Sohd State Commumcanons, Vol 38, pp 2 4 5 - 3 4 6 Pergamon Press Ltd 1981 Pnnted m Great Britain

0 0 3 8 - 1098 $ 1 ' 030245-02 $02 00,'0

A PARAMETRIC RESON-kNCE IN SEMICONDUCTORS

I E Aronov, E A Kaner and A A Slutsl, an

Institute for Radmph~s,cs and Electromcs, ,-Xcadem~, of Sciences of the L~ramian SSR, Kharkov 3 I0085, USSR

and

Phsslco-Techmcal Institute of Low Temperature, 4cademy of Sc,ences of the Ukrainian SSR, Kharko~ 310164, USSR

{Recened 20 October 1980 b~' E 4 Kaner)

A parametric instability and its associated negatwe electromagneuc absorption have been pred,cted tor semiconductors The stabthzmg mechamsm and the resonance line form have been analyzed

THE COOPERATIVE MOTION of current carriers m the serruconductor placed in a un,form d c magnenc field Ho ts characterized by the c}clotron rotation fre- quenc) ~o If the magnetic field changes penod,cal ly ~ t h time, ~ e

H(t) = Ho(l + a cosTt) (1)

where

0 < a ~ 1 and H(rl II OZ

then the c}c[otron frequency is modulated too, ~ l t ) = ~o( 1 + a cos'),t) This means the posslbdlty of a reso- nant parametnc instability m the system of the same type as the farruhar parametric resonance (PR) of mechamcal systems [ 1 ] h seems worth to note at flus pomt that the parametric mstabtht!, o fsohds to be dLs- cussed below is quahtanvely different from the para- metnc mstabthty well known for plasmas [2] To our knowledge, the effect has not been analyzed for the case of serruconductors

Let an external electric field E(t) (uniform but /,me dependent , pointing along the axas X) be apphed to the serruconductor beside the modulated magneuc field equation ( 1 ), viz

E(r) =E, +E2cos (~ , t+cO, g~ andE., = const (2)

The coordinate dependences of Ht t ) and E(t) can be neglected a pnon , provaded the skln la)er depths for the frequencies "r and ~o are much greater than the size of the specimen

The modulatton o f H ( t ) bnngs about a circmtal elec- tric field E~ obvmusly dependent on coordinates The form of that dependence ~s detern'uned by the excltatmn condmons of the field equation (1), e g , if the specimen hes at a crest of the magneuc field mslde a plane parallel wavegmde, then E_ (r, r) is a hnearl} polarized field,

E~(r, r) = H(t)y/c, wath E.-ll OX, (3)

v.h2ch depends hnearl~ on coordinates as a result of the weak mhomogenmt) of E~ (r, t') In the case of a circular wavegmde E ~ ( r . r ) = - t l / 2 c ) H a r In~,hat fol lowswe shall deal. for the most part, u,~th the hnear polanzat,on equauon (2) The appearance ot a PR m the morion equat,ons of an electron placed m the fields equation ( I ) and equanon13) was cons,dered m papers [ 3 - 5 ] Here we will d,scuss the results pertaining to the PR m sem,- conductors, the stabthzanon mechanism and the reso- nance hne form

For the polanzat ,on o f equanon (3) the pnnmpal PR occurs when the c}clotron )requenc) -Qo ,s close to T/2 (in the case of a circular polarization the PR con- dmon is -Qo "" ~; [5]) The natural stabthzmg mechamsm is related ~ l th the non-quadrant run of the energy. momentunt relauon ~tuch can be chosen m the form

e(p) = p'-."2.z + 6 . (p~ +p~)'-. 14)

where 6 Is the parameter descnbmg devaauon from the quadran t law The equanons of electron motion along the coord,nate v m the absence of the external field are

[2]

v - + 4 ( t ) c o s + vtt~ , (5) " /,7,) Q 0

.4 = l a ~ 4 s m 2 t , - - ~ q , v = ~2 + ~ c o s 2 b ' + / . z y 4 a

(6)

/.z~ = a~o, '7 g.. = fl,~ - "),/2, ~3 = 8 m 3 ~ 6 / 7

Here Px is the integral of mouon and ~. denotes the hnear fncnon coefficient proportional to the energy

(v)

relaxanon frequency The parameters ~,, ~t~,/1., and/.z 3 are

small Equanon (6) descnbe averaged mouons of slowly vaD'mg values (the time scales are those of a "'slow t ime") [6] At t¢ </1, the parametnc mstabth D can be stabthzed ov~ang to the non -Mean ty (,u3 :~ 0) The result- mg steady-state values 4 and v of the amphtude and

245

2-4.b

pha~e ,,,. hath c o r r e s p o n d ~o t',)c~ " on the 4 ' : ' phase

plane are gp, en b',

--I= = ~ 0 ~ l . ,'? = lu d - ' [ { u ~ - ,." )t " - u. , ~ u x ] ~S)

2:'1 = -- arcsm ik U i ) n"- "gO(~ 3 ) 2.,. = 7,1 + ,-,.r

Here 0q r ) is the unt t step func t ,on and sgn v deno te s the

sko--.n o( ~. ~3,e ,,,,ould hke to emphas ize tha t the ~tead~ -

state values e q u a n o n (e,) are reached af ter t~mes o ( the

order (p~ - ~ 1 -~ : + 1 ~.. ,t,2 ,hat ter ~thzch are the mlnaI

c o n d m o n s

A par t icular tea ture c,l the s tead, , -s ta te regm~e

e q u a u o n 15) ,s the e ' u s t ence ot t~so eqmpr, . ,bable ,,alues

ot the p h a s e . ' l z ; '~and c, Hence. the e n s e m b l e o t

c h a r g e c a r n e r s m ~ t m l l , , {~e a t r ~ ) d ~ s t n b u t e d a c c u r d

mg to the B o l t z m a n n Iat~ ~ t t h a t e m p e r a t u r e T. n,),, at

the PR acqu,res the d~smbut~on l'unctv-m

t l 2

- v , 5 ' , p , ~ - " ' ~ o [ ~ - " r q " F(: z PI 2(2r rmT)~ _. _ ¢,( l =1

I 0 1 , 6 [ p , - ",'~,(t)] e x p l - p ~ ' _ nTI .

,.,.here t, is the carr ier c o n c e n t r a t i o n and 5 ([) deno te s the

secund term m the RHS ot e q u a u o n 1 5 ) m , ' .hath 4 and

~' shall be su0~t~tuted ~,. their hm~tmg ~table "aluea

t rom e q u a u o n 13)

kea ' .m~ the der l ' , ' amm and d~scus~u3n ol e q u a u o n

I'a) unt i l a more deta , led p u b h c a u o n v,e will present the

u l tuna te result for the t ,me averaged specific pov.er

abso rbed b', the s e n m . u n d u c t o r under the PR conditv:,n~

l rom a ~ e a k s ~ a l t~eld, n a m e h

- " - - + a u'z ,37= = O ,,.r ~ 2 e - : ' ' 3 ~ 2

l ]} -

. ,'~ 2 t : ( y 212 + ( 0 1 -r- , . . , _ - 1 , 2 - b l v . j -t - I , _ )

. 4 : m e : , ,z2~4 2

T ~ e C t ) l J l b l t ) n f requent ' , t' a%ncla ted v.i th tile Fnt)n]ell-

turn tranafer 2enerall,. e'.,.h~blts a pov.'er lay. d e p e n d e n c e

P 4 R _ 4 \ I E T R I C R E S O N K \ C E Ix, S E \ I I C O \ D L ' C T O R S \ o l 38 \ o 3

on me ener.: '. : , r camers , ' . ' l z L v v ~ v ' [ ' ] W l t h q < 0

t h e d c absor-. ',2n l i e , ) - E i j ~ o u l d b e n e g a t p . e ~,~,Lth

q -'> '3 ne_,au.e ~.oub..t b e . o m e at - , -- -, 2 abso rp t ion ,_,t

the c o m p o n e - " E. in rk.: ,v,.mal field The a b ~ o r p n o n

l,ne '~,:,uld a l -_ , . s be ul a resonan t torm since 0 is pr,-,

port,..~naI to 'e In o the r , ,nrd~. 0 ~s essennal lv non-

ze roon I ' , nea- h e P R

The e,usten,.e or non-e',_lstence ol a pa ramet r i c

m~ta~fl~t,, that haa been .-l~acussed ab,)' e ~, de te r rmned

b- tn . ~ to rm o "2re stead'. ~tate d ts t r tbum' .n laa equa-

u, ,n ~:~1 The ,-,tabLllt,. alan ,,.an manlte~t tt~el( t h rouzh

phen -mwna ,-, ..~r than electr,r, m a 2 n e u , ab.,, ,rpth-m

wl'uch ~uggest~ ne ~ pu~ tb l l t t t e s b o t h t,',r ph, ~LCal in'.e~ti-

• ,atv-,ns ot ser:.-zonduct,ors and for apph,.:att,,n~ ,.ff the

PR

R E F E R E N C E S

L D La~J.au,_'k K '1 kalsh, tz. t fe~/ ,anz's , p 103 Xauka P'-~bl Co ' loscn, . : 1%'~ Im Russian) \ P StLn The Paramerr'c Etre~ r v / P ' 2 ~ e u u l Radla'Z ' Lpo;', Plasmas Nauka PuN Co . klosc3t~. 119"31 fm Russian) B kehn.>rt. DI na,'etc~ 'V Clmrged Parncles Nortf~- HoLland Publ Co , -kmsterdarn 11%41 (Russ Trans --ttomlsdat Fh.lb[ Co . Moscov.. [ 9 " 6 1 [ ~, MaL.an and V I Magnko D~namtc St mme,'~,es atz,~f C,~hercpg State~ oJ Q u a n t u m S~ste*,,~ p 40 Nauka Publ C,3 \ lo~cov 19~-~ ( In Rua-_l_,n I V B Shx.an. Z h L Y F " o I q u ( 1 9 " o p "4 \ B,:,;_',l,,ubo,. & '~u A M~tr,-~poEk,, 4~1o,p r,_,ttc Ile;,,,_)ds tit t "e T/',e.)r~ , 't 3,,~,n-hnear Osct;- ,'aqo,,~ z 2gO \ a , l k a h , hl C'o \l,:,sco',~ q l':m41 Is~c,)nd P , u ~ a n ed~a,_,n ) F G Ba>., & ~ u H C,~re,,a.:z H,,, E lec trons and P )~ eu . , ' Electro~',agmcrtc k alcs '," S e , w c n n d , , c r .~ a n J Ga~ D,scharze Plasmas. p 3~ Nauka Publ C, , \1 . . . . ,'~ ( l ' - ) - :