SURFACE MODES AT A FERRITE-SEMICONDUCTOR INTERFACE

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SURFACE MODES AT A

FERRITE-SEMICONDUCTOR INTERFACE

A. Boardman, A. Irving

To cite this version:

A. Boardman, A. Irving. SURFACE MODES AT A FERRITE-SEMICONDUCTOR INTERFACE.

Journal de Physique Colloques, 1984, 45 (C5), pp.C5-275-C5-284. �10.1051/jphyscol:1984541�. �jpa- 00224159�

JOURNAL DE PHYSIQUE

Colloque C5, supplément a u n04, Tome 45, a v r i l 1984 page C5-275

SURFACE MODES AT A FERRITE-SEMICONDUCTOR INTERFACE A.D. Boardman and A.K. Irving

Department o f Pure and AppZied P h y s i c s , U n i v e r s i t g o f S a l f o r d , SaZford M5 4WT, U . K .

Résumé - Nous déterminons l e comportement d'ondes m a g n é t o s t a t i q u e s de s u r f a c e à une i n t e r f a c e f e r r i t e - s e m i c o n d u c t e u r . Nous é t a b l i s s o n s des f o r m u l e s donnant l e comportement du champ é l e c t r i q u e dans l e semiconducteur, e t l ' i n f l u e n c e que l a d e n s i t é de p o r t e u r s l i b r e s du semiconducteur a s u r l a f e r r i t e . L ' é t u d e approchée des ondes de s u r f a c e se propageant dans une d i r e c t i o n f a i s a n t un grand an- g l e avec l e champ magnétique a p p l i q u é montre que l e s a f f i r m a t i o n s a n t é r i e u r e s p r é t e n d a n t l ' e x i s t e n c e d'onde h é l i c o n de s u r f a c e s o n t sans fondement, dans l e s c o n d i t i o n s normales.

A b s t r a c t

-

1 - INTRODUCTION

There have been many i n v e s t i g a t i o n s o f s u r f a c e waves on f e r r i t e - semiconductor b o u n d a r i e s /1/. Almost a l 1 o f t h e s e a r i s e f r o m an i n t e r e s t i n t h e i n f l u e n c e t h a t t h e s e m i c o n d u c t o r has on t h e b e h a v i o u r o f t h e c l a s s i c a l m a g n e t o s t a t i c s u r f a c e waves s u p p o r t e d by t h e f e r r i t e . A l 1 t h e m a g n e t o s t a t i c systems f r o m t h e l a t t e r p o i n t of v i e w have s u r f a c e waves t h a t p r o p a g a t e p e r p e n d i c u l a r l y t o an i n t e r n a l magnetic f i e l d t h a t l i e s i n t h e i n t e r f a c e between t h e f e r r i t e and t h e semiconductor. I n an a t t e m p t t o c o n f o r m t o expected e x p e r i m e n t a l p r a c t i c e i t i s a l s o u s u a l l y assumed t h a t t h e semiconductor i s c o l l i s i o n - d o m i n a t e d . O b v i o u s l y t h e V o i g t c o n f i g u r a t i o n d e s c r i b e d above i s v e r y i m p o r t a n t . I t l e a d s , f o r i n s t a n c e , t o n o n - r e c i p r o c a l s u r f a c e wave p r o p a g a t i o n and t h e p o s s i b i l i t y o f u s e f u l m a g n e t o s t a t i c d e v i c e s . I t i s , n e v e r t h e l e s s , a l s o i m p o r t a n t t o c o n s i d e r ' o f f - a x i s ' p r o p a g a t i o n i n w h i c h t h e wave v e c t o r t u r n s away, i n t h e i n t e r f a c e p l a n e , f r o m t h e V o i g t c o n f i g u r a t i o n towards a Faraday c o n f i g u r a t i o n p a r a l l e l t o t h e i n t e r n a l m a g n e t i c f i e l d . If such a f i e l d i s s t r o n g enough t h e n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984541

C5-276 JOURNAL DE PHYSIQUE

t h e s e m i c o n d u c t o r i s a l s o under i t s i n f l u e n c e . T h i s i n f l u e n c e vanishes i n t h e V o i g t mode b u t c o u p l i n g t o t h e f e r r i t e o f m a g n e t i c s e m i c o n d u c t o r e f f e c t s , a r i s i n g t h r o u g h t h e o f f - d i a g o n a l d i e l e c t r i c t e n s o r elements, appears as t h e s u r f a c e wave v e c t o r assumes a f i n i t e a n g l e t o t h e V o i g t d i r e c t i o n , T h i s c o u p l i n g i s p r o p o r t i o n a l t o t h e magnitude o f t h e c y c l o t r o n f r e q u e n c i e s and i n p r i n c i p l e s h o u l d be i m p o r t a n t .

P r e v i o u s l y p u b l i s h e d work on t h i s problem i s n o t d e s i g n e d t o s t u d y t h e f e r r i t e b u t t o f u r t h e r advance t h e i d e a t h a t s u r f a c e h e l i c o n s e x i s t / 2 / . The v e r y e x i s t e n c e o f t h e s e waves, however, i s i n some d o u b t /3/ and we w i l l show i n t h i s paper t h a t t h i s d o u b t i s c o n f i r m e d . I n d e e d t h e c i t e d work /2/ a l s o does n o t p r e d i c t t h e c o r r e c t f e r r i t e s u r f a c e wave b e h a v i o u r .

A f t e r s e t t i n g up a g e n e r a l t h e o r y of t h e i n t e r f a c e modes c e r t a i n s p e c i a l i s a t i o n s w i l l be made t o il 1 u s t r a t e c e r t a i n b e h a v i o u r and i n p a r t i c u l a r t o show t h a t s u r f a c e h e l i c o n s do n o t e x i s t , f o r an n - t y p e semiconductor. A l i m i t e d comparison o f s t r o n g and weak c o l l i s i o n cases i s made b u t t h e n u m e r i c a l work i s d e v e l o p e d f o r t h e weak c o l l i s i o n case. A n a l y t i c a l f o r m u l a e a r e p r e s e n t e d , wherever p o s s i b l e , and o n l y semi - i n f i n i t e media a r e c o n s i d e r e d .

II - FIELD COMPONENTS AND DISPERSION EQUATION

The f e r r i t e system i s shown i n F i g . 1 . I t i s assumed t h a t t h e f i e l d s i n t h e f e r r i t e a r e m a g n e t o s t a t i c so t h a t t h e f i r s t - o r d e r e l e c t r i c f i e l d a r e n e g l i g b y s m a l l . Hence w i t h i n t h e f e r r i t e

c u r l -0, d i v !=O where -- H i s t h e r f magnetic f i e l d .

F i g . 1 - The f e r r i t e - s e m i c o n d u c t o r system

The f e r r i t e i s c h a r a c t e r i s e d by t h e permeabi 1 i t y t e n s o r

where t h e elements a r e d e f i n e d as

Here a l 1 f r e q u e n c i e s a r e measured i n u n i t s o f mm, t h e m a g n e t i s a t i o n f r e q u e n c y w = p M so t h a t RH=Bo/Bm, R=w/wm, where p M =B i s t h e

m O O ' O 0 O

s a t u r a t i o n m a g n e t i s a t i o n and Bo i s t h e i n t e r n a 1 magnetic f l u x d e n s i t y which, i n t h e absence o f a d e m a g n e t i s a t i o n f a c t o r , i s e q u a l t o t h e a p p l i e d m a g n e t i c f l u x d e n s i t y . I f i t i s now assumed t h a t a l 1 rf components Vary as e x p i ( & . - w t ) . , and t h a t w i t h i n t h e f e r r i t e = -v$ where Q i s a m a g n e t o s t a t i c p o t e n t i a l , t h e n d i v

= O becomes

- axi a ( p i j j k $)=O i . e . p . I J .k.k.=O 1 J ( 2 . 5 )

and t h e y-component o f t h e wave v e c t o r i s g i v e n by

The c u r l e q u a t i o n o f ( 2 . 1 ) t h e n g i v e s t h e f i e l d components as

where t h e common f a c t o r e x p i ( k x x + k z z - u t ) has been suppressed.

I n t h e semiconductor i t i s n o t p e r m i s s i b l e t o n e g l e c t t h e f i r s t - o r d e r e l e c t r i c f i e l d s and t h u s o p t f o r a p u r e l y m a g n e t o s t a t i c s o l u t i o n . I t i s p e r m i s s i b l e , however, t o n e g l e c t t h e displacement c u r r e n t and t o c o n s i d e r t h e e f f e c t o f e l e c t r o n i c c o l l i s i o n s t o be e i t h e r s t r o n g (as i s o f t e n t h e case) o r weak. I n d i m e n s i o n l e s s terms, t h e r e f o r e t h e d i e l e c t r i c t e n s o r elements f o r a one-component semiconductor a r e

C5-278 JOURNAL DE PHYSIQUE

S t r o n g c o l 1 i s i o n case R<<Rc, Rp,q

Weak c o l l i s i o n case R<<Rc, R

p Y T 7

where 1 ^{c z z} 1 >> 1 ^cxy 1 >> 1 ^{c x x} 1 i n b o t h cases, R =U /U Rc=w /w

P P m' c m'

n =v/vm,v i s t h e c o l 1 i s i o n f requency and t h e microwave f r e q u e n c y range o f i n t e r e s t a l l o w s t h e n e g l e c t o f any phonon c o n t r i b u t i o n . I n t h e semiconductor /4,5,6/

c u r l

E

where ^E, i s t h e l a t t i c e p e r m i t t i v i t y , E i s t h e rf e l e c t r i c f i e l d

L

and g i s t h e d i e l e c t r i c t e n s o r . I t i s n o t d i f f i c u l t t o show w i t h t h e s e e q u a t i o n s t h a t t h e d i m e n s i o n l e s s f o r m o f k i s

Y

KZ=-$ _{Y X}+ ( 1 / 2 ~ ~ ~ ) [ R ~ ~ E 1 ~^E~~ ~1 2 + ~ ~ X l - ~ ~ ( ~ X X + ~ Z Z ) ~ ~ ] E ~ ~ - ( 2 . 9 ) where

The p r o o f t h a t s u r f a c e h e l i c o n s e x i s t depends upon a number o f a p p r o x i m a t i o n s /5,6/, One o f t h e s e i s t h e assumption t h a t Kx>>KZ so i t i s n e c e s s a r y now t o d e v e l o p s m a l l KZ expansions o f e q u a t i o n (2.10).

I n t h e l i t e r a t u r e two e x p a n s i o n parameters p r e v a i l /5,6/. These a r e

Now, as has a l r e a d y been s t a t e d , i t i s c l e a r t h a t f o r t h e l o w frequency r a n g e i n v o l v e d , t h e d i s p l a c e m e n t c u r r e n t can be n e g l e c t e d . T h i s means t h a t , as has been concluded b e f o r e / 3 / , x n l . T h i s can be seen f r o m

n "n ml2

P . - . > = 1 , R<<n<<Rc

n2eC nt n Q:

The e x p a n s i o n parameter y i s a c c e p t a b l e s i n c e

Since t h e assumption t h a t x<<l i s n o t c o r r e c t i t c a n n o t be used as an e x p a n s i o n parameter f o r t h e s i n g l e component dominated semi- c o n d u c t o r s , such as InSb and GaAs, t h a t a r e l i k e l y t o be used i n p r a c t i c e . I t i s a l s o t h e case t h a t n e i t h e r h o l e s o r c o l l i s i o n s a r e n e c e s s a r y t o t h e e x i s t e n c e o f h e l i c o n s / 7 / .

x i s n o t r e q u i r e d , however, because an e x a m i n a t i o n o f e q u a t i o n (2.10) shows t h a t , i f t h e displacement c u r r e c t i s n e g l e c t e d , and n o f u r t h e r a p p r o x i m a t i o n s a r e made, then

E X ~ - ~ E ~ ~ ~ ~ -

XY

so t h a t

2 i n3 t 3

DE --!? KZ [q<<n] ; DS 2 J ; ~ K ~ / (R Rcn ) L~>>Q] (2.16) mc

2 2 t h e n we o b t a i n 1: t h e s o l u t i o n s t o (2.9) a r e K' = - k1 K ⁼ -K2

y 1 y 2

Wea k c o l 1 is i o n s : K;

,g

P C Z P c (2.17)

S t r o n g c o l 1 i r i o n s : ~ i ~ % : - i ~ ' Q / n f i . ~ f i fi Q'KZliih (2.18)

P P c

These a r e q u i t e d i f f e r e n t forms t o p r e v i o u s l y p u b l i s h e d e x p r e s s i o n s . They d i f f e r b y b e i n g complex c o n j u g a t e s and b e i n g O ( K Z ) i n s t e a d o f O(KZ). 2 T h i s K, dependence w i l l a l l o w t h e O(Kz) t o be n e g l e c t e d i n e q u a t i o n (2.23) 2 i n a s m a l l ( K ~ / K ~ ) expansion and t h u s remove t h e p o s s i b i l i t y o f s u r f a c e h e l i c o n p r o p a g a t i o n .

The f i e l d components i n t h e semiconductor a r e , i n g e n e r a l l i k e TE and TM modes c o u p l e d t h r o u g h t h e m a g n e t i c f i e l d . L e t us, t h e r e f o r e , use t h e mode1 developed by t h e Russian i n v e s t i g a t o r s who have p r e d i c t e d s u r f a c e

c5-280 JOURNAL DE PHYSIQUE

h e l i c o n s and associate t h e s o l u t i o n s f o r KL i n e q u a t i o n (2.8) w i t h T M ( p ) and TE(H,=O) p a r t i a l waves. This leads t o magnetic f i e l d Y

components i n t h e semiconductor w i t h t h e form

The i m p o s i t i o n of the c o n t i n u i t y o f Hz, Hx and u H +p H a t t h e i n t e r f a c e Y X X Y Y Y

y i e l d s t h e equation

!-

where x " = R ~ E ~ ~ - K ~ - K ~ and t h e p a r t i t i o n s h i g h t l i g h t t h e uncoupled TE mode s o l u t i o n t h a t a r i s e s when KZ=O.

Equation (2.22) gives t h e general d i s p e r s i o n e q u a t i o n

I I I - APPROXIMATIONS TO DISPERSION EQUATION

Equation (2.231 reduces t o two known cases. The f i r s t i s

The s o l u t i o n of t h i s e q u a t i o n i s displayed i n Fig.2. The second case i s fi $ O, ^E = O, K, = O (weak c o l l i s i o n s )

P XY

..

t h a t i s t h e l i m i t o f p r e v i o u s work /9/ on a f e r r i t e slab, bounded by a semiconductor, as t h e s l a b t h i c k n e s s goes t o i n f i n i t y and weak

c o l l i s i o n s a r e assumed.

If zXY = O and K2 = O, as would be t h e case i f the magnetic f i e l d i s t o o small t o i n f l u e n c e t h e serniconductor, then t h e d i s p e r s i o n

e q u a t i o n becomes, f o r weak c o l 1 is i o n s ,

F i g . 2 . R as a f u n c t i o n o f Kx and KZ f o r RH = 1 and R = O. V e r t i c a l s c a l e x 100. P

I f E $0 and % = O t h e n , s i n c e t h e t e r m on t h e r i g h t hand s i d e o f e q u a t i o n (2.23) i n v o l v e s XY

g ,

2 2

by s e t t i n g KZ/Kx<<l. I f t h i s i s t h e case, and s i n c e KZ i s O(KZ), t h e d i s p e r s i o n e q u a t i o n reduces t o

K ; Y + K ~ ( V K ~ + ~ K , 1% O (2.27)

Now t h e e x i s t e n c e ~f s u r f a c e h e l i c o n s o u g h t n o t t o depend upon t h e e x i s t e n c e of c o l l i s i o n s o r t h e presence o f h o l e s / 7 / so t h e s o l u t i o n o f e q u a t i o n ( 2 . 6 ) w i l l be developed f o r t h e weak c o l l i s i o n case i . e .

T h i s f o r m o f K2 l e a d s t o a s o l u t i o n f o r Q t h a t i s t h e complex r o o t o f

JOURNAL DE PHYSIQUE

where

I V - SOLUTIONS TO DISPERSION EQUATION E q u a t i o n (2.25) has t h e s o l u t i o n

n = (1 +R,+%K,/V'KF~) t i +K~IR~T-~ (2.33)

T h i s shows t h e e f f e c t o f fi on t h e u s u a l m a g n e t o s t a t i c s u r f a c e mode t r a v e l l i n g p e r p e n d i c u l a r t o t h e i n t e r n a 1 m a g n e t i c f i e l d . N o t e P t h a t as K x + O , M H + l showing an e l e v a t i o n above t h e $+RH Damon- Eschbach r e s u l t . As KX-- t h e u s u a l $+RH r e s u l t i s r e c o v e r e d . I t s h o u l d be n o t e d t h a t t h e p r e v i o u s l y p u b l i s h e d s u r f a c e h e l i c o n r e s u l t /2/ i s d e f i ' c T e n t , l'n any case, s i n c e i t does n o t r e d u c e t o t h i s r e s u l t .

The weak c o l l i s i o n , weak m a g n e t i c f i e l d case g i v e n by e q u a t i o n (2.26) r e s u l t s i n a s u r f a c e mode c u t - o f f e q u a t i o n

KZ4+ ( (nH-1 )/fiH)KZ2+ Kx"((K2+R') (nH+l )/n,)~; = O

x P ^(2.34)

T h i s shows a d r a m a t i c i n f l u e n c e o f t h e c a r r i e r d e n s i t y on t h e c u t - o f f c h a r a c t e r i s t i c . I t i s p a r a b o l i c n e a r t o t h e o r i g i n and becomes t h e u s u a l s t r a i g h t l i n e KZ=Kx/hH as Kx-. A l s o by s e t t i n g Kx=KllcosQ and KZ=KI1sinQ i n e q u a t i o n (2.25) and t h e n t a k i n g t h e l i m i t KI,& we f i n d t h a t

These r e s u l t s a r e i l l u s t r a t e d i n F i g . 3 and Fig.4. I t i s o b v i o u s f r o m e q u a t i o n (2.29) t h a t t h e o f f - a x i s waves a r e damped and do n o t possess a s u r f a c e h e l i c o n c h a r a c t e r , as p r e v i o u s l y p r e d i c t e d / 2 / . The s o l u t i o n o f e q u a t i o n (2.29) can be g e n e r a t e d b y assuming t h a t t h e wave v e c t o r i s r e a l and t h a t t h e f r e q u e n c y i s complex and b y w r i t i n g t h e c o e f f i c i e n t s as

I f t h i s i s d o n e then, p r o v i d e d Q=t=RR+iRI, w i t h RR>>RI, we o b t a i n

Here RR i s j u s t t h a t g i v e n by e q u a t i o n (2.33) and fiIa% , as expected. The v a r i a t i o n o f t= 2(RI/RcKZ) reduces t o

F i g . - ' k a s a f u n c t i o n o f K, and K, f o r a H = l f o r 4 -937 c o r r e s p o n d i n g t o a c a r r i e r d e n s i t y o f 1016~171-3. 9 =O, gxy=O. ~ e r t i c a y - s c a l e x 10.

-

F i g . 4 - T h r e s h o l d c o n t o u r s i n t h e Kx-Kz p l a n e f o r ~ x y = O , l l H = l a ~ d fip w i t h t h e values ( a ) 93.7 ( b ) 937 ( c ) 9370 ( d ) 93700.

E q u a t i o n (2.38) shows t h a t t h e darnping weakens t o z e r o as Kx- and

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becomes RcKZ/R as Kx+O. Note t h a t f o r t h e expansion t o be c o n s i s t e n t i t i s necessary t o mai'ntain t h a t Rc<<Q P

P.

V

-

He have examined t h e behaviour o f s u r f a c e waves a t a f e r r i t e - s e m i c o n d u c t o r i n t e r f a c e , Such a system o f f e r s many o p p o r t u n i t i e s t o benchmark t h e r e s u l t s o n t 0 known f e r r i t e b e h a v i o u r and hence s e r v e s as a c o n v e n i e n t system t o t e s t any s u g g e s t i ~ n t h a t s u r f a c e h e l i c o n s e x i s t . I t i s shown t h a t t h e c u r r e n t c l a i m f o r t h e e x i s t e n c e o f s u r f a c e h e l i c o n s c a n n o t be s u s t a i n e d f o r normal c o n d i t i o n s . We p r o v e t h a t t h e p r e v i o u s use o f u n s u b s t a n t i a t e d expansion parameters has l e d t o a t h e o r y based upon an i n c o r r e c t KZ dependence o f t h e decay f u n c t i o n s . These f u n c t i o n s are, i n f a c t , l i n e a r i n Kz and complex. F i n a l l y , t h i s work forms t h e b a s i s f o r f u r t h e r d e t a i l e d n u m e r i c a l work on l a y e r e d media and a p p l i e d m a g n e t i c f i e l d s t h a t do n o t l i e i n t h e f e r r i t e - s e m i c o n d u c t o r i n t e r f a c e .

V I

-

1. SODHA M.S. and SRIVASTAVA N.C., "Microwave P r o p a g a t i o n i n F e r r i m a g n e t i c s " , Plenum Press (1981) 266.

2. OLEINIK I.N. and YAKOVENKO V.M., Sov.Phys.Solid S t a t e

2

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