6 January 1997

ELSEVIER

PHYSICS LEfTERS A

Physics Letters A 224 (1997) 203-208

Magnetic properties of a quasi-one-dimensional organic ferromagnet

Shu-Qing Bao a,b, Jue-Lian Shen ‘, Guo-Zhen Yang ’ a CCAST (World Laboratory), PO. Box 8730, Beijing 100080, China

h Institute of Applied Physics and Computational Mathematics, PO. Box 8009-28. Beijing 100088, China ’ c Institute of Physics. Chinese Academy of Sciences, P.0. Box 603-12. Beijing 100080. China

Received 30 July 1996; revised manuscript received 25 September 1996: accepted for publication 2 1 October I996 Communicated by J. Flouquet

Abstract

A quasi-one-dimensional alternating-spin Heisenberg model is used to describe the charge-transfer organic feeomagnet, which is studied by the Green’s function method. The magnetic properties of the charge-transfer organic ferrobagnet in different temperature regions are obtained. A formula for the critical temperature T, is obtained and is found to ibe related to the spin of donors, to the intrachain exchange interaction, and to the spatial anisotropy parameter. This concbusion can explain Tc of the organic ferromagnet. The spatial anisotropy parameter of the organic ferromagnet [FeCp;] [?/CNE] we obtained is consistent with the estimated value of Narayan et al. The Curie-Weiss temperature 0 for the organic fdrromagnet [ CrCp;] [TCNE] we obtained is 0 = 15.6 K which is near the experimental result of 0 = 22 f 1 K.

PAC.? 75.50.Dd; 75.10.Jm; 75.30.D~ Keywords: Organic ferromagnet; Critical temperature; Green’s function

In recent years, several kinds of organic ferromagnets [l-4] have been discovered, which have stimulated theoretical and experimental interest. Among these organic ferromagnets, the charge-transfer molecul /organic

ferromagnet [ 2,3] investigated and synthesized by Miller, Epstein and others is particularly GP inter ,sting and

is discussed in this paper. This material is a quasi-one-dimensional organic ferromagnet which cbnsists of

one-dimensional stacks of type . . .DADADA. . . formed by the alternating donors D and acceptors, A. It can be expressed as [ MCp;] [TCNE] (or [MCp;] [TCNQ] ), where M can be Fe, Cr, and Mn. Many~ magnetic

properties of organic ferromagnets have been investigated in experiments, but until now the critical tqmperature and the Curie-Weiss temperature have not been calculated theoretically.

In Ref. [5], Zhou et al. presented a study of magnetic properties of a molecular-based alterqating-spin chain. Experimental and theoretical results of low-field magnetization and susceptibility were given. They used modified spin-wave theory for one-dimensional (1D) Heisenberg alternating-spin chains and introduced the Dyson-Maleev transformation. However, the resulting Hamiltonian in the modified spin-wave $eory was not Hermitian for the alternating-spin chain, so they met the difficulty of diagonalization. Moreovdr, because

’ Mailing address.

0375~9601/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII SO375-9601(96)00812-2

204 S-Q. Boo et al./ Physics Letters A 224 (1997) 203-208

the experimental results [2,3] show the existence of the interchain exchange interaction J’ apart from the intrachain interaction J, they added J’ to the formula [ 61, x3D = X1,)/[ 1 - (2zJ’/C)xio]. They began with a ID Heisenberg Hamiltonian, and transformed the one-dimensional results to three-dimensional results by considering a mean-field correlation for the interchain exchange coupling J’. However, using this method, it is very difficult to discuss the magnetic properties near the critical temperature. Therefore they did not get a formula for the critical temperature T,, although they realized that T, is related to the intrachain and interchain exchange interactions as well as the spins of donors and acceptors [7]. The Curie-Weiss temperature of the organic ferromagnet was not discussed in Ref. [ 51.

In this paper, different from Ref. [ 51, we begin with a quasi-one-dimensional alternating-spin model (S,, Sp) with ferromagnetic exchange interaction. Here S, represents the spin of donors. It is arbitrary and can be 3, I, ;, . . ., while Sp represents the spin of acceptors, and equals l/2. We will use the linear spin-wave theory and thermodynamic Green’s function method to investigate the magnetic properties of the organic ferromagnet. In linear spin-wave theory, we will apply the Holstein-F’rimakoff transformation rather than the Dyson-Maleev transformation. By the thermodynamic Green’s function method, a formula for the critical temperature and the Curie-Weiss temperature will result.

The Hamiltonian of the quasi- ID alternating-spin Heisenberg model with ferromagnetic exchange interaction is

H=-C~J~~[S~~S~~+~(S,:S~~+S,;S~~)]-~,~B~CS;;~-~Z~LB~CS;;~, (1)

(4 i i

where Jij = J when i and j are the intrachain nearest-neighbour sites and Jij = J’ = 6J when i and j are the interchain nearest-neighbour sites. J is the exchange energy and J > 0. S is the spatial anisotropy parameter. We assume the lattice is a simple cubic lattice. The lattice is divided into two sublattices: sublattice 1 ( LY, i E sublattice 1) and sublattice 2 (/3,j E sublattice 2). Each sublattice is a face-centered cubic lattice. gt, g2 represent the values of the isotropic Land6 g factor for spin sites S, and Sp, respectively, and h is the external magnetic field. S$, Sli and Sii represent the three components of the spin-& operator at site i in sublattice 1 with S: = S$kiS$. Sij, Sij and Sij represent the three components of the spin-Sp operator at site j in

sublattice 2 with Sgj = S;jfiSij.

After applying the following Holstein-Primakoff transformation [ 81: $2( 2S,) ‘/2( 1 + p,pa$z,l)a,/,

S,-,-(2S,)“24[( 1 + pqOa;p,l), S$ = S, - uzp,l (where 40 is a sublattice label, p = LY, p, and 1 is a site label, I = i,j; prp = -l/4&,), and using the linear approximation, the resulting Hamiltonian in momentum space is

with Ho = -;JNZS& - ;g,,uBhNS, - &pghN$, and A, = 2SPJZ + &/LBh, A2 = 2S,JZ + &,uBh, A3 =

-2JZ ,,&&. Here N is the number of spins in the total lattice, Z = 2 + 46 is the effective nearest-neighbor number, and

Yk = 92 cosk,+2Scosk,+2Scosk,). (3)

The Hamiltonian in Bq. (2) can be diagonalized by the foIlowing UkDlk + Vkbkr apk = -Vk&k +Uk&k. with U;, V; = ;[1r(&-AI)/(

diagonalized Hamiltonian is H = HO + ~k(WklDlk+Dlk+ w&&k+&) with the excitation energies

wk1,2 = i{(gl +B)PBh+2JZ(Sa+Sp) f d[(& - &)/.Qh + 2JZ(S, - S,# + 16J2Z2S,Sgy;}. (4)

S.-Q. Baa et al./ Physics Letters A 224 (1997) 203-208 205

From the above equation, we know that there are two magnon branches of the energy spectrum. wk2 represents the acoustic branch, while OkI represents the optical branch. When S, = Sp, and gt = g2, then Wkr.2 = gpnh + 2JZS( 1 f Yk), where wk2 is the energy spectrum for ferromagnets.

The magnetization M(h) can be expressed as M(h) = grpn(S;) +g2pn(S$). (Si) and (Si) are the sublattice magnetizations which are given by

where nrk is the magnon occupation number, nrk = @L&k). For sublattice 1 and sublattice 2, res$ectively,

nrk = n(@k,) = [ exp(@kl) - I]-‘, p= l/T, with I = 1,2.

When T < T,, by expanding Eqs. (5) and (6) at very low temperature, we get

(7)

(8)

where Br = dm, F, (8) = (1 +26)(ar2 t i?z2S2), F2(S) = (1+26)3’2/& C’,D’ are parameters

composed of A,B,C. A,B and C are defined as A = [(gl +g2)&+2JZ(S, +Sp)]/2JZ, B E [(g2 - gl),uBh + 2fZ( S, - Sp)]/252, C2 = 4SJp. 7 is defined as r = T/JZ.

Since the energy spectra Wkl and 6Jk2 in Eq. (4) have gaps at k = 0 which are different from that of the pure spin-l/2 ferromagnetic chain, the above sublattice magnetizations decrease exponentially at’very low temperatures [9]. When h = 0, exp[ -(l/r)(A - Br)] equals 1, and expt-( l/r)(A + B)] equals e-2Jzs~/T, Thus the spontaneous magnetization is proportional to T3/2, and T’Pe-‘?-JZ~*/T. For T << T,, e-2JZsmlT aeproaches zero rapidly, so the main temperature behavior of the spontaneous magnetization is T312 at very low temperature.

To investigate the magnetic properties of quasi-1D organic ferromagnets at high temperatures, we! shall use the thermodynamic Green’s function method. This method was first used for the spin-l /2 ferro agnet by

“, Bogoliubov et al. [lo]. Later, Tahir-Kheli and ter Haar et al. developed it for the arbitrary-spin fe omagnet

[ 11,121. In this paper, we will use the thermodynamic Green’s function method for quasi-one-di ensional

alternative-spin system. To use this method, we should first define some necessary !l thermodynami Green’s

functions, then construct equations of motion for these Green’s functions, and finally make cutoff appro/ximations to get the results. From Hamiltonian ( 1) , after such a calculation, we get the sublattice magnetization (S;) and

(S;) as follows,

cm = I& -@(s*)lrl +@(s,>l2S-+’ + [S, + 1 +@(s,)][@(s,)]2s~+’

[l + @(Sa>]2S~f’ - [@(s&)]=fi+’ (9)

(Si) = ; - 2(S3@‘(f>, ( 10)

where

206 S.-Q. Bao ei al./ Physics Letters A 224 (1997) 203-208

- nW2(k))[E2(k) - g2pBh - 2-=(s:)1). (11)

and

{‘dfi’? (k)) [El(k) - &%h - 252($&l

- n(PE2Ck)) [Ez(k) - g~/-&d - 2JZ($j)l}. (12)

Here E,,z(k) are energy spectra given by replacing S, and S, by (Si) and (Si), respectively, in Eq. (4). Using the above results, by solving Eqs. (9), ( 11) and Eqs. ( 10) and ( 12), we can determine the sublattice magnetizations (Si) and (SL) at any nonzero temperature.

(i) Spontaneous magnetization at very low temperature: we can get the same results as that in Eqs. (7) and (8) obtained by the spin-wave theory.

(ii) Spontaneous magnetization near the critical temperature: as T -+ T,, h = 0, (Si) and (S$) approach zero.

By expanding @(S,), Q’(i) in Eqs. (11) and (12) in terms of (Sz) and (S$), we get

cc> = ( p:csa + I)2 1 112

$&Y + 1) - 2 + 4J(S,+1)F(S) (1 -VT,) ,

($3, = ( w2(S, + 1)

1

11.2

$A& + 1) - + + 4j/,oF(S) (1 -VT,) 1

(13)

(14)

where

?-, = [ &sa(Sa + l)pJz/F(s> (15)

and

F(6) = ; c L- k 1-yk2‘

(16)

Eq. (1.5) is a formula for the critical temperature Tc for the charge-transfer organic ferromagnet. S, is the spin

of the donors; the spin of acceptors is l/2 and has been absorbed into the equations. It is shown that T, is

related to the spin of the donors S,, to the intrachain exchange interaction J, and to the spatial anisotropy

parameter S. By increasing the spin of the donors [MCpz], a higher critical temperature T, is expected if the intrachain exchange interaction J and the spatial anisotropy parameter S are the same. This is consistent in part with experimental results [3]. Experimentally, the critical temperature T, of organic ferromagnets is found to

increase from 4.8 K of [ FeCp; 1 [TCNE 1 to 6.2 K of [MnCp; J [ TCNQ] , corresponding to an increase of S, from l/2 to 1. On the other hand, because of the importance of the spatial anisotropy parameter 6 and the intrachain exchange interaction J, it is not difficult to understand the decrease in T, of the organic ferromagnets

[CrCpzl [TCNEI G’, = 3.6 K) compared with that of [FeCp,*][TCNE] (T, = 4.8 K), although the spin of the donors S, increases from l/2 to 312. This gives an explanation of the Tc [7] of the organic ferromagnet

[CrCp;] [TCNE]. In Table 1, we list some values of F(S) and Z/F(S) corresponding to different 6 values. From the table we

see that F( 6) increases with decreasing S. Materials of organic ferromagnets composed of the same component donor D and acceptor A will have different T, values because of their different spatial anisotropy parameters 6.

Table I

S.-Q. Bao et d/Physics Leuers A 224 (1997) 203-208 201

Some F values for different 6 = J//J

I 0.5 0.1 0.02 0.01 0.005 0.001 0.0005

F(S) 1.516 I s74 2.35 I 4.67 1 6.49 1 9.08 I 19.945 27.9# I Z/F(s) 3.958 2.541 1.021 0.445 0.314 0.222 0.100 0.072

We can use the above T, formula for organic ferromagnets. For decamethylchromocenium tetracyanoethanide

[CrCp;][TCNE], 5, = 1, J = 9.0 K, by varying S values, we get T, = 3.60 K when the spatial anisotropy

parameter 6 = 0.01296. For decamethylferrocenium tetracyanoethanide [FeCp;] [TCNE], S, = $, J & 27.4 K

[ 131, we get T, = 4.8 K. corresponding to S = 0.0124. This value is near the estimated value 6 N & N 9.0133 of

Narayan et al. [ 141. For decamethylmanganocenium tetracyanoquinodimethanide: [MnCp;] [ TCNQ]!. 5, = 1, for ./ = 14 K [3], we get T, = 6.2 K corresponding to 6 = 0.0295.

(iii) Magnetic properties in the paramagnetic phase: when T>>T,, at T = T/JZ>>l, h - 0, by using the asymptotic expansion technique, we can obtain the susceptibility in the paramagnetic phase. defined as

x = dM( h)/&\h_~, as follows,

x= g,g21*.s2(& +A4) 4Ad4

T 7(A3 + A4) (17)

where & = glA1/%2, A4 = g2/4gl, At = $S,(S, + l), Fs = (2/N) Cyk2. Because T z+ T,, i/r - 0, the above equation can approximately be written as

(18)

where

@=TJ’(S,gl,g2), ( 19)

P( 6, gl , gz) = gtg#( S) m /(2g:A1 + g$). 0 is the Curie-Weiss temperature of the charge-transfer organic ferromagnet. Eq. ( 18) tells us that when T > T,, the susceptibility in the paramagnetic phase for the organic ferromagnets follows the Curie-Weiss law. This conclusion is in good agreement with experimental results

[2,3]. For the organic ferromagnet [CrCp;] [TCNE], S, = 5, gt = 1.95, g2 = 2.00, T, = 3.60 K, S * 0.01296,

from Eq. ( 19) we get the Curie-Weiss temperature 0 = 15.6 K. This result is near the experimental value of @=22fl K [7].

Summary: using linear spin-wave theory and the thermodynamic Green’s function method, we have discussed the magnetic properties of quasi-one-dimensional organic ferromagnets. We get that the spontaneous magneti-

zation is proportional to T312 and T’/2e-2JzsnlT at very low temperatures. The main temperature biehavior of the spontaneous magnetization follows a T 3/2 law A formula for T, results and is found to be related to the . spin of the donors and to the intrachain and interchain exchange interactions. This can explain the T, of the organic ferromagnet [ CrCpz] [TCNE]. For the organic ferromagnet [FeCp;] [TCNEI, we get the ianisotropy parameter 6 = 0.0124 which is close to the estimated value of Narayan et al. [ 141. For [CrCp;] [T/CNE], we get T, = 3.60 K when 6 = 0.01296. The Curie-Weiss temperature we calculated for [CrCp;] [TCNk] is 15.6 K which is near the experimental value of 22 f 1 K [ 71.

208 S.-Q. Boo et al./ Physics Letters A 224 (1997) 203-208

References

] I I Y.V. Korshak, T.V. Medvedeva, A.A. Ovchinnikov and V.N. Spector, Nature 326 ( 1987) 370. [2] S. Miller, A.J. Epstein and W.M. Reiff, Science 240 (1988) 40. [3] W.E. Broderick, J.A. Thompson, E.P. Day and B.M. Hoffman, Science 249 ( 1990) 401. [4] H.O. Stumpf, L. Ouahab, Y. Pei, D. Grandjean and 0. Kahn, Science 261 ( 1993) 447. [5] P Zhou, M. Makiric, E Zuo, S. Zane, J.S. Miller and A.J. Epstein, Phys. Rev. B 49 ( 1994) 4364. [6] D.B. Losee, J.N. McElearney, G.E. Shankle, R.L. Carlin, PJ. Cresswill and W.T. Robinson, Phys. Rev. B 8 (1973) 2185. [7] E Zuo, S. Zane, P Zhou, A.J. Epstein, R.S. Mclean and J.S. Miller, J. Appl. Phys. 73 (1993) 5476. [ 81 J. Callaway, Quantum theory of the solid state (Academic Press, New York, 1976). [9] E.E. Reinehr and W. Hgueiredo, Phys. Rev. 52 ( 1995) 310.

[ IO] N.N. Bogoliubov and S.V. Tyablibov, Dokl. Akad. Nauk, 126 (1959) 53; C. Domb and M.S. Green, eds., Phase transitions and critical phenomena, Vol. 5B (Academic Press, New York, 1976) p. 259

[ 111 R. Tahir-Kheli and D. ter Haar, Phys. Rev. 127 ( 1962) 88. [ 121 H.B. Callen, Phys. Rev. 130 (1963) 890. [ 131 S. Chittipeddi, K.R. Cromack, J.S. Miller and A.J. Epstein, Phys. Rev. Lett. 58 (1987) 2695. [ 141 K.S. Narayan, K.M. Chi, A.J. Epstein and J.S. Miller, J. Appl. Phys. 69 (1991) 5953.