PHYSICS OF THE SOLID STATE VOLUME 41, NUMBER 10 OCTOBER 1999

Field-induced magnetic phase transitions in the Yafet–Kittel modelN. P. Kolmakova, S. A. Kolonogi , and M. Yu. Nekrasova

Bryansk State Technical University, 241035 Bryansk, Russia

R. Z. Levitin

M. V. Lomonosov Moscow State University, 119899 Moscow, Russia~Submitted December 18, 1998!Fiz. Tverd. Tela~St. Petersburg! 41, 1797–1799~October 1999!

The Yafet–Kittel model for a two-sublattice ferrimagnet with an antiferromagnetic exchangeinteraction in one of the sublattices was developed to describe magnetic-field-induced phasetransitions in the isotropic and Ising cases. Depending on the relative values of the exchangeparameters of the inter-sublattice interaction and the intra-sublattice interaction in the isotropiccase, two types of magnetic phase diagrams with two types of second-order phase transitionsare possible: to the noncollinear phase and to the spin-flip phase, and, in the Ising case, three typesof magnetic phase diagrams with first-order phase transitions are possible. ©1999American Institute of Physics.@S1063-7834~99!01610-X#

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1. In a two-sublattice ferrimagnet with stable sublatticean external magnetic field induces a transition from therimagnetic state with antiparallel arrangement of the mnetic moments of the sublattices to the ferromagnetic swith parallel magnetic moments via two second-order phtransitions through a noncollinear phase or, in the preseof a large enough uniaxial magnetic anisotropy, via one fiorder transition. If there exists an antiferromagnetic echange interaction inside one of the sublattices of the femagnet, thanks to which that sublattice can decay intoequivalent sublattices, then the magnetic phase diagramsthe magnetization curves become substantially more comcated. The first attempt to describe magnetic systems of stype was the Yafet–Kittel model1 ~see also Ref. 2!, whichconsiders a two-sublattice ferrimagnet with an antiferromnetic exchange interaction in each sublattice. In the molelar field approximation the possible magnetic states of sucsystem were calculated in the absence of an external mnetic field in the exchange interaction. The idea of trianguordering was introduced, which corresponds to an orientaof the magnetic moments of all the sublattices at certangles with respect to each other, and it was shown thapresence entails the possibility of an increase in the sattion magnetization in the presence of an external magnfield even in the limitT→0 K.

The Yafet–Kittel theory was developed for ferrites wispinel structure. While still remaining relevant for thecompounds, if this theory is developed from the casespontaneous transitions to transitions induced by an extemagnetic field, it can be used to describe the magnetic perties of many other compounds, including intermetallwith rare earths, e.g., ternaries of the typeRMn2X2 , whereX5Si or Ge. The present paper is dedicated to such a deopment of the Yafet–Kittel theory.

2. Let us consider the two-sublattice ferrimagnet withantiferromagnetic exchange interaction inside one of the slattices, thanks to which that sublattice can decay into t

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equivalent sublattices with magnetic momentsm1 and m2 ,where m15m25m. We calculate the magnetic phase digrams and magnetization curves of such a ferrimagnet inmolecular field approximation, in which the magnetic mments in a state of thermodynamic equilibrium are assumto be aligned with the effective fields acting in them. We wanalyze the signs and magnitudes of these effective fiand chose the states corresponding to minimum values othermodynamic potential for a prescribed value of the exnal magnetic field.

In the exchange approximation~in the absence of magnetic anisotropy! the thermodynamic potential can be writtein the form

F5J1m1•m2/21J2M•m1/21J2M•m2/2

2H•~m11m21M !. ~1!

Here J1 and J2 are the parameters of the antiferromagne(J1 ,J2.0) exchange interaction inside the unstable subtice and between the sublattices, respectively;M is the mag-netic moment of the stable sublattice.

We list the phases realized in such a ferrimagnet. Thare four of them. Three of them are characterized by paraorientation ofm1 andm2 , which thus form a common sublattice and are similar to the phases of an ordinary ferrimnet: a ferrimagnetic phase with antiferromagnetic orientatof m11m2 and M ~phaseA), a ferromagnetic phase (F),and a noncollinear phase (N). The fourth and last phase ithe triangular phase, and we denote it asC ~canted!. In it themomentsm1 andm2 form an angleu with the direction ofthe field which is given by

cosu52H7J2M

2J1m. ~2!

This phase is the initial state of the ferrimagnet forM /2m,J1 /J2 ; the sign in the denominator is minus ifJ1.J2 andplus if J1,J2 . In the opposite case (M /2m.J1 /J2) the A

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1650 Phys. Solid State 41 (10), October 1999 Kolmakova et al.

phase is the initial state. All of the phase transitionssecond-order, and the critical fields are given by the folloing relations:

HC↔A5H J1m2J2M /2 for J1,J2 ,

2J1m1J2M /2 for J1.J2 ;

HC↔F5J1m1J2M /2;

HA↔N5J2uM22mu/2;

HN↔F5J2~M12m!/2. ~3!

The magnetic phase diagrams are shown in Fig. 1. Figudepicts the most interesting magnetization curve with thsecond-order phase transitions, which is realized forJ1,J2

FIG. 1. Magnetic phase diagrams of a ferrimagnet with antiferromagnexchange interaction in one of the sublattices~isotropic case!: for J1.J2 ~a!and forJ1,J2 , i 5J2 /J1 ~b!. The solid lines represent second-order phatransitions.

FIG. 2. Schematic magnetization curve in the case of the phase diagraFig. 1~b! for M /2m,J1 /J251/i .

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2eandM /2m,J1 /J2 . For values of the parameters not satisfing these relations, the magnetization curves contain eione transitionC→F or two transitionsA→C→F or F→N→F.

3. Going beyond the limits of the exchange approximtion and taking the finite anisotropy of both sublattices inaccount is a very relevant and interesting problem. We hextracted the results for this situation into a separate paand in the present paper consider only the limiting case oinfinitely large anisotropy~the Ising case!. For this case thescalar products of the magnetic moment vectors in the thmodynamic potential~1! should be replaced by products othe projections of the magnetic moments onto the Ising asince the remaining projections are equal to zero. In tcase, the noncollinear and triangular phases cannot beized, but the ferrimagnetic phase (A) and the ferromagneticphase (F) are joined by another collinear phase, in whichm1

FIG. 3. The same as in Fig. 1 for the Ising model for different relatvalues of the exchange parameters:J1.2J2 ~a!; J2,J1,2J2 ~b!; J1,J2 ,i 5J2 /J1 ~c!. The dashed lines represent first-order phase transitions.

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1651Phys. Solid State 41 (10), October 1999 Kolmakova et al.

and m2 are ordered antiferromagnetically. We denote tphase asAF. In this case three types of phase diagramsrealized, which are shown in Fig. 3 for different relativvalues of the exchange parametersJ1 and J2 . All of thephase transitions are first-order. The magnetization cuare characterized by one or two jumps, which occur atcritical fields

HA↔AF5~J2M2J1m!/2;

HAF↔A5~J1m2J2M !m

2~m2M !;

HAF↔F5~J2M1J1m!/2;

H↔F5J2m. ~4!

Note that the field of the transition between theA and AFphases depends on which of the phases is the initial onformulas~4! the expressions forHA↔AF andHAF↔A are dif-ferent. In the case when theAF phase is the initial phase, thcritical field is a nonlinear function ofM and m, as can beseen in the phase diagrams shown in Figs. 3b and 3c. Alremaining critical fields are linear functions of the magnezations of the sublattices. A peculiarity of the phase trantion AF→A is the fact that in it not one magnetic momenas in all the other phase transitions, but two magnetic mments change their orientation, with the total magnetizatchanging fromM to 2m2M .

4. Thus, in this paper, in the effective field model, whave calculated the magnetic phase diagrams of ferrimagwith an antiferromagnetic exchange interaction in one ofsublattices for the isotropic and Ising cases. A considera

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of these limiting~in the magnetic anisotropy! cases is a necessary step in the solution of the problem for a ferrimagnesuch type with arbitrary anisotropy. Ferrimagnets of this tyinclude intermetallic compounds with a rare earth and mganeseRMn2X2 , which for some pure and diluted rarearths in certain temperature intervals are characterized bantiferromagnetic exchange interaction, both inside the mganese subsystem and between the rare-earth and mangsubsystems. The rare-earth subsystem in these composhould be isotropic forR5Gd since Gd31 is an S-ion, andIsing, e.g., forR5Dy since such is the single-ion naturethe anisotropy of the Dy31 ion. The available experimentadata for the magnetic properties ofRMn2X2 compounds~Refs. 3–5! points to the need to consider triangular orderifor the magnetic moments of the rare earth and manganand, consequently, underscores the urgency of develothe Yafet–Kittel model further along this path.

This work was carried out with the support of the Rusian Fund for Fundamental Research~RFBR! ~Project No.96-02-16373! and with the joint support of INTAS–RFBR~Project No. 95-641!.

1Y. Yafet and C. Kittel, Phys. Rev.87, 290 ~1952!.2S. Krupicka, Physics of Ferrites and Related Magnetic Oxides, Vol. 1~Mir, Moscow, 1976!.

3N. Iwata, K. Hattori, and T. Shigeoka, J. Magn. Magn. Mater.53, 318~1986!.

4H. Kobayashi, H. Onodera, and H. Yamamoto, J. Magn. Magn. Mater.79,76 ~1989!.

5H. Kobayashi, H. Onodera, Y. Yamaguchi, and H. Yamamoto, Phys. RB 43, 728 ~1991!.

Translated by Paul F. Schippnick