First-principles study of bulk stacking, $J_{\rm eff}$ picture, magnetic Hamiltonian, $g$ factors, and structural distortions of $α$-RuCl$_3$

TL;DR

This work identifies $R\bar{3}$ as the low-temperature bulk stacking of α-RuCl$_3$ via constrained DFT, aligning with experimental observations, and demonstrates that the electronic structure near the gap adheres to a $J_{\rm eff}=1/2$ picture when the quantization axis is chosen along the Néel vector. It develops and fits an anisotropic $J_1$-$J_2$-$J_3$ model (with non-negligible second-nearest-neighbor terms) to DFT total energies, and accurately computes $g$ factors using a canting method and translationally equivariant Wannier interpolation of orbital magnetization. The study also shows that twist distortions of the Cl sublattice are crucial for reproducing magnetic anisotropy, highlighting the need to incorporate these distortions in microscopic models. Together, these results refine the microscopic understanding of α-RuCl$_3$’s magnetism and provide a robust framework for exploring Kitaev physics in real materials, including guidance for modeling and interpreting experiments.

Abstract

$α$-RuCl$_3$ is a candidate Kitaev material that exhibits zigzag antiferromagnetic ordering below 7 K. One contentious issue regarding this material is its bulk structure in the low-temperature phase. Recently, it has become generally accepted from experiments that the low- and high-temperature structures belong to the $R\bar{3}$ and $C2/m$ space groups, respectively. However, there was no theoretical study supporting the $R\bar{3}$-type structure as the low-temperature structure. In this study, we use constrained density functional theory to show that the $R\bar{3}$ structure is lower in energy than the $C2/m$ structure, in agreement with experimental observations. Then, we show that the conduction band minimum states are almost of the $J_\textrm{eff}=1/2$ and $m_\textrm{eff}=-1/2$ character, if we set the angular momentum quantization axis to be parallel to the Néel vector; this is the first analysis of the $J_\textrm{eff}$ picture for $α$-RuCl$_3$ from this perspective. In addition, we compute the anisotropic magnetic exchange parameters and $g$ factors of monolayer $α$-RuCl$_3$, thereby providing a comprehensive understanding of its magnetism. Our results demonstrate that both second-nearest-neighbor exchange interactions and magnetic moments not captured by the conventional atomic orbital projection method are necessary for accurate description of the magnetism in $α$-RuCl$_3$. Moreover, the calculated $g$ factors are in fairly good agreement with experimental measurements, especially the small anisotropy between their in-plane and out-of-plane components. Finally, we examine the effects of structural distortions from a perfect RuCl$_6$ octahedron, already present in bulk $α$-RuCl$_3$ without any external perturbation, on the magnetic properties. (The abstract is cut here due to the word limit; see the pdf file for the full abstract.)

Figures (9)

• Figure 1: Crystal structure of bulk $\alpha$-RuCl$_3$ with (a) $C2/m$ stacking and (b) $R\bar{3}$ stacking. (c) Crystal structure of monolayer $\alpha$-RuCl$_3$. The $xyz$ axes correspond to the local octahedron axes, while the $XYZ$ axes are the global Cartesian axes. The $abc$ axes represent the crystallographic axes.
• Figure 2: Magnetic configurations used for the computation of exchange parameters. Red and blue discs denote up and down effective spins, respectively, with respect to a given collinear direction. (a)-(d) correspond to zigzag, Néel, stripy, and ferromagnetic configurations of the honeycomb lattice, respectively.
• Figure 3: Schematic of the canting method for obtaining the $g$ factors. (a) Before canting, the magnetic moments are aligned along the $X$ direction, and the total magnetic moment ($\vb{M}_{\rm tot}^{[\text{zAFM-}X]}$) is zero. (b) After canting the moments slightly to the $Y$ direction (exaggerated in the figure), the $Y$ component of the total magnetic moment ($\vb{M}_{\rm tot, cant}^{[\text{zAFM-}X]}$) is proportional to the canting angle $\Delta \theta$. All structures are zigzag antiferromagnetic.
• Figure 4: Projected density of states (PDOS) of $\alpha$-RuCl$_3$ around the valence band maximum ($E = 0$ eV) and the conduction band minimum ($E = 0.8$ eV).
• Figure 5: Magnitude of $\mathbf{J}_\text{eff}$ moments of all four or eight Ru atoms in a unit cell in all 60 magnetic configurations. In total, there are $4\times 48 + 8 \times 12=288$ symbols (48 configurations with four Ru ions per unit cell and 12 configurations with 8 Ru ions per unit cell) in each panel. $\theta$ and $\phi$ represent the collinear moment direction $\vb{e}^{[\xi]} = (\sin\theta \cos\phi, \sin\theta \sin\phi, \cos\theta)^{\mathsf{T}}$.