ENERGY RENORMALISATION AND DAMPING OF SURFACE SPIN WAVES IN HEISENBERG FERROMAGNETS

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ENERGY RENORMALISATION AND DAMPING OF SURFACE SPIN WAVES IN HEISENBERG

FERROMAGNETS

D. Kontos, M. Cottam

To cite this version:

D. Kontos, M. Cottam. ENERGY RENORMALISATION AND DAMPING OF SURFACE SPIN

WAVES IN HEISENBERG FERROMAGNETS. Journal de Physique Colloques, 1984, 45 (C5), pp.C5-

329-C5-333. �10.1051/jphyscol:1984548�. �jpa-00224166�

JOURNAL DE PHYSIQUE

Colloque C5, supplbment au n04, Tome 45, avril 1984 page C5-329

E N E R G Y R E N O R M A L I S A T I O N A N D D A M P I N G O F S U R F A C E S P I N W A V E S IN HE I S E N B E R G F E R R O M A G N E T S

D.E.

Kontos and

M.G.

Cottam

Department of Physics, University of Essex, CoZchester

C 0 4 3SQ, U . K .

~'esur~16

-

En u t i l i s a n t l a t h z o r i e de f o n c t i o n s de Green, nous s t u d i o n s l ' e f f e t d'une s u r f a c e s u r l e s i n t e r a c t i o n s rilagnon-rilagnon dans un c o r p s ferromagn/etique de Heisenberg. On en d k d u i t 1' 6 n e r g i e renormalis6e e t l'amortisser~nent d'ondes de s p i n de s u r f a c e .

A b s t r a c t

-

A Green f u n c t i o n t h e o r y i s ernployed t o s t u d y t h e e f f e c t of a s u r f a c e on magnon-nlagnon i n t e r a c t i o n s i n a Heisenberg ferrornagnet. R e s u l t s a r e deduced f o r t h e renorrllalised energy and darnping of t h e s u r f a c e s p i n waves.

I t

i s

w e l l k n o w n t h a t under c e r t a i n c o n d i t i o n s l o c a l i s e d s u r f a c e s p i n waves a r e p r e d i c t e d t o e x i s t i n ~ e i s e n b e r g f e r r o n ~ a g n e t s . The p r o p e r t i e s of t h e s e rnodes, and t h e i r i n f l u e n c e on therrnodynan~ic behaviour, s p i n wave resonance, l i g h t s c a t t e r i n g , e t c . , have been surmarised i n v a r i o u s review a r t i c l e s , e.g. /1-3/. Since s p i n waves a r e n o t e x a c t e i g e n s t a t e s of t h e Heisenberg Hamiltonian, i n t e r a c t i o n e f f e c t s

w i l l

occur r e s u l t i n g i n an energy renorrnalisation and damping of t h e modes. I n i n f i n i t e ferromagnets t h e t r e a t m e n t of i n t e r a c t i o n s between bulk s p i n waves h a s been p u t on a r i g o r o u s b a s i s by Dyson / 4 / . However, t h e presence of a s u r f a c e

w i l l

g i v e r i s e t o much r i c h e r and roore complicated schemes of s p i n wave i n t e r a c t i o n s , s i n c e s c a t t e r i n g p r o c e s s e s may t a k e p l a c e i n v o l v i n g s u r f a c e s p i n waves a s w e l l a s b u l k s p i n waves (which i n any c a s e have modified p r o p e r t i e s c l o s e t o t h e s u r f a c e ) . I n t h i s paper we efi~ploy a Green f u n c t i o n forrtlalism t o s t u d y s p i n wave i n t e r a c t i o n s i n s e m i - i n f i n i t e Heisenberg ferromagnets a t low temperatures T < < T

.

S p e c i f i c a l l y we deduce e x p r e s s i o n s f o r t h e energy renorrnalisation and darnping o f C s u r f a c e s p i n waves due t o t h e i r i n t e r a c t i o n s e i t h e r w i t h bulk s p i n waves o r w i t h s u r f a c e s p i n waves. Previous r e l a t e d work h a s , f o r exanlple, i n c l u d e d a c a l c u l a t i o n of t h e s u r f a c e s p i n wave darnping i n a simple cubic Heisenberg ferromagnet w i t h a (001) s u r f a c e f o r t h e s p e c i a l c a s e of nearest-neighbour exchange and z e r o s u r f a c e a n i s o t r o p y /5/. I n t h e p r e s e n t a n a l y s i s we c o n s i d e r t h e energy r e n o r r l a l i s a t i o n a s w e l l a s t h e d a r ~ p i n g , and we examine e f f e c t s of next-nearest-neighbour exchange, modified exchange n e a r t h e s u r f a c e , s u r f a c e a n i s o t r o p y and a p p l i e d magnetic f i e l d . Only a b r i e f o u t l i n e

i s

g i v e n h e r e ; d e t a i l s w i l l be p u b l i s h e d elsewhere.

We c o n s i d e r a s e m i - i n f i n i t e ferromagnet occupying t h e h a l f - s p a c e z

d

0 and d e s c r i b e d by t h e Har~liltonian

where Si

i s

a s p i n o p e r a t o r , J ( i , j )

i s

an exchange i n t e r a c t i o n , and t h e summations a r e over a l l rllagnetic s i t e s . The q u a n t i t i e s H and D ( i ) denote r e s p e c t i v e l y an a p p l i e d rllagnetic f i e l d and a s u r f a c e a n i s o t r o p y f i e l d p e r p e n d i c u l a r t o t h e s u r f a c e ; we assume D ( i ) t o be z e r o e x c e p t i n t h e s u r f a c e l a y e r ( z = 0 ) where

i t

has t h e v a l u e D. We c o n s i d e r h e r e

a

simple cubic l a t t i c e with (001) c r y s t a l l o g r a p h i c o r i e n t a t i o n of t h e s u r f a c e ; more g e n e r a l l y we have a l s o d e r i v e d r e s u l t s f o r b.c.c. and f . c . c . ferromagnets and f o r (011) s u r f a c e s . For t h e exchange terrrls we r e s t r i c t a t t e n t i o n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984548

C5-330 JOURNAL DE PHYSIQUE

t o two sirnple rilodels:

Model 1: Exchange J ( i , j ) c o u p l e s n e a r e s t neighbours o n l y , having t h e v a l u e JS i f b o t h s p i n s a r e i n t h e s u r f a c e l a y e r and t h e bulk v a l u e Jl o t h e r w i s e .

Model 2: A l l exchange i n t e r a c t i o n s a r e t h e same a s i n t h e bulk specimen, w i t h J ( i , j ) equal t o J1 between n e a r e s t neighbours, J2 between n e x t - n e a r e s t neighbours and z e r o o t h e r w i s e .

Using t h e Holstein-Primakoff r e p r e s e n t a t i o n t h e Hamiltonian (1) may b e transformed t o boson o p e r a t o r s , keeping terms up t o Fourth o r d e r . A s

i s

w e l l known t h i s type of approach can l e a d t o i n c o n s i s t e n c i e s f o r s m a l l v a l u e s of t h e s p i n S , b u t t h e s e d i f f i c u l t i e s a r e avoided i f 1/S

i s

formally t r e a t e d a s a small parameter and c a l c u l a t i o n s a r e c a r r i e d o u t c o n s i s t e n t l y t o each power i n 1/S ( s e e / 6 / ) . The l i n e a r s p i n wave p a r t of t h e Harniltonian can t h e n b e d i a g o n a l i s e d by a f u r t h e r t r a n s f o r m a t i o n t o normal mode o p e r a t o r s , a n d 3 t t a k e s t h e form

Here

q

= (qx,qy)

i s

a wavevector p a r a l l e l t o t h e s u r f a c e , and !J l a b e l s t h e normal modes-of t h e s e m i - i n f i n i t e system which c o n s i s t of a quasi-continuum of bulk s p i n waves w i t h wavevectors Q = ( q , q z ) and a d i s c r e t e s u r f a c e s p i n wave branch c h a r a c t e r - i s e d by wavevector

2. A s

expected, t h e e i g e n v a l u e s E ( q U ) have t h e u s u a l form

(e.9. s e e /1/) f o r t h e e n e r g i e s EB

(2)

and ES

(2)

of bulE and s u r f a c e s p i n waves r e s p e c t i v e l y i n a l i n e a r approximation. The s u r f a c e mode d i s p e r s i o n r e l a t i o n

i s

-

1

E S ( q )

-

= g~ B H+S [u ( 0 ) -U ( s ) ] + ~ s v (0) +Sv

(3) [A (2)

^f

A (4) 1

( 3 ) w i t h

u ( q ) = 4J1y

(3)

V ( q ) = J1 (Model 1)

( 4 )

u

( q )

-

= 4 J y ₁

(3)

+4J2cos (qxa) c o s ( %a) v

(3)

= ~ ~ + 4 J ~ y

(4)

(Model 2)

(Model 1) (Model 2)

where y ( q ) = [cos (q a ) +cos ( q a ) ]/2, d = ~ ! J ~ D / S V ( 0 )

,

and a

i s

t h e l a t t i c e c o n s t a n t . The

existence

c o n d i r i o n s f o r Y s u r f a c e s p i n waves a r e

A ( q )

<

-1

f o r an a c o u s t i c mode and

A ( q )

^{z 1} f o r an o p t i c mode r e s p e c t i v e l y , and t h e s e r e s t r i c t t h e v a l u e s of t h e parameters.

The terrt1il;Yint i n ( 2 ) , which d e s c r i b e s t h e l e a d i n g e f f e c t of magnon-magnon i n t e r - a c t i o n s i n t h e s e m i - i n f i n i t e ferromagnet,

i s

q u a r t i c i n t h e o p e r a t o r s a + ( q u ) and a(q1l). Here we d i s c u s s t h e r e s u l t i n g energy r e n o r m a l i s a t i o n and damping of t h e s u r f a c e s p i n waves. The c a l c u l a t i o n s can be c o n v e n i e n t l y performed by e v a l u a t i n g Green f u n c t i o n s ((a (%!.I) ;a"

(9' u '

)

))

w i t h i n a diagrammatic p e r t u r b a t i o n expansion in)lint, and t h e n a n a l y s i n g t h e complex energy p o l e s of t h e s e Green f u n c t i o n s . I

-

ENERGY RENORMALISATION

To f i r s t o r d e r of p e r t u r b a t i o n t h e r e

i s

a s h i f t AE i n t h e s p i n wave e n e r g i e s . For t h e s u r f a c e rt~odes we w r i t e AES = AESS+AESB, where AESS and AESB a r e t h e c o n t r i b u - t i o n s due t o i n t e r a c t i o n s with s u r f a c e modes and bulk modes r e s p e c t i v e l y . We f i n d

where _k

i s

a two-dimensional wavevector p a r a l l e l t o t h e s u r f a c e , _K = (,k,kz)

i s

a three-dimensional wavevector, and n (E) = l / [exp (E/kgT)

-11 .

^NII and NI denote

r e s p e c t i v e l y t h e number of atomic l a y e r s p a r a l l e l t o t h e s u r f a c e and t h e number of magnetic s i t e s i n each l a y e r ( b o t h nurubers raacroscopically l a r g e ) . The i n t e r a c t i o n v e r t i c e s WSS and W i n t h e c a s e of Model 1 a r e

SB

+ 2[l+~-'(q)] 2 s i n 2 (kz/2)

+ [

^[l-d-'

(r)]

/ [ 1 - 2 ~ - ~

(3)

c o s (2kz)

(r)]]

( 8 ) x

[ [ a

(9," +A- 1

(3)

(l+A-'(%)

I] (9

c o s (8-2kz) -cos8]

where o = (1-J

s

/J 1 ) and

a (q,k)

= 4[l+y

( ~ 5 )

-y

(q_)

-y

(511.

The phase a n g l e 0 depends on t h e

reflection

p r o p e r t i e s a t t h e s u r f a c e of a b u l k s p i n wave w i t h wavevector

5

and

i s

d e f i n e d by

The q u a n t i t i e s WSS and WSB p l a y an analogous r o l e t o t h e Dyson i n t e r a c t i o n v e r t e x /4/ i n t h e r e n o r m a l l s a t i o n of bulk s p i n waves i n an i n f i n i t e ferromagnet. However, t h e y a r e v e r y much more complicated due t o t h e lowering of symmetry produced by t h e s u r f a c e ; t h e y i n c o r p o r a t e t h e s t a t i s t i c a l weighting of t h e v a r i o u s modes n e a r t h e s u r f ace.

Because of t h e Bose f a c t o r s t h e summations i n (6) a r e dominated by t h e behaviour a t s m a l l ) : and

5 ,

where WSS and WSB s i m p l i f y . I f t h e wavevector summations a r e r e p l a c e d by i n t e g r a l s , t h e y can be e v a l u a t e d a n a l y t i c a l l y i n v a r i o u s l i m i t i n g c a s e s . The replacement

i s

s t r a i g h t f o r w a r d f o r t h e components of _k s i n c e t h e r e

i s

t r a n s - l a t i o n a l i n v a r i a n c e p a r a l l e l t o t h e s u r f a c e , w h i l s t t h e c o r r e c t procedure f o r r e p l a c i n g t h e summation over kz by an i n t e g r a t i o n h a s bee3 qiven by M i l l s /7/. A s an example we d i s c u s s some r e s u l t s f o r long wavelenths ( a q <<

1)

and f o r z e r o s u r f a c e a n i s o t r o p y (d = 0 ) . I f 0 < a < l t h e r e

i s

a s u r f a c e branch i n t h e unrenor- malised s p i n wave spectrum o c c u r r i n g j u s t below t h e continuum of b u l k modes. The lower edge of t h e continuum h a s energy EB(q_,O)

%

gvBH+Sv(O) a2q2+0(q4) and t h e s u r f a c e mode with energy ES

(q) i s

s p l i t o f f below t h i s by an amount % a2a4q4. I n g e n e r a l t h i s c l o s e proximity can produce a s u b t l e i n t e r p l a y between e f f e c t s due t o t h e s u r f a c e modes and t h o s e due t o s u r f a c e p e r t u r b a t i o n of t h e b u l k modes, e.g. a s found i n c a l c u l a t i o n s of s u r f a c e thermodynamic p r o p e r t i e s / 7 , 8 / . On d e f i n i n g T = kBT/Sv(0) and h = gpBH/Sv(0), we o b t a i n t h e f o l l o w i n g l e a d i n g o r d e r c o n t r i b u - t i o n s , assunling a2q2 << T <<

1:

Here F ( n , h / ~ )

i s

t h e Bose-Einstein i n t e g r a l f u n c t i o n (e.g. s e e / 6 / ) , which s i m p l i - f i e s t o t h e Riemann z e t a f u n c t i o n < ( n ) i f h = 0 . The dominant c o n t r i b u t i o n t o AES i n t h i s c a s e

i s

provided by t h e f i r s t term i n

(11)

and comes from i n t e r a c t i o n s w i t h bulk s p i n waves. 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 i s l e a d i n g o r d e r c o n t r i b u t i o n t o AES h a s t h e same form a s t h e energy c o r r e c t i o n f o r a bulk s p i n wave with small wavevector (q,O) i n an i n f i n i t e ferromagnet /4,6/. T h i s r e s u l t seems reasonable i n view of t h e e a r l i e r comments concerning t h e proximity of t h e s u r f a c e branch t o t h e lower edge of t h e bulk continuum. Moreover t h e c o n d i t i o n a2q2 << T assumed i n d e r i v i n g (10) and

(11)

i m p l i e s t h a t t h e p e n e t r a t i o n d e p t h ( 2 . l / o a q 2 ) of t h e s u r f a c e

C5-332 J O U R N A L DE PHYSIQUE

mode

i s

r e l a t i v e l y l a r g e , e . g . it

i s

l a r g e r than t h e t h i c k n e s s (a a / ~

4

) over which t h e magnetisation

i s

a p p r e c i a b l y p e r t u r b e d by t h e s u r f a c e /8/. The r e s u l t s f o r l a r g e r q

w i l l

be d i s c u s s e d elsewhere.

A d i f f e r e n t behaviour f o r AES

i s

p r e d i c t e d i f t h e r e

i s

a s u r f a c e a n i s o t r o p y f i e l d (d

#

0 ) . We t a k e t h e case of d < 0 and 0 < a < 1, which e n s u r e s t h e e x i s t e n c e of

an

a c o u s t i c s u r f a c e s p i n wave (provided t h e a p p l i e d f i e l d

i s

n o t t o o s m a l l ) . For a n i s o t r o p y f i e l d s s a t i s f y i n g T << d 2 < <

1

and a2q2 << d2 we f i n d t h a t t h e dominant c o n t r i b u t i o n t o AES

(3) i s

approximately p r o p o r t i o n a l t o d3r'?

(1

,ES ( 0 ) /kBT)

.

^{I t}

comes mainly from i n t e r a c t i o n s w i t h s u r f a c e s p i n waves ( i . e . from A E S S ) , u n l i k e t h e p r e v i o u s example with d = 0. T h i s d i f f e r e n c e can be understood a s a r i s i n g p a r t l y due t o an approximate node i n t h e bulk s p i n wave amplitudes a t t h e s u r f a c e i n t h e p r e s e n t c a s e , and p a r t l y due t o a r e l a t i v e enhancement i n t h e nuluber of t h e r m a l l y e x c i t e d s u r f a c e s p i n waves ( s i n c e EB(0) -ES(0)

2

d 2 s v ( 0 ) f o r d2 << 1)

.

We have a l s o c a r r i e d o u t c a l c u l a t i o n s t o i n c l u d e t h e e f f e c t of n e x t - n e a r e s t - neighbour exchange i n t e r a c t i o n s according t o Model 2. When d = 0 we f i n d t h a t many of t h e r e s u l t s can be expressed i n t h e same form a s f o r Model

1

b u t with r e d e f i n e d c o e f f i c i e n t s . For example, i f a2q2 < < T << 1 with Jl > 0 and J2 > 0 t h e same q and T dependences a r e p r e d i c t e d a s i n (10) and (11) b u t t h e o v e r a l l c o e f f i c i e n t s i n v o l v e J 2 a s w e l l a s J1. The d i f f e r e n c e s between t h e two models a r e more s i g n i f i - c a n t f o r d

#

0 .

I1

-

^DAMPING

C o n t r i b u t i o n s t o t h e damping a r e o b t a i n e d on r e n o r m a l i s i n g t h e s p i n waves t o second o r d e r of p e r t u r b a t i o n . The mechanism i s t h e u s u a l low-temperature s c a t t e r i n g p r o c e s s i n v o l v i n g f o u r s p i n waves, e x c e p t t h a t t h e s e may now be bulk modes o r s u r f a c e modes. Hence t h e damping

rS(q)

of a s u r f a c e s p i n wave

w i l l

be t h e sum of f o u r c o n t r i b u t i o n s denoted by

TS (q;

4s)

, rS (9:

35,181

, TS (3;

2S,28) and

(2;

l S , 3B)

,

according t o how many s u r f a c e (S) and bulk ( B ) modes a r e involved. W e have o b t a i n e d e x p r e s s i o n s f o r a l l f o u r terms, b u t f o r s i m p l i c i t y we d i s c u s s here j u s t t h e

c o n t r i b u t i o n

rS

(q;4S) which

i s

given by

where i n t h e c a s e of model 1 we have

The sur~mations a r e dominated by t h e behaviour a t small wavevectors, and t h e y rnay be performed a n a l y t i c a l l y f o r c e r t a i n c a s e s . For example, i n t h e absence of s u r f a c e a n i s o t r o p y

(2

= 0 ) and f o r T << a2q2 < < 1 we e v e n t u a l l y o b t a i n

T h i s a g r e e s with t h e r e s u l t o b t a i n e d p r e v i o u s l y by Tarasenko and Kharitonov

/5/

i n t h e l i m i t s of h < < T and h > > 7

.

We f i n d t h e same formal expression h o l d s f o r

Model 2 provided

a i s

r e d e f i n e d a s J /(J +4J

1,

( 0 <

a

< 1 ) . Our t h e o r

2 1 2

f

d2

a p p l i e s when d

#

0. For example, i f d < 0 such t h a t

r

<< d 2 <<

1

and a q we e s t i m a t e t h a t

r

^(q;4S)

^{i s}

p r o p o r t i o n a l t o d 6 ~ ~ ( 1 , ~ S ( 0 ) / k B ~ ) f o r Models

1

and

2

provided k T << _B E _S

yqi. _-

The o t h e r c o n t r i b u t i o n s

r (2;

3S, 1 B )

, r

(q; 2S, 2B) and

r (2;

lS,3B) t o t h e s u r f a c e

s

S

s p i n wave damping a r e g i v e n by much mose-complicated e x p r e s s i o n s t h a n (12) and ( 1 3 ) , and i n g e n e r a l t h e y r e q u i r e n u r ~ e r i c a J e v a l u a t i o n How v

5 5 r

^{I .} i f t h e s u r f a c e a n i s o - t r o p y f i e l d

i s

s u f f i c i e n t l y l a r g e ( d >> T and d 2 >> a q )

~t

may be shown t h a t

r

(q;3S,lB) and

r

(q;lS,3B) a r e n e g l i g i b l y small. T h i s

i s

e s s e n t i a l l y because i n t g e s e p r o c e s s e s energy can b e conserved only a t l a r g e wavevectors, and t h e S combination of Bose f a c t o r s i n t h e surnrland

i s

t h e n very small.

Acknowledgements

-

We thank R. Loudon and D.R. T i l l e y f o r h e l p f u l d i s c u s s i o n s .

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