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THE LANDÉ g-TENSOR FOR d-HOLES IN Pd
T. Rahman, J. Parlebas, D. Mills
To cite this version:
JOURNAL DE PHYSlQUE
Colloque
C6, supplkment au no 8, Tome 39, aolir 1978, page
C6-768
T.S. Rahman, J . C . ~ a r l e b a s ' and D.L. M i l l s
Dept. of Physics University of CaZifornia, I d n e , CA 92717, USA.
-r
R6sumE.- Nous pr6sentons une Btude thgorique de l a dependance en k du f a c t e u r g de Land6 pour l e s t r o u s d du Palladium au v o i s i n a g e du p o i n t X de l a zone de B r i l l o u i n . Notre a n a l y s e e s t fondQe s u r l'approximation d e s l i a i s o n s f o r t e s pour l e s Q l e c t r o n s d . Nous avons trouv6 que l e t e n s e u r g e s t hautement a n i s o t r o p e e t depend fortement de
g.
Nous d i s c u t o n s brievement nos r 6 s u l t a t s en l i a i s o n avec l e s expdriences de TESR s u r Pd e f f e c t u e e s p a r Hurdequint e t Monod.A b s t r a c t . - We p r e s e n t a t h e o r e t i c a l a n a l y s i s of t h e wave v e c t o r dependence of t h e Land6 g-tensor f o r d h o l e s n e a r t h e X p o i n t of t h e B r i l l o u i n zone of Pd m e t a l . The t r e a t m e n t i s based on a t i g h t binding approximation f o r t h e d e l e c t r o n s . We f i n d a h i g h l y a n i s o t r o p i c g t e n s o r w i t h s t r o n g wave v e c t o r de- pendence. We c o m e n t on t h e r e l a t i o n s h i p of our r e s u l t s t o t h e TESR experiment on Pd r e p o r t e d by ~ u r d e q u i n t and Monod
.
INTRODUCTORY R.EMARKS.- This paper summarizes t h e r e s u l t s of our s t u d y of t h e wave v e c t o r dependence of t h e Land6 g t e n s o r of t h e d h o l e s i n Pd m e t a l . The i n v e s t i g a t i o n was motivated by t h e l a r g e appa- r e n t g - s h i f t observed i n Pd by Hurdequint and Monod
/ l / i n t h e TESR s t u d y of t h i s m a t e r i a l . The s i g n a l i s presumably a s s o c i a t e d w i t h t h e e x c i t a t i o n of t h e d h o l e s ; i f so, t h i s experiment i s t h e f i r s t obser- v a t i o n of an e l e c t r o n s p i n resonance s i g n a l from e l e c t r o n s w i t h s t r o n g d c h a r a c t e r .
Other a u t h o r s have considered t h e g t e n s o r n e a r t h e X p o i n t of t h e B r i l l o u i n zone. L e n g l a r t / 2 /
noted t h e presence of l a r g e wave v e c t o r dependent e f f e c t i v e g f a c t o r s f o r e l e c t r o n s i n P d , and calcu- l a t e d them along c e r t a i n high symmetry l i n e s i n t h e B r i l l o u i n zone. Somewhat l a t e r , t h e t o p i c was re-in- v e s t i g a t e d by Mueller and co-workers / 3 , 4 / . I n t h e i r f i r s t paper 1 3 1 , t h e y argue t h a t o r b i t a l e f - f e c t s a r e unimportant and f i n d an e f f e c t i v e g v a l u e averaged over t h e Fermi s u r f a c e e q u a l t o 1.65, a va- l u e i n s e r i o u s disagreement w i t h t h e v a l u e
%
2.2 re- p o r t e d by Hurdequint and Monod / I / . We f i n d , a s d i d L e n g l a r t i n h i s e a r l y work, t h a t o r b i t a l c o n t r i b u - t i o n s a r e very important. I n a second paper141.
Mueller e t a l . e v i d e n t l y i n c l u d e t h e o r b i t a l c o n t r i - b u t i o n t o c a l c u l a t e t h e g v a l u e a t a few poincs on t h e X c e n t e r e d h o l e s u r f a c e s . These e a r l y c a l c u l a - t i o n s do n o t p r e s e n t t h e information i n a manner
*
Supported by U.S.Air f o r c e O f f i c e of S c i e n t i f i c Research under C o n t r a c t No. F 49620+ Permanent Address, a f t e r May 1978 : L a b o r a t o i r e d e S t r u c t u r e E l e c t r o n i q u e d e s S o l i d e s , ~ n i v e r s i t e ' Louis P a s t e u r , S t r a s b o u r g France.
t h a t a l l o w s c o n t a c t with t h e new d a t a .
The o r i g i n of t h e l a r g e wave v e c t o r dependence and a n i s o t r o p y of t h e g t e n s o r i s e a s y t o appre- c i a t e . With s p i n o r b i t c o u p l i n g n e g l e c t e d , t h e Fer- m i s u r f a c e l e a v e s h o l e s i n t h e topmost d band, which i s two f o l d degenerate a t X. The next lowest band i s very c l o s e i n e n e r g y a t theX p o i n t
3
0.2 eV below t h e topmost band. These t h r e e s t a t e s a r e mixed by s p i n o r b i t coupling. Very c r u d e l y , a t X we expect g - s h i f t s of t h e o r d e r of l / A , w i t h X t h e s p i n o r b i t coupling s t r e n g t h and A t h e s p l i t t i n g a t X . We have ?,/A % l a t X and t h e s p l i t t i n g Achanges s t r o n g l y a s one moves away from X . Thus t h e g s h i f t s depend s t r o n g l y on wave v e c t o r .
THE COMPUTATIONAL PR0DEDURE.- We have approached t h e problem i n two ways, b o t h of which a r e based on t h e t i g h t binding d e s c r i p t i o n of t h e d band, w i t h S-d h y b r i d i z a t i o n i g n o r e d . Along t h e l i n e from
r
t o X, i f only t h e topmost t h r e e s t a t e s ( s t r o n g l y mixed by s p i n o r b i t coupling, a s mentioned above) a r e r e t a i n e d , a n a l y t i c e x p r e s s i o n s can be found f o r t h e s p l i t t i n g (E + ( k )-
E +(k)) between t h e Bloch l e v e l s produced by a magnetic f i e l d o f . a r b i t r a r y d i r e c t i o n . Near t h e X p o i n t , a s one moves i n any d i r e c t i o n , we can expand t h e g t e n s o r i n powers of+
-+ (k-
k ) . These c a l c u l a t i o n s p r o v i d e an i n s i g h t in- X t o t h e m a t r i x elements and t h e s t a t e s t h a t c o n t r o l t h e behavior of t h e g t e n s o r . -+ We a l s o c a l c u l a t e g i j (k) a t a g e n e r a l p o i n t by d i a g o n a l i z i n g t h e f u l l 10 X 10 m a t r i x t h a t des- c r i b e s t h e d l e v e l s i n t h e presence of s p i n o r b i tc o u p l i n g and i n t e r a c t i o n w i t h magnetic f i e l d o f t h e
+
+
+
form g )1 ( L + 25) H. We have done t h i s f o r an a r - 0 B r a y of p o i n t s a r r a n g e d on t h e d-hole p o r t i o n of t h e Fermi s u r f a c e , presuming t h e s e s u r f a c e s a r e s i m p l e i n t e r s e c t i n g c y l i n d e r s w i t h r a d i u s t h a t p r o v i d e s 0.36 h o l e s / P d atom. We found a l l e l e m e n t s of t h e g t e n s o r a t each of 252 i n - e q u i v a l e n t p o i n t s on t h e model Fermi s u r f a c e . T h i s i n f o r m a t i o n a l l o w s us t o c a l c u l a t e a n e f f e c t i v e g f a c t o r a v e r a g e d o v e r a l l s i x h o l e s u r f a c e s , assuming t h e e x t e r n a l f i e l d i s a l o n g a (100) d i r e c t i o n . The d i s t r i b u t i o n o f g va- l u e s i s g i v e n i n f i g u r e 1 , and we f i n d
g -
< g(;) > = 2.31 ( 1 )r e s p o n s e . The asyrmnetry observed i n t h e d a t a may have i t s o r i g i n i n f i n i t e wave v e c t o r e f f e c t s , pos- s i b l y from u n r e s o l v e d Fermi l i q u i d s p i n waves 161.
R e f e r e n c e s
/ I / H u r d e q u i n t , H. and Monod, P. ( t o be p u b l i s h e d ) . / 2 / L e n g l a r t , P., J . Phys. Chem. S o l i d s
28
(1967)2011.
/ 3 / M u e l l e r , F.M., Freeman, A. J., Dimmock, J.O., and Furdyna, A.M., Phys. Rev. B
1
(1970) 4617. / 4 / M u e l l e r , F.M., Freeman, A. J.. and K o e l l i n g , D.D.J. Appl. Phys.
2
(1970) 1229.i n r e a s o n a b l e accord w i t h t h e d a t a , and a l s o
151
Freedman, R., and F r e d k i n , D.R., Phys. Rev. B <(g(;)-g)2 > E 6 2 = 0.08.-
1 I (1 975) 4847. 161 T h i s p o s s i b i l i t y was r a i s e d by P. Monod i n r e - marks t o one of u s . F i g . I : D i s t r i b u t i o n o f g v a l u e s . 80 70 6 0r
5 0 -&
4 0 -CONCLUDING REPIARKS.-We have used t h e Fermi l i q u i d d e s c r i p t i o n o f Freedman and F r e d k i n / 5 / t o o b t a i n a
+
d e s c r i p t i o n o f t h e p o s i t i o n and ESR l i n e w i t h k de- pendent g f a c t o r . The t h e o r y g i v e s a L o r e n t z i a n li- ne c e n t e r e d a t
geff { l
-
6 2 ( 1 + ~ ) / ( l + r 2 ) 2 } (2) where t h e Fermi l i q u i d p a r a m e t e r ' Bi s
-T
of parama- gnon t h e o r y andr
(Q
T )-l w i t hno
t h e Larmor f r e -0 0
