Magnetic order in four-layered Aurivillus phases

We determine the viability of four-layered Aurivillius phases to exhibit long-range magnetic order above room temperature. We use Monte Carlo simulations to calculate transition temperatures for an effective Heisenberg model containing a minimal set of required couplings. The magnitude of the corresponding coupling constants has been determined previously from electronic structure calculations for ${\mathrm{Bi}}_5{\mathrm{FeTi}}_3{\mathrm{O}}_{15}$, for which we obtain a transition temperature far below room temperature. We analyze the role of further neighbor interactions within our Heisenberg model, in particular that of the second-nearest-neighbor coupling within the perovskitelike layers of the Aurivillius structure, as well as that of the weak interlayer coupling, in order to identify the main bottleneck for achieving higher magnetic transition temperatures. Based on our findings, we show that the most promising strategy to obtain magnetic order at higher temperatures is to increase the concentration of magnetic cations within the perovskitelike layers, and we propose candidate compounds where magnetic order could be achieved above room temperature.

Figure 1

Structure of ${\mathrm{Bi}}_5{\mathrm{FeTi}}_3{\mathrm{O}}_{15}$, an Aurivillius phase corresponding to $m=4$, at temperatures below the ferroelectric transition, viewed from two slightly different perspectives. Purple and red spheres indicate ${\mathrm{Bi}}^{3+}$ and ${\mathrm{O}}^{2-}$ ions, respectively. The ions inside the blue octahedra $(B$ sites) are ${\mathrm{Ti}}^{4+}$ and ${\mathrm{Fe}}^{3+}$ with concentrations $x\left(\text{Ti}\right)=3/4$ and $x\left(\text{Fe}\right)=1/4$.

Figure 2

Construction of the cell used to simulate a four-layered Aurivillius phase. Our basic unit cell contains 2 formula units (f.u.). An intermediate cell is constructed as $2\times 2\times 1$ supercell of the basic cell, and $x·32$ magnetic ions are randomly distributed over the $32B$ sites of this intermediate cell. The final simulation cell is then constructed by stacking several of such intermediate cells to form a supercell composed of $8n\times 8n\times n$ unit cells $(n=1$ in the figure).

Figure 3

A sketch of the model for the network of $B$ sites in the four-layered Aurivillius structure considered in our Monte Carlo simulations. Black (white) spheres represent magnetic (nonmagnetic) cations. The three types of coupling included in our model are indicated by double arrows: nearest-neighbor $J_{\mathrm{NN}}$ in blue, next-nearest-neighbor $J_{\mathrm{NNN}}$ in orange, and interlayer $J_{\mathrm{INTER}}$ in green. Not visible in the picture but included in the calculations are the $J_{\text{NNN}}$ couplings within the same in-plane perovskite layer. The red circle highlights a triangle of connected spins that can in principle give rise to partial frustration in the case of antiferromagnetic coupling.

Figure 4

Temperature dependence of (a) magnetization, (b) magnetic susceptibility, and (c) the Binder cumulant for the model described in Sec. 2 and magnetic ion concentration $x=0.25$. Several system sizes have been considered: $8n\times 8n\times n$ unit cells with $n=2$ $\left(\blacksquare \right)$ in orange, $n=3$ $\left(⧫\right)$ in green, and $n=4$ $(•)$ in blue. The vertical dashed line in (c) indicates $T_{\text{C}}$, obtained from the intersection point of the Binder cumulants for the three different cell sizes.

Figure 5

Dependence of the transition temperature, $T_{\text{C}}$, on $J_{\text{NNN}}$ $\left(J_{\text{INTER}}\right)$ for $J_{\text{INTER}}=J_{\text{NN}}=45$ meV $(J_{\text{NNN}}=J_{\text{NN}}=45$ meV). The red squares (black dots) indicate the calculated values whereas the black (red) line is just a guide for the eyes. Error bars are estimated from the standard deviation of the peak positions of the magnetic susceptibility obtained for several different ${\mathrm{Fe}}^{3+}$ distributions. The inset shows the same data on a linear scale.

Figure 6

Values of the numerical derivative ${[\partial {\chi }^{-1}/\partial T]}^{-1}$ at different temperatures for the cases where either $J_{\text{INTER}}$ (black points) or $J_{\text{NNN}}$ (red points) is varied. The vertical lines indicate the transition temperatures for the various cases as shown in Fig. 5.

Figure 7

Temperature dependence of the magnetic susceptibility, Eq. (3), (top row) and Binder cumulant, Eq. (2), (bottom row) for magnetic ion concentrations $x=0.5$, 0.75, and 1.0 in the $m=4$ Aurivillius structures. Different system sizes have been considered: $8n\times 8n\times n$ unit cells with $n=2$ in orange $\left(\blacksquare \right)$, $n=3$ in green $\left(⧫\right)$, and $n=4$ in blue $(•)$. Vertical lines represent the obtained $T_{\text{C}}$.

Figure 8

Magnetic transition temperature, $T_{\text{C}}$, as function of magnetic ion concentration. Different symbols specify temperatures for different (scaled) values of $J_{\mathrm{NN}}$ (with $J_{\mathrm{NNN}}=3\%J_{\mathrm{NN}}$ and $J_{\mathrm{INTER}}=1\%J_{\mathrm{NN}})$. The dashed horizontal line corresponds to room temperature.