Monte Carlo simulations on magnetic behavior of a spin-chain system in triangular lattice doped with antiferromagnetic bonds

Abstract

A three-dimensional Ising-like model doped with anti-ferromagnetic (AFM) bonds is proposed to investigate the magnetic properties of a doped triangular spin-chain system by using a Monte-Carlo simulation. The simulated results indicate that a steplike magnetization behavior is very sensitive to the concentration of AFM bonds. A low concentration of AFM bonds can suppress the stepwise behavior considerably, in accordance with doping experiments on Ca₃Co₂O₆. The analysis of spin snapshots demonstrates that the AFM bond doping not only breaks the ferromagnetic ordered linear spin chains along the hexagonal c-axis but also has a great influence upon the spin configuration in the ab-plane.

Graphical abstract

Keywords

Ca₃Co₂O₆ compound / spin-chain / step-like magnetization

1 Introduction

Many compounds with the general formula A′₃ABO₆ (A′ is Ca or Sr, while A and B are transition metal elements) have been synthesized and studied due to their quasi-one-dimensional structures and complex magnetic properties [1–5]. Among these, Ca₃Co₂O₆, as the only compound in which both A and B sites are occupied by the same metallic element, has attracted a lot of interest [6–11]. It is composed of parallel one-dimensional (1D) Co₂O₆ chains aligned along the hexagonal c-axis, separated by Ca²⁺ ions [7]. Each chain is surrounded by six equally spaced chains, forming a two-dimensional (2D) triangular (hexagonal) lattice in the ab-plane. Generally, the interchain distance is about double of the intrachain Co-Co distance, which ensures that the intrachain ferromagnetic (FM) interaction is much stronger than the interchain antiferromagnetic (AFM) one [6]. For this compound one of the most attractive features is the steplike magnetization (M) plotted against the magnetic field (h) applied along the chains [8, 9, 11]. As 10 K < T < 25 K, with decreasing T the M-h relation gradually exhibits a M₀/3 plateau below h ~ h_c ≈ 3.6 T, above which M jumps up to the saturated value M₀. When T < 10 K, the M₀/3 plateau decomposes into three nonzero and equidistant substeps separated at h_S1 ≈ 1.2 T and h_S2 ≈ 2.4 T below h_c ≈ 3.6 T.

In the field of experimental investigations, doping and substitution are effective methods to tune the interactions in the materials. Many experiments show that the physical properties of Ca₃Co₂O₆ can be modified by a suitable substitution at both the Ca [2, 12] and Co sites [1, 3, 4]. Generally, introducing foreign elements to the Co site has more direct influence on the 1D Co₂O₆ chains, and consequently smoothens the magnetization steps [1, 3, 4]. In the work of Flahaut et al. [1], the experimental results indicates that the Cr³⁺ magnetic moments are antiferromagnetically coupled to their Co³⁺ neighbors, and the magnetization jumps, observed at low temperature, tend to be suppressed as the Cr³⁺ content increases. Recently, Jain et al. reported that the iron substitution reduces the one-dimensional characteristic of Ising spin chains because the iron doping breaks the ferromagnetically ordered linear spin chains along the hexagonal c-axis and produced an AFM exchange interaction between the Fe³⁺ ions [5]. Many experiments suggest that doping or substitution will introduce an AFM interaction in the spin chains, and accordingly take great effect on the magnetic properties of the materials. In this paper, based on the previous simulative investigation on triangular spin-chain system [13, 14], we perform a simulation on the 3D Ising model doped with the AFM bonds along the c-axis in a triangle lattice in order to comprehend further the effect of the doping on the magnetic behavior of the Ca₃Co₂O₆ compound.

2 Model and simulations

For Ca₃Co₂O₆ the strong Ising-like anisotropy of the spin interaction has been repeatedly confirmed both experimentally [9,10,15] and theoretically [16], which ensures that Ising models can be employed to describe the magnetic behavior of this compound [13, 14, 17]. Considering the 3D anisotropic spatial arrangement and ignoring the phase difference between the neighboring chains in Ca₃Co₂O₆, we may assume that the Co³⁺ ions are co-planar and then the lattice structure is composed of triangular 2D lattices stacked along the c-axis. The spin-chain is along the c-axis and the intrachain interaction is FM-type, namely J_intra> 0. The interchain interaction J_inter is AFM-type, namely J_inter< 0. The system Hamiltonian is then expressed as:

\begin{array}{}\text{(1)}& H=-{\displaystyle \sum _{[m,l]}{J}^{\text{intra}}}{S}_m{S}_l-{\displaystyle \sum _{[m,n]}{J}^{\text{inter}}{S}_m}{S}_n-h{\mu }_{\text{B}}g{\displaystyle \sum _m{S}_m^{}}\end{array}

where h is the magnetic field applied along the direction of up-spin; g is the Lande factor and μ_B is the Bohr magneton; [m, l] denotes the summation over all the nearest-neighboring pairs in the chains; [m, n] means the summation over all the nearest-neighboring pairs in the ab-plane; S_m is the moment of a lattice spin. It was revealed that the crystalline electric field splits the energy level of Co³⁺ ions into the high-spin (S = 2) and low-spin (S = 0) states [9, 16, 18]. Only the ions of high-spin (S = 2) contribute to the magnetic properties, so S_m = ±2 is chosen in the present simulation. In addition, the magnetic inhomogeneity of the interchain interaction is an important ingredient to induce the steplike feature of the magnetization below 10 K [14]. Therefore, a random exchange term Δ_m,n is taken into account in the interchain coupling. Then J^inter is replaced by J_{m,n}^{\text{inter}},

\begin{array}{}\text{(2)}& {\Delta }_{m,n}=A\cdot R_{m,n}\end{array}

\begin{array}{}\text{(3)}& J_{m,n}^{\text{inter}}=J^{\text{inter}}+J^{\text{inter}}{\Delta }_{m,n}\end{array}

where R_m,n is the random number in [–1, 1], and parameter A represents the magnitude of random exchange interaction.

Based on the above model, considering the influence of substitution in materials, we randomly substitute a few AFM interactions with the value –| J^intra | for intrachain FM coupling, namely a triangular Ising model doped with AFM bonds along c-axis is employed. The concentration of AFM bonds is denoted by p, which means the amount of intrachain AFM bonds divided by the total of intrachain bonds. The values of the parameters for the simulation are shown in Table 1. The values of J_intra, J_inter and A are judged from a quantitative comparison between the simulated results and the experimental data [13].

The simulation starts from an L × L × L (L = 40) Ising lattice with periodic boundary conditions. In fact, the relaxation of the spins along the chains is much faster than that for the interchain spin relaxation in the ab-plane. Therefore, in our simulation, the mechanism of spin flipping includes flipping of several neighboring spins. The flip probabilities of one spin or several neighboring spins are calculated respectively according to the Metropolis algorithm, based on the change of energy resulting from the spin flips. Then the stochastic flipping of spins is approved based on these flip probabilities [13]. The procedure of simulation is described as follows. At a given T, the simulation starts from h=0 with a random spin configuration. The magnetization is evaluated after reaching equilibrium. Afterwards, h is raised and the simulation is performed on the state obtained before to reach a new equilibrium. This process is repeated until high field.

Figure 1 presents the simulated M as a function of h with p=0. When T is above 25 K, M as a function of h shows the paramagnetic behavior. In the T-range from 25 K to 10 K, the wide M₀ /3 plateau appears gradually with decreasing T. As T is below 10 K, three substeps emerge at regular intervals of h_int ≈ 1.2 T below h_c ≈ 3.6 T, which is illustrated in the insert of Fig.1. These M (h) curves without doping demonstrate the typical steplike magnetic behavior, in agreement with the experimental results of Ca₃Co₂O₆ [8, 9, 11]. The doping of the AFM bonds will influence these magnetic behaviors considerably.

The simulated M (h) curves with different p at T = 12 K and 2 K are presented in Fig. 2 (a) and (b) respectively. It is demonstrated that only a low concentration of the AFM bonds doping in spin chains have great influence on the magnetic behavior of this system at low temperature. At about 12 K, the 1/3 M₀ plateau smear out gradually as p increases and a gradual growth of M with increasing h is established when p is about 0.05. The magnetic field values, necessary to induce the ferrito-ferromagnetic transition, decrease as p increases. When T = 2 K, the increasing value of p also seems to melt the three substeps. Finally for the sample with p = 0.05, no steps can be observed. It is consistent with the experiments, indicating that the AFM coupling introduced by doping or substitution really smoothens the magnetization steps.

The effect resulting from the AFM doping can be studied by spin configuration. Since the AFM couplings are doped into the intrachain interactions, they will affect the intrachain FM ordering directly. The sectional snapshots along the c-axis at h = 0 are demonstrated in Fig. 3, where the snapshots in the left column are for T=2 K and those at T=12 K are correspondingly illustrated in the right column, with p = 0, 0.02, 0.04 from the top down. The black and grey-white solid squares represent spin-up and spin-down respectively, and the spin up along the direction of h. It is observed that at p = 0, all the spins in one chain align in the same direction at T = 2 K [Fig. 3 (a)], and for most of chains all spins aligned in the same direction at T=12 K [Fig.3 (b)]. When p = 0.02 the snapshots in Fig. 3 (c) and (d) show that the spins in some chains do not maintain the same orientation, which indicates that the doping of the AFM coupling break the FM ordering in some chains. As p=0.04 [Fig.3 (e), (f)], the spins of many chains have different spin orientations, namely the long-range intrachain FM ordering is destroyed, and consequently the one-dimensional character of Ising spin chains is suppressed partially.

Though the doping of the AFM bonds is along the c-axis, they also have a great effect on the spin configurations in the ab-plane. Figure 4 shows the sectional spin snapshots in the ab-plane with different p at T = 12 K, as h = 1.8 T on the 1/3 M₀ plateau of magnetization, where the black and grey-white solid circles represent spin-up and spin-down respectively. When p = 0, shown in Fig. 4 (a) the spin configuration in the ab-plane is mostly homogeneous, showing a regular ferrimagnetic structure, namely one of the three spin chains takes spin down, while the other two take spin up, leading to M~ M₀/3. With doping of the AFM interaction, when p = 0.02 in Fig.4 (b), some small disordered paramagnetic areas appear. When p = 0.04 [Fig.4 (c)], more regions in the spin configuration show disordered paramagnetic feature. Therefore it is indicated that the AFM coupling doping results in the appearance of disordered regions in the ab-plane.

3 Conclusions

In summary, the magnetic properties of a doped spin-chain system in a triangular lattice are investigated by Monte Carlo simulation in an Ising-like model with the AFM bonds doped. The results show that the low concentration of AFM bonds can modify the magnetic properties of the system, namely the stepwise magnetic behavior is suppressed by the doping considerably, consisting with experimental observations on the doped Ca₃Co₂O₆. Through the investigation of the spin snapshots, it is demonstrated that the doping of AFM bonds breaks the FM ordering along the spin chains and introduces disordered regions in the spin configuration of the ab-plane.

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