Moiré band theory for M-valley twisted transition metal dichalcogenides

Abstract

We propose twisted bilayers of certain group IV and IVB trigonal transition metal dichalcogenides (TMDs) MX$_{2}$ (M$=$Zr, Hf, Sn and X$=$S, Se) as moiré materials. In monolayer form these TMDs have conduction band minima near the three inequivalent Brillouin zone $M$ points and negligible spin-orbit coupling, implying six flavors of low-energy conduction band states. The flavor sectors decouple at the single-particle level and in twisted bilayers are accurately described by emergent moiré-periodic Hamiltonians that we derive from small-unit-cell density functional theory calculations. Because the valley-projected Hamiltonians have large valley-dependent mass anisotropies and are time-reversal invariant, spontaneous valley polarization is signaled in transport by anisotropy instead of by the anomalous Hall and magnetic circular dichroism signals commonly observed in graphene and $K$-valley TMD-based moiré multilayers.

Figures (4)

• Figure 1: (a) Side view of (AA-stacked) crystalline bilayer 1T-HfS$_{2}$. (b) Top view of a bilayer with displacement vector $\boldsymbol{\mathsf{d}}$. Red (blue) dots identify the positions of Hf (S) ions. (c) Band structure for $\boldsymbol{\mathsf{d}}=\boldsymbol{0}$ obtained from DFT (black) and Wannier tight-binding model(red). (d) and (e) Low-energy conduction bands from Wannier basis for $\boldsymbol{\mathsf{d}}=(\boldsymbol{a}_{1}+\boldsymbol{a}_{2})/3$ and $\boldsymbol{\mathsf{d}}=2(\boldsymbol{a}_{1}+\boldsymbol{a}_{2})/3$.
• Figure 2: (a) Intralayer potential $\Delta_{l}(\boldsymbol{M}_{1};\boldsymbol{\mathsf{d}})$ and (b) interlayer tunneling amplitude $\Delta_{T}(\boldsymbol{M}_{1};\boldsymbol{\mathsf{d}})$vs. stacking $\boldsymbol{\mathsf{d}}$ in 1T-HfS$_2$ AA-stacked crystalline bilayers. The parallelograms identify a unit cell of HfS$_2$. The interlayer tunneling is periodic in a doubled unit cell (along $\boldsymbol{a}_{1}$ for valley $\boldsymbol{M}_{1}=\boldsymbol{b}_{1}/2$) due to the $\boldsymbol{\mathsf{d}}$-dependence of the WFs discussed in the main text. The $\boldsymbol{\mathsf{d}}$-dependence of these terms is mapped to position-dependence on the moiré scale when modeling twisted bilayers.
• Figure 3: (a) Schematic illustration of the moiré BZ (black hexagon) in $t$HfS$_{2}$, where the red (green) hexagon illustrates the bottom (top) monolayer BZ. (b) Moiré ($\tilde{\boldsymbol{G}}_{i}$) and crystalline bilayer ($\boldsymbol{G}_{i}$) reciprocal lattice vectors. Points in mBZ defined relative to $\boldsymbol{M}_{1}$; $m^- = \boldsymbol{0}$, $m^+ = \tilde{\boldsymbol{M}}_{1}$, $\gamma =\tilde{\boldsymbol{G}}_1/2$, $\kappa = \gamma - (\tilde{\boldsymbol{G}}_1 + \tilde{\boldsymbol{G}}_2)/3$. (c) $\boldsymbol{M}_{1}$-valley moiré conduction bands for twist angle $\theta = 5^{\circ}$. (d) Contour plot of the lowest energy moiré band for $\theta = 5^{\circ}$ and (e) its bandwidth for various values of $\theta$ (blue) compared to those in a hypothetical twisted bilayer with isotropic mass equal to the light $m^{\boldsymbol{M}_{1}}_{\perp,l}$. The interaction energy scale $e^2/\epsilon a_M$ calculated with $\epsilon = 3.8,6.9,15$, for the out-of-plane and in-plane dielectric constants of bulk hBN and the out-of-plane HfS$_2$ dielectric constant, respectively, are plotted as dashed black curves.
• Figure 4: Spatial distribution of electron density in the lowest energy conduction band's moiré Bloch eigenfunction in $t$HfS$_{2}$ for various moiré BZ momenta and twist angles. Parallelograms identify a moiré unit cell in each case.