Trigonal symmetry breaking and its electronic effects in the two-dimensional dihalides $M{X}_2$ and trihalides $M{X}_3$

We study the consequences of the approximately trigonal ($D_{3d}$) point symmetry of the transition metal ($M$) site in two-dimensional van der Waals ${MX}_2$ dihalides and ${MX}_3$ trihalides. The trigonal symmetry leads to a 2-2-1 orbital splitting of the transition metal $d$ shell, which is best represented by $d$ orbitals that describe the trigonal rather than $O_h$ symmetry. The ligand-ligand bond length differences (rather than metal-ligand) take the role of a Jahn-Teller-like mode and in combination with interlayer distances and dimensionality effects tune the crystal field splittings and bandwidths. These effects, in turn, are amplified by electronic correlation effects—leading to crystal field splittings of the order of 0.1–1 eV between the singlet and the lowest orbital doublet—as opposed to the often assumed degenerate $3t_{2g}$ states. Our calculations explain why most of the materials in this family are insulating, and why these considerations have to be taken into account in realistic models of them. Further, orbital order coupled to various lower symmetry lattice modes may lift the remaining orbital degeneracies, and we explain how these may support unique electronic states using ${\mathrm{ZrI}}_2$ and ${\mathrm{CuCl}}_2$ as examples, and offer a brief overview of possible electronic configurations in this class of materials. By building and analyzing Wannier models adapted to the appropriate symmetry we examine how the interplay among trigonal symmetry, electronic correlation effects, and $p\text{−}d$ orbital charge transfer leads to insulating, orbitally polarized magnetic and/or orbital-selective Mott states, and we provide a simple framework and numerical tool for others. Our paper establishes a rigorous framework to understand, control, and tune the electronic states in low-dimensional correlated halides. Our analysis shows that trigonal symmetry and its breaking is a key feature of the two-dimensional halides that needs to be accounted for in search of novel electronic states in materials ranging from ${\mathrm{CrI}}_3$ to $\alpha \text{−}{\mathrm{RuCl}}_3$.

Figure 1

(a) Sequential illustration of symmetry breaking as a result of the layered structure in dihalides and trihalides. Placing an ideal octahedron within a monolayer will reduce its $O_h$ symmetry to trigonal $D_{3d}$ symmetry, for a dihalide octahedron, and $D_3$ for a trihalide: In both cases, threefold rotations along the $z$ axis are preserved, as well as inversion symmetry. Three mirror symmetry planes are preserved in the dihalides, but not the trihalides, due to the different octahedral connectivity. Sketch of (b) tetragonal and (c) trigonal symmetry breaking of an octahedron. Heavy magenta arrows indicate the vector normal to monolayer planes. Layer structure of ${MX}_6$ octahedra for (d) ${MX}_2$ dihalides and (e) ${MX}_3$ trihalides. Definitions for halide-halide lengths for octahedra in (f) ${MX}_2$ and (g) ${MX}_3$ halides. The red edges with length $L_1$ surround the face parallel to the monolayer planes; $L_2$ are edges that are not parallel to the material plane and that connect nearest-neighbor ${MX}_6$ octahedra. $L_3$ edges do not connect octahedra, are not parallel to the plane, and only appear in ${MX}_3$ compounds.

Figure 2

Electronic orbital splitting at the nonmagnetic DFT level for ${\mathrm{TiCl}}_2$ as a function of (a) Ti-Ti interlayer distance and (b) $L_2/L_1$ ratio, as quantified by crystal field splitting at the $\mathrm{\Gamma }$ point and orbital occupancy as obtained from the density matrix of the orbital projections. In (a), the octahedron is kept perfect ($L_2=L_1$), with a $X\text{−}X$ distance of 3.43 Å. In (b), the interlayer distance is kept fixed to the experimental value, while $L_2/L_1$ is varied. The experimental interlayer distance and $L_2/L_1$ ratio are indicated with broken magenta lines.

Figure 3

Projected density of states for the Wannier functions for ${\mathrm{TiCl}}_2$ in the tetragonal basis (left) and trigonal basis (center), and eigenstates of the density matrix, as obtained by diagonalizing the density matrix (right). The crystal field splittings shown here are greatly increased in the presence of correlations and lead to a band gap opening in the presence of magnetism and correlations.

Figure 4

(a) $\vec{R}$ connecting transition metals $M$, for which we show the hopping matrices of the Hamiltonian below in (b). The diagonal elements are shown in blue for $e_g^{\sigma }$, in green for $a_{1g}$, and in red for $e_g^{\pi }$.

Figure 5

Wannier Hamiltonian at $\vec{R}=(0,0,0)$ either after diagonalizing the on-site Hamiltonian or as obtained directly from wannier90 using an $e_g\text{−}{t}_{2g}$ model.

Figure 6

Single layer lattice structure for materials with point group symmetry $C_{2h}$, which breaks the degeneracy of the doublets: (a) a layer of ${MX}_2$, with the structure from a layer of bulk ${\mathrm{ZrI}}_2$, where the black rectangle represents the new unit cell, and (b) a layer of ${MX}_3$, with a similar pattern representing bulk layers, for example, ${\mathrm{RuCl}}_3$ and ${\mathrm{CrI}}_3$ with the ${\mathrm{AlCl}}_3$ structure type; all atoms in the sketch are within one unit cell. Distances between the atoms are indicated as follows: The red dotted lines are the longest, followed by purple, blue, then mustard yellow. The distortions characteristic of ${\mathrm{ZrI}}_2$ are much stronger than those of the ${MX}_3$ materials. The $M$ site symmetries are $C_{1h}$ for panel (a) and $C_2$ for panel (b). No $d$ orbital degeneracies are enforced by either site symmetry, which leads to new possible states (e.g., those shown in Fig. 10).

Figure 7

DFT band structure and Wannier “fat bands” for a full $p\text{−}d$ model for (left) ${\mathrm{TiCl}}_2$ and (right) ${\mathrm{NiI}}_2$, exemplifying materials that are in the Mott-Hubbard (${\mathrm{TiCl}}_2$) and charge-transfer (${\mathrm{NiI}}_2$) regimes. Blue circles correspond to transition metal $M$ $d$ state projections; red squares correspond to ligand $p$ states.

Figure 8

Electronic band dispersions and orbital projected density of states for ferromagnetic, high-spin ${\mathrm{TiCl}}_2$ at the $\mathrm{DFT}+U=2$ eV level: (a) experimental structure and (b) structure with $L_2$ extended to 4.02 Å and interlayer distance kept constant. Left: Bands, majority spin in magenta, minority spin in cyan. Right: Projected density of states (PDOS), minority spin rescaled by dividing by 5. Increasing $L_2/L_1$ closes the gap by reducing the orbital polarization of the lower three orbitals. (c) Dependence of the electronic band gap in ${\mathrm{TiCl}}_2$ with $L_2/L_1$ ratio for a fixed interlayer distance.

Figure 9

On-site $\vec{R}=(0,0,0)$ Hamiltonian for the majority (spin up) channel for ${\mathrm{TiCl}}_2$ for $U=2$ eV and FM state, obtained from wannier90 followed by a rotation that diagonalizes the $d$ orbital density matrix, similar to Fig. 4. Note that the Hamiltonian is nearly diagonal, and the crystal field splittings are strongly amplified by correlations.

Figure 10

Different possible electronic states for $d^1\text{–}{d}^9$ occupations as discussed in the main text. Configurations that break the doublet degeneracy would be associated with further lattice distortions, reducing symmetry below trigonal, including structural modes as described in Fig. 6. States with higher $a_{1g}/e_g^{\pi }$ occupation ratios are likely to have a higher $L_2/L_1$ ligand bond ratio.

Figure 11

(a) DFT band structure, and fat bands of the top two Wannier orbitals on the hypothetical ${\mathrm{CuCl}}_2$ monolayer discussed in the main text. (b) Real-space isosurfaces of the highest energy $d$ orbitals, with occupations 0.912 and 0.93 per spin channel. Due to the pseudo-1D nature of the symmetry broken lattice structure, there is relatively little dispersion along the $\mathrm{\Gamma }\text{−}X$ direction.