Emergence of quasi-one-dimensional physics in a nearly-isotropic three-dimensional molecular crystal: Ab initio modeling of ${\mathrm{Mo}}_3{\mathrm{S}}_7{\left(\mathrm{dmit}\right)}_3$

We report density functional theory calculations for ${\mathrm{Mo}}_3{\mathrm{S}}_7{\left(\mathrm{dmit}\right)}_3$. We derive an ab initio tight-binding model from overlaps of Wannier orbitals; finding a layered model with interlayer hopping terms $\sim 3/4$ the size of the in-plane terms. The in-plane Hamiltonian interpolates the kagomé and honeycomb lattices. It supports states localized to dodecahedral rings within the plane, which populate one-dimensional (1D) bands and lead to a quasi-1D spin-one model on a layered honeycomb lattice once interactions are included. Two lines of Dirac cones also cross the Fermi energy.

Figure 1

(a) Crystal structure of ${\mathrm{Mo}}_3{\mathrm{S}}_7{\left(\mathrm{dmit}\right)}_3$ projected along the $c$ axis and (b) the kagomene lattice. In (a), one of the Wannier orbitals (WOs) for ${\mathrm{Mo}}_3{\mathrm{S}}_7{\left(\mathrm{dmit}\right)}_3$ is also shown. Note the predominantly dmit $\pi$-orbital character, with notable ${\mathrm{Mo}}_{3}d$-orbital character. The other six WOs are related to this one by the ${\mathcal{C}}_3$ symmetry of the molecules and the inversion symmetry relating the two molecules in each unit cell. Sulfur atoms are shown in yellow, molybdenum in purple, and carbon in brown. In (b) two plaquette states, $A_{\mathbf{R}}$ and $B_{{\mathbf{R}}^{\prime }}$, and a topologically nontrivial loop state are shown.

Figure 2

DFT and tight-binding band structures and density of states for ${\mathrm{Mo}}_3{\mathrm{S}}_7{\left(\mathrm{dmit}\right)}_3$. The left panel shows the DFT band structure (solid black) along with bands from the tight-binding models constructed from the WOs with a large hopping real space cutoff $(r_c=35$ Å, dashed green) and medium cutoff $(r_c=20$ Å, dashed blue). The $r_c=35$ Å bands lie nearly exactly on top of the DFT bands. Left inset shows the bands in a larger energy window, highlighting the gaps between the frontier bands (green) and the rest of the states (black). The right panel shows the total DOS (solid blue), the partial-DOS for the dmit groups (dashed green), and for the ${\mathrm{Mo}}_3{\mathrm{S}}_7$ core (dashed red). Right inset shows the first Brillouin zone with high symmetry points.

Figure 3

Main panel: Band structure of the kagomite lattice with $t_k,t_g$, and $t_z$ determined from the parametrization of ${\mathrm{Mo}}_3{\mathrm{S}}_7{\left(\mathrm{dmit}\right)}_3$ [cf. Table S1 (Ref. [29])]. Note that the Dirac lines remain and the two “flat” bands are now strictly one dimensional. The insets show the kagomene band structure for, from left to right, $t_k\to \infty$, the DFT values, $t_g=3t_k/2$, and $t_g\to \infty$ (in each case, the other $t$ takes the DFT values and the filling is fixed at $2/3)$. It is clear from the $t_g\to \infty$ limit that the flat bands on the kagomene lattice are adiabatically connected to the flats bands in the two copies of the kagomé lattice [30].