PHYSICAL REVIEW B VOLUME 35, NUMBER 13 1 MAY 1987 Properties of a spin-1 system with competing interactions and reentrant magnetic behavior P. J. Jensen, K. A. Penson, and K. H. Bennemann Institute for Theoretical Physics, Freie Universtiit Berlin, Arnimallee 14, D-1000 Berlin 33, Federal Republic of Germany (Received 24 October 1986) We study a spin-1 lattice model with anisotropic competing interactions. The phase diagram obtained by use of a generalized molecular-field treatment exhibits various modulated phases and, in addition, a transition from a high-temperature ferromagnetic to a low-temperature modulated magnetic phase. The obtained reentrant magnetic behavior is compared with experimental results for Ce3Alii compound. During the last few years periodically modulated struc- tures have been studied intensively. The simplest model used to explore such systems is the S = 2 anisotropic next-nearest-neighbor Ising (ANNNI) model, which has been studied by several groups from different view- points. ' The model consists of nearest-neighbor (NN) ferromagnetic interactions in the xy plane and competing NN ferromagnetic and next-nearest-neighbor (NNN) an- tiferromagnetic interaction terms only in the z direction. The model displays a series of transitions between various modulated and uniform phases. This ANNNI model was first introduced by Elliot in order to describe the helical magnetic structure in erbium and later on it was success- fully used by Bak and Boehm for the cerium compound CeSb. In the following we would like to extend such studies to a spin system with competing interactions including S' =0, ~ 1 states. This is done by defining, for a simple cu- bic lattice, an S =1 ANNNI model with quadrupolar spin interaction terms added. The resulting Hamiltonian is ob- viously a generalization of the Blume, Emery, and Griffiths (BEG) model, which was previously used for the descrip- tion of the k transition and phase separation of He- He mixtures. We shall investigate this model, in particular, with respect to a reentrant transition from a high- temperature ferromagnetic to a low-temperature modulat- ed phase. Possibly, this model may simulate the general behavior of CesA111 (Ref. 6). In our model the (S') terms of the BEG system are taken to be only in the z direction and are chosen in such a way that the combination of S' =0 between two neighbor- ing i S'i =1 terms is energetically favorable (in the fol- lowing we shall omit the superscript z of S'). The Hamil- tonian used is given in symmetrized form by P = — Jo+S;S;+1 — Ji QSS;+1 — J2+S;S;+2 — Kg [S; (1 — S;+1)+(I — S; )S;i|1 . x,y Here, Jo, Ji, and J2 are the usual exchange constants and the constant K describes the combined effect of single-ion anisotropy and quadrupolar interactions (see inset in Fig. 1). As in the ANNNI model we get Jo = J1 & 0 and J2 ( 0; in addition K ) 0. As usual, in S =1 lattice models different phases of Eq. (1) will be characterized by different values of the magnetization m; =(S;) and the quadrupolar order parameter gt =(S; ). In addition to the competition between J1 and Jz interactions (producing modulated phases) the competition between J and K terms results in the appearance of S =0 states. Hopefully, these may imitate a Kondo-like singlet state. The ground state of Eq. (1) at T =0 consists of four phases (see Fig. 1). As expected, the ferromagnetic (FM) phase (I) is established for Jt » i J2 i, K, the (2) antiphase (II) of the ANNNI model for i Jz i » J, , K, and a modified (2) antiphase (!II) for K» J|, i J2i. Between these three phases one finds a modified (3) antiphase (IV). This phase acts as a "compromise" between the three oth- er phases. For T & 0 the system described by Eq. (1) is solved within the mean-field approximation. The solu- tion is — except for the determination of the paramagnetic 1. 0— IZ ( ~tii j 3, , K 0. 5 r (t~«j HI(lo &o) 1. 0 2. 0 I K/ji FIG. 1. Various ground-state spin configurations of Eq. (1). The open circles denote sites with S' 0. The exchange coupling constants Ji and J2 refer to nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic interactions, respective- ly. The coupling constant K defined in Eq. (1) is the driving force for states including S' 0. These couplings are presented in the inset. The dashed line indicates the location of the phase diagram in Fig. 2. 7306 1987 The American Physical Society