Journal of Magnetism and Magnetic Materials 67 (1987) 221-226 221 North-Holland, Amsterdam

CORRELATED EFFECTIVE FIELD THEORY FOR ORBITALLY UNQUENCHED MAGNETIC SYSTEMS

Subhasis MUKHOPADHYAY * and Ibha CHATTERJEE Saha Institute of Nuclear Physics, 92, Acharya Prafulla Chandra Road, Calcutta 700009, lndia

Received 22 April 1986; in final revised form 30 December 1986

Correlated effective field theory has been used to study the magnetic properties of orbitally unquenched systems. The effect of an axial distortion on the magnetic properties has been studied. It has been found that this distortion has negligible effect in the ordered phase. Therefore, without assuming the axial distortion, a comparison has been made between the theoretical and the experimental results for the three-dimensional antiferromagnet KCoF 3.

1. Introduction

That fluctuations play a dominant role in de- termining the critical properties of a thermody- namic system is now well established. People have been trying to incorporate fluctuations in an effec- tive field framework in order to be able to go beyond the crude mean field (MF) approach. The drawback of MF theory (with corrections at vari- ous levels) is the complete neglect of fluctuations and as a result, their agreement with the experi- ments is very poor in the critical region. The unambiguous wrong prediction of MF that mag- netic specific heat is zero in the paramagnetic phase, calls for a viable alternative to MF.

One improvement over the MF theory, which incorporates fluctuations by taking into account the static spin correlation was developed by Lines [1]. This approach, known as the correlated effec- tive field (CEF) theory, goes beyond MF theory and helps us to deal with a more realistic Hamilto- nlan. This theory successfully explains the static properties of magnetic systems of different mag- netic symmetries [2-7]. The dimensionality effects [3] and the crystal field effect for spin-only sys-

* Present address: The University of Burdwan, Central Library Building, Computer Centre, Golapbagh; Burdwan, West Bengal, India.

tems [5-7] have been studied and the static mag- netic properties have been explained fairly well over a wide range of temperatures. Critical indices in different dimensions for an Ising system have been estimated [8] from this theory and are found to be different from MF estimates and are com- parable with available estimates of series expan- sion (3D) and exact (2D)results. This theory turns out to be a fairly successful theory for spin-only magnetic systems and one can hope that this theory could also account for the magnetic prop- erties of orbitaUy unquenched systems.

CEF theory was first applied to the cubic KCoF 3 compound by Suzuki et al. [9]. They introduced a temperature dependent deformation parameter in the ordered phase and obtained unphysical results near the critical temperature. In this communica- tion we have shown that this distortion parameter has very little effect on the susceptibility in the ordered phase and therefore, can be neglected. Here, we have explained the experimental results without assuming the distortion parameter and no unphysical result is obtained in the critical region.

2. Summary of the available results

The group of compounds with KMF 3, where M is only of the 3d transition group of elements, has

0304-8853/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

222 S. Mukhopadhyay, lbha Chatterjee / CEF theory for orbitally unquenched systems

been studied extensively and is known to order antiferromagnetically [10] below a well defined Nrel temperature. Martin et al. [11] first sum- marised their properties and the Nrel tempera- tures reported there, are uniformly greater than those observed later using more refined techniques [10]. The compound KCoF 3 has a simple per- ovskite structure [12], where M 2+ ions are arranged on the simple cubic lattice with F - ions at the centre of each of the two adjacent M 2÷ ions [13].

The susceptibility measurements [14] with 19F N M R reveals a large residual value of parallel susceptibility even down to liquid He temperature, which was ascribed to the residual orbital mo- ments of Co 2+ ions. With 59Co NMR in a single crystal shows [15] a large shift as much as 34% for the nuclear g factor, which is rather expected mainly from the residual orbital moment of Co 2÷ ions. The total orbital angular momentum and total spin for a free Co 2÷ is L = 3 and S = 3/2 , respectively. The ground state 4F when subjected to a cubic crystalline environment (due to oc- tahedrally arranged F - ions in the lattice) splits into 4T1, 4T 2 and 4A 2 with 4T 1 lying lowest. The triplet 4T 1 is separated from the next higher state 4T 2 by approximately 215 THz (1 T H z = 33.3 cm - ] ) and therefore, only 4T] contributes. 4T 1 has the same symmetry as that of a P state and can be described as an effective 1 = 1 state, where l is related to the true orbital angular momentum L by L = KI and r becomes negative (K = -1 .5) .

Holden et al. [16] performed inelastic neutron scattering experiments at low temperatures and obtained the dispersion curve for magnons and phonons. They observed that at temperatures lower than Tra, the magnon frequencies are much more temperature dependent than the phonon frequen- cies. Okazaki et al. [12] observed that the tetrago-

nal distortion which is needed to remove the orbital degeneracy from an effective-field framework is very small at low temperatures. Buyers et al. [17] developed the theory to account for the low lying magnetic excitations as observed by Holden et al. [16]. They [17] asserted that this distortion of the crystal field by the phonon can give a significant contribution to the magnon-phonon interaction. The spin-wave spectra of this compound suggests a low-lying longitudinal mode. This branch is almost flat [16] and thus directly amenable to an effective-field theory. Although MF reproduces qualitatively more of the magnetic behaviours, it is, nevertheless, completely unreliable in the criti- cal region. On the otherhand CEF has proven superiority [2-8] over MF, so it would be im- perative to apply CEF to account for the static susceptibility for unquenched magnetic systems.

Suzuki et al. [9] tried to explain the observed static susceptibility with CEF approximation and assumed a specific form for the temperature de- pendence of the deformation parameter near the critical region. They had been able to reproduce the susceptibility results at all temperatures except near T~q. They also obtained some unphysical results (like double values of the magnetic order parameter at each temperature near TN). Though the temperature dependence of the deformation parameter has a thermodynamic basis, it has very little effect on the static susceptibility in the ordered phase because in this phase the suscept- ibility is mainly guided by the order parameter which has a stronger temperature dependence than the deformation parameter. The transition temper- ature, of course, gets shifted a little bit as the deformation parameter is changed and we do not get any unphysical results very close to T N.

3. Theory

CEF theory incorporates static spin correlations by replacing a spin S 7 in the equation of motion of a particular spin Si r (~,--, spatial components) by (Sj v> +A~j (Si r - <Sir)). Thus the Hamiltonian for magnetic systems in CEF approximation becomes [1].

~icv-r = - E J~ar(Sir) 2 - 2EJ,'~S~/((ST)--ar(sIr)), (1) r ,J r , J

where err is a temperature dependent correlation parameter to be determined self-consistently.

S. Mukhopadhyay, lbha Chatterjee / CEF theory for orbitally unquenched systems 223

The order parameter is given by

<S~> = Z [ ( +,lSVl~,)e-e"/kr]/Z, (2)

where Z -- E . e -E . /k r is the partition function. Here the E, 's are the eigenvalues obtained by diagonalising the CEF Hamiltonian and the I ff ,) 's will be linear combination of 12 states of 4T 1 which correspond to three m t values and four m, values for a l = 1 and S -- 3 /2 system. Thus [~ , ) = Y'.~a,,]li; Si), where I i can take values + 1,0 and S~ can take on values +_ 1 /2 and ___ 3/2.

The susceptibility in CEF approximation is given by

kTxV(q) = ( (g~ : g~)) + UV(q) (3)

with tt = (Ki~ + 2S~)t%, where #s is the Bohr magnetic moment and g is the orbital quenching factor (to ~ 0). UV(q) is derived as

2[JV(q)-avJv(O)]<<g'~; Sir)) z k T - 2[ J r ( q ) - aVv(0)] <<S,V : S~V)) ' (4)

Ur(q) --

where

J (q ) = J e ~ ' t ~ n n

For any two operators A and B

<<A: n ) ) = <A: B> _ <~)<n) Z Z 2

where

<A:II)=~n [A""B""+kTY'~A"mB"~n+A""B""]m Em='~n ] e -e" /k r

and

(5)

(6)

<,4> = Y'.<~.IAI~.> e -~. (7) n

According to fluctuation theorem

k T y, x~(q) = ( ( t g : t ~ ) ) (8) N q

which sets EqUV(q) to zero. Thus

err = '~ ( J V(q) kT/[ k T - 2 ( j r ( q ) _ avjv(0)) } ( < S v: S.v) ) F_~jv(0) (9)

This equation is solved self-consistently using eqs. (1), (2), (5), (6) and (7). The uniform (q = 0) static susceptibility is calculated from the equation

1 [ v 2JV(0)(1-av) ( (g ,v :S~V)) 2 ] ' :> ) + ) " ( 1 0 )

224 S. Mukhopadhyay, lbha Chatterjee / CEF theory for orbitally unquenched systems

KCoF 3 has a cubic perovskite structure, it has 6 nearest neighbours and the magnetic structure demands that ~ S j Y ) = - ~ S ; v) ( j = i + 1). Magnetic ordering is assumed to be along the Z direction. Thus ($7) 4= 0 = ~S~) = ~S y ) and a x = ot y =//= ot z in the ordered phase; ~$7) = (S x ) = ~S/y) = 0 and a:' = a y = a z in the paramagnetic phase. In this case the Hamiltonian takes the form

= )kli°$ i + C(1:) 2 - 6J [ a x ( ( $ 7 ) 2 + (S/Y) 2 } Jr o / z ( s z)2] Jr 12J(1 + a ~)( S[)S~, (11)

where ;k is the spin-orbi t coupling parameter for the 4 T 1 state, C is the tetragonal crystal field parameter and J (< 0) is the antiferromagnetic exchange constant between the nearest neighbours. The temperature dependent correlation parameters are given by

k T ( l ( l ( 1 (cos ~rx + cos ~ry + cos ~rz) d x d e dz (12)

a ' = -~- -10 J0 J0 k T - 4 J [ (cos ~rx + cos try + cos ~z) ~-3-'d'Q] ( ( Sir: S,V> ) "

The above self-consistency condition for a v 's, is computationally more convenient than the one derived by Lines [1] or Suzuki et al. [9]. With the model Hamiltonian [eq. (11)], the magnetic order parameter and the susceptibilities are calculated for 3D antiferromagnet KCoF 3 and the effect of tetragonal crystal field has been studied.

4. Results and discussions

Suzuki et al. [9] were the first to apply the CEF framework to the ordered magnetic system KCoF 3. In their approach, they considered a tetragonal distortion (C ~ 0) present in the compound in the ordered phase while in the paramagnetic phase, the system becomes cubic (C = 0). Moreover, C has a temperature variation of the form C = C0[3(T- TN) /TN] 1/2 for ~TN< T~< T N with C O = - 9 . 0 cm-2. Although at the N6el temperature a crystallographic phase transition takes place [12], it does not materially affect the static magnetic property of this system in the ordered phase. We have found that the deformation hardly has any consequential effect on the observed static sus- ceptibility. In their paper Suzuki et al. [9] admitted the unphysical nature of the order parameter from their calculations and they claimed that this na- ture of the order parameter was observed by them even for the spin-only system [9]. But our calcula- tions never displayed such behaviour for spin-only system as evident from the present calculation as well as from the earlier calculations [2-7].

The parameter C does not have any dramatic effect on the static susceptibility as the order parameter, as shown in fig. 1, is not very much affected by C except for a little shift in the transi- tion temperature. The values of the order parame-

ter at different temperatures for C = 0 are shown by solid circles while the full line curve above it, is for C = - 9 . 0 cm - ] and the one below it for C = 9.0 cm-1. In all the cases, J, ~, ~ were kept fixed at values -10 .0 cm -] , - 1 . 5 and 211 cm -1, respectively, as estimated by Suzuki et al. [9]. From the curve one observes that C hardly has any effect on the order parameter, and in no cases whether C = 0 or C 4= 0, we observe double solu- tions of the order parameter. Consequently the static susceptibility is very little affected by the

1.0

0.8

A o

/ ~ " 0 .4

0.2

Fig. 1.

- T - - I0.0 cn~ j k = - I . $ ~ . 211 cn~ I

i - - C=O

C= - 9 . 0 cn~ j

I

L I 1 I I I I

0 20 40 60 8 0 100 120 140

T(K) Variation of order parameter with temperature in

presence and absence of crystal field.

S. Mukhopadhyay, lbha Chatterjee / CEF theory for orbitally unquenched systems 225

10.0

~: 8.0

6.0 %

'~ 4.G

K Co F 3 TN=IIBK 3=-g.Scf~ I k=-l./..~-211crf~ t C=O

~ l x NMR • Torque

I I I 0 1.0 2.0 3.0

TIT N

Fig. 2. Magnetic susceptibility vs. temperature when T N = 118 K.

presence of tetragonal distortion (C~=0) be it temperature dependent or not, and neither the static susceptibility nor the order parameter dis- plays any unphysical nature near T~.

The best fit parameters corresponding to T N = 118 K are J = -9 .5 cm -1, K= - 1 . 4 and h = 2 1 1 cm-1. The susceptibility results are shown in fig. 2, where the full line curve represents our calcula- tions and experimental results are shown by dots and crosses. Fig. 3 shows the variation of order parameter with temperature, while fig. 4 shows the temperature variation of correlation parameter. From :fig. 2, it is evident that at low temperature,

1.0

KCoF 3

C=O .7=-9.5CtI~ I k = - l . 4 A , 2 1 1 c r n i

0.8

A o o.e

v

A " 0./, N

0.2

I I i I 20 40 60 80 100 120 140

T(K)

Fig. 3. Variation of order parameter with temperature when TN =118 K.

0.2

0.1

• KCoF~

Y 0 I I I

100 200 30O

T(K)

Fig. 4. Variation of correlation parameter with temperature when T N = 118 K.

there is fair agreement between the theoretical and the experimental results, but at high temperature the experimental results lie above the theoretical results. It is very hard to obtain both the ordered and the paramagnetic susceptibilities with the same set of parameters. In fig. 3, the variation of order parameter shows that it should vanish at tempera- tures higher than T N and T N is defined to be the temperature at which the staggered susceptibility diverges. This value of TN is very close to the value obtained in birefringence experiment [18]. As this investigation suggests one can have a very good estimate of TN from CEF theory and this theory gives a good quantitative estimate of the susceptibility near the critical temperature. Suzuki et al. [9] though obtained better agreement in the paramagnetic phase, their values underestimate the ordered magnetic susceptibilities.

References

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5133. [5] S. Mukhopadhyay and Ibha Chatterjee, Physica 125B

(1984) 33.

226 S. Mukhopadhyay, Ibha Chatterjee / CEF theory for orbitally unquenched systems

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[17] W.J.L. Buyers, T.M. Holden, E.C. Svensson, R.A. Cowley and M.T. Hutchings, J. Phys. C 4 (1971) 2139.

[18] J. Ferre, J.P. Jamet and W. Kleeman, Solid State Com- mun. 44 (1982) 485.