Physica B 155 (1989) 375-378

North-Holland, Amsterdam

MAGNETIC PHASE TRANSITION IN RbFeCI,-TYPE MAGNETS UNDER HIGH MAGNETIC FIELDS ALONG THE c-AXIS

Naoshi SUZUKI and Yukio TAGAWA Department of Material Physics, Faculty of Engineering Science. Osaka University, Toyonaka StW, Japan

The magnetic phase diagram in the T-H plane (H 11 ) d t c 1s e ermined for RbFcCl,-type antiferromagnets based on the

full level scheme within the ‘Tz multiplet of Fe”. It is shown that the existence of an ordered phase at high fields depends

crucially on the ratio of the trigonal crystal field to the spin-orbit coupling.

1. Introduction

RbFeCl,-type hexagonal antiferromagnets have attracted much interest from various points of view: (1) singlet-ground-state systems and (2) frustration effect characteristics to the triangular lattice. Recently renewed interest has been aroused since anomalous magnetization curves have been observed in CsFeCI, and RbFeCI, in high magnetic fields [l, 21.

Within the ‘T, multiplet of the Fe*’ ion in the cubic field, the effective single-ion Hamiltonian for the RbFeCl,-type magnet can be expressed in terms of spin S = 2 and fictitious angular momen- tum L = 1 [3]:

X’=A(Lf-2/3)+AL.S-F.,H(-L;+2S,),

(1)

where A(>O), h(>O) and H represent the trigon- al crystal field, the spin-orbit coupling, and the external magnetic field along the c-axis, respec- tively. In case of H = 0 the ground state is singlet and the first and the second excited states are doublet. In the low-temperature and low-field region only the ground singlet (GS) and the first excited doublet (FED) play a dominant role.

Hence, an S = 1 effective spin Hamiltonian has been used to describe the magnetic properties of the RbFeCl,-type systems. Recent observation of the magnetization curves in CsFeCl, and RbFeCl, up to 40T (H 11 c) shows the following characteristic features: a large linear increase for H < -lOT (from 0 to 2.5 pcLB), a gradual in-

crease above 10T (from 2.5 pH to 3 pu), and then a large meta-magnetic moment jump from

3 pn to 4 pB around 30T. Within the above

singlet-doublet model it is never possible to understand the metamagnetic behavior and a necessity ‘of taking into account higher energy states above the EFD is suggested.

The purpose of the present paper is to de- termine the phase diagram in the T-H (tempera-

,kure vs. magnetic field) plane by using the single- L&! . .

zp Ion Hamiltonian (1) instead of the S = 1 effective spin Hamiltonian. Calculations are performed for various values of r = A /A.

2. Magnetic phase diagram

We assume isotropic exchange interactions be- tween the real spins and express the total Hamil- tonian as follows:

x = c SY; - c 2J,,S, - S, > I (11)

where 2”’ denotes the single-ion Hamiltonian for the ith Fe’+ . Ion of the form of eq. (1). Fe” ions are located on linear chains along the c-axis which form a triangular lattice. We assume two kinds of exchange coupling, J, (intrachain n.n. coupling) and J2 (interchain n.n. coupling). In RbFeCl, and CsFeCl, J, is ferromagnetic while J, is antiferromagnetic. Furthermore, the mag- nitude of J, is smaller by one order of magnitude than that of J,.

We first examine the energy level structure of

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the single-ion Hamiltonian (I) for H = 0. The cncrgy separation II between the GS and the FED increases at first with increasing value of r - -\/A, but it depends vcrv weakly on r in the range 2 < r < IO. On the other hand, the energy separation II’ bctwecn the GS and the second excited doublet (SED) decreases monotonically as r increases.

If the magnetic field is applied along the (‘- axis, a level crossing between the GS and the lower state of the FED occurs at the field strength corresponding to the energy D. When the field is increased further, a level crossing between the lower state of FED and that of the SED occurs at a higher field if r > r, (-3.7), but this level crossing does not occur at any field if r<r.

Wi determine the magnetic phase diagram by examining the instability of the paramagnctic state or field-induced ferromagnetic (FIF) state in which there exists only a uniform moment

induced by the external field. For that purpose we calculate the static q-dependent transverse susceptibility x (q) in the molecular-field ap- proximation (MFA).

In the MFA the effective single-ion Hamilto- nian for the FIF state is expressed as follows:

(3)

where J,, = 25, + 63, denotes the q = 0 compo- nent of the Fourier transform of the exchange coupling. J(q). The induced spin moment (S;) at each site is determined from the usual self- consistency condition:

(s;) = Tr[S, exp(-pXi.,,)]/Tr exp(-PZ’;.,,) (4)

The static transverse susceptibility can be ca- culated after Suzuki 141. The result is expressed as follows:

(5)

where $,;,, (A. H = F, S) denotes the single ion transverse susceptibility calculated from the MF Hamiltonian It;,,. and F represents the total moment p = - & + 2j4,S. c$,:,,~ is calculated from the following general expression:

with p,, = exp(-/3E,,)/C cxp(-PE,,,). Here E,, (E,,)) and 111) (Imz)) represent the eigenvalues and the eigcnstates of X;,,..

Now. the instability of the FIF state is de- termined from the divergence of x (Q). i.c.

1 - JCQM:, = 0. where Q represents the wave vector which gives the maximum of J(q). In the present case J(q) takes the maximum value at the K point. In fig. I we depict the field depen- dence of d, & at T = 0 K. The full and the dotted curves show the results for r = 8.0 and r = 3.0,

respectively. Both curves show a divergence at a field corresponding to the crossing between the GS and the lower state of the FED. In case of r = 8.0, qb& diverges also at a higher field corre- sponding to the crossing between the lower state of the FED and that of the SED.

The instability field of the FIF state is de-

Fig. I. Single-wn transverse susceptibility 6 ;,, its ii function

of /-I at 7‘= 0 K The full curve is for r = X.0 and the dotted

curve for r = 3.0. The straight horizontal line corresponds to

I/J, (Jh = 2J, ~ 35,). The exchange coupling constants .I1

and J. arc taken to be J,!h = 0.005 and JUJ, = -0.1.

N. Suzuki und Y. Tagawa I Magnetic phase diagram of RhFeCI,-type magnets 377

termined by the crossing between the curve of 4&Y and the straight line corresponding to l/J,, where JK = 25, - 3J2 represents the value of J( q) at the K point. In case of r = 8.0, J, = 0.005A and Jz = -0.1 J, the curve of 4,‘, and the line of l/J, cross each other at three points, H,, Hz, and H 2,,, as shown in fig. 1. In the regions of 1 - JK~k&<O, i.e. for O<H<H, and H?,<H< H ?,,, the FIF state is unstable for an ordered state in which spontaneous moments modulated by the wave vector at the K point appear in the c-plane. In the present case the spontaneous moments are expected to form a triangular struc- ture. In the case of I = 3.0, or generally for r < rc, we have only one region in which the FIF state is unstable.

It is noted here that at T = 0 and H = 0, @l.ym takes a finite value &,. Therefore, at H = 0 and T = 0 K magnetic order exists if J&, > 1 and does not exist if J,&, < 1. It is considered that RbFeCl, corresponds to the former case and CsFeCI, to the latter case.

As the temperature is increased from 0 K, values of 4 I,- decrease and the divergences at the crossing fields disappear. Therefore the unstable region of the FIF state shrinks with increasing temperature and finally the ordered region disappears at high temperatures. The magnetic phase diagrams for r = 8.0 and r = 3.0 determined in this way are shown in fig. 2. The actual spin arrangement in the ordered state represented by TRI in fig. 2 is a cone structure because there is a field-induced uniform moment

.04 .- x

2 -- I /-'\

\

9 : \ FIF \

.02 .. I \

I TRI : I I I I

o* !: I 0 0.1 0.2 0.3

PBH/h

Fig. 2. Magnetic phase diagram obtained for r = 8.0 (full

curve) and for r=3.0 (dotted curve). FIF denotes the field- induced ferromagnetic state and TRI the ordered state in

which the spontaneous moments in the c-plane form a tri- angular structure.

in the c-direction and the spontaneous moments in the c-plane form a triangular structure.

3. Magnetization curve and discussion

In order to calculate the magnetization as a function of H we must determine the spontanta- neous moments in the ordered region. If we

assume the spontaneous moment of one of the three sublattices is along the x-direction, the MF single-ion Hamiltonian for that sublattice can be expressed as follows:

%‘:rR, = A(Lf -213) + ALaS + /+,HL,

- 2(p.,H + Jo@,)>% - 2J,(S,)& . (7)

The induced and the spontaneous moments, (S,) and (S,), are determined simultaneously from the following self-consistency conditions:

(S,) =Tr[SU exp(-P~~Rr)l/Trexp(-p~~RI) (CT = x, z) (8)

Then, the thermal average ( Lz) is calculated from the resultant effective Hamiltonian SY;,,,

and the magnetization along the c-axis per Fe2’ ion is given by M, = J_+- (L,) + 2(S,)).

We have calculated the magnetization at T = 0 K for various values of r by fixing the values of J, and J2 as 0.005A and -0.1 J,, respectively. As typical examples we show the results for r = 8.0 and r = 3.0 in fig. 3(a).

In the case of r = 8.0 the magnetization shows

the following characteristic field dependence: (a) large increase from 0 J+ to 2 J_L~ for 0 < H < H, ; (b) almost field-independent behavior for H, < H < Hz,; (c) rapid increase from 2 pB to 4 J.L~ for Hz1 < H < H,, ; and (d) very gradual increase for Hz, < H. Here H,, H,, and HI1( denote the criti- cal fields shown in fig. 1.

In the case of r = 3.0 the magnetization is very small before entering the ordered phase, in- creases very rapidly in the ordered phase, and then shows a very gradual increase up to ex- tremely strong fields. It should be emphasized

N. Suzuki urtd Y. Tagnwa I Magnetic phuse diqmm of RhFeCl,-type tnagnet.~

(a)

4 (b)

Fig. 3. (a) The magnetization calculated for I = 8.0 (full

curve) and for r = 3.0 (dotted curve). If A = X0 cm ‘, 0. I h ipLB

corresponds to 17T. (b) The observed magnetization of

CsFeCl, [l] (thin full curves) and of RbFeCl, [2] (thick full

curves).

that for r < rC there is no second rapid increase of magnetization at higher fields.

On comparing the magnetization shown in fig. 3(a) with the observations in CsFeCi, and RbFeCi, (fig. 3(b)) it is found that it is rather difficult to explain in a unified way the observed results at both low and high fields. The theoreti-

cal results for r = 3.0 can explain fairly well the observed magnetization below 3OT, but it can never explain a large moment jump around 31- 34 T. If we use a larger value of r, say r = 8.0, a large and rapid increase of the moment at higher fields is certainly predicted. However, that mo- ment increase occurs in a range of field Hzr <

” < “2rr (“,,, - “2, - IOT if h=8Ocm~~‘), which is too wide compared with the experi-

mental field width -1 T for the moment jump. Furthermore, the value of the moment in the intermediate field region of 2.0 pB is smaller than the observed value of 2.5-3.0 p,%.

The origin of the discrepancy between the theory and the experiment is not clear at pre- sent. As possible origins we may point out (a) insufficiency of the MF approximation and (b) other interactions besides the isotropic exchange interaction between the real spins, such as an orbital dependent anisotropic exchange interac- tion. Considerations of these points are left as problems for the future.

Acknowledgement

This work was supported by the Kurata Foun- dation.

References

[I] M. Chiba ct al.. Solid State Commun. 6.3 (lY87) 427

[2] K. Amaya ct al., J. Phys. SW. Jpn. 57 (1988) 38.

[3] N. Suzuki. J. Phys. Sot. Jpn. SO (1981) 2931.

[4] N. Suzuki. J. Phys. Sot. Jpn. 4.S (1978) 1791.