Spin-orbit driven $J_{eff} = 1/2$ magnetism in a d$^7$ triangular-lattice monolayer cobaltate

TL;DR

This work demonstrates that monolayer CoBr$_2$ on a triangular lattice hosts spin-orbit entangled $J_{ ext{eff}}= frac{1}{2}$ magnetism with bond-dependent Kitaev exchange, arising from a dominant ligand-assisted hopping channel $t_6$ that strengthens Kitaev interactions relative to the Heisenberg term. By combining first-principles DFT with Wannierization and exact diagonalization, the authors extract a $J_1$–$K$–$oldsymbol{\oldsymbol{\Gamma}}$–$oldsymbol{\Gamma'}$–$J_3$ model and map the magnetic phase diagram in the $J_1$–$J_3$ plane, revealing ferromagnetic, stripy, 120$^{\circ}$ AFM, incommensurate spiral, and $Z_2$ vortex crystal phases. Quantum fluctuations via ED additionally reveal a bond-nematic phase absent in the classical LT analysis, underscoring the importance of beyond-classical effects in this system. The results position monolayer CoBr$_2$ as a promising platform to explore Kitaev-like physics and long-range Heisenberg interactions in 2D triangular cobaltates, with potential strain-tunable transitions and connections to related cobaltates.

Abstract

Recent theoretical and experimental advances have identified cobaltates with a high-spin $d^7$ electronic configuration as promising hosts for spin-orbit entangled $J_{eff} = 1/2$ magnetism that can support bond-dependent exchange interactions. In two-dimensional triangular lattices, the coexistence of such exchange frustration along with geometric frustration gives rise to a rich landscape of competing magnetic phases, establishing monolayer triangular $d^7$ cobaltates as a compelling platform for frustrated magnetism. Here we investigate a representative triangular-lattice monolayer cobaltate CoBr$_2$, where first-principles density functional theory (DFT) calculations reveal a dominant nearest-neighbor $t_{2g}$-$e_g$ hopping channel that enhances the ferromagnetic Kitaev-type exchange interactions. In contrast, the nearest-neighbor Heisenberg term is highly sensitive to a direct $t_{2g}$-$t_{2g}$ hopping path and electronic correlations. The magnetic exchange parameters are evaluated using the hopping amplitudes obtained from DFT calculations within an exact diagonalization framework. We construct the first and third nearest neighbor Heisenberg exchange dependent $J_1$-$J_3$ magnetic phase diagram in the physically relevant regime and identify multiple competing ground states, including ferromagnetic, stripy, spiral, and $120^{\circ}$ antiferromagnetic orders. The Luttinger-Tisza analysis further predicts a Z$_2$ vortex crystal phase, while exact diagonalization reveals a bond-nematic phase stabilized by the longer-range couplings. Going beyond the conventional bond-independent XXZ picture typically applied to Co$^{2+}$ systems, our results on monolayer CoBr$_2$ establish d$^7$ cobalt dihalides as a promising platform to explore the interplay of long-range Heisenberg and bond-dependent exchange interactions that can stabilize diverse magnetic ground states on a triangular lattice.

Figures (4)

• Figure 1: Crystal structure and non-spin-polarized electronic properties of monolayer CoBr$_2$. (a) Crystal structure: Co atoms form a triangular lattice, while Br ligands create edge-sharing octahedra around each Co ion. Local coordinate axes ($x$,$y$,$z$) and the nearest-neighbor bond convention ($X$, $Y$, $Z$) are indicated. (b) Non-spin-polarized electronic band structure with Wannier-interpolated bands and partial DOS near $E_F$ obtained from ab initio DFT calculations. (c) MLWFs for the downfolded Co-$d$ orbitals. (d) Direct hopping path $t_3$ between Co-$d_{xy}$ orbitals, primarily contributing to the Heisenberg exchange in $d^7$ cobaltates. (e) Dominant ligand-assisted hopping path $t_6$, which enhances the Kitaev interaction in monolayer CoBr$_2$.
• Figure 2: Magnetic exchange parameters of monolayer CoBr$_2$ obtained from ED and perturbative calculations. (a) Variation of the exchange couplings with the on-site Hubbard interaction $U$ at fixed Hund’s coupling $J_H = 0.7$ eV. The physically relevant $U$ range for CoBr$_2$ is shaded in blue. (b) Dependence of the exchange couplings on the hopping ratio $|t_3/t_6|$, illustrating the competition between direct and ligand-assisted hopping channels for fixed $U = 3.25$ eV and $J_H = 0.7$ eV. Results from ED are shown by circles connected with solid black lines, while perturbative results are indicated by dashed dark-red lines. The $|t_3/t_6|$ ratios corresponding to the monolayer CoBr$_2$ is marked by dashed blue vertical line.
• Figure 3: Magnetic phase diagram of monolayer CoBr$_2$. The fixed parameters are $J_2=0.0$ meV, $K=-2.7$ meV, $\Gamma=-0.1$ meV and $\Gamma'=0.3$ meV. (a) Semiclassical phase diagram obtained from the LT analysis. The reciprocal-space energy distributions for the different phases are shown, with the lowest-energy points (marked by red circles) indicating the ordering wave vectors corresponding to each phase. Since the wave-vector minima of the spiral phase vary smoothly with the parameters, we show only a representative energy distribution. (b) Quantum phase diagram derived from ED calculations on finite triangular-lattice clusters. The static spin structure factors for representative parameter sets within each region are also presented, illustrating the characteristic magnetic orders.
• Figure 4: (a)--(c) Finite clusters used in ED calculations: two distinct 24-site clusters and a 19-site cluster preserving $C_{6}$ symmetry. (d) Bond anisotropy $A_{b}$ and the 120$^{\circ}$ AFM order parameter $\Phi_{120}$ for $J_{3}=2.0$ meV as functions of $J_{1}$.