Interplay between Kitaev interaction and single ion anisotropy in ferromagnetic CrI$_3$ and CrGeTe$_3$ monolayers

Abstract: Magnetic anisotropy is crucially important for the stabilization of two-dimensional (2D) magnetism, which is rare in nature but highly desirable in spintronics and for advancing fundamental knowledge. Recent works on CrI$_3$ and CrGeTe$_3$ monolayers not only led to observations of the long-time-sought 2D ferromagnetism, but also revealed distinct magnetic anisotropy in the two systems, namely Ising behavior for CrI$_3$ versus Heisenberg behavior for CrGeTe$_3$. Such magnetic difference strongly contrasts with structural and electronic similarities of these two materials, and understanding it at a microscopic scale should be of large benefits. Here, first-principles calculations are performed and analyzed to develop a simple Hamiltonian, to investigate magnetic anisotropy of CrI$_3$ and CrGeTe$_3$ monolayers. The anisotropic exchange coupling in both systems is surprisingly determined to be of Kitaev-type. Moreover, the interplay between this Kitaev interaction and single ion anisotropy (SIA) is found to naturally explain the different magnetic behaviors of CrI$_3$ and CrGeTe$_3$. Finally, both the Kitaev interaction and SIA are further found to be induced by spin-orbit coupling of the heavy ligands (I of CrI$_3$ or Te of CrGeTe$_3$) rather than the commonly believed 3d magnetic Cr ions.

• The paper demonstrates that the Kitaev interaction predominantly drives the anisotropic exchange coupling in CrI3 and CrGeTe3, challenging the conventional XXZ model assumptions.
• It uses first-principles calculations to develop a Hamiltonian integrating both exchange coupling and single-ion anisotropy, with key parameters like K = 0.85 meV and Azz = -0.26 meV.
• Findings indicate that spin-orbit coupling from heavy ligands is crucial for these magnetic behaviors, offering new pathways for tailoring 2D materials for spintronic applications.

Insights on the Interplay of Kitaev Interaction and Single-Ion Anisotropy in CrI$_3$ and CrGeTe$_3$ Monolayers

This paper explores the intricacies of magnetic anisotropy in CrI$_3$ and CrGeTe$_3$ monolayers, providing insights into the microscopic interactions responsible for their unique magnetic behaviors. The authors employ first-principles calculations to develop a Hamiltonian that accounts for the interactions at play, primarily focusing on the Kitaev interaction and single-ion anisotropy (SIA).

Summary of Findings

The study reveals that the anisotropic exchange coupling in both CrI$_3$ and CrGeTe$_3$ is predominantly of the Kitaev type. This finding is significant because it challenges previous assumptions that CrI$_3$ could be accurately described by the XXZ model, which posits isotropic in-plane coupling and different out-of-plane coupling. The results extend the understanding of Kitaev interactions to 3$d$ transition-metal-based systems, a field typically dominated by 4$d$ or 5$d$ systems. Moreover, the authors identify that both the Kitaev interaction and SIA are primarily induced by the spin-orbit coupling (SOC) of heavy ligands, namely iodine in CrI$_3$ and tellurium in CrGeTe$_3$, as opposed to the traditionally considered contribution from chromium ions.

The paper introduces a Hamiltonian that incorporates both exchange coupling and SIA, expressed as:

$$\mathcal{H} = \frac{1}{2} \sum_{i,j} J\bm{\rm S}_i {\cdot} \bm{\rm S}_j + KS_i^{\gamma}S_j^{\gamma} + \sum_{i} A_{zz}{\rm S}^z_i{\rm S}^z_i$$

where $J$ is the isotropic exchange coupling and $K$ is the Kitaev interaction parameter. The paper demonstrates that these interactions lead to noticeable magnetic behaviors: CrI$_3$ exhibits Ising-like behavior with out-of-plane easy-axis magnetization, while CrGeTe$_3$ displays Heisenberg-like behavior with negligible anisotropy in three dimensions.

Strong Numerical Results

The study provides numerical insights into the exchange coupling parameters and the SIA coefficients for both materials. Notably, the Kitaev interaction parameter $K$ in CrI$_3$ is found to be 0.85 meV, significantly affecting its magnetic anisotropy. Similarly, the contribution of SIA to the overall magnetic anisotropy is highlighted, with the CrI$_3$ SIA coefficient $A_{zz}$ being -0.26 meV, thereby reinforcing the out-of-plane magnetic anisotropy.

Implications and Future Directions

The implications of these results are twofold. Practically, understanding the interplay between Kitaev interaction and SIA opens avenues for designing novel 2D magnetic materials with tailored properties for spintronic applications. Theoretically, this work extends the class of materials known to exhibit Kitaev interactions, which is fundamental in the pursuit of realizing quantum spin liquids. Additionally, the identification of the SOC of ligands as the primary contributor to these interactions in 3$d$ systems suggests alternative pathways to modulate magnetic properties, such as ligand substitution.

Future research could focus on exploring strain effects or chemical modifications to further enhance Kitaev interactions in Cr-based monolayers, possibly driving them towards quantum spin liquid states. Moreover, extending these computational insights to experimental validations could solidify the understanding and applicability of these two-dimensional systems.

In conclusion, this paper enriches the discourse on 2D ferromagnetism by providing a nuanced examination of the mechanisms underlying magnetic anisotropy in CrI$_3$ and CrGeTe$_3$, thereby bridging a gap between theoretical predictions and material realizations in the field of low-dimensional magnetism.