Giant orbital magnetization in two-dimensional materials

TL;DR

This paper clarifies how unquenched orbital magnetization can arise in two-dimensional transition-metal compounds through a synergy of crystal-field splitting, spin-orbit coupling, and electronic correlations. It shows that a self-consistent treatment of SOC combined with a Hubbard correction is required to obtain correct orbital polarization and large magnetic anisotropy, particularly for partially filled $t_{2g}$ bands in octahedral or tetrahedral environments. By screening the C2DB, the authors identify dozens of monolayers with potentially giant orbital moments, significantly expanding known examples beyond FePS$_3$ and VI$_3$ and highlighting the impact on 2D magnetism and Néel temperature predictions. The work provides a practical framework for predicting and understanding orbital contributions to magnetism in 2D materials, with implications for designing high-temperature 2D magnets and anisotropy-driven spintronics.

Abstract

Orbital magnetization typically plays a minor role in compounds where the magnetic properties are governed by transition metal elements. However, in some cases, the orbital magnetization may be fully unquenched, which can have dramatic consequences for magnetic anisotropy and various magnetic response properties. In the present work, we start by summarizing how unquenched orbital moments arise from particular combinations of crystal field splitting and orbital filling. We exemplify this for the cases of two-dimensional (2D) VI$_3$ and FePS$_3$, and show that Hubbard corrections as well as self-consistent spin-orbit coupling are crucial ingredients for predicting correct orbital moments from first principles calculations. We then search the Computational 2D Materials Database (C2DB) for monolayers having tetrahedral or octahedral crystal field splitting of transition metal $d$-states and orbital occupancy that is expected to lead to large orbital moments. We identify 112 monolayers with octahedral crystal field splitting and 62 monolayers with tetrahedral crystal field splitting and for materials with partially filled $t_{2g}$ bands, we verify that inclusion of Hubbard corrections as well as self-consistent spin-orbit coupling typically increases the magnitude of predicted orbital moments by an order of magnitude.

Figures (7)

• Figure 1: Sketch of the first-order splitting of the $d$-orbital energy levels in an octahedral crystal field (middle) and with the spin--orbit interaction included (to the right).
• Figure 2: (a) Geometry and coordinate system of a transition metal atom (grey) inside the octahedral cage consisting of six ligand atoms (pink). (b)-(c) Respectively, the states $\left|\newline\psi_{\hat{\mathbf{z}}}\right>$ and $\left|\newline\psi_{\hat{\mathbf{u}}}\right>$ in the octahedral cage. The cyclic color gradient displays the winding of the wave function's complex phase which is necessary for nonzero orbital moments.
• Figure 3: Sketch of the band structure of a material like FePS$_3$ with an octahedral crystal field and six valence electrons occupying the $d$-shell. Pure LSDA yields a metallic ground state whereas LSDA+$U$ opens a gap, but with no orbital magnetization. Only the LSDA+$U$ with self-consistent SOC yields a ground state with the correct orbital magnetization.
• Figure 4: Band structures of FePS$_3$ (top) and VI$_3$ (bottom) with and without self-consistent SOC and Hubbard corrections. In both cases, the partially occupied $t_{2g}$ bands are indicated in the bare LDA calculations. Note that there are two magnetic atoms pr unit cell and the $t_{2g}$ bands thus have six states for each spin channel.
• Figure 5: Energy of the FePS$_3$ crystal and components of the orbital moment vector at the Fe atoms when the direction of the spins at the Fe atoms are constrained to point along the direction $\theta$. Each data point represents a constrained-spin DFT calculation, and the energy is measured w.r.t. the $\theta=0$ data point.