Chern-Selective multi-valley Flat Bands in Twisted Mono-Bilayer and Mono-Trilayer MoTe$_2$

TL;DR

This work reveals that twisted A-AB and A-ABA MoTe$_2$ host low-energy moiré flat bands arising from both $\Gamma$ and $K/K'$ valleys, with $C_s=0$ for $\Gamma$ and $C_{\uparrow/\downarrow}=\pm1$ for $K/K'$, enabling true multi-valley physics. By combining first-principles DFT with fitted and universal accurate continuum models, the authors show that interlayer-hybridization governs the valley-resolved bands and that layer number, stacking, and displacement fields tune valley energy alignment, Berry curvature, and quantum geometry. They construct valley-resolved continuum models and Wannier-based tight-binding representations that accurately reproduce DFT bands, and identify distinct Berry curvature and charge-density patterns across valleys and twist angles. The results provide a robust framework for exploring correlation-driven, valley-controlled phases (e.g., valley-charge-transfer insulators and valley-selective FCIs) and motivate studying other twisted multilayer TMDs to harness layer- and valley-degree freedom in moiré materials.

Abstract

The interplay between moiré flat bands originating from different valleys can give rise to a variety of exotic quantum phases. In this work, we investigate the electronic properties of twisted mono-bilayer (A-AB) and mono-trilayer (A-ABA) MoTe$_2$ using first-principles calculations and continuum models. Unlike previous studies on twisted bilayer systems, in which low-energy flat bands originate solely from the $K/K'$ valleys, in A-AB and A-ABA twisted MoTe$_2$ (\tmt) the moiré bands at low energies arise from both the $Γ$ and $K/K'$ valleys, with spin Chern numbers $C_s=0$ (for $Γ$) and $C_{\uparrow/\downarrow}=\pm1$ (for $K/K'$), respectively. We show that the multi-valley moiré flat bands are governed by interlayer-hybridization effects, and that different stacking configurations and thicknesses tune the relative energy alignment between the $Γ$ and $K$ valley moiré flat bands. By constructing valley-resolved continuum models and performing Wannierization for the low-energy moiré bands, we further uncover that the Berry curvature and quantum metric distributions can be effectively tuned by the layer number and stacking configuration. Unlike other moiré systems, where only one kind of valley influenced the low energy physics, the simultaneous appearance of two distinct types of valleys, with different symmetries, establish A-AB and A-ABA \tmt\ as ideal platforms for studying layer-controlled multi-valley physics.

Figures (33)

• Figure 1: Schematic diagrams of the interlayer-coupling mechanism at the $\Gamma$ valley ($d_{z^2}$,colored red) and the $K$ valley ($d_{x^2-y^2}\pm id_{xy}$ , colored blue/green) in untwisted multilayer MoTe$_2$: (a) AA bilayer, (b) AAB trilayer, (c) AABA tetralayer. To illustrate the trend, we label the energy shift of the maximum energy (upper solid lines) at $\Gamma$ and $K$ in each multilayer system relative to the monolayer case (dashed) without interlayer coupling. $t_\Gamma$ and $t_K$ denote the interlayer coupling strength of $d_{z^2}$ and $d_{x^2-y^2}+id_{xy}$ between adjacent layers, respectively. It satisfies $t_\Gamma > t_K$.
• Figure 2: Atomic and electronic structures of untwisted multilayer MoTe$_2$ with (red lines) and without (black lines) lattice relaxation. The calculations include SOC effects. (a)-(d) show the band structures of monolayer, AA bilayer, AAB trilayer and AABA tetralayer MoTe$_2$. The green numbers denote the energy difference between the maximum energies at $\Gamma$ and $K$ valley in relaxed band structures (red). The insets in (a)-(d) exhibit the atomic structures viewed along the $x$-axis. Lattice relaxation is performed using the Grimme DFT-D2 van der Waals corrections (IVDW=10) implemented in VASP.
• Figure 3: Moiré bands and irreducible representations at high-symmetry points $\Gamma_M$ and $K_M$. (a)-(c) are A-AB $t$MoTe$_2$, and (d)-(e) are A-ABA $t$MoTe$_2$ with twist angles ranging from 5.09$^{\circ}$ to 3.89$^{\circ}$. We use $\xi=e^{i\pi/3}$, $\xi^*=e^{-i\pi/3}$, and $\overline{1}=-1$ to represent the spinful $C_3$ eigenvalues. $C_3$ eigenvalue at $K'_M$ is related to that at $K_M$ by $\mathcal{T}$ symmetry as $\xi_{(K_M',\uparrow)}^{(n)}=\xi_{(K_M,\downarrow)}^{*(n)}$. Green and red lines represent the top two valence bands from the $K/K'$ and $\Gamma$ valley, respectively. The low-energy $\Gamma$-valley bands are nearly spin-degenerate due to Kramers' degeneracy at the $\Gamma$ valley.
• Figure 4: (a) and (b) show the continuum-model fitting to DFT bands calculated from VASP for A-AB and A-ABA tMoTe$_2$, (c) and (d) present the moiré bands calculated from accurate continuum models at different valleys and DFT bands calculated from OpenMX for A-AB and A-ABA $t$MoTe$_2$. The twist angle is $3.89^\circ$. Black dot lines are DFT bands, while the other solid lines are the moiré bands calculated by continuum models. Red lines are $\Gamma$ valley bands , blue/green lines are $K/K'$ valley bands of A-layers, and orange lines are $K/K'$ valley bands of B-layer. Spin Chern numbers of the several low-energy bands are labeled in (a)-(d). (e)-(h) show the Berry curvature distribution of the top moiré bands from the $K$ valley calculated by corresponding continuum models in (a)-(d).
• Figure 5: (a) and (b) show the Wannier function distributions $|\left<{\textbf{r}}|{W^K_n(\textbf{0})}\right>|^2$ of the two orbitals corresponding to the top two bands at the $K$ valley in A-ABA $t$MoTe$_2$, which are extracted from the accurate continuum model with a twist angle of $3.89^\circ$ and localized at the M-XMX and X-MXM regions, respectively. (c)and (d) show the comparison of energy bands calculated from the tight-binding models (black) and accurate continuum models at $K$ (blue) and $\Gamma$ (red) valley with twist angle $3.89^\circ$ respectively, which exhibit excellent consistency.