Magnetic Anisotropy in Spin-3/2 with Heavy Ligand in Honeycomb Mott Insulators: Application to CrI$_3$

Abstract

Ferromagnetism in the two-dimensional CrI3 has generated a lot of excitement, and it was recently proposed that the spin-orbit coupling 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 the Heisenberg and Kitaev interactions, which favor the magnetic moment along the cubic axis via quantum fluctuations. When a slight distortion of the octahedra is present together with the spin-orbit coupling, three additional terms, the off-diagonal symmetric interactions Γ and Γ , and on-site Ising interactions arise. They result in the magnetic anisotropy that pins the moment perpendicular to the honeycomb plane as observed in a single layer CrI3, revealing the strong impact of octahedra enviornment on the spin model. A gap in the spin-wave spectrum and comparison to the spin-orbit coupled J eff = 1/2 and S=1 models are also presented.

Theory of magnetism in the van der Waals magnet CrI3

“…The slight deformation of this octhaedron in the actual material, is the reason behind constants Γ z and Γ xy . This is in agreement with a recent calculation that parametrizes the deformation of the octahedral environment [14]. This deviation from the NN cubic symmetry on CrI 3 has been reported and incorporated in several other works [9,43,44].…”

“…However, a similar negative Kitaev coupling was recently found in Ref. 14, where the authors treated the electronic repulsion by means of the Hubbard-Kanamori model. There is also a nonzero DM interaction on NNN, and it is one order of magnitude smaller than Kitaev interaction, but it can't be neglected because, as we will show in the next section, it significantly contributes to the gap opening at K-point.…”

“…In order to include the repulsion between electrons on the same Cr site, we use the Hubbard-Kanamori model [32,33], which has been extensively applied to describe magnetism emerging from Hund's rules in transition metal compounds [14,32,33,[38][39][40]. This model includes the repulsion U between electronic densities on the same orbital, the repulsion of electrons on different orbitals U , and an intra-atomic exchange J H , known as the Hund's coupling.…”

“…Moreover, recently, analytical calculations suggested a negative Kitaev constant instead. [14]. Heisenberg exchange J, Kitaev coupling K, and symmetric off-diagonal exchange Γ are compatible with the crystal's symmetries and have also been used to describe magnetism in honeycomb Iridium oxides [15,16].…”

“…Heisenberg exchange J, Kitaev coupling K, and symmetric off-diagonal exchange Γ are compatible with the crystal's symmetries and have also been used to describe magnetism in honeycomb Iridium oxides [15,16]. Furthermore, next-nearest neighbors (NNN) Dzyaloshinskii-Moriya interaction (DMI) it is also allowed by symmetry [17], and the component of the DM vector which is collinear to the magnetization played the same role in magnons as the SOC in the Kane-Mele's model [18] Parallel to the development of electronic and magnetic models, spin excitations on CrI 3 has been also studied on the single-layer limit [14,17,19,20], on bilayers [21] and heterostructures [22]. Most prominent results in this aspect are the anisotropy gap ∆ Γ of the lower band at Γ-point, and the K-point gap, which also has been remarked as a signature of a non-trivial topology in the magnons.…”