Abstract
Ferromagnetism in the two-dimensional CrI$_3$ has generated a lot of excitement, and it was recently proposed that the spin-orbit coupling (SOC) in Iodine may generate bond-dependent spin interactions leading to magnetic anisotropy. Here we derive a microscopic spin model of S=3/2 on transition metals surrounded by heavy ligands in honeycomb Mott insulators using a strong-coupling perturbation theory. For ideal octahedra we find Heisenberg and Kitaev interactions, which favor the magnetic moment along the cubic axis via quantum fluctuations. When a slight trigonal distortion of the octahedra is present together with the SOC, three additional interactions arise, comprised of the off-diagonal symmetric $\Gamma$ and $\Gamma^\prime$, and single-ion anisotropy. The resulting magnetic anisotropy pins the moment perpendicular to the honeycomb plane as observed in a single-layer of CrI$_3$, suggesting the significance of SOC and trigonal distortion in understanding magnetism of two dimensional Mott insulators. Comparison to the spin-orbit coupled $J_{\rm eff}$= 1/2 and S=1 models is also presented.
Highlights
Transition metal trihalides (TMTs) are layered materials composed of transition metals (M) and halides (X) of group 9 in a 1:3 ratio
Ferromagnetism in two-dimensional CrI3 has generated a lot of excitement, and it was recently proposed that the spin-orbit coupling (SOC) in iodine may generate bond-dependent spin interactions leading to magnetic anisotropy
We focus on the Z bond of the honeycomb (Fig. 1), as the other two bonds are related by C3 symmetry
Summary
Transition metal trihalides (TMTs) are layered materials composed of transition metals (M) and halides (X) of group 9 in a 1:3 ratio. Based on a strong-coupling perturbation theory of the generic spin model [2,3,4], it was shown that α-RuCl3 described by the effective spin Jeff = 1/2 has dominant bond-dependent Kitaev and off-diagonal symmetric interactions [5,6]. Ground state is possibly a spin liquid with gapless excitations [24] These studies were mainly of theoretical interest, until a microscopic derivation of the S = 1 Kitaev-Heisenberg model in multiorbital systems was found [25]. IV, we study the spin model with trigonal distortion present in the R3 ̄ rhombohedral lattice This includes the distortion-induced hopping matrix elements and three additional spin interactions generated via combined effects of SOC and distortion.
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