Chandra, Ram

Verma, J S

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Indian J, Phya, 49, 044-949 (1976)Acoustic^wave amplification in semiconductorsR am Chandra  a n d  J. S. V ermaDepartment oI Physics, Birla Institute of Technology and Science, Pilani 333031(Received 2 June 1975)Tlic tliooretical analy,sis of the atnplifioadon of an acou,stic-M^ avo in a scmioondnotor which oxhibits both piczoekjctric and doforina- t-ion potential coupling i&i made The expression for the critical (ilectron drift velocity for the acoustic amplification is derived. The effect of the ratio of the l f^fectlve masses ('ll tlie charge carriers on the transverse phonon-helicon wave interaction giving jiso to the amplification is also studied.I I ntroduction \In their previous paper tJie authors (Ram Chandra & Vcri^a 1974) studii'd theoretically the interaction of the drifted electrons with the acoustic waves in piezoelectric semiconductor which shows the negative <lifferential mobility It was pointed out that acoustic waves can be amplified evim if tlieir velo(;it,i(\s arc less than the drift velocity of the electrons in such a medium However m the above analysis, the effecn of the deformation potential coupling u^ as negk'ct(‘d Since all solids exhibit deformation potential it seems nece.ssai’y to include tlu' effect of this potential Moreover, while studying tlje transverse acoustic-wate interaction generally the ratio of the effective massoK of the charge carriers is not included Since for semiconductors this ratio cannot be neglected and therefore its effect must be included Tli,e.se are the few points of the present investigation In sec 2 an expression for the gam is derived Avhicli acc.oums for both deformation and piezoelectric couplings. Tlu" oxpressioii for thti critical electron drift velocity for the acoustic-wave amplification is given in st^ c 3. Tlu' effect of the ratio of tli,e effective massos of the charge carriers on th(^  amplificai ion factor in transverse phonon-helicon wave interaction is given in sec. 4 11 isshown that under some conditions the results agree witji the earlier work2 T heoryWe assume that the acoustic-wave propagates along the a:-diroction of t]\(‘ semiconductor crystal assumed to be homogeneous, isotropic and infinite in exteut The dc. electric field is also applied along the x-axis. The analysis is cssiu- tially one dimensional,944The expressions for the sti- «(«>»'• V „ «  , „ ,  '“• ""”>- ^ , ,T  ^  c f i ~ p £ _  /  C,i(. \ ,^ J,J' <J > r7.r’D  =  e S - \ -e K -  (  '^<v \ <)St' <1 I r7.i-(J)(3)whore c is the elastic constant «tdeform..,„„  ^ .....<>. >l,eande , OhiOx  ^ f)x^f  w _  d"~u ^ 6x^' eolipliiijr equations can )„•'ATiltfMl 11,SOH5/ / flx^  ■ ... (3)rJA\fix ( (J / •• (4)dn^d f(•^ )i~ ■*■”' ' - f '  -I-/7,, '?!li -  0t a i / o L r  ”  " '" ">ll■™.-dom,„.,e,lh o l d .  I n  t h e  a b o v e  e q u a t i o n s ,  E  , s  t h e  ’ , /  I “  ^ ’ ' o  o l o c n  .cdensity, n,„ the equil,brmtn,i,.ii„,^,'^^^,,^ 'f " ' ' ' ' ' ’ r''*'displacement wltich is in the ^ dli’ectin « ‘he lattice•nd e , . I ' * : ; : . : ; ’     “ ““ “■vAssuming the variation of E,  and ,, „ f  n . . e  , oombinmgeqs (3) to (5), we obtain ' andwJiore f)Qj- -  G*]c\L i - j ^ r  11-1- — ____I-  _(«)(7)a p F ^ m a t i o r "  ^  attenuation constant, a, after making anA ^  (iu/Vt)+Jac _946 Ram Chandra and J. S. Vermai n  o q  ( 7 ) .  I n  t o r m w  o f  f l u *  f j o m p l ( ^ x  o l a s t i o  c o n s t a n i  c , t h e  a t t e n u a t i o n  c o n s i a m ,a  =  (o)lvg)o  ^ T i n ( o * ^ l )  . . .  ( !) )a n d  IS  s u e h  t h a t  | a |  < <  \wlVg\,  w h e n *  i s  t h e  s o u n d  w a v o  v e l o c i t y .A f t e r  c n m h i n i u g  ( ‘ q s .  ( 7 )  t o  ( 9 ) ,  w o  o l i t a i ul-Z=(tukO==}(corMZ(^) (10)w h e r e  1C -^  e j^cc < <  I, wc ^  (fnQ/iJf i s  t j i e  d i e l e c t r i c  r e l a x a t i o n  f r e q u e n c y ,  toj) — vCIDn d i f f u s i o n  f r e q u e n c y ,  Y — ^—{vJvs) a n d  L'^  (CJq)-(cjr.)T J i .0 a m p l i f i c a t i o n  m e c h a n i s m  i s  t h e  s a m e  a s  d i s c u s s e d  i n  t h e  e a r l i c J ’ w o r k  ( R a m  C h a n d r a  V e r m a  1 0 7 4 ) .  F r o m  t h e  e x p r e s s i o n  ( 1 0 ) ,  i t  i s  c l e a r l y  S ( '.e n  t h a t  t h e  i m s l n s i o n  o f  t h e  d e f o r m a t i o n  p o t e n t i a l  i n  t h e .  c o u p l i n g  e q u a t i o n s  h e l p  i n  t l u '  i n c r e m e n t  o f  t h e  g a i n  a n d  i t  i n c r e a s e s  m o n o t o i i i c a l l y  w i t h  t h e y  f r e q u e n c y  I ' o r  a  —  0 ,  i e  , in . t h e  l i m i t  w h e n  i J i e  d e f o r m a t i o n  p o t e i i n a l  i s  n e g J e . c t n . d ,  t h ,e  e x i i r o s s i o nfor a  redu(!0S t o  t h a t  obta ined  h y  R a m  CJtaudra (S: V(u-ma (1074)1F r o m  e q .  ( 1 0 ) ,  w e  c a n  e o m i > a n ' .  t h e  g a i n  p e r  u n i t  l e n g t h  o h t i a i n e d  h y  j n i r c ^ l y  p i e z o e l e c t r i c  c o u p l i n g  a n d  b y  d e f o r m a t i o n  p o t t n i i i a l  c o u p l i n g  W i '  h a v e(Gaiii)tff/orffl •(Gain)j,^e2o- K - v /cr«> Ml)3. Gritt( al EuorTHON Dii i ft Vflo( ttyA s  i t  i s  w e l l  k n o w n  t h a i  a e o i t s i i c  a m p l i l i e . a i i o n  o c c j t r s  m  p i e z o i ' l e c t r i e  s e m i -  c o u d u c l - o r s  w h e n  t h e  ( O e e t r o n  d i ’ i f t  v i ^ l o c i t y  e x c e e d s  a  e . r i t i e a ]  v i x l i i e  I I  t h e  m e d i u m  c x l u h i l s  t h e  n o g a t i r ^ i *  d i f f e r e n t i a l  m o l i i l i i y ,  th c ^  a m p l i f i c a t i o n  e o n d i t m i i  r e q u i r e s  t h a t  t l u ^  a c o u s i  i c - w a v i ^  v e j o c i t j "  m u s t  h e  g r e a t e r  t h a n  t h i s  ( i r i t i c a l  e l e c t r o n  d r i f t  v e l o c i t y  M o o r e  .S: d e P i a n  ( 1 0 0 7 )  p o i n t e d  o u t  t h a t  a n  e x a c t  s o h i l i o n  f o r  l i e *  c r i t i c a l  v e l o c i t y  c a n  h e  f o u n d  a s  a  f i u u i t i o n  o f  t h e ;  f r e q u e n c y  o f  t h e  a c o u s t i c  s i g n a l  T h e y  o b t a i n e d  a  c o n d i t i o n  f o r  w h i c h  c * ,  t h e  e f f e c t  i v ( \  e l a s t i c  c o n s t  a m ,  h e c o i n e n  a  r e a l  q u a j i t i r y  s o  a s  t o  o b t a i n  p u r e l y  r e a l  v e l n e s  o f  k T h e  e o i i d i i  i o n  s o  o h l a i n i ' d  d e t e r m i n e s  t h e  c r i t i c a l  e l e c t r o n  d r i f t  v e l o c i t y  A s  a  r ( ‘ S i i l t .  o f  i l u u i '  a n a l y s i s  t l i e y  o h t a i j u x l  t h e  f o l l o w i n g  e x p r e s s i o n  f o r  t h e  c r i i i c n l  d r i f t  v e l o c i t y  v,V - V s  +  ’whort'_  '««“(! +e*/f.c)“  ~ p : r(12... (in)Acoustic amplification in semiconductors 947Here /  dniotos tlui fraction of lh(‘ tno))ilc electrons If all tlu' .space eluirp;c is due to tl».u condnclioii electrons (in aUsence of lrni>s) t h e n / - -  1^’ollowing Moore & dePun (UH)7), m c ol)taui the rollowing expressioji lor Ihe critical drift velocitJ^iV. --- V,(fi lfo)A^! 1 ,,, V I ,14  - -  ______ . .  0-1)c q  ( 1 3 )  f o i -  f - '  1T h e  c r i t i c a l  e l e c t r i c  f i e l d  K i r e i i g l l i  E'o w  g i v e n  f i yE'» --- 1 ' , / / ' . . •I t  c a n  b e  s e e n  t h a t  f o r  a  - -  0 ,  i e  , t v h e u  D i e  c o u p i m g  t .  i . u r e l y  p i c / e e l o c t r i e ,  e x p r c 38i o u  ( U )  f o r  t h e  c n t . c a l  d r i f t  v e l o c i t y  e ,  r e d u c e s  t o  i l u a  o l m u i i e d  b y  M o o r e  &  d e P i a n  ( l ' J 6 7 )  f o r  / * „  - -  I'' ! ' '< «  < f e f o r i u a t i o u  p o t e i d u U  c o u p l i n g ,  e c i(14) reduces tor, -  i\(/'o//'')(“ ^ “ ' “tj)H - ■ ' f . , 1 I. . .  ( 1 5 )whtoh .I^o^ly ....... * .  .r  .1.. "  »  <«noted that I'l approaches o so long as We, is ol the on m o u.-1 Tkansver.sk Phonoh-H klk'on Wavl; iNi-nuAnioK ^ lot of rcsocu-cu has been reported oil the, subject ol heliconof the hoheou wave, lies m th . -  " (Steele &. Vural lOtifi). fuand can load to  the ,,U,.o is often ignored whilemetals and gaseous plasmas the. . ■ ^vith the sound waves,studying the Ite.licou wave propagation a n c l t,However, in scmieoiiductoi's tlie latio o Paiidev 1975). Browneplays an im portant role ‘ ^ . h  ( ^B) have L i e d  Ihe(1904). Solymar & Lashmore-Davies they havetransvcrso-aeonstie intcraelion m pitAoc ,,a,rriers on ihe gainnot included the effect of effective masses ol the ch.ug ^fn this section wo leave siudii'd the oiccUon-holttin pieKoeleetric .semiconductor which accounts lor the948 Kam Ohandra and S. 8. Vermaeffective mass ratio. Here, we are neglecting the deformation potential just for simplicity. The amplification occurs when the drift velocity of the electrons exceeds the sound wave velocity. We have shown that under certain conditions the resiili agree with the earlier work.We assume that the sound wave propagates as a shear wave along the s-axis which is also the direction of the applied d.c magnetic field, of a semiconductor (%-TnSb), assumed to ho homogeneous, isotropic and infinite in extent. The lattice displacement is along the x and y axis (transverse sound wave) and the electrons are drifted along the c-axis. Following the standard method (Steele & Vural 1969) and using all the necessary coupling equations together with the equation of motion for tlic elections accounting ior the ratio of the effective* masses of the charge carriers (Singh & Pandey 1975), we write the dispersion relation in the following form[A] , eiw(m+jD^B)K^k\ ... (16)DriftedheliconSound Couplingwhere__{(D — kVQo is d(o)—j v )  ’ {o )—j p yA -^  — l iIn above equations Vg is the sound velocity, 6/ the semiconductor lattice di­electric constant, c the velocity of light, the diffusion constant, cjp\=r{v^-l V'z)M semiconductor plasxna frequency, v^  ^ tlui drift vtfiocity of the elec­trons, V and wjt} iho collision frequeiKjy and the cyclotron frequency of the oleotroiis respectively, d = and K the dimensionless coupling coefficient Tlie4- and — *sign corresponds to the two polarizations In absence of diffusion and 111 the limit as d —► 0 eq. (16) reduces to that obtained by Steele & Vural(1969) and they have discussed its solution in detailIn terms of the effective elastic constant c*, oq. (16) can be rewritten as= .. (17)wherec* =  [ i - «« {a .+ jD ^¥)eiK^]1ATo obtain the solution of eq. (16), we neglect the fast electromagnetic branch {k^ c^  >  >  oi^ cj) and consider only slow-wavo branches of the dispersionAcoustic amplification in semiconductors 94Uz ;  . m i n t :t,h. s . i „ » i„  o Z Z l , ! ' ” " " ' "{<oclv.){v,leyN T ?  ™ T ' ) ’ ' (:■)’>■■('?)■] -Ailu-ic 0 - -  (fi/cj)* Iho vclocii.y o) hjriit in tju-imHlmin, oj (^ _ Wp i^v) ii- llicdielectric rclaxalioii ii eqxioiK'Y and T -- IThis is the tnodiftt^ (‘xpressioii for ijie gam ui absence oi diffusion ini I ucconm- mg for the parameter rf 1„ iju- |im„ ^(A '/-!)('•v/e.) h iT a v /c i) -1 ( l i t )S i n g h  ( 1 9 7 3 )  i n  lh ^ ^  M i n i i l a j ’ c o u d a i o i i s  h a s  n l ) i a i i n M l  i h o  r ( > l J ( n n n g  f s p r o i s s m n  l o r  t h e  g a i nL I-\ /iHwcloi]- J mw h e r e  jti -  Fn  wc approached to tJn' last oloelromagnetic hrauch (t «  v )^, eq. (H)) j'educes to the sam e expression \vt.\ (-JO)] as olnaijied hy  Smgh (1973). Ihn sima  ^ in i]ie .semiconductor d cannot be equal lo  zero and tJi,eivron‘ ibs alleoi on iJte acoustic- AA’av(* amplifieatiiui must 1h‘, iuelndedA c k n o w l b d g m k n tThe first auiltoj* (IIC) is rhaiikinl to the CSFR (India) for hnaueia) assistance lu the form of P()Si Doetoral K<'h<i\vsliiji. TJte assistance of N()AA(USA)/SRG/ PL-4-80 is also aokiiowJ(\dged u ith  tliaiiksR eFBREN( TCSBnnvnr M  E M d v .  P h y s .  A v ta  37.Moore R. M. &  dePian L  19«7 lE K l C  T r e w s .  S { f - U . 145 H a m  Chandra &  Verina J S 11)74 h ir h a } t  J .  P h y s . 48, SI):! t'^ nigh A. )97:i l t d .  J  E l e c i r r n 34, 127R  N. &  Paiidoy N. L. 1975 P h y s .  I lv v Bll 771 Solymor L. &  Loshmore-Davievt C N. 1997, I n l .  J  E le c t r o n .  2 2 , 540.Nieeki M. C, A: Viiral B. 1909 W d v v  J H fe ta t i io ih t  on  S o l i d  S to le  V fa a m a n ^ MrGraA^ -Hill, NowYork.10

Acoustic-wave Amplification in Semiconductor

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Acoustic-wave Amplification in Semiconductor

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