Self-consistent phonons in ferromagnetic crystals

Volume 32A. number 3

PHYSICS

LETTERS

T h e i n c r e a s e in the r e l a x a t i o n t i m e r a t i o with d e c r e a s i n g t e m p e r a t u r e m a y be a c o n s e q u e n c e of e n e r g y and m o m e n t u m c o n s e r v a t i o n in the s c a t t e r i n g p r o c e s s . An e l e c t r o n w i t h g r o a p v e l o c i t y l e s s than the v e l o c i t y of the phonons c a n n o t be s c a t t e r e d by e m i s s i o n of a phoaon, only by a b s o r p tion, E l e c t r o n s w h i c h in k - s p a c e a r e c l o s e to the s y m m e t r y a x i s of the e n e r g y e l l i p s o i d have s m a l l e r v e l o c i t i e s than o t h e r e l e c t r o n s with the s a m e e n e r g y . When the t e m p e r a t u r e is l o w e r e d t h e s e e l e c t r o n s w i l l be the f i r s t o n e s to c e a s e e m i t t i n g p h o nonso S i n c e 1/7 H is m a i n l y d e t e r m i n e d by s c a t t e r i n g of t h e s e e l e c t r o n s , 1/7~: w i l l d e c r e a s e s t r o n g e r them 1 / 7 and K r w i l l i n c r e a s e _ In s i l i con the l a r g e s t phonon v e l o c i t y is that of the l o n g i t u d i n a l p h o n o n s . E l e c t r o n s n e a r the s y m m e t r y a x i s wiI1 h a v e t h e r m a l v e l o c i t i e s of t h i s o r d e r of m a g n i t u d e a r o u n d 5°K. T h i s is j u s t in the t e r n -

SELF-CONSISTENT

PHONONS

IN

29 J u n e L970

p e r a t u r e r a n g e in w h i c h K~- c h a n g e s , T h e a u t h o r w i s h e s to a c k n o w l e d g e P r o f . Dr. K. S a e r m a r k for his v e r y s t i m u l a t i n g i n t e r e s t in this work

References [1] R. Ito. H. K a w a m u r a and M , F u k a i . 13 (1964) 26.

P hys . L e t t e r s

[2] R. A. Stradling and V. V. Zhukov. Proc. Phys. Soe, 87 11966) 263. [31 D. Longand J. Myers. Phys. Rev. 120 11960) 39. L. J. Neuringer and W. J. Little. Proe. Int. Conf. Phys. Semieond. Exeter (1962) 614. [41 M. Owner-Petersen and M. R. Samuelsen. Phys, Star. Sol. 28 (1968) 211, [51 M. Fukai. H. Kawamura. K. Sekido and I. Imai. J. Phys. Soe. Japan 19 ~19641 30.

FERROMAGNETIC

CRYSTALS

No M. P L A K I D A Laboralor_v of Theoreiical Physics, Joinl lnslilale.lbr NHclear Research, Moscow. (rSSR

Received 27 May 1970

An expression for the self-consislent phonon frequencies in a ferromagnetic anharmonic crystal is obtained.

A new a p p r o a c h to the t h e o r y of s p i n - p h o n o n i n t e r a c t i o n s which t a k e s into a c c o u n t in a s i s t e n t m a n n e r the a n h a r m o n i c i t y of l a t t i c e v i b r a t i o n s w a s p r o p o s e d in I l l and [21o In the it is shown that the d e c o u p l i n g p r o c e d a r e f o r the G r e e n f u n c t i o n s a d o p t e d in I l l and [21 is to the v a r i a t i o n a l a p p r o a c h p r o p o s e d in 131 f o r c o n s i d e r i n g the s e l f - c o n s i s t e n t phonons in crystals° We c o n s i d e r a f e r r o m a g n e t i c c r y s t a l w h i c h can be d e s c r i b e d by the H a m i l t o n i a n I2]: = H I + H s = ( 2 M ) - 1 ~ - 1 7 2 + U(R i) - H H ~ S f - ½ ~ J ( R . - R i

- z

i

Lj

z

y

.) Si° S j

self-conpresent paper equivalent anharmonic

(1)

w h e r e R i = li+ U i . U i is the d i s p l a c e m e n t and S i is the s p i n o p e r a t o r of the a t o m in the l a t t i c e s i t e I i =
H h = ( 2 M ) - I ~-. 1 7 2 + ½ .

~ U ' °z

ij ° V j

(! and K s h a s the s a m e f o r m a s H s in (1) w h e r e the e x c h a n g e i n t e g r a l is r e p l a c e d by the f u n c t i o n J"t; d e p e n d i n g only on e q u i l i b r i u m s e p a r a t i o n b e t w e e n a t o m s I.. = ! : - ! ; o T h e p a r a m e t e r s q~/j and "/ij a r e U

134

,

g

(2)

Volume 32A. number 3

PttYSICS

LETTERS

29 June 1970

d e t e r m i n e d v a r i a t i o n a l l y f r o m the s t a t i o n a r y c o n d i t i o n for the t r i a l f r e e e n e r g y

Ftr= Fo + (

- i[O )0

F°

'

-fi-llnWr/exp(-fi

o) ]-

U s i n g a T a y l o r e x p a n s i o n in p o w e r s of d i s p l a c e m e n t s for U( !; "+ U;)+ a n d t h e r m a l a v e r a g e (° • • )o with the H a m i l t o n i a n (2) we get for (3) : ( U ) o = expI½/~j ( U / ~ ) :

o:

e

(3)

=

J( li) +Ui -

[ ~ ) and t a k i n g the

17i 5 ~ Uo(li)

1

viSIJo(;;j)

(4)

.

V a r i a t i o n of the f r e e e n e r g y (3) with r e s p e c t to (Si ~ S j)_ and (Ui Uj ~ g i v e s Zii : (J)o and the m a t r i x .. w h i c h d e t e r m i n e s f r e q u e n c i e s w . _ and o o l a r i z a t i o n ~ e c t o r s e - - ~ ) f the tri~tl h a r m o n i c p h o n o n s in U ;g;t R,~ (2) by the e q u a t i o n

ekX¢O2x = (MN)-1~)

/exp (_ik ° ;ij

) 17i 17j ( V)o - (Si+ Sj)

[1 - exp ( - i k +I(] )1 17i 17i (Y)o ] ' ek~

,

(5)

The d i s p l a c e m e n t c o r r e l a t i o n f u n c t i o n in the t r i a l h a r m o n i c a p p r o x i m a t i o n (2) has the s t a n d a r d f o r m

(UiUj)o

:: ( M N ) - I k~ ebb vk3t exp ( - i k ° / j )(2 wk?t )-1 c°th(½1Jwk~t)

(6)

The s p i n c o r r e l a t i o n f u n c t i o n (S i°Sj> = - 26 Fo/5 ~.j and c a n be o b t a i n e d by the s t a n d a r d m e t h o d s of the q u a n t u m t h e o r y of m a g n e t i s m . The e q u i l i b r i u m ' p o s i t i o n s of a t o m s I i = ( R i> a r e d e t e r m i n e d by the e q u i l i b r i u m c o n d i t i o n s for the c r y s t a l u n d e r e x t e r n a l forces~ In the c a s e of the i s o t r o p i c p r e s s u r e P we get the following e q u a t i o n of s t a t e for the c r y s t a l with v o l u m e V [2l

Pv :

,;.

17+.o +

ol;j ° v. o

(7)

The s e l f - c o n s i s t e n t s y s t e m of e q u a t i o n s (2) - (7) c a n d e s c r i b e the t h e r m a l , m e c h a n i c and m a g n e t i c p r o p e r t i e s of f e r r o m a g n e t i c c r y s t a l s in a wide r a n g e of t e m p e r a t u r e and e x t e r n a l p r e s s u r e , The v a l idity of the a d o p t e d a p p r o x i m a t i o n s c a n be e v a l u a t e d by t a k i n g into a c c o u n t the d a m p i n g of the s e l f c o n s i s t e n t p h o n o n s a n d m a g n e t i c e x c i t a t i o n s , e+ g. by the G r e e n f u n c t i o n m e t h o d [4, 5 I,

References [1] s. v. Tyablikov and H. Konwent. Phys. Letters 27A (1968) 130. [2}H.Konwent ,andN.M.Plakida. Preprint JINR P4-4723. Dubna. 1969. [3] N. S. Gillis. N. R. Werthamer and T. R. Koehler. Phys. Rev. 165 1196~) 951. [4] H. Konwent. Phys. Letters 28A (1968) 237. [51 N.M. Plakida and T. Siklos. Phys. Stat. Sol. 33 (1969) 103.

135