${\mathrm{CrI}}_3$ revisited with a many-body ab initio theoretical approach

${\mathrm{CrI}}_3$ has recently been shown to exhibit low-dimensional, long-range magnetic ordering from few layers to single layers of ${\mathrm{CrI}}_3$. The properties of ${\mathrm{CrI}}_3$ bulk and few-layered systems are uniquely defined by a combination of short-range intralayer and long-range interlayer interactions, including strong correlations, exchange, and spin-orbit coupling. Unfortunately, both the long-range van der Waals interactions, which are driven by dynamic, many-body electronic correlations, and the competing strong intralayer correlations, present a formidable challenge for the local or semilocal mean-field approximations employed in workhorse electronic structure approaches like density-functional theory. In this paper we employ a sophisticated many-body approach that can simultaneously describe long- and short-range correlations. We establish that the fixed-node diffusion Monte Carlo (FNDMC) method reproduces the experimental interlayer separation distance of bulk ${\mathrm{CrI}}_3$ for the high-temperature monoclinic phase with a reliable prediction of the interlayer binding energy. We subsequently employed the FNDMC results to benchmark the accuracy of several density-functional theory exchange-correlation approximations.

Figure 1

Total energies of the rhombohedral structure predicted by FNDMC with LDA+$U$ trial wave functions with different $U$ values. The total energies are given as the relative differences from the lowest data point. The optimal $U$ value is estimated to be $U=2.4\left(4\right)\mathrm{eV}$ by quadratic fitting (dashed line).

Figure 2

Total energies per formula unit of the monoclinic structure predicted by FNDMC with different grid sizes of twist-averaging. The total energies are given as relative differences from that with a $4\times 4\times 4$ grid. This figure indicates the one-body finite-size error is sufficiently suppressed with a $3\times 3\times 3$ grid.

Figure 3

Total energies of the monolayer ${\mathrm{CrI}}_3$ predicted by FNDMC for the inverse of different simulation cell sizes.

Figure 4

Total energies of monoclinic bulk ${\mathrm{CrI}}_3$ predicted by FNDMC for the inverse of different simulation cell sizes.

Figure 5

The interlayer binding curves of the monoclinic structure predicted with vdW–DF–optB86b for two different geometries: experimental and relaxed with vdW–DF–optB86b.

Figure 6

FNDMC prediction of the binding curve of the monoclinic structure. The vertical axis indicates the relative energy differences from the lowest total energy. The dotted line indicates the quadratic fitting curve with the four lowest data points.

Figure 7

DFT prediction of the interlayer binding curves of the monoclinic structure with different functionals. The experimental separation distance is shown as a vertical line. The binding energy predicted by FNDMC is shown as a gray shaded region. The PBE result is just behind the LDA result so the PBE curve is not visible.

Figure 8

DFT prediction of the interlayer binding curves of the rhombohedral structure with different functionals. The experimental separation distance is shown as a vertical line. The binding energy predicted by FNDMC is shown as a gray shaded region. The PBE result is just behind the LDA result so the PBE curve is not visible.