Trigonal symmetry breaking and its electronic effects in the two-dimensional dihalides MX2 and trihalides MX3

Abstract

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 $3\phantom{\rule{0.16em}{0ex}}{t}_{2g}$ 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 $p\text{\ensuremath{-}}d$ 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 $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$.