Phase transition of two dimensional Heisenberg antiferromagnet of S = 12

680 PHASE TRANSITION OF TWO DIMENSIONAL HEISENBERG ANTIFERROMAGNET OF S = M. M A T S U U R A , Y. Y A M A M O T O , H. Y A M A K A W A , T. H A S E D A Faculty of Engineering Science, Osaka University, Toyonaka 560, Japan

and Y. AJIRO Department of Chemistry, Faculty of Science, Kyoto University, Kyoto 606, Japan

From the analysis of the susceptibility of Cu(HCOO).4H20 in which the Cu 2~ ions are on two inequivalent sites, the staggered susceptibility of a two-dimensional Heisenberg antiferromagnet is examined. From the results, and from proton NMR and heat capacity on the related salts, characteristic features of the phase transition of a two-dimensional Heisenberg antiferromagnet are inferred.

Ordering of a two-dimensional Heisenberg (2dH) spin system is a most interesting unsolved problem. A rigorous proof was put forward by Mermin and Wagner that no spontaneous magnetization appears at a finite temperature in a 2dH ferromagnet [1]. Stanley and Kaplan have conjectured that 2dH ferromagnets of s t> 1 may have a new type of transition, involving a divergence of the susceptibility at T~, but no spontaneous magnetization below Tc [2]. Many experimental studies have been done on these attractive problems, mostly in ferromagnets [3, 4]. In the present work, we investigate 2dH antiferromagnet of s = ½, which may be an even more interesting case than the ferromagnet. Since the staggered magnetization does not commute with the total hamiltonian, the antiferromagnet may be even more difficult to order than the ferromagnet. Cu(HCOO)2-4H20 (CuF4H) has a layer-structure along the c axis and approximates a 2dH antiferromagnet. From the symmetry, we know the g-tensors of Cu 2+ ions at (0, 0, 0) and ({, ½, 0) sites, g~ and g2 are not equivalent. In such a crystallographic two-sublattice (C2) system, not only a uniform magnetization but also a staggered magnetization is induced by applying a uniform field [5]. The reason is that under a field H each spin of Cu 2+ at a (0, 0, 0) site has a response to g,H while each at (2, ½, 0) site responds to g2H. This circumstance is quite equivalent to the following: the Cu 2÷ ion system with a g-tensor g[=½(g~+g2)] (hereafter C, system) feels a uniform field H and a staggered Physica 86--88B (1977) 680-682 O North-Holland

field H,(= g - l d / / ) where d is defined by ,.(g~ - g2). The corresponding uniform and intermode susceptibilities of the C2 system X (2) and X(,2]can be expressed by [6] "*,r(l) ] X(2) . =A. tag

--1~(I)

X(2)

~r(1)

,,,=x,g

I--

A~ g a,

/

(1)

I--

o,

(2)

where X(2) and X[' are the uniform and staggered susceptibilities of the C~ system. yc2) The temperature dependence of --U is shown in fig. I [7]. The principal values along L , ( - ¢ ) and L2(b) axes increases rapidly down to TN. The characteristic feature is certainly coming from X~~) because X~~) does not show any such anomaly a c r o s s T N as seen in Xc~(--,r X~~) is the linear response coefficient for H, which is staggered in the ab plane and uniform along the c axis. So we sign the field as H,(Tr, 0) and the corresponding X~~) as X(~')(Tr,0). We can * L,u(emU/mol} 4'-

xlO 2

6

" u {ernu/mol) ~' , i o 5

5 , L,2

2 -

×xo

o

I

I ×20

La

l

40

'

60

l

80

'

160

I

0

20

40

6~0

80

I00

T(K)

Fig. 1. Susceptibility of CuF4H along the principal directions [7].

681 define another staggered field Hs(Tr, 7r), which is staggered along the c axis too, and the corresponding susceptibility Xts'~(Tr,7r). If the interplanar interaction along the c axis J' is zero, these two quantities are equal to nxt, m~(~r)where n is the number of ab planes and X~'~(~-) is the staggered susceptibility of one ab plane. If J' is finite and ferromagnetic, H~(Tr,0) enhances the cooperative action of the system while Hs(Tr, 7r) suppresses it. So X~s'l(Tr,0) I> nX~')(~") 1> X~')(Tr, 7r). If J' is finite and antiferromagnetic, then x~l)(Tr, 0) ~ nx(sl)(7] ") ~ X(sl)('JT, T/'). As the increase of the measured Xt~) near TN is coming from Xts')(Tr,0) and J' is antiferromagnetic from proton NMR below TN [8] the following two cases will be considered with reference to the experiments. (1) If we could make J' smaller, Xtu2) should increase more significantly. (2) If we apply an external field H corresponding to H~ of the order of J', a nonlinear effect of staggered magnetization may be observed. The temperature dependence of Xtu2) of Cu(HCOO)2.2CO(NH2)2.2H20 (CuFUH) which can be seen as derived from CuF4H by partial substitution of H20 by urea molecules [9], is compared with that of CuF4H in fig. 2(a). The intraplanar interactions in both salts are about the same [9]. However, the interplanar interaction of C u F U H is estimated to be much weaker than that of CuF4H from the larger interplanar

separation [10]. The more significant increase of X 12~in C u F U H supports the first prediction. The temperature dependence of proton NMR frequency shift in CuF4D, a substituted form of CuF4H by D20, is shown in fig. 2(b). The shift depends on the applied field intensity and is different from ,~<2~ --u even at the field corresponding to an Hs of l0 5 times the exchange field. This remarkable field effect is evidence for the second prediction. Recent heat capacity measurement showed that the anomaly at TN was extremely small in CuF4H [ll] and C u F U H [9]. The entropy change associated with the anomaly was 0.01% of the total magnetic entropy (R ln2) for CuF4H and much smaller for C u F U H (fig. 3). J , i , I i , , , I

J mole K J ,':hoLeK I 6.0

x,

z

6.5

/

'/"

/I Cp

t

6.( A

cpl 5.5

i'l I16.5 I I i ~ ~IZO J T(K) (a)

/o

d

"

15.0

i

15.5

16.0 T(K)

(b)

Fig. 3. Heat capacity of (a) CuF4H [11] and (b) C u F U H [9] near TN.

- / -

I ) CuFUH I) 2) CuF4H

:5

~

2)H:O.63kOekoe

\\1)

3)H:64

From these results, we conclude that there is a phase transition at a finite temperature TN far above 0 K at which the staggered susceptibility shows a remarkable peak, but the heat capacity shows little anomaly, in such a quasi-twodimensional nearly Heisenberg antiferromagnet.

References

I

I6

I

I

I8 (a)

I

T (KI-

I

20

I

01'6

I

I

18

I

T (K)

I

20

I

(b )

Fig. 2. (a) Susceptibility of CuF4H and C u F U H in powdered form [9]. (b) NMR frequency shift normalized to 1.0 at 20.4 K at which the shift changes linearly with external field intensity up to 8 kOe.

[1] N.D. Mermin and H. Wagner, Phys• Rev. Lett. 17 (1966) 1133. [2] H.E. Stanley and T.A. Kaplan, Phys. Rev. L e t t 17 (1966). 913. [3] A.R. Miedema, P. BIoembergen, J.H.P. Colpa, F.W. Gorter, L.J. de Jongh and L. Noordermeer, Proc. 19th M.M.M. Conf. Boston (1973). [4] K. Hirakawa and H. Ikeda, J. Phys. Soc. Jap. 35 (1973) 1328. [5] M. Matsuura and Y. Ajiro, J. Phys. Soc, Jap. 41 (1976) 44.

682 [6] Generally, these include also intermode susceptibilities X~s'd and X~'~ due to Dzyaloshinsky-Moriya type interaction. From the analysis of angular dependence of proton NMR frequency, these contribution is found much smaller than that of X~" (Ajiro et al., to be published). [7] H. Kobayashi and T. Haseda, J. Phys. Soc. Jap. 19 (1963) 541.

[8] A. Dupas and J.P. Renard, Phys. Lett. 33A (1970) 470. [9] Y. Yamamoto, M. Matsuura and T. Haseda, J. Phys. Soc. Jap. 38 (1975) 1776. [10] H. Kiriyama and K. Kitahama, Acta. Cryst. B32 (1976) 330. [11] Y. Yamamoto, M. Matsuura and T. Haseda, J. Phys. Soc. Jap. 40 (1976) 1300.