Spin-orbit coupling in ${\mathrm{Mo}}_3{\mathrm{S}}_7{\left(\mathrm{dmit}\right)}_3$

Spin-orbit coupling in crystals is known to lead to unusual direction-dependent exchange interactions; however understanding of the consequences of such effects in molecular crystals is incomplete. Here we perform four-component relativistic density functional theory computations on the multinuclear molecular crystal ${\mathrm{Mo}}_3{\mathrm{S}}_7$(dmit)${}_3$ and show that both intra- and intermolecular spin-orbit coupling are significant. We describe a powerful Wannier spin-orbital construction technique and use it to determine a long-range relativistic single-electron Hamiltonian from first principles. We analyze the various contributions to this Hamiltonian through the lens of group theory, building on our previous work. Intermolecular spin-orbit couplings like those found here are known to lead to quantum spin-Hall and topological insulator phases on the 2D lattice formed by the tight-binding model predicted for a single layer of ${\mathrm{Mo}}_3{\mathrm{S}}_7$(dmit)${}_3$.

Figure 1

The two-dimensional lattice of ${\mathrm{Mo}}_3{\mathrm{S}}_7$(dmit)${}_3$, known as the decorated honeycomb lattice, the star lattice, and the kagomene lattice; the latter since it interpolates between the kagome lattice and the honeycomb lattice. The kagome-like hopping (black) is labeled $t_k$, while the graphene-like hopping (green) is labeled $t_g$. The full 3D lattice stacks layers of the kagomene lattice directly on top of one another in the $z$ direction. An example of the in-plane next-nearest-neighbor hopping $t_h$ (red) is also indicated; this hopping is chiral (it preserves inversion symmetry between the pair of molecules while breaking the reflection symmetry, as there is no reflection plane in the crystal).

Figure 2

A pair ${\mathrm{Mo}}_3{\mathrm{S}}_7$(dmit)${}_3$ molecules, with molybdenum atoms in purple, carbon in brown, and sulfur in yellow. The canting of the dmit ligands and the vertical asymmetry of the ${\mathrm{Mo}}_3{\mathrm{S}}_7$ core means that this molecule has ${\mathcal{C}}_3$ symmetry; it is symmetric only under rotations by $2\pi /3$. The molecules of this pair are related by an inversion center, mapping site $i$ to $i+3$.

Figure 3

One of the twelve Wannier spin orbitals (per unit cell) of the ${\mathrm{Mo}}_3{\mathrm{S}}_7$(dmit)${}_3$ crystal. The three Kramers pairs per molecule are related to each other by the ${\tilde{\mathcal{C}}}_3$ symmetry of the molecule, and the two molecules per unit cell are related by an inversion center between them. The real part of the spin-up component of one of the spin-up Wannier orbitals is shown (the other components are orders of magnitude smaller).

Figure 4

Comparison of the four-component DFT (green) with our model including SMOC (dashed red), showing very good agreement; on the right the same data are shown in a narrower energy window, with the addition of the scalar relativistic DFT (solid blue) and the model including all computed SO terms (dashed black) up to a cutoff radius of 35 Å, which shows excellent agreement. Spin-orbit coupling lifts the degeneracies at the Dirac ($K$) and $\mathrm{\Gamma }$ points, splitting the bands into Kramers pairs. This first-principles parametrization of ${\widehat{H}}_{SMO}$ captures all of the qualitative features introduced by relativistic effects in a very simple model. Including intermolecular SOC up to 35 Å gives excellent agreement with the DFT. ${\widehat{H}}_{SMO}$ is parametrized from the Wannier overlaps, giving ${\lambda }^z=4.91$ meV = $0.082t_k, {\lambda }^{xy}=2.50$ meV = $0.042t_k$. ${\widehat{H}}_0$ is the first-principles tight-binding model produced from scalar relativistic DFT with $r_c=35\AA$, and is in excellent agreement with the scalar relativistic DFT electronic structure [17].