LINEAR AND NONLINEAR SPIN WAVE EXCITATIONS IN SUPERLATTICES AND AT SURFACES

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LINEAR AND NONLINEAR SPIN WAVE EXCITATIONS IN SUPERLATTICES AND AT

SURFACES

R. Camley

To cite this version:

R. Camley. LINEAR AND NONLINEAR SPIN WAVE EXCITATIONS IN SUPERLAT- TICES AND AT SURFACES. Journal de Physique Colloques, 1984, 45 (C5), pp.C5-315-C5-323.

�10.1051/jphyscol:1984546�. �jpa-00224164�

JOURNAL DE P H Y S i v U t

Colloque C5, supplément au n04, Tome 45, a v r i l 1984 page C5-315

LINEAR AND NONLINEAR S P I N WAVE E X C I T A T I O N S I N SUPERLATTICES AND AT SURFACES

R . E . C a m l e y

Physics Department, University of CoZorado, CoZorado Springs, CO 80907, U.S.A.

Résumé - Nous d i s c u t o n s l e s ondes de s p i n s dans un super-réseau composé a l t e r n a t i - vement de couches magnétiques e t nonmagnétiques. Une couche magnétique i s o l é e a des ondes de s p i n s de s u r f a c e à ses l i m i t e s . Dans une s t r u c t u r e composée de couches, ces ondes i n t e r a g i s s e n t e t forment une bande d ' e x c i t a t i o n s volumique du super-réseau.

Sous c e r t a i n e s c o n d i t i o n s il e x i s t e a u s s i des modes de s u r f a c e du super-réseau s e m i - i n f i n i . Ces e x c i t a t i o n s peuvent ê t r e é t u d i é e s p a r d i f f u s i o n de l a l u m i è r e e t nous présentons l e s r é s u l t a t s de l a t h é o r i e e t l e s comparons avec ceux de

l ' e x p é r i e n c e . Nous d i s c u t o n s a u s s i comment l a n o n l i n é a r i t é de l ' é q u a t i o n de B l o c h p o u r l e s s p i n s c o n d u i t au couplage des ondes des s p i n s . Nous n o t o n s que l ' i n t e r a c - t i o n de deux ondes de s u r f a c e p e u t p r o d u i r e une t r o i s i è m e onde d e s u r f a c e a i n s i que l ' i n t e r a c t i o n de deux ondes de volume p e u t p r o d u i r e une onde de s u r f a c e . A b s t r a c t - We d i s c u s s t h e s p i n waves o f a s u p e r l a t t i c e composed o f a l t e r n a t e l a y e r s o f m a g n e t i c and nonmagnetic m a t e r i a l s . Each magnetic f i l m i n i s o l a t i o n has s u r f a c e

s p i n waves on i t s boundaries. I n t h e l a y e r e d s t r u c t u r e , t h e s e i n t e r a c t t o f o r m a band o f b u l k e x c i t a t i o n s o f t h e s u p e r l a t t i c e . Under c e r t a i n c o n d i t i o n s a s u r f a c e mode o f t h e s e m i - i n f i n i t e s u p e r l a t t i c e a l s o e x i s t s . These e x c i t a t i o n s can be probed by l i g h t s c a t t e r i n g experiments, and we g i v e t h e o r e t i c a l r e s u l t s and compare them w i t h experiment. We a l s o d i s c u s s how t h e n o n l i n e a r i t y o f t h e B l o c h s p i n e q u a t i o n l e a d s t o a m i x i n g o f s p i n waves. We n o t e t h e p o s s i b i l i t y o f two s u r f a c e waves i n t e r - a c t i n g t o produce a t h i r d s u r f a c e wave o r even o f two bu1 k waves i n t e r a c t i n g t o pro:

duce a s u r f a c e s p i n wave.

I n t h i s paper we f i r s t e x p l o r e t h e behaviour o f l o n g wavelength, m a g n e t o s t a t i c s p i n waves i n magnetic s u p e r l a t t i c e s . The p r o p e r t i e s o f t h e s e modes a r e governed, n o t by t h e s h o r t - r a n g e exchange i n t e r a c t i o n , b u t by macroscopic d i p o l e f i e l d s s e t up by t h e m o t i o n of t h e s p i n s p r e c e s s i n g around t h e magnetic f i e l d . These d i p o l e f i e l d s a r e c a l c u l a t e d t h r o u g h t h e use o f t h e m a g n e t o s t a t i c f o r m o f M a x w e l l ' s e q u a t i o n s . The e q u a t i o n s g o v e r n i n g t h e s p i n system, B l o c h ' s e q u a t i o n s , a r e l i n e a r i z e d whlch i s a p p r o p r i a t e f o r small ' a m p l i t u d e o s c i l l a t i o n s . I n t h e second p o r t i o n o f t h l s paper we c o n s i d e r some e f f e c t s which come f r o m t h e n o n l i n e a r terms i n t h e B l o c h e q u a t i o n s . A r e c e n t development i n r n a t e r i a l s c i e n c e i s t h e a p p l i c a t i o n o f e v a p o r a t i o n t e c h n i q u e s t o p r o d u c i n g modulated o r l a y e r e d s t r u c t u r e s . One t y p e o f s t r u c t u r e has a l t e r n a t i n g l a y e r s o f f e r r o m a g n e t i c and nonmagnetic m a t e r i a l s . The f e r r o m a g n e t i c l a y e r has a t h i c k n e s s dl and t h e nonmagnetic l a y e r has a t h i c k n e s s d2. The m a g n e t i z a t i o n i s p a r a l l e l t o t h e l a y e r s as i s t h e a p p l i e d f i e l d . T h i s geometry i s i l l u s t r a t e d i n F i g u r e 1. Because o f t h e p e r i o d i c i t y o f t h e s t r u c t u r e , i t i s sometimes c a l l e d a s u p e r l a t t i c e .

The l a y e r e d magnetic s t r u c t u r e has been i n v e s t i g a t e d now by s e v e r a l techniques;

s t a t i c m a g n e t i z a t i o n /1/, f e r r o m a g n e t i c resonance /2/, and r e c e n t l y by l i g h t s c a t - t e r i n g / 3 , 4 / . Some i n t e r e s t i n g and novel r e s u l t s have been found. I t i s t h e i n t e n - t i o n of t h i s paper t o b r i e f l y r e v i e w t h e t y p e s o f c o l l e c t i v e e x c i t a t i o n s w h i c h can o c c u r i n magnetic s u p e r l a t t i c e s and show how t h e unusual f e a t u r e s o f thesemodes may be seen i n a l i g h t s c a t t e r i n g experiment.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984546

JOURNAL DE PHYSIQUE

NONMAGNETIC MEDIUM

I

NONMAGNETIC MEDIUM _{I d 2} t Y

NONMAGNEFIC MEDIUM

1

VACUUM

F i g . 1 - Sample geometry c o n s i d e r e d i n t h e p r e s e n t paper. One has a s e m i - i n f i n i t e s t a c k o f f e r r o m a g n e t i c f i l m s each o f t h i c k n e s s dl, and t h e y a r e separated by a non- magnetic f i l m o f t h i c k n e s s d2.

We s t a r t b y s i m p l y d e s c r i b i n g t h e n a t u r e o f t h e c o l l e c t i v e modes o f t h e magnetic s u p e r l a t t i c e . F i r s t c o n s i d e r a s i n g l e i s o l a t e d magnetic f i l m . T h i s f i l m can s u p p o r t b o t h b u l k and s u r f a c e spinwaves. The b u l k modes have a s t a n d i n g wave c h a r a c t e r p e r - p e n d i c u l a r t o t h e f i l m s u r f a c e s w i t h i n t h e magnetic m a t e r i a l and have d i p o l e f i e l d s which e x t e n d o u t s i d e t h e m a t e r i a l , d e c a y i n g e x p o n e n t i a l l y w i t h d i s t a n c e f r o m t h e

s u r f a c e s o f t h e f i l m . The s u r f a c e spinwaves i n t h e f i l m a l s o have d i p o l e f i e l d s ex- t e n d i n g beyond t h e magnetic m a t e r i a l . When we b r i n g s e v e r a l f i l m s t o g e t h e r , t h e y i n t e r a c t t h r o u g h t h e s e d i p o l e f i e l d s and produce a c o l l e c t i v e mode.

The c o l l e c t i v e modes a r e t h u s made up o f b u l k o r s u r f a c e modes i n each l a y e r , modu- l a t e d by an envelope f u n c t i o n which d e s c r i b e s t h e r e l a t i v e a m p l i t u d e s between t h e d i f f e r e n t l a y e r s . We may have b u l k s u p e r l a t t i c e modes i n an i n f i n i t e s u p e r l a t t i c e where t h e envelope f u n c t i o n has a b u l k w a v e l i k e s t r u c t u r e . I n a s e m i - i n f i n i t e super- l a t t i c e , we may a l s o have s u r f a c e modes f o r which t h e envelope f u n c t i o n decays ex- p o n e n t i a l l y as one l e a v e s t h e s u r f a c e and p e n e t r a t e s i n t o t h e s t r u c t u r e . These two t y p e s o f modes a r e i l l u s t r a t e d i n F i g u r e 2.

We now o u t l i n e t h e b a s i c method used i n s o l v i n g f o r t h e d i s p e r s i o n r e l a t i o n f o r spinwaves on a s u p e r l a t t i c e and p r e s e n t some o f t h e r e s u l t s . The d e t a i l s o f t h e c a l c u l a t i o n a r e p r e s e n t e d elsewhere / 5 / . The method used i s s i m i l a r t o t h a t employed t o s o l v e t h e Kronig-Penney mode1 f o r e l e c t r o n p r o p a g a t i o n i n a p e r i o d i c p o t e n t i a l . We c o n s i d e r f i r s t t h e d e s c r i p t i o n o f spinwave e x c i t a t i o n s i n a i n f i n i t e l y extended s u p e r l a t t i c e , t h e n we t u r n t o t h e s e m i - i n f i n i t e a r r a y i l l u s t s a t e d i n F i g u r e 1. I n t h e m a g n e t o s t a t i c 1 i m i t we c o n s i d e r t h e demagnetizing f i e l d hd(X,t) generated b y t h e s p i n m o t i o n , which has v a n i s h i n g c u r l

and so one may w r i t e

BULK WAVE

SURFACE WAVE

nonmagnetic _film

i /

-\i i /

I 1

F i g . 2 - I l l u s t r a t i o n o f b u l k and s u r f a c e waves on a s u p e r l a t t i c e . I n each f i l m t h e r e i s a s u r f a c e wave. F o r t h e b u l k modes t h e r e i s a s i n u s o i d a l envelope f u n c t i o n ; f o r t h e s u r f a c e wave t h e r e i s an envelope f u n c t i o n which decays e x p o n e n t i a l l y as one moves away f r o m t h e upper s u r f a c e .

* +

where @,(;,t) i s t h e magnetic p o t e n t i a l

.

+ V - b = O

* ³ ( 3

I n t h e magnetic medium b and h a r e r e l a t e d by t h e magnetic s u s c e p t i b i l i t y t e n s o r which, i n t h e l o n g wavelength v i m i t , depends o n l y on t h e f r e q u e n c y Q o f t h e s p i n m o t i o n . Thus

* W . - *

b = hd + 4nx(Q) hd ( 4

where t h e n o n v a n i s h i n g elements o f t h e t e n s o r x i n t h e f e r r o m a g n e t a r e

and

JOURNAL DE PHYSIQUE

The s u s c e p t i b i l i t y t e n s o r can be o b t a i n e d f r o m t h e l i n e a r i z e d B l o c h e q u a t i o n s .

Throughout t h e paper we measure f r e q u e n c ÿ i n u n i t s o f magnetic f i e l d and r i s a pheno- m e n o l o g i c a l s p i n damping t i m e . I t i s easy t o show t h a t i n t h e magnetic f i l m s i n t h e c o o r d i n a t e system o f F i g u r e 1, gm s a t i s f i e s

an a n i s o t r o p i c f o r m of L a p l a c e ' s equations,sometimes r e f e r r e d t o as t h e Walker equa- t i o n . I n t h e nonmagnetic f i l m and i n t h e vacuum, we must have

We must ç o l v e t h i s s e t o f e q u 9 t i o n s f o r 0, s u b j e c t t o t h e boundary c o n d i t i o n s t h a t t h e t a n g e n t i a l components o f hd and t h e normal components o f 6 a r e c o n t i n u o u s a t each i n t e r f a c e .

The s o l u t i o n f o r b u l k waves on a s u p e r l a t t i c e has t h e form o f a Rloch wave p e r p e n d i - c u l a r t o t h e l a y e r i n g and a p l a n e wave p a r a l l e l t o t h e l a y e r i n g . Thus we e x p e c t a s o l u t i o n o f t h e f o r m

Here Q, governs t h e s p a t i a l v a r i a t i o n o f t h e envelope f u n c t i o n , n indexes t h e l a y e r s ,

0 ( y - n L ) i s t h e s o l u t i o n o f L a p l a c e ' s E q u a t i o n o r t h e Walker E q u a t i o n f o r t h e appro- p r i a t e l a y e r and i s a f u n c t i o n o f t h e d i s t a n c e w i t h i n each f i l m , and f i n a l l y L =dl+d2 i s t h e p e r i o d o f t h e s t r u c t u r e .

I f we w i s h t o c o n s i d e r s u r f a c e waves on a s e m i - i n f i n i t e s u p e r l a t t i c e , Eq. ( 1 0 ) no l o n g e r h o l d s s i n c e t h e i n t r o d u c t i o n o f t h e s u r f a c e e l i m i n a t e s t h e p e r f e c t p e r i o d i c i t y . We t h e n l o o k f o r s o l u t i o n s w i t h a surface wave c h a r a c t e r . Thus we t a k e

Now cx governs t h e e x p o n e n t i a l decay o f t h e envelope f u n c t i o n as one moves away f r o m t h e s u r f a c e .

By u s i n g t h e e q u a t i o n s f o r t h e b u l k and s u r f a c e wave s o l u t i o n s g i v e n i n Eq. ( 1 0 ) and Eq. ( 1 1 ) i n L a p l a c e ' s e q u a t i o n o r i n t h e Walker e q u a t i o n and a p p l y i n g t h e a p p r o p r i a t e boundary c o n d i t i o n s as mentioned e a r l i e r , we o b t a i n t h e d i s p e r s i o n r e l a t i o n f o r t h e frequency R as a f u n c t i o n o f QI , ^Q,, , dl and d2. T h i s i s done f o r b o t h t h e b u l k and s u r f a c e s p i n waves. The complete e x p r e s s i o n s f o r t h e r e s u l t s a r e l e n g t h y , and we do n o t reproduce them here.

We w i l l i l l u s t r a t e t h e r e s u l t s w i t h some n u m e r i c a l examples. As a mode1 system we c o n s i d e r a l t e r n a t i n g l a y e r s o f f e r r o m a g n e t i c N i (M, = 480 G ) on nonmagnetic Mo. The a p p l i e d f i e l d H i s 1000 G. T h i s system has been s t u d i e d r e c e n t l y t h r o u g h B r i l l o u i n s c a t t e r i n g . We P i m i t o u r a t t e n t i o n t o p r o p a g a t i o n p e r p e n d i c u l a r t o t h e appl i e d f i e l d . I n F i g u r e 3 we p r e s e n t r e s u l t s f o r t h e frequency o f t h e v a r i o u s modes v e r s u s t h e r a t i o d l / d 2 . R e s u l t s a r e p l o t t e d f o r t h r e e d i f f e r e n t v a l u e s o f Q,, d l . The g e n e r a l

f e a t u r e s o f a l 1 t h r e e s e t s o f curves a r e as f o l l o w s : 1 ) There i s a band o f b u l k s t a t e s ( b u l k s t a t e s f o r t h e s u p e r l a t t i c e - i n each f e r r o m a g n e t i c f i l m t h e r e i s a s u r f a c e w a v e - l i k e mode) w i t h a maximum range i n f r e q u e n c y f r o m K B t o .5(Ho + ^{B )}

where B = Ho + 4nMs, 2 ) I n general as Q,L i n c r e a s e s , t h e f r e q u e n c y o f t h e mode de- creases. For t h e v a l u e s of Q I , d2 used here, t h e d e n s i t y o f s t a t e s i s l a r g e s t near Q,L = T. As Q I , dl i s i n c r e a s e d , t h e d e n s i t y o f s t a t e s becomes more u n i f o r m o v e r t h e a l l o w e d frequency range. 3 ) There i s a s u r f a c e mode f o r which t h e f r e q u e n c y i s i n d e - pendent of t h e r a t i o d l / d 2 and equal t o t h a t o f t h e Damon-Eshbach f r e q u e n c y o f t h e s e m i - i n f i n i t e ferromagnet. T h i s mode e x i s t s however

2

F i g . 3 - Frequency o f v a r i o u s modes vs t h e r a t i o d /d2. B u l k modes o f t h e s t a c k a r e shown w i t h a shaded r e g i o n t h e s u r f a c e modes a r e skown by a s o l i d l i n e .

One method t o probe t h e i n t e r e s t i n g f e a t u r e s o f t h e c o l l e c t i v e e x c i t a t i o n s i s t h r o u g h a l i g h t s c a t t e r i n g experiment. The geometry f o r such an experiment i s i l l u s t r a t e d i n F i g u r e 4. The i n c i d e n t l i g h t has f r e q u e n c y wo and wavevector ko. The i n c i d e n t wave may be s c a t t e r e d by t h e i n t e r a c t i o n w i t h a spinwave o f f r e q u e n c y Q and wavevector Q,,

.

Wo - W s = R

( 1 2 )

F i g . 4 - Geometry o f t h e l i g h t - s c a t t e - r i n g e x p e r i m e n t . The i n c i d e n t l i g h t , w i t h wavevector ko and frequency(;io, s t r i k e s t h e s u r f a c e a t an a n g l e 8, w i t h r e s p e c t t o t h e s u r f a c e normal. The s c a t - t e r e d l i g h t has wavevector k s and f r e - quency 0 S.

C5-320 JOURNAL DE PHYSIQUE

Thus i n a l i g h t s c a t t e r i n g experiment i f one p l o t s i n t e n s i t y o f t h e s c a t t e r e d l i g h t v e r s u s f r e q u e n c y s h i f t , and a peak a t a p a r t i c u l a r f r e q u e n c y s h i f t i s seen, t h i s corresponds t o s c a t t e r i n g f r o m a p a r t i c u l a r spinwave mode o f f r e q u e n c y R.

I n F i g u r e 5 we p r e s e n t a t h e o r e t i c a l c a l c u l a t i o n f o r t h e l i g h t s c a t t e r i n g spectrum f r o m a magnetic s u p e r l a t t i c e s t r u c t u r e . We i n v e s t i g a t e i n t h e t o p f i g u r e d2 = 3 d l so no s u r f a c e wave i s expected. We see a broad peak a t f r e q u e n c y j u s t above Rb. T h i s peak corresponds t o t h e s c a t t e r i n g f r o m t h e b u l k spinwave band o f t h e s u p e r l a t t i c e as can be seen by comparison w i t h F i g u r e 3. I n t h e l o w e r f i g u r e we g i v e t h e l i g h t s c a t t e r i n g spectrum i n t h e case d l = 3d2. I n t h i s case we see a new peak a t h i g h e r f r e q u e n c i e s i n a d d i t i o n t o t h e b r o a d band seen above. T h i s new peak i s due t o t h e s c a t t e r i n g f r o m t h e s u r f a c e spinwaves o f t h e s u p e r l a t t i c e . We n o t e t h a t t h e s c a t t e r - i n g f r o m t h e s u r f a c e spinwave peak i s s t r o n g l y n o n r e c i p r o c a l . The s u r f a c e spinwave peak appears o n l y on one s i d e o f t h e spectrum. T h i s f e a t u r e i s c o n s i s t e n t w i t h t h e r e s u l t s f o r t h e s c a t t e r i n g o f l i g h t f r o m s u r f a c e spinwaves on a s e m i - i n f i n i t e f e r r o - magnet and i s due t o t h e n o n r e c i p r o c a l n a t u r e o f t h e s u r f a c e spinwave mode.

We m e n t i o n t h a t r e c e n t l y t h e s e t h e o r e t i c a l c a l c u l a t i o n s have been v e r i f i e d by e x p e r i - ments /3,4/. The experiments show c l e a r l y t h a t t h e s u r f a c e spinwave mode e x i s t s f o r d l > d2 and does n o t e x i s t f o r dl < d2. The d e t a i l s o f t h e l i g h t s c a t t e r i n g c a l c u l a - t i o n a r e g i v e n i n Ref. 5. Other t h e o r e t i c a l c a l c u l a t i o n s a r e a l s o i n agreement / 3 / . As n o t e d i n t h e i n t r o d u c t i o n , t h e B l o c h e q u a t i o n s which govern t h e s p i n system a r e n o n l i n e a r . I n many problems, as i n t h e d i s c u s s i o n above, we have l i n e a r i z e d t h e s e e q u a t i o n s t o l o o k o n l y a t small a m p l i t u d e o s c i l l a t i o n s . However, i n t e r e s t i n g e f f e c t s a l s o o c c u r i f t h e n o n l i n e a r terms a r e r e t a i n e d . I n t h i s s e c t i o n we r e v i e w some r e - c e n t work on t h e n o n l i n e a r m i x i n g o f b u l k and s u r f a c e m a g n e t o s t a t i c spinwaves i n i n f i n i t e and s e m i - i n f i n i t e ferromagnets.

We d e a l h e r e w i t h t h e m i x i n g o f two waves t o produce a r e s o n a n t l y enhanced t h i r d wave. We b e g i n o b t a i n i n g t h e c o n d i t i o n s under which a m a g n e t o s t a t i c wave d e s c r i b e d by a wavevector k and f r e q u e n c y w(k) can i n t e r a c t n o n l i n e a r l y w i t h a second magneto- s t a t i c wave o f wavevector k ' and f r e q u e n c y w ( k ' ) t o produce a wave w i t h wavevector k+k' and f r e q u e n c y w ( k ) + w ( k l ) , whose a m p l i t u d e i s r e s o n a n t l y enhanced. I n general

F i g . 5 - L i g h t - s c a t t e r i n g spectrum f r o m a s t r u c t u r e o f a l t e r n a t i n g l a y e r s o f N i and Mo. Here dl f d . We see a peak a t Rs i n t h e case dl > d2 due t o s c a t t e r i n g from s u r f a c e waves o f f h e l a y e r e d s t r u c t u r e . F o r dl < d 2 t h e r e i s no peak a t as.

t h i s r e q u i r e s t h a t t h e f r e q u e n c y and wavevector correspond t o a p o i n t on t h e d i s p e r - s i o n c u r v e o f t h e l i n e a r medium. Thus t h e resonance c o n d i t i o n i s

Because o f t h e unusual p r o p e r t y o f t h e m a g n e t o s t a t i c spinwaves c o n s i d e r e d h e r e -

t h e f r e q u e n c y depends o n l y on t h e d i r e c t i o n o f p r a p a g a t i o n - t h i s c o n d i t i o n may be s a t i s f i e d i n an unusual way. I f t h e d i r e c t i o n s o f k and k ' a r e f i x e d , v a r y i n g t h e magnitude o f k r e l a t i v e t o t h a t o f k ' v a r i e s t h e d i r e c t i o n o f t h e o u t p u t wave. Then by v a r y i n g t h e d i r e c t i o n o f t h e o u t p u t wave, one v a r i e s t h e f r e q u e n c y u n t i l t h e r e - sonance c o n d i t i o n i s s a t i s f i e d .

An a l t e r n a t i v e method t o f i n d a geometry f o r resonance i s t o f i x t h e d i r e c t i o n o f one o f t h e i n p u t waves and o f t h e generated wave, and t o f i n d t h e d i r e c t i o n o f t h e second i n p u t wave f o r which t h e resonance c o n d i t i o n h o l d s . I n t h i s case, t h e d i r e c - t i o n o f t h e generated wave may be f i x e d because t h e magnitudes o f t h e p r o p a g a t i o n wavevectors may be a r b i t r a r l y changed u n t i l t h e v e c t o r sum l i e s i n t h e p r o p e r d i r e c - t i o n . We w i l l s t u d y n o n l i n e a r m i x i n g u s i n g t h i s geometry.

The geometry i s i l l u s t r a t e d i n F i g u r e 6. An e x t e r n a l , c o n s t a n t magnetic f i e l d i s a p p l i e d a l o n g t h e z a x i s , and t h e s a t u r a t i o n m a g n e t i z a t i o n i s a l s o a l o n g t h i s a x i s . F o r b u l k waves we c o n s i d e r a f e r r o m a g n e t i c medium t h a t occupies a l 1 o f space. For s u r f a c e waves, we c o n s i d e r a s e m i - i n f i n i t e geometry where t h e f e r r o m a g n e t o c c u p i e s t h e r e g i o n y > 0.

We f i r s t c o n s i d e r two b u l k waves p r o p a g a t i n g i n t h e xz p l a n e as shown i n F i g u r e 6.

The g e n e r a l e q u a t i o n f o r t h e f r e q u e n c y of b u l k spinwaves as a f u n c t i o n o f t h e a n g l e o f p r o p a g a t i o n f3 w i t h r e s p e c t t o t h e x a x i s i s

F O

I p r o p a g a t i o n a l o n g t h e x a x i s t h e f r e q u e n c y i s t h u s ( ~ ~ 8 ) ~ ' ~ where B = Ho + ^4nMs.

For convenience, we s e t t h e geometry o f t h e two i n c i d e n t waves so t h a t t h e o u t p u t wave i s always d i r e c t e d a l o n g x. Thus we l e t kz = -k; ( b u t n o t k, = k i ) . The c o n d i - t i o n r e l a t i n g 0 and 8 ' may then be found from Eqs. ( 1 3 ) and ( 1 4 ) .

T h i s c o n d i t i o n may be e a s i l y achieved f o r t y p i c a l v a l u e s o f Ho and MS as we w i l l see l a t e r .

Ho,MS, z

F i g . 6 ; The geometry c o n s i d e r e d i n t h i s paper. The two i n c i d e n t waves w i t h wave- v e c t o r k and 1' propagate a t angles 8 and 0 ' w i t h r e s p e c t t o t h e x a x i s . The gen- e r a t e d wave w i t h wavevector &+Io i s always d i r e c t e d a l o n g t h e x a x i s .

C5-322 JOURNAL DE PHYSIQUE

The c o n d i t i o n f o r resonance i n t h e m i x i n g o f s u r f a c e waves may be d e r i v e d s i i n i l a r l y . I n t h i s case, t h e f r e q u e n c y o f t h e s u r f a c e waves as a f u n c t i o n o f 0 i s g i v e n by

The s u r f a c e modes a r e r e s t r i c t e d t o propagate o n l y f o r a n g l e s l0l<8 where cos0 ⁼ ( H / B ) I / ~ . W i t h t h i s r e s t r i c t i o n , t h e resonance c o n d i t i o n r e l a t i n g t f i e a n g l e s 0 2nd 0 ' o f t h e two s u r f a c e waves i s g i v e n b y

Again, t h i s c o n d i t i o n may be achieved f o r t y p i c a l v a l u e s o f Ho and MS.

We a l s o n o t e t h e p o s s i b i l i t y o f u s i n g two b u l k waves t o g e n e r a t e a r e s o n a n t l y en- hanced s u r f a c e wave. I n t h i s case t h e resonance c o n d i t i o n i s

w b ( k ) + w b ( k l ) = ws(k + k ' ) (18)

I n o r d e r t o o b t a i n f e e l i n g f o r t h e e f f i c i e n c y o f t h e g e n e r a t i o n o f t h e new wave, we c a l c u l a t e t h e t i m e average o f t h e square o f one o f t h e components o f m a g n e t i z a t i o n f o r b o t h t h e generated wave and t h e sum o f t h e i n p u t waves. The d e t a i l s o f t h e c a l - c u l a t i o n a r e p r e s e n t e d elsewhere /6/ ; here we g i v e o n l y t h e r e s u l t s . I f we t a k e t h e component o f r n a g n e t i z a t i o n a l o n g y, t h e n t h e t i m e average o f M$ f o r t h e gene a t e d wave we w r i t e as M$ t h e t i m e average o f M$ f o r t h e i n p u t waves we w r i t e as Ml.

E

We c o n s i d e r t h e geometry shown i n F i g u r e 6 f o r b o t h b u l k and s u r f a c e waves. F o r t h e ferromagnet we t a k e y t t r i u m i r o n g a r n e t w i t h Ms = 140 G and a p p l y a f i e l d Ho = 50 G.

We f i x 8 = 500 and p l o t t h e r a t i o M2/b12 as a f u n c t i o n o f 8' i n F i g u r e 7. I n o r d e r t o t o c a l c u l a t e t h e s e c u r v e s we have NU>

-

aides

2 2

F i g . 7 - M /M. vs. 8'. 8 i s f i x e d a t 50'. Note t h a t t h e resonance f o r t h e s u r f a c e 'waves occu$s A t a d i f f e r e n t f r e q u e n c y f r o m t h e r e j o n a n c e o f .the b u l k wave. I n t h e u n i t s used A i s t h e a m p l i t u d e of h x ( x , t ) = -a$(x,t)/axx f o r one o f t h e i n c i d e n t waves.

2

N

>

4500

N-'

3000

\

N O 1500 E

O

\ .

BULK WAVES

66. 67. 68') 63- 70. 71- 72.

8'

P -

SURFACE WAVES 1

- -

-

-

t h a t f o r 8 ' = 67.6U t h e r e i s a c l e a r resonance f o r t h e g e n e r a t i o n o f b u l k waves. For surface waves t h e r e i s a resonance a t 0 ' = 71.4O. For s u r f a c e wave b o t h 8 and 0 ' a r e w i t h i n t h e c r i t i c a l a n g l e which i s 8 0 . 4 ~ . I t i s i n t e r e s t i n g t o n o t e t h a t t h e r e s o - nance f o r t h e b u l k waves i s c l e a r l y b r o a d e r t h a n f o r t h e s u r f a c e waves. The p o s i t i o n s f o r t h e resonances f o r t h e b u l k and surTace waves agree w i t h Eq. ( 1 5 ) and Eq. ( 1 7 ) .

I n summary, we have seen t h a t t h e n o n l i n e a r terms i n t h e e q u a t i o n s o f m o t i o n can l e a d t o t h e m i x i n g o f b o t h b u l k and s u r f a c e spinwaves. The generated waves f r o m t h i s mix- i n g can be r e s o n a n t l y enhanced i f some s i m p l e c o n d i t i o n s on t h e d i r e c t i o n s o f t h e two i n c i d e n t waves a r e met.

1. Zheng, J.Q., K e t t e r s o n , J.B., F a l c o C.M., and S c h u l l e r I.K, J.App1 .Phys. 53 3150

(1982) -

2. ? h a l e r , B.J., K e t t e r s o n J.B., and H i l l i a r d J.E., Phys.Rev.Lett.

o,

4. G r i m s d i t c h , M., Khan, M., Kueny A., and S c h u l l e r I . K . , ( u n p u b l i s h e d ) 5. Camley, R.E., Rahman T.S., and M i l l s , Phys. Rev. B27, 261 (1983) 6. Camley R.E. and Maradudin A.A., Phys. Rev. L e t t . 49, - 168 (1982)