Electric coupling properties of acoustic and electric surface waves

P ~ S i C 8 R E P O R T S ,,~e©tion C of l?h£sic~ L e t ~ r s ) I , no°S (197702 ~ - 2 6 8 . N O R T H - H e L l . A N D

ELECTRIC

COUPLING

PROPERT!~:S

OF

SURFACE

WAVES

ELECTRIC

C. A. A.J. G ~ E B E ,

ACOUSTIC

A N i)

P.A. van D A L E N ,

T . J . B. S W A ~ ~ U R G

and J.

W

O L,T E~ R

Ph~Hps R e s e a ~ h Laboratories, Iv'. V. Philip,' ~ , l o e i l a m ~ b , ~ e ~ . Ei~dhovcn, T~e Ne~be~,,z~s R e c e i v e d 1 M a r c h 19 ~'1

Contents:

235

1. Introduction 2. E f f e c t i v e d i e l e c t r i c constants 3. Dispersion relations

4. Relev~once of th ~ concept of ~eff to the excitation problem Appendix

2~4

i;~::.i Abstract. A discussion is given of electricc o u ~ a c ~ s an Imc:*-c ~ "-.~w~,~nvaries k~uds of s u r f a c e wave~ on s." ~ids. T h i s l e ~ d s to n u n l f l ~ tre~: r,, ,:" of th,~ p r o ~ t i o n ~.~r~d coupling p r o p e r t i e s of s :ch p h e n o m e n a a s e l e c t r , ~ t a ~ c , ~, r t:~. c ~ r r t e r ~ v e ~ o~:~ s e m i c o n d u c t o r s , Rayleigh ~ ~ves on p i e z ~ | e c t r i c m ~ i t a ~ d :,~:~ ~ e i n - G t t i y a e v naves. 2he influence of an ~ e e t r i c d r t R field and of a tr~n~,~,:~ ~, 'm~$rnetfc field i~ included and much a t t e n t i o n ~. paid tc a t n p l i ~ c a t i o n z~.~:~ : ~ , 1 ~ , ' ~,~,o~,.~mena. ~ e c ¢ ~ t : a l quantity ~. the t r e a t m e n t ~ ~ the e f f e c t i v e dietcet~'i c~'*~':~'~"n*~ ¢c,~ with wh}ch a ha:,f-spsce of m ~ r i t l e a r be c h ~ n c t e r i t e d . I n ~ e b r t ~ e n ' s .,~~pro:~ima~ion. of t h i s ~ t ~ n t i t y for the ,case of l ~ y l e i g h w~ves is a n a l y ~ and ~,,~.~,t,:~ m be r e a l | ~ t t c in man:: in,.cresting situa0dons. New effects, d e r i v e d in this p a p e r a ~ : 1) a tende~cy of Bleustetn-C~lyae~, ~mves to t u r n ~ l e ~ =, i.e., to r a d i a t e a c o u s t i c e n e r g y a ~ y f r o m the interface,, when ,elec~ t r i c a l l y coupled to a semicom~ucting a d j a c e n t m M t u m which is ~mbjccted to an e.Iect r i c d r i f t field and to a trm,~vert~e m a g n e t i c field; 2) a t e n d e n c y of a ~ t l e surface w a v e s to a c q u ~ low p h a s e v d o e i t i e ~ ~f t h ~ m ~ e field b e c o m ~ l a r g e and ~ s the p r o p e r d i r e c t i o n ; 3) an i n t e ~ i g ~ t a l e l e c t ~ e s t r u c ~ r e on a n o n - p ~ e z ~ I e c t r i c sem~c o n d u c t o r which is s u b J e c ~ d to a d.c. e l e c t r i c drift field may exhibit a n e g a t i v e a.c. conductance. The rele~,'ance of e~ff for the e l e c t r i c ~ exci~ation of s u r f a c e waves is a l s o ~b~,,~n to be c o n s i d e r a b l e a n ~ ' ~ o m e d ! L f ~ c ~ . s ~ , , n e i a ~ l ~ i ~ n~o_,,i~s t ~ , ~ t , ~ ~,~ ~ , e x c i t a t i o n p r o b l e m a r e d t s c u s s ~ in ~ t ~ s of t e f f . -

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$,'.~g~e orders f o r this ~ssue PHYSICS R E P O R T S (See*ion C o=" PHYSICS I ~ ~ }

! • ~ No. 5 (1971~ 23.5-; 6S

i

Copies of this i s s u e may ~ obts~ned a t tl e p r i c e g~ve~ ~ l o ~ , . AU o r d e r s ~h - ~ d be sent d i r e c t l y to the P u b l i s h e r . O r d e r s must ~ ~ c c c , . ~ l ~ i e d by check. } Stogie i s s u e p r i c e HI"t. 12.50, $ 3.50, p o s t a g e t n c h ~ d ~ ,

i

I, ~ T R O D U C T I O N Recently m u c h a t t e n t i o n h a s b e e n l;aid by r.tumerou~ .~L~thors to v a r i o u s k i n d s of ,v.;.rface w a v e s w h i c h a r e , at l e a s t p a r t J y , of an e l e c t r i c a l n a t u r e . P e r h a p s t h e m o s t . i m p o r t a n t e x a m p l e s of t h e s e a r e ac.~ustic s u r f a c e wa ,es it, piezoelect.ric s u b s t a n c e s . , which c a r r y e l e c t r i c f i e l d s i ~ s i d e as welt a s ou~.side ~he p i e z o e l e c t r i c m e d i u m . T h e m e c h a n i c a l l y p u r e l y t r a n s v e r s e a c o u s t i c s u r f a c e ~ave.n r e c e n t l y the,~,retically d e s c r i b e d by B t e u s t e i n [1] a s well as by Gulya.~.v. I '~..~ may be m e n t i o n e d ,~er.e a s a specl:~tl case.. F u r t h e r m o r e , e s s e n t i a l l y e l e c [ r o s t a t i c w a v e s m a y p r o p a g a t e a l o n g the ~. rfac'e 0[ s e m i c o n d u c t o r s . An i n t , e r e s t i ~ g e x a m p l e of t h e s e v a s d e s c r i b e d by Kine, i''] and o b s e r v e d by B ~ r k e and K i l o [4], who found t h a t ia a h e a l ly i n t r i n s i c s < m , ~.~.~,~,~,~, = ~ m : ~ ~ ~a e i e a : ~ c r~.-"c. ~ r ~ f i e l d par.~.iiei, tc the s a r f a c e , su~ i~ ~'aves may p r o p a g a , ee gain if a d.c. ~~.¢' e.r~.~,-e . , wIt]~ tow io "s ~,, w h i c h c£~anges into ._. . . . ~ ,~: magnetic field of stfffici.en~ s t r e n g t h ~,:, '¢ a p g t i e d . All t h e s e wav e~ c a n I n t e r a c t wi~h o n e a n o t h e r by m e a n s o5 the e x t e r n a l e l e c tric :fields w h i c h t h e y c a r r y . T h i s g i v e s r i s e to a s e r i e s of i a t e . r e ~ i a g ~h~ n~-v erlll stlch a s a) a m p a f i c a t i o n of a c o u s t i c s u r f a c e w a v e s by m e a n s of a s e m i c o n d u c t o r subjec+~ d to a d.c. e l e c t r i c d r i f t f i e l d [5]; b} e n h a n c e m e n t of t h i s a m p l i f i c a t i o n by m e a n s of a t r a n s v e r s e m a g n e t i c f i e l d I t - ~,!: c) a d r a s t i c c h a n g e of the p h a s e v e t o c i ~ af B ! e u s t e i n - O u l y a e v . . . .wav~..s.. .. [. 10], ~in_.~..

by means M a t r a v e r s e magnetic field (see S.4 ~ ~ i s ~ e r ) ; d) coupling of acoustic s u ~ a c e waves from one p i e z ~ , l ~ t r i c medium into another, without mechanical cow,tact [ 11]. It is the purpose of this paper to give a simple nut[led f~m~ll sig~Rl) descr~ptlon of a certain class of these waves and their interactions° This class ~s defined by two rectuirements:

:) the ~henome~m to be c o n s i d e ~ d must pr~,agate so slowly that the electric fieldf, ~. c~_u be described as the g r ~ d i e ~ of a s c a l a r electric potential ~ , ~.e., the pimse velocities must be much less fl'~an the velocity of light; 2) the phenomona should occur at the l n t e ~ a c e between two media which are o~dy electmc~ly coupled, i.e., only the electrical b o u n ~ r y c o ~ i t i o n s s h ~ l d be permitted to involve both media. According to these requirements w e shall, for instance, not discuss electro. magnetic surface waves (too high phase velocity) and Love wav3s (mechanically coupled medi'i). The above requiren,ents perrnR us to give a cot )lete descripttov of the coupling between two media by means of two simple b, und:..-y co~dition:~: I) continuity across the inter~ace of the electric poten' tal ~'; ~ ~ ~ across the interface of the normal corn:..~. ,~t c,tthe dielectric di~ ~/::! ~t~Dy !( i ~ asmm~e the interface to be p.~ ~ e~.~,,cularto the y-axis). I,, a liuear theory the propagation properties of a ,:,:,re do not depend on the amplitude of the variables involved but only o~ their ratio, l~herefore the key quantity [12] whi~:h describes the coupling bet~e¢~n ~,,o media ~atlsfyiug our requ~rem~nts is the.,ratio of D v and <~. In a ~ave phe~omeuon on a~ i~ter[ace, al! wave vectors inv,)Ived must'have t:,e s a m e component a~.~ng the interface. If we choose the x-clirection ~ a l l e l to this c o m m e n t , its magnitude c ~ be written as kx and we m a y consider as a key a UraUtityany combination ox Dy/~ ~ d kA~ W e shall in ~act use the quantity

kxr kx where the subscript r stands for "real part". This quantity has the dimension of a dielectric constant. F r o m the above it should be clear t~t, as far as propa~tion properties are uncoupled media are taken into accoun~ by demancimg the continuity of the qua.~tity (1). Accordingly all interface phenomena to b,~ discussed in this paper can be treated by m e a n s of the following ~ o l:asic steps

1~ Characterizing the half-spaces on both sides on the interface by a suitable expression for the quantity (I). A consistent choice should be m a d e as to the •

~

~.

positiv~ y-direction, otherwise (1) is not vniquely d~ined. Since w e arc char-

ELECTRIC COUPLING PROPEIt:[IES

239

act~,rtzing h a l f - s p a c e s h e r e , we shall (for each half-space) a s s u m e the y - a x i s to ~ d i r e c t e d in~,nrds. In the c a s e of a simple ,lielectric it then t u r n s out (see 2.t.1) that (1) is equal to the d i e l e c t r i c constant. We si~all a c c o r d i n g l y call (1) the cffect~ve d i e l e c t r i c constant ~,ff of a h a l l - s p a c e . It shouid be empl'asized hore that in (1) D v and ~ a r e to be taken just outside tt~e haaf-space, i.e., ~eff m~st ir chide the ~tnIlueace of any charge which may have a~:~umulated ~car .I~c ~mrface. Of c o u r s e %ff will in g e n e r a l be a function of k x and of the angutar frequency w. ~} Demanding the continuity of the quantity (1~, which yields the disl:,ersior: equation

eeff l(W,

-, eff2( o,

= 0.

(2)

Here ~eff 1 a ~ ~$ff2 a r e the two effective die.~ectric constants of the hallspaces on ~ s i d e s of ~ e i n t e r f a c e . T h e i r s u m and not t h e i r d i f f e r e n c e should be z e r o b e c a u s e of the sign conventio;~ that we made ~s to the punitive y - d i r e c t i o n in delining eeff. WRh the exception of p a r t 4, w h e r e excitation p r o b l e m s a r e d i s c u s s e d , t:~is paper is o r g a n i z e d a c c o r d i n g to the above t w o - s t e p s c h e m e . F i r s t , in part 2, we ~hall derive s o m e expressions f o : the eeft to be a s s i g n e d to various h a l f - s p a c e s , ,,;uch as d i e l e c t r i c and p i e z o e l e c t r i c s u b s t a n c e s (2.1) and semiconducting sub~tances which m a y be ~ubjected to d.c. : r a n s v e r s e mag~etic and longit~:,dinal electric fields (2.2). The effect of d i e l e c t r i c l a y e r s on top of t h e s e sub.,:tances will a l s o be considered (2.3). Subsequently. from the r e s u l t s Hms ~,btained, we shall c o n s t r u c t and investigate d i s p e r s i o n relations of the form (2~ and d i s c u s s t h e s e (part 3). We shall thus p r o d u c e a unified p i c t u r e of the p!,~momena mentioned m the beginning of this i n t r o d u c t o r y section in t e r m s of ~ quantity ~eff" In this r e s p e c t one r e s t r i c t i o n pertaining to ~teu,_~em-Gulyaev waves ~,h,:,~!~ atre:~dy be made at th!.s e a r l y s t a g e of the a r g u m e n t . It is well known [!, 2] that these waves can have v e r y g~-eat p e n e t r a t i o n depths, which depend on the e l e c t r i cal p r o p e r t i e s (i.e., on the C-elf ~-) of the adjacent medium. In fact the penetration depth of t h e s e w a v e s is finite only b e c a u s e of t h e i r e l e c t r i c i n t e r a c t i o n with the $~djacent m e d i u m . T h i s means that B l e u s t e i n - G u l y a e v waves ahvays iha~'e penetration depths which a r e not only p e r t u r b e d by the e l e c t r i c a l p r o p e r t i e s of the adjacent medium, but which a r e m a i n l y d e t e r m i n e d by t h e s e p r o p e r t i e s . It will be ~;hown (3.4) that even negative p e : ~ t r a t i o n depths, i.e., growth of the a m plitude into the p i e z o e l e c t r i c , a r e possible. This obviously implies that the piezoelectric s u b s t a n c e cannot be d e s c r i b e d any m o r e as a h a ! f - s p a c e with one surface. P l a t e s of finite t h i c k n e s s aheuld be c o n s i d e r e d in thin case. Because of

~he general two-step program outlined above. In addition, l:he~z penetrati¢~ depth, for which a simple equation involving the ~eff of the adjacent m e d i ~ (':~:~ be derived, should be positive and sufficiently sm~ll to w a r r a n t the use of the half-space concept for s a m p l e s of p r a c t i c a l size. Otherwise, the finite th~ck,~.~ of the p i e z o e l e c t r i c should e n t e r the argument.

2. Bf FEC I'l E D t B

2A. ~

,;CTF~JC

#

tit~ ~

tto~ed, that. for a n I $ ~ l e - d i @ t

~-m

.

lns[d~ the d i e l e c t r i c which, in a c c o ~ t a n c e ~ t t h o ~ r eo~,v,:~'|,~ ~ t;h~ v-direction, m a y be assumed to oc,c'~y ~ r ~ o n y > O. w ~ ~ v ~ the f O | l ~ equations

w$

E = -grad ~. D:~E. div D - - O .

fn the ,rS~ace we are considering, throug~out ,this ~ r, •, u r n that p : ~ r , e d i r e c t i o n along i n t e r f a c e s which a r e ~ ~ , n d ~ c u ~ a c to the y-~Jirectio:n w~ may ..... as f~mctions of x , y , .~n,d t m~t,y':

w h e r e k~,- ' ~ r + Jkxi,

~v = k y r ÷ J#yt-

With the aid of (7) we easily d e r i v e f r o m (4). (5~. ~ad (5~ the ~'olume ~d~s~ersi~m ~ rela~on

,,.,;.,,-"

=

o

or

t,,,

=

&t:

f o r al! values of ~, In o r d e r to o b ~ i n decay into the m ~ t u m , i.e. t o ~ r f l s ~ s i t i r e .v - v a l u e s , we. can only r e .t a i n n e g a t i v e. ~ l u e s of the i m a g i n a r y p a r t kyi of k~,~ i . e . , the choice of sign to ~ m a d e i:n ( 8 ) s ~ 1 6

I Accordingly (7) c ~ n g e s in this case to

P = Po e x p [ j ( ~ t - z,xx) ] e ~

k~

hxy

.

~1o)

We s h ~ l denote ~ e mode (10) a s the Lapia.ce m o d e 1,ecause it is a soluti~on of the L a p l a c e e q ~ t i o n (8), tt will f r e q u e n t l y turn ~p m the. m o r e c o m p l i c a t e d cases to be c o n s i d e r e d below. With the :aid of (9) we get f r o m (4) j "kxr----L. ~!

(tl)

indicating that E x and E~, a r e i,~ thi~ m~,de 90 d e ~ e e s out of p h a s e ( r o t a t i a g - ' ~ t~ic field v e c t o r s ) . Fro/in, the e q u a t i o n s (4), (5), and (9) we see that ins:,,de the die l e c t r i c we have

o 5 ::

,;. N f

|~;~, ~ ~,i~ ~

~.,f,'~ay

e~t.se~,~ Mv'e, ~|~? tttee ~ t t t f a c e ,=~,~rge (12) m u s t a l s o hold j u s t d|eI~e~r~e, ~ h i c h fmN~e.~ that t o t ~ d ~ e l e c t r i c m e d i u m (3) is valid a s i ~ | ' | e | ~ | ~ . |P* e"fmneetlO+l~ with ,the d e l l s i t ! t i n of ~ ~ff, w h i c h wn..:i given in

t~ tI~|:l~ l::|O~ ~ e c o n s i d e r ' t l ~ [~, :~,,O) ~ u H a e e of a C 6 v (or C~ov) s u b s t a n c e , lh:e ~d~t~):~ '[or rc~tation, ai} ali,~ l:~.~t~g trt the z~.direction It h a s b e e n showu 11,2] * ~ tM~ Nighty ~ymmetrte:a.l g e o m e t r y N r m i t s the p r o p a g a t i o n of e l e c t r o a c o u s t i , c B~e~.~e,~n-G~flvaev w a v e ~ in the x-direct],,on and c o n s e q u e n t l y m e c h a n i c a l p r o p er~,~:.~ m.uN ~-~:ter into tete expre,,~sto~,, to be d e r i v e d f o r eeff- % e s h a l l re/r~.~ce tt:e l l r g t ~ m ~ t of r~ft~, [ I ] and [2 ! in a t o r m adapted: to the p r e s e n t c o n t e x t , and deri~re ~ttt ~ x p r ~ s | o r t .for c a l f a . w e l l a s f o r the p e n e t r a t i o n d e p t h , a s d i s c u s s e d in the ~trf~oetltm, The w a v e ~ u n d e r c o n s i d e r a t i o n ~ r e m e c h a n i c a l l y p u r e l y t r a n s v e r s e , i.e., they ~n~olve o n l y d l ~ t p l a c e m ~ t s u ~r~ the z - d i r e c t i o n , in the p r e s e n t g e o m e t r y t h e s e -are coitpled o n l y ~o the s t r e s s e s 7 r e. T v e ; and to E x , E v , D r , and Dy by the r e l e ~aet p i e z o e l e c t r i c e q u a t i o n s of s~ate: T lfz

=

CSXZ

"

~"

~X,

Dx ~ e . ~ x z * ( p E x '

T~oz =: C S v z - e E v ,

.,116)

Dv = e S y z + ,~p E y ,

,~heve, b e c a u s e of the a b s e n c e of any z ~ e p e n d e n c e ,~u ~z '

Sxz

as r e f l e c t e d by (7) we have

au. ~y

-yz

tn (1~) and (I6} t h e d i e l e c t r i c c,o n s t a r t t of the p i e z o e l e c t r i c h a s b e e n p r o v i d e d ,:,'it!: a s u b s c r i p t p in o r d e r to a v o i d c o n f u s i o n l a t e r on. As L su.a! c and e d e n o t e (he relevant e ~ s t i c and p i e z o e l e c t .,i(. constan, ts, F a r t , h e r m o r e the a ~ r o p r i a t e ,N.uation of m o t i o n r e a J a 3 2:~

~ Tx z

P 8t 2""

.~x

~y

'

and of c o a r s e e q u a t i o n s (4) and (()' a r e s t i l l v a l i d , (5) bein~:~ rcp~.ac~,.~ I" ~d ~v~ <14! and (19). From (4), (7~,,, (13), and (15) we o b t a i n ' '~.y• s u b s t i t u t i o t : into (i8)o - (k$ , k

) c ] u : (k

and from (4)~ ~4~, aa~ (t6) we obtain by s,.abstRution into (6)"

( "t9

Equations (19) and (20) can be c o ~ b i n ~ t to yb:ld the complete volurne d iw~rsto,s relation [~)

.2

:!

Equation ( 2 1 ) e x p r ~ s ~ . the fact tlmt the bulk, ol the mKlium can both tr&nsver#e sound and the I,aplace ~mde,. The z e r o in the rilcht=ha~l m ~ ' , b e r ,of. (2L) show~ that h~ ~ e prestmt~ symm~trteatt th~ ~ : v e s , t~e: not coupled in the bulk. Couplt.g t l y b ~ , t ~ t - / e o , t ~ t t o n : t t m u ~ therefore :provide the influence o f mecltanical prelmrttes , ~ ¢ ~ f WV,ieh w e a r e looking for. ~ o r e ~ ~: can apply these boundary ctmdi~ontt the ~ , p 0 ~ , t b l e m o d e s m u s t | l r ~ be altaiyzed separ&tely.

(D The I~T#lace mode For this mode (to be denoted with superscrti;t t "';. ,tnd (9) ag'atn apply. T h er,ef o r e (19): in di cares: ,tlmt ,,I = O,

{~2)

i.e., tLe Laplace mode does not c a r r y mech~nical movement. Equations (17), (15),

(4) ~ yz

y

=

(9):

~,x'r k r e

(23)

and

lk.,-r,

"

" # x r kx ~p eI"

(24)

The quantities Tyz, I ¢I, and D I will have to obey boundary conditions. (If) The sound w a v z This wave (to be denoted with superscript If) is characterized by (see (21))

[.~2

-

(k2 kr:2)(1 ~

e2 "

+ k;~-~)! = 0,

(2,5)

frora which we obu~in v

k

C- . . . .

Xl

whe 're

(27) is known as the piezoelectrically stiffened ~lastic e~stRnt. Since, accord&rig to (25) k2;v + k yH 2 ~ 0, we infer from (20) ~p (pII = e

(28)

~ L g C T ~ . | C ,'.,OUPMNG P R O P E R T I E 3

l~lem

243

(~$) I ~ d $ , wtth {17) and ~4) m,bstituted in (14,,) and (16), to

OII - D !"t ~, 0

!'~9~

~.e. °:~ ~ound wave does v~ot c a r r y d~electric displacement. ~k2uation ( i 5 ) y i e l d s in a like m a n n e r yz ~

y

,

,

(30)

~ e r e ~ ~2 t~ the s q u a r e of the e l e c t r o m e c h a m c a l coup:in.~ constant

r 2 = (1.

-1.

Adhering to the r e q u l r e m e n t s m a d e in the introducti, m the mechanical boundary conditions m a y only involve the r d e z o e l e c t r i c mediura. We shall a s s u m e that it has no m e c h a n i c a l contact with tht: outside world at y = 0. Consequently the S r e e s t r v s s components Tvx, Try, and Ty z a r e equal to zero. Only Ty z is r e l e vant for the p r e s e n t discussion ~ifid ~hus ~e" have

TI z + T~z

(32)

= 0

at y ~ 0. T h i s e q ~ t i o n d e t e r m i n e s via (23) and (30) the relative magnitude and phase of (pI and (pII. With the aid of (24), (79), and ( 2 6 ) ~ e then obtain ~p

(33a)

(eft = (¢pI +
,,2 II"

kxr ~y which l:ogether with (26) beco~aes ep ~eff =

(33b)

Equation (33b) is s a t i s f a c t o r y to the extent that it shows that Iim C e f f = ep 7,2 - , 0 and ~ s o that c o n s i d e r a b l e deviation~ of eeff f r o m ~:u can only o c c u r if i~0/kxi is c o m p a r a b l e to the velocity of t r a n s v e r s e ' s o u n d ~/@/p. Oo the other hand (33b) contains a :L sig~_ and thus is ambiguous. This point which i~ rebated ~,~ ~he sign to be chosen for in (26) d e s e r v e s c a r e [ u l considera~'.oa as a~reaay d[h,cus~ed in the:~introductfSn. ~ To b ~ i n with we consider a plot of ~eff as ~ v e n by (33b) as a ~unc~ioa 'el phase velocity v = ¢o/k x. It is s c h e m a t i c a l l y sho~a;~ i , ;ig. ~ for r e a t vaiues of ,". The broken p a r t of the curve c o r r e s p o n d s to k II. > 0 (i.e., growth of acoustic a m piitude into the p i e z o e l e c t r i c ) and the drawn pa~rt c o r r e s p o n d s ':o k ~ < 0 (decay into the p i e z o e l e c t r i c ) . The i n t e r s e c t i o n of the asymptote ~th t h e : ' - a x i s is denoted by Vo and the z e r o of ~eff is denoted by v¢o to facilit:ate c o m p a r i s o n witi~ the

9C~

~.

3r.,

"C~ \ C

%

"sc~,

F i g . ]. E.,ff a s a f,, . . . .

I~ ~

[

:

~ ::.

•

/

,4

•

•

#

'

he8 .bye Pune/:ration depth D Curve: pos2 ~ u u s - g r e a t e r t h a ' n "~" ~ U O s e c t i o n Whe,-.. ~ _ .lOSs . " u Voo, E " - ~ arl~|o o ,. Y Character •. eft as given ~, g u s q u a n t " • ,.,ound ~ w , , ^ _ W h i c h the ~ , , ~ b y (3~'Jb) h , , , , ^ _ z t i e s Occu . _ ve~,.- --~-¢u can be ,...-,. - ~'y,~ace dis-~---: -'.y,-umes co,,--,r. F o r real ..... - o u ~;ound -,~a/at #,,, . . . . ~ ¢ ; y s at nh,~ . _ ",~-ex. 'I~#.. _ - " ,~-val_. e:/-,.,.u ~ne . ~'.,,~e ve • -',,~ e _ It ca~, be ~--medium v ,~ loc~ties a~- .--~- X p r e s s e s the e~?ectiv..-~ .~. ? ~ e n a l r e a d y f. . . . . . ' ~o o e i n g the b,.,,-" ~ . m c a bulk '

c o, This t

•

2Lu

~u mat

,~aa ~ e g a H v e ~"~'~ may be ..~, ~ - e s ~ C t t o ~th medium to: :, ~eff 2 w o u l d o " '-~3"~ ~arge. ' r "~'e

.~. / ; = " ~ -tea] ,, .~_ " ~"Y ' ) . . :~/(-~--~2i " i.~-~ .00 ' .

uld

We

,

hay

e a B ,

~k..,.

"th a re

eff as given e veloc~t a -_ w n Part of fl- - O y (33b)~is ,,,~,, y of the ~ ~~eg,4tive v:~ .......~--~.J ~s ~ e ~ e , & ~ . ; ~ " ~ and ne~..-

o r f £ ~ I ..,~ "~e P i e ~ o e l e . ~ , : - " b e t ~ e e . ~'nzch C O r r e s ; n ~ ' ~ z c

oho

or w~

. - -,u dra

t~,

h e P~ezoele,,,.,~ ' .,!a i m P l ~ e s of £,I.r irl ~'- ,~.~x'zc. St):in ~ = .

um

....

lYaev Wav~.:, e l f 2 W i l l al~ _ e world e co b e " ~- ...... r a y s c

-means

. ~~, ~. ., , ~

~~ ,-,~, . ,

"/east

-

d~ag to fi~

.

• ,,

In ,

,~~-_..,~~, a~, , ~ ~

,.

..

n

,,~=L:

: ~.i

" °~mila B l e USte~ ..... ,. , .. . .

$.

~

,._~

.... ~ . - ~ - ""y")_ ~,~.,A _ / - " ' ~.-. -~.vver . • -,-ceth '

=~renc/=

.. . . . .. n . - G u l.

~

Yaev Wave,-~.,

. Y

~

., Le~

ee W

f r o.~ m ...

'~eff2 ~ -e

"'.

.

. . .-. . it g,l

. . .

.

. . . ... . . ..e.f.t .that

,~P/(t +

-z

.alue ....

any

.ee

~

-

.the ( r e

.... ..~..~P P I ~ n

-

~

:~' a~

Una

-

ff e

......

,...-,,

3.4.

' ....

~,-...

-,..... ~...~..:..:.~. 0,., I f t h . end a ' mbi~ous.-e-..-. ~.~.,...~.e.a~ce t ." ~i'c-ent m e d i u --" •• r .... n reed. m zS ---- m

cc,,~..-.~..~ ,

~u a/~~-.~

s

.....

2

~. s. . . . .-.: a l d b e e: , n t o . the . • . gain .T h e a b o....v e - ~ e - n t h ~-. l o w, ,wa.~e , ....... . . m a,s . t b e......... ,......... ., .....~ n ~..... ~ ......... . . . . . . . X t r e ,.~: .rnelY . S l o , • .. P z e z o. e , lectr , s •~,.-..-,.In zero. ......• - C a r e t . . . . . . .......ho~, th . more d ,. • . .~ ~ . - f f ¢ ~, .,e d ~ actu..-. . . . . . u,.l ~ a . a . . .. a t e v e n .... . ~ ~e ~ i l m ~ - "~. " ~ f 2 . ~, " ~ , S~der

,. S4'~

~

/(I. ,2 "

- ~,-~-. '

,_:....~eff~... one

, ,u aetermifie

, ..

"

E~ECTR]:C COUPL::NG PROP ER 'r I E ':'it

245

~ e sign o f , ~ ' : " W * lean ~ i y ' i f l n d thiS .expr~ssion b y using again the e l e c t r i c a l ~ u n d a r y conditions, a s e x p r e s s ¢ ~ by (:2). So our ~ s i t i o n is as follows: 1} We have an e x p r e s s i o n (33b) for Ceff, which is not unambiguous, but from

which a neat-looking dispersion relation ,'an be obtained!, by combining (33b) and (2):

where t e f f 2 is the effective d i e l e c t r i c constant of the adj~ cent medium. 2) A~,plication of ~hts d i s p e r s i o n r e l a t i o n ts d a n g e r o u s b e c a u s e it does not tell anything about the actual sign oi kIIi. $) T h e r e f o r e ( 3 4 ) c a n only b e used t t ~ n unambiguous expres sion for kyIIi is a v a i l able and y i e l d s k~IIi < 0, T h i s e x p r e s s i o n is e a s i l y derivec by substituting (33b) for Ceff 1 in (2). We o t t a i n at once

Y

.~xr

Cp + eeff 2 '

anticipated, the ,sign of ~'k~[ is unambiguous. Equation (35) h a s a ~ e a d y been s t a t e d in somewh;Lt different form in reL [2].

i

1:2,!.3. Acoustic surface waves at the surface of a piezoele, iric of unspec'/.fled Symmet~y

I n thiS c a s e o f g e n e r a l g e o m e t r y , an exac~ analytic calculatio,~ o~ %fi in the style of the p r e v i o u s subsection is not practicai~ if o,ae wi~hes. ~¢,_~_-_-,~*,,~'~.~-~. propagation s i n c e in combination with (2) it would give r i s e to a d i s p e r s i o n r e l a tion from which actual (perhaps complex) v - v a l u e s can not be calculated au~lyrically. It h a s , however, been sho~-n by Ingebrigtsen [12] "~:,'~.,:a very sin~p~e small coupling approyAmation can be mane. In this subse,-tion we r e p e a t h:s a r ~ m e n t i n siightty r a m i f i e d f o r m in o r d e r to adapt :Lt to the p r e s e n t conte~-t. We ~ e a t ~ B 1 e u s t e I n - ~ t y : a e v ~vaves s e p a r a t e l y i!a the p r e v i o u s subsection s~nce :I n g e b r ' i ~ e ~ ' s . ~ e a ~ e n t i s n o t ~ i d [13] for ~hat ~ a r t i c u l a r kind of s u r f a c e :Waves. The ::reas~on:for this wilt b e indicated below. ::: We ~ i n c o n s i d e r a pier~oclectric h a l f - s p a c e y > 0. On the (y = 0) surface a wave will be a s s u m e d to p r o p a g a t e in the ~'-direction like b e f e r e w i ~ ~:.~sular frequency ~ I n n p i e z o e l e c t r i c four w a v e s a r e always p o s s i b l e [ 12] at specified ~, ~bx, and ~ , k z being z e r o in the p r e s e n t c a s e . In the limit of z e r o p i e z o e l e c t r i c coupiiu,~ three of t h e s e a x e mechanica, ~ (two p s e u d o - t r a n s v e r s e , one pseudo -longitudiuat~ a ~ :one Is e l e c t r i c a l . In an i s o t r o p t c m e d i u m ~he l a t t e r would be the L t p l a ~ e ~ o ~ e as defined b e f o r e . We shall indicate these four m o d e s with s uper~( r i p , s ~ ~. ~!:~. IV. Accordingly the m e c h a n i c a l boundary ,conditions on the y :: 0 pla~:,,e a r e

,

v. =0, T!xy÷ Tx[Iy + T!~T~ + TIxv

~ v +

÷ Tzv

~v

[

by _III +DyIV ] __k%rk%[el., @Ii'+ (g + ~o, ] % f f I +DH y +#y •

..,~

(37) " !-

'~

a t y =0.

ii

n

..

--'09 where f, g, and h are, possibly complicated, luncttons ~ v. determined by the piezoelectric. Similarly (37) b e c o m e s "-

xr- ~t(9 +~-

<,.. ÷~I~

eeffl.

~/>.

(37a}<:.

as exp:.~esse ~ L': 136a~ determine the'rel~

DI(V)-+ ~eff 1Ds(v)

(sst

= O,

where Dl(V) and D2(v) are, possibly complicated, functions o~ v. The central point of Ingebrigtsen's reasoning i s ~ u appro:~mation of (38)for

~lhich
be, fultyi des.cr~i~:i ,l~.~,~,!~"~!:~~ ~~~..>~

~)

where eel f 2 i s the effective dielectric constant of the adjacent m e , urn. Iu,~t~e:~,,.

eiec~ic ~s~ace~ves

propagating ouil,t h e , i n ~ a c e , b e ~ w e e n

of ~e two substances

.~p+eeff 2=0,

{40),

-

the piezOelectriC m e . u r n in ~zoetectrle..:~cottpll~:,Di and! a n ac:eustic s arfac)e Wave

~ d ~at~tlteir s l o p e s a t that,point should ~have a ra~io

ELECTRIC

2~7

COI~PLIN G P R O P E R T I E S

i O21v}

. a

~lg.2a. Schematic ptot of D 1 (v)~~nd D2(v) , s u function of v in ~ non-piezoelectric medium. ,s equals - Cp. medium.

Slightly differing zeros. Ratio of slopes

~city f o r Rayleigh waves in a piezoelectric me.iium.

~ .... ep , :?:?,,

.....

]

,

:):Thists s c h e m a t i c a l . y :piezoeleC~ic couplir~:

~ , s t : ' a ~ in fig. 2a. We may infer that in c a s e of small ~.~,~ ~ d D2(v) have almost the same z e r o s , which "~e d e -

~0te by t)~ a n d :o rc i: ..cti~el~.

f~r r e a s o n s w h i c h will b e c o m e c l e a r shorLly.

~)bviously v ~ ~ :'~c" l~.~'~.~reover, ~r~ Lhe neighbourhood of these z e r o s Dl(V) :rod ~.~(v~)sho~d ~tilJ ~ ,,~'e slcpe~ ~,ith a :ratio a l m o s t equal to -ep. This s~.tuation is ."i.:!,~e!8:en{~'iby th,~ dra,~na cur~";,:,~ ~:~ fig. 2b. If w e now a s s u m e that l l n e a r : , p p r o x -

~.,f,.~ons ~ D l ( v ) a n d D2(v) can be made in the neighb-ur~ood of the~.r .-~'=poo*~"~ ze: os (broken curves in Dg. 2 b ) ( 3 8 ) takes Lhe form - ~ p ( V - v ~ ) ÷ ¢e~ 1 -v -Vo) = 0 ,

~ :

and eel f 1 is given by

:i~:~v i s in t h e n e l g h b o u r h o o d of v o '~ voo. T h e p h y s i c a l m e a n i n g of t h e s e paran':e-

ters b e c o m e s c l e a r [12] ff we substitute (42) in (2)"

.- q),, '

%V

"V 0

+~eff2 =0.

-~viou,sly v o i s the phase velocity of an acoustic surface wave for teeff21 . . . . . . . . . . . . . . . . . . . .

~ m

~

I,

-

"t~v~

= - ~ ,

~ %~l~,j[..J~g'e~,'E

dtum and v ~ iS the limiting value of the phase velocity for eel f 2 << ~p. if:As to the approximation (42), the following r e m a r k s s e e m ~vpropr]ate: ~!!):Iathe!~case of leaky ,surface w a v e s v, o and v~, are complex. In the folIow~ng ,~'e ::~;i:~i~Sh~:!~u~e: ~LhatVo and v , are real and known from either numerical ¢aic~,tmeasurements. 42) was derived using a line~u" approximation of Ol (v) around v = ~:'~ ann o, u2W) around v = Vo, where for small coupling v o ~ v~. Cons~2uenfiy, ...., ~

if at all, (42) s h o u l d b e a g 0 ~ ~proximation~ fi v l i e s between-Vo a n d ~ ® . . .....: : since in l a t e r sections we shall be constde~l~t~g ;complex values Of V (not c,~ v~o and v o) we have to expand this ' p e r m t R e d ' region of ~-values into tl~e complex plane. The logical thing to do is to take a c i r c l e with radius I v ~ - V o I/2 and centre (v~, +vn)/2. It will b e indicated in 3.3 that v r e m a i n s in this circle iff ~ W t ~ an e x t e r n a l semiconductor, if ~ ~ pDro~mation breaks d o w n for Bleu3) It] steln,_~lyaev:waves. T h e r e a s o n for ~Is is that the linear expansion of/)l(V) around voo!i s i n ~ a l i d in that c a s e . An ~ t e r n a t i v e e z p r e s s i o u for ~eff I which is applicable in th~s c a s e was derived in the pre~,iods subsection and i l l u s t r a t e d in fig. 1 where we have also i n d i c a t e d t h e anal~vgU:ev of Vo a n d v ~. F 0 r c o m p a r i s o n a sketch ot ~ e f f l as g ~ ~ en by (42) is given in fig. 2c. " .... ~ Eq. (42) ~ does n o t reflect, the l o s s y e ~ r a c t e r (Complex f.eff)/:whieh ~the:surf~ce ; 4) should display a t v-values ~¢Lich a r e g r e a t e r than the velocity o f SOund waves in the bulk ~ d where, consequently, volum~ waves ,~n be t r a n s m i t t e d i n ~ ~ e bulk. On the o t h e r hand (42) and thus fig. 2c should b~ qualitatively c o r r e c t for ~' < Vo. Comparison with ~ . I. t h e r e f o r e suggests the pos~ib:e e x i s t e n c e of a ~.~ slow acoustic surface wave in the present geometry.

ic constant

o f non-p~ezoe;~

.tic s e m i c o n d u c t o r s

:nce of fre,e charge carrier's ou c~ff of a non~Plezoele~ -: t r i c m e d i u m w t H b~,. discussed.: Conseq~bntliv: e l e c t r i c t r a n s - ~ r f ~pr@erties will ~~ e n t e r the argument. F o r the sake of conciseness we shall adopt f o r these prope r t i e s a simple theoretical picture, However, it should be cleat- that extensions of the present t r e a t m e n t to m o r e complicated situations a r e p o s s i b l e in principle, though in m~ny c a s e s they may introdace too g r e a t a n r m b e r of (adjustable) parameters. : ~ . . :~ To be specific w e shall consider a semiconductor with one kind of charge car'~,: riers (electrons). In the absence of a magnetic field ~ s h ~ d e s c ~ i b e these w~ h an isotropic scalar mobility ~ and a diffusion co~ficlent D, In the presence of a ~ d.c. magnetic field B inthe z-direction these will be, assumed to,acquire tensor character-

( \ v

' v

~zz/

=

, \ v

v

(43)

~'zz/ 4

where ~xy = ~ ~ xx, Dxv = c~Dxx with c~ ~ ~tB and whe" e ffxx and Dxx will be • roughly p ~ r t i o n a l t ~ (1+ (~'i "I'r ( m a g n e t o r e s i s ~ n c e ) . r r a p p t n g phenomena and a l s o the effect of band bending will be ignored. T h e l a t t e r r e s t r i c t i o n , however, can p a r t i y b e r e l a x e d b e c a u s e in p r i n c i p l e the r~ f e ~ d i e l e c t r i c ~c~nst~ut ~ a ~;; se~0~d~tc'~r '~0Vbrc~ With':a:se~0iaductiug l a y e r e a u ~ e~l~ulat~ with i~eth~d~ simihr to those described ir~ this paper, An' ...... examoie,. .of'~thm: ... .........tec~ique~ ... ~'|=anbe found in ref. [14]. . . . . .

ELECTR:IC:.COUPLI~IG PROPERTIES

~ ,~

249

Mainly in order to c o m m e n t on s o m e pofmts in the literature w e shall firs~ calculate ~eff neglecting diffusion: and subseq,,=n~y p r e s e n t a m o r e complete ~ e a t m e n t in which diffusion is included. In both calculations we i g n o r e the d.c. space charge a c c u m u l a t e d near the surface of the s e m i c o n d u c t o r by the combined action of the e l e c t r i c drift field and the t r a n s v e r s e mag~etic field. Some attention is paid to this effect in the appen-

2~:2.2. O$ff~ion neglected In ac,~0rdance with o u r couvention we c o n s i d e r a s e m i c o n d u c t o r occupying the half-space~,y > 0. A d.c, m a g n e t i c induction B in the z - d i r e c t i o n will be a s s u m e d as well a s a d.c. e l e c f r i c d r i f t f i e l d in the xodirection along which the waves t . be iC~nsider~l propag~.te. We shall again have to use (4) and (5). Instead of (6) we ::::~ust w r i t e : div D ' = -qn 1 ,

(44)

where q is the e l e m e n t a r y c h a r g e , in o r d e r to a l l o ~ for a p e r t u r b a t i o n ~ i on ~:he ~ui~ibriuxn e l e c t r o n d e n s i ~ n o. Evidently n 1 obeys a continuity equation

where tim c u r r e n t density J , in the absence of di fusion, must be written as

'Ix = ¢rxxEx + ¢r~TEy - q n I Vd ,

(46) J y = -¢zxyE x + CrxxEy. Here vd is the drift velocity i m p a r t e d to the e l e c t r o n s by the d.c. e l e c t r i c field. tn (46)ihe conductivity t e n s o r e l e m e n t s o:ij a r e r e l a t e d to the mobility tensor elements of (43) by :

aij=q~ij.

(47'~

A ~ i n u s t n g , ~ e convention (7) we obtain f r o m (4), (5), (44), (45), and (46~ the b u ~ d i s p e r s i o n relation

bY simply e q u a t i n g the s e c u l a r d e t e r m i n a n t to z e r o . The d i e l e c t r i c con~tant has been ~ v e n a Subscript / in o r d e r to indicate that it p e r t a i n s to the c r y s t a l lattice It should be noted that Crxy does not occur in (48). in fact the ,,er,vation c, , show's that it should contain Crxy + Cryx, but this quantity ~.s z e r o iu the ,ure~e~t c a s e , !aS!can b e seen f r o m (47~ and (43). ~ Equation ( 4 8 ) s h o w s that t v o volume modes a r e possible:(1)The: L a p l a c e mode c h a r a c t e r i z e d by k2 + k 2 = C which we nave atreadv e!~.countered. In this mode n 1 = 0 as can be s e e n ~ ¢ combining (4). (5), m:d (44)"

..

..~b

~

, '

.:

i,

'

:L,

"a

+' •

,,ei.

(Oi

~X~

q +

_ xVd) = O,

~ g ~ t,.hat i~.ei,s.~c ~ charge mode (50) cannot be of ',tm~: portauce;Iv~.thepresent: co~ie~,R,beca~seit does not satisiy the boundaxy co'ndl" .... '.................. " it d ~ r s from all w a v e s hRherto:,.consldered in thls ':

' ....

' ........ '~: "

"

e

.,,

e "

k Rri~t h , , ~

~It w h l be'~shown, in , ky".-.enters~,itito~.,:(50); .,..,,,

.s~ ~.. a

Eat

as id.i as #y is atiowed to

adjust ItseR,;~.~d. t h e r e f o r e b u ~ :m~!es llke (50). c~n. indeed sat!s{y me: ......-, .~:,,. • ecnditions ~imp0sedby a surface .w~ve Phenomenon. .!n~..~t,./i~t,;WHI become"~i.cI~.:at the ehd ~ol 2"S.-.30~at" the sUrlace eharge int.r~C~~:in ' " r e f . [3].i~(and.:~.~!;.:used.iath¢.fp!!,owing di.scnssioa) to replace the, mqde~ (50,) e m e r g e s quite naturaiiy fr0m the~lculation~s i f o n e fi: _.. includesdi~fUSiov~nd subsequently takes the limR of zero diffusion c0efficie, :~. AS fm,~:,as .~e bulk is concerned, Wethus only W e to c~ns~der the Laplace m~e-,..in which we again have " •t k , . , 1

m)

mriace into the ~H-sl.~_ce .;; > 0 iS to be a~oided,,With the aid of (9)we: write d?wn an expression for J y , to be used below in connection with l:~oundary conditionL~. Equations (46.), (B), and (9) yiehl:

Usually the boundary con~tion to. be imposed is that .no current shall flow through the surface: J y = 0 at O. I n o r d e r to ~ s f y this boun~y~coudition~one needs ~ a t l e a s t ~vo m o d e s i n s t e a d o f only the L a p ~ c e m ~ e WhichWe are,considering ~i. ~o-,',~."~.-~~o we w e ~have t .h. . .e. r i , g 'ht ..,.. : .......... .. . .i~ore.d'f~,~, . . . . . moa,, (5 0 ) ~ l a i a e .d ~ o w , :~ ...... .... to in~. troduc.e ~ma l t e r ~ t i v e .he~e,~:Accordingly we a s s ~ e , :{oliowlngr~, .[3],/a sur~ face c,harge densRy '-qns ~(with-~mensl0n ~ , ' z } . w e ~.liso a s s m e this surface charge to drift with a drift velocity u d which gives ris,.~ t~,)a sUrfa:ce current den. sity in the x-direction r

. . . .

"

J~S = - q ' ~ s V d

: .

i'

. ,

!

',~:

i: : : ~

-

(with ~iImenSi.~ Am "1).

;

.

"

. ..... . . . . . . . : . . :

.~

~i:::.,~:

Note. that tt i s not . . . . drift Velocity m'~dequa~sl..the: b u .". lk drHt velocity Vd, which occurred in (50). The ~all treatment in 2.2.3 :~11 ,show

ELECTRO C ~ P L I N G PRoPER:tIES

,:,::.<.

251

'tO +,::,

~:.!~at v d = v d (thin c o m e s about: b e c a u s e we n e g l e c t trapping p h e n o m e n a in the :surface). Obviously, a s w a s staten in ref. [3], n s and 3s murat obey a continuity equation

~.s :

•

J

= J

e n s Vd - J y ,

::~here dy is W e n by: ~5~i),,: ThlS~yields the r e l a t i o n between sur:~ace charge d e n s i : ~]aud ~ttal w : h i c h : w e u e ~ in o r d e r to w r i t e dram an e x p r e s s i o n for ~eff"

"%~X " 7 7

:

"qns='

"

"

, "

.kx r "

•

~o .

.

.

.

.

.

(52)

•

The final s t e p ,now is to say t ~ t Dy just outside the s e m i c o n d u c t o r and : tnsidethei:sem~.couductor d i f f e r by an a m o u n t q~:s:

Dy

just

iTogether with (52) and ( 9 ) t h i s yields

1 ¢off "= "~xr kX~ y=-9

~x'x

-je~

kxr J

(53)

J(~ -kxVd)

Equation (53) has s o m e i m p o r t a n t f e a t u r e s which will be p a r t l y obscured Later on by the introduction of diffusion, but r e m a i n of interest. They carl be most ~ s i l y . d i s c t t s s e d , a t this stage. ~:~The conduct/vity: of t h e m e d i u m contributes a complex (and t h e r e f o r e non ~'eactive) t e r m tO Cell, which changes sign ff ¢~ = kxv d. Thzs t e r m for instance causes ~ e absorption and the amplification [5] of acous, ttc s u r f a c e waves Ut mter~ci~ ~ L h a s e m l c o n d u c h n g ad)acent m e d m m (see 3.3). 3) If a ~ t r ~ V e ~ S ~ ' magnetic fieId iS p r e s e n t (a , 0 ) the conductivity, also c o n t r i b utes a r e a l t e r m to e e l f. T h e s i g n of this t e r m depends on • - kxvd and on a,. F o r l a r g e a - ~ a l u e s aria f 0 r s m a l l values of ~, - kxV~ it can d o m i n a t e the right. hand i%ember Of (53). It should thus be p o s s i b l e to obtain e s s e n t i a l l y negative values f o r Ceil' T h e s e a r e expected to p r o d u c e exotic b e a ~ v i o u r of acoustic T

4

,e

Neg~afive contributions to the r e a l p a r t of e e l f as d e s c r i b e d by (5~) also result in e n h a n c e m e n t of the interaction between o r d i n a r y acoustic ~ r f a C e ' w a v e s and a s e m i c o n d u c t o r [ 6 - 9 ] a s w~ll be d i s c u s s e d in 3.3. The physlcal r e a s o n for this effect, which is typicaI of s u r f a c e w a v e s . ~ecomes :..... clear f r o m the fact that the field lines in the L a p l a c e mode a r e curved ~ such a way (fig. 3) that a (spatial) Hall angle a r c t a n ~ between fieids and c u r r e n t s r e s u l t s in a p h a s e angle in t i m e between t h e s e quay_titles (in fig. 3 we r e p r e -

v d = vd •

,,~,,,

This shows that in ref. [3] the s u r f a c e -ll~:~eecl ~ . a s s u m e d t o d r ~ wi~h the bulk dri.~t velocity v d. F u r t h e r m o r e , i t c a n be s e e n f r o m (62) to~ether with (~6) that we s t i l l have' crx x . ~ ~o

It may-be useful to note that Irk'the.case of real k x, no. magnetic fie!d :(a=O), and no dr~t veloc~+..(._v, d --.;0);(6~)!~,¢o~resp0., nds,tO the a dmlttanc~, of .the,cl,reb~t ~f..,.,s,fig.."4, Whex'~e .... 'per

unR a r e a ,

"

per.unlt a r e a ,

C~! =..e,Ikr]

e e l f a-s g~ven by (62) ta not an a n a l y t i c function of k x.

.........

.

; ! .i:>4t

F r o m a p r a c t i c a l point of v , e w ~ R , i s n e c . e s s a r y t o ev:t,: -, ... t h e c h a n g e in,the ef~ f e e t ] r e d i e l e c t r i c c o n s t a n t of a h ~ - S n a c ~ _ if ~ d { ~ : , , : - - , . : ].aver i s s i t u a t e d 0 n top e " r hieh is v e r y thin. com. ch :,~aybe prac,ticaHy in-

[fects which One would .exis especially of::~importance if one wants to study the case of pal,,e y.electric interaction between ~m ace~lstic surface wave ,rod an adjacent semiconductor, where mechanical contact must be avoided [5]. The equation to be constructed has already been given in ref. [12] in slightly d i f f e r e n t f o r m . W e consicler~.-,adielec~ic:~sot~opic ~ er-witl),dielectric ~ ,,'on s t a n t cd,, o c c u p y ing the re~o~i'.~0< y < ~..For y<->>.fd,i~.,a,sSume.i a s.ul~s.tances,~itheffe.ctlvb:di~u,iec-, .,,~ tric consent e e l s . .in the laye~,iWe,i~an:,~!~.~e.,the ~plac,~ m~e-; :iibutsix~cew e : bare two boundaries~n0w;-We-0 ~ , to .use b ~ i possible val,.;.esof ky at fgiven kx., and accordin tWo"~"~ ' ' ' . g IY w, e/ ~~have :~ .... ! ~ .en~Lals ""

,.

~

"

._(~I ~ [ j ( f d O t :

tlt

'--"

"

J"

""

"" ~ a

'

] ~'~"

'Y

~. . . . . . .

" $

'

r'

'

~XrX) .:'~i)Xy] )

So at x = d.the total-..potenti~,is ~#(a) = ( ~ o i e - , - x a + ~o" e + ' ~ x ~ e ~ [ ) ( e t : ' k X x ) ] a n d the y - c o m p o n e n t 0 f t h e d i e l e c t r i c d~splacernen't i s

kxx)] , B o u n d a r y c o n d i ~ o n s of c o u r s e d e m a n d that a t y .

~ ' ~ x r k~:

= '~eff,

d we h a v e

ELECTRIC' COUPLING :~'ROPERTIES

255

aild SO the ratl0 of #~I and ~Iol isflxed" elo( 6 d/~'XT " ¢ eff I k.xT I)e "lexd = <#loI(~:dkxr + Ceff

l~,~-rl)@x d.

(63~

A t y = O w e b~vo

¢(0)

=

' I

tI

~...+ ~,o.,

-Job-

y(m

From (<%) the effective d i e l e c t r i c constant Ceff of the whole s y s t e m can at once be obtained:

~d

af

: 6(0)

kxr kx

-

[kxr I tgh ~x d ~xr

+

~eff

]@xr] d + ~ eft - - k ~ r tgh ~x d

where of Course .Y.

i s independent of the sign of k x. Note that i~ a c c o r d a n c e with e x p e c t a t i o n !

lira eeff = % i f , d-~0 $

lira

C e f f = '~d"

3. DISPERSION ~ L A T I O N S

~.1. Propagation of charge' carrier waves along the surface of,: sem~.+(.~%:ctc./ Both f o r the s a k e of c o m p l e t e n e s s as well a s by way of e×~m~:le we ~-" ~qv,~ , consider in t h i s s e c t i o n the w a v e s ,tescr~.bed in re£s. [3] and [4]. The c a s e under c , m s i d e r a t i o n is that of a s e m i c o n d u c t o r .conzaining holes as we11 as ~ e c t . r - =~, which h a s an i n t e r f a c e with v a c u u m . A p a r t f r o m diffusion, recom!+mation ",v,~s also n e g l e c t e d in r e f s . [3] a n d [4[. T h i s a l i o w s one to r e g a r d the ~ ,ntributions o~ e l e c t r o n s aria h o l e s to e e f f a s additive. We c o r r e s p o n d i n g l y obtain the dispers:~o,,~ an additional t e r m to t a k e ~c,~ount of h o l e s , and for Ceffz fl~e d~,:~ectr,c cous~a,,,~t ~.... , a t t is % of v a c u u m . The ."~-

]

__7

kxr - eO

+ el +

j(¢o _ k r , , d n )

.

"

k:rr _I

.l-

•

j(~, _ k~Vdw~

-0,

168~

where electron- and hoie contributxons have been denoted by subscripts n and p

~.~

. • •

•~ ~ .

It should.be~noted tlmta similar, equation:e~s~,:
iAsdisct/~sed at length,,in.vefs,[3].:~nd[4],equation(68) allows of Strongly amp!ffying.waves~urlder-cer~,in .~:ondtti0ns,"Since this a m p l i f i c a t i o n h a s b e e n ob.s e ~ v e d ~ e r i m e n t a ! ! y [4], .one<~ma:~ conclude that it may: o c c u r als,-: in ~e. pres.. e n c e of diffusion and. recombi~tation. 3.2. On the ~ t e r a c t i ~ v o)" two ,~coustic " ' surfac.e waves in two adjacent but meVh~nicaZly f r e e , identic.#t p i e z o e l e c t r i c media The p h e n o m e n o n . , ~ C o ~ p h n g a e o u S t i c e n e r g y f r o m one piezoel e t h i c medium a~r.oss ~n a ! r g a p into a s i m i ! a r m e d t u m . h a S . ! ~ n d e m o n s t r a t e d e x p ~ r i m e n t a U y ! 11]. We shall now s h o w t ~ t in the: :case ~:ofRaY!eig,h wave s thi s phenomenon can be quite well d e s c r i b e d with the Rid~!0f t U g e b r i g t s e n ' s apl~ o:,.tmation f o r e eff given by (42), white the interaction of Bleusteir~-Gulyaev wave~.. ~hv~:id be c o r r e c t l y . de:sctlb~:~iby(,S~b~, ~....." "• ~:

Write do~.the gene~l form. of the dispersion,relP,e,onfor two:tdenfi~ cal m~ia,: ~-ep~ratedby a ~electric ~yer of thlclcness d --,:~.,an airgap).~Equa" r~on,(2) canbe used-here I~iw e : ~ e thetW0 pertine~,tt-,~.-spacesto be tW0Ide~t~[ layer, ~,",t t the aid of (~iS)we • we_fir!st

• •~#r ~ ~: e f f + e d

] kxr] Cd + ~

Th~s

= O.

(69)

c eff tgh kx d

equation i s of second o r d e r in eeff, the solution being m

'~eff = ~ d

~

1

a

'

!kxr I ,

,

.

"

'

(70)

If •Ee~f : ~'. i. s. .r.e. ~ a t. . . . r e a t values of k, whicl~ ~s the. c a s e ~n ( 4 2 ) a n d a l s o in :(33b), Men w e - s e e from (70): that two adjacent identical m e d i a s e p a r a t e d by a d i e l e c t r i c layer m e •case of ~ y l e i g h ~ a v e s the:res~Ring~ p h a s e velocities a r ~ situated in the region vo < v < Voo, w h e r e (4"~) can be t r u s t e d if tLe pLe~oelectri~ ccuplhlg is s m a l l , white in the case of B i e u s t e i n - G u ! y a e v w~.ves ~e do not get ~ n~z-~ 'ire penetraUon depth;, w h i c h m e a n s that tn the~limit of t h i c k sam~4es (33b'~ ca: 0e used. .... ~ The following.remarker muse ba:~made:. 1) It shou!id be noted f r o m (70):that in:-the timit d ~ ~ t h e Valu~.~s of eef f f o r which propagation t~ces p l a c e both a r e lir~ ¢0:ff = -ed, a s expech.~l. quenuy,m

.~(:

F.,L,ECT~IC C O ~ L I N G PROPERTIES

257

!:~'~!In ihe :limit d -~ 0 the v a l u e s O f f e r f for w~ich p r o p a g a t i o n t a k e s p l a c e b e c o m e i.e. d--,O

d-~0

lira v = v ~ d-,O

and

(7 ~) 2

lira d-~O

d'~O

[kXr{

'

= -~.

i.e.

l i m v = vo .

d-~O

~3) Of :c6urse: ~ e ~ 0 p6ssieii~ttes allowed by the ~ sign in (70) a r e solati:~ns ~f o p posite s y m m e t r y , allowing t o t a l e n e r g y t x a n s f e r between the two media. ~[:~4)The coupling b e t w e e n d i f f e r e n t media (or i d e n t i c a l media of d i f f e r e n t o r i e n t a tion) r ~ u t r e s a m o r e : ~ f u l ~ r e a t m e n t b e c a u s e it may turn out that p r o p a g a tion t'akes p l a c e at v - v a l u e s w h e r e (42) or (33b) cannot be t r u s t e d . V'or instance, the l o s s of e n e r g y of a (fast) Rayleigh wave in one m e d i u m to a (slow) acouStle bulk wave in a n o t h e r medium cannot be d e s c r i b e d with the use of e x pre~iiidns lille :~:(42)~fo~ B b ~ : m edia: 3.3, Interaction between R a y l e i g h waves in a p i e z e e l e c t r i c and electrons i~'. a

semiconductor ;n d e m o n s t r a t e d s o m e t i m e a gx~ [5] and field on t h i s -)henomenon has been discussed [6, 7] and e ~ r i m e n t a l ! y ~' .,~served -[8, 9]. In t h i s , s e c t i o n we shall c o n s i d e r s o m e t h e o r e t i c a l a s p e c t s of ~his phenom ~. e,a, !an the b a s i s of (42). To this end we s u b s t i t u t e (42) for Ceff I in (2), while for ~eff 2 we s h a l l u s e for the t i m e being an u n s p e c i f i e d quantity es denoting v.o:: e relevant e x p r e s s i o n , such a s (5~i) o r (62) for the effective diei.ec~ric cons~am of a ~ e m i c o n d u c t o r , which again m a y be c o v e r e d by a d i e l e c t r i c k, y e r . Wc ~ . . . . rain V - Voo

e P v_- v o +.....

ES=

0

or[x ] •

(¢S+e.p)(V-V o) =Ep(Voo-vo).

(72)

Note that (72) h a s the f o r m

k)

k) =

k),

(73

which often e n s u e s ff two w a v e s with disper.,don relati~:~s = o,

g(w,

= o

are subjected to a finite co~.~pling 6(w,k)[*]. At this s t a g e we can c o n v e n i e n t l y d i s c u s s the distinction be~,weea r~sonan:: and n o n - r e s o n a n t a m p l i f i c a t i o n which has been raadc "~:: . . .:. ~.. [~..,, ~ s s u ~ i n g smv.li .... . o~;piing, ~i . e . , s m a l l 'values of 8(¢o, k). ii (73) is s a t i s f i e d for 6 = 0 at a n ~ l a r f r e iquency ~oo by a w a v e n u m b e r ho, the wave n u m b e r k l which sat:sfi~ v ~'v?: t!or ~:¢ 0 at: the s a m e a n g u l a r f r e q u e n c y wo can be obtain~.,d f r o m an e,~p~rsion of :(:73~ba r o u n d hO: [*] % + ~p = 0 is of the form (2).

.

.

.

.

~ ~,~:

.

t

+/(%, ko)g~(~o, ~o)1 = ~('°o,~o1::~,,~

..

~,

.

. . . . .

:,

,...,,.~t~

.

,

~

.

,

. , ,

.

.

.

.

.

.

.

(~4)

.

in:~b unc0upied ,case We::in~:ust ~ T e e i t h e r We mayassume without l o s s of. g e n e r N i t y

Sine f(~o(

)

.... •.

.

,, ,

. ~-

....

,~:,

,

~, .:

-

. :

Since t~e left~no,:me~er::i,~fi(751 cRses depending on which of ~ese note •:the cases aS, !'resonant,, ~ d a) R e s o n ~ t , : i n t e r a c t i o n . :

.......

~o) z(%,~o;], ~(~o, ~). (~a) .....,

'i4

'

,~qn ~ r e s ~ t " .

~....... , . : , ,

.....

•

the

Ib~ .iL-'.:::':.

'

'-_.~e P ~ S e

velocity ts clom~ant,~nd: conse......

:. . . . . .

(76)

The condition for 8(~Jo,~o) under which (76) is wdld is derived by demanding that substitution of (76) into the left.-hand m e m b e r of (75) yields indeed a dominant second term. The result is

[f~(~o, ~.o) z(~o, ko)] ~ T h i s condition,

~s anticipate,

i s f u l f i l l e d if g(COo, k o) is::small, e n o u g h .

interuction

f (r~k)=O glLo,k|=O

Fig. ,~. Sche~ttc:: : : ' ~:~ t l lou:~sf t "r~:~:r'~ ~ :o n'il: !;~:;:;i::~":resomnlt ~:~'~ d : : ;:'..... ::~':~ ' n o n . r e s o"n a~n t '~ ' ":interaction . . . . . . . . . . bet"ween, .... ':~ two:' . . . waves, . :: ' ' which ff uncoupled have dispersion relationsf(tO, k) ~Oand g ( ~ , k)= 0 (drawn curves)and which if coupled show the behavtour indicated by the. broken curves.

-

i!b.)N o n - r e s o n a n t : i n t e r a c t i o n ~-We ex~,c~-this, i n t e r a c t i o n to take place if (Cao,ko) is safficiently far away from any point in ( ~ s p a c e w h e r e DOth uncoupled waves have the s a m e phase ~l,~ci~, i , e . , whereg(a~o,b,o) i s l a r g e enough and where a c c o r d i n g l y the f i r s t term in the l e f t - h a n d m e m b e r of ("/5) dominates. Consequently we have here

~ d the condltion:for 8(w o, ko) under which this type of interaction takes place is ~sily. seen: to,.be. 2

1

"~:

I[~t~hould i ~ : n o ~ t l i a t ................

(78)

in the r e s o n a n t c a s e {k1 -~o) is p r o p o r t i o n a l ~o e-~ 2, w~ile )rtional to ~. F o r s m a l l { i - ~ l u e s 51/:'->5 t e n d s to produce g r e a t e r deviations ot :i~'. : T h i s is i l l u s t r a t e d s c h e m a t i c a l l y in {}:and g ( w , k ) = 0 a r e both satisfied at -

,

:~:~'A c o n c l u ~ i o n ~ t ~ e : ~ w n

f r o m {~8) iS that the interaction can b e c o m e non~isonant:~ the coupling i s m a d e s m a l l enough, a p a r t from points in (a~,k)-sl~ce : ~ r e : b O ~ : ] ( ¢ ; ~ ) ~ d g ~ , k ) a r e equal to zero. : ~ t e r this d i g r e s s i o n of a g e n e r a l nature we retur,~ t~ ~ e c a s e in hand, i.e.. the interaction b~tween a R a y l e i g h wave and a sem~,.:o~,ductor as d e s c r i b e d bv ~2). FOr the s a k e o f E ~ I t y w e h e r e substitute for ¢s the e x p r e s s i o n (53). Xi~e thus neglect both the influence of a finite a i r g a p and diffusion. The result is r

lkxrl]

(79) ~--f(t~,kX)-~ * ........g(W,k.V)

"

-~+-- O(W,kx)- "

Since w e are considering the irdluence of the semiconductor on the propagation of Rayleigh w a v e s we mus{ i d e n ~ y ] ( u ' , k x) with ~/le x - v o, a s shown in (79), w h e r e g ( w , k x ) and 8 ( ~ , k x) b a r e alSo-been indicated. It is at once c i e a r fr,~m (79) that ](W, bx) = 0 i m p l i e s r e a l k x at r e a l w, while g ( ~ , k x) = 0 imp!~es complex k x. at real a~ u n l e s s we have l a r g e m a g n e t i c fields, i . e . , large ~ - v a l ~ e s . %'e thus exnpe.~:t t o find no i n t e r a c t i o n of the r e s o n a n t type unless ~ is iar.~e ep.ough. F r o m our 2 r e :~ious r e s e t s it should be c l e a r that :..the:magnitude which c~ n~cst have in o r d e r Io :ii$~ a t r a n s i t i o n f r o m r e s o r t e r to n o n - r e s o n a n t will be the g r e a t e r t~,e s m a l l e r :v,¢:- v o is, i . e . , the s m a l l e r the coupling is. Consequent:ty ~ <'- I is i:~ itself aol .L ~ e c e s s a r y condition for n o n - r e s o n a n t i n t e r a c t i o n , as might be inferx ekd from ~ef,, [?]..

.:~C~:

,

• .~,A...~,

j,,ORIIi~g et ~.

One important f e ~ t u r e o f (~9)-remains to be discussed, We b a ~ H e n in 2.1 that (4~), on which (72) was based, m a y n~t ~ be ~truet~o~,y ~ t s i d e the' circle in the complex v-plane with radius l~,e - Vo]/2 and c e n t r e ( t , , e + v ) / S "t (:an be shown from (79) that for a ,.~ I tho (complex) sound velocity does not vie,. late thls,xestrlctton for any value of the semiconductor parameters. S ~ c ~ diffusion and. a finite airIrap oz~ly reduce the excursions of v Into the complex v-plane, a s will be. intui~vely clear, we ~ that it makes s e n s e to d e s c r i b e the Interaction beb~eeti Raylelgh waves and a s e m i c o n d u e t o r ' w i ~ l ~ i g e b r t g t s e ~ ' s ~pproxIu~ation, We emphasize, however, that for strong magnetic fields the ~t~-cn~h of the i n t e r a c t i o n i n c r e a s e s iS]., whtch may [ n ~ l i d a t e (4~!). The influence of a t r a n s v e r s e magnetic field on the interaction beLvceen ~tyieI~h waves and a semiconductor was experimentally studied by one of the authors [8] ~ -tW.L The theoretical curves a t w e l l . a s the fact that for aI1 drift field,vah:.e,s the interact~0n.was non-res0nant w e r e ,established w l t h t h e ~tuatlons presented in this l~Per. In t h e expei,!m~nt 0~ r ~ . [g]'~he d.~. ~h~r,T,~ tn t h e sur"* | face conP,entratlon of the e~.ectrm s in the semiconductor, which is caused by the combined effectoft he electric drift fisld and the m~,E , ~ c field, did not satisfy the condition fozmulated in the appendix at drift velo,.',:~ ~.~ch significantly exc e ~ i ~ e l , ye!~itty of: sound' As: ~ i a ~ n e d in the apl~mdix tt is difficalt to n t t mate~t0:~t e~nt.~ w/oudd i n ~ l i d a t e the p r e s e n t theol'y~.

3.4. ~ n t ~ ~ n

bet~e~.~ BleustePn-~lyaev

wo~e~

"~. ~ ~

semiconductor

~~;'~'~ ~ : ~~'~ - .~". -~, ~" ~..... !

' t e i r t '--Gulyaev wavt u ' ~,,.u" a semiconductor should ~: .... ~ Bleus !'i~~:~rib~by substituting (33b) and (53) or (6~) i~ (2). This specifies (3,1). This equation, however, can only be used, as discussed in 2.1,2, under the condition that the imaginary p,trL of the y-component of the acoustic wave vector as given by (35) is negative. Fi[~res 6 and 7 provide ~ illustration of this difficulty.

1

~

I! ~.

1

-1

Fig. 6. Amplification of Bleustein-Gulyaev wave as a function of drift velocity as calculated for a pi,.~zoelectrich~Af space from (34) for several values of cz = /~B.

E L E C T R I C COUPLING P R O P E R T I E S

261

In fig. 6 the. i m a g i n a r y p a r t of k,v as o b t a i n e d for B l e u s t e i n - G u l y a e v waves from ( 2 ) a f t e r substitution of (33b) and (62) is plotted v e r s u s d r i f t v e l ) c l t y for various v a l u e s o[ a d.c. tran~t,e,~,~ .i,~g,ie~ic f'......~'-' ~"~,,,. . . . ..~., . . ,--'----'~,~,. . . . . . . . . . --~ta~es ~sed perta, tn to ! 5 0 ~?em n - t y p e Si o~ PXE 5 "p' i e ,- ~ceramtc, at a f r e q u e n c y of 5 N;Hz. We see :~[rom the figure ~luL[ i~ ,t ~v~iu. o£ dr.~.t t.ieki--vames w h e r e wi~mout m a g attic field: w e h a v e a m p l i f i c a t i o n , t h e r e is " a b s o r p t i o n " at s u f f i c i e n t l y s t r o n g magnetic f i e l d s , What r e a l l y h a p p e n s in this r e g i o n is that e n e r g y is ~still Zed from the s ~ m i c o n d l m t o r into t h e p i e z o e l e c t r i c but that at the s a m :; tin::e the ~n~ry c o n d i t i o n s can only b e s a t i s f i e d if the B l e u s t e i n - G u l y a e v wa'~e r a d i a t e s e n e r ~ i n t o the b u ~ , in o t h e r w o r d s it h a s b e c o m e "leaky"° In f a c t m c r e e n e r g y is radiated into the bulk of t h e p i e z o e l e c t r i c than the s e m i c o n d u c t o r can p r o v i d e and decay ,letup" the s u r f a c e o c c u r s . F i g u r e 6 sh~ws as well that the r e v e r s e phenomenon a l s o e x i s t s at d r i f t v e l o c i t i e s and m a g n e t i c f!e!ds of opposite sign. Here we o b t a i n f r o m the c a l c u l a t i o n a,~ " a n t i - l e a k y " wave, which of c o u r s e is o17 questionable p h y s i c a l r e a l i t y . T h e c o m p l i c a t i o - ~ I l l u s t r a t e d in fig. 6 a r e . of course, r e m o v e d in the c a s e of a p i e z o e l e c t r i c plate of finite t h i c k n e s s , p r o v i d e d on one f a c e with a s e m i c o n d u c t o r . E n e r g y r a d i a t e d into the m e d i u m hs thep r e [tectod at the opposite f a c e el '.he plate. The r e s u l t s of a c a l c u l a t i o n [10] whici~ does not d U t e r f r o m B l e u s t e i n ' s [17], a p a r t f r o m the fact that (62) was used. for • e d i e l e c t r i c c o n s t a n t on one f a c e of the p l a t e while at the o t h e r face ~ = e o was assumed, a r e shown in fig. 7 f o r the f u n d a m e n t a l [17] mode. In t h i s f i g u r e , for which the s a m e p a r a m e t e r walues w e r e used a s for fig. 6, and w h e r e the t h i c k ness of the P ~ 5 plate w a s t a k e n to be 0.2 m m , we s e e indeed that the " l e a k y " and " a n t i - l e a k y " p h e n o m e n a h a v e vanished. °~-

I |¢;' ,;.: ...........

---~-va_- ~E.

J.

-1

Fig. 7. Same as fig. 6, c a l c u l a t e d f o r p i e z o e l e c t r i c plate of finite t h i c k n e s s (tundament~',! mode), The units i n d i c a t e d on the a x e s a r e the s a m e as in fit; C

In 2.1.2 a n o t h e r phenome~btZ ~Was: antieilmted~ ::,:the~,~0ssible.. e x i s t e n c e , of a slow variety of the BleuE:tein-t~ulyaev.Wave:. which could"~come,'about byai~propriate negative values of eeff of t he adjacent medium. I n this p a p e r ,w'e only give ;m idealized p i c t u r e of a posSible'way to o b s e r v e this wave, in o r d e r to illustrate:, also the flnRe thicknes$ ~ the ; te2;oeteetrie:,is~::rtakeuAuto~account, is to :be Pub lished,:!btse~her~ [IO],: ~.~,~:.: . . . . . ~ . ' .. •. . . . . .., H e r e we shall~ use i n s t e a d : , o f ~ ($2),~a~ ;~tmplffied version of ( 5 3 ) , assuming a , semlconductorwhlchls s~flCienfly conductive towarrant the neglect of e I and :~' withlsuffleiently large mobility to,warrant the ~ssump, lon c~ = .aB ~> 1. Under thez, e:e,)ndittons:~ (53): ~ e s t h e fo.:m I

if we a s s u m e v d = 0. Because of m a g n e t o r e s i s t a n c e a x x is roughly p r o p t :~"onal to (1 + a2) "1, and t h e r e f o r e we w r i t e for a >> 1: Eel f =

(8o):

OfcO~xr

wh

the absence of a ma::a~ ~ field. If v,,e substitute

"

F °Ik'r!12 L .

- ~

1 - 7 4 L[eP - c~.~x r a °tkxwt]2

v, hile at the s a m e time (35) yields

(s2) Cpa a~.bv r

It should be noted that in contradistinction to the ease of Rayleigh waves, !

,

Ji

Fi,g. 8. S c h e m a t i c plo/. of exlm~eted dispe rsion c u r v e of a Bleu~,~ein-Gulyaev wave which is

rendered "slow" by semiconductor subjected to strong transverse magnetic field,

::

ELECTRIC COtrPLING PROPERTIES

263

where we had to use a s m a l l coupling a p p r o x i m a t i o n , the validRy of (~.l) and (82) is not r e s t r i c t e d to s m a l l v a l u e s of ~2 F u r t h e r m o r e it o:~c:-.,, "?e ,nen,~onec] th.~t (81) cannot be w r i t t e n like {73). I~t is of the m o r e complicated , o r m tg,~ : 5. We d i s c u s s (81) and (82) by w r i t i n g t h e s e equations as .

'~

(81a) and

II = +~,2

]kxr]

~.

(82~.)

with

akxr~

Q

Let us f i r s t : c o n s i d e r :negative values of ,~kxr, which m e a n s that at fixed m a g netic field we r e s t r i c t o u r s e l v e s to p r o p a n e : i o n along the x - a x i s in a p a r t i c u l a r direction, e . g . , the negative d i r e c t i o n (krr~ 0,. ~ >0). C l e a r l y we have negative ~-values now and so we s e e f r o m (82a) t{hat kI~i r e m a i n s n e g a t i v e . Accordingly (8~a) can be u s e d and s h o ~ s us that the phas~ ;¢elocity w/k cannot deviate m o r e than a f r a c t i o n of the o r d e r of ~,4 f r o m the v e l o c i t y of sound. ~ h i s is s c h e m a t i cally i l l u s t r a t e d in the l e f t - h a n d p a r t of fig. 8 by t ~ ~;tr=ight ~,ne Which is mc~,~t to indicate approximate';y .4~.,--~-¢""~"n'°~¢--*~'-'~-.~-propaga~ ,a. For p o s i t i v e values ef ~ k x r , i.e., for p r o p a g a t i o n in the c,p p c s i t e direction, if rema~ms fixed, we have p o s i t i v e ~ arid a c c o r d i n g to (81a! the pha:;e vel(~cit~- ~, :~'~ varies f r o m c-(1 _,/4)/p to z e r o if 77 v a r i e s f r o m infinity to C p ( 1 -~ 2), :,~-!:.~i~ : : ~ responds to a v a r i a t i o n of w f r o m z e r o to 1 - .t2 ~ o l k x r I Cp '

~ ;exr

This is i l l u s t r a t e d in the rJ.ght-hand p a r t of fig. 8. We m a y a s s i g n physical real-ity to this d r a s t i c change of the phase velocity sin,:-e a c c o r d i n g to (82a) kt.!i r e mains n e g a t i v e and even a c q u i r e s an infinRely l a r g e absolute value at = e p / ( 1 ~,2), which c o r r e s p o n d s to an i~ffiniteY~y small p e n e t r a t i o n dept;~. If 77 d e c r e a s e s still f u r t h e r (i.e. if w i n c r e a s e s still f u r t h e r ) we have a c cording to (81a) i m a g i n a r y k x (cut-off) until 77 =tip~(1 +V2). Subsec, uent!y, i.e. for values of ~ which a r e s m a l l e r th~.n c p /,( 1 + Vz): (82a~ i n d i c a t e s L~osi~'e ~,~ . which i n v a l i d a t e s (81a) and (81). Corr".spondingly the r i g h t - h a n d part .)f ~.~,~,~ may not be extended to l a r g e r ~ - v a t ~ e s u n l e s s one takes i~to acc,~unt the i~n~t~. ~ c k u e s s of the s a m p l e [ 10]. We thus obtained a d i s p e r z i o n c u r v e (fig. 8) which at fixed m ~ e t t c i:~eid :~ ditferent f o r w a v e s t r a v e l l i n g to the r,.ght a s c o m p a r e d with w a v e s trave!lin~: ~,. the left, D e p e r d i n g on the s i r a of ~ o r e wave is c o n s i d e r a b l y slowed down and the other p r o p a g a t e s n o r m a l l y . ~'he question w h e t h e r the slow ~ v e actually can be seen is a s u b j e c t of f u r t h e r i:,vestigation. F o r w/kx-~O it b e c o m e s purciy e~ec~'~c

:+++~:

.:C, A+.~A+J, ORgI~BE et aL O F T H E CONCF,PT OF-~ef f T O T H g g X C ~ A T I O N P R O B L ~

4. R I ¢ ~ V A N C g

In .t ~ls p a r t we d i s c u s s the excitation of (at l e a s t partllr) e l e c t r i c s u r f a c e waves by m e a n s of a prune e l e c t r o d e s t r u c t u r e on the reb~,,ant s u r f a c e . The t r e a t m e n t m u s t net be re rrarded a s a full t h e o r y , which can s e r v e to perforn~ pra,~t!cal c a l c u l a t i o n s , but a s a m e a n q to d i s c u s s e x i s t i n g t r e a t m e n ~ s . The tea sontng should b e valid f o r a l l wave~ c o n s i d e r e d in thts p a p e r , in p a r t i c u l a r for Rayleigh w a v e s , which have a l r e a d y r e c e i v e d m u c h attention in t h i s r e s p e c t [12.18-211. We again c o n s i d e r a h a l f - s p a c e 3' > 0, in which we have a m a t e r i a l , c h a r a c t e r i z e d by a c e r t a i n e e f f ( ~ , k), in which all r e l e v a n t p r o p e r t i e s of the m a t e r i a l a r e included. On this s u r f a c e we a s s u m e an i n t e r d i g i t a l t r a n s d u c e r to be present. With an a.c. v o l t a g e g e n e r a t o r a voltage ¢o exp (just:) is m a i n t a i n e d between the two s e t s of e l e c t r i c a l l y c o n n e c t e d f i n g e r s . S i n c e the v o l t a g e g e n e r a t o r directly d e t e r m i n e s only the v o l t a g e on the f i n g e r s , a n d not b e t w e e n t h e m o r inside th~ m e d i u m , we do not know the d i s t r i b u t i o n of the p o t e n t i a l ~p(x. y ) a t the s u r f a c e , ¢(.v. 0). On the f i n g e r s t h e r e will also be a ( s u r f a c e - ) c h a r g e d i s t r i b u t i o n O(.v) which i s a,.~ y e t unknown. No c h a r g e will be a l l o w e d to c r v ~ s tho y = 0 plane. We shall f i r s t pay attention to th~ F o u r i e r c o m p ~r~ts ~(kx) and P(kx) o~ ¢(±', 0) and p(x). T h e s e a r e of c o u r s e c o n n e c t e d w ~ ~ - , ~ - ~ of the f o r m e x p [ j ( t o t - kxX)]. Tht n o r m a l c o m p o n e n t of t h e d ~ e l e c t r t c d i s p l a c e m e n t cartload these w a v e s m u s t have a d i s c o n t i n u i t y p(k) at the s u r f a c e .,, = 0:

"Oy(kx) ]

..

= p(k )

(sin

If we h a w vacuum for y < 0 (this r e s t r i c t i m is only ::t~ade f o r c o n v e n i e n c e and is not e'.;sential a t all) (83) can a l s o be writte,1 a s

Ceff + Eo = -~T~x) k x r k v ,

(84)

~.,ilere ~o is the d i e l e c t r i c coast-ant of ~'acuun~. B e c a u s e in the c a s e u n d e r c o n s i d e r a t i o n (excitation) k x is r e a l , we can. change (84) into

p( x) ":

+ Co]Ikxl

We no~ ~ have the following s i t u a t i o n : !) The p o t e n t i a l oo

coix, 9) = f

~(kx)exp[-jkxX]dkx

is :fully known on tLe f i n g e r s . Between them and outside the t r a n s d u c e r region it is ,lot kno~m. 2) The c h a r g e d e a ~ l t p(x) = f -*oO

p(kx)exp[-Jkxx]dkx

(87)

E %ECTRIC COUPLING PROPERTi:,;b

265

i S t u l l y known (~0) bet'ween the f i n g e r s a s well as o u t s i d e the t r a n s d u c e r r e glon. On the f i n g e r s tt le~ not known. 3) ~,(kr) and p(k x) a r e r e l a t e d by (85). Cons,~uentl~' ,p(v,O) and p(x) a r e fu!ly determim,,d au a function ¢~i lhe aml~'ih~de ,.,~ ,ff ~he ;~.c. ~'~,ltage a.pp}ied to the e l e c ~ r o d c ~ . Though f o r a r b i t r a r y ~eff(~o, k) no p r o c e d u r e has been i n d i c a t e d to obtain a n a lytic e x p r e s s i o n s f o r 99(x, O) (or ~(kx)), an a n a l y t i c s o l u t i o n h a s been o b t a i n e d by Engart [~2] f o r the s ~ c i a l c a s e o~ tn i n f i n i t e l y e x t e n d e d p e r i o d i c t r a n z ducer on a d i e l e c t r i c s u b s t a n c e ( w h e r e ~eff is i n d e p e n d e n t of k). H o , ' e v e r , his (and s i m i l a r ) r e s u l t s h a v e been u s e d [ 1 2 . 1 8 - 2 0 ] f o r the c a l c u l a t i o n of :he b e havtour of t r a n s d u c e r ~ t r u c t u r e s on p i e z .... ~,~c~:'!~ zuhs~,,nce¢~, w h e r e '~eff d e p e n d s strongly on ~, a s can a l r e a d y be seen in the a p p r o x i m a t i o n (4~'):

eeff

eP v - ~o

- ~p ~

- ~'o

.

(42)

~ v i o u 8 l y the e r r o r t h u s i n t r o d u c e d h a s to do with the f a c t that the d i s t r i b u t i o n of the applit:¢l a . c . v o l t a g e o v e r v a r i o u s F o u r i e r c o m p o n e n t s i s m o s t f r e q u e n c y depend~nt at thorpe f r e q u e n c i e s w h e r e a c o u s t i c w a v e s a r e g e n e r a t e d . T h i s m e a n s that at t h o s e f r e q u e n c i e s w h e r e the b e h a v i o u r i~ m o s t i n t e r e s t i n g the d e v ' a t i o n of the F o u r i e r s p e c t r u m f r o m E n g a n ' s r e s u l t s is t.~:e g r e a t e s t . F r o m the a b w e it s h o u l d be c l e a r t h a t an i d e l c a l c u l a t i o n of the p e r t o r m a n c e 0f a t r a n s d u c e r would r u n a s f o l l o w s : l} C a l c u l a t e eeff{w , k) f o r the :~ubstance(s) to be :-.~msidere:!. Th~'~ ~,,.y bc d:'~nc (without a p p r o x i m a t i o n b u t at the c o s t of cons ~ i r u b l e c o m p u t a t i o n a l ~roub.~e) even f o r a p i e z o e l e c t r i c s u b s t a n c e . 2) i, rom t h i s c a l c u l a t e o{~'x), using (85) and the g e o m e t r i c a l propertie~:~ ,! ~!~,:: t r a n s d u c e r . The d]f~ c u l t i e s a r e c o n c e n t r a t e d in t~,~s ~;t p. Obv~.,:~,:~'~:: ~}:~:~: ~ t r i b u t m n p(x) wit.t be f r t . ~ u e n c y d e p e n d e n t . 3) I n t e g r a t e p(x) o v e r o n .~ of the two s e t s of t r a n s d u c e r f i n g e r s which a r e c o n nected to the v o l t a g e g e n e r a t o r . T h e t i m e d e r i v a t i c e of the r e s u l t is the ~.:~rrent, f r o m which th~ a d m i t t a n c e of tire t r a n s d u c e r can be obtained. .~ ~ d c ~ a t l o n a c c o r d i n g to this s c h e m e (ii ~f e a s i b l e at all) weuid not be r~'strutted to s m a r t coupth~g; reflect}.ons of e x c i t e d waves at t~-ansducer f~ngers, u~.:.~:.:~l ~mgec widths and f i n g e r s p a c i n g could be a c c o u n t e d f o r , too. ?-also the ~':e~ eratior of bulk w a v e s c o u l d be included. F u r t h .-rmo~e the s c h e m e is not restricted to a p a r t i c u l a r kind o~ waves. By way of e x a m p l e we m e n t i o n the c a s e of an interdi~ta~, e l e c t r o d e on ~ ~ - ~ o n - p i e z o e l e c t r i c

semiconductor

subjected

to an ~ . : e c t r i , : c l r ~

stru~:~,~}:~

i L~:i;i. T}-~:

a , c im:aedance of t h i s s y s t e m h a s b e e n n u m e r i c a l l y c a l c u l a t e d by one of us (S.). The e l e c : r o d e s t r u c t u r e w a s a s s u m e d to be p e r i o d i c in /be x-~.rectio~- wi' h ,.~riod 2~,!h o and to e x t e n d f r o m x = _~o to x = +~. C o n s e q u e n t l y the F o u r i e r in~,?grals (86) a n d (87) r e d u c e d to F o u r i e r s e r i e s . Due to the f a c t that a d ! a c e a t f;.ngets have o p p o s i t e p o t e n t i a l , t h e s e s e r i e s only c o n t a i n e d odd t e r m s :

286

~

~(x)

C,-.&;~:;.J-,:GREEBE e t iL

[q)(nk o) exp(-j nkoX)

-

:- -

+ ~(..nk O) e x p ( j n l ~ o x )

] ,

odd o

.

:o

......

.o

•

o - . :~. i¢

odd

e c a u s e of the d r i f t veiocRy v d. ~n [85). With the aid of tl~ese equacalculated as a f i r s t step i~ a con-: as:

whet

(as) tions

verg a) b) c) d)

~eff in (85)~S proiJ~irl~ d e s c r i b ~ b y (62)=. ) . equations (88) and (89) can be truncated a f t e r the Nth t e r m . ~(x) equals a value of ¢o exp (jo)t) at N d i f f e r e n t sites on each finger. P(D ~
Subsequently p(x) was evaluated using (89) and Integrated over one finger. This pro~Ss:,~asrepeateclf0r::~i~crea~lng values of N, until ~ae i n t e g r a l o f p ( x ) o v e r a finger had converged s ~ i c i e n t l v . This i n t e g r a l , ~,.2ttplied by joJ/~ o ~ s then identified with t h e a d m i t t a n c e of a finger pa~.r : ~ 0::,as turned out that the conre )c~e n e s negative if the two con-•

,~/ko < v d

Crxx, . / ~ el k o < Vd

and

a r e satisfied. These conditions a r e ~ntuith'ely clear" as us~al the drift velocitv should excee~l the phase velocity of the d r i v e n waves on the a t r u c t u r e and the : , ~ . ~ t ~irne of bunched c h a r g e between two f i n g e r s (v d k o ) - : should be less thin ~.~ d,~,,~,, ic relaxation t i m e Et/orx, x . •

'

.

.

.

.

:

~

~

i

ACKNOWLEDGMEN~

Valuable c o m m e n t s on the m a n u s c r i p t of this paper by P r o f e s s o r D. Polder, Dr. H'-G,, J u n g i n g e r and Dr. M. T. V l a a r d i n g e r b r o e k a r e g r a t e f u l l y acknowledged.

APPEND~

In a s e m i c o n d u c t o r which is subjected to both ~ d.c. e l e c t r i c field E and a d.c. •~ , ~ , ~ ~=~u u , u.~, ~p~c~ Charge (which mus~ pro:Tide me ~tan field) accumulates n e ~ the surface. In the context of 2.2 it is ~nl:~rtant to investigate whether the change of the s u r f a c e conductivity, caused b "~,u~ " effect, is s e r i o u s . I n o r d e r t o d o so we a ~ e r e to the g e o m e t r i c n l conventions m a d e In the text. The s e m i c o n d u c t o r is a s s u m ~ to occupy the r e ~~'m n y > 0, the e l e c t r i c field is in :~)This assumption is not eorreet oecause due .~othe pres.ence of m e electrodes vd is not necessarity uniform along the ~,~-direction[23]. However, this does not affect the poiJ~ to be made here.

ELECTRIC COUPLING PROP ~RTIES

267

the x+d~,recdon and B p a r a l l e l to the z - a x i s , In this c o n f i g u r a t i o n t b e r e is only a y+dcpendence of the v a r i a b l e s involved. In fact the p h e n o m e n a wiil col~sist c~f a R ,q uui[orm p a r t , indicated with a ,+,ub~crip[ o and a y-depende~Jt p a r t i[~,".:~](,++~+[i,e(,+~~V~h +[ a;,bmcr,t.pt: . . . . . . . . . . . . . . . .] In tertl+ls o( +~!~e~notation, used ~n the text we thus havc ~_):+ + ' +, ~,..... ,+ +++++t~++,+++ c u r r e n t in the y - d i r e c t i o n :

J y o = " ~x'~ E x o + CrxxEyo = 0.

(A.1)

H e r e Exo is the d r i f t f i e l d and Evo i s the Hall f i e l d quantities w e wr+.te d o w n the (lineaxZized) rel~ion JY'.l = a x x E . v l + q D x x

dn 1 dy

-

F o r the n o n - u n i f o r m

0,

(A.2)

where the m o d u l a t i o n n 1 of the e l e c t r o n d e n s i t y should sat~ s l y the equation dEyl E

--dy

~~ ~

q nl

because of (5) and (44). Dxy does not e n t e r (A.2) b e c a u s e n I d o e s not depend on x, the d r i f t of Pt 1 does not e n t e r the e q u a t i o n s b e c a u s e it cannot c o n t r i b u t e to a d.c. c u r r e n t in the v - d i r e c t i o n , and E~, 1 d o e s not o c c u r b e c a u s e Ex+ ! ¢ 0 would i m p l y curl E 1 * 0. F r o m (A.2) a n d (A.3~ we e a s i l y obtain the y - d e p e n d e n c e of n 1" h i 0 : ) : P e Ky + Qe"~Y,

(A.4

where a" is the i n v e r s o of the Deb j e - H t i c k e i ~ .+:+~,

Dxx e 1 ' and w h e r e we put P = u m o r d e r to obtain d e c a y ~f n I int,:, the s e m i c o n d u c t o ~ . Now we h a v e on ~tccount of (A.3): 0

~

+°= +%vo o;

i

co

d~ +

dy = f o

cn1(y ) cQ ~, dy =+", ¢-'-~: +

With (A.1) we thus obtain f o r Q:

Q = ~i Crxy

1/r~xx

" -d ~.~--7..Exo V + , D .... "

if +re now a s s u m e , in a c c o r d a n c e with the text+ ~ha~

~

c~.x

- 4 IIXXnO

g'~X B ,

- ~'xx Exo

= Vd,

+

no ~'l

-a'v

1/

Tn order to judge the relevance of (A.7) s o m e ~ssumptions which have been made +Sh~Id+be tees! led,.;+~ +

I

+t

.

1~ Equatlon (A.7) has been derived ~ t h the aido~ the linearized ex~presSi~ {A.2)~i~ C ~ s e q u e n f l y (A.7)only indicates the c o r r ~ t order of m a g n i ~ d e ot "I(Y) ~s long as "I(0) is smaller thav about ~ / ' 2 . 2) A tge h~ the band bending of the or ~ bared b ~ d ~ n ~ r~ I c h mostly is prestmt and which w e l g . 0 r ~ ~from the beginning. +,

In other words, in cases where (A.7) indicates a se~ ious change in the con-

Y +

Appi. Phys. L~tters 13 ~1968) 4J 2 | ~ P Letters 9 (1969) 37. P h v ~ + Letters 12 (1968) 31~L G.S.Klno. AI~pI. Phys. Letters 12 (!9~) 310. M . L s ~ n . C . F . Q u s t e and H.J. S~3w, Appl. Phys. Letters 13 (t9~;8) 314. [6] C. A. A. J. Greebe. Phys. L e t t e r s 31A (1970) ] 6. [7] A. Bers and B. E. Burke, Appl. PI~vs. Lette-s t6 (~97o) :~00 [8] J.Wotter, Phys. Letters 34A (1971) 87. [9] C.Krtscher and A.Bers, Appl. Phls. Letters t8 (197I)~49. [10] P.A.van Da[en, to be published in Philips Res. Rept+s. [~I] W.L=Boud, J . H . C [.~ns, H.M.Gerard, T,M. Ree~.r ~¢d H.J ~ w , Ap~L Ph)~. L~;1~ r s 14 (1969)122. ,.~.... ~ : [12] K, A. lngebrigtsen, J . AppL Phys. 40(1969) 2681, , [13] C . A . A . j . Greebe s n d K . A . I n g e b r l ~ e n , Phys. L e t t e r s 30A (1969) 364, [14] B.E. Burke, A.Bers, H.I. Smith, R,A. Cohen aud R. W. Mountain, Proc. IEE~] 58 (1970) 1775. [15] C . A , A . J . Greebe ~nd P.A. ~an Daten, PhiLips F~es. Repts. 24 (t969) 168. [16] H.Oksmoto sad H.Ohnumn, Jap. J. Appl, P~vs. 9 (1970) 1113. [!7] J. L, Bleustein, J, Ac. Soc. Am. 45 (1969) 614. [18] S. G,Josht and i~LbI.WhRe, j . Ac. ~ c . Ar~. 46 (1969) 17. [19] G.A,C0qttin a n d H. F. Tlerslvn, J. Ac. So~,~.Am. 41 (1967) 921. [20] W.R. S~alth, H.M. Gerard, ,L H.+Cothns, T.M. Reeder m~d H. J. Shaw, IEEE T r ~ n s a ~ + tions ~ T - 1 7 (1969) 8 5 6 . t=-,q t;. ~ . r s e n g ,

~r;r;r; -transactions E ' D - i 5 ~ 9 6 8 ) 586.

[22] IL Engan, IEEE Transsctlons ED-16 (1969) 1014. [23] T . J . ]~. Swanenburg, to be published.

,new i oRn-Holland pUbllCal:lOn Solid State Physics. Theotetical F:,~ysics, Mathematical Physics, M~thema :tcs

i,q

The El,ectron Theory of Solids By G. C. FLETCHER, Monr, s~= University, C;ayto;=, Australia.

1971. 275 pages, 61 illustrations, 15 graphs, 3 tables. Hfl. 5 4 . ~ (ca. S 15.00) ISBN 0 7204 0207 7 Summarizes the basic theory of electrons in ideal, sim3le solids in a manner intelligibl~ to experimentalists, while at the .,ame time indicating the mathematical problems to theoretical physicists commencing research in the field. No specific knowledge of solid state physics or sophisticated mathematics is assumed at the beginning, and the material is presented in such a way t h ~ a basic understanding of band theory should have been achi~,,~-,d b~fore more difficult concepts are introduced. CONTENTS" One-,=~lectron equation. Crys~a! ;attic,.~s and reciprocal lattices. Bloch functions. Eigen values, density of states and con~ta:=t energy surfaces. Mathematical methods. One-electron potential. Freeelectron approximation. Nearly-free-electron aporoximaticn. Ceitut~.tr method. Tight-binding method and Wannier functions. ,~.uc,mented p!84~e wave method. Scatering matrix and Green's functio,~ methods. Orthogona!ized pla,ne wave method. Symmetry and degeneracy. Group theory and Bloch functions. Symmetrization of 0-~-sethods of band calculation. E~ectron spin and relativity. P.~ri,apotentials in sotid,s Quantum defect methed and model potentials. Parametrization o,~ General references on band theory.

*""'°'"NORTH P.O.IIOX~

,

Sole d~ttributorw for the U.SJ~ ~nd Canad# : ArneticBn Elnvier Pvblithing C.~mpmny,Inc., 52 Vanderbift Avenue, ?#ew Vc~k, ~%Y. tOOl 7

httTHERLAN~S

ELECTRIC COUPLING PROP ~RTIES

267

the x+d~,rect~on a n d B p a r a l l e l to the z - a x i s . In this c o n f i g u r a t i o n t h e r e is only a y+dcpendence of the v a r i a b l e s involved. In f a c t the p h e n o m e n ~ wiil col~sis~ ~f :~ R ~q uui[orm p a r t , i n d i c a t e d with a ,+,ub~crip[ o and a y - d e p e n d e ~ t p a r t iD.,.:~ict[[,e(~ ~Vi~h +~ m,b++~.rip~+., + . . . . . . ~ 0 In . t e.r m s. o( . the. notation u s e d in the [ex/ we thus ht~vc [~:_):~''~,,+ ......... :+,~,~ ~:,+++ c u r r e n t in the y - d i r e c t i o n :

Jyo =

" ax~

Exo + ':rxxEyo = 0.

(A.1)

H e r e E x o is the d r i f t f i e l d and Evo i s t h e H a l l f i e l d q u a n t i t i e s we wr+.te down the (linealZized) r e l ~ i o n

F o r the non-uni+~orm

dn 1

axxE.vl

JY'.l =

+

q Dxx dy

-

0,

(A.2)

where t h e m o d u l a t i o n n 1 of the e l e c t r o n d e n s i t y should sat~ s l y the equation dEyl E

--dy

~~ ~

q nl

because of (5) and (44). Dxy d o e s not e n t e r (A.2) b e c a u s e n I d o e s not de,/.,end on x, the d r i f t of n l d o e s not e n t e r the e q u a t i o n s b e c a u s e it c a n n o t c o n t r i b u t e to a d . c . c u r r e n t in the v - d i r e c t i o n , and E., 1 d o e s not o c c u r b e c a u s e Ex+! ¢ 0 would i m p l y curl E 1 * 0. F r o m (A.2) a n d (A.3~ we e a s i l y obtain the y - d e p e n d e n c e of n 1" hi0: )

:

P e Ky + Qe"~Y,

(A.4

where a" i s the i n v e r s o of the Deb j e - H i J c k e i ~ .+:+~'

Dxx e 1 ' and w h e r e we put P = u m o r d e r to o b t a i n d e c a y ~f n I int,:, the s e m i c o n d u c t o ~ . Now we h a v e on ~tccouat of (A.3): 0

~o

+°= +%vo ,-+f

i

co ~+

dy = fo

cn1(y ) ~.+

cQ dy =+- +---~ +

With (A.1) we thus o b t a i n f o r Q: Q =

~i Crxy 1/r~xx " -d ~---UExo V~-,D .... "

if +re now a s s u m e ,

c~.x

in a c c o r d a n c e with the text+ ~ha~

g'~XB ,

- ~'xx E x o = V d ,

nl(Y~

=

,'dB~/~BT

(A.,)~-

•

In order to judge the relevance ol (A.7) some ,~uumptions which have bee. made ~Sh~!di! )e reca! led~.: ~ .

I) Equat, on (A.7) has been derived ~ t h the aid of the llnearlzed expression (A.2!)~i~ Consequently (A.7)only indicates the correct order of magni~de o[ "I(Y) as long as "1(0) is smaller than about ,~/'~. 2) A ~ge t~, the band bending of the or the band bending ~ I e h mostly is p i - e s ~ t and which we ignored from the:beginning. in o t h e r w o r d s

in c a s e s w h e r e (A.7) i n d i c a t e s a se, tons change In the con-

?hys. L~tters 13 ~1965) 412 Letters 9 (1969) 37. L e t t e r s 12 (1968) 31". Ln~, AI:PI. Phys. L e t ~ r s "~2 (!9~O 310. :in. C . F . Q u s t e and H, J, Shaw, Appl. Phys. L e t t e r s 13 (1968) 314, s. L e t t e r s 31A (197t~) 16, [7] A. Bers and B. E. Burke, Appl. Pl~vs. Lette-s 16 (t97~) :~00 [8l J.Wotter, Phys. Letters 34_~ (1971) 87. [9l C . K r i s c h e r and A.Bers. Appl. PhS"s, Letters t8 (197I) ~49. [101 P.A.van Daten, to be published in Philips Res. Rep~,s. [Ul W.L.Bond, J.H. C tans li.M. Gerard, T. I~I.Reec~;r ~ d H.J. ~ a w , Ap~i. Ph~. Lc~I~rs 14 (1969)122. : [121 K.A.lngebrigtsen, J, APP|. Phys. 40 (1969) $681. [lal C . A . A , j . Greebe s n d K : A . I n g e b r l ~ e n . Phys. Lette~rs 30A (1969) 364, [141 B,E. Burke, A.Bers, H.I. Smith, R,A, Cohen and R. W. Mount:~in, P r e c . IEEE 58 (I97~)) 1775. [15] C . A , A , J . Greebe and P, A. van Dalen, PhiHps Res. Repts. 24 (rt9~;9) 168. [16] H.Okamoto snd H.Ohuum•, Jap. J. Appt, P ~ ' s . 9 (1970) 1113. [!7] J. Lr, Bieustein. J, Ac. Soc. Am. 45 (1969) 614. [18] S. G , J o s h t and I~.M.Whl~, j . Ac. ~ c . AM. 46 (1969) 17. [19]G.A, C ~ n and HI. F. Tlersl~en, J. Ac. ~¢.. Am. 41 (1967) 921. r20] ~V, R. S;alth, H, M. Gerard, J . H . Coihns, T, M, Reeder a~d H. J. Shaw, IEEE Transa~ t i o n s ~ T - 1 7 (1969) 856 t ~ t t~. ~, Isvng, I ~ T r a n s s c ~ o n s ~D-i~ fi968) 586, [22] H, Engan, IEEE Transactions ED-16 (1969) 1014. [23] T . J . I~. Swanenburg, to be published. O

T"

J

~