Two-Dimensional Charge Order in Layered 2-1-4 Perovskite Oxides

Monte Carlo simulations are performed on the three-dimensional (3D) Ising model with the 2-1-4 layered perovskite structure as a minimal model for checkerboard charge ordering phenomena in layered perovskite oxides. Because of the interlayer frustration, only 2D long-range order emerges with a finite correlation length along the $c$ axis. Critical exponents of the transition change continuously as a function of the interlayer coupling constant. The interlayer long-range Coulomb interaction decays exponentially and is negligible even between the second-neighbor layers. Instead, monoclinic distortion of a tetragonal unit cell lifts the macroscopic degeneracy to induce a 3D charge ordering. The dimensionality of the charge order in ${\mathrm{La}}_{0.5}{\mathrm{Sr}}_{1.5}{\mathrm{MnO}}_4$ is discussed from this viewpoint.

Figure 1

(a) ${\mathrm{K}}_2{\mathrm{NiF}}_4$ tetragonal lattice structure and the interactions in the Hamiltonian $\mathcal{H}$. The interlayer coupling $J_{\perp }$ is partially drawn for clarity. (b) Frustrated stacking of two adjacent planes. Solid and gray lattices are for neighboring planes projected to the stacking direction. Open and filled circles denote up and down spins, respectively. (c) Second-neighbor interlayer coupling $J_{\perp }^{\prime }$ (blue solid line) and longer-range interactions between the second-neighbor layers (light-blue dashed lines). (d) Monoclinic lattice distortion. This introduces two different interlayer couplings $J_{\perp }^{(\pm )}$.

Figure 2

Temperature dependences of (a) $C$ and (b) $g$ in the case of $J_{\perp }=0.5$. The lines are guides to the eye.

Figure 3

The 3D spin structure factor $S(\pi ,\pi ,k_z)$ divided by the square dimension $L^2$ is plotted against $k_z$ below $T_{\mathrm{c}}$ in the case of $L=64$. The inset shows that $S(\pi ,\pi ,0)/L^3$ linearly goes to zero with $1/L$.

Figure 4

The best-fit result of the finite-size scaling for correlation functions obtained by our MC calculation based on the histogram algorithm in the case of $J_{\perp }=0.5$.

Figure 5

$J_{\perp }$ dependences of (a) $T_{\mathrm{c}}$ and (b) $\nu$ and $\eta$. Dashed (dotted) lines denote the values of the 2D (3D) Ising model. Filled (open) symbols are for $L^3$ ($L^2\times 2$, namely, bilayer) systems for $\mathcal{H}$. Error bars are omitted when they are smaller than the symbol sizes. Note that the values of $J_{\perp }$ for bilayer systems are divided by two for convenience.

Figure 6

The finite-size scaling for $F$ obtained by MC calculations in the case of $J_{\perp }=0.5$. We have used the values of $T_{\mathrm{c}}$ and $\nu$ obtained from the analysis given in Fig. 4.