The hyperfine field on impurities in Gd

Volume 30A, number 9

PHYSICS LETTERS

THE

HYPERFINE

FIELD

29 Decomber1969

ON I M P U R I T I E S

IN Gd

I. A. C A M P B E L L

Physique des Solides, Facult~ des Sciences, 91-Orsay, France Received l0 November 1969

Predictions are given for the hyperfine fields on s-p impurities in Gal.

A s i m p l e v a r i a n t of the D a n i e l - F r i e d e l c o n duction e l e c t r o n p o l a r i z a t i o n m o d e l [1] has b e e n shown to give a p l a u s i b l e explanation of hyperfine field s y s t e m a t i c s for i m p u r i t i e s in t r a n s i t i o n f e r r o m a g n e t i c hosts [2]. The two p r i n c i p a l a d j u s t able p a r a m e t e r s that had to b e c h o s e n in o r d e r for the c a l c u l a t e d c u r v e to be a r e a s o n a b l e fit to e x p e r i m e n t w e r e : f i r s t , that the effective i n t e r a c t i o n between d and conduction e l e c t r o n e J s d i s n e g a t i v e in t h e s e hosts; secondly that the n u m b e r of conduction e l e c t r o n s is r a t h e r low (~ 0.5 e l e c t r o n s / h o s t atom). It is p o s s i b l e to u s e exactly the s a m e model to c a l c u l a t e the h y p e r f i n e fields on s - p i m p u r i t i e s i n a f e r r o m a g n e t i c Gd host by s i m p l y changing the r e l e v a n t p a r a m e t e r s of the model. We have done the c a l c u l a t i o n , a s s u m i n g j u s t as b e f o r e that the conduction e l e c t r o n s a r e u n i f o r m l y p o l a r ized in the host m e t a l , and that the i m p u r i t y can be r e p r e s e n t e d by potential w e l l s V t , ~ which a r e d e t e r m i n e d s e l f - c o n s i s t e n t l y f r o m two conditions: (i) that the i m p u r i t y charge d i f f e r e n c e is s c r e e n e d out, and (ii) that the localized e l e c t r o n m o m e n t (which would b e f in this c a s e ) o n the i m p u r i t y is zero. D e t a i l s of the c a l c u l a t i o n can be found in ref. 2. In going o v e r to a r a r e e a r t h host, we have changed the two p r i n c i p a l p a r a m e t e r s of the model. J s f has b e e n taken as positive, and the n u m b e r of conduction e l e c t r o n s p e r host atom is t a k e n as 3, as the m a g n e t i c m o m e n t of a Gd atom in Gd m e t a l is a p p r o x i m a t e l y equal to that of a Gd 3+ ion, so 3 conduction (or d) e l e c t r o n s a r e needed to give c h a r g e n e u t r a l i t y . T a k i n g t h e s e v a l u e s , the c u r v e c a l c u l a t e d for the conduction e l e c t r o n p o l a r i z a t i o n p ( Z ) at an s - p i m p u r i t y site is given as a function of Z, the n u m b e r of s - p e l e c t r o n s on the i m p u r i t y , in fig. 1. The host conduction e l e c t r o n p o l a r i z a t i o n Ph will be p r o p o r t i o n a l to J s f and is independent of the impurity. The h y p e r f i n e fields at the different i m p u r i t i e s should be equal to p(Z) m u l t i p l i e d by the a t o m i c

0.1 2

t

6

Z ~8

-0.2

-0 .i Fig. 1. Calculated polarizations at the site of an s-p impurity as a function of the number Z of s-p electrons. Ph is the polarization at a host site. The scale of Z is approximate. h y p e r f i n e c o n s t a n t a p p r o p r i a t e to the i m p u r i t y e l e m e n t (table AI of ref. 2). It can b e s e e n that the c u r v e is r a t h e r d i f f e r ent from the equivalent c u r v e for a t r a n s i t i o n m e t a l host. F r o m it, we would p r e d i c t that fields for noble m e t a l i m p u r i t i e s in Gd would be s m a l l and positive, t h o s e for i m p u r i t i e s in the middle of the s - p s e r i e s l a r g e and negative, with s m a l l e r fields for i m p u r i t i e s towards the end of the s e ries. The only s - p i m p u r i t y whose field in Gd has so f a r b e e n m e a s u r e d is Sn(Z =4) [3]. Here the field H = - 329 kOe is indeed negative, and l a r g e r than fields at Sn in t r a n s i t i o n f e r r o m a g n e t i c hosts. No field on a noble m e t a l i m p u r i t y in Gd has b e e n m e a s u r e d , but the fields on Pt and Pd a r e + 140 (~ 40) kOe and + 350 (± 70) kOe r e s p e c t i v e l y [4], p o s i t i v e and s m a l l as the m o d e l p r e dicts for the n e i g h b o u r i n g noble m e t a l s . If we u s e the field for Sn in Gd to c a l i b r a t e the conduction e l e c t r o n p o l a r i z a t i o n , we find j ~ d ~ _ 0.15 j F ~ , which is the right o r d e r of magnitude. 517

Volume 30A, n u m b e r 9

PHYSICS

W e h a v e i g n o r i z e d t h e d e l e c t r o n s in t h i s c a l c u l a t i o n ; t h i s is r e a s o n a b l e a s l o n g a s only s - p i m p u r i t i e s a r e c o n s i d e r e d , but t h e p r e s e n c e of d polarization will complicate the situation conside r a b l y f o r t r a n s i t i o n i m p u r i t i e s . F u r t h e r d a t a on s - p i m p u r i t i e s in Gd w o u l d b e of g r e a t i n t e r e s t .

EFFECT

OF DUE

SCATTERING TO CARRIERS

LETTERS

29 December 1969

References 1. E. Daniel and J. Friedel, J. Phys. Chem. Sol. 24 (1963) 1601. 2. I.A. Campbell, J. Phys. C. 2, (1969) 1338. 3. V. Gotthardt, H. S. M~ller and R. L. M~ssbauer, Phys. Letters 28A (1969) 480. 4. D. Murniek, L. Grodzins, R. Kalish. R. R, Borchers, J. Bronson and B. Herskind, in: "Hyperfine structure and nuclear radiations", eds. E. Matthias and D.A. Shirley (North-Holland. 1968).

ON OPTICAL NONLINEARITIES IN SEMICONDUCTORS

K. C. R U S T A G I and S. S. J H A Tata Institute of Fundamental Research, Bombay-5, India Received 17 November 1969

A general formula for the t h i r d - o r d e r nonlinear uptical susceptibility due to c a r r i e r s is obtained to ascertain the relative importance of nonparabolicity and the energy-dependent scattering.

In r e c e n t y e a r s , t h i r d o r d e r o p t i c a l f r e q u e n c y m i x i n g e x p e r i m e n t s [1,2] have b e e n p e r f o r m e d to m e a s u r e n o n l i n e a r o p t i c a l s u s c e p t i b i l i t y ×(3)(¢o1,¢Ol,- w 2) due to c a r r i e r s in H I - V s e m i c o n d u c t o r s . The c o n t r i b u t i o n due to bound e l e c t r o n s is s e p a r a t e d f r o m that due to c a r r i e r s by m e a s u r i n g the n o n l i n e a r s u s c e p t i b i l i t y a s a f u n c t i o n of the c a r r i e r c o n c e n t r a t i o n n. If 2¢o1 and w 2 a r e s m a l l e r than the b a n d - g a p , J h a and B l o e m b e r g e n [3] h a v e shown that it is p o s s i b l e to c o n s i d e r a one band m o d e l [4] to c a l c u l a t e the c a r r i e r c o n t r i b u t i o n to ×(3)(Wl, ¢ol,- 0)2) due to the n o n p a r a b o l i c i t y of the c o n d u c t i o n band in a n - t y p e s e m i c o n d u c t o r . H o w e v e r , the n o n p a r a b o l i c i t y of the band i s not the only s o u r c e of n o n l i n e a r i t y . In a m i x i n g e x p e r i m e n t , it i s q u i t e p o s s i b l e that w 1 - w 2 is c o m p a r a b l e to the c o l l i s i o n f r e q u e n c y v of the c a r r i e r s ; a l t h o u g h w 1 and ¢o2 m a y i n d i v i d u a l l y be l a r g e c o m p a r e d to v. S i n c e , in g e n e r a l , v d e p e n d s on the c a r r i e r e n e r g y E k i.e. v = v(]k[) =- v k w h e r e k is the w a v e v e c t o r of the e l e c t r o n , t h i s l e a d s to an a d d i t i o n a l c o n t r i b u t i o n to the n o n l i n e a r s u s c e p t i b i l i t y . T h i s w a s f i r s t d i s c u s s e d by K o l o d z i e j c z a k [5] and l a t e r e m p h a s i z e d by Kaw [6] in the p r e s e n t c o n t e x t *. T o a s c e r t a i n the r e l a t i v e i m p o r t a n c e of the n o n p a r a b o l i c i t y and the e n e r g y d e p e n d e n t s c a t t e r i n g , u s i n g the B o l t z m a n n t r a n s p o r t e q u a t i o n in the r e l a x a t i o n t i m e a p p r o x i m a t i o n f o r the s c a t t e r i n g , we obtain, in t h i s note, a g e n e r a l f o r m u l a f o r the t h i r d o r d e r o p t i c a l s u s c e p t i b i l i t y due to c a r r i e r s in a n o n p a r a b o l i c band. In the i n f i n i t e w a v e l e n g t h l i m i t f o r the e l e c t r i c f i e l d ~(t), the d i s t r i b u t i o n f u n c t i o n f ( k , t ) s a t i s f i e s the equation ~f ~t - ~e ~ ( t ) - ~~f : - v k f f - f o )

(1)

w h e r e f O ( E k) i s the F e r m i d i s t r i b u t i o n f u n c t i o n with ~ k f ° ( E k ) = n. T h e d e p a r t u r e f r o m l i n e a r i t y b e i n g s m a l l , to s o l v e eq. (1) in s u c c e s s i v e o r d e r s in the i n c i d e n t e l e c t r i c f i e l d ~ i n c ( t ) we u s e the e x p a n s i o n s f = f ( 0 ) +f(1) + f ( 2 I + f ( 3 ) + . . . , ~ _ ~ (1) + ~(2) + . . . . e t c . , w h e r e [~(n)[ and f(n), n ¢ 0, a r e of o r d e r

* Apart from s e v e r a l algebraic e r r o r s in ref. 6, the final expression and the main conclusion of this paper are not quite c o r r e c t .

518