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The effect of exchange enhancement on the field dependence of the paramagnetic spin susceptibility of metals
Volume 22, number 3
THE
PHYSICS LETTERS
EFFECT OF EXCHANGE OF THE PARAMAGNETIC
15 August 1966
ENHANCEMENT ON THE SPIN SUSCEPTIBILITY
FIELD DEPENDENCE OF METALS
E. P. WOHLFARTH*
Institute for Atomic Research Iowa State University, Ames. Iowa, USA Received 7 July 1966
The field dependence of the spin susceptibility has been shown to be of the form X(/D = X(0) (1 +fill2) and the coefficientfi to be strongly exchange enhanced. An estimate of/3 for palladium gives /3 --<0.99 × 10 -11 Oe -2 so that a marked field dependence of the susceptibility is here expected in quite practicable fields.
In an e a r l i e r c o m m u n i c a t i o n [1] the p h e n o m e non of i t i n e r a n t e l e c t r o n m e t a m a g n e t i s m was d i s c u s s e d . It was shown t h e o r e t i c a l l y that if a high m a g n e t i c field i s applied to a p a r a m a g n e t i c m e t a l at low t e m p e r a t u r e s , then at a c r i t i c a l field s t r e n g t h H c a t r a n s i t i o n to a f e r r o m a g n e t i cally o r d e r e d state should occur u n d e r specified conditions r e g a r d i n g the topology of the density of s t a t e s curve N(e) n e a r the F e r m i e n e r g y E F and the s t r e n g t h of the e l e c t r o n i n t e r a c t i o n s . To i l l u s t r a t e the magnitude of the effect when these conditions a r e satisfied, a rough e s t i m a t e of H c was made for p a l l a d i u m where the o c c u r e n c e of the effect was thought to be p a r t i c u l a r l y likely. It was found that the lower bound on H c was a p p r o x i m a t e l y 500 kOe. Even in the p r e s e n t stage of the high m a g n e t i c field a r t it would s e e m to be difficult to c a r r y out an u n a m b i g u o u s test of this t h e o r e t i c a l p r e d i c t i o n . However, although the effect itself m a y thus not yet be o b s e r v a b l e , c o n s i d e r a t i o n s s i m i l a r to those of ref. 1 can, u n d e r specified conditions, lead to the p r e d i c t i o n of a p r o n o u n c e d n o n l i n e a r i t y of the m a g n e t i z a t i o n , field c u r v e , or, e q u i v a l e n t l y , to a m a r k e d field dependence of the spin s u s c e p t i b i l i t y in fields c o n s i d e r a b l y lower than Hc. The p r e s e n t l e ~ e r p r e s e n t s the t h e o r e t i c a l background to this effect and e s t i m a t e s the degree of the n o n - l i n e a r i t y for palladium. Such an e s t i m a t i o n has only now been made p o s s i b l e with any a c c u r a c y as a r e s u l t of the r e c e n t c a l c u l a t i o n of the N(E) curve for palladium [2]. Using this calculation the magnitude of the exchange e n h a n c e m e n t [3] of the low field s u s ceptibility, defined in r e l a t i o n (3), can be obtained and i s found to b e l a r g e . Since, a s will be shown, this exchange e n h a n c e m e n t i n f l u e n c e s the field 280
dependence of the s u s c e p t i b i l i t y to a much l a r g e r extent than it does the low field s u s c e p t i b i l i t y itself, a v e r y m a r k e d effect may be p r e d i c t e d to occur in quite p r a c t i c a b l e fields for p a l l a d i u m . It is a s s u m e d that (i) the e l e c t r o n i n t e r a c t i o n i s via a short range potential leading to a constant i n t e r a c t i o n e n e r g y /eft which is a s s u m e d to have been c a l c u l a t e d by taking account of c o r r e l a t i o n effects [3], (ii) the r e l a t i v e m a g n e t i z a t i o n ~ is s m a l l in the p r e s e n c e of the applied field H so that t e r m s of o r d e r ~5 may be neglected, (iii) the d e n s i t y of s t a t e s c u r v e i s differentiable n e a r E F. It then follows from ref. 1 that m i n i m i z i n g the total e n e r g y of the e l e c t r o n s at 0°K leads to the following r e l a t i o n between ~ and H: n~
n
22
t
Here n i s the n u m b e r of p a r t i c l e s per atom and u i s given in t e r m s of the f i r s t and second d e r i v a t i v e s of N(e) as ~,,2n
The low field s u s c e p t i b i l i t y at 0°K is thus given by [3] X(0) =nPB~/H = 2•2BN(EF) [1-leffN(EF)] "1, (3) where the f i r s t factor is the usual P a u l i spin s u s ceptibility and the second factor r e p r e s e n t s the exchange e n h a n c e m e n t . On expanding (1) it i s then found that the field dependence of the s u s c e p t i b i l i ty is given by * Visiting Distinguished Professor; on leave of absence from Department of Mathematics, Imperial College, London, England (May-July, 1966).
Volume 22, number 3
PHYSICS LETTERS
x(H) = X(0) (1 + ~n2),
(4)
3 = ½ ~2B[1-/effN(EF)]-3.
(5)
where
For palladium the exchange enhancement of ×(0-) is, according to ref. 2, equal to as much as 12.4, so that the enhancement of ~ amounts to 1.91 x 103. The value of v can not be calculated at present with any accuracy as the topology of N(¢) is not known sufficiently well. An upper bound on v may, however, be obtained from (2) by takingN'(EF) = 0, i.e. by replacing v by Uo where ~o = (N"/N)¢ = E F.
(6)
F o r then it i s p o s s i b l e to make use of the r e l a t i o n [1] for the t e m p e r a t u r e dependence of the unenhanced low filed s u s c e p t i b i l i t y ×o(0) by again putting N ' (E F) = 0, so that Xo(0) = 2~2BN(EF) (1 +~ vo 7r2k2T2),
15 August 1966
and magnitude of ~ should give some i n d i c a t i o n s r e g a r d i n g the topology of N((). However, the exchange e n h a n c e m e n t is u n u s u a l l y l a r g e in p a l l a dium and many of its alloys and the p r e d i c t e d effect in other m e t a l s may by much s m a l l e r . Related m e a s u r e m e n t s for ferromagnetic m e t a l s have r e c e n t l y been r e p o r t e d [5]. In the foregoing d i s c u s s i o n the o r b i t a l c o n t r i bution to the o b s e r v e d low field s u s c e p t i b i l i t y , which i s s m a l l for p a l l a d i u m [6], and its high field dependence, which is at p r e s e n t unknown, have been ignored. F u r t h e r , the effects of Landau l e v e l s have also not been taken into account; at t e m p e r a t u r e s still r e a s o n a b l y low c o m p a r e d to E F / k and for p o l y c r y s t a l l i n e s p e c i m e n s such effects should, however, not occur. I a m grateful to Dr. F. H. Spedding and all my c o l l e a g u e s at A m e s for t h e i r g r e a t k i n d n e s s and hospitality d u r i n g my stay.
(7)
giving for p a l l a d i u m [1] v o = +1.18 × 10 26 e r g -2. Hence f r o m (5) --< 0.99 x I0 - I I Oe -2.
(8)
This value of ~ is to be compared with the much lower one obtainable if exchange enhancement is ignored, [4] namely /~ --< 5.2 x 10-15 Oe -2.
(9)
From (4) and (8) the lower bound on the field required to produce as much as a 10~o change of X(H) at low temperatures is as low as 101 kOe. For palladium the indications are that the coefficient fl in (4) is positive. If for some other metal E F lies in a region of the N(~) curve where the curvature is relatively small or negative then the susceptibility is predicted to decrease at higher fields. Hence measurements of the sign
References 1. E.P.Wohlfarth and P.Rhodes, Phil. Mag. 7 (1962} 1817. 2. A.J. Freeman, A.M. Furdyna and J. O. Dimmock, J.Appl. Phys. 37 (1966} 1256, and to be published. 3. C.Herring, Magnetism, eds. G.T.Rado and H.Suhl (Academic Press, New York}, Vol. 4 (1966}. 4. M.Shimizu, T.Takahashi and A.Katsuki, J. Phys. Soc, Japan 17 (1962} 1740. 5. J.H.M. Stoelinga and R. Gersdorf, Physics Letters 19 (1966} 640; A. J. Freeman, N.A. Blum, S. Foner, R.B. Fraenkel and E. J. McNiff, J. Appl. Phys. 37 (1966} 1338. 6. H. Kimura, A. Katsuki and M. Shimizu, J. Phys. Soc. Japan 21 (1966} 307.
281
THE
PHYSICS LETTERS
EFFECT OF EXCHANGE OF THE PARAMAGNETIC
15 August 1966
ENHANCEMENT ON THE SPIN SUSCEPTIBILITY
FIELD DEPENDENCE OF METALS
E. P. WOHLFARTH*
Institute for Atomic Research Iowa State University, Ames. Iowa, USA Received 7 July 1966
The field dependence of the spin susceptibility has been shown to be of the form X(/D = X(0) (1 +fill2) and the coefficientfi to be strongly exchange enhanced. An estimate of/3 for palladium gives /3 --<0.99 × 10 -11 Oe -2 so that a marked field dependence of the susceptibility is here expected in quite practicable fields.
In an e a r l i e r c o m m u n i c a t i o n [1] the p h e n o m e non of i t i n e r a n t e l e c t r o n m e t a m a g n e t i s m was d i s c u s s e d . It was shown t h e o r e t i c a l l y that if a high m a g n e t i c field i s applied to a p a r a m a g n e t i c m e t a l at low t e m p e r a t u r e s , then at a c r i t i c a l field s t r e n g t h H c a t r a n s i t i o n to a f e r r o m a g n e t i cally o r d e r e d state should occur u n d e r specified conditions r e g a r d i n g the topology of the density of s t a t e s curve N(e) n e a r the F e r m i e n e r g y E F and the s t r e n g t h of the e l e c t r o n i n t e r a c t i o n s . To i l l u s t r a t e the magnitude of the effect when these conditions a r e satisfied, a rough e s t i m a t e of H c was made for p a l l a d i u m where the o c c u r e n c e of the effect was thought to be p a r t i c u l a r l y likely. It was found that the lower bound on H c was a p p r o x i m a t e l y 500 kOe. Even in the p r e s e n t stage of the high m a g n e t i c field a r t it would s e e m to be difficult to c a r r y out an u n a m b i g u o u s test of this t h e o r e t i c a l p r e d i c t i o n . However, although the effect itself m a y thus not yet be o b s e r v a b l e , c o n s i d e r a t i o n s s i m i l a r to those of ref. 1 can, u n d e r specified conditions, lead to the p r e d i c t i o n of a p r o n o u n c e d n o n l i n e a r i t y of the m a g n e t i z a t i o n , field c u r v e , or, e q u i v a l e n t l y , to a m a r k e d field dependence of the spin s u s c e p t i b i l i t y in fields c o n s i d e r a b l y lower than Hc. The p r e s e n t l e ~ e r p r e s e n t s the t h e o r e t i c a l background to this effect and e s t i m a t e s the degree of the n o n - l i n e a r i t y for palladium. Such an e s t i m a t i o n has only now been made p o s s i b l e with any a c c u r a c y as a r e s u l t of the r e c e n t c a l c u l a t i o n of the N(E) curve for palladium [2]. Using this calculation the magnitude of the exchange e n h a n c e m e n t [3] of the low field s u s ceptibility, defined in r e l a t i o n (3), can be obtained and i s found to b e l a r g e . Since, a s will be shown, this exchange e n h a n c e m e n t i n f l u e n c e s the field 280
dependence of the s u s c e p t i b i l i t y to a much l a r g e r extent than it does the low field s u s c e p t i b i l i t y itself, a v e r y m a r k e d effect may be p r e d i c t e d to occur in quite p r a c t i c a b l e fields for p a l l a d i u m . It is a s s u m e d that (i) the e l e c t r o n i n t e r a c t i o n i s via a short range potential leading to a constant i n t e r a c t i o n e n e r g y /eft which is a s s u m e d to have been c a l c u l a t e d by taking account of c o r r e l a t i o n effects [3], (ii) the r e l a t i v e m a g n e t i z a t i o n ~ is s m a l l in the p r e s e n c e of the applied field H so that t e r m s of o r d e r ~5 may be neglected, (iii) the d e n s i t y of s t a t e s c u r v e i s differentiable n e a r E F. It then follows from ref. 1 that m i n i m i z i n g the total e n e r g y of the e l e c t r o n s at 0°K leads to the following r e l a t i o n between ~ and H: n~
n
22
t
Here n i s the n u m b e r of p a r t i c l e s per atom and u i s given in t e r m s of the f i r s t and second d e r i v a t i v e s of N(e) as ~,,2n
The low field s u s c e p t i b i l i t y at 0°K is thus given by [3] X(0) =nPB~/H = 2•2BN(EF) [1-leffN(EF)] "1, (3) where the f i r s t factor is the usual P a u l i spin s u s ceptibility and the second factor r e p r e s e n t s the exchange e n h a n c e m e n t . On expanding (1) it i s then found that the field dependence of the s u s c e p t i b i l i ty is given by * Visiting Distinguished Professor; on leave of absence from Department of Mathematics, Imperial College, London, England (May-July, 1966).
Volume 22, number 3
PHYSICS LETTERS
x(H) = X(0) (1 + ~n2),
(4)
3 = ½ ~2B[1-/effN(EF)]-3.
(5)
where
For palladium the exchange enhancement of ×(0-) is, according to ref. 2, equal to as much as 12.4, so that the enhancement of ~ amounts to 1.91 x 103. The value of v can not be calculated at present with any accuracy as the topology of N(¢) is not known sufficiently well. An upper bound on v may, however, be obtained from (2) by takingN'(EF) = 0, i.e. by replacing v by Uo where ~o = (N"/N)¢ = E F.
(6)
F o r then it i s p o s s i b l e to make use of the r e l a t i o n [1] for the t e m p e r a t u r e dependence of the unenhanced low filed s u s c e p t i b i l i t y ×o(0) by again putting N ' (E F) = 0, so that Xo(0) = 2~2BN(EF) (1 +~ vo 7r2k2T2),
15 August 1966
and magnitude of ~ should give some i n d i c a t i o n s r e g a r d i n g the topology of N((). However, the exchange e n h a n c e m e n t is u n u s u a l l y l a r g e in p a l l a dium and many of its alloys and the p r e d i c t e d effect in other m e t a l s may by much s m a l l e r . Related m e a s u r e m e n t s for ferromagnetic m e t a l s have r e c e n t l y been r e p o r t e d [5]. In the foregoing d i s c u s s i o n the o r b i t a l c o n t r i bution to the o b s e r v e d low field s u s c e p t i b i l i t y , which i s s m a l l for p a l l a d i u m [6], and its high field dependence, which is at p r e s e n t unknown, have been ignored. F u r t h e r , the effects of Landau l e v e l s have also not been taken into account; at t e m p e r a t u r e s still r e a s o n a b l y low c o m p a r e d to E F / k and for p o l y c r y s t a l l i n e s p e c i m e n s such effects should, however, not occur. I a m grateful to Dr. F. H. Spedding and all my c o l l e a g u e s at A m e s for t h e i r g r e a t k i n d n e s s and hospitality d u r i n g my stay.
(7)
giving for p a l l a d i u m [1] v o = +1.18 × 10 26 e r g -2. Hence f r o m (5) --< 0.99 x I0 - I I Oe -2.
(8)
This value of ~ is to be compared with the much lower one obtainable if exchange enhancement is ignored, [4] namely /~ --< 5.2 x 10-15 Oe -2.
(9)
From (4) and (8) the lower bound on the field required to produce as much as a 10~o change of X(H) at low temperatures is as low as 101 kOe. For palladium the indications are that the coefficient fl in (4) is positive. If for some other metal E F lies in a region of the N(~) curve where the curvature is relatively small or negative then the susceptibility is predicted to decrease at higher fields. Hence measurements of the sign
References 1. E.P.Wohlfarth and P.Rhodes, Phil. Mag. 7 (1962} 1817. 2. A.J. Freeman, A.M. Furdyna and J. O. Dimmock, J.Appl. Phys. 37 (1966} 1256, and to be published. 3. C.Herring, Magnetism, eds. G.T.Rado and H.Suhl (Academic Press, New York}, Vol. 4 (1966}. 4. M.Shimizu, T.Takahashi and A.Katsuki, J. Phys. Soc, Japan 17 (1962} 1740. 5. J.H.M. Stoelinga and R. Gersdorf, Physics Letters 19 (1966} 640; A. J. Freeman, N.A. Blum, S. Foner, R.B. Fraenkel and E. J. McNiff, J. Appl. Phys. 37 (1966} 1338. 6. H. Kimura, A. Katsuki and M. Shimizu, J. Phys. Soc. Japan 21 (1966} 307.
281