Microscopic origin of magnetism in monolayer $3d$ transition metal dihalides

Motivated by the recent wealth of exotic magnetic phases emerging in two-dimensional frustrated lattices, we investigate the origin of possible magnetism in the monolayer family of triangular lattice materials $M{X}_2$ ($M$=V, Mn, Ni and $X$=Cl, Br, I). We first show that consideration of general properties such as filling and hybridization enables to formulate the trends for the most relevant magnetic interaction parameters. In particular, we observe that the effects of spin-orbit coupling (SOC) can be effectively tuned through the ligand elements as the considered $3d$ transition metal ions do not strongly contribute to the anisotropic component of the intersite exchange interaction. Consequently, we find that the corresponding SOC matrix elements differ significantly from the atomic limit. In the next step and by using two ab initio based complementary approaches, we extract realistic effective spin models and find that in the case of heavy ligand elements, SOC effects manifest in anisotropic exchange and single-ion anisotropy only for specific fillings.

Figure 1

(a) Common structure of the triangular $M{X}_2$ materials, for which monolayer structures are investigated in this work. Relevant bonds, the crystallographic $\left(abc\right)$ coordinate system and the cubic $\left(xyz\right)$ coordinate system are defined. (b) Filling for ${\mathrm{VX}}_2, \mathrm{Mn}{X}_2$, and $\mathrm{Ni}{X}_2$ with $X=\text{Cl,}\text{Br,}\text{and}\text{I}$. A darker color illustrates higher filling (grey) or stronger SOC (blue). (c) Dominant hoppings $t_3$ and $\tilde{t}$ on a nearest-neighbor ${\mathrm{Z}}_1$ bond.

Figure 2

(a) Cubic crystal-field splitting (CFS) of $d$ orbitals into $t_{2g}$ and $e_g$ levels, (b) ab initio (GGA) density of states (DOS) of ${\mathrm{VCl}}_2$, (c) $p\text{−}d$ hybridization, i.e., integral of $3d$ DOS in the energy region dominated by $p\text{−}d$ bonding [illustrated by the grey box in (b)], and (d) $U_{\mathrm{avg}}$ calculated with cRPA for each $M{X}_2$ material.

Figure 3

Illustration of predominant nearest-neighbor spin-dependent hopping ${\lambda }_z$, arising from spin-orbit coupling effects on a ${\mathrm{Z}}_1$ bond (defined in Fig. 1). It results from the hopping along the upper (${\lambda }_z^{\left(u\right)}$) and lower (${\lambda }_z^{\left(l\right)}$) paths: ${\lambda }_z={\lambda }_z^{\left(u\right)}+{\lambda }_z^{\left(l\right)}$.

Figure 4

(a) $3d$-orbital MLWFs in ${\mathrm{VCl}}_6$ octahedral coordination in ${\mathrm{VCl}}_2$ monolayer. [(b) and (c)] Combination of MLWFs for $t_3$ and $\tilde{t}$ hopping paths in ferromagnetic configuration. Isosurface levels were set at $\pm 0.8a_0^{-3/2}$. Blue and red colors show opposite signs of MLWF.

Figure 5

DOS projected onto $M\text{−}d$ and Cl-$p$ orbital states for (a) ${\mathrm{NiCl}}_2$, (b) ${\mathrm{VCl}}_2$, and (c) ${\mathrm{MnCl}}_2$. Fermi level is set at energy origin.