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Specific heat of the two dimensional antiferromagnet K2NiF4
Volume 36A, number 2
SPECIFIC
HEAT
PHYSICS LETTERS
OF
THE
TWO
DIMENSIONAL
16 August 1971
ANTIFERROMAGNET
K2NiF 4
M. B. SALAMON *
Institute for Solid State Physics, Roppongi, Minato-ku, Tokyo, Jalmn and I. HATTA
Tokyo Institute of Technology, Ohkayama, Meguro-ku, Tokyo, Japan Received 2 July 1971 The specific heat of the two dimensional antfferromagnet K2NiF 4 h~s bern measured near Tc(2). A small peak, superimposed on a linearly increasing contribution above Tc(2) , was observed.
W h i l e it i s r i g o r o u s l y [ I ] t r u e that a two d i m e n s i o n a l i s o t r o p i c H e i s e n b e r g m a g n e t cannot s u p p o r t long r a n g e o r d e r , Stanley and Kaplan[2] (SK) and L i n e s have d e m o n s t r a t e d the p o s s i b i l i t y of a d i v e r g e n t s u s c e p t i b i l i t y at ~c2) = 1.2 JS(S+ 1)/k. In L i n e s ' m o d e l the d i v e r g e n c e r e s u l t s f r o m the p r o g r e s s i v e " d e - r e n o r m a l i z a t i o n " of s p i n waves with k > ~ / L , w h e r e L i s a t e m p e r a t u r e d e p e n dent c o h e r e n c e length, d i s t i n c t f r o m and s h o r t e r than the u s u a l c o r r e l a t i o n length. T h e a c c o m p a n y i n g shift of the fluctuation p o w e r s p e c t r u m to l o w e r k i n c r e a s e s both the s h o r t r a n g e o r d e r and L a s the t e m p e r a t u r e i s l o w e r e d , f i n a l l y leading to the SK t r a n s i t i o n . T h e p r e s e n c e of a finite a n i s o t r o p y a f f e c t s only the low-k s p i n w a v e s , s t a b i l i z i n g the long r a n g e o r d e r , but does not a f f e c t the n a t u r e of t h e t r a n s i t i o n g r e a t l y [3]. We r e p o r t h e r e t h e o b s e r v a t i o n of the a n o m a lous p a r t of t h e s p e c i f i c heat of K2NiF 4 n e a r the t r a n s i t i o n . S e v e r a l p r o p e r t i e s of t h i s m a t e r i a l have a l r e a d y b e e n m e a s u r e d [4, 5] at T~2) and t h e y t e n d to c o n f i r m the two d i m e n s i o n a l n a t u r e of the t r a n s i t i o n . A c c o r d i n g to L i n e s , the n e a r e s t n e i g h b o r c o r r e l a t i o n function can b e e x p r e s s e d as
cos(k.a)
4J(So'Sa} 1 ~ 2-c°s(kxa)-c°S(kya)+(4J×o)-I 3kT - N k <~/L
(1) w h e r e (4JXo)- 1 = 0.018 a 2/L 2 = 1.45 E2 a b o v e the * Alfred P. Sloan Research Fellow on leave from the Department of Physics. University of Illinois, Urbana, Illinois, USA.
t r a n s i t i o n and w h e r e E = (T/T(c2) -1). T h e s e v a l u e s w e r e obtained by L i n e s f r o m a fit of his e x p r e s s i o n s to the high t e m p e r a t u r e expansion [2]. N e a r T(c2) , w h e r e L is l a r g e , we m a y e x pand (1) and p e r f o r m the i n t e g r a l s to find 1
CH=0 =-~NzJ(d/dT) ~So.Sa} = 70Re ,
(2)
to lowest o r d e r in L -1. Outside the c r i t i c a l r e g i o n , L i n e s ' m o d e l p r e d i c t s a constant c o n t r i bution of ~ 2R extending to ~1.4T(c2) . The r e s u l t s o f an a c m e a s u r e m e n t [6] of the s p e c i f i c heat a r e shown in fig. 1. We have s u b tracted a parabolic lattice term, extrapolated f r o m low t e m p e r a t u r e s . The l i n e a r p o r t i o n r i s e s f r o m T(c2) = (98.7±0.2)K at a r a t e of 2.SRE, much m o r e slowly than p r e d i c t e d by (2). At h i g h e r t e m p e r a t u r e s , the s u b t r a c t i o n of the l a t t i c e b a c k g r o u n d b e c o m e s i n c r e a s i n g l y difficult, but t h e r e a p p e a r s to b e a b r o a d m a x i m u m of ~ 1.5R at T ~ 160K. T h e p r e s e n c e of a p e a k at T~2) is to b e e x p e c t e d [7] in an a n i s o t r o p i c s y s t e m when --
Volume 36A, number 2
~°Jl
0.~ -.~ 0.161-
PHYSICS
A
! ~.
/
98,0 ~.2 98.4 96.G~8 99.0~.299.
T ('K)
j ,
,
/
"
d
~
0.:
82
86
90
94 98 I02 I06 TEMPERATURE ('K)
110
114
Fig. 1. Specific heat of K2NiF4 relative to a parabolic lattice background. The dashed line is the prediction of Lines' theory. coinciding within 0.2K. Since the a n o m a l o u s specific heat is a s m a l l f r a c t i o n of the l a t t i c e background (Clat(100) = 15.8 c a l / m o l e . K ) , s e p a r a t i o n of the m a g n e t i c p a r t is somewhat ambiguous. U s e of a l i n e a r background, for example, gives s i m i l a r r e s u l t s , but with the l i n e a r p o r t i o n r i s i n g from 97K, a p r o v o c a t i v e but a p p a r a n t l y s p u r i o u s r e s u l t . A c o m p a r i s o n of the specific heat of K2NiF 4 with the i s o s t r u c t u r a l compound .K2MgF 4 will be r e p o r t e d s e p a r a t e l y .
The authors acknowledge the hospitality of the I n s t i t u t e for Solid State P h y s i c s w h e r e t h e s e measurements were performed.
References [1] N.D.Merrnin and H.Wagner, Phys. Rev. Letters 17 (1966) 1133. [2] E. Stanley and T. A. Kaplan, Phys. Rev. Letters 17 (1966) 913. [3] M.E.Lines, Phys. Rev. B3 (1971) 1749. [4] R.J.Birgeneau, J. Skalyo and G.Shirane, Phys. Rev B3 (1971) 1736. [ 5] E.P. Maarschall, A.C. Botterman, S. Vega and A.R.Miedema, Physica 41 (1969) 473. [6] M.B.Salamon, Phys. Rev. B2 (1970) 214. [7] M.Suzuki, to be published.
** We are grateful to N. Sakamoto, of the Electrotechnical Laboratory, and to M.Kajiura of the Broadcasting Science Laboratory, NHK, for supplying samples of K2NiF4. *****
86
16 August 1971
We note that the c r i t i c a l exponent for the specific heat, a = - 1 , does not satisfy the s c a l i n g law r e s u l t du = 2- a when L i n e s ' value v= 1 is used. The a g r e e m e n t is no b e t t e r with the e x p e r i m e n t a l [4] value v=0.57, although in this case, the exponent of the peak (which we cannot e x t r a c t with accuracy) should be u s e d in the s c a l i n g relation. Thus, while L i n e s ' model a p p e a r s to d e s c r i b e the q u a l i t a t i v e b e h a v i o r of the SK t r a n s i t i o n , it gives too r a p i d a r i s e in Cp. It r e p r e s e n t s an i m p r o v e m e n t over the RPA r e s u l t , which p r e d i c t s a step i n c r e a s e in Cp, but leaves open the exact n a t u r e of the SK t r a n s i t i o n .
/
~OF~'°O.I] .~ DETAIL OF "~ l O ~ , . ~ PEAK 0.~
LETTERS
SPECIFIC
HEAT
PHYSICS LETTERS
OF
THE
TWO
DIMENSIONAL
16 August 1971
ANTIFERROMAGNET
K2NiF 4
M. B. SALAMON *
Institute for Solid State Physics, Roppongi, Minato-ku, Tokyo, Jalmn and I. HATTA
Tokyo Institute of Technology, Ohkayama, Meguro-ku, Tokyo, Japan Received 2 July 1971 The specific heat of the two dimensional antfferromagnet K2NiF 4 h~s bern measured near Tc(2). A small peak, superimposed on a linearly increasing contribution above Tc(2) , was observed.
W h i l e it i s r i g o r o u s l y [ I ] t r u e that a two d i m e n s i o n a l i s o t r o p i c H e i s e n b e r g m a g n e t cannot s u p p o r t long r a n g e o r d e r , Stanley and Kaplan[2] (SK) and L i n e s have d e m o n s t r a t e d the p o s s i b i l i t y of a d i v e r g e n t s u s c e p t i b i l i t y at ~c2) = 1.2 JS(S+ 1)/k. In L i n e s ' m o d e l the d i v e r g e n c e r e s u l t s f r o m the p r o g r e s s i v e " d e - r e n o r m a l i z a t i o n " of s p i n waves with k > ~ / L , w h e r e L i s a t e m p e r a t u r e d e p e n dent c o h e r e n c e length, d i s t i n c t f r o m and s h o r t e r than the u s u a l c o r r e l a t i o n length. T h e a c c o m p a n y i n g shift of the fluctuation p o w e r s p e c t r u m to l o w e r k i n c r e a s e s both the s h o r t r a n g e o r d e r and L a s the t e m p e r a t u r e i s l o w e r e d , f i n a l l y leading to the SK t r a n s i t i o n . T h e p r e s e n c e of a finite a n i s o t r o p y a f f e c t s only the low-k s p i n w a v e s , s t a b i l i z i n g the long r a n g e o r d e r , but does not a f f e c t the n a t u r e of t h e t r a n s i t i o n g r e a t l y [3]. We r e p o r t h e r e t h e o b s e r v a t i o n of the a n o m a lous p a r t of t h e s p e c i f i c heat of K2NiF 4 n e a r the t r a n s i t i o n . S e v e r a l p r o p e r t i e s of t h i s m a t e r i a l have a l r e a d y b e e n m e a s u r e d [4, 5] at T~2) and t h e y t e n d to c o n f i r m the two d i m e n s i o n a l n a t u r e of the t r a n s i t i o n . A c c o r d i n g to L i n e s , the n e a r e s t n e i g h b o r c o r r e l a t i o n function can b e e x p r e s s e d as
cos(k.a)
4J(So'Sa} 1 ~ 2-c°s(kxa)-c°S(kya)+(4J×o)-I 3kT - N k <~/L
(1) w h e r e (4JXo)- 1 = 0.018 a 2/L 2 = 1.45 E2 a b o v e the * Alfred P. Sloan Research Fellow on leave from the Department of Physics. University of Illinois, Urbana, Illinois, USA.
t r a n s i t i o n and w h e r e E = (T/T(c2) -1). T h e s e v a l u e s w e r e obtained by L i n e s f r o m a fit of his e x p r e s s i o n s to the high t e m p e r a t u r e expansion [2]. N e a r T(c2) , w h e r e L is l a r g e , we m a y e x pand (1) and p e r f o r m the i n t e g r a l s to find 1
CH=0 =-~NzJ(d/dT) ~So.Sa} = 70Re ,
(2)
to lowest o r d e r in L -1. Outside the c r i t i c a l r e g i o n , L i n e s ' m o d e l p r e d i c t s a constant c o n t r i bution of ~ 2R extending to ~1.4T(c2) . The r e s u l t s o f an a c m e a s u r e m e n t [6] of the s p e c i f i c heat a r e shown in fig. 1. We have s u b tracted a parabolic lattice term, extrapolated f r o m low t e m p e r a t u r e s . The l i n e a r p o r t i o n r i s e s f r o m T(c2) = (98.7±0.2)K at a r a t e of 2.SRE, much m o r e slowly than p r e d i c t e d by (2). At h i g h e r t e m p e r a t u r e s , the s u b t r a c t i o n of the l a t t i c e b a c k g r o u n d b e c o m e s i n c r e a s i n g l y difficult, but t h e r e a p p e a r s to b e a b r o a d m a x i m u m of ~ 1.5R at T ~ 160K. T h e p r e s e n c e of a p e a k at T~2) is to b e e x p e c t e d [7] in an a n i s o t r o p i c s y s t e m when --
Volume 36A, number 2
~°Jl
0.~ -.~ 0.161-
PHYSICS
A
! ~.
/
98,0 ~.2 98.4 96.G~8 99.0~.299.
T ('K)
j ,
,
/
"
d
~
0.:
82
86
90
94 98 I02 I06 TEMPERATURE ('K)
110
114
Fig. 1. Specific heat of K2NiF4 relative to a parabolic lattice background. The dashed line is the prediction of Lines' theory. coinciding within 0.2K. Since the a n o m a l o u s specific heat is a s m a l l f r a c t i o n of the l a t t i c e background (Clat(100) = 15.8 c a l / m o l e . K ) , s e p a r a t i o n of the m a g n e t i c p a r t is somewhat ambiguous. U s e of a l i n e a r background, for example, gives s i m i l a r r e s u l t s , but with the l i n e a r p o r t i o n r i s i n g from 97K, a p r o v o c a t i v e but a p p a r a n t l y s p u r i o u s r e s u l t . A c o m p a r i s o n of the specific heat of K2NiF 4 with the i s o s t r u c t u r a l compound .K2MgF 4 will be r e p o r t e d s e p a r a t e l y .
The authors acknowledge the hospitality of the I n s t i t u t e for Solid State P h y s i c s w h e r e t h e s e measurements were performed.
References [1] N.D.Merrnin and H.Wagner, Phys. Rev. Letters 17 (1966) 1133. [2] E. Stanley and T. A. Kaplan, Phys. Rev. Letters 17 (1966) 913. [3] M.E.Lines, Phys. Rev. B3 (1971) 1749. [4] R.J.Birgeneau, J. Skalyo and G.Shirane, Phys. Rev B3 (1971) 1736. [ 5] E.P. Maarschall, A.C. Botterman, S. Vega and A.R.Miedema, Physica 41 (1969) 473. [6] M.B.Salamon, Phys. Rev. B2 (1970) 214. [7] M.Suzuki, to be published.
** We are grateful to N. Sakamoto, of the Electrotechnical Laboratory, and to M.Kajiura of the Broadcasting Science Laboratory, NHK, for supplying samples of K2NiF4. *****
86
16 August 1971
We note that the c r i t i c a l exponent for the specific heat, a = - 1 , does not satisfy the s c a l i n g law r e s u l t du = 2- a when L i n e s ' value v= 1 is used. The a g r e e m e n t is no b e t t e r with the e x p e r i m e n t a l [4] value v=0.57, although in this case, the exponent of the peak (which we cannot e x t r a c t with accuracy) should be u s e d in the s c a l i n g relation. Thus, while L i n e s ' model a p p e a r s to d e s c r i b e the q u a l i t a t i v e b e h a v i o r of the SK t r a n s i t i o n , it gives too r a p i d a r i s e in Cp. It r e p r e s e n t s an i m p r o v e m e n t over the RPA r e s u l t , which p r e d i c t s a step i n c r e a s e in Cp, but leaves open the exact n a t u r e of the SK t r a n s i t i o n .
/
~OF~'°O.I] .~ DETAIL OF "~ l O ~ , . ~ PEAK 0.~
LETTERS