Collective excitations in itinerant ferromagnets

Collective excitations in itinerant ferromagnets

Volume 38A, number 5 P HY SI C S L E T T E R S COLLECTIVE EXCITATIONS IN ITINERANT 28 F e b r u a r y 1972 FERROMAGNETS U. B R A N D T Univer...

99KB Sizes 0 Downloads 46 Views

Volume 38A, number 5

P HY SI C S L E T T E R S

COLLECTIVE

EXCITATIONS

IN

ITINERANT

28 F e b r u a r y 1972

FERROMAGNETS

U. B R A N D T

Universit~lt Dortmund, Institut Jar Physik, 46 Dortmund, W.-Gerrnany Received 14 January 1972

A Hubbard model containing as an additional part long range Coulomb forces is discussed as a model for a ferromagnet. Two types of collective modes are found in the longitudinal susceptibility - the plasma oscillations, which are no pure density fluctuations, and a decaying mode, which is almost magnetic.

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 n v o l v e d in the l o n g i t u d i n a l m a g n e t i c s u s c e p t i b i l i t y of an i t i n e r a n t f e r r o m a g n e t a r e c a l c u l a t e d . We u s e the H u b b a r d m o d e l in H a r t r e e - F o c k a p p r o x i m a t i o n - f o r f u r t h e r d e t a i l s and d e f i n i t i o n s of the f o l l o w i n g q u a n t i t i e s s e e r e f . [1] - w i t h the m o d i f i c a t i o n of t a k i n g into a c c o u n t the long r a n g e p a r t of the C o u l o m b i n t e r a c t i o n in g e n e r a l i z e d R P A in the s e n s e of r e f . [2]. T h i s is n e c e s s a r y b e c a u s e in the p o l a r i z e d s t a t e , d e n s i t y and s p i n d e n s i t y f l u c t u a t i o n s a r e m i x e d up. T h i s p r o b l e m h a s b e e n t r e a t e d p r e v i o u s l y [3], but not c o n s i s t e n t l y in a c o n s e r v i n g and c o v a r i a n t f o r m in t h e s e n s e of r e f . [1], t h u s the r e s u l t s do not a g r e e , but t h e i r s t r u c t u r e is s i m i l a r . In t h i s g e n e r a l i z a t i o n we o b t a i n f o r the s u s ceptibilities: Xeta = (XOa - Uq X°+X ° - ) D -1, ×+D

= X-+ = (V+Uq) X°+X°-D -1

(a : +, -) with

(1)

= 1 - Uq(X°+ + X° - ) - V2X°+XO- - 2VUqX°+X ° - .

H e r e Xa/3 is the r e s p o n s e f u n c t i o n c o r r e s p o n d ing to the ( h a h # > - e o r r e l a t i o n - f u n c t i o n , w h e r e h a is the d e n s i t y o p e r a t i o n of the e l e c t r o n s of s p i n a . Uq is the F o u r i e r - t r a n s f o r m of the l o n g r a n g e p a r t of the i n t e r a c t i o n , b e h a v i n g l i k e 47r e2/q 2 f o r s m a l l q, and V i s the s h o r t r a n g e i n t e r a c t i o n c o n s t a n t X° a = X° a a i s the r e s p o n s e f u n c t i o n of the n o n - i n t e r a c t i n g s y s t e m , but t a k i n g into a c c o u n t the s p l i t t i n g e n e r g y in t h e s e l f - e n e r g y p a r t . T h e l i m i t Uq --. 0 g i v e s the r e s u l t of r e f . [1], c o n t a i n i n g n o n - p h y s i c a l s o u n d - l i k e e x c i t a t i o n s , but f o r s m a l l q and 00 the l i m i t Uq --' oo i s a p p r o p r i a t e , i . e . , f o r q ' = q / k ~ <~0.1 and w' = w/lz o <~0.1, w h e r e k ~ i s the

F e r m i m o m e n t u m of the n o n - i n t e r a c t i n g s y s t e m and ~to the c o r r e s p o n d i n g F e r m i e n e r g y . T h i s l i m i t can b e o b t a i n e d a l s o by non a l l o w ing d e n s i t y f l u c t u a t i o n s by c o n s t r a i n t s . In the c a s e X°+ = X° - , i . e . , in the n o n - m a g n e t i c c a s e , o u r r e s u l t a g r e e s with that of r e f . [1]. In the l i m i t T ~ 0 and u s i n g the p a r a b o l i c m o d e l band s t r u c t u r e t h e s e q u a n t i t i e s can b e c a l c u l a t e d e a s i l y , and we o b t a i n two t y p e s of e x c i t a t i o n s f o r s m a l l q: one f o r r e a l w at the p l a s m a f r e q u e n c y , and one f o r p u r e l y i m a g i n a r y f r e q u e n c i e s in the u n p h y s i c a l p a r t of the c o m p l e x p l a n e , i . e . , the f u n c t i o n s ×a/3(¢o) c o n t i n u e d a n a l y t i c a l l y f r o m the u p p e r to the l o w e r half of the c o m p l e x p l a n e h a v e p o l e s at s o m e i m a g i n a r y f r e q u e n c y ¢o = - i v (q). T h e n o r m a l m o d e s and its s u s c e p t i b i l i t i e s can b e o b t a i n e d by d i a g o n a l i z i n g the s u s c e p t i b i l i t y m a t r i x (×)aft; the n o r m a l m o d e s a r e 1 (1 +/31,2)n" + ( i - ~ i , 2 ) ~ w h e r e h is the d e n s i t y o p e r a t o r and ~ the spin o p e r a t o r (= n+ - h _ ) , and ~ 1 , 2 = (~1,2 - x ° + + Vq xo÷ xO-){(v + Vq) x°+ x ° - } -1 ; the c o r r e s p o n d i n g s u s c e p t i b i l i t i e s a r e Xl ,2 = X°+ X°'(/12,1 )-1 .

(2)

g l , 2 a r e g i v e n by ~1,2 = ½ (X°+ + X° - - 2UqX°+X°-) ± {¼(X°+ - X°-) 2 + (V+ Uq) 2 (X°+ X°-)2} 1/2. By t h i s p r o c e d u r e we a l s o h a v e s e p a r a t e d the p l a s m a o s c i l l a t i o n s and the d e c a y i n g m o d e , b e c a u s e e a c h of the X1,2 h a s only one of t h e s e p o l e s . F o r the p l a s m a e x c i t a t i o n s w e o b t a i n the p l a u s i b l e r e s u l t f32 = n_/n+ f o r s m a l l q ; h e r e we h a v e u s e d at an e s s e n t i a l p o i n t that in o u r m o d e l the s p i n up and down e l e c t r o n s h a v e the s a m e effective mass. 361

Volume 38A, n u m b e r 5

PHYSICS

LETTERS

28 F e b r u a r y 1972

F o r t h e d e c a y i n g m o d e we h a v e HI ~ -1, s o t h i s i s a p u r e l y m a g n e t i c e x c i t a t i o n . It h a s a d i s p e r s i o n r e l a t i o n L,'(q') = 7 q ' w h e r e V d e p e n d s on t h e r e l a t i v e m a g n e t i z a t i o n M ( s e e fig. 1). In t h e l i m i t M - ~ 1 t h i s r e l a t i o n i s n o t a p p l i c a b l e , b e c a u s e t h e v a l i d i t y c o n d i t i o n q / k ~ << 1 i s n o t f u l f i l l e d . In t h e l i m i t M ~ 0 ~ t e n d s to 0; in t h i s c a s e we o b t a i n v'(q') = q'3/3~. It s h o u l d b e mentioned that for the decaying mode the limit Uq ~ ~ ( i . e . , p l a s m a f r e q u e n c y ~ co) i s a v e r y good a p p r o x i m a t i o n f o r a l l r e a l i s t i c c a s e s .

I0"I

10-;

i0-3

References [1] u. Brandt, H. Lustfeld, W. P e s c h and L. Tewordt, J. Low Temp. Phys. 4 (1971} 9. [2] L. P. Kadanow and I. I. Fatko, Phys. Rev. 136 (1964} All70. [3] O. I. Kim, H. C. Praddaude and B. B. Schwartz, Phys. Rev. L e t t e r s 23 (1969} 419.

1ff~

i

i

,

,

~

t

q

i

i

0,5

Fig. 1. The n o r m a l i z e d i m a g i n a r y velocity of the decaying mode in dependence of the relative m a g n e t i z a tion M.

362