase d iagram . | | M any of the p ro p e rtie s of th e a llo y s v e ry sm o o th ly and I I ! m o n o to n ically w ith com position fro m th o se of p u re Ge to those of i ! j p u re Si. T h ese include p u rely a v e ra g e p r o p e rtie s of the c ry s ta ls su ch a s d e n sity and la ttic e p a ra m e te r, a s w ell a s e le c tr ic a l p ro p e rtie s su ch as band gap and io n iz atio n e n e rg ie s of shallow d o n o rs o r a c c e p to rs . The conduction band m in im a in Si a re along <100> a x es; in Ge they a re at the zone edge in a < 111 > d ire c tio n . The tra n s itio n fro m G e-lik e to S i-lik e co n d u ctio n bands o c c u rs at i about Ge , -Si oc , producing a change in slo p e in the re la tio n sh ip | • 15 • o5 I b etw een band gap and c o m p o sitio n ( Jo h n so n and C h ristia n 1954; j B ra u n stein , M oore and H e rm a n 1957 ). T he tra n s itio n in the j conduction bands is a lso seen in m a g n e to -re s is ta n c e m e a su re m e n ts < ( G licksm an 1955 ). The s tru c tu re of the v a le n c e band h as been | in v e stig a te d by in fra re d a b so rp tio n stu d ie s in p -ty p e alloy i sp e cim en s ( B ra u n stein 1963a ). The s p in -o rb it sp littin g is se en to I v e ry lin e a rly w ith com position, and th e h o le effectiv e m a s s e s v a ry i ! sm oothly. j The v ib ratio n a l s p e c tra of Ge an d Si a re quite s im ila r; the i fre q u e n c ie s of m o st of th e m o d es s c a le a p p ro x im a te ly as j 1 ( M0 ./M _ ) , as is show n in F ig . 1 . The v ib ra tio n a l s p e c tra of i O X G6 | Ge^Si^ ^ allo y s a re q u a lita tiv e ly d iffe re n t, sin ce the d is o rd e r in the | allo y s h as a pronounced e ffect. M any of th e m odes becom e sp a tia lly lo c a liz e d o r q u a si-lo c a liz e d . The p eak s in the d e n s ity -o f-s ta te s ten d to be at freq u e n cie s m o re n e a rly c h a r a c te r is tic of Ge o r Si, . ra th e r than a t an a v e ra g e freq u en cy ( T a y lo r 1967 ). A lloys n e a r the m iddle o f the ran g e hav e la rg e n u m b ers of m odes both n e a r G e-lik e fre q u e n c ie s and S i-lik e fre q u e n c ie s. Such fe a tu re s a re c h a ra c te ris tic of a la rg e n u m b e r of m ix ed c ry s ta l sy s te m s ( Chang and M itra 1968; L u co v sk y , B ro d sk y and B u rste in 1968 ), but u n til it was rec o g n ize d th a t c h a r a c te r is tic freq u e n cie s (/o'* H i ) V £ (a) Si C oot) 1 0— I —r- ~i i i r - i - i —i ' - - - . / - - / ------Tf\ " _i—1 _ _ i—i . i .1 i . P / P . (b) & e C>oo) H.O 9,0 M.O U.O F r e q u e n c y (/& ' * H f ) z 10 .0 F ig u re 1. V ibrational sp e c tra o f Ge an d Si. a , b: D isp ersio n cu rv e s along a (100) d ire c tio n fo r Si (B rockhouse 1959) and Ge (B rockhouse 1958). c, d: Phonon d e n sity -o f-sta te s fo r Si and Ge (Dolling 1966). of both Ge and Si should be p re s e n t, s e v e r a l e x p e rim e n ts th at a re se n sitiv e to the phonon s p e c tra w e re in te r p r e te d as indicating s h o rt-ra n g e o rd erin g in the a llo y s. T h ese include m e a su re m e n ts ; of p h o n o n -assiste d o p tic al tra n s itio n s ( B ra u n ste in , M oore and H e rm a n 1957 ), p h o n o n -a ssiste d e le c tr o n tu n n elin g a c ro s s p -n ju n ctio n s ( Logan, R ow ell and T ru m b o re 1964 ), and tw o-phonon ' la ttic e ab so rp tio n ( B ra u n ste in 1963b ). S ubsequent th e o re tic a l w o rk h a s shown th at no o rd e rin g n eed be in v o lv ed to ex p lain the j o b se rv e d re s u lts ( T ay lo r 1967 ). As a re s u lt of the effect of d is o rd e r on the la ttic e v ib ra tio n s , : Ge Si. allo y s have a f a irly low th e rm a l co nductivity. T h is, along X X "X i w ith a rea so n a b ly high th e rm o e le c tric p o w er and high m eltin g j te m p e ra tu re , h as m ade allo y s in th e c o m p o sitio n ran g e 0. 2 < x < 0 .4 i j u se fu l fo r th e rm o e le c tric g e n e ra to rs fo r p ro d u cin g e le c tr ic a l pow er | ( D ism ukes e t. al. 1964 ). Ge Si a llo y s a r e a lso u se fu l fo r I " 1 3C X ■ * X in fra r e d d e te c to rs. The p rin c ip a l u tility of th e allo y s is th a t the | band gap and the io n izatio n e n e rg ie s of im p u ritie s v a ry sm oothly | i w ith co m position so th a t the w av elen g th of p e a k re sp o n se can be v a rie d by varying the alloy c o m p o sitio n ( M o rto n , Schultz and H ard y 1959 ). 2. V ib ratio n al M odes of Im p e rfe c t C ry sta ls The n o rm al m odes of v ib ra tio n of a p e rfe c tly p e rio d ic c r y s ta l a r e plane w aves; the m odes a r e u n ifo rm in am p litu d e fro m one u n it 8 1 c e ll to a n o th e r, v a ry in g only in p h a se . H ow ever, a ll r e a l c ry s ta ls : have d efects; th ese m ay be la ttic e d efects such as v a c a n c ie s, d islo c a tio n s, etc., o r im p u ritie s , e ith e r in te r s titia l o r s u b s ti tu tio n a l. A defect, w hich b re a k s down the p e rio d ic ity , m ay a lte r the fo rm of the v ib ra tio n a l m o d es, m aking th e m n o n -u n ifo rm in a m p litu d e. Since a ll of the u n p e rtu rb e d m odes of an N -a to m p e rfe c t c r y s ta l have only 1/N of th e ir e n e rg y a t any given s ite , it would : se e m rea so n a b le th a t a defect involving n s ite s w ould have an I e ffe c t on the o rd e r of n /N on the p ro p e rtie s of any one m ode. T his j I : is not n e c e s s a rily tru e ; e x tra o rd in a ry v ib ra tio n a l m odes m ay be | p ro d u ced . T hese m aybe lo c a liz e d m o d e s, w hich have a freq u en cy i | o u tsid e the allow ed bands of fre q u e n c ie s of the n o rm a l la ttic e and ; an am plitude decaying ex p o n en tially w ith d istan c e fro m the defect. I i ! A lso , th e re m ay be m odified in -b a n d m odes w hich have n e a rly the t | sam e freq u e n cie s a s in the u n p e rtu rb e d c ry s ta l, but the am plitude j ■ a t the defect site is an o m alo u sly la rg e fo r th e se m odes in som e range j of fre q u e n c ie s; the d efect am p litu d e v s. freq u e n cy in th is range is I { c h a ra c te riz e d by a re s o n a n c e -lik e peak. T his b e h a v io r is g e n e ra lly r e f e r r e d to a s an in -b a n d re so n a n c e o r a re so n a n c e m ode; the la tte r nam e is le s s a p p ro p ria te sin c e m an y m o d es of the la ttic e a re involved. Both types of e x tra o rd in a ry la ttic e v ib ra tio n s , the lo c a liz e d !m o d es and the in -b a n d re s o n a n c e s , a re of c o n s id e ra b le in te r e s t. In I both c a s e s , the fre q u e n c ie s a re quite se n sitiv e to th e im m ed iate I en v iro n m en t of the d efect. The lo c a l m odes a re p a rtic u la rly u se fu l ! sin ce th e ir fre q u e n c ie s can be c a lc u la te d w ith a f a i r d eg ree of I | a c c u ra c y and c an e a s ily be m e a s u re d e x p e rim e n ta lly . i T hus, the stu d y of lo c a l m odes can be u s e d to probe sy m m e try i • • i of d e fe c ts, p a irin g in te ra c tio n s betw een im p u ritie s , and s im ila r | e ffe c ts. In C h a p te r IV the stu d y of lo c a liz e d m o d es is ap p lied to the p ro b le m of d e te rm in in g o rd e rin g in G e-S i a llo y s. T he in -b an d i ! re so n a n c e s a re of c o n sid e ra b le th e o re tic a l in te r e s t, since c a lc u la te d !fre q u e n c ie s of in -b a n d re so n a n c e s a r e v e ry s e n s itiv e to the d e ta ils | j of the m odels u s e d fo r the la ttic e and fo r the d e fe c t. C o m p ariso n i | betw een th eo ry and e x p e rim e n t can th en p ro v id e a good te s t of the ;th e o ry . In m o s t of the c a s e s involving s e m ic o n d u c to rs , the th e o ry i. does not ag ree w ith the e x p e rim e n t. | L a ttic e D ynam ics T h is se ctio n in clu d es a d isc u ssio n of la ttic e d ynam ics of p e rfe c t c ry s ta ls a s a g e n e ra l b ack g ro u n d an d to define the notation to be u se d , then d is c u s s e s th e g e n e ra l c a se of a la ttic e w ith d e fe cts, and v a rio u s a p p ro x im atio n sc h e m e s fo r c a lc u la tin g lo c a liz e d m ode fre q u e n c ie s. a . P e rfe c t L a ttic e In the h a rm o n ic a p p ro x im atio n , th e c la s s ic a l equations of 10 m otion of an atom in a p e rfe ct c ry s ta l of N u n it c e lls w ith s atom s ; p e r c ell can be w ritte n W . x ' u ' * ' ) = 0 (1) The u ( ! ) a re d isp lacem en ts of a to m s fro m e q u ilib riu m ; th e index u, H ! , ru nning fro m 1 to N, la b e ls the c e ll, K la b e ls one of the s site s in the c e ll, and a , e ith e r x, y, o r z, is a c a rte s ia n com ponent of the ; d isp lacem en t. The $ / ( ! , ! * ) a r e h a rm o n ic fo rce co n stan ts o c , h; p, h ! coupling atom s w ith eq u ilib riu m p o sitio n s and in th e p e rfe c t ■4 *4 ! c ry s ta l the fo rc e constants depend o n ly on (R^~ a n ^ n° t on th e individual values of JL and JL' . T h ese equations a re a s e t of 3sN coupled equ atio n s; th ey can j be decoupled into 3sN independent h a rm o n ic o s c illa to r equations by ; tra n sfo rm in g fro m the co o rd in ates d e s c rib in g individual ato m ic I d isp lacem en ts into th e n o rm al c o o rd in a te s o f th e c ry s ta l. F o r a i p e rfe c tly p erio d ic c ry s ta l, w ith p e rio d ic b o u n d a ry co n d itio n s, the j ! tra n sfo rm a tio n is quite sim p le, sin c e th e n o rm a l m odes hav e the i " ! . . j fo rm of planet- w av es. The 3sN e q u atio n s a r e tra n s fo rm e d into N j se ts of 3s equations by the su b stitu tio n of i I • u(»,xU ) = dj (’ > (2) j*q w here d.(q) is a n o rm al c o o rd in ate w h ich s p e c ifie s the e x cita tio n of 1 a n o rm a l m ode, and X«u(q» is th e s p a tia l am p litu d e of the O C H n o rm a l m ode 11 ^ ( i i | = |NM |ir i «*, (q )« 'l|q ' S<1 < 3> ; w hich is n o rm a liz e d so th at ! n » y * > i 2 = 1 < 4 ) C C .K J & ! The re s u ltin g se t of 3s e q u atio n s, fo r e ac h of the N v alu es of q in ! -♦ i the f ir s t B rillo u in zone, can be so lv ed fo r the fre q u e n c ie s uu.(q) and J O C K ; the c o rre sp o n d in g e ig e n v e c to rs C T . (q) . The index j, running fro m J j 1 to 3 s, sp e c ifie s a b ra n c h of the phonon sp e c tru m . T he freq u en cy | of th re e of th e b ra n c h e s is z e ro f o r J q J = 0; th e se a r e the acoustic i i j m o d es. T he o th er 3 s -3 b ra n c h e s have n o n -z e ro fre q u e n c ie s when | q | = 0; th e se are th e optic m o d es. In d ire c tio n s of high sy m - j m e try in q -sp a c e the b ra n c h e s m a y be fu rth e r d istin g u ish e d as being I (xh j tr a n s v e rs e o r longitudinal m o d e s. The C T . (q) sp e c ifie s the atom ic l J j — ♦ j d isp la c e m e n ts w ithin a unit c e ll f o r a m ode sp e c ifie d by j and q . Gi fa) (‘ l) = b l i ' fa) a ,h J 3 T h ese e ig e n v ec to rs a r e o rth o g o n al and m ay be n o rm a liz e d so th at JJ ? , i T , 5 , r ,5 = i op v < 6> 3 F o r any given m ode th e d isp la c e m e n ts u ( A ) in two u n it c e lls w ith O C H # p o sitio n s R and R / d iffe r only b y a p hase fa c to r, e*^ J L A W hen the n u m b er of u n it c e lls , N, is v e ry la rg e , as is the case fo r any m a c ro sc o p ic c ry s ta l, the allo w ed v a lu e s of q a re so c lo se ly ! - 12 i sp a ce d th a t q c a n be c o n s id e re d a continuous v a ria b le fo r m any p u rp o se s. The sim p le st exam ple of la ttic e v ib ra tio n s is given by the i m onatom ic lin e a r chain w hich h as only one d eg ree of free d o m p e r i | ato m . The m odes a re plane w av es e*^ ^X -U } fa)*] . eig en v ecto r i fo r e a c h m ode is sim p ly 1. T he re la tio n b etw een freq u e n cy and wave v e c to r is uj(q) = (4 $ /M )^ sin (q a /2 ), w h e re $ is the fo rce constant : and a is the la ttic e spacing. A n o th er sim p le exam ple is the diatom ic i lin e a r chain. Since th is la ttic e h a s two a to m s p e r u n it c e ll the : phonon sp e c tru m h as both a n optic an d an a c o u stic b ra n c h . T h ere [ I a re two tw o-com ponent e ig e n v e c to rs fo r e a c h value of q, specifying j ! j the am p litu d es of the two a to m s . T he r e s u lts a re su m m a riz e d in j F ig . 2. ' T h e re a re no sy ste m s in n a tu re quite as sim p le as lin e a r i i ! c h a in s. The v ib ra tio n a l fre q u e n c ie s of r e a lis tic m o d els of re a l I ! -+ c ry s ta ls , except fo r v alu es of q fo r w hich the eq u atio n s can be sim p lifie d by the use of s y m m e try , a re found th ro u g h tedious c a l- j . c u latio n s b e st done w ith an e le c tro n ic c o m p u te r. b. Im p e rfe c t L a ttic e If th e re a r e any d efects in the c r y s ta l the tra n s la tio n a l sy m m e try of the la ttic e is rem o v e d . T he n o rm a l m odes a re not quite plane w av es, as w as the c a se w ith th e p e rfe c t c ry s ta l, so th e re is g e n e ra lly no sim ple tra n s fo rm a tio n th a t g iv es the n o rm a l c o o rd in - 13 p l - H -OK>dK>iXK>UWK>^- tn p h— 2a —► ! w, U ,-* - u 2 - * - £ c > J [2^3 (-jtr, * ] [2 f/S j [ 2 /V X /2 (b) o £~tjertv ector ( U n H ) ( *>2 ■ \K 2 ♦ w2 2 ja {o,i) (1, 0 ) 5 > - "sta * (m j ’v t ) F ig u re 2. Sim ple o n e -d im en sio n a l m o d e ls fo r la ttic e v ib ratio n a l s p e c tra . a) One dim ensional m o natom ic la ttic e m o d el, w ith only f ir s t * neighbor fo rc e s . b) D isp ersio n cu rv e (uj v s. q) f o r the m o d el in a). c) One dim ensional d iato m ic la ttic e m o d el, w ith only f ir s t neighbor fo rc e s. d) D isp ersio n c u rv e, and e ig e n v ic to rs fo r zone c e n te r and zone edge m odes, fo r the m o d e l in c). _ J • .............................. '.....14' ! a te s . The f i r s t d e ta ile d tre a tm e n t of th e v ib ra tio n a l p ro p e rtie s of i d efects in c ry s ta ls w as given by L ifsc h itz (1943 a ,b ; 1944); th e re I have been a n u m b er of m o re re c e n t p a p e rs extending th is w ork ; (M ontroll and P o tts 1955; D aw ber and E llio tt 1963 a ,b ; M aradudin 1963, 1966) (M arad u d in 's rev iew s both contain e x ten siv e b ib lio - ; g ra p h ie s). T h ese stu d ies h av e show n th a t it is in p rin c ip le p o ssib le j to calcu late the e ig e n freq u e n cies and o th e r p ro p e rtie s of a la ttic e I containing d e fe cts in te r m s of an expansion in th e n o rm a l m odes of | the p e rfe c t la ttic e ; th is n o rm a l m ode expansion is u su ally developed i : in a G re e n 's function fo rm a lis m . In g e n e ra l, such calcu latio n s re q u ire a m o re e x ten siv e know ledge th an is u su a lly a v ailab le of the i m ode fre q u e n c ie s and e ig e n v e c to rs of th e p e rfe c t la ttic e . In som e i sim p le c a se s of in te r e s t, su ch as th e sin g le iso to p ic su b stitu tio n ! | m odel, the e x p re ssio n s beco m e su fficien tly sim p le th a t useful | j re s u lts a re e a sily obtained . T his m ethod of calcu latio n is review ed I h e re ; except as noted, the d e riv a tio n is e sse n tia lly th at of D aw ber i ' , j and E llio tt (1963a). F o r th e c a se th at the d e fe ct is one o r m o re su b stitu tio n a l im p u ritie s , w hich m ay d iffe r fro m th e h o st both in m a s s and fo rc e c o n stan ts, th e equations of m o tio n (Eq. 1) fo r s ite s k , on w hich the m a s s has b een changed to M - AM and the fo rc e co n stan ts H K changed to " A §aH g K • ^ *) b ecom e p, H , I p 15 n The ato m ic d isp la c e m e n ts can be e x p re s s e d as a sum of n o rm al m o d es, as in E q. 2, so th at I (8) w h ere X (£ » ^) is th e am p litu d e of ato m ic d isp lac em e n ts in the Ct K i n o rm a l m ode, and d(f) is the n o rm a l c o o rd in a te fo r th e p e rtu rb e d ! la ttic e . The index f, w hich la b e ls th e m ode, u rn s fro m 1 to 3sN. I F o r each n o rm a l m ode th e equation of m otion is i W ' * 4 ' 4 ' ’ X b k ' V ’ 1 " * - \ - ‘M K W u Z % a K ( i , l ) + (9) ; The changes in th e equation have b een c o llected on the rig h t hand side j i ‘ • j as a d riv in g te r m fo r an o th e rw ise unchanged se t of h a rm o n ic | o s c illa to r eq u atio n s. i | C o n sid er fo r a m o m en t a sim p le h a rm o n ic o s c illa to r w ith an I e x te rn a l d riv in g fo rc e a t a freq u e n cy ou. The equation of m otion is j I 2 - to m x + kx = F(ou) (10) The re sp o n se of th e o s c illa to r is 1 F x " m 2 2 (11) ( 0 o - ( V 2 w h ere iu q = k/m. F o r the c a se of a s e t of coupled o s c illa to rs , the re sp o n se to a d riv in g te r m is s im ila r in fo rm , except th a t the » 16 p o sitio n co o rd in ate m u st be e x p re ss e d a s a su m over th e n o rm a l co o rd in a te s of the o s c illa to rs , and th e d riv in g fo rce on each n o rm a l co o rd in ate is obtained fro m a F o u rie r e x p an sio n in th e n o rm a l co o rd in a te s of the d riv in g te rm . T he G re e n 's function (re sp o n se to d e lta function ex citatio n ) fo r the la ttic e w h ich re la te s th e re sp o n se of the co o rd in ate a n in th e 4th c e ll to an e x c ita tio n of the 0H ' c o o rd in ate of the 4 7 th c e ll is ... . i > < * • (q)Sj ( q ) e g u > ( u / ) = l - r £ - J ----------- 3 -5 -5-------*------------------ (12) a k B k N(M M /)2 q, j u ). (q) - o u n * H j w hich sa tisfie s the equation -W Z M K gw {l,l")+ Z ^ l") (13) * « » T « " e x 'i ' ^ B hh The equations of m o tio n of the n o rm a l m o d es of th e p e rtu rb e d la ttic e can be solved in te rm s of th is G r e e n 's function to give X„„ (£. « = 2 , 2 m B k' I ' Y K - i m B K B K Y * Y H (14) The condition th a t so lu tio n s e x ist to the s e t o f equations is th at th e s e c u la r d e te rm in a n t be z e ro , so th a t th e eig en v alu e equation th a t gives the freq u e n cie s of the p e rtu rb e d la ttic e is p > * > W l u l 1 = 0 (15> w h e re *')+ A W ' U i ')- (16) 17 In the g e n e ra l c a s e , a c o m p lete know ledge of the eig en v ecto rs a ? * (q) and of the re la tio n sh ip betw een u ) and q fo r the p e rfe c t J la ttic e is needed to ev alu ate th is d e te rm in a n t. (1) Iso to p ic Im p u rity in th e D iam ond L a ttic e T h ere a re som e in sta n c e s w h e re th e G re e n 's function m ethod p ro v id es a p ra c tic a l m ethod to e v alu ate th e fre q u e n c ie s of a p e rtu rb e d la ttic e . Of p a rtic u la r in te r e s t in th e p re s e n t w ork is the i diam ond la ttic e . If a su b stitu tio n a l im p u rity in th is c ry s ta l is ap p ro x im ated as an iso to p ic su b stitu tio n , changing only the m a s s but | not the fo rce c o n sta n ts, only th e eq u atio n s of m o tio n for the im p u rity j s ite a re p e rtu rb e d . D aw ber and E llio tt have c o n sid e re d this p ro b lem ! in c o n sid era b le d e ta il. F o r th is c a s e , no G re e n 's functions coupling ; d iffe re n t la ttic e s ite s a r e n e c e s s a ry ; only th e g u l _ /(0) 4 0 a re C t t H j d $ K ; needed. F o r a site w ith cubic o r te tr a h e d r a l sy m m e try , g® _ /(0) 0 , Hi p» H j I 4 0 only if a = 0 ; th e d e te rm in a n t (Eq. 15) th en h a s only the th re e I diagonal te rm s , a il of w hich a r e e q u al. A lso , fo r a cubic c ry s ta l j “ ■ ' * . ; w h e re a ll atom s h av e th e sam e m a s s and eq u iv a le n t s ite s , su ch as Ge o r Si | C T ? ’ H (q) (from Eq. 5) fo r a ll m o d e s , c o n sid e r- J j S ably sim plifying the G re e n 's fu n ctio n . If a m a s s d efect p a ra m e te r M -M ' -AM is defined as e = — ^ ^ , th e eig en v alu e equation (Eq. 15) K H fo r the new la ttic e fre q u e n c ie s b e c o m e s ’ the index f w hich la b e ls th e new m odes ru n s fro m 1 to 3sN. I T he e x te n t to w hich th e m ode is lo c a liz e d can be d eterm in ed by calcu latin g th e m ode in te n s ity a t the im p u rity site . When the ; defect is a sin g le s u b s titu tio n a l im p u rity , w hich we sh all assu m e | to be su b stitu te d on a site in the c e ll a t & = 0, the n o rm aliz a tio n j condition (Eq. 4) fo r th e p e rtu rb e d la ttic e becom es 1 ' a * 1 W f ’ 0) |2 = * ' <18) i If we use the re la tio n s h ip b etw een \ (f* & ) an d Xa ' ( * > ) (IH > pH | given by su b stitu tin g E q. 12 and E q. 16 into E q. 14, we obtain an i i , ,2 j e x p re s s io n fo r J x (f, 0) | T h is c an be sim p lifie d by the use of Eq. 5 an d by the o rth o n o rm a lity of the e 1^ ^ te rm s to obtain, fo r the iso to p ic su b stitu tio n m odel, S I Xa,H(f’ 0) I = M 3sN ^ ? 2 2 " ® (19) jq W . (q) - tt)-(f) J If aj > U > , th e n th e su m m atio n in E q s. 17 and 19 m ay be m a x . . 1 9 1 j stra ig h tfo rw a rd ly c o n v e rte d to in te g ra ls o v e r U ); this sim p lifies a c tu a l n u m e ric a l e v a lu a tio n of th e se e x p re ss io n s . If V (|i) is a i 2 | d en sity of u n p e rtu rb e d s ta te s p e r u n it ran g e of = iu , n o rm alized ; so th a t J V(n) dn = 1 , Eq. 17 c an be w ritte n as l-l m M - 1 + ez' J* .f l i . = o (20) o n - Z ' (w here /Z* = u j i s the fre q u e n c y of the lo ca liz ed m ode, and X j |A im = U ) m ax is th e m ax im u m phonon freq u en cy of the u n p ertu rb ed I j la ttic e . The e x p re s s io n fo r th e am p litu d e of the im p u rity atom s im ila rly b eco m es Ix(°) I2 = ± t e2z ' 2 j ” . c J - 1 (2 1) o ( z ' - | i ) The in te g ra ls in E q . 20 an d E q. 21 w e re ev alu ated by DE fo r the Si lattic e s p e c tru m show n in F ig. 3a. The solutions a re show n in F ig . 3b fo r the fre q u e n c y an d im p u rity am plitude of the th re e -fo ld deg en erate lo c a l m o d e of a n iso to p ic im p u rity v s. the m a s s defect. If U ) is w ith in the ran g e of the ban d m odes the denom inator of the su m has m an y c lo s e ly sp a ce d z e ro e s , so th at the co n v ersio n to an in te g ra l is m o re involved. D aw ber and E llio tt (1963) a rriv e at F ig u re 3. P r o p e r tie s of a lo c a l m ode in silic o n fo r an iso to p ic im p u rity m o d e l, a s c a lc u la te d by D aw ber and E llio tt. a) H isto g ra m of th e d en sity of v ib ra tio n a l m odes p e r unit fre quency in S i, fro m w hich the in te g ra ls w e re evaluated. b) C alcu lated lo c a l m ode p ro p e rtie s : I. R elativ e freq u en cy / - 1 of lo c a liz e d m ode in Si a s a function of m a s s d efect m = (M -M )/M ; II. R e la tiv e am plitude of lo c a liz e d m ode a t the d e fe ct a to m . 21 3, O 100 200 300 400 500 (cm"1) 0.5 E 1 .0 03 0.6 0.2 6 an in te g ra l re p re s e n ta tio n by co n sid erin g in d etail the contributions 2 -4 2 of the te rm s of th e stu n fo r w hich [cu. (q) - id X jls n e a r z e ro . A J m o re in te re s tin g a p p ro a c h , developed by P ro f. J. Sm ith is as follows] T he g e n e ra l a p p ro a c h w ill be to c o n sid e r the effect of a d efect on th e su sc e p tib ility of the la ttic e . In g e n e ra l, a su sc e p tib ility S is the ra tio of a re s p o n se x of a sy s te m to a driving fo rce F. If the to ta l fo rc e on th e sy s te m c o n sists of an e x te rn ally applied fo rc e and an in te rn a l fo rc e F . , the su sc e p tib ility is S = F + F. e i The a p p a re n t su s c e p tib ility as se en by an e x te rn al o b se rv e r, h o w ev er, is S = ~ app F As w as se e n in E q. 9, the u n p e rtu rb e d la ttic e could be re p re s e n te d by a s e t of h a rm o n ic o s c illa to r s , and the change in th e equations of m o tio n in tro d u c e d by a d e fe ct is equivalent to a d riv in g te rm applied to th a t s e t of h a rm o n ic o s c illa to r s . In th e c a se of a sim p le m a s s change w ith no fo rc e c o n stan t change (the iso to p ic su b stitu tio n 2 m odel) th e d riv in g te r m in Eq. 9 b e co m es AM (j6)u> u ( Z ) . F o r t a a n th is s y s te m , th e re s p o n se to an e x te rn a l fo rc e is u = S (F + F . ) = S (F + AM o j2u) (22) 6 . 1 6 so th at th e a p p a re n t su s c e p tib ility is T he s u s c e p tib ility of a sin g le h a rm o n ic o s c illa to r of freq u en cy ii)j is ! s < ” > = H - - T T (24) I U O j - U ) ] | The n o rm a l m o d es of a c r y s ta l la ttic e containing sN atom s a re a | se t of 3sN h a rm o n ic o s c illa to r s . If the n u m b er of m odes a t fre- j Iquency U ) is pfuu.) , the s u s c e p tib ility fo r a driving fo rce on one ! | ato m of an u n p e rtu rb e d silic o n la ttic e is i o o ; c / / \ 1 1 p P(u>.) dm I * * = 3sN M i , 1 , 1 (25) O C . C U ij - U ) |T h is is , h o w e v e r, only the r e a l p a rt S/ of a com plex su sc e p tib ility S = S ' + iS " . S 7 an d S7 7 a re re la te d by the K ra m e rs-K ro n ig re la tio n » S77(u) ) u u d uo S'< “ > ■ I J 2 2 < 26> O U U j - U ) F ro m a c o m p a ris o n of e q u atio n s 25 and 26, we see th a t s ' » - Z B S S ^ <2 7 > We a re now in a p o sitio n to e v alu ate the v ib ra tio n a l p ro p e rtie s of the im p u rity fo r fre q u e n c ie s in the band m ode reg io n . One p ro p e rty of in te r e s t is the e x te n t to w hich the im p u rity m o d ifies the b an d m o d e s. A m e a s u re of th is is the m ode in te n sity 2 2 | u | a t th e im p u rity s ite re la tiv e to the in te n sity |uq | when AM = 0. If we s q u a re Eq. 23, w e obtain u Js F |2 (I - AM(u2S ')2 + (AM u ,2S " ) 2 (28) In th e u n p e rtu rb e d c a se (AM = 0) th e d en o m in ato r is one, so th a t the re la tiv e in te n sity of the m ode b eco m es u u , AM 2 ^ p 1-3535n ‘ " p -r p({ju:i ) d i u 1 U )j - ( ! ) + (29) i T his e x p re s s io n allow s the evaluation of the inband v ib ra tio n a l | in te n sity in te r m s of an in te g ra l o v e r the d en sity of s ta te s r a th e r I th an th e su m m atio n re q u ire d by E q. 17. i | The in f ra r e d a b so rp tio n of th e im p u rity can a lso be obtained i I ; fro m the a p p a re n t su sc e p tib ility . The im p u rity .is a ssu m e d to have a sin g le e le c tro n ic c h a rg e e, coupled to the in cid en t ra d ia tio n field w hich has an e le c tr ic field E(u)). The a b so rb e d pow er is P = = (eE) ulS''p (30) F ro m th is we can obtain the s p e c tra l dependence of the in f ra r e d a b so rp tio n . T he a b so rp tio n c o efficien t a(ou) is p ro p o rtio n a l to th e e x p re s s io n fo r th e a b so rb e d pow er; fro m E q s. 25, 27 and 29 w e can e v alu ate S / / ap p and obtain | a(u)) « P = (eE)2 uj(n/2M) p(uu) (31) AMui 3M sN 6MsN T he a g re e m e n t betw een th e o ry and e x p e rim e n t fo r im p u rity - : in d u c e d inband a b so rp tio n in sem ico n d u cto rs h a s not b een v e ry good ( se e c h a p te r III ). It is not su rp ris in g th at d isc re p a n c ie s should i e x is t, h o w ev er, sin ce the inband ab so rp tio n should be e x p ec te d to be I q u ite s e n s itiv e to the d e ta ils of th e coupling of the im p u rity to th e i I la ttic e , and the iso to p ic su b stitu tio n m odel u se d to d e sc rib e the d im e n sio n a l G re e n ’s fu nction calcu latio n is a t a ll sim p le . H ow ever, c a lc u la tio n s have b een done fo r m o re re a lis tic m odels of d e fe c ts , i E llio tt an d P feu ty have com puted freq u en cies of lo c a liz e d m o d es fo r ! s e v e r a l d e fe cts in silic o n , including a su b stitu tio n a l im p u rity w ith fo rc e c o n sta n t c h an g es, p a ir s of su b stitu tio n a l im p u ritie s w ith a n a d ju s ta b le fo rc e c o n sta n t b etw een them , an iso la te d in te r s titia l im p u rity , and an in te rs titia l-s u b s titu tio n a l p a ir ( L i-B ) w ith fo rc e c o n sta n t ch an g es. T h e ir m o d els a re s till o v e r-s im p lifie d . In o r d e r to keep th e d im e n sio n ality of the p ro b le m w ith in p r a c tic a l j d e fe ct a n d the sin g le v ib ra tin g change m o d el u se d to d e s c rib e th e | in fra r e d a b so rp tio n a r e p o o r enough ap p ro x im atio n s to r e a lity th a t ! it is n o t s u rp ris in g th a t the th e o ry does not fit the e x p e rim e n t. (a) M ore G en eral M odels T h e iso to p ic su b stitu tio n is the only m o d el fo r w hich th e th r e e 26 lim its , th ey have to le a v e out som e of the fo rce co n stan t changes th a t sh o u ld be included in re a lis tic m odels as fo r in sta n c e , th e im p u r ity -la ttic e fo rc e c o n stan ts of im p u rity -im p u rity p a ir s . H o w ev er, th e re s u lts of th e calcu latio n s do give som e in fo rm a tio n on th e s o r t of effects to be expected for som e c a se s of re a l p h y sic a l i n te r e s t. The a p p lica tio n of th e ir re s u lts to the c a se of P -G a p a irs i jin Ge is d isc u sse d in C h a p te r V. C alcu latio n of fre q u e n c ie s of inband m odes fo r r e a lis tic m o d els jof d e fe c ts is a re la tiv e ly m o re difficult p ro b lem . The G re e n 's 2 ■ + 2 -1 fu n ctio n s contain a fa c to r (to. (q) - w ) w hich m ak es th em v e ry J [se n sitiv e to m in u te d e ta ils of the la ttic e sp e c tru m when to is w ithin ian a llo w e d range of fre q u e n c ie s of the u n p e rtu rb e d la ttic e . A p p ro x - i jim ate G re e n 's functions w hich give good re s u lts fo r o u t-o f-b a n d i lo c a l m o d e s do not n e c e s s a r ily give good re s u lts fo r inband [re so n a n c e . ! I i j j 4. P re v io u s O b serv atio n s of L o calized V ib ratio n al M odes ! “ “ “ “ S e v e ra l e x p e rim e n ta l techniques have been used to o b se rv e lo c a liz e d m o d es. The sim p le st and m o st d ire c t m ethod is m e a s u r e m e n t o f in fra re d a b so rp tio n of the frequency of th e lo cal m o d e . T his r e q u ir e s a h o st c r y s ta l w hich is tra n s p a re n t to in fra re d ra d ia tio n a t th a t fre q u e n c y , a condition sa tisfie d by a n u m b er of io n ic and c o v a le n t c ry s ta ls including a lk a li and alk alin e e a rth h a lid e s and i 27 | e le c tric a lly co m p e n sa te d se m ic o n d u c to rs. In frared - re fle c tiv ity j and R am an s c a tte rin g m e a su re m e n ts have a lso been u se d in som e I ! in sta n c e s , p a rtic u la rly fo r high c o n c e n tra tio n s of the d efect giving i j th e lo c a l m ode. A nother m ethod, w hich h a s a lso been u se d w ith ; m e ta ls , is th ro u g h m e a su re m e n ts of p h o n o n -a ssiste d tunneling • p r o c e s s e s . T h is h a s been done both in se m ic o n d u cto r p -n ju n ctio n s t i ( T a y lo r 1967 ) and in su p erco n d u ctin g ju n ctio n s ( C h a n d ra se k h a r ! and A d le r 1968 ). Low freq u e n cy inband re so n a n c e s have b e en se e n | by the above m eth o d s a s w ell a s by n e u tro n s c a tte rin g ( M acintosh ! an d B je rru m -M o lle r 1968 ) and by in d ire c t m ethods su ch as the m e a su re m e n t of the te m p e ra tu re v a ria tio n of sp ecific h e a t o r th e rm a l co n d u ctiv ity a t low te m p e ra tu re s ( P ohl 1968; C u lb ert I an d H ubener 1968 ). j T he f i r s t o b se rv a tio n s of lo c a liz e d v ib ra tio n a l m o d es th a t i j w e re in te rp re te d in te rm s of the concept of a lo c a l m ode being one | of the n o rm a l m o d es of a p e rtu rb e d c ry s ta l w e re of h y d rid e ion im p u ritie s in a lk a li h alid e c ry s ta ls ( S c h a efer I960). In fra re d a b so rp tio n bands of oxygen v ib ra tio n s in Si h a d been id en tifie d p re v io u s ly ( K a is e r, K eck an d L ange 1959 ), but th ey w e re i n te r p re te d in te r m s of a m o le c u la r m odel, tre a tin g the Si la ttic e as being to ta lly rig id , r a th e r th an a s a lo c a liz e d m ode. M e a su re m e n t of the in fra re d a b so rp tio n stre n g th of th e 1107 cm " * oxygen band h a s becom e the sta n d a rd m eth o d of m e a su rin g oxygen c o n ten t of Si. 28 j I A s im ila r oxygen v ib ra tio n is a lso seen in Ge a t 855 cm "* (K a ise r j 1962). | The f ir s t o b se rv a tio n s of lo cal m odes in se m ic o n d u c to rs to ; ] b e in te rp re te d as such w e re th o se of b o ro n in silic o n , done n e a rly j sim u lta n e o u sly by fo u r d ifferen t g ro u p s. S m ith and A n g ress(1 9 6 3 , j : j a ls o A n g re ss e t. a l. 1965) o b serv ed the lo c a l m ode of iso la te d b o ro n ! j ato m s in Si w hich w as com pensated by co u n ter doping w ith p h o sphorus j du rin g grow th of the c ry s ta l follow ed by e le c tro n irr a d ia tio n to I ' I jachieve final co m p en satio n . The o th ers (B alkanski and N a z a re w itz j 1964, 1966; S p itz e r and W aldner 1965; C hrenko e t. a l. 1965) ! jp erfo rm ed th e co m p en satio n by diffusing L i into B -doped c r y s ta ls , jthe L i p a irs w ith th e B and sp lits the lo c a l m ode in to a x ia l and ! I jtra n s v e rs e v ib ra tio n s w ith c o n sid era b ly d iffe re n t fre q u e n c ie s . L i jm odes a r e a ls o o b se rv e d . Since b o ro n and lith iu m each have two |sta b le iso to p e s e a sily o btainable in fa irly p u re fo rm , a ll the lo c a l 'm odes can be unam biguously id en tified as being p rin c ip a lly b o ro n j jor lith iu m v ib ra tio n s by th e ir iso to p e sh ifts. An ex am p le of the in fra re d sp e c tru m of Si containing L i is given in F ig . 24, C h ap ter V. L o cal m odes of b o ro n p a ire d w ith su b s titu tio n a l do n o rs (P, A s, Sb) in Si have been o b se rv ed by T sev to v e t. a l. (1968) and by N ew m an and S m ith (1968). The b o ro n m o d es a r e s p lit by th e p a ire d d o n o r, but w ith s m a lle r sp littin g th an in th e c a se of L i-B p a irs Inband a b so rp tio n h as a ls o been o b se rv e d in B -doped Si c o m p e n sa te d w ith a su b stitu tio n al donor ( A n g re s s e t. a l. 1965 ), and j a ls o in A l-d o p ed Si ( Devine and Newm an 1970 ) and n e u tro n ir r a d ia te d Si ( B alkanski 1964 ). Some of the fe a tu re s in a b so rp tio n s p e c tra a p p e a r to be c h a ra c te ris tic of the p a r tic u la r im p u ritie s , ; w hile o th e rs a p p e a r to be c h a ra c te ris tic of the S i la ttic e . T his , topic is d isc u sse d in m o re d e ta il in C h a p te r IV. T he lo c a l m ode of c arb o n in Si was r e p o r te d by N ew m an and i W illis (1965), who id en tified it by the iso to p e sh ift. Since C is not , an e le c tr ic a l dopant in Si, th ere is no sim p le m eth o d to m e a su re r its c o n ce n tra tio n . Newm an and W illis c a lib ra te d the a b so rp tio n j b and s tre n g th v s. carb o n c o n cen tratio n by doping w ith ra d io a ctiv e 14 ! C, th u s providing a m ethod of m e a su rin g the am o u n t of s u b s ti- I 12 ; tu tio n a l c a rb o n in Si. U nfortunately, the lo c a l m ode band of C, i | the m o st com m on iso to p e, is at the sam e fre q u e n c y a s the s tro n g e s t ! tw o-phonon band in Si; but even so it p ro v id e s th e e a s ie s t m e a su re I of su b stitu tio n a l c arb o n content of Si. i j T he convenient com pensation tech n iq u es th a t w o rk a t a ll w e ll fo r Si a re m a rg in a l ( L i diffusion ) o r do not w o rk ( e le c tro n o r n e u tro n irra d ia tio n ) fo r Ge. In addition, if Ge is im p e rfe c tly c o m p e n sa te d by L i diffusion it w ill be p -ty p e . T h is m ean s th a t low te m p e ra tu re m e a su re m e n ts w ill be difficult b e c a u se of stro n g a b so rp tio n fro m in te rv a le n c e band tra n s itio n s in p -ty p e Ge ( K a ise r e t. a l. 1953 ). C onsequently, the f i r s t re p o rte d o b se rv a tio n of a 30 i ; I lo c a l m ode in Ge w as th at of the Si lo ca l m ode, o b se rv e d in R am an sc a tte rin g by F eld m an et. a l. (1966). T his lo c a l m ode h a s sin ce been found to be w eakly in fra re d a c tiv e (see C h a p te r. IV). The lo cal m odes of P and of L i p a ire d w ith Ga (see C h a p te r. VI) w e re the f ir s t in Ge fo r w hich in fra re d a b so rp tio n w as re p o rte d . (C osand and S p itz e r 1967). Since then th e lo c a l m odes of b o ro n , both iso lated and in B -L i p a irs have been re p o rte d (N azarew icz and Ju rk o w sk i 1969); th e lo ca l m ode s p e c tra a r e v e ry s im ila r to the c a se of B in Si. :In fra re d a b so rp tio n of lo ca l m odes h as b een ex ten siv ely istu d ies in IH -V se m ic o n d u c to rs. H ayes (1964) f i r s t re p o rte d the i lo c a l m odes of Al and P in GaSb. L o rim e r e t. a l. (1966) re p o rte d th e Al and P lo c a l m odes in GaAs; S m ith e t. a l. (1966) then re p o rte d ithe lo c a l m ode of Al in InSb. Al and P im p u ritie s a r e not e le c tr i c a lly a c tiv e in th e s e c ry s ta ls , and the lo c a l m odes give a convenient i >and se n sitiv e m e a s u re of th e c o n c e n tra tio n s of th e s e im p u ritie s . i 'L o c a l m odes of s e v e ra l im p u ritie s w e re se e n in G aP by S p itze r |e t. a l. (1969) by in fra re d a b so rp tio n and by Hon e t. a l. (1970) in I |R am an s c a tte rin g ; the lo ca l m ode of n itro g e n in G aP w as seen by T hom as and H opfield (1966) as a sideband on th e e m issio n fro m excitons bound to n itro g e n a to m s. L ocal m odes of L i com plexes in GaAs have y ield ed c o n s id e r ab le in fo rm a tio n on th e co n fig u ratio n of and conditions fo r fo rm atio n :of p a irs and o th er d e fe c ts. One of the outstanding exam ples of the u se of lo cal m odes as a p ro b e to study d efects in c ry s ta ls has b een the v a rio u s stu d ies of Si in GaAs . Si is an a m p h o teric dopant; S i^ g is an a c c e p to r and SiQa is a d o n o r. Both sp e c ie s give a lo ca lized m ode, but a t d iffe re n t fre q u e n c ie s, so the stre n g th of the in fra red a b so rp tio n bands d ire c tly g iv es a m e a s u re of th e co ncentration of each sp e cie s (L o rim e r and S p itz e r 1966). T his h as been exploited to m e a s u re re d is trib u tio n of Si betw een th e two types of j s ite s in th e p re s e n c e of L i a t high te m p e ra tu re s (S pitzer and A llred S 1968). A lso, it h as b e en u sed to m e a s u re the tendency of Si to co m p en sate o th e r d o p an ts, e ith e r do n o rs o r a c c e p to rs , during the j (growth of GaAs c ry s ta ls . j At the tim e of th is w ritin g , th e m o st co m p lete sin g le so u rce jof in fo rm atio n on in fra re d a b so rp tio n due to high freq u en cy lo calized m o d e s is th e rev ie w by N ew m an (1969), w hich quite thoroughly |c o v e rs th e w ork th at h as b een done in io n ic and sem ico n d u cto r ic ry s ta ls . P ro b ab ly th e b e s t sin g le s o u rc e on o th e r a sp e c ts of i lo c a liz e d m odes and inband m o d es is L o c a liz e d E x citatio n in Solids, (edited by R. F. W allis, w li ch is the p ro ce ed in g s of the In tern a tio n al | C onverence on L o ca liz ed E x c ita tio n s in Solids held at U. C. Irv in e in S e p tem b er 1967. CH A PTER II A pproxim ate C a lc u la tio n M ethods fo r L o calized M odes A. P re v io u s W ork In p rin c ip le , th e G re e n 's fu n ctio n technique gives an exact : solution fo r the fre q u e n c ie s and e ig e n v e c to rs of a la ttic e w ith a ; defect. In p ra c tic e , e v e n fo r a s sim p le a defect as a substitutional j im purity, the r e s u lts a re not u su a lly v e ry good, both because of | im p erfect know ledge of the la ttic e sp e c tru m of the u n p ertu rb ed 1 j c ry s ta l and b ecau se th e re is g e n e ra lly no reliab le way to p red ic t the I fo rce constant ch an g es a s s o c ia te d w ith a su b stitu tio n a l im purity. j I Thus for the p u rp o se s of in te rp re ta tio n of e x p erim e n tal re s u lts , it i j is useful to c o n sid e r a p p ro x im a te m eth o d s, w hich m ay be com puta- I tionally s im p le r th a n a rig o ro u s c a lc u la tio n of lo calized m ode : freq u en cies. | i Since the in -b a n d m o d es a r e sp a tia lly extended and involve ! m otion of a ll of the a to m s of the la ttic e to about the sam e extent th e re is no good w ay to c a lc u la te th e ir freq u e n cie s and eig en v ecto rs without c o n sid e rin g the e q u atio n s of m otion fo r the e n tire c ry s ta l. H ow ever, fo r lo c a l m o d es th e re a r e sim p le app ro x im atio n m ethods available since the m ode is sp a tia lly lo c a liz e d and only a rela tiv e ly sm a ll num ber of a to m s p a rtic ip a te . As an exam ple, it w ill be shown that o v er 99% of th e e n e rg y of the lo ca l mode of iso lated su b stitu tio n al b o ro n in s ilic o n is co n tain ed in the m otion of the b oron 32 33 and its f ir s t and second n e ig h b o rs. One should th e re fo re expect to be ab le to c a lc u la te th e freq u en cy , given a m odel fo r the fo rce j c o n stan ts, to w ithin 1% by co n sid erin g only the m otion of th e se a to m s. :An a lte rn a tiv e m ethod is to a p p ro x im a te th e G re e n 's function. In th e exam ple above, th e b o ro n ato m and its f i r s t and second n eig h b o rs include a to ta l of 17 ato m s; such a co llectio n of atom s c o n sid e re d by 1 iitse lf can hav e a t m o st 51 d istin c t fre q u e n c ie s. It is th e re fo re p o ssib le to a p p ro x im a te th e G re e n 's functions for th e la ttic e w ith no Im ore than 51 te r m s and obtain th e sa m e re s u lt as is obtained by [C onsidering th e m o tio n of th e d efect and its f ir s t and second j jneighbors. The erro r cau sed by c o n sid e rin g a fin ite defect reg io n con- isistin g of th e d e c e c t and som e su rro u n d in g ato m s h as been d isc u sse d j in th e lite r a tu r e (Dean 1968 a, b). The lo c a l m ode freq u en cy |c alcu lated in a fin ite reg io n is a lo w e r bound for the re s u lt th a t jwould be o b tain ed in an in fin ite la ttic e fo r the sam e m odel of the jdefect and the la ttic e . T hat th is should be so is fa irly obvious, i sin ce allow ing only th e a to m s in th e d efect region to m ove is equivalent to le ttin g th e m a s s of the’ ato m s o u tsid e th e defect reg io n becom e in fin ite . An u p p e r bound can a lso b e obtained, e x p re ss e d in te r m s of th e a m p litu d e of th e e ig e n v e c to r fo r the defect region and the te r m s coupling m otion in sid e the d efect region to the r e s t of the la ttic e . 34 The siz e of the defect reg io n re q u ire d fo r a given a c c u ra c y depends on the d eg ree of lo c a liz a tio n of the m ode. A m ode w ith a : freq u e n cy ju s t b a re ly above the m axim um phonon freq u e n cy of the | u n p e rtu rb e d la ttic e w ill not be highly lo c a liz e d and a fa irly la rg e I d e fe ct re g io n is re q u ire d to ad eq u ately ap p ro x im ate the m ode. A v e ry high freq u e n cy lo c a l m ode m ay be so w ell lo c a liz e d th a t the ' la ttic e h a rd ly p a rtic ip a te s a t a ll, and good re s u lts m ay be ob tain ed j by c o n sid e rin g only the d efect its e lf a s a defect reg io n . C o n sid e rin g I th e se sam e e x tre m e c a s e s in a G re e n 's function c a lcu latio n , a I r a th e r la rg e n u m b er of te rm s m u st be included in the G re e n 's i i | functions to d e sc rib e m odes w ith fre q u e n c ie s close to the fre q u e n c ie s i i of the b and m o d es. F o r,lo c a liz e d m odes w ith fre q u e n c ie s f a r r e - ] I m oved from the band m o d es, none of the te r m s in the G re e n 's j j | function depends v e ry stro n g ly on the u n p e rtu rb e d m ode fre q u e n c ie s | and re a so n a b le re s u lts a re obtained by rep la cin g the su m m atio n o v e r i J the la ttic e m odes in the G re e n 's function by a single te r m of the form I 2 2 1 / ) , w hich is eq u iv alen t to an E in ste in ap p ro x im atio n w h ere e a c h ato m in the la ttic e is an independent o s c illa to r w ith a fre q u e n c y i U ) . o A sim p le m eth o d of th is s o rt th a t is u se fu l fo r p re d ic tin g fre q u e n c ie s of lo c a l m odes in som e c a se s is to a p p ro x im ate the la ttic e sp e c tru m by d elta functions a t the fre q u e n c ie s of the p rin c ip a l p eak s in the d e n s ity -o f-s ta te s of the h o st la ttic e ( S p itz e r 1966 ). In th e iso to p ic su b stitu tio n m odel, th e equation fo r the lo ca l m ode b e co m es Z e w ( ~Z T + ~~Z 2 + ~2 T + ~ ~ 2 T ) + 1 = 0 (3 2 ) \ u,T A “ w u jL A ‘ “) u,L O - tw U )T O ’ 1 " / * The a p p ro p ria te la ttic e freq u e n cie s fo r u se in th is e x p re s s io n can be found fro m fits to th e m ultiphonon in fra re d la ttic e a b so rp tio n (S p itzer 1967). T able 1 shows freq u e n cie s c a lc u la te d by th is 'm ethod fo r a v a rie ty of im p u ritie s in s e v e ra l se m ic o n d u c to rs; th e se iare c o m p a red to the e x p erim e n tally o b se rv ed fre q u e n c ie s. F o r c a s e s such as Ge o r H I-V com pounds w h e re the two m a s s e s a re !n early equal, th e v ib ra tio n a l sp e c tru m of the c ry s ta l, w hen (n o rm alized to the sa m e m axim um phonon freq u en cy , w ill be s im ila r i to th e v ib ra tio n a l sp e c tru m of Si. F o r th e se c ry s ta ls , th e c a lc u la tio n s of D aw ber and E llio tt, w hich give (o u T - c u ) / iv v s. the J j IH 3 .X ITXcLX i I jm a ss d e fe ct e , can be u sed to find to T by re p la c in g m of Si by j Xj 1X12.X i to fo r the h o st being c o n sid ered . T hese v alu es a r e a ls o show n in i m a x I T ab le 1. The d ire c t solu tio n of the equations of m otion fo r a d e fe c t re g io n h as a ls o b een exploited fo r calcu latin g lo cal m ode fre q u e n c ie s. F o r a sim p le d efect such as a sin g le im p u rity on a s ite of high sy m m e try th e equations of m otion can be decoupled th ro u g h u se of su m m e try c o o rd in a te s. T h ere a r e exam ples in th e lite r a tu r e of su ch c a lc u la tio n s fo r d efect regions c o n sistin g of a su b stitu tio n a l 36 Table 1 . Comparison o f l o c a l mode f r e q u e n c ie s c a lc u la t e d by two d i f f e r e n t methods w ith th e e x p e rim en ta l v a l u e s . The " d e n s it y - o f - s t a t e s " column g i v e s v a lu e s i o b ta in ed from th e G reen 's fu n c tio n c a l c u l a t i o n o f Dawber and E l l i o t t (1 9 6 3 b ), s c a le d t o th e proper f o r th e h o s t . The v a lu e s in th e " c h a r a c t e r is t ic fr e q u e n c ie s" column are th o se o b ta in ed from e q u a tio n 1 8 , by u s in g i c h a r a c t e r i s t i c phonon fr e q u e n c ie s o b ta in e d from m u lti- : phonon in fr a r e d a b s o r p tio n , a s g iv e n by S p it z e r ( 1 9 6 7) . TABLE 1 Density Characteristic Impurity Host of States Frequencies Experiment S i 682 cra“^ 662 cnf * 644 cm”l U B Si 664 637 620 12C S i 644 618 608 13C 14 13C S i 625 599 586 C Si 610 584 572 10B Ge 600 583 571 1:lB Ge 582 563 547 Si Ge 387 374 392 P Ge 372 360 343 Al GaAs 374 354 362 P GaAs 368 334 355 Si GaAs 378 349 3 8 4 ,3 9 9 Al InSb 318 298 296 Al GaSb 297 314 P GaSb 323 321 38 im p u rity w ith f ir s t n eig h b o rs in a N aC l la ttic e , ( Ja sw a l 1965 ) w ith f ir s t o r f i r s t and seco n d ( K rish n a , M urthy and H a rid a sa n 1966 ) n eighbors in a zinc blende la ttic e , o r fo r a vacancy and m any sh e lls of n eighbors in Cu ( L and and G oodm an 1962 ). F o r a le s s ; sy m m e tric a l d efect, su ch a s two o r th re e im p u ritie s, it m ay not be I p o ssib le to decouple the eq u atio n s of m otion, and the lo c a l m ode ; freq u en cy m u st be found fro m the 3NX3N sy ste m of equations fo r an JN ato m d efect re g io n . T his m ay be quite unw ieldy; N = 17 fo r a ! su b stitu tio n a l im p u rity and its f i r s t and second neighbors in a ■diamond la ttic e . j B. A M new " m eth o d fo r c a lcu latin g lo cal m ode fre q u e n c ie s i | I and e ig e n v e c to rs . The u se of a la rg e d efect re g io n p re s e n ts m a jo r d iffic u lties if i I lone tr ie s to solve the e n tire s y s te m of eq u atio n s. F o r a sy ste m | w here only one im p u rity p ro d u ce s lo c a liz e d m odes, out of a ll the I j m odes of the d efect re g io n th e r e a re only th re e lo c a liz e d m odes; it I . j is a ssu m e d th a t only th e s e th re e m odes a re re a lly of in te r e s t. T h ese I m odes have the h ig h e st fre q u e n c ie s of any of the m odes of the d efect region; the re m a in in g m o d es, w hich a re the band m o d es, a re u su a lly a t a su b sta n tia lly lo w e r freq u en cy . If the eq u atio n s of m o tio n of the d efect reg io n a re e x p re s s e d in i - - m a trix fo rm , the w ell-k n o w n m eth o d of ite ra te d m u ltip lic a tio n of a v e c to r by a m a trix to find the dom inant eigenvalue of the m a trix (H igm an 1964) can be u sed to c alcu late both th e lo c a l m ode fre q u e n cies and e ig e n v e c to rs. T here is no m ention in the lite r a tu r e o f th is m eth o d having been p rev io u sly applied to lo c a l m o d e p ro b le m s, lit is p a rtic u la rly su ite d fo r calculations of fre q u e n c ie s and e ig e n v e c t o r s of com plex d e fe cts as it m e re ly re q u ire s th a t th e lo ca l m o d es b e w ell s e p a ra te d in frequency fro m the lo w e r-fre q u e n c y band m o d e s . It does not depend on the u se of sy m m e try to red u c e th e co m p lex ity of th e p ro b lem . We w ill th e re fo re c o n s id e r th is i m eth o d in som e d e ta il. T he eig en v alu es X^ and eig en v ecto rs- of an N XN h e rm itia n jm a trix [M ] s a tis fy th e re la tio n sh ip [M ] V. = X.V.. The can b e !c h o sen so th at any N com ponent v e c to r t |f can be w ritte n ij ; = | SO^V. . M ultiplying \|i by [M ] gives If it is d e s ire d , s u c c e s s iv e can be found by u sing a t r i a l v e c to r (33) |If X^ is the la r g e s t eigenvalue and 0 then (34) and (35) o rth o g o n al to th o se a lre a d y found. T his a lso w ill give\s in d ep en d en t 40 V. 'a w ith d eg en erate e ig e n v alu es. T his m ethod is of value in the c a se of lo c a l m odes sin ce th e re a re a few eig e n v alu es of in te re s t, th o se re p re se n tin g the lo ca l m o d es, and they a re su b s ta n tia lly la r g e r th an the r e s t of the e ig e n v alu es, w hich re p re s e n t the band m o d es. The s im p le st m o d el to w hich th is can be a p p lied is the lin e a r c h a in w ith f i r s t neighbor fo rc e c o n sta n ts. The e q u atio n s of m otion : a re - M .u A i . = - ( u. -u. . ) - $*+1 ( u. -u.,, ) i i l- l i l - l l i i + l ' (36) |w h ere u. is the d isp lac em e n t of the ith atom fro m its e q u ilib riu m jposition, and is the fo rc e c o n stan t betw een the ith and jth ato m s i ^ ;( §? = §* ). The equations can be re w ritte n as ^ 1 I 1 i-1 M. $ + § § , i-1 i+1 i+1 2 u - i + w r j — u -, i = w l - l M. l M. l+l i (37) |By defining the NxN m a trix ( i + ^ i ) / M - l - l l+l i M .. = ij / M - l+l l if i = j if j = i - H 1 othe rw ise (38) an d the colum n v e c to r /u 1 \ u . u = (39) i the s e t of equations m ay be w ritte n a s a m a trix eigenvalue eq u atio n ; is o la te d fro m the o th e r m odes its eigenvalue and e ig e n v e c to r can be i : found by ite ra te d m u ltip lic a tio n of a su itab le t r i a l v e c to r U by the ■ m a trix [M ] . | The ra te of co n v erg en ce of th is p ro c e s s depends on the | s e p a ra tio n of the lo c a l m ode freq u e n cy fro m the h ig h e st band m ode i j freq u e n cy and on the num ber of m odes w ith fre q u e n c ie s n e a r the | | m ax im u m band m ode freq u en cy . C onvergence to w a rd the c o rr e c t eig en v alu e is quite rap id . The [M ] U = U ) 2U . (40) I In th is fo rm the p ro b lem is independent of the c h o ices of M. and 1 ' . A s long as th e re is a sin g le h ig h e s t fre q u e n c y m ode fa irly w ell : ra te of co n v erg en ce can be d e te rm in e d b y c o n sid e rin g the a p p ro x i- | m ate e ig e n v e c to r obtained a fte r k+1 ite ra tio n s . F ro m e q .3 ^ , i N I we see th at T his can be expanded to give (42) If we le t 6^ be the differen ce b etw een an e x a c t eig en v alu e X . and the la r g e s t eigenvalue X^ , th en X • X a . ■ x N 6 . 42 (43) o r X. ~ = 1 - 6. / X., N 1 N We can th en w rite £ (4 2k 1 + 1 - i Krl j v 2k 1 + 1 4 ; ( < X . 1 a N/ w hich sim p lifie s to x<k+1> x 2k /a. i l a. 6 . 2X N fb. ] XA T , 1 + S f r , i^N V 2k - . 2 fa. i al_. 4 N / (44) (45) (46) M odes fo r w hich — — is la rg e have v e ry little e ffe c t a fte r k+1 2k b eco m es quite sm a ll. b eco m es a t a ll la rg e , sin ce the te r m f t A 2 The te r m -— in the su m in eq.h-6 can be n e g le c te d a fte r the f ir s t W 6 i . few ite ra tio n s . The te rm s fo r w hich -r— is sm a ll, so th a t r - A N 6. V/N) does not ra p id ly becom e sm a ll, a r e m u ltip lie d by ; th ey have N re la tiv e ly little e ffect on the eig en v alu e. C onvergence to the c o rre c t e ig e n v e c to r is not as fa st; the % -\k . The kth a p p ro x im a tio n to the e r r o r s d e c re a se only as e ig e n v e c to r is 43 V . (k> _ [m ] a N | CM]k * w hich re d u c e s to (47) r(k) _ N N H i*N f . -v 2k - , 2 fX. \ fa.. i v. n ; a, T r + i^ N v > y a. V. i i 1 + £ . 2k , . 2 rX. A /a. i i i i ^ N V Nj i a. NJ (48) A fter the c a lc u la tio n s p re s e n te d h e re had b een co m p leted , it i | w as pointed out by P ro f e s s o r J . Sm it th a t the ite ra tiv e p ro c e d u re i ! could be m o d ified to o b tain f a s te r convergence by su b tra c tin g a | c o n sta n t fro m e ach of the e le m e n ts on the diagonal of the m a trix . If the c o n stan t is ch o sen to be h a lf of the m axim um band m ode e ig en v alu e, the band m ode eig en v alu es of the m odified m a tr ix w ill ran g e fro m to + £ o f the o rig in a l m ax im u m band m ode eig en v alu e. The d ifferen c e b etw een the lo c a l m ode eigenvalue an d the next la r g e s t eigenvalue re m a in s the sa m e , h o w ev er, so th a t the ra tio of th a t d ifferen ce and the m agnitude of the eig en v alu es of the new m a trix is im p ro v ed by a fa c to r of tw o. C o n sid e r the e x p re s s io n w h e re [IT is a.u n it m a trix . If we define u i = X , we o b tain J m ax o ( CM] - - r - [I] ] > j t = °N VN (50) i F ro m th is w e can ev alu ate th e e x p re ssio n fo r th e kth a p p ro x im atio n ' to th e e ig e n v e c to r. x_ ♦ 00 _ - ( c m ] ) [i] 1 * ( y and ob tain V l f (k) - N i + E 2k a. i?fN \ XN " Xo *N (51) (52) F o r the c a se of-a sin g le lo c a l m ode w ith ( freq u en cy ) = X^ and only slig h tly la r g e r th a n X , the u n d e sire d m ode th a t is e lim in a te d i m o st slow ly by the ite ra tiv e p ro c e s s is the one w ith eigenvalue X q . | The am p litu d e of th is m ode d e c re a s e s as r x - x \ k I o o X - * N o T J (53) If we w rite X„T = X + e , th is becom es N o V T + V f 1 2e 1 + t — V K J (54) j Thus we see th a t fo r s m a ll e the m odified ite ra tiv e p ro c e ss i c o n v e rg e s tw ice a s fa s t a s the o rig in a l ite ra tiv e p ro c e s s , w h e re the i a m p litu d e of the sa m e m ode d e c re a s e d as k r N; (55) 1. Sim ple L in e a r C hain The s im p le s t ex am p le is th a t of a single lig h t im p u rity of m a s s aM ( 0 < a < 1 ) in the m iddle of a ch ain of ato m s of m as s M, an d w ith a ll the a to m s c o u p led only by a f ir s t neighbor fo rce c o n sta n t $ . T h is p a rtic u la r e x am p le is so sim ple th at it c an be so lv e d in c lo s e d fo rm ( M o n tro ll, 1955; K itte l 1966 ), but it w ill s e rv e to 46 'illu s tr a te th e m ethod. i C o n sid e r, fo r ex am p le, a c h ain of one h u n d red a to m s, w ith the ! |im p u rity a t th e fiftie th s ite . If the v alu es M = 1 and §= 0. 25 a r e used,j i . ith en u ) = 1 , w h ere u ) is the m ax im u m freq u en cy of the m ax m ax iu n p e rtu rb e d chain. The m a trix b eco m es 48 -.25 .5 -.25 49 -.25 .5 -.25 T t v y f T cn »25 .5 »25 , [ M ] = 50 (56) 51 -.25 .5 -.25 52 -. 25 .5 -.25 F o r the in itia l t r i a l v e c to r , le t a ll the com ponents \|r = 0, e x ce p t = 1 . F o r a v e ry lig h t im p u rity m a s s , such a s a = 0. 1, the lo c a l | m ode fre q u e n c y is c o n sid e ra b ly la r g e r th an w , and c o n v erg en c e is IXlctX quite ra p id . T able 2 show s the fre q u e n c ie s and e ig e n v e c to rs fo r s u c c e s s iv e ite ra tio n s fo r a = 0. 1, co m p a rin g th e m to th e e x a c t v alu es o b tain ed fro m the e x p re s s io n s given by K itte l ( 1966 ). C a lc u la tio n of the fre q u e n c y fo r v alu es of a b etw een 0. 1 and 1. 0 in ste p s of 0. 1 u sin g 100a ite ra tio n s gave the c o rr e c t freq u en cy to at le a s t five sig n ific a n t d ig its ex cep t fo r a = 1. 0, fo r w hich th e re is no lo c a l m o d e. 2. D iso rd e re d L in e a r C hain A n ex am p le fo r w hich the ite ra tiv e calcu latio n m eth o d is m o re T able 2 . I n i t i a l t r i a l v e c t o r , and n orm alized v e c t o r s r e s u l t i n g from th e i t e r a t e d m atrix m u lt i p l i c a t i o n f o r an impurity mass of 0.1 of the host mass. The resulting « a p p roxim ation t o th e freq u en cy a t each s t e p i s a l s o g iv e n . The e x a c t r e s u l t s i n th e l a s t column are c a lc u la t e d from th e c lo s e d form e x p r e s s io n s g iv e n by K i t t e l (1 9 6 6 ). TA BLE 2 N u m b er of Ite ra tio n s : In itia l T ria l E x act R esu lt F req u en cy 2.23886 2.29157 2.29391 2.29416 2.29416 w H 2 W o c J & co ■ B A • H d u V o •H •» ^ 0 } O a. 45 0 0 0 0 -.0 0 0 0 0 0 -.0 0 0 0 0 0 46 0 0 0 0 .000008 .000008 47 0 0 0 -.0 0 0 1 1 3 -.000145 -. 000145 48 0 0 .002374 .002707 .002762 .002762 49 0 -.049875 -.0 5 2 2 3 8 -.0 5 2 4 5 6 -.052486 -.0 5 2 4 8 6 50 1.000000 .997509 .997262 .997238 .997235 .997235 51 0 -.0 4 9 8 7 5 -.0 5 2 2 3 8 -.0 5 2 4 5 6 -.052486 -.052486 52 0 0 .002374 .002707 .002762 .002762 53 0 0 0 -.0 0 0 1 1 3 -.0 0 0 1 4 5 -.0 0 0 1 4 5 54 0 0 0 0 .000008 .000008 55 0 0 0 0 -.0 0 0 0 0 0 -.0 0 0 0 0 0 49 a p p ro p ria te is the c a lc u la tio n of the freq u en cy of the lo ca l m ode of a lig h t im p u rity in a d is o rd e re d la ttic e . C o n sid e r, fo r in sta n c e , a lin e a r chain w ith a lig h t im p u rity of m a s s oM (0 < a < 1) as in th e p rev io u s exam ple, except th a t th e "u n p e rtu rb e d " chain contains a random m ix tu re of ato m s of m a s s e s M and 2M. T h ere is no closed fo rm solu tio n fo r th is p ro b le m . The lo ca l m ode freq u en cy depends on the con fig u ratio n of th e n e ig h b o rs of th e im p u rity as w ell as on |th e im p u rity m a s s , so th a t fo r a given im p u rity m a s s th e re m ay be m any d iffe re n t lo ca l m o d e fre q u e n c ie s. To a p p ro x im a te ly o b tain th e d istrib u tio n of fre q u e n c ie s fo r an iim p u rity of a given m a s s , th e lo c a l m ode freq u en cy w as evaluated jfor each p o sitio n in a ran d o m chain. By using s e v e ra l sh o rt chains i ra th e r than a sin g le long one, the com putation tim e fo r a given jnum ber of s ite s can b e re d u c e d c o n sid e ra b ly . R esu lts a re shown fo r ia to ta l of 99 s ite s ; th e c a lc u la tio n s w e re done by using th re e chains r jof 33 ato m s each . The im p u rity w as su b stitu ted on a site in a chain and the freq u en cy ev alu ated fo r s e v e ra l v alu es of a betw een a = 0. 1 and a - 0. 6. T his w as re p e a te d fo r each s ite , w ith p erio d ic boundary conditions being u se d to avoid effects of the ends of the c h a in s. The r e s u lts , given in T ab le 3 and F ig u re 4, c le a rly show the in c re a s in g se n sitiv ity of the lo c a liz e d m ode to its en v iro n m en t a s a is in c re a s e d and the m ode b e co m es le s s lo c a liz e d . In T able 3 the 50 Table 3 . Atomic d is p la c e m e n ts f o r a o n e-d im en sio n a l l o c a l mode o f a l i g h t im p u rity o f mass m in a c h a in o f m asses m and 2m. D isp la cem en ts o f th e l i g h t atom and i t s i f i r s t f i v e n e ig h b o rs t o e i t h e r s id e are shown f o r = 0 . 1 , !o . 3» and 0 .6 f o r th r e e d i f f e r e n t c o n f ig u r a t io n s o f n e ig h b o r s . The c o n f ig u r a t io n o f n e ig h b o r s o f th e l i g h t im p u rity jis i l l u s t r a t e d w ith open c i r c l e s f o r atom s o f mass m and jshaded c i r c l e s f o r atoms o f mass 2m. The c i r c l e w ith an X i jin d ic a te s th e im p u rity s i t e . ! a Frequency C o n fig u r a t io n 0.1 0.3 0.6 4.559 2.747 2.110 Displacements 000 .000 .002 000 .000 .0 1 5 000 .003 .034 001 .0 1 4 .068 025 .076 .133 998 o 00 OS • .847 053 .183 .437 002 .0 3 4 .2 2 5 000 .006 .1 1 5 000 .001 .0 5 7 000 .000 .0 2 4 u r a t io n TABLE 3 0 .1 0 .3 0 .6 4 .5 5 9 2 .7 4 7 2 . I0i D isp la cem en ts .000 .000 .002 .000 .000 .016 .000 .003 .0 3 5 .001 .0 1 4 .071 .0 2 5 .076 .136 .9 9 8 , .978 .866 .053 .183 .434 rs o o . .033 .190 .000 .003 .0 2 8 .000 .000 .0 0 4 .000 .000 .002 Config u r a t io n 0.1 0.3 0.6 4 .5 5 9 2 .7 4 5 2 .0 8 8 Displacements .000 .000 .003 .000 .000 .019 .000 .003 .041 .001 .0 1 4 .079 .0 2 5 .076 .145 .998 x-i 00 Os • 0895 .053 .180 .407 .001 .0 1 4 .0 6 5 .000 .0 0 3 .0 2 9 .000 .000 .005 .000 .000 .002 52 F ig u re 4. A p proxim ate d istrib u tio n of lo ca l m ode fre q u e n c ie s fo r lo c a l m odes of an im p u rity of m a s s M in a ra n d o m ch ain of m a s s e s M and 2M, obtained fro m a to tal of 99 s ite s . The d istrib u tio n of freq u e n cie s w as calcu lated by su b stitu tin g the im p u rity on each site in tu rn . The "ran d o m c h ain s u sed in this c a lcu latio n happened to contain 20 s ite s w ith tw o lig h t f ir s t n e ig h b o rs, 25 site s w ith two heavy f ir s t n e ig h b o rs, and 54 site s w ith one lig h t and one heavy f ir s t n e ig h b o r. S im ila r deviations fro m a p u rely uniform d istrib u tio n o c c u r a lso for second and fu rth e r neighbors, so the freq u e n cy d istrib u tio n show n h e re only q u alitativ ely re p re s e n ts th a t o f lo c a l m o d es in an in fin ite random chain. I I 53 FREQUENCY i a=o.l 4.52 4.60 i n 3.35 3.25 u. a =0.3 2.67 2.81 d =0.4 250 2.40 <t= 0.5 224 i a = 0.6 2.20 i 2.10 2.00 i 54 j fre q u e n c ie s and d isp la c e m e n ts obtained fo r differen t values of a a re show n fo r th re e d efect site co n fig u ratio n s w hich differ only in i secon d an d fu r th e r n e ig h b o rs. F o r a = 0 .1 , the m ode is quite w ell | lo c a liz e d , and a ll th re e s ite s give the sam e frequency. F o r a = 0 .3 , i j the m ode is le s s w ell lo c a liz e d and the s ite s differing in second | n e ig h b o rs y ie ld d iffe re n t fre q u e n c ie s. F o r a= 0. 6, the lo ca l m ode involves c o n sid e ra b le m o tio n of m any n eig h b o rs, and the th re e s ite s I a ll give d iffe re n t fre q u e n c ie s. The d istrib u tio n of freq u e n cie s i ; ob tain ed fo r a ll 99 s ite s th a t w e re c o n sid e re d is shown in F ig u re 4, i w h e re th e n u m b er of m odes a t a given freq u en cy is plotted v s . I I fre q u e n c y fo r v a lu e s of a fro m 0. 1 to 0. 6 . F o r a = 0. 1 the m ode | is se n sitiv e only to its two f i r s t n eig h b o rs; th re e d istin c t i i \ j fre q u e n c ie s a r e se e n , re s u ltin g fro m s ite s fo r w hich the f ir s t I I I n e ig h b o rs a re both lig h t, one lig h t and one heavy, o r both heavy. I ! F o r la r g e r v a lu e s of a the c o n fig u ratio n of second and fu rth e r ! I | n e ig h b o rs b e co m es im p o rta n t and m any d ifferen t freq u e n cie s re s u lt. 3. E x te n sio n to T h ree D im ensions a . F o rm u la tio n T he ite ra tiv e c a lc u la tio n m ethod can be re a d ily ex ten d ed to m o re c o m p lic a te d s y s te m s , su c h a s the diam ond la ttic e . T his is of in te r e s t sin c e th e re h a s b een c o n sid e ra b le e x p e rim e n ta l study of lo c a l m odes of im p u ritie s and im p u rity co m p lex es in se m ic o n d u cto rs w ith the diam ond s tru c tu re o r the c lo s e ly -re la te d zincblende s tru c tu re . An exam ple w ill be given in C h a p te r V of the ap p lica tio n of th is technique to the case of a B-Lii p a ir in Si w h e re the B has a Ge second n eighbor; th is is a r a th e r co m p licated p ro b le m to tr e a t by I any o th e r m ethod. F o r a m odel of th e la ttic e w h e re only s h o rt-ra n g e in te ra c tio n s jare included, m o st of th e m a tr ix ele m e n ts a r e z e ro . F o r efficient u se of th e ite ra tiv e c a lc u la tio n m eth o d , it is d e s ira b le to index the m a tr ix e lem en ts so th a t th e n o n -z e ro m a trix ele m e n ts a r e grouped i ito g eth er in so m e sim p le m a n n e r. R e p rese n tin g th e m a trix in the iform of a conventional tw o -d im e n sio n a l a r r a y is e x tre m e ly i linefficient fo r the c a se being c o n sid e re d , as th e n o n -z e ro elem en ts jwould be p re tty w ell s c a tte re d th roughout an a r r a y th a t is m o re than 198% z e ro s . If m o re th a n two in d ic es a r e u sed to sp ecify a m a trix I e le m e n t, a notation th a t is m uch m o re a p p ro p ria te to the diam ond la t- | itice can be developed. i | The silico n (diam ond) la ttic e is f. c. c. w ith two ato m s p e r i | rh o m b o h ed ral p rim itiv e c e ll. The p rim itiv e tra n s la tio n s a r e in [1 1 0 ] d ire c tio n s. A s ite in th e la ttic e can b e sp e cified by (ji, m , n, k) w h ere th e in te g e rs Z , m , n give th e p rim itiv e tra n s la tio n s fro m the o rig in to the c e ll, and H sp e c ifie s one of the two s ite s in th e c e ll. A c a rte s ia n com ponent of th e d isp la c e m e n t of any atom can then be w ritte n a s u a (^> n, K), w h e re a = x , y, o r z. 56 T able 4 S ite N e a re s t N eighbors D irec tio n to N eighbor A ,m , n, 1 [III] A + l,m ,n , 1 [1 11] A ,m , n, 2 i A ,m +1, n, 1 [111 ] A ,m ,n + l, 1 [111 ] A, m , n, 2 [111] A - l ,m ,n ,2 [111] I A , m , n, 1 _ A ,m - l ,n , 2 [111] | A , m , n» 1,2 [1 1 1 ] | I If a g e n e ra liz e d diam ond la ttic e not n e c e s s a rily having the sam e ! m a s s a t each site is c o n sid e re d , fo r h a rm o n ic tim e dependence, and I | c o n sid erin g only f i r s t n eig h b o r c e n tra l f o rc e s , th e equations of ! m o tio n fo r an ato m on the s ite A ,m , n, h , w ith m a s s M (A ,m , n, h ) a re i 2 * ♦ A m n n w U Amnn j -M (u u = V -§*'m' nV k Vl jin h v l / A m n n Amnh Amh H )• r^ m n n I * A /m /n /H/ Amnn (57) The sum on th e rig h t hand sid e extends o v e r th e fo u r n eig h b o rs of , . , . x 'm 'n V , A J& 'm 'n'n'. th e s ite A ,m , n, h . The fo rc e co n stan t is $ j f a i r i K ’ a Amnx, 18 a unit v e c to r in th e d ire c tio n of the n e ig h b o r. It is m o re convenient to sim p lify the n o tatio n , re fe rrin g to a n e ig h b o r X /m /n /H/ by an index i , so the equations becom e ■ M „ iw^u.. = ^ - $. {(u. - u.)» (58) Amnn Amnn 1 Amnn i i 1 57 D efining (59) ; and u . = — u „ _ , op j& m nK p a com ponent of eq. b eco m es (60) im n n a Amnn (61) T his is the d e s ire d fo rm of the m a trix eigenvalue equation, w ritte n ; in a co m p act fo rm containing only the n o n -z e ro e le m e n ts of the i w hich the t r i a l v e c to r is not orthogonal ). S u ccessiv e ite ra tio n w ill d efects su ch a s im p u rity p a ir s , it is n e c e s s a ry to know w hat the e ig e n v e c to rs of the lo c a l m odes look lik e . To get the e ig e n v e c to rs fro m a G re e n 's function c a lc u la tio n re q u ire s ev alu atin g G re e n 's functions coupling d iffe re n t s ite s , w hich is som ew hat co m p lica te d . The ite ra tiv e c a lc u la tio n m eth o d is quite convenient h e r e , sin ce it g e n e ra te s an e ig e n v e c to r in the p ro c e s s of c a lc u la tin g the lo c a l m ode j m a trix . If the u. a a re com ponents of a tr ia l e ig e n v e c to r, the le ft ! Ip | side of the equation w ill y ield com ponents of a v e c to r w hich is ' c lo s e r to the dom inant e ig e n v e c to r ( o r the h ig h est e ig e n v e c to r to I give c lo s e r ap p ro x im atio n s to the eig e n v ec to r. b. E xam ple s To have a good in tu itiv e u n d erstan d in g of lo c a liz e d v ib ra tio n a l m o d es, p a rtic u la rly of the e ffe c ts that a re o b se rv e d fro m com plex 5 8 fre q u e n c y . Since the c a lc u la tio n is fea sib le fo r ra th e r la rg e defect ! re g io n s , m odes th a t a re not too w ell lo calized can be c o n sid ere d , although c o n v erg en ce b eco m es a p ro b lem if the m ode freq u en cy is not v e ry h igh. As an ex am p le, it is p o ssib le to solve fo r the e ig e n - : v e c to r of a lo c a liz e d v ib ra tio n a l m ode of a light im p u rity and then see how it a c q u ire s m o re of the c h a r a c te r of th e z o n e -c e n te r optic : m odes a s the im p u rity is m ade h e a v ie r and the m ode b ecom es le s s lo c a liz e d . Once the c a lc u la tio n is s e t up fo r the lo ca liz ed m ode of I | a lig h t im p u rity , the e ffect of fo rc e co n stan t changes on both the i j fre q u e n c y an d e ig e n v e c to r of a lo c a liz e d m ode can e a s ily be d e te r- 1 I i m in e d , sin c e changing a m a s s o r a fo rce co n stan t does not in c re a s e i i j th e d iffic u lty of th e c a lc u la tio n . i. F o rc e co n stan t changes fo r the j i B lo c a l m ode | I F o r the f i r s t ex am p le, we w ill co n sid er the effects of fo rce I | c o n stan t ch an g es in a m o d el fo r the b o ro n lo cal m ode in Si. F o r the Si la ttic e , u se only f i r s t n eig h b o r c e n tra l fo rce co n stan ts betw een the a to m s. F o r su ch a m o d el the m ax im u m freq u en cy of the u n p e rtu rb e d la ttic e is uj = (8 /3 )§ /M If the m a s s e s a re se t eq u al to m a x u n ity a n d th e fo rc e c o n sta n ts to 0 .3 7 5 , the freq u e n cie s obtained fo r th e lo c a l m o d es w ill a u to m a tic a lly be sc a le d re la tiv e to the m ax im u m phonon fre q u e n c y of th e u n p e rtu rb e d la ttic e . The b oron is r e p r e s e n te d by changing th e m a s s on one of the site s to . 357 (. 392) fo r j...................................................... ~ .................................................................. ’........ 59 11 : ( B). The fo rc e c o n sta n ts coupling th is site to its n eighbors w ere I ! m ade ad ju sta b le . A F o r tr a n p ro g ra m w as w ritte n to p e rfo rm the calcu latio n fo r I a 5 x 5 x 5 a r r a y of p rim itiv e c e lls . The b o ro n site w as in the c e n te r I c e ll of the a r r a y on th e f i r s t su b la ttic e . A tom s a t the su rfa c e s of the j a r r a y w ere c o n s id e re d to be bound to non-m oving ato m s outside the | j a rr a y . The in itia l t r i a l v e c to r w as a d isp lac em e n t only of the i im p u rity . The fre q u e n c ie s and e ig e n v e c to rs w ere c a lc u la te d for ! | s e v e ra l v alu es of B -S i fo rc e c o n sta n ts, ranging fro m 1. 5 to 0.45 of i | a S i-S i force c o n sta n t. F o u rte e n ite ra tio n s w e re u se d fo r the in itia l | calcu latio n w ith th e la r g e s t fo rc e co n stan t s ta rtin g w ith b o ro n m otion i j only fo r the in itia l t r i a l v e c to r. W hen the fo rc e co n stan t w as ! changed the e ig e n v e c to r fro m the p rev io u s c a lcu latio n w as u sed as I the in itia l t r i a l r a th e r th an s ta rtin g ag ain fro m only an im p u rity m otion; nine ite r a tio n s w e re m ade a fte r e ac h change in the force I . c o n stan ts. | The c a lc u la te d lo c a l m ode fre q u e n c ie s a re show n in F ig u re 5, along w ith the r e s u lts of a G re e n 's function c a lcu latio n by E llio tt an d P feu ty for the dependence of the b o ro n lo c a l m ode freq u e n cy on the f ir s t neighbor c e n tr a l fo rc e c o n stan t. The ite ra tiv e calcu latio n gives a m uch la r g e r v a ria tio n of the lo c a l m ode freq u e n cy fo r a given change in the fo rc e c o n sta n t.th a n does the G re e n 's function calcu latio n . A n im p o rta n t d iffe re n c e b etw een the two calcu latio n s is 60 F ig u re 5. C alcu lated lo c a l.m o d e freq u en cy for b o ro n in Si v s. re la tiv e fo rce c o n stan t, ^ g^/ g^ g^ . T he cu rv e s ( -------- ) and ( ----------- ) a r e the r e s u lt fo r and of the ite ra tiv e c a lcu latio n using a f i r s t n eig h b o r c e n tra l fo rce m odel fo r the la ttic e : ( ............) and ( ..................... ) a r e the re s u lts for **B and l^B obtained by E llio tt and P feu ty fro m a G re e n 's function calcu latio n . 61 I 1 \ \ \ \ \ \ \ 0.5 i • l 5 ' ■ - ’ \ . . . ' i i \ [ ‘ " 62 th a t the G re e n ’s function c a lc u la tio n im p lic itly co n tain s a m o d el fo r the Si la ttic e th a t in clu d es c e n tra l and n o n -c e n tra l fo rce co n stan ts ; fo r s e v e ra l sh e lls of n eigh bors and only the f ir s t neighbor fo rce c o n sta n t is v a rie d , w hile in the calcu latio n re p o rte d h e re the la ttic e i m o d el h as been r e s tric te d to f i r s t neighbor c e n tra l fo rc e s , so th a t th e to ta l coupling of the b o ro n to the la ttic e is being v a rie d . The d isp lac em e n ts c a lc u la te d fo r the b o ro n and its f ir s t ; n e ig h b o rs fo r s e v e ra l v alu es of the B -S i fo rce co n stan t a re show n j in T able 5; it is in te re s tin g to note th at the v a ria tio n w ith freq u e n cy : of the ra tio of im p u rity d isp la c e m e n t to f ir s t neighbor d isp la c e m e n t ; in th is c a se is su b sta n tia lly d ifferen t fro m the case w here the fre q u e n c y is changed by changing the m a ss of the im p u rity . In a | sim p le iso to p ic su b stitu tio n m o d el th is ra tio d e c re a s e s ra p id ly as i ! th e im p u rity m a ss is in c re a s e d and the lo ca l m ode freq u en cy ) a p p ro a c h e s u j , but fo r th is c a se the ra tio changes v e ry little I T lc L 2 C o v e r a w ide range of fre q u e n c ie s. i j ii. Isotopic S ubstitution i In the c a se of an iso to p ic su b stitu tio n , w h ere the im p u rity -S i fo rc e c o n sta n ts a re the sam e a s the S i-S i fo rc e c o n sta n ts, the lo c a l m ode should m erg e into the h ig h est freq u en cy band m ode in the lim it th a t the im p u rity m a s s a p p ro a ch e s the h o st m a s s and the lo c a l m ode fre q u e n c y a p p ro a ch e s the m ax im u m in -b an d freq u en cy . F ig u re 6 show s an a to m w ith its fo u r n eig h b o rs, along w ith the 11 T able 5 . Frequency and d isp la c e m e n ts f o r th e l o c a l mode o f B in S i v s . the B -S l fo r c e c o n s t a n t . N orm alized B -S i Force C onstant N orm alized Frequency D isp la cem en ts ( x , y , and z components) Boron Im p u rity F i r s t 1 N eighbors 2 (s e e F igu re 3 6) 1.50 1 .5 8 5 .9^0 0 0 - .0 9 8 - .0 9 8 - .0 9 8 - .0 9 8 .098 .0 9 8 - .0 9 8 - .0 9 8 .098 - .0 9 8 .098 - .0 9 8 1 .2 0 1.^21 .937 0 0 - .1 0 0 - .1 0 0 - .1 0 0 - .1 0 0 .100 .100 - .1 0 0 - .1 0 0 .100 - .1 0 0 .100 - .1 0 0 0 .9 0 1 .2 3 6 .931 0 0 - .1 0 3 - .1 0 3 - .1 0 3 - .1 0 3 .103 .10 3 - .1 0 3 - .1 0 3 .1 0 3 - .1 0 3 .10 3 - .1 0 3 0.6 0 1.026 .897 0 0 - .1 1 1 - .1 1 3 - .1 1 3 - .1 1 1 .1 1 3 .1 1 3 - .1 1 1 - .1 1 3 .1 1 3 - .1 1 1 .113 - .1 1 3 X F ig u re 6. Atom in a diam ond la ttic e w ith its four n e a re s t n e ig h b o rs, and th e ir re la tio n sh ip to th e cubic axes of the c ry s ta l. The d ire c tio n s of th e bonds fro m th e c e n tra l atom to the v a rio u s n eig h b o rs a r e [ i l l ] to 1, [ i ll ] to 2, [ f l l ] to 3, [ I I I ] to 4. I.............. ’.................. * .............. • 65 ! | cubic ax es of the diam ond s tru c tu re . The h ig h est freq u en cy band i m ode in a diam ond la ttic e is e s s e n tia lly a rig id v ib ra tio n of the ; ! ! two su b la ttic e s a g a in st e ac h o th e r. If the d isp lac em e n t of the c e n te r j i i a to m is along the x a x is , the d isp la c e m e n ts of its neighbors a re a lso I along the x a x is but in the opposite d ire c tio n . If the c e n te r a to m is 1 I a lig h t im p u rity giving a h ig h freq u e n cy lo c a liz e d m ode w ell above J i I i ; the m ax im u m band m ode freq u e n cy the d isp lacem en t of the n e ig h b o rs i I w ill be m o re n e a rly along the bonds b etw een the im p u rity and the I i ! n eighb ors r a th e r th an a n ti- p a r a lle l to the im p u rity m otion as w ith ! the band m o d e. i I a . C e n tra l F o rc e M odel I J The e ig e n v e c to rs and fre q u e n c ie s w ere c alc u la te d fo r s e v e ra l I i lvalues of im p u rity m a s s by u sin g a la ttic e w ith only f i r s t neighbor ! c e n tra l fo r c e s . The c a lc u la tio n w as done w ith a 5 X 5 x 5 a r r a y of ! t ip rim itiv e c e lls a s p re v io u sly d e sc rib e d , w ith the exception th a t i j p e rio d ic b o u n d ary conditions w ere u se d so a s not to c o n stra in the e ig e n v e c to r to z e ro a t the s u rfa c e s of the a r r a y . The d isp lac em e n ts of the im p u rity and its f i r s t n eighbors fo r a n in itia l im p u rity d isp lac em e n t along the x a x is a re given in T able 6 ', along w ith the fre q u e n c ie s. F o r th is c a s e , ev en a s the im p u rity becom es n e a rly as m a ssiv e a s a h o s t ato m , the im p u rity d isp lac em e n ts re m a in n e a rly along the bond d ire c tio n s-a n d the lo ca l m ode does not begin to re se m b le the h ig h e s t freq u e n cy band m ode; as w ill be shown, T able 6 . L ocal mode o f a l i g h t im p u rity i n a diamond l a t t i c e , f o r a model o f th e l a t t i c e w ith o n ly f i r s t n eig h b o r c e n t r a l fo r c e c o n s t a n t s . The freq u en cy and th e d isp la c e m e n ts o f th e im p u rity and i t s f i r s t n e ig h b o rs are g iv e n f o r s e v e r a l v a lu e s o f th e im p u rity m ass. N orm alized Im p u rity Mass N orm alized Frequency D isp la cem en ts ( x , y , and z components) Im p u rity F ir s 1 t N eighbors 2 (s e e F igu r 3 e 6) .3 1 . ^ 0 .960 0 0 - .0 7 8 - .0 7 8 - .0 7 8 - .0 7 8 .0 7 8 .078 1 1 . . . 0 0 0 0 0 0 0 00 C O C O 00 0 0 0 . . . 1 1 .5 1 .1 9 5 .890 0 0 - .1 2 7 - .1 2 7 - .1 2 7 - .1 2 7 .127 .127 - .1 2 7 .1 2 7 - .1 2 7 - .1 2 7 - .1 2 7 .127 • 7 1.081 .783 0 0 - .1 6 6 - .1 6 7 - .1 6 7 - .1 6 6 .167 .167 - .1 6 6 .167 - .1 6 7 - .1 6 6 - .1 6 7 .167 .9 1.0 2 0 .653 0 , 0 - .1 8 5 - .1 9 2 - .1 9 2 - .1 8 5 .192 .192 - .1 8 5 .192 - .1 9 2 - .1 8 5 - .1 9 2 .192 •95 1 .0 0 9 .625 0 0 - .1 8 6 - .1 9 6 - .1 9 6 - .1 8 6 .196 .196 - .1 8 6 .196 - .1 9 6 - .1 8 6 - .1 9 6 .196 how ever, th is fe a tu re depends stro n g ly on the p a rtic u la r fo rc e co n stan t m o d el. b. Inclusion of N o n -C en tral F o rc e s If c a lc u la tio n s a r e to be m ade for lo c a l m odes w ith fre q u e n c ie s n e a r th e m ax im u m u n p ertu rb ed phonon freq u en cy , the I m odel fo r th e u n p e rtu rb e d la ttic e should be fa irly good. The c e n tra l fo rc e -o n ly m o d el is not a p a rtic u la rly good d e scrip tio n of th e Si la ttic e ; n o n -c e n tra l fo rc e s should a ls o be included. j The s im p le s t fo rm fo r adding n o n -c e n tra l fo rc e s to E q. 58 |is to f ir s t c o n sid e r th e c a se w h ere the c e n tra l and n o n -c e n tra l fo rc e i co n stan ts a r e e q u al. The fo rc e on an atom (Jt>, m , n, k) cau sed by a re la tiv e d isp la c e m e n t (u . -u .) is in the d ire c tio n of the re la tiv e r j&mnH i t d isp la c e m e n t fo r th is high sy m m e try , ra th e r than along f. a s w as j :the c a s e w ith only a c e n tra l fo rc e constant. A fo rc e of th is s o r t i jplus a c e n tra l fo rc e can be u se d to e x p re ss the to ta l fo rc e w hen the ! c e n tra l and n o n -c e n tra l fo rc e constants a re not equal. E quation 58 i j can thus be re w ritte n as i 4 " ^ jG m n n 1 0 U i m n n ~ “ 0-p ) fu_ g t n n x “ u ^ X ri + P * i 3i > ’ w h e re p is the ra tio of n o n -c e n tra l to c e n tra l fo rc e c o n sta n ts. F o r Ge o r Si, p is about 0. 3 (P h illip s 1970). e . 3 (62) 68 | T he e ig e n v e c to rs and freq u e n cie s of the lo cal m ode fo r an ; iso to p ic s u b s titu tio n w e re c alcu late d including n o n -c e n tra l f ir s t j 'n eig h b o r f o r c e s . The fre q u e n c ie s and the d isp lac em e n ts of the ;im p u rity an d its f ir s t n eig h b o rs a r e shown in T able 7. The re s u lts a r e s im ila r to th e c e n tra l-fo rc e -o n ly c a se , but th e re a re s e v e ra l |obvious d iffe re n c e s . The m otion of th e neighbors is n e v e r d ire c tly along the im p u rity -n e ig h b o r bonds, as was the c a se w ith only c e n tra l I fo rc e s , but alw ays h a s a com ponent a n ti-p a ra lle l to the m otion of j jthe im p u rity . T his a n ti-p a r a lle l com ponent a p p e a rs to a p p ro a ch a j fixed fra c tio n of the d isp la c e m e n t of the neig h b o rs as th e lo c a l m ode !freq u en cy b e c o m e s la rg e . As the lo c a l m ode frequency b eco m es j | s m a lle r, a p p ro a c h in g the m ax im u m u n p e rtu rb e d la ttic e freq u e n cy , jthe a n ti- p a r a lle l com ponent b eco m es la rg e r, so th e lo ca l m ode |m e rg e s sm o o th ly into th e h ig h e st freq u en cy band m ode. A nother fe a tu re is th a t a g r e a te r d iffe re n c e in m a s s betw een the im p u rity and h o s t is re q u ire d to p ro d u ce a lo ca l m ode w hen the n o n -c e n tra l fo rc e s a r e in clu d ed . In th e c e n tra l-fo rc e -o n ly m odel, th e re is a w ell i s e p a ra te d lo c a l m ode w hen the im p u rity m a ss is 0. 95 of a h o st m a s s , b u t, when n o n -c e n tra l fo rc e s a r e included, th e lo c a l m ode h as a lm o st m e rg e d in to th e band m o d es when th e im p u rity m a s s is 0. 90 of a h o st m a s s . The re a s o n fo r the d iffe re n c e in the re s u lt of the lo c a l m ode c a lc u la tio n s w ith and w ithout n o n -c e n tra l fo rc e s is th a t th e diam ond la ttic e is s h e a r -u n s ta b le if th e re a re no n o n -c e n tra l fo rc e s w hen T able 7 . L ocal mode fr e q u e n c y , and im p u rity and f i r s t n eig h b o r d is p la c e m e n ts , f o r a l i g h t s u b s t i t u t i o n a l im p u rity in a diamond l a t t i c e . The l a t t i c e model in c lu d e s f i r s t n eig h b o r c e n t r a l and n o n -c e n tr a l fo r c e c o n s t a n t s . N orm alized Im p urity Mass’ N orm alised Frequency D isp la cem en ts (x , y , and z components) Im purity F i r s t 1 N eighbors 2 (s e e F igu re 3 6) 4 .3 1 .3 7 1 .977 0 0 - .0 9 1 - .0 3 9 - .0 3 9 - .0 9 1 .039 .039 - .0 9 1 .039 - .0 3 9 - .0 9 1 - .0 3 9 .039 .5 1 .1 1 8 .912 0 0 - .1 7 0 - .0 6 7 - .0 6 7 - .1 7 0 .067? .067 - .1 7 0 .067 - .0 6 7 - .1 7 0 - .0 6 7 .067 .7 1 .0 1 2 .701 0 0 - .2 4 1 - .0 7 1 - .0 7 1 - .2 4 1 .071 .071 - .2 4 1 .071 - .0 7 1 - .2 4 1 - .0 7 1 .071 .75 1 .0 0 2 .560 0 0 CMN- O- C\1 C M O O . . . I l l - .2 2 9 .057 .057 - .2 2 9 .057 - .0 5 7 - .2 2 9 - .0 5 7 .057 * T h is c a s e had n ot y e t converged w e ll a f t e r 25 i t e r a t i o n s s t a r t i n g w ith th e e ig e n v e c to r o b ta in ed f o r M * = 0 .7 a s a t r i a l v e c t o r . ; 70 | ■ only f i r s t neighbor fo rc e s a re c o n sid e re d . If the n o n -c e n tra l fo rc e s i I a re left out, the tra n s v e rs e a co u stic m odes of the la ttic e w ill have ; z e ro freq u e n cy . The c o rre sp o n d in g tra n s v e r s e o p tic m o d es w ill a ll have freq u e n cy u ) . The re s u lt is a la rg e d en sity of s ta te s a t u ) , 7 m ax ® 7 m ax w hich, as is show n in se c tio n C of th is c h a p te r, allow s a lo c a l m ode to e x is t fo r an in fin ite s im a l m a ss defect. A lso, as the m a s s defect ! is m ade s m a ll and the lo c a l m ode freq u en cy a p p ro a ch e s the m ax im u m I i band m ode freq u e n cy , the lo c a l m ode e ig e n v e c to r w ould be e x p ec te d j to re s e m b le the h ig h est freq u en cy la ttic e m ode. F o r the c a se of 1 n o n -c e n tra l fo rc e s only, the tra n s v e rs e optic m odes fo r a ll q have i i freq u e n cy u o , w h e re a s only the z o n e -c e n te r optic m ode h a s th is { rxictx | freq u e n cy if n o n -c e n tra l fo rc e s a re included. T his acco u n ts fo r the i | d ifferen c e in the lo cal m ode e ig e n v ec to r in the two c a s e s . 4. C o m p ariso n of the Ite ra te d M a trix M u ltip li- j c atio n M ethod w ith O th er M ethods ! ' " ~ ~ ~ ”” ~ ~ ~ ~ I j The ite ra tiv e calcu latio n m ethod th at h as been developed h e re p ro v id e s a convenient com putational technique fo r finding both the fre q u e n c ie s and e ig e n v e c to rs of lo c a liz e d v ib ra tio n a l m odes fro m the eq u atio n s of m otion fo r the defect and a reg io n of th e la ttic e su rro u n d in g the d efect. Unlike o th e r calcu latio n s u sin g a defect re g io n th a t have a p p e a re d in the lite ra tu r e , th is m eth o d can be u se d to solve fo r ju s t the local-.m odes, ignoring the band m o d es, and does not depend on the u se of sy m m e try to red u ce the co m p lex ity of the .................................................................7i ; p ro b le m . T his m eth o d m ak es fe a sib le the c a lcu latio n of e ig e n v e c to rs of th e lo c a l m odes o v e r extended reg io n s, w hich is of c o n sid e ra b le lv alu e in obtaining an in tu itiv e g ra sp of the effects of in te ra c tio n s of I d e fe c ts on lo c a l m ode fre q u e n c ie s. G re e n 's function c a lc u la tio n s l w ill, in p rin c ip le , give e ig e n v e c to rs fo r lo c a l m o d es, but it p ra c tic e the re q u ire d G re e n 's functions a re not e a s ily e v alu ated , a s they : re q u ire a d e ta ile d know ledge of the d isp e rsio n c u rv e s an d e ig e n v e c t o r s of the h o st la ttic e . V arious ap p ro x im ate G re e n 's function I c a lc u la tio n c a lc u la tio n s ca.n give the freq u e n cie s but not the | e ig e n v e c to rs . I The p rin c ip a l lim ita tio n of the m ethod is in the fo rc e c o n stan t ! im o d e lu s e d to d e s c rib e the la ttic e . The calcu latio n is g re a tly i ‘sim p lifie d , p a rtic u la rly in th re e d im en sio n s, if only f i r s t neig h b o r I I in te ra c tio n s betw een ato m s a re included. T his does not give a p a rtic u la rly good d e s c rip tio n of m o st c ry s ta ls . F o rc e c o n sta n ts |co u p lin g seco n d an d fu rth e r n eighbors a re sim p le to in clu d e, but j s e rio u s ly in c re a s e the co m p u tatio n al tim e involved. Anothe r lim ita tio n , w hich is not u su a lly too s e rio u s , is th at the co n v erg en ce is v e ry slow if the lo c a l m ode freq u en cy is clo se to the m ax im u m band m ode freq u e n cy . C. C ondition fo r the O ccurence of a L o c a liz e d M ode: The T h re sh o ld V alue fo r the M ass D efect A high freq u e n cy lo c a liz e d m ode m ay re s u lt fro m th e s u b s ti- 72 tu tio n of an a to m of lig h te r m a ss fo r one of the a to m s of the h o st c ry s ta l. H ow ever, even in the case th at the fo rc e c o n sta n ts a re not . changed, th e re is g e n e ra lly som e m in im u m value of the m a s s defect i I e = ( M -M ' )/M th a t is re q u ire d b efo re a lo c a l m ode w ill a p p e a r. S e v e ra l a u th o rs have m ade c alcu latio n s of lo ca l m ode fre q u e n c ie s in ; Si, and have o b tain ed values fo r e . ran g in g fro m 0. 075 ( D aw ber, j m in o o \ i 1963b ) to 0 .2 5 ( A n g re ss, 1964 ), depending on the e s tim a te d la ttic e I s p e c tru m fo r the u n p e rtu rb e d Si la ttic e . i The m in im u m value of the m a ss d efect can be found fro m j E q. 11 by le ttin g the lo c a l m ode freq u en cy ap p ro ach the m ax im u m i | la ttic e freq u e n cy . One obtains ! - 1 i e . =--------------------------------------- m m 2 U ) m ax fflm ax ! » U ) - U ) ; 0 m ax | I w h e re the in te g ra l h as b een w ritte n in te rm s of fre q u e n c ie s, r a th e r ! th an the sq u a re d fre q u e n c ie s u se d in E q. 16. It h as b een noted by L ifs c h itz ( 1956 ) th a t th is in te g ra l is fin ite fo r a r e a lis tic d e n sity - o f -s ta te s g(u)) fo r a th re e d im en sio n al c ry s ta l, but d iv e rg e s fo r the o n e -d im e n sio n c a se so th a t a lo c a liz e d m ode m ay o c c u r fo r a r b itr a r ily s m a ll e . The b e h a v io r of the in te g ra l depends v e ry stro n g ly on the n a tu re of g(uu) n e a r u u , sin ce the d e n o m in ato r of ° ° m ax the in te g ra l goes to z e ro at u ) . Some in sig h t into the re la tio n sh ip mEX b etw een the phonon s p e c tru m of the h o st and Sm ^n can be g ain ed by / <**> 73 c o n sid e rin g rea so n a b le m o d els of g(u)) n e a r id i ° m ax In the v ic in ity of U ) , g(uj) g e n e ra lly h a s the fo rm g(uu) = rnsix iA ( 0) - U ) )^ , w h ere the co n stan t A is a m e a s u re of how ra p id ly ' m ax g(uj) in c r e a s e s as u j is d e c re a se d . A m e a su re of the dependence of i : e . o n the e x te n t to w hich the sta te s a re bunched n e a r the top of : m m £ ;the phonon s p e c tru m can be obtained by a ssu m in g g(uu) = A(ou -uu) m2>x i fo r id . < u j < u u and g(uj) = 0 fo r 0 < u u < u ) . a s show n in F ig . 7. mxn — — m ax ° — m in ! Dim | The g(uu) is th en n o rm a liz e d to C g(u>) d(uu) = 1 by se ttin g 0 _ 3. A = ■ § • ( U ) -(A ) . ) 2 . The in te g ra l in E q. 63 b eco m es 2 ' m ax m m ' 5 i I D i m ax . A I ' V 1 ^ / ( J U 2 2 m in u ) - I D m ax (64) 1 T his can be s e p a ra te d into U ) ( A ) m ax i m ax x ( u ) -a) )*duu r ( id -u) )^duu ' — 1 m ax f ( V a x - ^ , f J ( U ) + U U ) J 2 ( J U J ( U ) + U U ) J U J -U J m ax ' m ax m ax u j U ) . mxn mxn and re d u c e d to (65) ’ ^ m ax^ m in ^ , V a x ’ V i n r (x+2V a x ) d x + r j x J X J 2u) J x ) -g m a x *' J x a 2 u u m ax The re s u ltin g value one o btains is ( 66) 8 3 € . = ■ ? C mxn 3 [ i n f r § - ] (67) . * " > '■ N 3 3 cr> u w mln m a x C J ^mln max (a) (b) F ig u re 7. A pproxim ations to the phonon d e n sity of s ta te s fo r a re a l c ry s ta l. a) P a ra b o la b) P a ra b o la and d e lta function. w h ere 2 1 j, ^min ~ z 1 ------- - S-1 <68> V, m a x The re s u lt show n in F ig . 8 is th at the th re s h o ld e fo r a lo ca l m ode to a p p e a r is z e ro if g(uu) is bunched up n e a r u u , and m&x in c re a s e s as g(co) is s p re a d out so th at it is s m a lle r in the v icin ity of u u m ax The d e n s ity -o f-s ta te s of any r e a l c y rs ta l is not w ell a p p ro x i m a te d by the p a ra b o lic fo rm of the above ex am p le, a s can be se en 1 by c o m p a riso n w ith the c u rv e s in F ig . 1, w hich have b een d e riv e d ! fro m e x p e rim e n ta l m e a su re m e n ts of d isp e rsio n c u rv e s . F o r Si o r i Ge, the d e n s ity -o f-s ta te s does s ta r t fro m the top w ith a s m a ll I I p a ra b o lic p a rt, but then h a s a pronounced peaked s tru c tu re . A j b e tte r ap p ro x im atio n th an a sim ple p a ra b o la fo r g(uu), b ut w hich is i s till sim p le enough to w ork w ith, is ! - - h j g(uu) = a f (u a -uu . ) 2(u u - u u ) + p 6 (u u . ) (69) & * m ax m m m ax r m m ' ' I fo r u u . < u u < u u , and g(uu) = 0 fo r u u < u u . . m m — — m ax m m To m a in ta in the n o rm a liz a tio n of g(uu) , a + ( 3 = 1. F o r p = 1, 2 ^m in g(uu) = 6 (u u . ) so th a t e . = 1 ------ ------ as is show n by the so lid line ' m m m m -2 m ax in F ig u re 8. Sm in ^o r function is alw ays la r g e r th an th a t given by th e p a ra b o lic g(uu) . If we le t be the value o b tain ed w ith the (2) p a ra b o lic g(uu) (a = l) and f^a-t obtained w ith the 6 -fu n ctio n 76 F ig u re 8. e v s. ( u j v - «J _ _ * _ ) / . Solid c u rv e is fo r m in m ax m in m ax g(ou) = 6(a,m in ) . D ashed curve is fo r the p a ra b o lic g(«j) . 0.8 0.6 min 0.4 0.2 Q2 0 0.4 L0 0.6 0.8 o j - c j , max min CJ max I ; (P=l) , th en the in -b etw e en c a se s a re given by -1 77 m m a s '1'. e(2> m m m m (70) t _ j E ven b e tte r ap p ro x im atio n s can be d ev ised w hich do not have so | m uch ten d en cy to o v e r-e m p h a siz e the h ig h e r fre q u e n c y s ta te s , but ! th ey w ould not re a lly add anything new to the q u a lita tiv e re s u lts ! a lre a d y o b tain ed . A s m a ll o r v an ish in g value fo r e . is a s s o c ia te d w ith a i m m I d e n s ity -o f-s ta te s th a t is stro n g ly peaked n e a r u u o r fin ite at J m ax i u ) : la rg e v a lu e s of e . re s u lt w hen th e re a re v e ry few sta te s ! m ax m m n e a r u u m ax | CH A PTER III I E x p e rim e n ta l P ro c e e d u re s The e x p e rim e n ta l w o rk involved in th is r e s e a r c h in clu d ed i ; grow th of c ry s ta ls of sp e c ifie d com position, c h a ra c te riz a tio n of the i i c ry s ta ls w ith re g a r d to co m p o sitio n and hom ogeniety , p re p a ra tio n of ; sa m p le s fro m the c ry s ta ls , and in fra re d tra n s m is s io n m e a s u re m e n ts ! of th e sa m p le s. A. C ry s ta l G row th The c ry s ta ls u se d in th is study w ere a ll grow n fro m a m e lt, j j e ith e r by the C z o c h ra lsk i technique o r the v e rtic a l B rid g e m a n j tec h n iq u e . F o r the stu d ies th a t w ere done, p o ly c ry sta llin e sa m p le s | a re su ita b le ; som e of the ingots w e re single c ry s ta ls and o th e rs j j p o ly c ry s ta llin e . The p rin c ip a l re q u ire m e n t fo r the sa m p le s w as j i th a t th ey be hom ogenous and be grow n w ith a sp e cified co m p o sitio n 1 an d sp e c ifie d doping le v e l. | In g e n e ra l, a doped c ry s ta l w ill not have the sa m e c o m p o sitio n a s th e m e lt fro m w hich it is grow n. As a m e lt is so lid ifie d , im p u ritie s m ay be e ith e r p re fe re n tia lly in clu d ed in o r ex clu d ed fro m th e so lid . If, a t e q u ilib riu m , an im p u rity h a s a c o n c e n tra tio n C s in th e so lid w hen the c o n c e n tra tio n in the liq u id is C^, th e n C g/C ^ = kQ is defined a s the e q u ilib riu m d istrib u tio n co efficien t. An im p u rity th a t r a is e s the m e ltin g te m p e ra tu re h as k > 1; one th a t lo w e rs the o m e ltin g te m p e ra tu re h a s k < 1. T his can be illu s tr a te d b y the phase ° 78 79 1 i d ia g ra m fo r the Ge Si allo y sy ste m in F ig u re 9. A t low Si X X • * X c o n c e n tra tio n s, c o n sid e rin g Si to be the so lu te o r im p u rity , the ; d istrib u tio n c o efficien t of Si in Ge is about 4 o r 5; a t low Ge concen- : tra tio n s the d istrib u tio n co efficien t of Ge in Si is about 0. 2 - 0. 4. In p ra c tic e , c ry s ta ls do not grow at e q u ilib riu m , an d k in etic e ffe c ts m ay be im p o rta n t. In any sy ste m w ith a n o n -z e ro ra te of ! so lid ific a tio n , a c o n c e n tra tio n g rad ie n t of the so lu te is g e n e ra te d a t ; the in te rfa c e if the d istrib u tio n co efficien t is not u n ity , sin ce the ; so lu te is e ith e r re je c te d into the m elt o r se le c tiv e ly re m o v e d fro m ; the m e lt a t the grow th in te rfa c e . T his c o n stitu te s a n e t flux of solute I e ith e r into o r out of the in te rfa c e . In the b o u n d ary la y e r of the m e lt | a d ja c e n t to the in te rfa c e , th e re is little m ix in g w ith the bulk of the i 1 i m e lt. The so lu te is tra n s p o rte d by diffusion, so th is flux re q u ire s i | a c o n c e n tra tio n g ra d ie n t of solute in the m e lt a d ja ce n t to the in te rfa c e , i ' I One r e s u lt of the c o n c e n tra tio n g ra d ie n t is th a t the o b se rv e d i ra tio of solute c o n c e n tra tio n s in the so lid an d the m e lt w ill d iffe r I fro m the e q u ilib riu m d istrib u tio n co efficien t, as the so lid liq u id in te rfa c e h as a d iffe re n t so lu te c o n ce n tra tio n fro m the a v e ra g e o v e r the m e lt. A nother is th a t th e c o n ce n tra tio n g ra d ie n t of so lu te gives r is e to a v a ria tio n of the fre e z in g te m p e ra tu re in the m e lt su ch th a t the fre e z in g te m p e ra tu re of the m e lt in c re a s e s w ith d ista n c e f ro m the in te rfa c e . S upercooling of the m e lt, c a lle d c o n stitu tio n a l s u p e r co o lin g , m ay th e n o c c u r if the te m p e ra tu re g ra d ie n t of the in te rfa c e 80 F ig u re 9. a. P h a se d iag ra m for the G e-S i sy ste m . The liq u id u s c u rv e is th a t d eterm in ed e x p erim e n tally by S tb h r and K lem m . T he dash ed lin e is the solidus cu rv e c a lcu late d by S te in in g e r fro m the e x p e rim e n ta l liquidus; the points a r e the e x p e rim e n ta l v a lu e s fo r the solidus d eterm in ed by S tb h r and K lem m . b. Si in Ge; a n exam ple of an im p u rity w hich r a is e s th e m eltin g te m p e ra tu re . The d istrib u tio n c o efficien t fo r Si in Ge is g r e a te r than unity. c. Ge in Si; an im p u rity w hich lo w e rs the m e ltin g te m p e ra tu re . T he d istrib u tio n coefficient of Ge in Si is le s s than unity. 81 1300 o o _ £ 1200 3 5 E 1100 a. 2 UJ H 1000 900 .20 .40 ,60 .80 Ge Si ATOM FRACTION Si (a) Si in Ge Ge in Si % <l (b) (c) 82 I is not s te e p e r th an the g ra d ie n t of the fre e z in g te m p e ra tu re of the j m e lt. B oth of th e se phenom ena a re illu s tra te d in F ig u re 10. In the grow th of doped se m ic o n d u cto r c ry s ta ls , changes in the ; 7 i I i ) effectiv e d istrib u tio n co efficien t of dopants m ay be se e n a s the grow th ra te is v a rie d . F lu c tu a tio n s in the grow th ra te , su ch as th o se j w hich m ay a r is e fro m te m p e ra tu re v a ria tio n s , m ay give ris e to in - ' hom ogenous doping. i If c o n stitu tio n a l su p e rco o lin g o c c u rs , the grow th in te rfa c e | g e n e ra lly b eco m es u n sta b le , re s u ltin g in uneven grow th w hich e n tra p s . | p o rtio n s of th e m e lt an d p ro d u ce s an inhom ogenous so lid . D ism ukes ; and E k stro m ( D ism ukes 1965 ) have in v e stig a te d the conditions fo r i | j hom ogenous grow th of Ge Si a llo y s. T hey show th a t a sim p le I <X X “• X i ! m o d el fo r d iffu sio n -c o n tro lle d grow th g iv es a re la tio n sh ip betw een j the grow th ra te R an d te m p e ra tu re g ra d ie n t G to avoid c o n stitu tio n a l su p e rc o o lin g . The re la tio n sh ip is G /R > m (C -C )/D , i t S . w h ere C . an d C a re the c o n c e n tra tio n s of Ge in the liq u id and so lid , i i t s m is the slope of the liq u id u s c u rv e a t C , and D is th e diffusion i t c o efficien t of Ge in the .m elt. T h e ir e x p e rim e n ta l d ata fo r grow th I conditions giving hom ogenous ingots is fit re a so n a b ly w ell a ssu m in g 4 2 - 1 a c o n stan t D of 1. 3 x 1.0 cm sec fo r the e n tire range of m e lt o co m p o sitio n . F o r a te m p e ra tu re g ra d ie n t of about 50 C /c m , th is re q u ire s grow th r a te s le s s th an one inch p e r day fo r Ge Si. allo y s X X " X w ith 0. 4 < x < 0. 7; la r g e r ra tio s of grow th ra te to te m p e ra tu re : F ig u re 10. S teady sta te d iffu sio n -lim ited grow th fro m a m e lt containing an im p u rity . : a. If k Q < 1, the im p u rity is re je c te d fro m the so lid a t the in te rfa c e . ; b. If k > 1, the im p u rity is e x tra c te d fro m the m e lt a t the ! in te rfa c e . T he re s u ltin g im p u rity g ra d ie n t gives an im p u rity c o n c e n tra tio n a t the in te rfa c e th at is d iffe re n t fro m C j , i i the a v e ra g e o v e r the m e lt. The effective d istrib u tio n co ef- i fic ie n t b e co m es ( C . / C . )k . ; ' 1 1 ' o | c. T he im p u rity g ra d ie n t p ro d u ce s a free zin g te m p e ra tu re j g ra d ie n t in the m e lt, such th at the fre e z in g te m p e ra tu re in c re a s e s w ith d ista n c e fro m the in te rfa c e . If the te m p e ra tu re I | g ra d ie n t in the m e lt is le s s than the c ritic a l valu e, p a r t of the j I m e lt a d ja c e n t to the in te rfa c e is su p erco o led ( show n by the i shaded re g io n ) and the in te rfa c e is u n sta b le . D ISTA N C E SOLID— 4 — MELT F R E E Z I N G T E M P E R A T U R E \ V A \ A* i o XI -1 vvV „—-O > > 1 — / " xp- 1 P 4 O ' \ XI V -H : V ' \ o IMPURITY IMPURITY C O N C E N T R A T I O N C O N C E N T R A T I O N 0 0 j ^ I i 85 1 g ra d ie n t a re allo w ed fo r a llo y s com posed m o stly of one of the | c o n s titu e n ts . Stable te m p e ra tu re c o n tro l at the grow th in te rfa c e is n e c e s s a ry fo r hom ogenous grow th of a llo y s, sin ce a d e c re a se in te m p e ra tu re w ill cau se a te m p o ra ry in c re a s e in the grow th ra te w hich m ay be ! j enough to c au se c o n stitu tio n a l su p erco o lin g to o c c u r. As w as 'm e n tio n e d above, te m p e ra tu re changes m ay a lso re s u lt in inhom o- 1 genous doping. T h ere a re s e v e ra l c a u se s fo r te m p e ra tu re v a ria tio n s jin c ry s ta l grow th s y s te m s . One is la c k of a good te m p e ra tu re se n sin g land c o n tro l s y s te m fo r the c ry s ta l-g ro w th fu rn a c e . A nother, w hich j |is im p o rta n t in C z o c h ra lsk i grow th sy ste m s w h ere the ingot is t ! j ro ta te d a s it is grow n, is a la c k of a x ia l sy m m e try in the te m p e ra tu re I p ro file . If th e re is th is la c k of sy m m e try , then p a rts of the ingot |a re a lte rn a te ly h e a te d and cooled as it is ro ta te d . C onvection I I c u rr e n ts , e ith e r in the a tm o sp h e re a ro u n d the c ry s ta l o r w ithin the j jm e lt, can p ro d u ce te m p e ra tu re flu ctu atio n s th a t c a n a ffect grow th. ! jit is to be noted th a t slow d rifts in te m p e ra tu re a r e u su a lly of little consequence a s r e g a rd s h om ogeneity of the c ry s ta ls , but ra p id flu ctu atio n s a re to be av o id ed w h en ev er p o ssib le . 1. S i-R ic h Ge Si, x 1-x E x cep t fo r one ingot containing 2% Ge, obtained fro m B ell T elephone L a b o r a to rie s , the Ge Si ingots w ith x < 0. 12 w ere X X “ X grow n in an in d u ctio n h e a te d C z o c h ra lsk i p u lle r m a n u fa c tu re d by 86 'L e p e l H igh F re q u e n c y L a b o ra to rie s . The m elt w as contained in a !fu sed q u a rtz c ru c ib le w hich in tu rn w as inside a g rap h ite s u s c e p to r. iT he te m p e r a tu re c o n tro l sy ste m w as a L eeds and N o rth ru p | "R ayotube" ra d ia tio n s e n s o r focused on the bottom s u rfa c e of th e !g ra p h ite s u s c e p to r to se n se its te m p e ra tu re . The R ayotube output w as c o m p a red to a s e t v o ltag e, and the d ifferen ce w as u se d to d riv e ja L eed s N o rth ru p "M -L in e" th re e -a c tio n c o n tro lle r. The c o n tro lle r | jin tu rn o p e ra te d an SCR c irc u it to c o n tro l the c u rre n t through th e t js a tu ra b le r e a c to r in th e r. f. g e n e ra to r. The in g o ts w e re on < 111 > j Si se e d s; th e se e d w as ro ta te d at 20-60 rpm during g row th. M ost |of th e a llo y in g o ts w e re grow n a t a ra te of 0.25 in c h /h r, w ith so m e i jp o rtio n s having b e e n grow n as slow as 0. 1 in c h /h r. The o rig in a l jm o to r in th e p u lle r w as re p la c e d w ith a g e ared m o to r to allow th e |slo w pulling r a te s u se d . | E v a p o ra tio n of oxides fro m th e m e lt w as a p ro b le m b e c a u se of th e len g th of tim e involved in pulling a c ry s ta l a t 0. 25 in c h /h r. If th e rim of th e q u a rtz c ru c ib le stu c k up above th e g ra p h ite s u s c e p to r as m uch a s 0. 1 in ch , it w ould be cool enough th at th e oxide w ould co n d en se on th e in sid e edge of th e rim . E ventually, a th in la y e r of oxide w ould c o lle c t, extending inw ard fro m the rim of the c ru c ib le and blocking th e v iew of th e grow th in te rfa c e . T his could be p re v e n te d by c u ttin g off the c ru c ib le .so that it p ro tru d e d only slig h tly above th e top of th e su s c e p to r; if th e cru cib le w as c u t down below 87 'th e top edge of th e s u s c e p to r the m elt would be carb o n doped. It I 'w as found th a t th e seed b ecam e coated w ith oxide th a t in te r f e r e d | w ith w etting of th e se e d by the m e lt if it w as left e x p o sed above the 'c ru c ib le du rin g th e in itia l p erio d of m eltin g and te m p e ra tu re a d ju s t m e n t. A p iece of alu m in u m tubing was m ounted in sid e th e p u lle r on Ithe b o tto m of th e pulling rod gland, such that the se ed an d se e d | chuck could be r e tr a c te d into th e tubing. This w as su fficie n t to keep j jthe se ed cle an u n til it w as put into the m e lt. i The s ta rtin g m a te ria ls w e re polycrystallirie in tr in s ic g ra d e j g e rm a n iu m obtained fro m E a g le -P ic h e r In d u stries and h ig h p u rity i i | poly c ry s ta llin e silic o n obtained fro m W acker C hem ical C om pany. | T he seed and th e p ie c e s of Si and Ge w e re w ashed w ith tric h lo ro - j eth y len e to rem o v e any g re a s e o r wax, rin se d in a c e to n e o r m e th a n o l, land th en etch ed in C P4, a 5:3:3 m ix tu re of HNO,, H F, and HA c. i 5 I I T he Si w as etched b e fo re th e Ge; a clean-looking, shiny s u rfa c e I w as not obtain ed on th e Si if the solution had been u se d f i r s t to e tc h th e G e. T he etching w as follow ed by se v e ra l rin s e s in d eio n ized w a te r. The q u a rtz c ru c ib le w as cleaned in the sa m e w ay as th e Si and G e. F o r th e b o ro n doped c ry s ta ls , finely pow dered e le m e n ta l b o ro n w as added to th e c ru c ib le w ith th e Si and Ge b efo re m e ltin g . The n o rm a l b o ro n w as 99. 99% b o ro n obtained fro m U nited M in e ra l and C h em ical C o rp o ra tio n . B oron iso to p icly enriched to 96. 5% w as ob tain ed fro m O ak R idge N ational L a b o ra to rie s . 88 S ilicon is se le c tiv e ly rem o v ed fro m the m e lt a s it so lid ifie s, I so th a t the Ge c o n c e n tra tio n in c re a s e s along the ingot. The volum e 3 ; of the m e lt w as u su a lly about 20-25 cm , the d ia m e te rs of the | p u lled ingots w ere le s s th an 2 cm , and no m ore than o n e -th ird of the ; m e lt w as fro z e n . F ro m th e se d ata and th e o b se rv e d d istrib u tio n I c o e ffic ie n t of about 0 .4 fo r Ge in Si, the m axim um v a ria tio n of the Ge c o n c e n tra tio n in a sp ecim en 2 m m th ic k should be only about . 02 ! of th e Ge content, o r about . 25% Ge in th e Ge 10Si 00 c r y s ta ls . In i ( 1 2 t oo | fa c t, m o st of the sa m p le s w ere th in n er th an 2 m m and cut fro m | se c tio n s of the ingot le s s th an 2 cm in d ia m e te r, so th a t v a ria tio n j of c o m p o sitio n w ithin a sam ple w as not a p ro b lem . i 2. G e-R ich Ge Si, ! x 1 -x ! M ost of the G e -ric h Ge Si sam p les w e re cut fro m ingots j X ± **x I th a t h a d been grow n m any y e a rs ago at B e ll T elephone L a b o ra to rie s ; | the d e ta ils of th e ir grow th is not known. They w ere grow n in a ! h o riz o n ta l boat by e ith e r a zo n e-lev e lin g o r g ra d ie n t-fre e z e i I te c h n iq u e . The ingots containing m ore th an 10% Si w e re c ra c k e d , in d ic a tin g th at th e re is pro b ab ly co n sid erab le in te rn a l s tr a in . The c a r r i e r c o n ce n tra tio n s w ere a ll low enough th at th e re w as no s ig n i fic a n t in fra re d a b so rp tio n due to fre e c a r r i e r s . One ingot w ith a m ax im u m Si content of 12% w as grow n du rin g the p re s e n t study so th at the re s u lts obtained fro m it could be c o m p a re d to the re s u lts given by the B ell Telephone L a b o ra to rie s 89 s a m p le s. F o r a v a rie ty of re a so n s the ingot w as grow n by the v e rtic a l B rid g em an tech n iq u e, w h ere the m e lt is en clo sed in an elongated c ru c ib le w hich is lo w ered fro m a hot zone of a fu rn a c e in to a c o o le r zone. S o lid ificatio n begins at the bottom of th e c ru c ib le and p ro c e e d s to w ard th e to p . The sy ste m h as th e advantage of being ; sim p le enough in o p e ra tio n th a t it can be le ft unattended, so th at ■ v e ry slow grow th ra te s can be used o v e r long tim e s . The te m p e r- | a tu re g ra d ie n ts a r e such th a t th e re a r e no convection c u rre n ts to i c a u se te m p e ra tu re flu ctu atio n s. A nother fe a tu re w hen th is m ethod I |is u se d w ith G e -ric h Ge S i, is th a t th e re is som e g ra v ity s e g r e - j X JL “ X gation of the m e lt, so the f ir s t p a rt to fre e z e h as a h ig h e r Ge ; c o n c e n tra tio n than th e a v e ra g e fo r the m e lt; th is som ew hat d e c re a s e s th e ra p id change in co m p o sitio n along th e ingot th a t re s u lts fro m the ■ la r g e d istrib u tio n co efficien t of Si in Ge. The p rin c ip a l d isad v an tag e i ! of th e B rid g em an m ethod is th a t the m e lt is so lid ified in sid e a rig id j j c ru c ib le ; m o st se m ic o n d u c to rs, including Ge and Si, expand on j fre e z in g so th a t a rig id c ru c ib le m ay in tro d u ce s tra in into the c ry s ta l. T he ingot w as grow n in a q u a rtz c ru c ib le , m ad e fro m a p iece of 1 cm i. d. fused q u a rtz tubing w ith th e b o tto m end ta p e re d to a p o in t. The Ge and Si w e re etched as d e sc rib e d in th e p rev io u s se c tio n and se a le d in to th e c ru c ib le u n d er vacuum . The ingot w as fro ze n a t about one inch p e r day in a 10 C /cm tem p era tu re gradient. j j The m e lt co n tain ed about 5 a t % S i. The phase d ia g ra m of S to h r and l ; K lem m (1939) gives a so lid co m p o sitio n in eq u ilib riu m w ith a 5 a t % j m e lt a s 20 a t % Si; the re c e n tly c a lcu late d phase d ia g ra m of i ! S te in in g e r (1970) gives the e q u ilib riu m so lid as about 15%. The f i r s t ; p a rt of the m e lt to fre e z e co n tain ed about 12 at % Si; th is value i in clu d es the e ffe c ts of g ra v ity se g re g a tio n and a n o n -z e ro grow th | r a te , but ten d s to. su p p o rt the c a lc u la te d phase d ia g ra m , i The ingot w as p o ly c ry s ta llin e . The su rface a d ja c e n t to the i s lic e cu t fo r IR m e a su re m e n ts w as etch ed w ith C P 4 ( 5 HNO^: 3 H F: j 3 HAc ). M uch of the su rfa c e show ed no evidence fo r inhom ogeneity i I u n d e r e x am in a tio n th ro u g h a 30X m ic ro sc o p e; p a rt of the su rfa c e | show ed evenly sp a ce d c o n c e n tric rin g s w hich a p p ea re d to be a ! c ro s s - s e c tio n th ro u g h grow th s tria e d e sc rib e d by D ism ukes and i E k s tro m (1965) a s c h a ra c te ris tic of hom ogenous grow th of G e-S i a llo y s. T h ere w e re som e sm a ll a re a s th at etched aw ay v e ry ra p id ly ; th e se m ay have b een reg io n s of inhom o ge nous grow th. The c r ite r ia of D ism ukes and E k stro m would reco m m en d a s te e p e r te m p e ra tu re g ra d ie n t, a t le a s t 2 5 °C /cm , but th e ir grow th s y s te m w as le s s fav o ra b le b e ca u se of convection-induced flu ctu atio n s; the grow th conditions u se d h e re a p p a re n tly w ere m a rg in a lly sta b le . 3. G erm an iu m The p rin c ip a l d ifficu lty in the m e a su re m e n t of in fra re d iabsoption lo c a liz e d v ib ra tio n a l m odes of e le c tric a lly activ e ; im p u ritie s in Ge is th a t of e le c tr ic a l co m p en satio n to rem ove ;fre e c a r r i e r a b so rp tio n . L ith iu m diffusion can be u se d to com pen- jsa te p-type g e rm a n iu m , but w o rk s only if the c a r r i e r co n ce n tra tio n 1 8 - 3 is le s s th an about 5 x 10 cm " and w o rk s w ell only fo r c a r r i e r 1 8 - 3 I c o n ce n tra tio n s le s s th a n 1x10 cm " . F o r the dopant c o n c e n tra - 18 20 3 Itions com m only u s e d in lo c a l m ode stu d ies ( 1 0 - 10 c m ” ) it is ■ necessary to grow fa irly w ell co m p en sated c ry s ta ls . To o b se rv e the p h o sp horus lo c a l m ode in Ge, galliu m was I jused to co m p en sate the p h o sp h o ru s. P h o sp h o ru s and g alliu m have I n e a rly the sam e d is trib u tio n co efficien t ( k = 0. 080 fo r P , 0. 087 j o jfor Ga ) so th a t the re la tiv e c o n c e n tra tio n v a rie s slow ly as the m e lt | |is so lid ifie d . T h is dopant p a ir is a lso v e ry convenient, since an j iin itia l 1:1 c o n c e n tra tio n ra tio can e a s ily be ach iev ed by doping w ith j |g a lliu m phosphide. The c ry s ta ls w e re grow n by the C z o c h ra lsk i m ethod in a re s is ta n c e h e a te d p u lle r. The b a sic fra m e w o rk and pulling m e c h a n ism is a c o m p lic a te d sy s te m of w ire s and pulleys w ith no n a m e p late; ru m o r ( A llre d 1970 ) a ttrib u te s its m an u factu re to A llen - Jo n e s E le c tro n ic s C o rp o ra tio n . The p u lle r w as m odified so th at its fin a l c o n fig u ra tio n w as a s show n in F ig . 11. The V ycor tube w as, u n fo rtu n ately , slig h tly too s m a ll to a c c e p t a sta n d a rd tw o -in ch d ia m e te r q u a rtz c ru c ib le , so the c ru c ib le s h ad to be sh ru n k by the 92 TO VACUUM PUMP AND HYDROGEN SUPPLY TO POLLING AND S E E D ROTATING MECHANISM t BUSHING M E T A L FLANGE O-RING S E A L TU B E FOR COOLING WATER T E F L O N EPOXY STA IN L E SS S T E E L ROD GRAPHITE SEED CHUCK / Ge S E E D Ge NGOT VYCOR TUBE KANTHAL HEATING QUARTZ CRUCIBLE FIBERFRAX MELT INSULATING BLOCK FROM „ TEMPERATURE CONTROLLER C H R O M E L - ! ALUMEL THERMOCOUPLE! TO j T EM PERATURE i CONTROLLER j F i g u r e 11. R e s i s t a n c e h e a t e d C z o c h r a l s k i g r o w t h s y s t e m f o r G e c r y s t a l s . 93 • g la ssb lo w e r. ^ ■ C o m m e rcia l hy d ro g en , c le an e d up w ith a " D e-O xo " c a ta ly tic p u rifie r follow ed by a cold tr a p cooled w ith liq u id n itro g e n , w as jused a s an a tm o sp h e re . A ty p ic a l c ry c ib le c h arg e w as about 50 gm of Ge. In itially , a < 111> se e d w as used; the doping w as not v e ry hom ogenous, pro b ab ly b e ca u se of the fo rm a tio n of a fa c e t on the bottom of the ingot. W ith a se e d o rie n te d a p p ro x im a te ly along a ;<311> a x is, the doping w as m o re n e a rly hom ogenous. P u llin g ra te s ran g ed fro m 1 to 6 c m /h r . The seedcw as ro ta te d a t 15 rp m . The f i r s t good G aP doped c r y s ta l had 60 m g of G aP added to :53 g. of Ge giving [G a ] = [ P ] = 3 .6 x lO ^ c m ^ in the m elt; th is 18 -3 i should give [ P ] 2. 9 X 10 cm a t the top of the ingot. The ingot I iwas not p e rfe c tly hom ogenous but did co n tain p-type reg io n th a t could be co m p en sated by L i diffusion. A gallium doped c ry s ta l t J m o stly co m p en sated w ith a rs e n ic and antim ony w as grow n by doping i . . I w ith GaAs and e le m e n ta l Sb. Two ingots w ere grow n w hich w ere | m ore h eav ily doped w ith G aP. The f i r s t w as e n tire ly p-type and none of it could be co m p e n sa te d w ith L i; a p p a re n tly phosphorus w as being lo s t fro m the m e lt by e v a p o ra tio n . The second one w as doped by adding 10 m g of In P and 106 m g of G aP to 49 gm of Ge. The In P effec tiv e ly only adds p hosphorus to the ingot being grow n sin ce the d i s t r i bution co efficien t of In is quite s m a ll ( < 0. 001 ). The ex act d e g re e of c o m p e n sa tio n obtained during grow th is 7 Inot know n. The s lic e s th at could be c o m p letely c o m p e n sa te d by i L i diffu sio n had re s is tiv itie s of 0. 02 to about 0. 06 fi - cm ; fo r sin g ly doped p -ty p e Ge th ese re s is tiv itie s c o rre s p o n d to c a r r i e r 1 7 - 3 : c o n c e n tra tio n s of 1 - 5 x 10 cm . H ow ever, in h ig h ly co m p en sa te d m a te r ia l the m o b ility should be lo w e r, so the c a r r i e r ! c o n c e n tra tio n m ay be h ig h er fo r a given re s is tiv ity . An a tte m p t to grow an alum inum doped c ry s ta l in the sam e i p u lle r w as u n su c c e ss fu l because a la rg e am ount of s la g , p ro b ab ly ; an oxide of alum inum , fo rm ed on the su rfa c e of the m e lt and | in te r fe r e d w ith seed in g and grow th. A p p aren tly , the alu m in u m I re a c te d w ith the q u a rtz cru cib le o r w ith re s id u a l oxygen in the i j s y s te m . | S e v e ra l a tte m p ts w ere m ade to grow b o ro n doped Ge by u sin g | the B rid g e m a n m eth o d w ith ap p aratu s s im ila r to th a t u s e d fo r G e- j | ric h G e-S i a llo y s. G re a t difficulty w as e n c o u n te re d in rep ro d u c ib ly i I doping w ith b o ron, a p p aren tly b ecau se m o lte n Ge does not re a d ily i w et b o ro n . In one e x tre m e exam ple, a la rg e lum p of b o ro n w as put in th e c ru c ib le w ith the Ge. The lum p ended up em b ed d ed in the 18 -3 ingot, but the c a r r i e r co n cen tratio n w as only about 5 x 10 cm im m e d ia te ly a d ja ce n t to the b oron and c o n sid e ra b ly lo w e r e ls e w h e re . B . C om pensation by L ithium D iffusion L ith iu m is a shallow donor in both Si and Ge; its so lid so lu b ility depends quite stro n g ly on the am ount of o th e r im p u ritie s 95; p re s e n t. The L i sits in an in te r s titia l s ite . It can ionize a cc o rd in g • f * — to L i -* L i + e ; the e q u ilib riu m fo r th is re a c tio n , and thus the ' re la tiv e c o n c e n tra tio n s of n e u tra l and io n ized L i, is c o n tro lle d by the F e r m i le v e l, w hich is the c h e m ic a l p o ten tial fo r e le c tro n s . The to ta l c o n c e n tra tio n of L i is the sum of the co n ce n tra tio n s of n e u tra l L i and io n iz ed L i. The c o n c e n tra tio n of the n e u tra l L i is d e te rm in e d by an e q u ilib riu m w ith a n e x te rn a l p h a se , ty p ic ally a n allo y of L i w ith the Ge o r Si. Thus the to ta l so lu b ility of L i in e q u ilib riu m w ith an e x te r n al p h ase depends on the F e rm i le v e l, w hich in tu rn depends on the c o n c e n tra tio n of o th e r im p u ritie s . A q u an titativ e tre a tm e n t of the above a rg u m e n t by R eiss e t. a l. in w hich it is a ssu m e d th a t the only o th e r im p u rity is an a c c e p te r A gives the so lu b ility as -n+ A ~ . f ^ D = ---------------------------------------------- - + . 1 + y / l + ( 2n./D + ) I 1 + t / 1 +(2n /D +)2 + < Do > 2 } * • (71) + + w h e re D is the re s u ltin g so lu b ility fo r the io n ized donor, Dq is the io n iz ed donor so lu b ility in the ab sen ce of any o th e r dopants, A~ is the io n iz e d a c c e p to r c o n c e n tra tio n , and n ., w hich depends on te m p e ra tu re , is th e in trin s ic e le c tro n co n ce n tra tio n . F o r any doping 17 20 -3 le v e ls com m only u se d fo r lo c a l m ode stu d ies ( 1 0 - 10 cm ) the lim itin g c a se of A" n^ is ap p ro a ch e d at room tem po r a - + tu re fo r lith iu m d o n o rs, so D » A . L i diffuses quite re a d ily into Si and Ge, so th is e q u ilib riu m is a tta in e d fa irly e a sily . D iffusion I of L i donors th u s gives an e a s y m ethod of co m p en satin g a piece of ip -ty p e m a te ria l, sin c e the L i so lu b ility is c o n tro lle d by the doping i I lev e l. In p ra c tic e , th is w o rk s quite w ell a t any doping le v e l fo r Si. 2 0 - 3 S am ples doped to 10 cm w ith b o ro n w hich c o rre sp o n d s to a - 3 ; re s is tiv ity p < 10 fi- cm , c a n be co m p e n sa te d to a re s is tiv ity of I s e v e ra l h u n d red Q- c m o r m o re by L i diffusion. The com pensation is m o re c o m p lica te d in Ge, as the in c re a s e of n^ w ith te m p e ra tu re j is su ch as to in tro d u c e a m in im u m in L i so lu b ility a t te m p e ra tu re s of i Q i 100-200 C; the diffusion is u su a lly done above th is te m p e ra tu re and i I the sam p le m u st be ra p id ly quenched to avoid p re c ip ita tio n of L i i | d u rin g cooling. It is d ifficu lt to a c h ie v e r co m p en satio n to the degree j re q u ire d to rem o v e in f r a r e d fre e c a r r i e r a b so rp tio n in Ge w hen the 1 7 - 3 in itia l doping le v e l is m o re th a n a few tim e 10 cm , although it 18 -3 ih as been done a t a c c e p to r c o n c e n tra tio n s of up to 5 X 10 cm i | !( N azarew icz 1969 ). ! The L i diffusion is done by alloying a la y e r of L i to the su rfa ce of the sa m p le , p lacin g the sa m p le in a c ru c ib le along w ith g ra n u la r SiC to su p p o rt the sam p le aw ay fro m the w a lls of the c ru c ib le , and th en h e atin g the c ru c ib le in a tube fu rn a c e , u su a lly w ith an arg o n a tm o sp h e re , at the d e s ire d te m p e ra tu re fo r the d e s ire d length of tim e . A fter the diffusion, the sam p le co u ld be e ith e r slow ly cooled by m e re ly pushing the c ru c ib le to the end of the fu rn a c e , o r it could j be quenched, u su a lly w ith m in e ra l o il. F o r low diffusion te m p e r a tu r e s ( 400-500°C ) a q u a rtz c ru c ib le is su ita b le ; a t h ig h e r j : te m p e ra tu re s , w here the lith iu m h as an a p p re c ia b le v a p o r p r e s s u re , m olybdenum c ru c ib le s w ith clo se fittin g lid s w e re u se d . F o r diffusion te m p e ra tu re s above 850°C , it w as found d e sira b le w ith the ! la r g e r m olybdenum c ru c ib le s ( in sid e d im en sio n s 1 .1 " d ia m e te r by 1. 2M o r 2. 5" length ) to p r e - s a tu r a te the SiC packing w ith L i by ; p lacin g sm a ll lum ps of L i in sid e a g a in st the sid e s of the c ru c ib le and i o I h eatin g it to 900-1000 C fo r 20-30 m in u te s. T h is se e m e d to red u ce j I the lo ss of L i fro m the su rfa c e of the sa m p le . A fte r the sam p les j w e re cooled th ey w ere im m e rs e d in w a te r. The rem a in in g allo y i ! la y e r would re a c t w ith the w a te r to p ro d u ce h y d ro g en b u b b les. If the ! sam p le did not " fiz z ," . the alloy la y e r h a d b e en lo s t by e v ap o ratio n | ! o r oxidation, and g e n e ra lly the sam p le w as not co m p en sated . | | 1. A lloying of L ith iu m i | L i w as allo y ed onto the su rfa c e of the sa m p le s on a sm a ll s tr ip i J h e a te r in an a rg o n a tm o sp h e re . A s m a ll am ount of L i in the fo rm of j a su sp en sio n of fine p a rtic le s in m in e ra l o il w as s p re a d on one s u rfa c e of the sam p le; the sam p le w as th e n h e a te d u n til the L i re a c te d v isib ly w ith the su rfa c e ( 4 0 0 -5 0 0 °C fo r Ge, 500-600°C fo r Si ). A fter the sam ple had cooled, the allo y in g p ro c e d u re w as re p e a te d on the opposite side of the sa m p le . One d ifficu lty th a t w as i 98 | often e n co u n tere d w as th a t the L i w ould m e lt and fo rm into b a lls on the su rfa c e of the sam p le b efo re it a llo y ed w ith the sam p le; w hen it fin a lly did re a c t, the b a lls w ould c re a te deep p its of a llo y in the ' su rfa c e . The p ro g lem could be avoided to som e ex ten t by lapping ; the sam p le w ith 600 m e s h SiC b e fo re allo y in g . The n a tu ra l lith iu m ; w as a c o m m e rc ia lly p re p a re d su sp e n sio n w ith p a rtic le siz e < . 001 6 in ch , obtained fro m U nited M in e ra l and-C hem ical C om pany. The L i 6 | su sp en sio n w as p re p a re d fro m L i m e ta l, obtained fro m O ak Ridge N atio n al L a b o ra to rie s , by m ixing lum ps of the m e ta l to g e th e r w ith | o il in a b le n d e r. 2. Ge Si. A lloys x 1-x 3 j I A ttem p ts to co m p en sate b o ro n -d o p ed Ge Si w ith L i by usin g I X — 3 C | the s ta n d a rd tech n iq u e, w o rk ed p o o rly fo r allo y s w ith x 0. 06 and v e ry p o o rly o r not a t a ll fo r allo y s w ith x » 0. 12. If a sa m p le w as a llo y ed h e a v ily enough th a t th e re w ould be su fficien t L i a c tiv ity at the end of the diffusion to re a c t w ith w a te r, th e re w ould be tin y veins ( ty p ic a lly , a few te n th s of a m illim e te r a c ro s s ) of L i a llo y p h ase running th ro u g h o u t the sa m p le . T his w as tru e fo r diffu sio n te m p e r a tu r e s ranging fro m 400 to 900°C . D iffusion fo r s h o r te r tim e s o r w ith le s s L i p re s e n t did not y ield w e ll-c o m p e n sa te d sa m p le s. O c ca sio n al sa m p le s w hich w e re fa irly w e ll-c o m p e n sa te d and not too badly a tta c k e d by the L i could be ob tain ed by u sin g a fa irly s m a ll am ount of L i on the sa m p le in a c ru c ib le p r e - s a tu r a te d w ith L i, an d w ith O 99 idiffusion te m p e ra tu re s above 700 C. It w as found th a t w e ll-c o m p e n sa te d sam p les could be |c o n s is te n tly p ro d u ced by a two step diffusion p ro c e s s . In the f ir s t diffusion, a v e ry lig h t coating of L i w as used, so th a t a fte r a th re e - h o u r diffusion a t 850-950°C , th e re w ould not be enough su rfa c e L i a c tiv ity to re a c t w ith w a te r. T y pically, the re s is tiv ity of the sam p le i w ould in c re a s e fro m an in itia l value of about . 006 -c m to . 01 - ,. 1 Q- cm , in d ic atin g a sig n ific an t am ount of co m p en satio n . A fter th is , th e sa m p le could be re -a llo y e d w ith L i, w ith a s heavy a co atin g as w ould be u se d fo r L i diffusion of Si, and d iffused ag ain a t | 800-900°C . Good c o m p en satio n ( p > 500 fi- c m ) w ould re s u lt. The i jre a s o n fo r the su c c e s s of th is p ro c e d u re is not u n d ersto o d ; a p p a re n t l y som e s o r t of an n ealin g tak e s place w hen only a s m a ll am ount of L i j is p re s e n t. | 3. G erm an iu m j A t ro o m te m p e ra tu re and below th e so lu b ility of L i in p -ty p e ! jGe is c o n tro lle d a lm o s t e n tire ly by the net a c c e p to r c o n c e n tra tio n 15 -3 ( N -N ) if th is is g r e a te r th a n about 10 cm . A t h ig h e r X X X J o te m p e ra tu re s ( 100-200 C ) the in trin s ic c a r r i e r c o n ce n tra tio n in c re a s e s enough to becom e im p o rta n t and the so lu b ility m ay be le s s th a n ( R e is s 1956 ). A t the te m p e ra tu re s u su a lly u se d fo r d iffu sio n ( 400°C ) the L i so lu b ility is ag ain g r e a te r th an N .- N —. To X X J J • o b tain good co m p e n sa tio n , sa m p le s m u s t be quenched rap id ly fro m i . 1°° j the diffusion te m p e ra tu re to p re v e n t p re c ip ita tio n of the L i. Some sa m p le s to be quenched w e re diffused in m olybdenum j c ru c ib le s, but ra p id ly rem oving the lid fro m a hot c ru c ib le is ! difficult. Since the Ge sa m p le s a re n o rm a lly diffused at 550°C or I le s s , the L i v a p o r p r e s s u r e is low enough th a t q u a rtz c ru c ib le s a re ! not rap id ly a tta c k e d . A q u a rtz c ru c ib le and h o ld e r th at can be rap id ly rem o v ed f ro m the fu rn a c e and e m p tie d w e re m ade a s shown j in F ig. 12. F o r ra p id quenching, its co n tan ts w e re dum ped into a i b e a k e r of oil w ith in a few seco n d s of being rem o v e d fro m the fu rn ac e. I A piece of 1 /4 " m e s h w ire s c re e n w as p la c e d in the b e a k e r to catch j the sam p le, but allow the hot SiC su rro u n d in g the sam ple to fa ll to I | the bo tto m of the b e a k e r, thus allow ing the sam p le to cool f a s te r . I i i It w as d ifficu lt to c o m p en sate Ge w ell enough to allow IR I i i | tra n s m is s io n m e a s u re m e n ts if the in itia l re s is tiv ity w as le s s than t ! ! . 02 Q- cm . H o w ev er, N a z a re w ic z and Ju rk o w sk i ( 1969 ) did, w ith j | difficulty, co m p e n sa te an d m ake IR m e a s u re m e n ts on sam p les w ith an in itia l r e s is tiv ity of only . 005 Q -cm . To o b se rv e the p h o sp h o ru s lo ca l m ode in Ge, it was n e c e s s a ry I Jto o v e r-c o m p e n sa te th e p h o sp h o ru s w ith g a lliu m during grow th, since L i diffusion c an only co m p en sate p -ty p e m a te ria l. Sam ples fro m the P and Ga doped in g o ts w ere s e le c te d fo r being u n ifo rm ly p-type w ith a r e s is tiv ity > . 02 fi-cm . T h ese w e re allo y ed w ith L i, diffused a t 500 o r 550°C fo r 15-25 h o u rs , and th en quenched to room 101 INNER CRUCIBLE OUTER CRUCIBLE WITH HANDLE S AM PLE 8 0 M E S H SIC F ig u re 12. Q uartz c ru c ib le a rra n g e m e n t fo r rap id ly quenching sam p les a fte r L i d iffusion. The h andle extends to th e end of the fu rn ac e tube. te m p e ra tu re . The sam p le w ould u su a lly be u n ifo rm ly n-type a fte r i ‘ quenching but would becom e high re s is tiv ity p -ty p e a fte r a few h o u rs i to a few days a t room te m p e ra tu re . The sa m p le s w e re s to re d in | liq u id n itro g e n to p rev en t fu rth e r change. C a re w as tak e n not to ; h e a t the sa m p le s during lapping and p o lish in g . None of the sa m p le s : w as co m p en sated w ell enough to rem ove a ll m e a s u re a b le fre e : c a r r i e r a b so rp tio n . I C. R e sistiv ity and Type M e asu rem en t R e sistiv ity of sa m p le s w as m e a s u re d w ith a fo u r-p o in t probe i sy s te m , by u sin g a D um as fo u r-p o in t p ro b e h e a d m o d el P 8 -8 5 w ith ; . 025" probe spacing ( Dum as In st. Co. , C o sta M esa, C alif. ), an j ad ju sta b le c u rre n t so u rc e and a H e w le tt-P a c k a rd m o d el 425 DC I i Im ic ro v o lt-a m m e te r. The fo u r p ro b e p o in ts a re ev en ly sp aced in a i j i ! s tra ig h t lin e; c u rre n t is a p p lied b etw een th e o u te r two p ro b es and the ! voltage betw een the in n e r p ro b es m e a s u re d . The re la tio n sh ip betw een | | the m e a s u re d tr a n s fe r r e s is ta n c e Y /I and th e re s is tiv ity depends on jthe sam p le g eo m etry and is given in the lite r a tu r e f o r s e v e ra l i j g e o m e trie s ( Sm its 1958 ). F o r the lim itin g c a se of a th in infinite I sh e e t of th ic k n e ss t, p 4. 5 V /tl if th e p ro b e sp a cin g is g re a te r th an about 2t. F o r a se m i-in fin ite sam p le ( d im en sio n s m uch V g r e a te r th an the probe spacing ) p = n a y , w h ere a is the probe sp acin g . C onductivity type is d e te rm in e d by a h o t-p o in t p ro b e, c o n s is - .103 : tin g of a s m a ll so ld e rin g iro n and a g alv an o m eter. The sam p le is ! co n n ected to one side of the g alv an o m eter; the o th e r side is * co n n ected to the so ld e rin g iro n tip , w hich is th en p la c e d in c o n tact | w ith the sa m p le . The conductivity type at the point of c o n tact of the ; h o t p ro b e d e te rm in e s the sig n of the th e rm o e le c tric p o w er an d thus ! th e d ire c tio n of c u rre n t flow through the g a lv a n o m e ter, w hen, a s is | the c a se h e re , the th e rm o e le c tric pow er of the m e ta l hot p ro b e is j sm a ll. i ; D. D e te rm in a tio n of C om position T he d e n sity and la ttic e p a ra m e te r of Ge Si a llo y s v a ry X X " X j a lm o s t lin e a rly w ith x fro m the values fo r p u re Ge to th o se of ! I p u re Si. The deviations a re s m a ll but have b een ta b u la te d | ( D ism ukes 1964 ), so th a t e ith e r d en sity of la ttic e p a ra m e te r ! I I m e a s u re m e n ts p ro v id e a u se fu l m ethod of d e te rm in in g co m p o sitio n i of Ge Si a llo y s. | X X " X D ensity m e a s u re m e n t is the sim p le st technique. The sam p le is w eighed in a ir , and th en w eighed in w a te r by suspending it on a fine w ire in a b e a k e r of w a te r to w hich a tra c e of so ap w as ad d ed to a id in w etting the sa m p le . The sp ecific g rav ity is th e n ( w eight in a i r ) / [ ( w eight in a ir ) - ( w eight in w a te r ) ] . Since the d e n sity of w a te r depends on te m p e ra tu re and d isso lv ed im p u ritie s , the m e a s u re m e n t is sta n d a rd iz e d by m e a su rin g the sp e c ific g ra v ity of a p ie c e of p u re Ge o r Si. An a n a ly tic a l balance w ith a s e n s itiv ity of 104] i 1 mg w as u se d fo r m o st of the m e a s u re m e n ts . F o r the s m a lle s t ; i ! sa m p le s ( 0. 1 X 0 .7 x 1 .5 c m ) the se n sitiv ity of the m e a su re m e n t ; is not v e ry high; it is only about + . 01 fo r x. In m o st c a s e s , la r g e r ■ ( | sa m p le s w ere u se d , so th a t the s e n sitiv ity w as . 003 o r b e tte r. Some ! m e a s u re m e n ts w e re done w ith a b alan ce w ith a re so lu tio n of 0. 1 m g. I H ere it w as found th a t the d e g re e to w hich the w a te r w et the w ire ; a ffe cte d the m e a su re m e n t, so e ffe ctiv ely , the s e n sitiv ity was only I about 0. 3 m g. i t X -ra y d iffra c tio n m e a s u re m e n ts w e re m ade on som e of the ; G e -ric h sa m p le s w h e re it w as su sp e c te d th at the co m p o sitio n of the | ingot w as v a ry in g ra p id ly thus m aking it u n d e sira b le to u se la rg e ! J sa m p le s. The x - r a y m e a su re m e n t a ls o w ill check the hom ogeneity i I of the allo y , sin ce a n inhom ogenous allo y w ill give b ro ad en ed d if- | fra c tio n lin e s . F o r th e se m e a s u re m e n ts , th in s lic e s w ere cut fro m I j the ingots a d ja c e n t to the s lic e s u se d fo r the in f ra r e d m e a su re m e n ts . ! i | The sa m p le s w e re ground to a pow der and m ixed w ith.pow dered Ge a s a re fe re n c e . The m ix ed pow der w as s p re a d on a g la ss m ic ro scope slid e and h e ld in place w ith a dilute so lu tio n of Duco cem en t in a c e to n e . M e a su re m e n ts w e re th en m ade of the d iffra c tio n peaks of seve r a l h ig h -an g le re fle c tio n s by u sin g C uK ra d ia tio n . The 0 1 a n g u la r s e p a ra tio n s b etw een the allo y lin e s and the lin es fro m pure Ge w e re u se d to d e te rm in e the la ttic e p a ra m e te r of the alloy. It w as u su a lly n e c e s s a ry to a d ju st the m ic ro sc o p e slid e in the sam ple I 105 ! h o ld e r to obtain about eq u al in te n sitie s fro m the Ge and the allo y . ! The sam e technique w as u se d to c o m p a re la ttic e p a ra m e te rs 2 0 - 3 ! of pure Si, Si doped to about 4x10 cm w ith b o ro n , and som e of I the sam e B -doped Si th a t h a d b een lith iu m d iffused. T hese ; m e a su re m e n ts w ere m ade to d e te rm in e how m u ch of the la ttic e ; s tr a in in tro d u c ed by b o ro n w as rem o v e d by lith iu m diffusion. E . P o lish in g S am ples w ere ground to the d e s ire d th ic k n e ss by grin d in g the ; sam p le w ith a w a te r s lu r r y of 600 m e sh SiC g rin d in g pow der on a t i g la ss p la te . ! The sam p le w as h e ld by m ounting it on the end of a so lid b r a s s I J Q | c y lin d e r w ith sa lo l ( phenyl s a lic y la te ), w hich m e lts a t about 40 C. I I j The b r a s s plug w as h e ld in a s ta in le s s s te e l sle e v e w hich could'be j a d ju ste d to p ro tru d e beyond the end of th e b r a s s plug by the d e s ire d I i j th ic k n e ss of the sa m p le . The g rin d in g w as follow ed by lapping i ' j a g a in st a g la ss plate w ith 1200 m e sh a b ra s iv e s lu r r y to rem o v e j about . 002" and by 3200 m e sh s lu r r y to rem ove a n o th e r . 001". The sa m p le , s till m ounted in the p o lish in g jig , w as th en p o lish ed w ith 600 m e sh SiC on a se lv y t ( co tto n ) p o lish in g clo th on a ro ta tin g w heel. The cloth w as u su a lly dam pened w ith w a te r and liq u id d e te rg e n t, although so m e tim e s ju s t w a te r w as u se d . The slig h t h aze of s c ra tc h e s le ft by p o lish in g w ith 600 m e sh a b ra s iv e could m o stly be rem o v ed by a fin al p o lish w ith 1200 m e s h a b ra s iv e . 106 F . In fra re d M e asu rem en ts The in fra re d tra n s m is s io n m e a su re m e n ts w e re m ade by using g ra tin g s p e c tro m e te rs . A ll m e a su re m e n ts below 250 c m - * and i som e in th e ran g e 20 - 670 c m - * w e re m ade on a P e rk in -E lm e r m o d el 301 double b e am s p e c tro m e te r. The re s t of the m e a s u re - im e n ts w e re m ad e on a sin g le b e a m sp e c tro m e te r designed and b u ilt by P r o f e s s o r W. G. S p itz e r, u tilizin g a P e rk in -E lm e r m odel 210B ; m o n o c h ro m a to r. In th e 301 double b e am sy ste m , the two beam s a r e chopped ! out of p h a se , com bined w ith a b eam s p litte r, and fed th ro u g h th e | m o n o c h ro m a to r to a sin g le G olay d e te c to r. The am p lified output of i • 'th e d e te c to r is ru n into two s e p a ra te synchronous re c tif ie r s , each one in p h a se w ith one of th e c h o p p ers. T hus, two independent I outputs a r e obtained, one fro m the sam pling b eam (I ) and one j • j fro m th e re fe re n c e b e a m (I ). The tra n s m is s io n of a sp e cim en j ! in th e sam p lin g b e a m is given by the ra tio I T /I . The c o n stru c tio n ! ° of th e s p e c tro m e te r p re s e n ts som e d ifficu lties in m aking a c c u ra te h ig h -re s o lu tio n m e a s u re m e n ts . To keep the sig n a l-to -n o is e ra tio m o re o r le s s c o n stan t d u rin g a sc an , a se rv o -m e c h a n ism v a rie s th e m o n o c h ro m a to r s lit w idth to keep the output of the IQ channel c o n sta n t. T his c a u se s th e re so lu tio n to v a ry co n sid era b ly th ro u g h a sc a n . The G olay d e te c to r is quite se n sitiv e to m ec h an ica l v ib ra tio n s ; the ch o p p er and synchronous re c tifie r a sse m b ly p ro - 107 I duces a c o n sid e ra b le am ount of v ib ra tio n , w hich re s u lts in a co n stan t false sig n al in both 1,^. and I q c h a n n e ls. A t high a m p lifie r gain se ttin g s, the fa ls e sig n a l m ay be not sm a ll c o m p a red to the d e sire d signal; th is c an c a u s e p ro b le m s. The in d ic ate d tra n s m is s io n is then ( I— + c )/( I + c ). The c o n stan t fa lse sig n als v a ry w ith 11 0 6 ' gain, so the in d ic a te d tr a n s m is s io n b eco m es a function of gain in ! the constant I m o d e. The in d ic a te d tra n s m is s io n v a rie s w ith the o ! I b eam in te n sity if th e s p e c tro m e te r is ru n w ith co n stan t s lits , j T hese d ifficu lties a r e not b o th e rso m e a t lo w er gain se ttin g s u se d fo r I lo w -re so lu tio n m e a s u re m e n ts , and can in g e n e ra l be avoided by j I m aking sin g le -b e a m m e a s u re m e n ts . | The 301 s p e c tr o m e te r h a s the sam p lin g p o sitio n a t a focal i i point betw een the s o u rc e and the m o n o c h ro m a to r. F ilte rin g to I | rem ove h ig h e r o r d e r ra d ia tio n fro m the g ra tin g s is done w ith | | re s tra h le n p la te s , s c a t t e r p la te s , o r w ire g rid s u se d as re fle c tio n i I j f ilte r s , c ry s ta l c h o p p e rs , and, in som e w avelength ra n g e s, t r a n s m issio n f ilte r s . T h e o p tic a l lay o u t is show n in F ig u re 13; recom m ended f ilte r in g is given in T able 8. The re c o m m e n d e d filte rin g w as g e n e ra lly used; but in som e c a se s w here s e v e r a l sc a n s w e re to be m ade in a p a rtic u la r w ave length ran g e, s lig h tly d iffe re n t filte rin g w as u se d to avoid f ilte r changes d u rin g the s c a n . F o r e x am p le, a N aF re fle c tio n f ilte r w as u se d o v er the ran g e 425 - 350 cm V the P e rk in -E lm e r re c o m - source area bea m combining a rea I deti detector chopper grating monochromator F ig u re 13. O ptical layout of the double b e am f a r in fra re d sp e c tro p h o to m e te r. Table 8 . Recommended filtering for the Perkin-Elmer Model 301 far infrared spectrometer. W avelength or F req u en cy Sange M 2 and M 2* on F i l t e r Wheel F l and F l' M 4 - Chopper and M 4 -* M10 on F i l t e r W heel F2 Ml 2 F3 F i l t e r W heel M onochromator G r a tin g B la z e R u lin g 1 s t L/mm o r d e r 665-4-OOcnT1 1 5 -2 5 P o s . 1 M irror BaP2 F in e S c a t t e r P la t e L iF P o s . 1 F in e - S c a t t e r P la t e Open P o s. 2 444cm” 1 4-0 2 2 .5 ^ 0 0 -320cm- 1 2 5 - 3 1 .3 P o s . 2 NaF SaP2 F in e S c a t t e r P la t e L iF P o s . 1 F in e S c a t t e r P la t e Open P o s . 2 4-4-4-cm ” 1 4-0 2 2 .5 £ h c c Q " J - i J 3* 9 320-220cm “ 1 3 1 .3 -4 -5 -5 P o s. 2 NaF BaP2 F in e S c a t t e r P la t e BaF_ P o s f 2 F in e S c a t t e r P la t e Open P o s . 2 222cm” 1 20 4-5 4 CO o £ o C D 2 2 0 -l7 0 cm “ 1 4 -5 .5 -5 9 P o s . 3 NaCl BaF2 i F in e S c a t t e r P la t e BaF £ Pos • 2 F in e S c a t t e r P la t e Open P o s. 2 222cm” 1 20 J*5 T a b le 8 — C on tin u ed 170-130cm '': L 5 9 -7 7 P in e S c a t t e r P la t e P o s. k C sl C oarse S c a t t e r P la t e KC1 P o s. 3 C oarse S c a t t e r P la t e Open P o s. 2 111cm"1 10 90 130-98cm _1 7 7 -1 0 2 P in e S c a t t e r P la t e P o s . ^ C sl C oarse S c a t t e r P la t e KBr P o s. C oarse S c a t t e r P la t e Open P o s . 2 111cm"1 10 90 98-80CHT1 1 0 2 -1 2 5 P in e S c a t t e r P la t e P o s. k C sl C oarse S c a t t e r P la t e CsBr P o s. 5 C oarse S c a t t e r P la t e Open P o s . 2 111cm"1 10 90 8O-69C 1 1 1 " 1 1 2 5 -1^ 5 C oarse S c a t t e r P la t e P o s . 5 C sl C oarse S c a t t e r P la t e CsBr P o s . 5 C oarse S c a t t e r P la t e Open P o s . 2 55.5cm 1 5 180 6 9 -^ lcm ” ^ 1^5-2^0 C oarse S c a t t e r P la t e P o s . 5 C sl C oarse S c a t t e r P la t e KRS-5 P o s. 6 C oarse S c a t t e r P la t e P o ly e th y le n e F i l t e r #1 P o s . k 55.5cm 1 5 180 o H g A rc S ou rce -1 11 J j m ended filte rin g is a m ir r o r above 400 cm and the N aF fro m ■ 400 to 333 cm"’*. O cca sio n ally , the g lo b ar so u rc e w as u se d in th e1 I 100 - 170 cm * range in ste a d of the Hg a rc so u rc e . No changes in i the o b se rv e d sp e c tra re s u lte d fro m th e se su b stitu tio n s. Low te m p e ra tu re m e a su re m e n ts w ith the 301 w e re m ade w ith j th e sam p le m ounted on a co p p er cold fin g e r in the vacuum space of ; a g la ss d ew ar. The d ew ar u s e d plyethylene w indow s; an eq u al j th ic k n e ss of polyethylene w as put in the re fe re n c e b eam fo r double b e a m m e a s u re m e n ts , i , j The single b e a m s p e c tro m e te r h as the sam p lin g a r e a a fte r I th e m o n o ch ro m ato r. It u se s a g lo b ar so u rc e and th erm o co u p le ! I | d e te c to r. M u ltila y er tr a n s m is s io n f ilte r s m a n u fa c tu re d by P e rk in - i ! E lm e r C om pany a re u se d to rem ove h ig h e r o rd e r ra d ia tio n fro m the i | g ra tin g s. A m e ta l d ew ar equipped w ith C s l windows is u se d fo r ! j liq u id n itro g e n te m p e ra tu re m e a s u re m e n ts . A flexible bellow s and i pivot allow s the n itro g e n r e s e r v o ir and co ld fin g e r a sse m b ly to be I m oved, so th at the sa m p le , m ounted on the c o p p er cold fin g e r, can be m oved in o r out of th e o p tic al path. A m e ta l d ew ar m an u fa ctu red by S u lfrian C ry o g en ics In c o rp o ra te d is u se d fo r liq u id h e liu m te m p e ra tu re m e a s u re m e n ts . T he d ew ar h a s a ro ta ry jo in t so th at the sa m p le , m ounted on a c o p p e r cold fin g e r, can be m oved in o r out of the o p tic al path. It is equipped w ith K RS-5 w indow s. The liq u id n itro g e n te m p e ra tu re d e w ars on both s p e c tro m e te rs 112 I lea k ed enough th at the in su latio n p ro v id ed by the vacuum could be s e rio u s ly d e g ra d ed during a ru n if th ey w ere not continuously I pum ped. B oth IR s p e c tro m e te rs a re som ew hat se n sitiv e to the v ib ra tio n p ro d u ced by a m ec h an ica l pum p, so a V a ria n " V a c so rb " i s o rp tio n pum p w as u sed . T hese pum ps have an ad d itio n al advantage, ; in th a t th ey do not have any b a c k stre a m in g oil v a p o rs th a t m ay co at ! the sam p le o r the windows w ith oil. M e a su re m e n ts on the single b e am s p e c tro m e te r w ere m o stly j tak en a t a re so lu tio n of 0. 5 - 1 .0 cm \ The m e a su re m e n ts on the ; 301 d o u b le-b e am s p e c tro m e te r w ere tak en at re so lu tio n s of 2 to ! 5 cm *. The p o o re r re so lu tio n w as obtained a t s h o rte r w av elen g th s, I I ; w h ere s m a lle r s lit w idths should have been u se d . H ow ever, the | I G olay c e ll is a p p a re n tly le s s effic ie n t at sm a ll slit w idths w hen only i i j a s m a ll fra c tio n of its ta r g e t is filled . j The a b so rp tio n co efficien t (a) w as d e te rm in e d fro m the tra ils - | m is s io n (T) and the sam ple th ick n ess (x) w ith the re la tio n sh ip T = i ( 1-R )^e aX/( 1-R ^ e ). F o r the G e-Si allo y s the re fle c tiv ity (R) w as tak en a s eq u al to the re fle c tiv ity of the pure m a jo r c o n s ti tuent; a s long as ax is not too sm a ll, th is w ill not in tro d u ce m uch e r r o r into the v alu es c a lc u la te d fo r a , and the re la tiv e v alu es fo r a w ill be quite good. . The e r r o r in the d ifferen ce in a b so rp tio n coefficients, is d [ ( a1- a 2) x ] a p p ro x im a te ly — ^ ------------ AR . F ro m the e x p re s s io n fo r T, I fo r w hich one o b tain s j ^ e -2a,]X e -2a 2x dR ( a r 0C 2) x = 2R . . 2 -2 a xx . . 2 -2 a 2x ( 1-R e 1 ) ( 1 -R e * ) i F o r a w o rst c a se e s tim a te , a ssu m e th a t the v a ria tio n in i re fle c tiv ity is s im ila r to the v a ria tio n in band gap, w ith a lm o st h a lf of the to a l change o c c u rin g b etw een 0 and 15% Si; a Ge Q O Si • oo .12 | allo y w ould th en hav e R = . 346 a s c o m p a re d to . 36 for. Ge, I ! | T y pically, the s m a lle s t value of a x e n c o u n te re d would be 0. 2 . F o r I i th ese n u m b er, A( ) x < . 007, w hich c o rre sp o n d s to | . I Z — • A( a i ~a 2 ) 5 °* 1 c m - 1 . i Some d iffe re n tia l tr a n s m is s io n m e a su re m e n ts w ere m ade on t I i | the s in g le -b e a m s p e c tr o m e te r. T h ese w e re done by m aking a sc a n ! | f ir s t w ith one sa m p le in the b eam , th en w ith the o th er, then taking j jthe ra tio of the re c o rd e d in te n s itie s . If the sa m p le s a re the sam e ! th ic k n ess and ax is not too sm a ll, the d ifferen c e in ab so rp tio n co efficien ts is o b tain ed to a good a p p ro x im atio n fro m ( a^-a^) x = jfc n ( I2 / I1 ). • CH A PTER IV In fra re d A b so rp tio n of L a ttic e M odes and th e S ilicon L o c a l M ode in Ge Si, x 1 -x T his c h a p te r p re s e n ts a stu d y th ro u g h the u se of in fra re d ! I a b so rp tio n m e a su re m e n ts of som e a s p e c ts of the la ttic e dynam ics of I Ge S i1 alloys in the range 0 < x < 0. 12 an d 0. 88 < x < 1. The X i “X " 1 ' 1 in fra re d a b so rp tio n of undoped Ge Si show s fe a tu re s not se en in X X mX ; e ith e r Ge o r Si; som e of th e se fe a tu re s a p p e a r to be c h a ra c te ris tic | ! of single-phonon p ro c e s s e s . The in te r e s t in the phonons of such | i I a llo y s has been g re a tly in c re a s e d b ecau se of the th e o rie s developed I fo r lo c a liz e d m ode and band m ode a b so rp tio n . The th e o re c tic a l | d isc u ssio n s of C h a p te r I and II c a n be re la te d to the in fra re d i . . . j a b so rp tio n fo r c a s e s w h e re the la ttic e s y s te m is one of a n e a rly pure 1 I m a te ria l and the im p u ritie s can be tr e a te d a s is o la te d point d efects. j | In the p re s e n t w ork, th e se th e o rie s should apply if the value of x is I I . su fficien tly clo se to e ith e r z e ro o r one. W hile in fra re d la ttic e a b so rp tio n in Ge Si allo y s h as been X J. — X p re v io u sly m e a su re d , the study w as not su ffic ie n tly d e ta ile d to p e rm it a co m p reh en siv e c o m p a riso n w ith c u rr e n t th e o rie s . In the w o rk shown h e re , in fra re d a b so rp tio n m e a s u re m e n ts a t d ifferen t te m p e ra tu re s fo r allo y s w ith x < 0.12 and x > 0 . 88 a re re p o rte d fo r a freq u en cy range of 100 < v < 700 c m "* . T his fre q u e n c y ran g e is su fficien t to en co m p ass m o s t of the sin g le phonon e n e rg y range fo r both x = 0 and x = 1 as w ell as m uch of the o b se rv e d tw o-phonon : a b so rp tio n in th e se c a s e s . The m e a su re m e n ts show the a b so rp tio n due to the lo c a liz e d v ib ra tio n a l m ode of th e is o la te d su b stitu tio n a l I Si in a p red o m in an tly Ge la ttic e . The freq u en cy is clo se to th at c alc u la te d w ith Eq. 32. The a b so rp tio n c ro s s - s e c tio n is sm a ll, ; in d icatin g th at th e a p p a re n t c h a rg e s a s s o c ia te d w ith th e Si a r e a j s m a ll fra c tio n of an e le c tro n c h a rg e . In th e S i-ric h m a te ria ls I (sm a ll x) a b so rp tio n bands a r e o b se rv e d w hich a p p e a r to c o r r e s - I i pond to a sin g le phonon d e n s ity -o f-s ta te s sp e c tru m ra th e r th an th e | ! th e o re tic a lly p re d ic te d band m ode a b so rp tio n . T e m p e ra tu re i ; dependence m e a su re m e n ts in d icate th a t th e a b so rp tio n is of the sin g le i , | phonon v a rie ty . T hese la tte r re s u lts a r e q u a lita tiv e ly s im ila r to >those re p o rte d fo r n e u tro n -irra d ia te d S i. S am ples w hich a r e G e -ric h i show , in addition to th e Si lo ca l m ode, an en h an cem en t of so m e of I ; th e tw o-phonon bands p re s e n t in p u re G e. It is not c le a r fro m the I ! te m p e ra tu re dependence w h eth er th e in c re a s e in a b so rp tio n is of the sin g le-p h o n o n o r tw o-phonon type. The a b so rp tio n in th is c a se does ! not c o rre sp o n d to peaks in the sin g le-p h o n o n d e n s ity -o f-s ta te s sp e c tru m fo r Ge. A. V ib ratio n al A b so rp tio n of D efects in th e D iam ond Type L a ttic e We have d isc u sse d the influ ence of d efects on th e v ib ra tio n a l m odes of the p e rfe c t d iam ond-type la ttic e in C h a p te rs I and II. Of p a rtic u la r in te r e s t h e re is the d isc u ssio n of d e fect-in d u ced in f r a r e d ' a b so rp tio n in the freq u en cy range of the band m odes; i . e . , a t ! fre q u e n c ie s le s s th an (« . D aw ber and E llio tt ( h e r e a f te r DE ) ^ m ax ; c o n sid e re d the m odel of an isotopic su b stitu tio n ( no fo rc e c o n sta n t changes ) of an im p u rity of d ifferen t m ass w ith an io n ic c h a rg e e, p laced a t the c e n te r of an u n ch arg ed la ttic e . F o r th is m o d el, DE ; give the a b so rp tio n co efficien t as 2 2 ! oc(cu) = - 3 ^ I X(f, 0) I2S(oj) , (74) j I w h ere D is the defect d en sity , T ) is the re fra c tiv e index, A is a lo c a l 2 2 j fie ld te r m , — = A, and S(uu) is the d e n s ity -o f-s ta te s p e r u n it • 2 2 1 freq u en cy . The S(uu) is re la te d to v(uu ) by S(uu) = 6sNu)V(uj ), w h ere | I s is the n u m b er of a to m s /u n it c e ll and 3sN is the to ta l n u m b e r of I m o d es. The sq u a re of the absolute value of the im p u rity a m p litu d e j ( X(f, 0 ) J , w here f la b e ls the m ode and 0 the im p u rity p o sitio n , i I is given by I x '£’ °> I2 ■ m W ("2 « W < z > + [ 1 - z ^ 0 ] 2 ) • • (75) T his e x p re s s io n is s im ila r to Eq. 21 , w hich applies to the lo c a l m ode. Some of the q u alitativ e c h a ra c te ris tic s of b an d m ode a b s o r p tio n a re deducible by in sp e ctio n of th ese e x p re ssio n s. F o r e-*0, n I M ^ M , | X(f, 0 ) J — an d o c(u o ) -♦ S(uj) . Thus f o r an im p u rity of the sam e m a s s as the h o st ato m it re p la c e s , the a b s o rp tio n should have the sam e a p p earan ce as S(o j). The d e n sity -o f-sta te s h a s fo u r 117' i m ain peaks ( Jo h n so n 1965 ) roughly c o rre sp o n d in g to the d en sity | m ax im a of the TO, LO , LA and TA b ra n c h e s of the phonon sp e c tru m . ! The TO and TA peaks a re p a rtic u la rly pronounced fo r both Si and Ge b ecau se of the fla tn e s s of th e se b ra n c h e s o v e r m uch of the zone. F o r ■ e -* 1 (M 7« M) o r fo r e -* la rg e negative value (M7» M ) , the f ir s t 2 ; te r m in the d en o m in ato r of | X(f, 0) j should dom inate w henever 2 2 2 4 Z v (Z) is la rg e . Since Z = u ) » the high freq u e n cy a b so rp tio n n e a r TO is e x p ec te d to be m uch w e a k e r th an the low freq u e n cy peak n e a r ! TA(uu_r. 500 c m - * and (JO ,-* ~ 130 cm * in Si). In th o se freq u en cy X U 1 A . ; 2 I reg io n s w h ere V (z) is s m a ll the seco n d te r m in the d en o m in ato r of i | X(f, 0) | ^ m ay have a m a jo r e ffe c t on a . W h e n l-e Z ■ V^ (*K = q; I we have a re so n a n c e co n d itio n w h ere the a b so rp tio n m ay be larg e i i I even though S(uo) is not la rg e . i 2 | It is a p p a re n t th a t the a ssu m e d S(u j) o r v(u) ) m ay have a m a jo r ! ! e ffect on the c a lc u la te d n u m b er of re so n a n c e s and th e ir fre q u e n c ie s. I ! B ellom onte an d P ry c e ( 1968 ) c a lc u la te d the band m ode a b so rp tio n j | fo r b o ro n im p u ritie s in the Si la ttic e and u se d a d e n s ity -o f-s ta te s i I i I e s tim a te d fro m the c r itic a l point a n a ly sis of L oudon and Johnson (1966) w ith the n e u tro n d iffra c tio n d ata of D olling (1963). T his d e n s ity -o f-s ta te s d iffe rs sig n ific a n tly fro m th o se of A n g re ss, Goodwin and Sm ith (1965) o r Jo h n so n (1965) and w ith it, B ellom onte and P ry c e p re d ic t a n e a r-re s o n a n c e a b so rp tio n n e a r 225 c m - * w hich is not p re d ic te d by o th e r c a lc u la tio n s. T his p re d ic tio n w as in a g re e m e n t w ith the o b se rv a tio n of A n g re ss, e t. a l. of a sm a ll sh a rp peak n e a r 227 cm "^ . B ellom onte and P ry c e point out th e se n sitiv ity I of th e ir calcu latio n s to th e v(ji) function. I ; i : I In a ll of th e above d isc u ssio n it h as b een a ssu m e d th a t th e fo rce I co n stan ts a r e unchanged by the su b stitu tio n of th e im p u rity and th at ! th e ch arg e m o d el is one of an ionic o r s ta tic c h arg e e on the im p u rity . ! The la c k of v a lid ity of th e f i r s t a ssu m p tio n w ill c e rta in ly a ffect th e I c a lc u la tio n s, ho w ev er, u n le ss th e changes a re m a jo r ones they a re not expected to a lte r the g e n e ra l n a tu re of th e c o n clu sio n s. M ore I re c e n t calcu latio n s (E llio tt and P feu ty 1967, P feuty 1968) have taken j both fo rc e co n stan t changes and d e fe ct p a irin g into account in | p red ic tin g lo c a l m ode fre q u e n c ie s. | The a ssu m p tio n th at the to ta l coupling of th e rad ia tio n field to ! th e phonons, including th e lo ca l m o d e, a r is e s fro m the s ta tic c h arg e i | e c a rrie d by th e im p u rity is a s e rio u s one fo r the p re s e n t c a s e . Since | Si im p u ritie s in Ge a r e not ex p ected to have an ionic c h a rg e the ! a b so rp tio n a t th e lo c a l m ode freq u e n cy should b e z e ro . The in a d e quacy of th is a ssu m p tio n is vividly d e m o n stra te d (Newman and W illis 1965, N ew m an and S m ith 1969) by th e a b so rp tio n of carb o n in Si at th e C lo ca l m ode freq u en cy . A gain C is not expected to have a sta tic c h arg e y e t th e a b o srp tio n by C is a p p ro x im a te ly five tim e s stro n g e r than th at by B ” in co m p en sated Si w h e re th e b o ro n is an ionized a c c e p to r. In a s e r ie s of th e o re tic a l p a p e rs L eig h and S zig eti (1967a, A X 7 ; b, 1968 ) have pointed out th at the a p p a re n t c h a rg e is d e te rm in e d 1 la rg e ly by sh o rt range effects in v alen cy c ry s ta ls , and the s ta tic ; ch arg e c a r r ie d by the im p u rity m ay play a re la tiv e ly m in o r ro le in , the a b so rp tio n . B. P re v io u s E x p e rim e n ta l O b serv a tio n s T h ere have been s e v e ra l o b se rv atio n s ( N ew m an and S m ith 1969, B alkanski and N azarew icz 1966, W ald ner, H ille r and S p itz e r | 1965, C osand and S p itzer 1967, N azarew icz and Ju rk o w sk i 1969 ) I of in fra re d a b so rp tio n a ttrib u te d to lo c a liz e d v ib ratio n a l: m odes of i im p u ritie s in Si and Ge. As p rev io u sly m entioned, h o w ev er, the t | ; lo ca l m ode of Si in Ge h as been re p o rte d only in R am an sc a tte rin g I | e x p e rim e n ts. A R am an line a ttrib u te d to the Si lo c a l m ode w as j j o b se rv e d by F e ld m a n et.. a l. (1966) fo r w hich V = 389 to 402 c m " ' as ; x = 0. 99 to 0. 67 in Ge Si, allo y s. At x < 0. 95 lin e s w e re a ls o I x 1 -x 1 — j o b se rv e d fo r 448 and 476 cm "* both of w hich a r e above the top of the | . I phonon s p e c tru m of u n p e rtu rb e d Ge. T hey w e re in te rp re te d a s the i ; ! R am an activ e m odes of S i-S i n e a re s t neig h b o r p a ir s , w hich have sy m m e try . In sa m p le s of sm a ll x ( S i-ric h ), P a r k e r e t .a l . o b se rv e d a t le a s t th re e peaks in the R am an s p e c tra w hich lie below the Si o p tic a l m ode line n e a r 520 cm * and a re a ssu m e d to be re so n a n t o r n e a r - r e sonant m o d es. By o b serv in g the R am an lin e s a s x v a rie s a c ro s s the fu ll ran g e, the follow ing c o rre la tio n s w e re o b se rv ed : 120 (Ge S i. ) ' x 1-x ' S m a ll x L a rg e x Si o p tical.m o d e “► Si p a ir bands in Ge c ry s ta l I j L o w est freq u en cy re so n a n t ■ + Ge o p tical m ode j m ode I -1 ~ 400 cm re so n a n t m ode -» Si lo cal m ode | | U n fo rtu n ate ly , the re p o rt of th e w o rk at sm a ll x did not give two of f ’th e th r e e re so n a n t m ode fre q u e n c ie s, j A v a rie ty of d e fe cts have been shown to p ro d u ce band m ode | a b so rp tio n in Si. T h ese include both su b stitu tio n al and in te r s titia l j I im p u ritie s and dam age in tro d u c ed by e le c tro n o r n e u tro n ir ra d ia tio n , i A n g re ss e t. a l. (1965, 1968) have o b serv ed band m ode a b so rp tio n in s a m p le s containing b o ro n and a su b stitu tio n a l donor, e ith e r P , A s, o r Sb, w ith th e fin al co m p en satio n achieved by e le c tro n irr a d ia tio n , and in a P -d o p e d sa m p le co m p en sated by n e u tro n irra d ia tio n . The m o s t p ro m in e n t band, a t 441 cm "^ , was a ttrib u te d to a p h o sp h o ru s re s o n a n c e m ode. T h e re w e re a n u m b er of w eak er b an d s; so m e w e re a ttrib u te d to B, A s, o r P re s o n a n c e s . O thers a t 491, 468, 423, 347 331, and 315 cm""* w e re a s s o c ia te d w ith sp e cific im p u ritie s , but o c c u re d n e a r c ritic a l poin ts o r reg io n s of high d e n s ity -o f-s ta te s and w e re a ttrib u te d to a c tiv a tio n of la ttic e m odes by the im p u ritie s . T hey found th a t th e a b so rp tio n due to b o ro n a g re e d fa irly w ell w ith th e c a lc u la tio n of D aw ber and E llio tt (1963b) fo r in -b an d a b so rp tio n , but th a t th e r e w as little a g re e m e n t betw een the p re d ic te d and o b se rv e d • s p e c tra fo r P o r A s. Devine and N ew m an (1970) stu d ied the in -b a n d j a b so rp tio n of A1 in Si w hich w as co m p en sated e ith e r by L i diffusion j o r e le c tro n irra d ia tio n . The L i co m p en sated sam p les show a L i I v ib ra tio n a t 520 cm * and two new bands a t 469 c m * an d 431 c m S | it is su g g e ste d th a t th ey m ay be A1 reso n an c e m odes of L i^ -A l" p a ir s . | No o th e r s h a rp fe a tu re s a re se e n in the L i-c o m p e n sa te d sa m p le s. In the e le c tro n ir r a d ia te d sa m p le s the com pensation is + + j p rim a rily due to th e fo rm a tio n of A1 in te r s titia ls , m o st of w hich j a re p a ire d w ith su b stitu tio n a l A l; o th e r ra d ia tio n dam age m ay a lso j j be p re s e n t. S e v e ra l bands a re o b se rv e d , som e of w hich o c c u r a t th e j ! fre q u e n c ie s of c r itic a l points d e n s ity -o f-s ta te s m ax im a in the Si i I | phonon s p e c tru m . T h ese include a la rg e peak of about 480 cm " i and s m a lle r peaks a t 419, 335, 235, 212, and 150 c m ” *. Two j a d d itio n al p eak s a r e o b se rv e d w hich cannot be a s s o c ia te d w ith any ob- i i j vious fe a tu re s of the phonon s tru c tu re ; th ey su g g est th a t the one a t i i -1 ++ j 242 cm m ay be a reso n an c e m ode of in te r s titia l A l and the o th e r 1 - 1 | a t 455 c m , m a y be a reso n an c e m ode of su b stitu tio n a l A l. j D am age fro m fa s t n e u tro n irra d ia tio n of Si ( B alkanski 1964, A n g re ss e t,a L 1963 ) p ro d u ces a b so rp tio n peaks a t som e of the sam e fre q u e n c ie s a s a re o b se rv e d fo r the doped sp e cim en s. In p a rtic u la r, a la rg e b and n e a r 480 c m s m a lle r bands n e a r 415 cm * and 340 cm \ an d a b ro a d band w ith som e s tru c tu re betw een 100 and 200 cm * a re se e n . The o v e ra ll shape of the a b so rp tio n b e a rs 122 ■ c o n sid e ra b le re s e m b la n c e to the Si d e n s ity -o f-s ta te s . It m ay be noted th a t d iffe re n t m e a s u re m e n ts of the a b so rp tio n of neutron ; ir r a d ia te d Si d iffe r in so m e d e ta ils; B alk an sk i show s a band a t about i 250 cm - * quite clo se to th e 242 cm * band re p o rte d in e le c tro n - i | irr a d ia te d A l-doped Si. T he prev io u s a b so rp tio n m e a su re m e n ts w hich have a d ire c t b e a rin g on the p re s e n t w o rk a r e th o se of B ra u n ste in (1963b) on the | a b so rp tio n in Ge S i1 a llo y s. This w o rk c o n sists of a study of the j tw o-phonon a b so rp tio n and its a n a ly sis to deduce c h a ra c te ris tic ] i phonon e n e rg ie s fo r th e a llo y s. Two new a b so rp tio n bands, not p re - i I ! se n t in e ith e r the x = 0 o r x = 1 m a te ria ls , a r e o b serv ed . F o r sm all i ! -1 -1 jx , a band is found n e a r 485 cm and fo r la rg e x, one n e a r 214 cm . i i | T h ere w as no evidence fo r a b so rp tio n by a Si lo ca l m ode. Both new j bands w e re a ttrib u te d to tw o-phonon p ro c e s s e s . T he te m p e ra tu re I _ i dependence of the 485 cm band w as c o n siste n t w ith a tw o-phonon I p ro c e s s , and it w as a ls o a ssu m e d th a t c ry s ta l sy m m etry would still fo rb id th e p re s e n c e o f a lin e a r e le c tric m o m en t. T his assum ption is b ased on a s ta tic c h a rg e m o d el and th e inadequacy of th is m odel h a s a lre a d y b e en m en tio n ed . C. E x p e rim e n ta l R e su lts and D isc u ssio n 1. Ge S i. w h e re 0 < x < 0. 12 x 1 -x — — In F ig u re s 14 and 15 th e m e a su re m e n t of the in fra re d a b so rp tio n of sa m p le s w ith x = 0 . 02 and x = 0 . 1 2 is given for both 8 7- 6 _ 5 E o 3 3- 2 - * * * * o\ x O O V . - X V V v V * * „ * N , * ° ° ..X X * x x x x x x ‘ x ^ x x " ° x O o*xxx 3 # P % O O O o yO O oo® ® ® ® ® ® a O o A O • dO O X X XX X * o o +° %°o0 oo o 00-00° °° + 4- o ° X 0 0 X > o o x. + " T • +.+ ++-H-+* ^ n - + + x x X X '« o + •• + + + + ++ • • • • • * ,++++ • • • • -* H -+ + + » • H i* * * ' O o J - 350 4 0 0 4 5 0 5 00 . 550 v ( cm"') 600 650 700 F ig u re 14. A b so rp tio n c o efficien t v s. w avenum ber for S i-ric h GexS ij allo y s a t room te m p e ra tu re and liquid n itro g e n te m p e ra tu re . Ge 12^ 8 8 ^ : x = 0 = ^N T ; Ge Q 2 * S i qg : + = R T , o = LN T . 124 F ig u re 15. A b so rp tio n co efficien t v s. w avenum ber for room te m p e ra tu re Si, (+); and fo r Ge 12® * 88 a * " roorn te m p e ra tu re i (x) and liquid n itro g e n te m p e ra tu re (o). D ashed cu rv e is ; a b so rp tio n p re d ic te d by D aw ber and E llio tt fo r a heavy im p u rity (As) in Si. The solid a rro w s in d ic ate p o sitio n s of p eak s in the l | v ib ra tio n a l d e n s ity -o f-s ta te s of Si ( F ig u re 1 ) and p o sitio n s of i 1 re so n a n c e and n e a r re so n a n c e m o d es a s c alcu lated by M aradudin. x " x X + X \ V » c 2 v o o 0 XxxXx X C 9 v ^ V o „ 4 / ' ^ X o O 0 0 0 0 0 O W 0 CP0 ° % 3 ° . o x 0 ° * ° r „ o % o + o X o? ° ° o „ *> ° o O ® o v X vXXXXXX X X O o O O O ® o X X X X X X X X X X xX x X » < X X X X x + 0 xxX ++ v + + + o o f - y r a r b i t r a r y a s c a l e + +++ + + i \ + + + + + / \ + / \ . + + '+ + ___ - i _____------------------------------------------------- i __________ i __________ 10 100 200 300 400 500 600 . v ( c m - 1 ) 700 1 y A! 300°K and ~ 80°K and c o n tra ste d w ith th a t fo r p u re Si (x=0). The in tro d u c tio n of Ge h a s pro d u ced a nu m b er of ch an g es in the a b so rp tio n I ; s p e c tru m . In p a rtic u la r, pronounced a b so rp tio n b ands a re in tro d u c ed i n e a r 485, 405, and 125 cm "* in both the 300°K an d 80°K c u rv e s . In ! ad d itio n , a sh o u ld e r in the a b so rp tio n is o b se rv e d n e a r 2 0 0 cm *. i The dependence of the a b so rp tio n of the two h ig h e r freq u e n cy bands i 1 on the c o n c e n tra tio n of Ge is show n in F ig . 14. In F ig . 15 th e re is ; a b ro a d band, extending fro m about 220 to 340 c m " * , s e e n only in the \ 80°K c u rv e . T hat sam ple w as p-type w ith a r e s is tiv ity of about 10 ; ficm ; b o ro n w as the m o st lik e ly contam inant. The b and c o v e rs the j | fre q u e n c y ran g e of the s e v e ra l a b so rp tio n bands o b s e rv e d fo r tr a n s i- f j tio n s b etw een e le c tro n ic s ta te s of h o les bound to b o ro n a c c e p to rs in i | Si. The in te g ra te d a b so rp tio n is a p p ro x im a te ly th a t w hich is ! | o b se rv e d at 80°K in Si doped to the sam e re s is tiv ity . Thus it is j ■ i c o n s is te n t to a ssu m e th a t th is band is due to in h o m o g en eo u sly | b ro a d e n e d le v e ls of hbles bound to b o ro n a c c e p to rs . 1 ! ■ ’ - 1 | The sh a rp band n e a r 516 cm " in the x = 0. 02 sa m p le as w ell a s the b ro a d e r sm a ll bands n e a r the sam e fre q u e n c y in the o th e r tw o sa m p le s a re a ttrib u te d to the p re se n c e of oxygen. Since a ll of th e se sa m p le s w ere grow n fro m q u a rtz c ru c ib le s , the a p p e a ra n c e of th is b an d w as not unexpected. A ll th re e sa m p le s a ls o show ed the c o rre sp o n d in g high freq u en cy oxygen band n e a r 1106 c m *. The a b so rp tio n bands in d icate a d e c re a sin g oxygen c o n te n t w ith in c re a s in g 127 Ge co n ten t in ingots grow n u n d er a p p ro x im a te ly the sam e conditions I I of c ry s ta l ro ta tio n and p u ll ra te . T h is d e c re a s e m ay be la rg e ly the re s u lt of thw lo w er grow th te m p e ra tu re of the ingots w ith la r g e r g e rm a n iu m content. The m ain bands n e a r 485 to 405 cm * c le a rly show a s u b s ta n tia l in c re a s e w ith in c re a s in g x. The 485 cm~* band is in a g re e m e n t w ith the p rev io u s re s u lts of B ra u n ste in (1963b). The band n e a r 508 cm * m en tio n ed by B ra u n ste in fo r sa m p le s of low x is not o b se rv e d in th e se sa m p le s. If p re s e n t it m ight be o b sc u re d by the stro n g band a t 485 cm~* and the oxygen band. The la rg e band n e a r 600 cm * in the x = 0. 012 allo y is the j j eq u iv alen t of the band n e a r 610 cm * in the pu re Si(x=0) sa m p le . T his I j is a tw o-phonon band a ttrib u te d to the sim u lta n eo u s e x c ita tio n of a j j tr a n s v e r s e optic and a tr a n s v e r s e a c o u stic phonon. The te m p e ra tu re | dependence in the tra n s itio n p ro b ab ility co m es fro m the fa c to r | | (n,j.Q +l)(n1 j 1 ^ + l) - n ^ n ^ w hich m u ltip lie s the m a trix e le m e n ts i I b etw een the in itia l and fin a l s ta te s . The n 's a re the B ose fa c to rs fo r < the phonons; the phonon e n e rg ie s a re a p p ro x im a te ly th o se of the peaks in the d e n s ity -o f-s ta te s a ris in g fro m the TO and TA b ra n c h e s . T h e re is c o n sid era b le d isc u ssio n of th e se points in the lite ra tu r e (S p itze r 1967 ). The 600 cm * band in the x = 0 . 012 sa m p le show s a te m p e ra tu re dependence w hich is s im ila r, although som ew hat le s s th a n th e tw o- a(300°K ) 128 j phonon band in p u re Si. F o r p u re Si, the ■ '^ g Q oj ^ 1 9 w hile in ! the x = 0. 12 c a se , —- 999 -I9 ., ~ 1 . 5. The a v a lu e s a r e the peak a(80°K ) v alu es of the b ands. T his te m p e ra tu re b e h av io r m ay be c o n tra s te d ; to th a t of the new bands of the alloy a t 125, 405, and 485 cm *. The i new bands a re m uch le s s te m p e ra tu re dependent; .ffl. ~ l . o a(80°K ) : to 1 .2 depending upon the band c o n sid ere d . The te m p e ra tu re I I dependent fa c to r is even c lo s e r to 1. 0 if one a s s u m e s th a t the new I bands a re su p e rim p o se d on a fe a tu re le s s , te m p e ra tu re -d e p e n d e n t ; b ack g ro u n d a b so rp tio n d e te rm in e d by e x tra p o la tin g a c r o s s the b ase I of the band. It is re a d ily a p p a re n t fro m F ig s . 14 and 15 that “ b a c k g r o u n d 3 0 |)oK 1 . 0 fo r a ll th re e new b an d s, w hich w ould a ““background g0° K | be the c a se if th e se w ere single phonon b an d s. The te m p e ra tu re | dependence of the background is th en a p p ro x im a te ly 1 .3 - 1 .5 . As i I I p re v io u sly in d icated , P a r k e r e t. a l. ( 1969 ) re c e n tly re p o rte d the I I j o b se rv a tio n of th re e peaks below the top of the Si phonon s p e c tru m in I the R am an s p e c tra of S i-ric h allo y s. One of the p eak s lie s n e a r 400 cm * and m ay be the sam e m ode re sp o n sib le fo r o u r 405 cm * band. The fre q u e n c ie s of the o th e r two R am an p eak s w e re not given. The fre q u e n c ie s of reso n an c ec an d v n e a r-re so n a n ce m odes fo r Ge in Si have been c a lc u la te d b y M aradudin (1966) an d th e ir fre q u e n c ie s a re in d ic ate d by the dashed a rro w s of F ig . 15. The d a sh ed a rro w s n e a r 453 and 490 cm "* indicate reso n an ce m o d es w hile the o th e rs 129 | a r e n e a r-re s o n a n c e s . In the p re s e n t c a se , the bands we o b se rv e show som e a g re e m e n t w ith the c alcu late d p o sitio n s ( d a sh e d a rro w s ) I in F ig . 15. T hese c alcu latio n s u se d a d e n s ity -o f-s ta te s c u rv e fo r ' p u re Si c a lc u la te d by M aradudin and the c o m p a riso n s a re w ith : e x p e rim e n ta l c u rv e s fo r allo y s w ith s e v e ra l p e rc e n t G e. H ow ever, i the o b se rv e d bands do not show any la rg e freq u e n cy sh ifts b etw een ; 2% and 12% Ge. The c a lc u la te d fre q u e n c ie s of in -b an d re s o n a n c e s , h o w e v er, I i a re v e ry se n sitiv e to e x a c t d e ta ils of S(u)) th at one u s e s to c h a r a c te r - i ; iz e the pure m a te ria l. The calcu latio n s of B ellom onte an d P ry c e 1 I ‘ ( (1969) can be ap p lied to the case of Ge in Si. F ig u re 16 show s th e ir I hm ax . | p lo t of §(Z) = f — , on w hich the c u rv e 1 /eZ h a s b een | d Z -^ | draw n fo r e= -1 .5 8 , the a p p ro p ria te value fo r Ge in Si. The i t j in te rs e c tio n s of the two c u rv e s a re solutions of E q. 11. T h e ir data t p re d ic ts re so n a n c e m o d es a t 480 and 455 cm~* and a n e a r-re s o n a n c e a t 120 c m ’ \ a s c o m p a re d to the values of 490, 453, and 125 cm~* th a t M a ra d u d in 's c a lc u la tio n p re d ic ts . T h e-n ear re s o n a n c e s in M a ra d u d in 's c a lc u la tio n a t 330 and 385 cm * a r e re p la c e d by re so n a n c e s a t 350 and 370 cm *. In the o b se rv e d in f r a r e d a b so rp tio n th e only s tru c tu re n e a r th e se fre q u e n c ie s is one band a t 405 cm *. In one of th e ir p a p e rs (1963b), DE c a lc u la te th e in -b a n d a b so rp tio n to be ex p ected fo r s e v e ra l im p u rity m a s s e s in silic o n ; a com m o n fe a tu re fo r heavy im p u ritie s is the a lm o st to ta l la c k of ♦fe), l /e z (cm*) 130 +6 +4 +2 400' .300 iOO 200 100 -2 -4 F ig u re 16. I. (z) a s c alcu lated by B ellom onte and P ry c e . II. 1 / e z fo r Ge in Si (e = -1 . 58 ). _ 1 ' J L j J L t j a b so rp tio n n e a r 500 cm " , the p o sitio n of the stro n g TO peak in j ! • i i ? f j !S(u)). T his is the p re d ic te d re s u lt d isc u sse d p re v io u sly ( E q. 75 ) J iw here M / » M . The a b so rp tio n c a lc u la te d fro m E q. 74 is illu s tra te d j b y the d a sh ed c u rv e in F ig . 15 fo r su b stitu tio n a l a rs e n ic im p u ritie s i in Si. A rse n ic h as a m a s s clo se to th a t of Ge and both a re m o re th an tw ice th a t of Si. The o b se rv e d bands do not a p p e a r to c o rre sp o n d I to the p re d ic te d a b so rp tio n ex cep t p o ssib ly th at a low freq u e n cy band I is o b se rv e d . In p a r tic u la r, the c a lc u la te d c u rv e does not show the i -1 | m a jo r band n e a r 485 cm I In sp e ctio n of the DE c a lc u la te d band m ode a b so rp tio n s p e c tra , show n in F ig . 17, show s th a t the bands of F ig . 15 have a stro n g i | re s e m b la n c e to th o se c a lc u la te d fo r su b stitu tio n a l im p u ritie s w ith jm a s s e s c lo se to th a t o £ S i. As we have pointed out, in th e se c a se s i j jthe c a lc u la te d c u rv e s a re s im ila r to the S(uo) c u rv e . The p o sitio n s I . i of the fo u r m a jo r p eak s in S(uu) fo r Si a re given in F ig . 15 by the so lid a rro w s . A ll th re e of the new bands a re v e ry c lo se to peak p o sitio n s. H ow ever, th e re does not a p p e a r to be any a b so rp tio n p eak a s s o c ia te d w ith th e d e n s ity -o f-s ta te s p eak n e a r 340 c m "* . The e n tire re g io n fro m 215 c m * to 390 cm * a p p e a rs to be devoid of any fe a tu re s th a t could be a ttrib u te d to v ib ra tio n a l a b so rp tio n , w h e rea s s e v e ra l bands w e re found in th is reg io n fo r o th e r dopants ( A n g re ss Goodwin, and Sm ith 1968, D evine and N ew m an 1970). A c o m p a ris o n of th e p re s e n t m e a su re m e n ts w ith th o se re p o rte d 132 025 100 200 300 400 500 V (cm-1 ) ~ e 0.75 u 0.50 0.25 100 200 300 4 0 0 500 2 /(crrf *) F ig u re 17. O ptical a b so rp tio n co efficien ts p re d ic te d by DE for 1 9 - 3 10 cm im p u ritie s of v a rio u s elem en ts in group III and group V. a. F u ll c u rv e , b o ro n ; th is cu rv e is an a v e ra g e of the two curves fo r and w eighted a cc o rd in g to th e ir abundances. F o r ^ B , = 0 .6 4 3 ; fo r ^*B, = 0 .6 0 8 . B roken cu rv e , p h o s p horus; = -0 .1 0 3 . D otted c u rv e , antim ony; = -3 . 334. b . C ontinuous c u rv e , alu m inum ; = 0 .0 3 9 . B roken cu rv e , a rs e n ic ; = -1 .6 6 6 . D otted c u rv e , b ism u th ; = -6 .4 3 9 . 1 ____ J L O O fo r n e u tro n -irra d ia te d Si is show n in F ig . 18. The c u rv e fo r the i n e u tro n -irra d ia te d c a se is th a t of B alk an sk i (1964). W hile the c u rv e s d iffer in d e ta il, th e re a re s e v e ra l obvious s im ila r itie s . The o rig in of the a b so rp tio n in the ir r a d ia tio n c a se h as been a ttrib u te d to the in tro d u ctio n of d is o rd e r w hich p e rm its coupling of the ra d ia tio n field to phonons of a ll w ave v e c to r v a lu e s. In the p r io r d isc u ssio n it w as t ; noted th a t A n g re ss e t. a l. o b se rv e d in -b a n d a b so rp tio n n e a r 490 and j 404 cm * in B and P doped Si. T h e re fo re , the o rig in of the a b so rp - | tion in a ll of th e se c a s e s m ay be s im ila r and re fle c t S(yj) fe a tu re s as I w ell as the c h a r a c te r is tic s of the d efect involved. A som ew hat i ' !analogous c a se to the a b so rp tio n se e n n e a r d e n s ity -o f-s ta te s peaks | is se en in R am an s c a tte rin g fro m GaAs . ( S tra h m and M c- jW h o rter 1969 ). L in e s a re se e n a t fre q u e n c ie s of zone-edge o p tical j ■ : phonons of G aP; th e a llo y in g of As into G aP h as allow ed s c a tte rin g i ■ ;fro m m odes th a t a r e n o rm a lly R a m a n -in a c tiv e . 2. Ge Si, w h ere 0. 88 < x < 1 x 1 -x — — j ! The a b so rp tio n in a n allo y sam p le w h ere x = 0. 88 is show n in F ig . 19 fo r th re e d iffe re n t te m p e ra tu re s an d c o m p a re d w ith the a b so rp tio n of p u re G e. C o m p a riso n s of the a b so rp tio n of Ge ggSi ^ and pu re Ge a t long w av elen g th s a re show n in F ig s . 20 and 21 fo r ro o m te m p e ra tu re and liq u id n itro g e n te m p e ra tu re . One of the m o re s trik in g fe a tu re s of F ig . 19 is the new band n e a r 390 cm T his ban d is not o b se rv e d in p u re Ge but w as one of X = 0.12 a neutron irradiated 5 '-\ 100 200 300 400 450 500 550 V (cm"') F ig u re 18. A b so rp tio n c o efficien t v s. w avenum ber for n e u tro n irra d ia te d Si and Ge. 12Sl. 88 24 liquid He te m p e ra tu re (------ ) liquid N^, te m p e ra tu re ( ) room te m p e ra tu re ( — — ) Ge a t liquid N_ te m p e ra tu re (••••) 20 5 12 o o o < 4 200 300 250 350 Wavenumbers (cm’ 1) 400 450 500 F ig u re 19. A b so rp tio n co efficien t v s. w avenum ber for Ge 88^ . 12 liquid He te m p e ra tu re , liq u id N£ te m p e ra tu re , and ro o m te m p e ra tu re , and for Ge a t liq u id N2 te m p e ra tu re . acm 1 0 ' 150 J - I L. Ge..Si .,103 .88 .12 i_______t 200 250 300 -I v cm i i i 1 1 ...i 350 4 0 0 F ig u re 20. A b so rp tio n co efficien t v s . w avenum ber for p u re Ge and Ge^ ggSi room te m p e ra tu re . A rro w s in d ic ate d e n s ity -o f-s ta te s p eak s fo r Ge. U > O' } I 1 i I ! 5 I 1 5 0 200 250 300 350 400 ‘ ' 1/ cm" F ig u re 21. A b so rp tio n co efficien t v s . w avenum ber © ft liquid N te m p e ra tu re fo r G e, Ge q5Si and Ge Si ^ .88 .12 i i i 95 137 r ” ............. .. . . . . . . . . . . . . . . :.................... '.. ......... iss" th o se o b se rv e d in the R am an e ffe c t fo r Ge Si. allo y s w hen x > 0. 75. 3 C X -X l ~ U J i It is the band a ttrib u te d to the lo c a liz e d v ib ra tio n a l m ode of Si im p u ritie s . The o b se rv e d freq u e n cy of 390 c m * is c lo se to 394 cm * w hich is p re d ic te d fro m the th e o ry of DE fo r iso la te d , su b stitu tio n a l Si in the Ge la ttic e . The la tte r c a lc u la tio n s w e re done o rig in a lly fo r the Si la ttic e but they can be extended d ire c tly to the Ge la ttic e sim p ly I by a re -n o rm a liz a tio n of the m ax im u m phonon freq u e n cy . The ju stific a tio n of this p ro c e d u re h a s been d is c u s s e d p re v io u sly i ( L o rim o r and S p itze r 1966 ). The la rg e te m p e ra tu re dependence of th e stro n g tw o-phonon i _ i j band n e a r 360 cm m ak es the te m p e ra tu re dependence of the Si band ! d ifficu lt to d e te rm in e . H ow ever, a t liq u id n itro g e n te m p e ra tu re i w h e re the tw o-phonon background is red u c ed , the Si lo c a l m ode band i w as m e a s u re d fo r s e v e ra l v alu es of x and the re s u lts a re show n in | F ig . 22a and 22b. The is the m e a s u re d a b so rp tio n co efficien t, j and is the background a b so rp tio n e s tim a te d by a lin e a r e x tr a - ! p o latio n a c ro s s the b a se of the band in F ig . 22a. The dependence of J (a m "a |5)< ^v ° £ ^he 390cm * band of F ig . 22b is given a s a function of Si co n ce n tra tio n (1-x) in F ig . 23. B ecau se of the u n c e rta in tie s in , the re s u lts a re not su fficien tly a c c u ra te to e s ta b lis h the p re c is e n a tu re of the dependence of J * (a rn "a j3) ^ on h o w ev er, the in c re a s e in band s tre n g th w ith in c ra s e in Si c o n c e n tra tio n is c le a rly ev id en t. I X --X - 6 410 375 380 410 400 •I 390 400 390 v c m F ig u re 22-. A b so rp tio n co efficien t v s . w avenum ber fo r the Si lo c a l m ode in G e -ric h Ge S ij_ x a llo y s, a. O b serv ed a b so rp tio n , b. L o cal m ode band w ith background su b tra c te d . o Ge 9948 ^ oo6, Ge_ 985 S i# 015, x G e. 975S l. 025* 0 G e. 955S l. 045’ 0 G e. 880S l. 120' 0.40 S y m b o l S a m p l e a 1 0 1 * 1 0 2 • 1 0 4 o 1 0 3 .01-02 .02-03 .04-05 .12 .006 0. 10 l - X s l o p e I 0 .0 1 ®-0 0.005 100 F ig u re 23. C o n cen tratio n dependence of the in te g ra te d a b so rp tio n stre n g th for the Si lo c a l m ode band. The a b so rp tio n c r o s s - s e c tio n of th e Si lo c a liz e d m ode, J (am-ab)dv i------------------ is one to th re e o r d e rs of m agnitude s m a lle r than . (l-x )(5 x 1022) th o se p rev io u sly o b se rv e d fo r im p u ritie s in Ge, Si, and III-V | com pound sem ic o n d u cto rs ( N ew m an and W illis 1965, B alk an sk i and ! N a zarew icz 1966, C osand a n d S p itz e r 1967, N a za rew ic z and Ju rk o w - sk i 1969). In th is e stim a te of the a b so rp tio n c ro s s - s e c tio n it is j a ssu m e d th a t a lm o st a ll of the m e a s u re d Si content e x is ts as a n e a rly ; random ly d istrib u te d s u b s titu tio n a l Si sp e c ie s fo r [ S i ] ~ 5 x 102*cm 2, | (1-x) ~ 0. 01. If an y of the s ilic o n e x is ts in the fo rm of n e a re s t I ! neighbor p a irs , the site sy m m e try is and the p a ir h a s six m odes i jfo r w hich th e re a re fo u r v ib ra tio n a l fre q u e n c ie s. M odes a t two I / | fre q u e n c ie s should be in f ra r e d a c tiv e . Such Si p a irs have been j ! id en tified in GaAs w h ere the s p e c tru m is m o re co m p lica te d b e ca u se i of the lac k of in v e rs io n sy m m e try ( L o rim o r and S p itz e r 1966 ). I ! _ j I F eld m an e t. a l. (1966) re p o rt R am an lin e s n e a r 448 an d 476 cm " in I “ “ * i | the G e -ric h allo y s w hich th ey te n ta tiv e ly a ttrib u te to Si p a ir s . In the j p re s e n t c a se w eak a b so rp tio n bands a re ob tain ed n e a r 440 and 454 cm "^ w hich could be the in f r a r e d a c tiv e v ib ra tio n s of the sam e defect. In any c a s e , if the id e n tific a tio n is c o rr e c t, th a t bands a re v e ry w eak, in d icatin g a s m a ll c o n c e n tra tio n of p a irs re la tiv e to iso la te d Si. No o th e r stro n g lo c a l m ode bands a re o b se rv e d . The sm a ll a b so rp tio n c r o s s - s e c tio n fo r the Si lo c a l m ode in Ge c o n tra s ts sh a rp ly w ith the m o st n e a rly analogous c a s e , th a t of C in :Si ( N ew m an and W illis 1965, N ew m an and Sm ith 1969 ). E ach is j ; j ! an is o e le c tro n ic im p u rity in a ho m o p o lar c ry s ta l, and th e m a s s ! d efect e is s im ila r fo r both. H ow ever, the c o n fig u ra tio n of the bonding e le c tro n s is ex p ected to be quite d iffe re n t fo r th e two c a s e s . The e le c tro -n e g a tiv ity v alu es fo r Ge and Si a re a lm o st e x a c tly the i sa m e , w hile the e le c tro -n e g a tiv ity fo r c arb o n is c o n sid e ra b ly i d iffe re n t. Ge and Si fo rm a u n ifo rm solid so lu tio n o v e r the e n tire i j co m p o sitio n range of m ix tu re s; carb o n is only slig h tly soluble in -3 ; Si (<10 %) and in la rg e q u an tities fo rm s a com pound, SiC. A lso, | the te tr a h e d r a l co v alen t ra d ii a re m uch m o re c lo se ly m a tc h e d fo r Si I in Ge th an fo r C in Si; r_ /r_ . = 0. 65 and r_ . / r „ = 0. 96. i C Sx Sx Ge ) | S ilico n h as th re e iso to p es of m a s s e s 28, 29, an d 30 in re la tiv e i abundance of 92. 5%, 4. 5% and 3%. U nder m o re fa v o ra b le conditions I | th re e c lo se ly sp aced lo ca l m ode bands should be o b se rv e d ; but w ith i i j the s tro n g tw o-phonon background"and the re la tiv e ly w eak a b so rp tio n j ' | of the lo c a l m ode, no s tru c tu re is o b se rv ed th a t c an u n am b ig u o u sly | be id e n tifie d a s belonging to the d ifferen t iso to p e s of S i. In additon to the Si lo ca l m ode, two o th e r ch an g es a re o b se rv e d in th e se G e -ric h a llo y s, and they a re evident in F ig s . 20 an d 21. At liq u id n itro g e n te m p e ra tu re th e re is a b ro ad band n e a r 200 cm V in the a llo y sa m p le s w hich is not p re s e n t in the p u re Ge sa m p le . T his b and is p re s e n t in the room te m p e ra tu re m e a s u re m e n ts of both the a llo y an d p u re Ge. The pure Ge a p p a re n tly h as tw o -p h o n o n a b s o rp - i _1 144 jtio n n e a r 200 cm and th is band is su b sta n tia lly en ch an ced in the 1 |a llo y s a m p le s . B ra u n ste in re p o rte d the freq u en cy of a b ro a d band j in s im ila r allo y sa m p le s as being about 214 cm * a t 300°K . T here is a n o v e ra ll te m p e ra tu re dependence of the a b so rp tio n in the region of th is band, but in the allo y the band does not d e c re a s e as m uch a t low te m p e ra tu re s a s it does in p u re Ge. This is a p p a re n t fro m F ig . 20 and 21. A s im ila r b e h av io r is o b se rv e d fo r the doubly-peaked b ro ad jband c e n te re d n e a r 290 c m ” *. P u re Ge h as a tw o-phonon band at j about th is p o sitio n . The TO p eak in the phonon d e n s ity -o f-s ta te s is | about 282 c m ” *. The re so n a n c e m ode c alcu latio n s of M aradudin, of i I B ellom onte and P ry c e , and of D aw ber and E llio tt w hen sc a le d to the I Ge phonon fre q u e n c ie s w ill p re d ic t a w eak reso n an c e n e a r th is p o sitio n fo r a lm o st any lig h t im p u rity ; how ever, th is reso n an ce is p re d ic te d to have v e ry litte a b so rp tio n . W hether the o b se rv e d band J j is a n in c r e a s e of th e tw o-phonon a b so rp tio n o r w h e th er the bands | grow in the a llo y s w ith a te m p e ra tu re independent a b so rp tio n cannot | | be a n sw e re d u n am biguously fro m the data. The bands in the alloys I a re n e a rly te m p e ra tu re independent if one a ssu m e s the bands a re su p e rim p o s e d on a te m p e ra tu re dependent background. H ow ever, the change in to ta l a a t th e p eak p o sitio n betw een 300°K and 80°K a re in re a so n a b le a g re e m e n t w ith th a t e x p ec te d fo r tw o-phonon p ro c e s s e s . T he a rro w s in F ig . 20 in d ic ate th e peak p o sitio n s in the phonon d e n s ity -o f-s ta te s c u rv e fo r p u re Ge. T h ere is not any obvious single phonon a b so rp tio n th a t is c o rr e la te d to the phonon d e n sity -o f - sta te s as w as the c a se in the S i-r ic h a llo y s. CHA PTER V i L o c a liz e d V ib ratio n al M odes of B o ro n -L ith iu m P a ir s in S i-R ich G e-Si A lloys As h as b een d isc u sse d in C h ap ter I and II, the fre q u e n c ie s and ■ d e g en e ra c y of lo c a liz e d v ib ra tio n a l m odes of light im p u ritie s a re j v e ry se n sitiv e to the im m ed iate en v iro n m en t of the im p u rity . T his i p ro p e rty has p re v io u sly been exploited to probe p a irin g in te ra c tio n s ! betw een im p u ritie s in sem ico n d u cto rs; th is c h a p te r d e s c rib e s the i I a p p lic a tio n of lo c a l m ode stu d ies to the p ro b lem of d e te rm in in g j s tru c tu re on an ato m ic sc a le in a se m ic o n d u cto r alloy, j M e a su re m e n ts a re p re se n te d h e re of th e in fra re d a b so rp tio n i i i of the lo c a liz e d v ib ra tio n a l m odes of b o ro n p a ire d w ith lith iu m in j | Ge Si a llo y s. The freq u e n cie s of b o ro n w ith fo u r Si n e a r e s t j « J C X " « X » | n eig h b o rs and b o ro n w ith th re e Si and one Ge n e a re s t n e ig h b o rs a re j | found to be w ell se p a ra te d . The fre q u e n c ie s re su ltin g f ro m d iffe re n t | second n eig h b o r co n fig u ratio n s a re not quite w ell enough s e p a ra te d to be re s o lv e d into individual lin e s. The su p e rp o sitio n of th e ir a b so rp tio n a p p e a rs a s an a s s y m e tric broadening of the bands w hich a r is e fro m d iffe re n t f i r s t neighbor c o n fig u ra tio n s. H o w ev er, enough s tru c tu re is p re s e n t th a t the o b se rv e d bandshapes c an be re p ro d u c e d by c a lc u la tio n and the d istrib u tio n of second n eig h b o rs can be in fe rre d ; The e ffe cts of th ird and fu rth e r n eighbors a re not re s o lv e d a t a ll an d a p p e a r only as inhom ogenous line bro ad en in g . 145 The o b se rv e d a b so rp tio n s p e c tra a re c o n siste n t w ith a random i I d istrib u tio n fo r the seco n d n eig h b o rs of b o ro n ato m s in S i-ric h i Ge Si ; no evidence is found fo r second neighbor o rd e rin g . The X X " X f i r s t n e ig h b o rs, h o w ev er, m ay have som e o rd e rin g ; it a p p e a rs th at i i j b o ro n a to m s have fe w e r Ge f i r s t n eig h b o rs th an w ould be ex p ected ! fro m a ran d o m d istrib u tio n . j A. B ackground The lo c a l m odes of an is o la te d im p u rity in a c ry s ta l w ith a | T ^ (te tra h e d ra l) point g roup sy m m e try a t the im p u rity site w ill have ja trip ly d e g e n e ra te in fra re d -a c tiv e v ib ra tio n a l freq u en cy . Any J p e rtu rb a tio n w hich lo w e rs the sy m m e try m ay sp lit the d eg en eracy | | an d sh ift any o r a ll of the v ib ra tio n a l fre q u e n c ie s. T y p ically , ! j h o w ev er, the m odes we have c o n sid e re d , such a s the B lo c a l m ode in | Si, a re so w e ll lo c a liz e d th a t the p e rtu rb a tio n m u st have sig n ifican t i i . {am plitude a t the im p u rity s ite , o r a t le a s t a t a f i r s t n eig h b o r s ite , in i ! i o rd e r to p ro d u ce an o b se rv a b le e ffe c t. O b serv a tio n s of lo c a liz e d v ib ra tio n a l m odes of an im p u rity can thus be u s e d to study the im m e diate e n v iro n m en t of the im p u rity . T h e o re tic a l stu d ies have been m ade of the v ib ra tio n a l fre q u e n c ie s to be e x p ec te d of p a ire d im p u ritie s . T h ere h a s b e en ex te n siv e e x p e rim e n ta l w ork on s p lit tin g s of lo c a liz e d m o d es c a u se d by a n o th e r im p u rity p a ire d w ith the im p u rity giving r is e to the lo c a liz e d m ode. L o c a l m o d es of both is o la te d b o ro n a to m s and of b o ro n p a ire d w ith o th e r im p u ritie s have b een o b se rv e d fo r both Si and Ge c r y s ta ls . f I B o ro n -lith iu m p a irs have b een se e n in Ge; in Si, m o d es have been i o b se rv e d fo r b o ro n p a ire d w ith L i, P , A s, o r Sb, o r a n o th e r boron i ato m . In a ll of th e se c a s e s , w ith the ex ce p tio n of th e b o ro n -b o ro n j p a irs , the o b se rv e d lo c a l m odes a re e s s e n tia lly b o ro n v ib ra tio n s ; p e rtu rb e d by the s tr a in fie ld an d / o r coulom b fie ld of th e p a ire d ion. B oron is an e le c tr ic a l dopant in Si o r Ge; to av o id having the i lo c a l m ode in fra re d a b so rp tio n m a sk e d by the fre e c a r r i e r a b so rp - ; tio n , the sa m p le s u se d in th is stu d y w e re co m p e n sa te d by L i I diffusion. I n te r s titia l lith iu m do n o rs p a ir quite re a d ily w ith the j j su b stitu tio n a l b o ro n a c c e p to rs ; the in te ra c tio n of the lith iu m w ith the j ' ’ | b o ro n sp lits the trip ly d e g en e ra te m ode of iso la te d ^ B ( ^ B ) at i 620(644) c m " * into a sin g ly d e g e n e ra te m ode along the L i-B ax is i ! _ i i a t 564(584) cm an d a doubly d e g e n e ra te tr a n s v e r s e m ode at j 655(681)cm~* ( S p itz e r and W ald n er 1965, E llio tt and P fe u ty 1967 ). ! i j The fre q u e n c ie s of th e se m o d es a re a lm o st independent of the L i j iso to p e, in d ic atin g th a t th e se a r e a lm o st e n tire ly b o ro n lo c a l m odes 7 6 w ith little m o tio n of the L i. The band a ttrib u te d to L i( L i) at 522(534) c m * show s no o b se rv a b le sh ift in freq u e n cy w hen the b o ro n isotope is changed; it th en is a lm o s t e n tire ly a L i lo c a l m o d e. The in te ra c tio n betw een L i an d B should a ls o s p lit the d e g e n e ra c y of the lith iu m v ib ra tio n s . The seco n d L i fre q u e n c y m a y be below the m axim um phonon freq u e n cy of Si and m e rg e into the band m odes so I th a t it does not p ro d u ce a sh a rp a b so rp tio n band. T h ere h a s b een | | som e re c e n t c o n je c tu re th a t since the o b se rv ed L i freq u e n cy is v e ry \ c lo se to the Si R am an freq u en cy the freq u en cy sp littin g is s m a ll and | the o b se rv e d band in clu d es a ll th re e m odes ( E llio tt and P fe u ty 1967). I M odels have b een su g g e ste d fo r the in te ra c tio n b etw een the B and ; L i th a t w ould sh ift the tra n s v e rs e m odes h ig h e r, so they should i acco u n t fo r the o b se rv e d band. H ow ever, in Ge, L i-G e p a irs : p roduce two L i bands w ith a stre n g th ra tio of ~ 2:1 and the s tro n g e r . I I b and is lo w er in freq u e n cy ( C osand and S p itzer 1967 ). T hus it is I not yet c le a r how the b o ro n and lith iu m in te ra c t to s p lit the | ' ! d e g en e ra c y . | In Si, a s m a lle r sp littin g is o b se rv ed fo r a su b stitu tio n a l donor j j i P , A s, o r Sb p a ire d w ith the b o ro n th an fo r B -L i p a ir s . The I a ssig n m e n t of the o b se rv e d bands is not definitely a s c e rta in e d . The m o d el of N ew m an and Sm ith (1968) gives the sp littin g the o p p o site sig n of w hat is o b tain ed w ith L i p a irs in Si. In th e ir m o d el, the a x ia l m ode lie s h ig h e r fo r P , A s, o r Sb p a irs . H ow ever, in la te r w o rk , N ew m an (1969) h as su g g ested th a t th is m o d el m ig h t be in e r r o r . The m agnitude of the sp littin g is only about o n e -th ird of the sp littin g c a u se d by L i; i . e . , about 30 c m " 1. The sp littin g is s u p e r p o se d on a sh ift in freq u e n cy w hich is a p p ro x im ate ly lin e a r in the co v alen t ra d iu s of the donor. T h e re have b een no re p o rts of the effects of a m ix ed c r y s ta l I 149 ! host on im p u rity lo c a l m o d es in se m ic o n d u c to rs. H ow ever, the | effects of a m ix ed c r y s ta l h o s t on h y d ro g en lo c a l m odes in a lk a li ; h alid es have b een o b s e rv e d by M e rlin and R eshina ( 1966). T h eir c ry s ta ls w ere p rin c ip a lly KC1, a llo y ed w ith a few p e r se n t of e ith e r 1 8 - 3 ' Na, C s, o r Rb o r F , B r, o r I, and doped to about 10 cm " with i hydrogen. The re s u ltin g s p e c tr a a re sim p le to in te r p r e t since the I H" sits on the anion s u b la ttic e in place of a C l", and a catio n su b sti- j tution can be only an odd n eig h b o r ( f ir s t, th ird , e tc . ) w hile an i | anion su b stitu tio n can be only a n ev en n eig h b o r. The c o n ce n tra tio n ' of the alloying e le m e n t w as low enough th at th e re w as not m uch I likelihood of m o re th a n one su b stitu tio n in the im m ed iate v icin ity j of any H ion. They id e n tifie d bands c o rre sp o n d in g to one f ir s t i i | neighbor o r one th ird n e ig h b o r fo r the Na, C s, and Rb a llo y s. In i I ! the alloys w ith F , B r, a n d I, b ands w e re o b se rv e d fo r one second neighbor. The o b se rv e d sp littin g s w e re found to v a ry a lm o st } lin e a rly w ith the ionic ra d iu s of the alloying e le m e n t. The effects of ! i i . | m ore d ista n t n eig h b o rs w e re to inhom ogenously b ro a d e n the o b serv ed localm ode a b s o rp tio n lin e s an d to sh ift the o b se rv e d fre q u e n c ie s, including the fre q u e n c y of the is o la te d H~ lo c a l m ode, in in v e rse p ro p o rtio n to th e change in a v e ra g e la ttic e p a ra m e te r. They conclude th a t the p r in c ip a l e ffe c t of a neighboring im p u rity on the lo cal m ode is th at the s t r a in in tro d u c ed by the im p u rity changes the fo rce c o n stan ts se e n by th e H . 150 The ra tio of lo c a l m ode a b so rp tio n s tre n g th fo r iso la te d H ; and H w ith one Na f ir s t neighbor is v e ry c lo se to w hat w ould be I p re d ic te d fo r a ran d o m d istrib u tio n of the Na in the KC1:H. H ow ever, i fro m the m e a s u re d a b so rp tio n , the a p p a re n t c o n c e n tra tio n of Rb o r j Cs is h ig h e r th an p re d ic te d , in d ic atin g a p re fe re n tia l p a irin g betw een : the H and the Rb o r C s. It is a lso p o ssib le th a t a t le a s t p a rt of the j i ' d isc re p a n c y in a b so rp tio n stre n g th m a y re fle c t a change in the H j I lo ca l m ode o s c illa to r stre n g th c au se d by the p a ire d im p u rity ; such ! changes have been o b se rv e d in o th e r sy s te m s ( N ew m an 1969 ). B. Expe rim e ntal j ! F o u r Ge S i1 ingots w ere u s e d in th is study; in e ac h case ! X 1 -X i | no m o re th an one th ird of the m e lt w as c ry s ta lliz e d in o rd e r to lim it | the change in c o m p o sitio n along the ingot. The n o m in al co m p o sitio n s i j w ere Ge Q^Si and undoped, Ge Q^Si ^ an d doped w ith ~ 2 x ' 1 9 - 3 ' 10 cm of n o rm a l b o ro n , Ge _Si _ and undoped, an d Ge ,- S i Q O i . 1 2 . oo .1 2 . oo [ 1 9 - 3 10 I and doped w ith ~ 1. 5 x 10 c m ” w ith B(96. 5% iso to p ic p u rity ). j The undoped Ge _ ,S i and the doped Ge 0Si O Q w ere single * Oo i /4 • 1 u i oo c ry s ta ls ; the o th e r two ingots w e re p o ly c ry s ta llin e w ith ty p ic al c ry s ta llite dim en sio n s of s e v e ra l m m . The b o ro n doping le v e ls w ere e s tim a te d fro m re s is tiv ity m e a s u re m e n ts and the d ata of B ra u n ste in (1963a ) fo r hole m o b ilitie s in Ge Si . The c o m p o si- Jl tio n w as d e te rm in e d by d en sity m e a s u re m e n ts , and h lso by e le c tro n m ic ro p ro b e a n a ly sis. M icro p ro b e m e a s u re m e n ts to ch eck the I hom ogeniety taken, a t d ifferen t p laces on one of the sa m p le s containing 12% Ge a g re e d w ithin . 02 of the Ge c o n c e n tra tio n . A line sc a n o v e r a length of 1 m m w ith a re so lu tio n of 1 jim show ed : no v a ria tio n la r g e r th an 0. 01 of the Ge c o n c e n tra tio n . The b o ro n doped Si w as co m p en sated by diffusion of lith iu m fro m a su rfa c e allo y la y e r by the p ro ced u re p re v io u sly d e sc rib e d . S e v e ra l u n su c c e ssfu l a tte m p ts w ere m ade to u se L i diffu sio n to 19 20 ■ co m p en sate som e v e ry h eav ily b o ro n -d o p ed ( 5 x 1 0 -2x1 0 -3 ! c m ) Ge Si sa m p le s w ith h ig h e r Ge content ( x = 0. 2 - 0. 3 ). X 1 •* X T hese a tte m p ts w e re not su c c e ssfu l as the sa m p le s w e re e ith e r j h e av ily p itte d by the L i o r w ere not co m p en sated . In fra re d tra n s m is s io n w as m e a su re d w ith the sin g le b e a m I g ratin g s p e c tro m e te r. The s p e c tra l re so lu tio n w as ty p ic a lly 0. 5 - ! 1 cm G e n e ra l fe a tu re s of the s p e c tra w e re tak e n fro m continuous tra c e s ; d e ta ile d 80°K m e a su re m e n ts of som e bands w e re m ade I ' ■ p o in t-b y -p o in t w ith the sa m p le -in , sa m p le -o u t tech n iq u e. Som e i j d iffe re n tia l m e a su re m e n ts w ere m ad e, w here an undoped allo y j i sam p le of the sam e th ic k n e ss as the doped sam p le w as p la c e d in the i S b e am w hen the I m e a su re m e n t w as m ad e. The v a lu e s of ax fo r I o th e se sa m p le s w ere la rg e enough that the a p p ro x im atio n I / I » i C d e a 2 )x i s re a so n a b ly a c c u ra te . C. R e su lts In F ig . 24, the a b so rp tio n s p e c tra a re show n in the re g io n of • \ 1 \ ' / © / I / \ / I u " J □ £ i - j <0 J l_ I _ Q Q ■ ~ » I Q Q o ■ ~ i o Z CD c n = _ W □ V ■cP.sF ___ _ ^ - " A J I L. 500 550 600 650 WAVENUMBER (cm -1) 700 F ig u re 24. A b so rp tio n co efficien t v s. w avenum ber fo r b o ro n doped, lith iu m com pensated sa m p le s a t 80°K. A: Si, n o rm a l b o ron, ^Li. B: Ge Q^Si Q 4 » n o rm al boron, ^Li. C: Ge 12Si 8g, 10 B, 6L i. i 1 3 0 ; | the b o ro n and lith iu m lo c a l m odes fo r Si and Ge ^ S i ^ w here j | both a re doped w ith n o rm a l b o ro n ( 81% **B, 19% *^B ) and fo r | Ge to Si O Q doped w ith 96. 5% *^B. A ll th re e sa m p le s have b een j i 1Z i oo i I c o m p en sated by lith iu m diffusion; the lith iu m is in in te r s titia l s ite s , ; m o st of it in the fo rm of L i* -B p a ir s . The a b so rp tio n c u rv e fo r S i:B -L i given in F ig . 24, show s the I bands a ttrib u te d to B and B -L i p a ir s , a s h as been p rev io u sly I ; d isc u sse d . The a ssig n m e n ts a r e in d ic a te d on the fig u re . The I p rin c ip a l fe a tu re s of the lo ca l m ode s p e c tra a re se e n a ls o in the I | a llo y s, w ith som e notable c h an g e s. T h ere is a g e n e ra l sh ift to I j lo w e r fre q u e n c ie s, a s show n in T able 9, and a b ro ad en in g of the 1 | m ain b an d s. In ad d itio n , som e w eak new bands a p p e a r. In the j : Ge , -S i O Q sa m p le , w h e re the e ffe c ts a re m o st pronounced, th e lin e | 0 X 2 i O O w idth of the tr a n s v e r s e ***B v ib ra tio n a t 677 cm * is se en to be n early ! tw ice as la rg e as in p u re Si, and the shape is som ew hat a s s y m e tr ic , being b ro ad e n ed m o re on the high fre q u e n c y sid e . The a x ia l m ode band is even m o re a s s y m e tric a lly b ro ad en ed ; h o w ev er, the low freq u en cy side of the band is v e ry n e a rly a s ste e p as the side of the sam e band in pu re Si. T his band is se en m o re c le a rly in F ig . 25, w hich show s the d iffe re n tia l a b so rp tio n ( ) betw een b o ro n - doped and undoped Ge Si ot) sa m p le s a t ~ 10°K . The band is se e n f l u i OO to have no re so lv a b le s tru c tu re . T h e re is a new band n e a r 553 cm "* th a t is m uch w e a k e r th an the 580 cm * band but h a s the sam e ABSORPTION COEFFICIENT (cm" i 450 500 550 WAVENUMBER (cm") 600 650 700 750 i i i e 25. D iffere n tia l a b so rp tio n b etw een Ge j^S i g8 * L i and undoped Ge . 12S a t liq u id He te m p e ra tu re . D ata w as tak en fro m continuous sc an s; le v e l sh ifts a re the r e s u lt of m isa lig n m e n t of the d ew ar. 154! “155;' a s s y m e tr ic sh ap e. The o th er new band in the allo y is a t 651 c m ; it is of s im ila r stre n g th to the 553 c m ”* band but it o v e rla p s so m e w hat both the iso la te d band at 638 cm * and the 677 c m * band. T hus its stre n g th and line shape a re difficult to d e te rm in e . TABLE 9 L o c a liz e d Mode F re q u e n c ie s at 80°K Si G e.0 6 S i.94 Ge . 12 10B tr a n s v e r s e 682. 0 679. 0 677. 0 a x ia l 586. 1 582. 3 5 8 0 .3 7t • L i 523. 5 521.5 5 1 8 .2 D. D isc u ssio n | F o r th e S i-ric h allo y s stu d ied h e re , the s y s te m is b a s ic a lly j i th e sa m e a s S i:B -L i, ex cep t th a t it is p e rtu rb e d by the G e. If the I ! j s y s te m is n e a r ran d o m , m o st B s ite s should have fo u r Si f ir s t j j n e ig h b o rs, som e w ill have one Ge and th re e Si, few w ill have two Ge, a lm o s t none m o re th an two. S im ila rly , ex p ected c o n fig u ra tio n s can be c a lc u la te d fo r seco n d n eig h b o rs, th ird n e ig h b o rs, e t c . . The e ffect one e x p e c ts to o b se rv e fro m the su b stitu tio n of a Ge fo r one of the Si a to m s n eighboring the B is a sp littin g o r sh ift of a B lo c a l m ode. The e ffe c t m a y be la rg e fo r f ir s t neighbors but1 is e x p e c te d to fa ll off ra p id ly w ith in c re a s in g d istan ce to the p e rtu rb in g a to m . Since the 580 and 677 cm * bands develop d ir e c tly fro m the _1 156 | 586 and 682 c m bands of B -L i in Si, they m u st include the 'a b s o rp tio n due to B w ith fo u r Si n e a r n eig h b o rs. The 553 c m ” * band h as a lm o st ex ac tly the sam e a s s y m e tric | sh ap e a s th e 580 c m ” * band. It se e m s re a so n a b le to a ssu m e th a t j the sam e m e c h a n ism gives ris e to th is s tru c tu re in both c a s e s , and ■any m o d el m u st a ttrib u te the s tru c tu re to som e m ec h an ism o r in te r - :a c tio n th a t is e s s e n tia lly unchanged by w h a tev e r sp lits th e .553 c m ” * I band off fro m the 580 cm * band. The la rg e s t effe cts due to adding j jGe th a t a r e se e n a r e the sp littin g s betw een the 553 c m ” * and 580 jcm * b ands and betw een the 651 c m ” * and 677 c m ” * bands. T h ese j jsp littin g s have to be f i r s t neig h b o r in te ra c tio n s u n less it is a s s u m e d i jthat none of the B h a s any Ge f i r s t n eig h b o rs. If the sp lit off bands I I we re due to seco n d n eig h b o rs, it would be d ifficu lt to e x p la in why i . . jonly one s p lit off band is se en fo r each p rin c ip a l B -L i band w hen jth ese a re m any n o n -eq u iv alen t p o ssib le co n figurations of second i n e ig h b o rs. E . M odels F ig u re 26 show s an ato m (1) in a diam ond la ttic e w ith its fo u r ' f i r s t n e ig h b o rs (2 -5), tw elve second n eig h b o rs (6-17), and a t e t r a h e d ra l in te r s titia l s ite (18). T hroughout the r e s t of th is p a p e r, m e n tio n of la ttic e s ite s by n u m b er w ill r e f e r to th is fig u re . The boror. s ite w ill alw ays be p o sitio n 1. A ssig n m e n ts could be p ro p o se d a ttrib u tin g both the sp lit off F ig u re 26. A n atom (1) in a diam ond la ttic e w ith its four f ir s t n e ig h b o rs (2 -5), tw elve second n eig h b o rs (6-17), and a t e t r a h e d ra l in te r s titia l site (18). T h ere a r e th re e se ts of second n eig h b o r s ite s (6 - 8 , 9-14, 15-17) such that th e .s ite s w ithin a s e t a r e eq u iv alen t re la tiv e to a s u b s titu tio n a l-in te rs titia l 158 j : bands and the a s s y m e tric b ro ad e n in g to f i r s t n e ig h b o rs. C onsidering ! j only the 553 and 580 c m ” * b ands one could su g g e st th at a Ge f irs t ! n eighb or on site 5, o p p o site th e L i of the B -L i p a ir , gives a la r g e r ; sp littin g than a Ge on any of th e o th e r th re e f i r s t n eig h b o r s ite s , i _ i | 2 -4 . The assig n m en ts could th en b e m ade th a t the 553 cm band is | due to site s with a Ge o p p o site th e L i, and th e b ro ad en in g on both i I bands is caused by having one o r m o re Ge f i r s t neig h b o rs on the ! o th e r th re e s ite s . T his m o d el w ould im ply a la rg e tendency fo r a s s o - | ciation betw een Ge and B . H ow ever, the e ffe cts of Ge f ir s t neighbors | in the th re e sites off the B -L i a x is should b e v e ry pronounced on the : tr a n s v e r s e m odes; th e 651 and 677 cm ” * bands a re too sim p le to j su p p o rt th is m odel. Any o th e r m o d els th at m ay be p ro p o sed s till | m u st tak e into account th e s im ila r ity in sh ap e b etw een th e 553 and i ; 580 c m ” * bands, and th a t th e s tr u c tu r e is too co m p licated to be i re so lv e d into a su p e rp o sitio n of only two o r th re e bands w ith sim p le shape. A m odel that s a tis fa c to rily fits a ll th e o b se rv e d d ata is to a ssig n the 580 and 677 c m ” * bands to the a x ia l and tr a n s v e r s e m odes of B -L i w here the B has fo u r Si f i r s t n e ig h b o rs. The 553 and 651 c m ” * bands a re the c o rre sp o n d in g m odes fo r b o ro n w ith one Ge atom am ong the f ir s t n e ig h b o rs. W hen one Ge f i r s t n eig h b o r is p re s e n t the L i e ith e r sits o p p o site th e Ge o r a d ja ce n t to th e Ge, but only one of the configurations o c c u rs to any a p p re c ia b le ex ten t. The lin e 159 shape of any in d iv id u al o s c illa to r is sy m m e tric , but its freq u e n cy ■depends on the seco n d neighbor co n fig u ratio n . The o b se rv e d b ro ad en ed a s s y m e tric bands a re the su p e rp o sitio n of the a b so rp tio n fro m boiton a to m s w ith d ifferen t second neig h b o r c o n fig u ra tio n s, i One of the obvious fe a tu re s of the 580 c m * and 677 c m * bands is the g re a te r b ro ad en in g of the lo w er freq u e n cy band. In both c a se s : the b ro ad en in g is a ttrib u te d to second n e ig h b o rs. One p o ssib le | ex p lan atio n is b a se d on the p ro p e rty of lo c a liz e d v ib ra tio n a l m odes ! th a t a h ig h e r freq u e n cy m ode is a tte n u a te d m o re ra p id ly w ith i d istan c e th an is a lo w er freq u e n cy m ode. The low fre q u e n c y m ode i I | should involve m o re m otion of the seco n d n e ig h b o rs, an d thus be jm o re a ffe c te d by su b stitu tio n of a Ge fo r a Si. T his ex p lan atio n is i ! u n s a tis fa c to ry fo r two re a s o n s . The e ffect of the m a s s change should i j be to lo w e r the m ode freq u en cy c au sin g b ro ad en in g on the low ‘freq u e n cy sid e s of the ban d s, and, a s is show n in the appendix, the | ! e ffect is too sm a ll. The change in freq u e n cy of the a x ia l b o ro n m ode of a B -L i p a ir w hen a Ge is su b stitu te d fo r a Si seco n d neig h b o r w as c alcu late d ; the value depended on w hich n eig h b o r w as su b stitu te d , and ran g ed fro m - 0 . 1 to 01 . 1 cm "* fo r an iso to p ic su b stitu tio n m odel. If re a so n a b le changes of the G e-S i fo rc e c o n stan t a re c o n sid e re d , the la r g e s t change is -1 .5 c m "* , w hich is s till too sm a ll. F . L in esh ap e C alcu latio n s ! 160 ! The shape of the bands does not uniquely give the seco n d n e ig h b o r d istrib u tio n w ithout som e ad d itio n al as su m p tio n s, but a | sim p le m o d el can be d ev ised w ith w hich a p ro p o sed d istrib u tio n can : be te s te d to se e if it is c o n siste n t w ith the o b se rv ed band sh ap e. In I fa c t, it is found th a t the o b se rv e d lin e shape can be obtained fro m a : p u re ly ran d o m d istrib u tio n of seco n d n eighbors and a few e m p iric a l I but p h y sic a lly re a so n a b le n u m b e rs. C o n sid er f ir s t the a x ia l m ode I a t 580 cm it show s the la r g e s t e ffe c ts fro m the second n e ig h b o rs. I j It is a sin g le n o n -d eg e n era te m ode and the p e rtu rb a tio n s due to s e c - j I ond n e ig h b o rs a re a ll s m a ll c o m p a re d to the sp littin g due to the b o ro n i • ; b ein g p a ire d w ith L i. It should th e re fo re be a re a so n a b le a p p ro x i- j m a tio n to c o n sid e r the e ffe cts of d iffe re n t second n eig h b o rs as being | | in d ependent. The. e ffe c t of a given co n fig u ratio n of seco n d neig h b o rs j is ta k e n a s the su m of the e ffe c ts of the individual second n eig hbors | i c o n sid e re d se p a ra te ly . j j R e fe rrin g to F ig . 26, it is se e n th at in te rm s of the e ffect on i — ’ f’ j the a x ia l m ode of a B -L i p a ir th e re a re th re e d istin c t s e ts of se co n d n e ig h b o rs. One s e t c o n sists of the th re e site s 6-8 th at a re + n e ig h b o rs of the in te r s titia l L i , 18. The second s e t c o n sists of th e six s ite s 9-14 w hich a re in a plane w hich is p e rp e n d ic u la r to th e B -L i a x is and p a s s e s th ro u g h the b o ro n site . The th ird s e t c o n sists of th e th re e re m a in in g s ite s , 15-17. The th re e d iffe re n t se ts of s ite s a re d esig n ated by a n index i = 1, 2 o r 3, re s p e c tiv e ly . A Ge a to m in one of the s ite s of the itn se t is then a ssu m e d to sh ift the fre q u e n c y of the lo c a l m ode by som e ; c h a ra c te ris tic am ount 6.. If the n u m b er of Ge ato m s in the ith set l is n^, then a boron ato m w ith a c o n fig u ra tio n ( n ^ n ^ n ^ ) of Ge second neighbors w ill have its lo c a l m ode freq u e n cy sh ifte d by S n .6 . rela tiv e to th a t of a b o ro n w ith no Ge seco n d n e ig h b o rs. If a . i i l i random d istrib u tio n of Ge a to m s th ro u g h o u t the la ttic e is a ssu m e d , : then the allo y co m p o sitio n d e te rm in e s the p ro b a b ility of any config- 1 u ra tio n . F o r Ge Si the p ro b a b ility of a c o n fig u ra tio n (n .,n , n_) X i ~ X . > X w < 5 ! is p (n x) p(n2) p(n3) ; p(m ) = V;- ---j; * 1 (1-x) w h ere k = 3 i ' “ r i ' I fo r i = 1 o r 3 and k = 6 fo r i = 2. If the a b so rp tio n a(v) fo r a single | o s c illa to r can be w ritte n a s a lin e -sh a p e fu nction c e n te re d about a : freq u en cy V q , a(v) = L(V-\>o ), th e n th e re s u ltin g inhom ogeneously : b ro ad en ed line in the allo y is e x p re s s e d a s a su m of the a b so rp tio n I of individual o s c illa to rs , e a c h w ith its fre q u e n c y d e te rm in e d by the , second neighbor c o n fig u ratio n , w eighted by the p ro b a b ility fo r the | o c c u rre n c e of th at c o n fig u ra tio n ; a(v) = N E i a ll(n , n , n ) L(V-V - E n .6 .) p(n.) . F o r the o . 1 1 . r 1 i i S i-ric h allo y s being c o n sid e re d , the su m m atio n w as tak e n only o v er the tw enty co n fig u ratio n s fo r w hich E n^ < 3. The p ro b a b ility that E n^ > 4 is le s s th an . 01 fo r a n a llo y w ith 6 % Ge and le s s th an 0. 05 fo r an allo y w ith 12% Ge. In c o m p a rin g the p re d ic te d a b so rp tio n band shape to an e x p e rim e n ta lly o b se rv e d band, the n o rm a liz a tio n fa c to r N and the unshifted freq u en cy vQ a r e d e te rm in e d fro m the in te g ra te d in te n sity and the frequency of the o b se rv ed band. If a I L o re n tz ia n lin e -sh a p e is a ssu m e d fo r a sin g le o s c illa to r, th e w idth : p a ra m e te r m u st be d e te rm in e d . The effects of th ird and fu rth e r n eig h b o rs a r e lum ped into th e linew idth, so th e lo w e r bound on the w idth is th a t o b se rv ed fo r th e boron lo c a l m ode in Si. It w ill be seen ; th at a good fit can be obtained by sim p ly using th e w idths o b se rv ed ■ fo r B in Si, in d icatin g th at the effect of th e m o re d is ta n t neig h b o rs is I sm a ll. T h e re a r e then th re e fre e p a ra m e te r s , th e 6 ., to d e te rm in e the shape of th e lin e . > The o rig in of the sh ifts of 6 . . is d e sc rib a b le in te r m s of the i n a tu re of th e in te ra c tio n betw een th e b o ro n and its Ge second | i n e ig h b o rs. The freq u en cy of the b o ro n lo c a l m ode is d e te rm in e d i a lm o st e n tire ly by the m a s s of th e b o ro n , the fo rc e c o n stan ts b etw een | b o ro n and o th e r a to m s, and the m a s s e s of th e f ir s t n e ig h b o rs. The m ode is su fficie n tly w ell lo ca liz ed th at changing a fir s t- to - s e c o n d n eig h b o r fo rc e co n stan t o r a second n eig h b o r m a s s h a s v e ry little e ffect on th e freq u en cy . The fo rce constant b etw een b o ro n and a second n eig h b o r Ge should not be su b sta n tia lly d iffe re n t fro m th at w hich o c c u rs fo r a Si second n eig h b o r. H ow ever, the a s s y m e tric a l b ro ad e n in g s p re a d s out o v e r ~20 c m ” *. T h ere a r e two in te ra c tio n s betw een a b o ro n ato m and a Ge second neig h b o r th a t a r e lik e ly to be stro n g enough to account fo r the o b se rv e d sh ifts. One is th e la ttic e 164 due to p o la riz a tio n induced on the neighbors by the Li . A Ge ato m | is both la r g e r and m o re p o la riz a b le than the Si it re p la c e s . If th e i re p u lsiv e in te ra c tio n due to th e in c re a se d siz e of the Ge d o m in ates, ! J . ; a Ge w ould push and in te r s titia l L i c lo s e r to the boron, in c re a s in g | the e le c tr o s ta tic field seen by th e boron and in cre asin g th e sp littin g ; of th e b o ro n m o d es, thus lo w erin g the axial m ode freq u en cy . If the j a ttra c tiv e in te ra c tio n due to the in c re a se d p o lariza b ility d o m in a tes, i | » ^ I then the L i w ill b e pulled away from the boron, which w ill r a is e the f . i ■ a x ia l m ode freq u e n cy . The la tte r is w hat is required to fit the ! i # | o b se rv e d lin e -s h a p e . A ssum ing th at th e changes in the b o ro n m ode i (fre q u e n c ie s fo r a change in th e L i+ - B~ in te ra c tio n a re in the sam e | ra tio as the to ta l sp littin g fro m the isolated^B frequency, the e le c tr o - j s ta tic p a rt otf.the freq u en cy sh ifts of th e 580 and 677 cm * bands |sh o u ld b e re la te d by 6l e (677) » 6l e (580) = -0 . 6 6l e (580). ! 1 • • + j The e ffe ct of the s tro n g Coulom b in te ra c tio n betw een th e B and L i should b e to pull th e L i+ along th e < 1 1 1> ax is tow ard th e b oron and aw ay fro m th e second neig h b o r s ite s . This should favor the p o la r iz a tio n in te ra c tio n and red u ce th e o v erlap in te rac tio n b etw een th e L i+ and a Ge in th e f i r s t s e t of s ite s , giving the d e sire d p o sitiv e v alu e f o r 6. (580). T he 6 . (677) w ill then be n eg ativ e, w hereas th e s tr a in l e 1 e sh ifts fo r th e 677 band should be positive, thus accounting fo r th e l e s s e r b ro ad e n in g of th e 677 cm""* band. T he s im p le s t m odel to d e te rm in e a p p ro p ria te v a lu e s for th e 165 i ; freq u en cy sh ifts is to a ssu m e th a t a Ge on any of the second neighbor ; s ite s gives the sa m e s tr a in shift; 6 = 6 = 6 , and 6 = 6 + 6 . 1 S u X X S X G : The v alu es fo r th e se p a ra m e te rs w hich give a fia rly good fit to the \ shape of the 580 cm "* band a re 3. 3 c m ’ * fo r 6 ^ and 9. 5 cm~* fo r ; 6 , so th at 6 is 6 .2 c m *. As is se e n in F ig . 27, th is gives a X X 6 i re a so n a b le fit to the g e n e ra l shape of th e a x ia l m ode. A b e tte r fit can be o b tain ed by u sin g a som ew hat m o re r e a lis tic | m o d el fo r th e e ffe c ts of s tr a in induced by Ge second n eig h b o rs. The I p rin c ip a l m a n n e r in w hich a seco n d neighbor Ge can affect the b o ro n I m ode fre q u e n c y is to in c re a s e the fo rc e c o n stan ts coupling thei-boron i I j to its n e ig h b o rs by d isp lac in g one of the n eig h b o rs to w ard the bo ro n . | A Ge ato m in the th ir d s e t of seco n d neighbor s ite s , nos. 15-17 in i ! j F ig . 26, w ill d isp lac e the n eig h b o r (no. 5) th a t lie s on the B -L i a x is, i ' p rin c ip a lly a ffe c tin g the fo rc e c o n stan ts coupling the b o ro n to th at n eig h b o r. A Ge a to m in e ith e r the f i r s t o r seco n d s e t of second j n eig h b o rs w ill d isp la c e one of the o ff-ax is f i r s t neighbours, changing J m ain ly the fo rc e c o n sta n ts b etw een the n eig h b o r and the boron. Since the a x ia l m ode a t 580 c m * is se n sitiv e m o re to the fo rce co n stan t b etw een the b o ro n and the o n -a x is f ir s t n eig h b o r th an to any of the o th e r fo rc e c o n sta n ts, the freq u e n cy sh ifts 6^ should sa tis fy 6^(580) < = « 6_(580) < 6,(580); 6 is s till given by 6 = 6 + 6 . The v alu es C t <j X X X 8 X G fo r th e se p a r a m e te r s th a t a re found to give a good fit to the o b se rv e d band shape ( see F ig . 27 ) a re 6 , = 5. 0 c m ’ *, 6 = 2 .7 cm "*, and ABSORPTION COEFFICIENT (cm" O O o "540 560 580 600 V640 W AVENUMBER (crrf) 660 680 700 F ig u re 27. L o ca liz ed v ib ra tio n a l m ode a b so rp tio n of b o ro n in B - -L i+ p a irs in Ge j^S i gg- C irc le s a r e e x p e rim e n ta l p o in ts. D ashed lin e is the fit obtained w ith ju s t one sh ift ( , = - = o )• Solid lin e s a re fits obtained w ith two s tr a in sh ifts. Is 2 3 ' I F o r th e tr a n s v e rs e lo c a l m ode, the seco n d n eig h b o rs in the ' f i r s t and second se t of site s w ill give the la r g e r s tr a in sh ift, so I 6 . » 6 _ > 6, . U sing the ra tio 6 . (677)/6. (580) = - 0 .6 and the sam e 1 S Ct J XG XG i n u m e ric a l values fo r the s tr a in s h ifts , one o b tain s 6 ^ = 5 .0 - 4 . 1 = ; 0 . 9 , = ^3 = c m "*« - A - good fit c an be m ade u sing v alu es | quite clo se to th ese n u m b e rs, w ith the b e s t v alu es being 6 ^ = 1 . 0 , ; = 5 .4 , and 6, = 2. 9 cm * ( see F ig . 27 ). F o r the tra n s v e r s e u < 3 I lo c a l m ode the sim p le m o d el w ith only one s tr a in sh ift does not give i I a rea so n a b le fit to the o b se rv e d band sh ap e. i It is not s u rp ris in g th at the above m odel w orks w ell fo r the | a x ia l m ode; it is a n o n -d eg e n era te m ode, and the sp littin g induced i * I ! by the L i w hich defines the m ode is la rg e c o m p a re d to the e ffe cts of i I : the second n e ig h b o rs. The sim p le f i r s t o r d e r p e rtu rb a tio n t r e a t- I | m en t should w ork w ell. The tr a n s v e r s e m ode, ho w ev er, is a doubly i d e g en e ra te m ode. The sim p le d e sc rip tio n of co n fig u ratio n s of second n eighbors is not ad equate to d e sc rib e th e ir e ffe ct on sp littin g the d eg en eracy of the m ode. In te rp re tin g the freq u e n cy sh ift p a ra m e te rs 6^ a s additive e ffe c ts of in d iv id u al Ge seco n d n eighbors does not se em re a so n a b le ; the m o d el is too sim p le . F o r the t r a n s v e rs e m ode the 6^ m u st be ta k e n a s being an a v e ra g e e ffect, w ith the im p o rta n t point being the p a rtia l c a n c e lla tio n of the s tr a in and Coulom b sh ifts. The fits fo r both bands a ssu m e d ran d o m d is tr i- : _ 168 ! bution of the seco n d n eig h b o rs, and no e v id e n ce is found to ; c o n tra d ic t th is . The 553 c m ” * band is n o m in ally the sa m e shape and w idth as i ' ■ ■ i - 1 1 the 580 c m ” band. The m o d el w hich fits th e 580 band q u a lita tiv e ly | ! fits th is b and a ls o . The 553 band is w eak enough th a t it is h a rd to I get a re a lly q u an titativ e c o m p a riso n . : + The e ffe c t of a f i r s t n e ig h b o r Ge on th e a x ia l m ode of B - L i j should depend on w h eth er the L i is in the s ite along th e B -G e ax is I o r in one of the th re e o ff-ax is s ite s . H o w ev er, only one band is i ! se e n sp lit off fro m the 580 c m ” * band; th is is the 553 cm "* band. ! The 677 c m * band a lso a p p e a rs to have o n ly one s a te llite , the : _ 1 | 651 cm band. If the c o n je c tu re about the L i-B d ista n c e being ! n o ticeab ly le s s th an a bond len g th is c o r r e c t, then, sin ce the Ge is j la r g e r th an the Si, the p r e f e r r e d p osition f o r the L i should be the I ! i site opposite the Ge. T his does not split th e tr a n s v e r s e m ode | d e g en e ra c y , and only one sh ifte d a x ial m ode is se en . It is , how ever, | ■ | p o ssib le th a t th e re a re c e n te rs fo r which th e tr a n s v e r s e m ode d e g en e ra c y is lifte d and the se co n d band is h id d en u n d e r the la rg e 677 cm * band. . No band h as yet been s e e n th a t can be a ttrib u te d to B ” w ith a f i r s t n eig h b o r but no p a ire d L i. A t room te m p e ra tu re o r below , the - + u n p a ire d B band is its e lf quite sm a ll, but th e B - L i p a irs d is so c ia te re a d ily at h ig h e r te m p e ra tu re ( 100-200°C ) a n d the is o la te d 169 b o ro n band b ecom es quite la rg e . H ow ever, the L i begins to diffuse I out of the sam ple in a few m in u te s a t th ese te m p e ra tu re s and the fre e I c a r r i e r background a b so rp tio n in c re a s e s . T he IF tr a n s m is s io n I m e a su re m e n t m u st be m ade quickly. S e v e ra l sa m p le s w ere i | m e a s u re d a t e le v a te d te m p e ra tu re s , but the noise le v e l and b a ck - ; ground a b so rp tio n w e re such th a t no side band of the is o la te d b o ro n ■ band could be id en tified . G. B -L i P a irin g D istan ces j - - | In th e d isc u ssio n of both the f ir s t and second n e ig h b o r e ffe c ts, j i the a ssu m p tio n w as m ade th a t the d istan ce fro m the b o ro n to the ! + ! in te r s titia l L i should be le s s th an the d istan ce to the c e n te r of the i \ te tr a h e d r a l in te r s titia l s ite , w hich is the sam e a s a S i-S i bond length, ! ab o u t 2. 35 X. P re v io u sly p u b lish ed values (P e ll I960, P o lia k an d W att 1965) fo r B - L i p a ir d ista n c e s in Si range fro m 2 .4 - 2 . 9-&. T h ese v a lu e s, how ever, w e re d e riv e d fro m m e a s u re m e n ts of th e rm a l d isso c ia tio n of L i* -B p a ir s . R ie ss e t;a l. (1956) c a lc u la te d the f r a c tio n of ions p a ire d a s a fu nction of te m p e ra tu re fo r a m o d el w hich + c o n s id e rs the in te ra c tio n betw een B and L i as only a C oulom b in te r a c tio n s c re e n e d by the bulk d ie le c tric co n stan t of the c ry s ta l. T he B- L i d ista n c e is th en a p a ra m e te r to define the p a irin g e n e rg y . It h a s been noted (K ro g er 1964) th a t th is m o d el n eg lects e la s tic s tr a in e n e rg y fo r a L i* ion d isp lac ed fro m the c e n te r of an in te r s titia l s ite , a s w e ll a s n eg lectin g the red u c tio n in effectiv e d ie le c tric sc re e n in g 170 w hen th e d ista n c e betw een the p a ire d ions is sm a ll. T hese te rm s a r e j ; i j of o p p o site sig n and w ill tend to can cel if the p a irin g d istan c e is le s s j th an a bond len g th . The m odel h as been supported by n u m b ers I obtained fo r p a irin g d ista n c e s fo r L i+ p aired w ith Al, Ga, o r In in Ge | (R iess et a l. 1956) and Al in Si (M aita 1958); the values obtained a r e i I 1. 6 - a . 8 k , c lo se to th e su m of the covalent rad iu s of the a c c e p to r land th e ionic ra d iu s of th e L i+. A low er p airin g energy is o b se rv e d ! fo r L i+ - B~ p a irs in both Ge and L i, and th is is in te rp re te d as ! in d ic atin g a la r g e r p a irin g d istan c e. i j | T he m e a s u re d p a irin g energy is the d ifferen ce in en erg y b e - I tw een a L i+ in a n o rm a l la ttic e s ite and the L i+ in a site a d ja ce n t to i i jth e a c c e p to r. The c h a rg e on th e L i p o la riz e s its n eig h b o rs, so the I j d iffe re n c e in p o la riz a tio n en erg y m u st also be included if th e p o la r- I i iz a b ility of th e a c c e p to r is d ifferen t fro m th at of a silico n ato m . The e n erg y is p ro p o rtio n a l to the p o la riz a b ility of the n eig h b o rs; B e ll- m o n te and P ry c e (1968) have e stim a te d it as about 0. 3 eV fo r each of the n e a r n e ig h b o rs fo r L i+ in a te tra h e d ra l in te rs titia l s ite in Si. As a rough e s tim a te , th e p o la riz a b ility of an atomccaaabe tak en as being p ro p o rtio n a l to its volum e. A boron atom h as le s s than h a lf th e volum e of a Si ato m , hence a s m a lle r p o la riz a b ility . O ther su b stitu tio n a l a c c e p to rs a r e as la rg e o r la r g e r th an Si, so th a t th e change in p o la riz a tio n en erg y m ight be expected to reduce the o b se rv e d L i+ - B~ p a ir en erg y in Si by 0. 1 - 0. 2 eV as c o m p a red to 171 1 ,J- - i Li - Al p a irs . The o b se rv e d d iffe re n c e is about 0. 1 eV. The j | ag re em e n t of th e quoted n u m b ers is probably ju s t fo rtu ito u s, j D isto rtio n of th e la ttic e is not n eg lig ib le around a su b stitu tio n al j boron sin ce the b o ro n is sm a ll c o m p ared to th e Si it re p la c e s. This i ! m ay m odify the in te r s titia l s ite su fficien tly that th e sim p le m odel j i ! above is not re a lly v a lid . i I It has b een re p o rte d th at th e su b stitu tio n of boron fo r silicon contacts the la ttic e by about 0 . 8 of the volum e of a silico n atom fo r | each su b stitu ted b o ro n a to m (Horn 1955). In th e p re s e n t w ork x -ra y la ttic e p a ra m e te r m e a s u re m e n ts w e re m ad e com paring Si, Si heavily 10 -3 doped w ith B ( ~ 5 x 10 cm ), and lith iu m diffused heavily B- doped Si. It w as found th a t the L i d iffusion r e s to re s about 40% of the c o n tra c tio n c a u se d by the bo ro n , so that B -L i p a irs m ay involve a c o n sid e ra b le am ount of e la s tic en erg y . A dditional ev id en ce fo r a s m a ll L i-B p a ir d istan c e com es fro m the sp littin g of th e b o ro n lo c a l m ode in Si. P a ire d s u b s ti tutional donors (P, A s, Sb), w hich m u st be clo se to a bond length away sin c e th ey a r e c o v alen tly bonded to th e la ttic e , only give about o n e -th ird th e sp littin g given by a p a ire d L i*. Since the splitting is assu m ed to be c o n tro lle d by the Coulom b in te ra c tio n betw een B ” and the p a ire d d o n o r, th e d is c re p a n c y is conveniently explained if the Li* - B" p a ir d ista n c e is s m a lle r than a bond len g th . T hus, the p rev io u sly re p o rte d p a irin g d ista n c e s fo r Li* - B ~ p a irs a re not to I 172 ! be tak en too se rio u sly . H. S u m m ary and C onclusions j L o c a liz e d v ib ra tio n a l m odes of B - L i p a irs in S i-ric h I 1 i Ge Si, w ere stu d ied w ith the in te n t of u sin g the lo c a liz e d m ode a s a x 1 -x j p ro b e fo r p o ssib le s tru c tu re in the a llo y . It w as found th a t the ab- | so rp tio n stre n g th of the bands due to b o ro n w ith a Ge f i r s t neighbor i w as lo w er than m ight be e x p ec te d fro m a ran d o m d istrib u tio n .o f Ge | and B throughout the la ttic e . T h is m ay be c a u se d by a ten d en cy fo r | B and Ge to avoid fo rm in g f i r s t n eig h b o r p a ir s , a s m a lle r o s c illa to r 1 j s tre n g th fo r B m odes when a Ge f i r s t n eig h b o r is p re s e n t, o r both of ; th e se effe cts to g e th e r. I j The effect of the seco n d n eig h b o r Ge a to m s on the b o ro n i | m odes w as to produce a p ro nounced a s y m m e tric b ro ad en in g of the t i | o b se rv e d bands. A m o d el is p ro p o se d w hich fits the band sh ap es I | and does not re q u ire invoking any o rd e rin g am ong the seco n d neig h b o rs of the bo ro n . It is a ssu m e d th a t the p rin c ip a l e ffe c ts of a seco n d neighbor Ge atom a re a change in the fo rc e c o n sta n ts coupling the b o ro n to the la ttic e and, on som e s ite s , m o d ifica tio n of the B -L i p a irin g in te ra c tio n b e ca u se the Ge d iffe rs in p o la riz a b ility fro m the Si. The a sy m m e tric shape of the bands c a n th e n be fit quite w ell as a su p e rp o sitio n of the a b so rp tio n of b o ro n atom s- on s ite s having d iffe re n t second neighbor c o n fig u ra tio n s by u sin g th re e a d ju sta b le i . p a ra m e te rs d e sc rib in g the s tre n g th of th e s tr a in and p o la riz a tio n j in te ra c tio n s , and assu m in g th at the d istrib u tio n of Ge in the la ttic e J i i I is random . : j C aution m u st be e x e rc is e d in the in te rp re ta tio n of th e se | m e a su re m e n ts since th e re is the p o s s ib ility th a t the b o ro n m ay be j ; p re fe re n tia lly se g re g a te d e ith e r into o r aw ay fro m any G e -ric h o r : S i-ric h o r o th erw ise anom alous reg io n s th a t m ay e x ist; a ls o , the ( e ffect, if any, of the L i diffusion on the sh o rt ran g e o rd e rin g is not | known. If the low stre n g th of the bands fro m f i r s t neig h b o r B -G e i ! p a irs is c au se d by a low c o n c e n tra tio n of th e se p a ir s r a th e r th an a ; low o s c illa to r stre n g th , th en th is is an exam ple of a se g re g a tio n ! j phenom enon. The seco n d n eig h b o r d istrib u tio n m ay be m o re r e p r e - j se n ta tiv e of the bulk of the c ry s ta l; the r e s u lt h e re is c o n siste n t I j w ith a ran d o m alloy, b u r does not by its e lf p ro v e th at the allo y is I I ! random . ! The lo c a l m ode sp e c tru m in the allo y does give in fo rm a tio n on | | the s tru c tu re of the B -L i p a ir th a t w as not o b tain ed fro m m e a s u r e m en ts in Si o r Ge. The m o d ificatio n of the B -L i p a irin g in te ra c tio n th a t is re q u ire d to fit the o b se rv e d line sh a p es stro n g ly su g g e sts th a t the c o rr e c t value fo r the B -L i p a irin g d ista n c e should be le s s th an 2. 35 X, the S i-S i bond len g th , r a th e r th an the p re v io u sly re p o rte d v alu es of 2 .4 - 2 .9 X . CH A PTER VI L o c a liz e d V ib ratio n al M odes of P h o sp h o ru s and of L ith iu m I , P a ir e d w ith G allium in G erm anium T h e re a re s e v e ra l lig h t e le m e n ts w ith a la rg e enough so lu b ility in Ge th a t in fra re d a b so rp tio n fro m lo c a liz e d v ib ra tio n a l m o d es ‘ should be o b se rv a b le . T h ese include b o ron, alum inum , silic o n , and I p h o sp h o ru s am ong th o se dopants th at occupy su b stitu tio n a l s ite s , and | lith iu m an d oxygen am ong th o se dopants th at a re p rim a rily in te r - i s titia l. G erm an iu m is one of the two m o st w idely stu d ied se m ic o n - I i ' ! d u c to rs , the o th e r being silic o n . The la ttic e dynam ics of p u re Ge j h a s b een stu d ie d both e x p e rim e n ta lly and th e o re tic a lly in c o n s id e r able d e ta il, ho w ev er, p r io r to the p u blication of the m a te ria l ! c o v e re d in th is c h a p te r, the only re p o rte d in fra re d a b so rp tio n of | im p u rity v ib ra tio n s w as th at of in te r s titia l oxygen ( K a ise r 1962). i i j The only o th e r re p o rt of a lo c a liz e d m ode w as th a t of the Si lo c a l ! i m ode as o b se rv e d by R am an sc a tte rin g ( F e ld m a n e t. a l. 1967 ). T h ere w e re , by th a t tim e , n u m erous re p o rts of lo c a l m odes o b se rv e d in IR a b so rp tio n in o th e r se m ic o n d u cto rs such as Si, G aA s, GaSb, InSb, CdS, ZnSe, and in s e v e ra l ionic c ry s ta ls ( se e , fo r in sta n c e , the rev iew by N ew m an 1969 )• The la c k of o b se rv atio n s in Ge w as p rin c ip a lly due to the d ifficu lty of e le c tric a lly co m p en satin g h e a v ily doped Ge to s u p p re ss fre e c a r r i e r a b so rp tio n . W ith the e x ce p tio n of Si and O, a ll the lig h t e le m e n ts w ith la rg e so lu b ilitie s 174 | 175 a re e le c tric a l d o p an ts, and the silic o n lo c a l m ode (see C h ap ter III) i is only v e ry w eakly IR a c tiv e . Since th e p u b licatio n of the re s u lts I p re s e n te d in th is c h a p te r, N a zarew icz and Ju rk o w sk i (1969) have i o b se rv e d th e lo c a l m odes of iso la te d b o ro n and of b oron in B~ - Li* ! p a irs in Ge co m p en sated by L i diffusion. T h e ir co m pensation w as not : good enough to allow o b se rv atio n of the L i m odes of the B ” - Li* p a ir s . As of th is tim e , th e re is s till no re p o rt of the Al lo c a l m ode. i The situ a tio n w ith Ge c o n tra s ts stro n g ly w ith th a t of Si, w h ere | co m p en satio n is re la tiv e ly e asy . The lo c a l m odes of a ll of the !highly so lu b le lig h t im p u ritie s have been o b se rv ed and ex ten siv e I j stu d ie s have b een m ad e of th e lo ca l m odes of th e s e im p u ritie s p a ire d i t | w ith o th e r im p u ritie s . s I j A quite c lo se d e g re e of co m pensation is needed, sin ce the fre e I c a r r i e r a b so rp tio n c ro s s se c tio n is quite la rg e . In the freq u en cy i ’ - 1 ran g e of in te r e s t (300 - 600 cm ) the a b so rp tio n c ro s s se c tio n of a fre e h ole in Ge is about 5 x 10” ^ cm ^ a t 300°K, and it in c re a s e s w ith d e c re a s in g te m p e ra tu re (K a iser e t. a l. 1953). The a b so rp tio n c ro s s se c tio n of th e b o ro n lo ca l m ode in both Ge and Si is only -18 2 1 - 2 x 1 0 ” cm , sm a ll enough th at th e band is not d e te ctab le above the fre e c a r r i e r a b so rp tio n w ithout a fa ir d e g re e of co m pensation. Q uite good co m p en satio n is needed if q u an titativ e m e a su re m e n ts a re to be m ad e of the lo c a l m ode band sh ap e and s tre n g th . F o r p h o sp h o ru s th e situ a tio n is not quite as u n fav o rab le. The a b so rp tio n | 176 ! _ 2 ! c ro s s se ctio n of fre e e le c tro n s is only about 2 x 10 " cm a t the ; phosphorus lo c a l m ode freq u e n cy , w hile the phosphorus lo cal m ode -18 2 ! c ro s s se c tio n is about 5 x 10" cm , a little la r g e r than fo r b o ro n . | It probably would b e p o ss ib le , given p r io r know ledge of the frequency of th e lo c a l m o d e, to o b se rv e th e IR a b so rp tio n of the p horphorus lo c a l m ode in an uncom pensated sa m p le . Q uantitative i m e a su re m e n ts , h o w e v er, re q u ire fa irly good com pensation. ! The in itia l m a te r ia l u sed w as doped so th a t [ G ] ^ 18 -3 • 4 x 10 cm . S lic e s w e re cut th a t had reg io n s th at w ere p-type i w ith p > . 02Q -c m . T h ese w e re co m p en sated by L i diffusion. The i i i ! sam p les w e re th en ground and p o lish ed and the IR tra n s m is s io n w as I m e a su re d a t 80°K . I I -1 -1 In the s p e c tr a l reg io n fro m 270 cm to 510 cm four a b s o rp - j tion bands not p re s e n t in high p u rity Ge a re found in L i com pensated j j Ga and P doped Ge a t 80°K (sam ple 1, F ig . 28). Two of the b ands, I at 356, 380 c m "* , change freq u en cy to 379, 405 cm * if ^L i is u sed in ste a d of n a tu ra l L i (93% ^Li). T h ese two bands also a p p e a r a t th e sam e freq u e n cy in Ga doped Ge w hich has been n e a rly co m p en sated w ith As and Sb and th en diffused w ith L i (sam ples 2, 3, F ig . 28); the two b ands in th e Ga and P doped m a te r ia l at 343 and 350 cm - * w hich do not show a L i iso to p e sh ift do not a p p ea r in th is m a te ria l. The p re s e n c e of tw o L i bands im p lie s th a t th e in te rs titia l L i is 177 ! F ig u re 28. A b so rp tio n c o e ffic ie n t a t liq u id -n itro g e n te m p e ra tu re v s. I 7 w avenum ber fo r g e rm a n iu m sa m p le s: (1) P and Ga doped, L i l ! 7 | diffused; (2) Ga, A s and Sb doped, L i diffused; (3) Ga, A s ! and Sb doped, ^L i diffused; (4) p u re G e. V arying am ounts j j of background a b so rp tio n a r e the r e s u lt of im p e rfe c t com p en satio n . 40 30 20 4 0 0 3 8 0 360 340 179 in a p a ire d configuration, pro b ab ly as L i-G a p a ir s . The lo w er , freq u en cy band of the two is a lm o st ex actly tw ice a s stro n g as the ! h ig h er frequency band. I n te rs titia l L i p a ire d w ith an a c c e p to r ! should have C^ sy m m e try , w hich w ill s p lit th e lo cal m ode into a j sin g le a x ial m ode and a doubly d e g e n e ra te tr a n s v e r s e m ode. The ■ s im p le st m odel fo r the a b so rp tio n fro m a lo c a liz e d m ode is th a t of j a sin g le ch arg ed im p u rity v ib ra tin g in a u n ifo rm d ie le c tric m edium . i I T his m odel w ill give tw ice th e a b so rp tio n s tre n g th to a doubly degen- j e ra te m ode as to a sin g le m ode, so th a t it w ould a ssig n the band a t : 356 (379) c m “ * to the doubly d e g e n e ra te m ode and th e 380 (405) cm * I 7 6 j band to the singly d e g e n e ra te m ode of L i( L i). The m odel is o v e r- 1 | sim p lified , h o w ev er, and th e re a re c a s e s , su ch as p a irs betw een | b o ro n and su b stitu tio n a l donors in Si, w h e re the s tro n g e r of two | bands is pro b ab ly not th e doubly d e g e n e ra te band (T sevtov et. a l. i 1968, N ew m an and Sm ith 1968). The m o d el u se d fo r B - L i p a irs in Si, th at of co m p letely lo c a liz e d B and L i v ib ra tio n s in te ra c tin g w ith a sim p le Coulom b field , p re d ic ts th e doubly d e g e n e ra te v ib ra tio n s to be the h ig h e r freq u en cy ones (W aldner et a l. 1965). In Si, the B .v ib ratio n s q u a litativ e ly fitted th is m o d el, but only one L i m ode w as o b se rv e d , w ith v, , the lo c a l m ode freq u e n cy , c lo se to V - j j llld> X th e m ax im u m phonon freq u en cy of the u n p e rtu rb e d la ttic e . In the p re s e n t c a se , although th e vT values fo r both L i m odes > jj m ax the v L (doubly d e g e n e ra te ) < vL (singly d e g e n e ra te ) and the ! 180 ! sp littin g Av is m uch s m a lle r th an th a t p re d ic te d by the coulom b m odel. It is not obvious th at th e re should be two P bands as th e re is I only one abundant P iso to p e. A n o th er sp e c im e n w as p re p a re d w ith I about tw ice the doping lev e l of the f ir s t P and Ga doped sp ecim en . The la r g e r band a t 343 cm * a lm o st doubled in sten g th fro m a = 23 c m * to a =39 cm * w hile the s m a lle r band a t 351 c m ” * m o re than ; trip le d fro m a = 5 c m ” * to a = 16 c m ” * ( m e a s u re d in te rm s of j peak a b so rp tio n stre n g th ). A p o ssib le m odel is th a t the s tro n g e r band is a ttrib u te d to the i | trip ly d e g en e ra te m ode of is o la te d su b stitu tio n a l P and the w e a k e r one is due to P -G a p a irs . It is not u n re a so n a b le th a t only a sin g le j ad d itio n al band be se en fro m p a irs . T sev to v , e t. a l. (1 9 6 8 ) have i I o b se rv e d p a irin g of B w ith P , A s, and Sb in silic o n ; in e ac h c a se j | two a d d itio n al bands w e re o b se rv ed , but in one c a s e , B - P p a ir s , one j i | of the bands ap p ea re d only as a s m a ll sh o u ld e r on the band due to j ' j is o la te d B w hile the o th e r band w as sp lit off by a s u b s ta n tia l am ount. i ! . • j If one of the two freq u e n cie s of P p a ire d w ith Ga is the sam e a s the I is o la te d P freq u en cy , only one a d d itio n al band w ould be o b se rv e d . In a given h o st la ttic e , the fre q u e n c ie s of lo c a l m odes depend both on the im p u rity m a ss and on changes in fo rc e c o n sta n ts, In the c a se of a P -G a p a ir, the effect on the P lo c a l m ode fre q u e n c ie s of changing the m a s s of one of the f ir s t n eig h b o rs fro m th a t of Ge 181 ; i ( a t. w t. 72. 6 ) to th a t of Ga ( a t. w t. 69. 7 ) can be f a ir ly e a s ily ! c a lc u la te d . E llio tt and P feu ty ( 1968) have done su ch c a lc u la tio n i fo r b o ro n in silic o n , w hen the boron is p a ire d w ith a n o th e r im p u rity : i of a r b itr a r y m a s s . W hen the a r b itr a r y m a s s is slig h tly lig h te r th an i j the h o st la ttic e a to m s, the case analogous to P -G a p a ir s in Ge, th ey I ! find th a t the a x ia l m ode of the p a ir is r a is e d in fre q u e n c y , w hile the ! doubly d e g e n e ra te tra n s v e rs e m ode is a lm o st unchanged. The | re a s o n fo r th is can be se e n q u a litativ e ly by exam ining th e e ig e n - j ! v e c to rs fo r the lo c a l m ode of an iso la te d im p u rity . It is sim p le st I to u se the e ig e n v e c to rs fo r the c e n tra l-fo rc e -o n ly m o d e l ( T able 6 , | C hap. 2 ), fo r w hich the neighbors of th e im p u rity m o v e along the 1 bonds b etw een the im p u rity and the n e ig h b o rs. F o r an im p u rity I I v ib ra tio n along one of the cubic axes the m agnitude of th e m o tio n is | | th e sa m e fo r a ll the n e ig h b o rs, but two of the n e ig h b o rs a r e m oving in w a rd to w a rd the im p u rity w hile two a re m oving o u tw ard . The d ire c tio n of m o tio n of the neighbors fo r a v ib ra tio n of th e im p u rity j ■ along e a c h of the cubic ax es is shown in T able 10. D ire c tio n D ire c tio n of m otion of the n eig h b o rs of im p u rity v ib ra tio n (1 ) (2 ) (3) (4) (1 0 0 ) A - r l A - r 2 A r 3 A r 4 (0 1 0 ) A - r l A r 2 A - r 3 A r 4 (0 0 1 ) A - r l A r 2 A r 3 A " r 4 ( L ab e lin g of n eig h b o rs and ax es is a s show n in F ig . 6 , C h ap ter 2 .) ' ! I & is a v e c to r giving the d ire c tio n fro m the im p u rity | to the ith n eig h b o r. ! T able 10 ; i ! T he a p p ro p ria te m o d es fo r a p a ir su ch as P - Ga a re an a x ial ! m ode along a [111 ] d ire c tio n and two tr a n s v e r s e m o d es, su ch as f ! along [110 ] o r [ l l Z ] d ire c tio n s; th e se m ay be obtained fro m lin e a r i | com binations of the m o d es in T able 9. F o r the [111] m ode the ! n e ig h b o r along th e a x is h as th re e tim e s as la rg e a v ib ratio n as each j j of th e o th e r th re e n e ig h b o rs. In the tr a n s v e r s e m odes the o n -ax is i j n eig h b o r does not v ib ra te at a ll. T hus, fo r th is m o d el, th e a x ia l j m ode of a P -G a p a ir should be ra is e d in freq u en cy slig h tly , sin ce | th e Ga is le s s m a s s iv e th an th e Ge it re p la c e d , w hile the tr a n s v e r s e | m o d es a r e not a ffe c te d . If the m o re r e a lis tic e ig e n v e c to rs obtained I ‘ j fro m a m odel in cluding n o n -c e n tra l fo rc e s a r e u se d (T able 7, | j C h a p te r. 2), th e tr a n s v e r s e m odes a re not to ta lly in se n sitiv e to th e i Ga, and the a x ia l m o d e is not quite as se n sitiv e as in the c e n tra l- fo rc e -o n ly m o d el, but th e effect of th e lig h te r m a s s of the Ga w ill s till be to ra is e th e a x ia l m ode freq u en cy c o n sid e ra b ly m o re th an it r a is e s th e tr a n s v e r s e m ode freq u en cy . The changes in m ode freq u e n cie s p re d ic te d by c o n sid erin g only th e m a s s change q u a lita tiv e ly a g re e w ith th e o b se rv e d re s u lt of -One_band_split_off_to_higher_frequency.__HQ.weLve£,_since_no_ajCCOimt__ : 183 ! hg.s b een ta k e n of the changes in fo rc e c o n stan ts, even this q u a lita - ! tiv e a g re e m e n t m u st be c o n sid e re d fo rtu ito u s. S e v e ra l fa c to rs ■ a ffe c t the fo rc e c o n sta n ts. The P is s m a lle r and the G a slig h tly i la r g e r th an the Ge ato m s th ey re p la c e , and th e ir bonds to the re s t i ' of the c r y s ta l a r e d iffe re n t fro m the n o rm a l Ge-Ge bonds. T he | G a -P bonding is a lso d iffe re n t fro m th e se bonds, and th e re is a ! ■ coulom b in te ra c tio n b etw een the Ga and the P. I The phonon s p e c tra of Ge and Si a re sufficiently s im ila r that a i j j G re e n 's fu n ctio n c a lc u la tio n fo r the lo c a l mode freq u e n cie s in silic o n j should be a p p lica b le to g e rm a n iu m if th e freq u en cies a r e sc a le d a c c o rd in g to the ra tio of the m ax im u m phonon freq u e n cie s of Ge and j • ! Si, p a rtic u la rly if the lo c a l m ode freq u e n cy is not too close to the i m ax im u m phonon freq u e n cy ( see C hap. II, sec. A ). F ro m the c a lc u la tio n of D aw ber and E llio tt fo r a m a ss su b stitu tio n w ith no fo rc e c o n sta n t c h a n g e s, the lo c a l m ode freq u e n cy of P in Ge should be 1. 27 v , w hile the o b se rv e d fre q u e n c y of 343 c m ’ * is only about zn3>x 1. 14 v . T hus, the P -G a fo rc e c o n stan ts a re su b sta n tia lly w eak er m ax 1 th an G e-G e fo rc e c o n sta n ts. A n a p r io r i e stim a te of th e e ffe c ts of p a irin g a G a w ith the P is d iffic u lt. A c o m p a riso n can be m ad e w ith th e an alogous c a s e s of b o ro n p a ire d w ith su b sta n tia l donors in Si ( T sev to v 1968; N ew m an 1968 ). The c a s e s m o st c lo se ly analogous to P -G a p a ir s in Ge a re B -A s and B -S b, w here the im p u rity giving th e lo c a l m ode is s m a lle r th an a h o st a to m but is p a ire d with an | 184 ! ; im p u rity la r g e r than a h o s t ato m . In th e se c a s e s one boron i I I freq u e n cy is above an d one is below th e freq u e n cy of iso lated boron; ; the sp littin g is about 30 c m "* , c o m p a re d to the 8 cm "* sp littin g fo r I P -G a p a ir s . One sh o u ld not ex p ect the re s u lts to com pare too j c lo sely , sin ce the la ttic e m is m a tc h fo r b o ro n in Si ( covalent ra d ii i | of 0 . 88 X and 1 . 17 X ) is g r e a te r than th a t fo r P in Ge ( 1 .1 0 X i and 1. 22 X ), an d the coulom b in te ra c tio n is m o re effectively j i sc re e n e d in Ge than in Si. It is im p o ssib le to p re d ic t the freq u en cies j j of such a p a ir su ffic ie n tly a c c u ra te ly to d efin itely a s s e r t that the | m issin g band of the P -G a p a ir m u st lie at the sam e frequency a s the j iso la te d P lo c a l m ode. i In su m m a ry , th e lo c a liz e d m ode of is o la te d phosphorus in Ge j j is o b se rv e d in in f r a r e d a b so rp tio n a t 343 <im~*. The lo ca l m odes ! 6 7 -1 | of Li( L i) p a ire d w ith Ga a re o b se rv e d at 379(356) cm and 405(380) cm "*; the lo w e r freq u e n cy band h a s about tw ice the a b s o rp tio n stre n g th of the h ig h e r freq u e n cy band, so it is probably the doubly d e g e n e ra te tr a n s v e r s e m ode of the p a ire d L i. A nother a b so rp tio n b and at 351 cm "* is te n ta tiv e ly id e n tifie d as being one of the m odes of P p a ire d w ith Ga; the o th e r P m ode is assu m ed to have the sam e fre q u e n c y a s does is o la te d P . D ifficu lties in a c c u ra te ly p re d ic tin g fre q u e n c ie s of such p a irs a re d isc u sse d . C H A PTER VII ! S u m m ary A. C a lc u la tio n of L o c a liz e d V ib ra tio n a l Mode F re q u e n c ie s The m ethod of finding th e h ig h est eig en v alu e of a m a trix by i i ! ite ra te d m u ltip lic atio n of a v e c to r by the m a tr ix w as u se d as a te c h - ! nique fo r calcu latin g lo c a l m ode fre q u e n c ie s, u se fu l p a rtic u la rly fo r i : d iso rd e re d c ry s ta ls w ere sy m m e try c o n sid e ra tio n s cannot be u se d ! to sim p lify the calcu latio n . T he technique w as u se d to calcu late i j the change in freq u e n cy of the b o ro n lo c a l m ode in Si to be ex p ected j fro m su b stitu tio n of a Ge fo r one of the seco n d n eighbors of the I | | b o ro n . The su b stitu tio n w as a p p ro x im a te d by a sim ple change in m a s s , and by both a change in m a s s and a G e-S i fo rc e constant ! ! d iffe re n t fro m the S i-S i fo rc e c o n sta n t. A sim p le f ir s t neighbor | I fo rc e constant m odel w as u se d to a p p ro x im ate the la ttic e . The j ! c a lc u la te d changes in freq u e n cy a re quite s m a ll c o m p a red to the I I ■ . | freq u en cy sp re a d of s tru c tu re o b se rv e d in the b o ro n lo cal m ode bands in stu d ies of b o ro n lo c a l m o d es in Ge Si a llo y s. T his X X “ X r e s u lt is la te r u se d in su p p o rt of a m o d el fo r the effe cts of second n eighbors on the b o ro n lo c a l m ode w hich ta k e s s tr a in effects into account but n e g le cts the e ffect of m a s s ch an g es. B. In fra re d A b so rp tio n of L a ttic e M odes and the Si L o c a l Mode In S i-ric h Ge S i, ( x < . 12 ) th e re a re th re e in fra re d x 1 -x ' — 186 | a b so rp tio n bands a t fre q u e n c ie s below the m ax im u m phonon I freq u en cy of Si. The a b so rp tio n a s s o c ia te d w ith th e se bands a p p e a rs ; to be independent of te m p e ra tu re , in d icatin g a sin g le-p h o n o n p ro c e s s . ■ The fre q u e n c ie s of 485 cm "*, 405 cm "*, and 125 cm "* do not | c o rre sp o n d w ell to a b so rp tio n peaks p re d ic te d by the th e o ry of ; D aw ber and E llio tt, n o r to p re d ic tio n s of fre q u e n c ie s a t in -b an d le so n an ces. T hey do, h o w ev er, c o rre sp o n d fa ir ly c lo se ly to peaks ! in the single-phonon d en sity of s ta te s of Si. | In G e -ric h allo y s ( x > . 88 ) th e re is a lso in -b a n d a b so rp tio n , | but it o c c u rs a t fre q u e n c ie s of m u lti-p h o n o n la ttic e bands of Ge. i I I I The a b so rp tio n is te m p e ra tu re dependent, but not a s stro n g ly a s is i I e x p ec te d fo r a tw o-phonon p ro c e s s . As is the c a se w ith the S i-ric h i l j a llo y s, the a b so rp tio n does not c o rre sp o n d to th a t p re d ic te d by the j ! D aw ber and E llio tt th eo ry ; in th is c a se it does not c o rre sp o n d to the sin g le-p h o n o n d en sity of s ta te s e ith e r. i The Si lo c a l m ode is se e n in in fra re d a b so rp tio n in the G e -ric h a llo y s. Its freq u e n cy v a rie s w ith the allo y c o m p o sitio n fro m 388 c m * a t x = . 99 to 394 c m * a t x - . 8 8 , in good a g re e m e n t w ith p rev io u s R am an sc a tte rin g m e a su re m e n ts . The a b so rp tio n c ro s s 20 2 se c tio n ( J adv / [S i] ) is only 10" cm , two to th r e e o rd e rs of m agnitude le s s than fo r o th e r lo c a l m odes in se m ic o n d u c to rs. C . L o c a liz e d V ib ratio n al M odes of B o ro n -L ith iu m P a ir s In S i-R ich G e jS i, A lloys | 187 The in fra re d a b so rp tio n bands due to boron lo ca l m odes of B" - L i+ p a irs in Ge S i. w e re o b se rv ed for 0 < x < 0. 12. In Si X 1 “ X | (x = 0) th e re a r e two lo c a l m odes of B” - L i+ p a irs th at a r e a lm o st j e n tire ly b o ro n v ib ra tio n s; one is along th e B -L i a x is, and the i o th e r, doubly d e g e n e ra te m ode, is tr a n s v e rs e to the a x is . In ! sp e c im e n s containing Ge the boron bands a re a sy m m e tric a lly I b ro a d e n e d and two new bands a p p e a r. The new bands a r e a ttrib u te d ! to th e lo c a l m o d es of b o ro n on s ite s w ith a Ge f ir s t n eig h b o r. The I a s y m m e tric b ro ad en in g of the b o ro n bands is a ttrib u te d to in te ra c tio n of th e B - L i p a irin g in te ra c tio n . A m o d el is p ro p o sed w h ereb y th e ! a s y m m e tric lin e sh ap es a re fit by a su p e rp o sitio n of th e a b so rp tio n j fro m b o ro n ato m s w ith d ifferen t second neighbor c o n fig u ra tio n s. The j o b se rv e d lin e sh ap es can be fit quite w ell by assu m in g a ran d o m ; d istrib u tio n of Si and Ge on the second neighbor s ite s . It is n e c e s - | s a r y to a ssu m e a value fo r the d istan c e betw een p a ire d B and L i : th a t is sig n ific a n tly s m a lle r than th a t p rev io u sly re p o rte d ; p o ssib le re a s o n s fo r th e d isc re p a n c y a r e d isc u sse d . D. L o ca liz ed V ib ratio n al M odes of P h osphorus and L ith iu m in G erm an iu m L o c a liz e d v ib ra tio n a l m odes of iso la te d p h o sp h o ru s, lith iu m p a ire d w ith g a lliu m , and w hat is te n ta tiv e ly identified a s p h o sp h o ru s p a ire d w ith g alliu m w e re o b se rv ed in g erm an iu m . The Ge c ry s ta ls I 188 I w e re grow n w ith the p h o sp h o ru s slig h tly o v e r-c o m p e n sa te d w ith | g a lliu m so th a t L i d iffu sio n could be u se d fo r fin al co m p en satio n . ; Two bands w e re id e n tifie d fro m isotope sh ifts as being i | p rim a rily L i v ib ra tio n s . The two bands o c c u r w ith a stre n g th ra tio | v e ry n e a rly 2:1 and a re p re s u m e d to be the tr a n s v e r s e and a x ial ! v ib ra tio n s of in te r s titia l L i p a ire d w ith su b stitu tio n a l Ge. If the i stre n g th ra tio is tak en a s in d ic atin g th a t the lo w e r freq u e n cy band is ! indeed the doubly d e g e n e ra te tr a n s v e r s e m ode, th is c a se conflicts ! w ith th e o re tic a l m o d els of B - L i p a irs in Si w h ere the tra n s v e rs e | m ode is a ssu m e d to lie h ig h e r in freq u en cy . ! i | The band a ttrib u te d to P p a ire d w ith Ga in c re a s e s in stre n g th j |f a s te r th an lin e a rly w ith p h o sp h o ru s c o n c e n tra tio n in sa m p le s in t I w hich the p hosphorus an d g a lliu m c o n ce n tra tio n s a re n e a rly equal. I ! i No e x p e rim e n ts w e re done to d e te rm in e how its stre n g th v a rie s w ith ] jh eat tre a tm e n ts th a t w ould be ex p ected to change the c o n ce n tra tio n ! . of P -G a p a ir s . P a ire d p h o sp h o ru s should be ex p ected to give two bands; it w as a ssu m e d th a t the seco n d band m u st have the sam e freq u en cy as does is o la te d p h o sp h o ru s. i 189 R E F E R E N C E S ; W .P . A llre d , 1970, p riv a te co m m u n icatio n . ; J . F . A n g re s s , A .R . Goodw in, and S. D. S m ith, 1965, P ro c . I Roy-: Soc. A 287, 64. 1 1 | J . F . A n g re ss , S. D. S m ith, a n d K .F . R enk, 1964, L a ttic e D ynam ics , ed ited by R .F . W allis ( P e rg a m o n P r e s s , New Y o r k ) , p. 467. I M. B alk an sk i, 1964, 7th In te rn a tio n a l C o n feren ce on the P h y sic s i of S em ico n d u cto rs, P a r is ( A c a d e m ic P r e s s , New Y ork ) , j j p. 1 0 2 1 . I | M. B alk an sk i and W . N a z a re w ic z , 1964, J . P h y s. C hem . Solids j . 25, 437; 1966, Ib id . , 27, 671. A .S . B a rk e r, J r . , 1968, L o c a liz e d E x c ita tio n s in S olids, edited by | R .F . W allis ( P len u m P r e s s , New Y o rk ) p. 581. j L . B ello m o n te and M .H .L . P ry c e , 1966, P ro c . P h y s. Soc. 89, 967; 1968, L o c a liz e d E x c ita tio n s in S olids, edited by R . F . W allis ( P len u m P r e s s , New Y o rk ) p. 203. R. B ra u n ste in , 1963a, P h y s. R ev. 130, 869; 1963b, I b id , 130, 879. R . B ra u n ste in , A .R . M o o re , and F . H e rm a n , 1957, P h y s. R ev. 109, 695. B .N . B ro ck h o u se, 1959, P h y s. R ev. L e tt. 2 , 256. B .N . B ro ck h o u se and P .K . Iy e n g a r, 1958, P h y s. R ev. I l l , 747. I . F . Chang and S. S. M itra , 1968, P h y s. R ev. 172, 924. ; A .E . C osand and W. G. S p itz e r, 1967, A ppl. P h y s. L e tt. 11, 279. ; P . D aw ber and R . J . E llio tt, 1963a, P r o c . R oy. Soc. A 273, 222; 1963b, P ro c . P hys. Soc, 81, 453. ; P . D ean, 1968a, L o calized E x c ita tio n s in S olids, e d ited by R .F . ‘ W allis ( P lenum P r e s s , New Y o rk ) p. 109; 1968b, J . P h y s. i C hem . 1,22. i i S .D . D evine and R. C. Newm an, 1970, J . P h y s. C hem . Solids 31, I 685. i \ J . P . D ism ukes and L . E k stro m , 1965, T r a n s . M etal. Soc. AIM E ; 233, 672. i | J . P . D ism ukes, L . E k stro m , a n d R .J . P a ff, 1964, J. P h y s. C hem . | 6 8 , 3021. ! I j J . P . D ism ukes, L . E k stro m , E .F . S te ig m e ir, I. K udm an, and ! | D .S . B e e rs, 1964, J . A ppl. P h y s. 35, 2899. i r | G. D olling, 1963, In e la stic S c a tte rin g of N e u tro n s in Solids and i L iq u id s, Vol. II, IAEA, V ienna, p. 37. | G. D olling and R . A. Cowley, 1966, P r o c . P h y s. Soc. 8 8 , 463. , R .J . E llio tt, 1963, P ro c . of In te rn a tio n a l C o n feren ce on L a ttic e D ynam ics, edited by R .F . W allis ( P e rg a m o n P r e s s , New Y o rk ), p. 459. R .J . E llio tt and P . P feuty, 1967, J . P h y s . C hem . S olids 28, 1789. D. F e ld m a n , M. A shkin, and J .H . P a r k e r , J r . , 1966, P h y s. R ev. ; M. G licksm an, 1955, P hys. R ev. 100, 1146; 1956, I b id , 102, 1496. i M. G lick sm an and S .M . C h ristia n , 1956, P h y s. R ev. 104, 1278. W. H ayes, 1964, P h y s. R ev. L e tt. 13, 275. : B. H igm an, 1964, A pplied G ro u p -T h e o re tic and M a trix M ethods ( D over, New Y ork ). : F .H . H orn, 1955, P hys. R ev. 97, 1521. D .T . Hon, W .L . F a u st, W .G . S p itz e r, a n d P . F . W illia m s, 1970, P hys. R ev. L e tt. 25, 1184. I S. S. Ja sw a l, 1965, P hys. R ev. 140, A 687. i E .O . Johnson, a n d S .M . C h ristia n , 1954, P h y s. R ev. 95, 560. | j F .A . Johnson, 1965, P ro g re s s in S em ic o n d u cto rs, ed ited by A . F . j G ibson and R . E . B u rg e ss ( CRC P r e s s , C leveland ), vol. 9, j P- 1?9- j F .A . Johnson and R . Loudon, 1966, P ro c . Roy. Soc. 281, A 274. i W. K a is e r, 1962, J . P hys. C hem . Solids 2 3 , 255. i W. K a is e r, R .J . C ollins, and H .Y , F a n , 1953, P h y s. R ev. 91, i I 1380. i W. K a is e r, P .H . K eck, a n d C .H . L ange, 1959, P h y s. R ev. 101, 1264. C. K itte l, 1966, In tro d u ctio n to Solid S tate P h y s ic s ( W iley, New Y ork ), p. 156. N. K rish n a m u rth y and T .M . H a rid a sa n , 1966, P h y s. L e tt. 2 1 , 372; 1969, J . Indian In st. Sci. 51, 1 . > 192 i F .A . K ro g e r, 1964, C h em istry of Im p e rfe c t C ry s ta ls ( In te rsc ie n c e , New Y o rk ), p. 267. ; R .A . L ogan, J .M . R ow ell, and F .A . T ru m b o re , 1964, P h y s. R ev. 136, A 1751. i i P .L . L and and B. Goodm an, 1962, J . P h y s. C hem . Solids 28, 113. I ■ • i j R .S . L eig h and B. S zigeti, 1967a, P ro c . R oy. Soc. 301 A, 211; 1967 b, P h y s. R ev. L ett. 19, 566; 1968, L o c a liz e d E x c ita - ! tio n s in S o lid s, edited by R .F . W allis (P le n u m P r e s s , New I j Y ork ), p. 159. | I.M . L ifs c h itz , 1956, Nuovo C im ento Suppl. 4, 716. i ! O .G . L o rim o r and W. G. S p itz e r, 1966, J . A ppl. P h y s. 37, 3687. i ! O .G . L o rim o r, W .G . S p itze r, and M. W ald n er, 1966, J. A ppl. | j P h y s. 3>7, 2509. ! G. L u co v sk y , M .H . B rodsky, and E . B u rste in , 1968, L o ca liz ed | E x c ita tio n s in S o lid s, ed ited by R .F . W allis ( P len u m P r e s s , New Y ork ), p. 592. J . P . M aita, 1958, J . P h y s. C hem . Solids 4, 6 8 . A . A . M aradudin, 1966, Solid S tate P h y s ic s, edited by F . S eitz and D. T urnbull (A c a d e m ic P r e s s , New Y o r k ) vol. 18, p. 273. A. A . M aradudin, E .W . M ontroll, and G. W e iss, 1963, Solid S tate P h y s ic s , Suppl. 3, edited by F . S eitz and D. T u rn b u ll (A c a d e m ic P r e s s , New Y o r k ) . D .N . M e rlin and 1.1. R eshina, 1966, S oviet P h y s. Solid S tate G .A . M orton, M .L . S chultz, and W .E . H a rty , 1959, RCA R eview | 20, 599. ! W. N azarew icz and J . Ju rk o w sk i, 1969, P h y s. S tat. Sol. 3 1 , 237. ; R . C. New m an, 1969, A dvances in P h y scs 18, No. 75, 545. R .C . N ew m an and R . S. S m ith, 1967, P h y s. L e tt. 2 4 A , 671; 1968, L o c a liz e d E x cita tio n s in S o lid s, edited by R .F . W allis ( P len u m P r e s s , New Y ork ), p. 177; 1969, J. P h y s. C hem . | Solids 3j0, 1493. I R .C . N ew m an and J . B. W illis, 1965, J . P h y s. C hem . S o lid s 26, i ! 373. I E .M . P e ll, I960, J . A ppl. P hys. 3J,, 1675. ! ! P . P feu ty , 1968, L o c a liz e d E xcitations in S o lid s, ed ited by R . F . j W allis ( P len u m P r e s s , New Y ork ), p. 193. i M. P o lia k and D .H . W att, 1965, P h y s. R ev. 140, A 87. | j H. R ie s s , C .S . F u lle r, and F , J. M orin, 1956, B ell S y stem T ech. J . 3 5 , 535. G. S c h a effer, I960, J . P h y s. Chem . Solids 12, 233. J . S m it, 1970, p riv a te com m unication. S .D . Sm ith, R .E .V . Chaddock, a n d A .R . Goodwin, 1966, J . P h y s. Soc. Ja p an , Suppl. II, J21, 67. F .M . S m its, 1958, B ell S ystem T ech. J. 37, 711. W .G . S p itz e r, 1966, p riv a te com m unication; 1967, S e m ic o n d u cto rs and S e m im e ta ls , vol. 3, edited by R .K . W illa rd so n and A . C. B e e r ( A cad em ic P r e s s , New Y o rk ), p. 17. W .G . S p itz e r and W. P . A llre d , 1968, J . A ppl. P h y s. 39, 4999- W .G . S p itz e r, W. A llre d , S .E . B lum , and R .J . C hicotka, 1969, J . A ppl. P h y s. 40, 2589. W .G . S p itz e r and M. W ald n er, 1965, J . A ppl. P h y s. 36, 2540. J . S te in in g ^ r, 1970, J . A ppl. P h y s. 41^, 2713. H. Sttihr and W. K lem m , 1939, Z. A norg. C hem . 241, 313. N. D. S tra h m and A . L . M cW horter, 1969, L ig h t S c a tte rin g in S o lid s, ed ited by G .B . W rig h t ( S p rin g e r-V e rla g , New Y o rk ), p. 455. D. T a y lo r, 1967, P h y s. R ev . JJ56, 1017. D .G . T hom as and J. J . H opfield, 1966, P h y s. R ev. 150, 680. V . T sev to v , W .P . A llre d , an d W .G . S p itz e r, 1968, L o c a liz e d E x c ita tio n s in S o lid s, edited by R .F . W allis ( P le n u m P r e s s , New Y o r k ) , p. 185. F .A . T ru m b o re , I960, B e ll S ystem T ech. J . 39> 205. M . W ald n er, M . A . H ille r, and W. G. S p itz e r, 1965, P h y s. R e v . 140, A 172. R .F . W allis, 1968, e d ito r, L o calized E x c ita tio n s in S olids ( P le n u m P r e s s , New Y o rk ) . A PPE N D IX 1 C a lc u la tio n of the Change in the L o ca l Mode F re q u e n c y of B o ro n in S ilicon c a u se d by the A ddition of G erm an iu m Second N eighbors In C h a p te r V it w as s ta te d th a t changing the m a s s of a seco n d i n eig h b o r of a b o ro n ato m by su b stitu tin g Ge fo r Si h ad v e ry little | e ffe ct on th e B lo c a l m ode fre q u e n c y . T his w as c a lc u la te d by u sin g i th e ite ra tiv e c a lc u la tio n sc h em e developed in C h a p te r II. i j A c o m p u te r p ro g ra m in itia lly w as s e t up to be able to ev alu ate ! lo c a l m ode fre q u e n c ie s of a B -L i p a ir, including the L i lo c a l m o d e. | The ^L i m ode is clo se to the m a x im u m band m ode freq u e n cy , and is ! 7 i not v e ry s h a rp ly lo c a liz e d ; L i p a ire d w ith b o ro n gives a v ib ra tio n a l i j | m ode th a t m ay be a c tu a lly slig h tly below the m ax im u m phonon t I fre q u e n c y of Si and thus not a tru e lo c a l m o d e. A 5 x 5 x 5 a r r a y of j . | p rim itiv e c e lls w as u se d to avoid having to c o n sid e r e ffe c ts of the j j b o u n d a rie s. T his is c o n sid e ra b ly la r g e r th an is needed fo r the b o ro n t I ! m o d es, but it is not too unw ieldy a s the co m p u tatio n al d ifficu lty I i in c r e a s e s only lin e a rly w ith th e n u m b er of ato m s c o n sid e re d w hen the m o d el is r e s tr ic te d to s h o rt-ra n g e in te ra c tio n s . In the m o d el u se d , the b o ro n d iffe re d fro m the Si in m a s s and fo rc e c o n sta n ts, as d id the g e rm a n iu m . Since it is known th a t the lith iu m does not p a rtic ip a te m uch in the b o ro n lo c a l m o d es, the L i w as a p p ro x im a te d a s being s ta tio n a ry . 195 196 The tr a n s v e r s e b o ro n lo c a l m odes of the B - L i p a ir a re the h ig h e st fre q u e n c y m odes of the s y s te m , and thus the freq u e n cy is e a s ily found. The a x ia l m ode e ig e n v e c to r and freq u en cy a re found by u sin g a d isp la c e m e n t of the b o ro n along the B - L i ax is a s the in itia l t r i a l v e c to r; th is is o rth o g o n al to the e ig e n v ec to rs of the tr a n s v e r s e m o d es. To fit the o b se rv e d freq u e n cy fo r iso la te d *^B, it w as found ; th a t the B - Si fo rc e c o n stan t should be about 0. 85 of the S i-S i j fo rc e c o n sta n t. To get the o b se rv e d fre q u e n c ie s fo r the B - L i p a ir j m odes the B - Si fo rc e c o n stan ts h ad to be about 0. 95 of a S i-S i j | fo rc e c o n sta n t. The L i - B fo rc e co n stan t w as about -1 . 2 of a Si - Si ! fo rc e c o n stan t. T his value is s im ila r to w hat E llio tt and P feuty i j found to be n e c e s s a ry fo r the to ta l re d u c tio n in a x ia l fo rc e c o n stan ts I I in a G re e n 's fu n ctio n calcu latio n ; th e ir fin al m o d el had the red u c tio n i i i in a x ia l stiffn e s s d istrib u te d am ong s e v e ra l d iffe re n t fo rce c o n stan t ch an g es. The e ffe ct of a seco n d n eig h b o r su b stitu tio n w as c alc u la te d fo r the a x ia l m ode. S ixteen ite ra tio n s w e re m ade fo r an in itia l t r ia l v e c to r of b o ro n m o tio n only. T his gave a freq u en cy changing by -5 le s s th an 10 p e r ite ra tio n . One of the seco n d n eighbors w as changed and six m o re ite ra tio n s m ade to ev alu ate the change in fre q u e n c y . C hanging one of the seco n d n eighbors changes the sy m m e try , so th a t the m ode ob tain ed w ith an in itia l a x ia l d isp la c e - 197 j m en t is not ex ac tly o rth o g o n al to the h ig h e r freq u e n cy tr a n s v e rs e j m o d e s, and re p e a te d ite ra tio n w ill ev en tu ally y ield one of the i I tr a n s v e rs e m o d es. T h is p ro b le m w as avoided by c o n stra in in g the ! j i b o ro n to m ove only along the L i - B a x is . The p e rtu rb e d m ode is i c lo se enough to being p u re ly a x ia l th a t th is is not too s e rio u s an a p p ro x im atio n . T h ere a re th re e d is tin c t ty p es of seco n d neig h b o r s ite s : re la tiv e to a L i - B p a ir, a s a r e show n in F ig . 26, C hap. IV. F o r the f i r s t and second s e ts of se co n d n eig h b o r s ite s the changes in the b o ro n I a x ia l m ode freq u e n cy re s u ltin g fro m the su b stitu tio n of one Ge i i second neighbor w ere -0 . 13 c m an d -0 . 17 cm fo r a sim p le m a s s j su b stitu tio n , and -0 . 08 c m and -0 . 04 cm if the G e-S i fo rc e co n stan t ; w as a ssu m e d to be 20% la r g e r th a n the S i-S i fo rc e c o n stan t. The i ; su b stitu tio n of a Ge on the th ir d s e t of seco n d n eig h b o r s ite s gave a ! c o n sid e ra b ly la r g e r sh ift, - 1 .1 c m “ * fo r the m a ss su b stitu tio n , j and -1 .5 cm * fo r the m a s s an d fo rc e c o n stan t change. T hese la s t n u m b ers a re p ro b ab ly o v e r- e s tim a te s . T he tr a n s v e r s e m ode freq u a n cy w as fit by in c ra s in g a ll the B - Si fo rce c o n sta n ts, but the fo rc e co n stan t b etw een the b o ro n and a to m 5 ( F ig . 26 ) should p ro b ab ly be no la r g e r th an the fo rc e c o n stan t fo r iso la te d boron. The la r g e r fo rc e co n stan t r e s u lts in an in c re a s e d am p litu d e fo r the m otion of the f ir s t and seco n d n e ig h b o rs along the L i - B a x is in the a x ial m ode, in c re a s in g the a p p a re n t e ffe c t of a se co n d n eig h b o r m a ss change. F u r th e r re fin e m e n ts of the m o d el w e re not c o n sid e re d w o rth w h ile, sin ce the p rin c ip a l p u rp o se of the c a lc u la tio n w as to show th a t the change in seco n d n eig h b o r m a s s did not c o n trib u te sig n ific an tly to the o b se rv e d e ffe c ts d is c u s s e d in C h a p te r V.

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Creator Cosand, Albert Edmund (author)

Core Title Infrared-Absorption Studies Of Localized Vibrational-Modes And Lattice-Modes In Germanium - Silicon Alloys

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Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,physics, condensed matter

Language English

Advisor Spitzer, William G. (committee chair), Smit, Jan (committee member), Whelan, James M. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-682319

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