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Simplified expressions for spin wave energies in the band model of ferromagnetism
Volume24A, number 7
PHYSICS
SIMPLIFIED IN T H E
EXPRESSIONS BAND MODEL
LETTERS
27 M a r c h 1967
FOR SPIN WAVE ENERGIES OF FERROMAGNETISM
D. M. EDWARDS
Department of Mathematics, Imperial College, London, England Received 13 February 1967
The random phase approximation expression for D, the coefficient in the energy Dq2 of a long wave-length spin wave, is simplified. It reduces to a simple energy integral for a tight-binding form of band and some consequences are discussed. In the s i m p l e s t i t i n e r a n t - e l e c t r o n model of f e r r o m a g n e t i s m a single band, with Bloch e n e r gies ek, is p a r t i a l l y filled with e l e c t r o n s , n per atom ~ h e r e n < 1, which have an effective i n t e r action e n e r g y I when on the s a m e W a n n i e r function. An a p p r o x i m a t e d e s c r i p t i o n of the ground state and s i n g l e - p a r t i c l e excitations of this s y s tem is given by Stoner theory and s p i n - w a v e excitations may be included c o n s i s t e n t l y by m e a n s of the r a n d o m phase a p p r o x i m a t i o n [e.g. 1]. The energy of a spin wave of s m a l l wave vector q in a cubic lattice at 0OK is of the form Dq 2 where, within the r a n d o m phase approximation, D
I tEk
3N-
(V2Ek + i Vek 12 ,
~k< ~_ Here A, the exchange splitting, is given by n~l, where ~ is the relative magnetigation and N i s the number of atoms. The summations are over k with the restrictions indicated. The Fermi energies E+ in the sub-bands of spin + (; may be written % = ~ -~ ~ a ,
(2)
where ~t is the c h e m i c a l potential of S t o n e r ' s theory at 0°K. E x p r e s s i o n (1) m a y be s i m p l i f i e d by noting that use of G r e e n ' s t h e o r e m gives the r e sult
lWkl2=
~
(e~-~k)v2~k •
(3)
Ek< e¢7
On combining eelS. (i), (2) and (3) one obtains
D= ~ ~ (ek- ~t) V2Ek , 3NA2 E_ <~k
= ~ JR exp (ik'R) ,
¢(/~-E)N(e)de ,
(6)
where N(e) is the density of s t a t e s per atom. This r e s u l t has the following c o n s e q u e n c e s : (i) If ~t = 0, D < 0 so that the f e r r o m a g n e t i c state is unstable. In p a r t i c u l a r if N(e) is s y m m e t r i c about the c e n t r e of the band D < 0 for a h a l f - f i l led band (n = 1) with any ~. This extends the wellknown g e n e r a l r e s u l t for ~ = 1. (ii) Calculations of D by Katsuki and Wohlfarth [3] for a s i m p l e cubic s - b a n d could be s i m p l i f i e d and may be r e a d i l y extended to other tight-binding bands for which the density of s t a t e s is known. (iii) In the case of a very weak f e r r o m a g n e t (~ << 1), with a r e g u l a r N(~) c u r v e n e a r the par a m a g n e t i c F e r m i energy E F, D = -(n12R2/36)[g(cF) + eFg'(cf)]~ + O(~3).
(7)
Thus two necessary criteria for very weak ferromagnetism, the Stoner criterion and D > 0, may be written
IN(6 F) > 1, (4)
(5)
where the s u m m a t i o n is over n e a r e s t neighbours with I/~1 = R. The origin of energy is chosen at the ' c e n t r e of gravity' of the band so that no constant t e r m a p p e a r s in eq. (5). Hence V2ek = -R2Ek and eq. (4) may be w r i t t e n D =3A2~
+
ek< e~
and c l e a r l y Dcc ~ as ~ ~ 0, as shown e a r l i e r by Doniach and Wohifarth [2]. This f o r m of D, which has not been noted p r e v i o u s l y , is p a r t i c u l a r l y s u i t a b l e for application to a t i g h t - b i n d i n g form of band with
N(c F ) + e FN'(E F ) < 0,
(8)
where E F is measured from the 'centre of gravity'
Volume24A, number 7
PHYSICS LETTERS
G of the band. Hence v e r y weak f e r r o m a g n e t i s m in a t i g h t - b i n d i n g f o r m of band is m o s t likely, a c c o r d i n g to the r a n d o m phase a p p r o x i m a t i o n , if the F e r m i l e v e l is w e l l - r e m o v e d f r o m G and l i e s in a r e g i o n w h e r e the density of s t a t e s is high and i n c r e a s i n g in the d i r e c t i o n of G. The work d e s c r i b e d h e r e m a y be extended to f i n i t e t e m p e r a t u r e s and this will be r e p o r t e d l a t e r by Mathon and Wohlfarth. I am g r a t e f u l to P r o -
ELECTRICAL OF
27 March 1967
l e s s o r E. P. W o h l f ar t h f o r his e n c o u r a g i n g i n t e r e s t in this p r o b l e m . References
1. T. Izuyama and R. Kubo, J. Appl. Phys. 35 (1964) 1074. 2. S. Doniach and E. P. Wohlfarth, Phys. Letters 18 (1965) 209. 3. A. Katsuki and E. P. Wohlfarth, Proc. Roy. Soc. A295 (1966) 182.
RESISTANCE AND MAGNETIZATION TYPE II SUPERCONDUCTORS
J. M. RAYROUX, D. 1TSCHNER and P. MI3LLER P h y s i c s Laboratory, Brown, B o v e r i and Co., Ltd. Received 17 F e b r u a r y 1967
The voltage U across a type II superconductor under constant transport current and linear field sweep conditions is shown to be proportional to B dB/dt. The relation of U to the magnetization and the a. c. losses is pointed out.
L o s s e s o c c u r in type H s u p e r c o n d u c t o r s when they a r e e x p o s e d to t i m e v a r y i n g f i e l d s or c u r r e n t s . In this l e t t e r we give the r e s u l t of m e a s u r e m e n t s m a d e on a s u p e r c o n d u c t i n g w i r e c a r r y ing a c o n s t a n t c u r r e n t while being e x p o s e d to a perpendicular magnetic field increasing linearly with t i m e and p r o p o s e to u s e the m i c r o s c o p i c v o r t e x r e p r e s e n t a t i o n of a type II s u p e r c o n d u c t o r in the m i x e d s t a t e f o r the i n t e r p r e t a t i o n of the results. The e x p e r i m e n t a l s e t up is e s s e n t i a l l y the s a m e as the one u s e d by T a q u e t [1]. T h e w i r e probe is wound b i f i l a r l y and p l a c e d p e r p e n d i c u l a r in the field of a s u p e r c o n d u c t i n g solenoid. T h e potential d i f f e r e n c e U a c r o s s the b i f i l a r p r o b e is m e a s u r e d by m e a n s of a Keithley type 150 m i c r o v o l t m e t e r and r e c o r d e d on a M o s e l e y type 2 D R - 2 M X - Y r e c o r d e r . All m e a s u r e m e n t s w e r e p e r f o r m e d at 4.2°K. The t r a n s p o r t c u r r e n t I t being set, the m a g n e t i c f i e l d is i n c r e a s e d l i n e a r ly f r o m z e r o up to s o m e m a x i m u m f ie ld (15 kOe for N b - Z r and N b - T i , a few kOe f o r Nb) kept constant f o r a while and s e t back to z e r o at about the s a m e r a t e of change. Let us c o n s i d e r the r e s u l t s of 2 s u c c e s s i v e up and down s w e e p s on a Nb-25 % Z r w i r e ( $ =
0.127 ram; no Cu plating) (c.f. fig. 1). T h e i r g e n e r a l c h a r a c t e r i s t i c s ( d i s a p p e a r a n c e of U f o r d H / d t = 0, p r o p o r t i o n a l i t y to It, H and d H / d 0 a g r e e with T a q u e t ' s m e a s u r e m e n t s in the h i g h e r f i el d r e g i o n . Two new ef f ect s a p p e a r , h o w e v e r , on our m e a s u r e m e n t s : f i r s t while the f i el d is inc r e a s e d , the v o l t a g e r e m a i n s n e g l i g i b l e up to a t h r e s h o l d f i el d v al u e and s e c o n d l y the e v i d e n c e of the h y s t e r e s i s . In our view the m e a s u r e d v o l t ag e has its o r i gin in a c o m p l e x i n t e r a c t i o n m e c h a n i s m b et w een the t r a n s p o r t c u r r e n t It, the e x i s t i n g v o r t i c e s , whose d en si t y at a t i m e t is nL(t) = B(/-/)/~o, and the newly c r e a t e d ones whose b i r t h r a t e is f~L(t) = / } ( H ) / ¢ o (¢o = 2 × 10 -7 g a u s s , cm2). Let us a s s u m e that the r e s i s t i v i t y p c o r r e s p o n d i n g to U is p r o p o r t i o n a l to the product of the s a m p l e induction B m u l t i p l i e d by its t i m e d e r i v a t i v e / ~ and let us w r i t e it as follows: P ¢c B B = S ( d B / d I t ) ( d H / d t )
.
(1)
C o n s i d e r i n g d H / d t constant it is e a s y to show that the product B • d B / d H f o r a t y p i c a l non i d eal s u p e r c o n d u c t o r of s e c o n d kind is indeed a c u r v e s i m i l a r in shape to our e x p e r i m e n t a l one. Ano t h er c o n f i r m a t i o n of our a s s u m p t i o n can be 351
PHYSICS
SIMPLIFIED IN T H E
EXPRESSIONS BAND MODEL
LETTERS
27 M a r c h 1967
FOR SPIN WAVE ENERGIES OF FERROMAGNETISM
D. M. EDWARDS
Department of Mathematics, Imperial College, London, England Received 13 February 1967
The random phase approximation expression for D, the coefficient in the energy Dq2 of a long wave-length spin wave, is simplified. It reduces to a simple energy integral for a tight-binding form of band and some consequences are discussed. In the s i m p l e s t i t i n e r a n t - e l e c t r o n model of f e r r o m a g n e t i s m a single band, with Bloch e n e r gies ek, is p a r t i a l l y filled with e l e c t r o n s , n per atom ~ h e r e n < 1, which have an effective i n t e r action e n e r g y I when on the s a m e W a n n i e r function. An a p p r o x i m a t e d e s c r i p t i o n of the ground state and s i n g l e - p a r t i c l e excitations of this s y s tem is given by Stoner theory and s p i n - w a v e excitations may be included c o n s i s t e n t l y by m e a n s of the r a n d o m phase a p p r o x i m a t i o n [e.g. 1]. The energy of a spin wave of s m a l l wave vector q in a cubic lattice at 0OK is of the form Dq 2 where, within the r a n d o m phase approximation, D
I tEk
3N-
(V2Ek + i Vek 12 ,
~k< ~_ Here A, the exchange splitting, is given by n~l, where ~ is the relative magnetigation and N i s the number of atoms. The summations are over k with the restrictions indicated. The Fermi energies E+ in the sub-bands of spin + (; may be written % = ~ -~ ~ a ,
(2)
where ~t is the c h e m i c a l potential of S t o n e r ' s theory at 0°K. E x p r e s s i o n (1) m a y be s i m p l i f i e d by noting that use of G r e e n ' s t h e o r e m gives the r e sult
lWkl2=
~
(e~-~k)v2~k •
(3)
Ek< e¢7
On combining eelS. (i), (2) and (3) one obtains
D= ~ ~ (ek- ~t) V2Ek , 3NA2 E_ <~k
= ~ JR exp (ik'R) ,
¢(/~-E)N(e)de ,
(6)
where N(e) is the density of s t a t e s per atom. This r e s u l t has the following c o n s e q u e n c e s : (i) If ~t = 0, D < 0 so that the f e r r o m a g n e t i c state is unstable. In p a r t i c u l a r if N(e) is s y m m e t r i c about the c e n t r e of the band D < 0 for a h a l f - f i l led band (n = 1) with any ~. This extends the wellknown g e n e r a l r e s u l t for ~ = 1. (ii) Calculations of D by Katsuki and Wohlfarth [3] for a s i m p l e cubic s - b a n d could be s i m p l i f i e d and may be r e a d i l y extended to other tight-binding bands for which the density of s t a t e s is known. (iii) In the case of a very weak f e r r o m a g n e t (~ << 1), with a r e g u l a r N(~) c u r v e n e a r the par a m a g n e t i c F e r m i energy E F, D = -(n12R2/36)[g(cF) + eFg'(cf)]~ + O(~3).
(7)
Thus two necessary criteria for very weak ferromagnetism, the Stoner criterion and D > 0, may be written
IN(6 F) > 1, (4)
(5)
where the s u m m a t i o n is over n e a r e s t neighbours with I/~1 = R. The origin of energy is chosen at the ' c e n t r e of gravity' of the band so that no constant t e r m a p p e a r s in eq. (5). Hence V2ek = -R2Ek and eq. (4) may be w r i t t e n D =3A2~
+
ek< e~
and c l e a r l y Dcc ~ as ~ ~ 0, as shown e a r l i e r by Doniach and Wohifarth [2]. This f o r m of D, which has not been noted p r e v i o u s l y , is p a r t i c u l a r l y s u i t a b l e for application to a t i g h t - b i n d i n g form of band with
N(c F ) + e FN'(E F ) < 0,
(8)
where E F is measured from the 'centre of gravity'
Volume24A, number 7
PHYSICS LETTERS
G of the band. Hence v e r y weak f e r r o m a g n e t i s m in a t i g h t - b i n d i n g f o r m of band is m o s t likely, a c c o r d i n g to the r a n d o m phase a p p r o x i m a t i o n , if the F e r m i l e v e l is w e l l - r e m o v e d f r o m G and l i e s in a r e g i o n w h e r e the density of s t a t e s is high and i n c r e a s i n g in the d i r e c t i o n of G. The work d e s c r i b e d h e r e m a y be extended to f i n i t e t e m p e r a t u r e s and this will be r e p o r t e d l a t e r by Mathon and Wohlfarth. I am g r a t e f u l to P r o -
ELECTRICAL OF
27 March 1967
l e s s o r E. P. W o h l f ar t h f o r his e n c o u r a g i n g i n t e r e s t in this p r o b l e m . References
1. T. Izuyama and R. Kubo, J. Appl. Phys. 35 (1964) 1074. 2. S. Doniach and E. P. Wohlfarth, Phys. Letters 18 (1965) 209. 3. A. Katsuki and E. P. Wohlfarth, Proc. Roy. Soc. A295 (1966) 182.
RESISTANCE AND MAGNETIZATION TYPE II SUPERCONDUCTORS
J. M. RAYROUX, D. 1TSCHNER and P. MI3LLER P h y s i c s Laboratory, Brown, B o v e r i and Co., Ltd. Received 17 F e b r u a r y 1967
The voltage U across a type II superconductor under constant transport current and linear field sweep conditions is shown to be proportional to B dB/dt. The relation of U to the magnetization and the a. c. losses is pointed out.
L o s s e s o c c u r in type H s u p e r c o n d u c t o r s when they a r e e x p o s e d to t i m e v a r y i n g f i e l d s or c u r r e n t s . In this l e t t e r we give the r e s u l t of m e a s u r e m e n t s m a d e on a s u p e r c o n d u c t i n g w i r e c a r r y ing a c o n s t a n t c u r r e n t while being e x p o s e d to a perpendicular magnetic field increasing linearly with t i m e and p r o p o s e to u s e the m i c r o s c o p i c v o r t e x r e p r e s e n t a t i o n of a type II s u p e r c o n d u c t o r in the m i x e d s t a t e f o r the i n t e r p r e t a t i o n of the results. The e x p e r i m e n t a l s e t up is e s s e n t i a l l y the s a m e as the one u s e d by T a q u e t [1]. T h e w i r e probe is wound b i f i l a r l y and p l a c e d p e r p e n d i c u l a r in the field of a s u p e r c o n d u c t i n g solenoid. T h e potential d i f f e r e n c e U a c r o s s the b i f i l a r p r o b e is m e a s u r e d by m e a n s of a Keithley type 150 m i c r o v o l t m e t e r and r e c o r d e d on a M o s e l e y type 2 D R - 2 M X - Y r e c o r d e r . All m e a s u r e m e n t s w e r e p e r f o r m e d at 4.2°K. The t r a n s p o r t c u r r e n t I t being set, the m a g n e t i c f i e l d is i n c r e a s e d l i n e a r ly f r o m z e r o up to s o m e m a x i m u m f ie ld (15 kOe for N b - Z r and N b - T i , a few kOe f o r Nb) kept constant f o r a while and s e t back to z e r o at about the s a m e r a t e of change. Let us c o n s i d e r the r e s u l t s of 2 s u c c e s s i v e up and down s w e e p s on a Nb-25 % Z r w i r e ( $ =
0.127 ram; no Cu plating) (c.f. fig. 1). T h e i r g e n e r a l c h a r a c t e r i s t i c s ( d i s a p p e a r a n c e of U f o r d H / d t = 0, p r o p o r t i o n a l i t y to It, H and d H / d 0 a g r e e with T a q u e t ' s m e a s u r e m e n t s in the h i g h e r f i el d r e g i o n . Two new ef f ect s a p p e a r , h o w e v e r , on our m e a s u r e m e n t s : f i r s t while the f i el d is inc r e a s e d , the v o l t a g e r e m a i n s n e g l i g i b l e up to a t h r e s h o l d f i el d v al u e and s e c o n d l y the e v i d e n c e of the h y s t e r e s i s . In our view the m e a s u r e d v o l t ag e has its o r i gin in a c o m p l e x i n t e r a c t i o n m e c h a n i s m b et w een the t r a n s p o r t c u r r e n t It, the e x i s t i n g v o r t i c e s , whose d en si t y at a t i m e t is nL(t) = B(/-/)/~o, and the newly c r e a t e d ones whose b i r t h r a t e is f~L(t) = / } ( H ) / ¢ o (¢o = 2 × 10 -7 g a u s s , cm2). Let us a s s u m e that the r e s i s t i v i t y p c o r r e s p o n d i n g to U is p r o p o r t i o n a l to the product of the s a m p l e induction B m u l t i p l i e d by its t i m e d e r i v a t i v e / ~ and let us w r i t e it as follows: P ¢c B B = S ( d B / d I t ) ( d H / d t )
.
(1)
C o n s i d e r i n g d H / d t constant it is e a s y to show that the product B • d B / d H f o r a t y p i c a l non i d eal s u p e r c o n d u c t o r of s e c o n d kind is indeed a c u r v e s i m i l a r in shape to our e x p e r i m e n t a l one. Ano t h er c o n f i r m a t i o n of our a s s u m p t i o n can be 351