Moiré-modulated $Γ$ valley in twisted bilayer and twisted double-bilayer MoTe$_2$

Abstract

Twisted MoTe$_2$ hosts intriguing correlated quantum phenomena including the fractional quantum anomalous Hall effect in twisted bilayer (t-BL) MoTe$_2$ near 3.7$^\circ$, which is sensitive to the twist angle and moiré superlattices. Here, we directly visualize the twist-angle-modulated electronic structure of t-BL and twisted double-bilayer (t-DBL) near this critical angle. We find that the moiré superlattice not only modifies the relative energy between $Γ$ and K valleys in t-BL MoTe$_2$, but also strongly reconstructs the $Γ$ valley for both t-BL and t-DBL. Specifically, the deep $p_z$-derived band at $Γ$ exhibits a distinct splitting that systematically varies with increasing twist angle. Theoretical analysis suggests that this modulation arises from the twist-angle-dependent lattice relaxation, especially interfacial corrugations. Our work directly visualizes the moiré-modulated electronic structure and provides key spectroscopic information of lattice relaxation and interlayer interactions underlying the physics of twisted MoTe$_2$.

Figures (5)

• Figure 1: (a) Schematic of NanoARPES measurements. (b) Optical image of t-BL MoTe$_2$ with $\theta = 5.0^{\circ}$ (sample S1). (c) L-AFM image showing a moiré period of $\lambda$ = 4.1 $\pm$ 0.3 nm. (d),(e) Dispersion images measured on t-BL MoTe$_2$ (sample S1) and 1 ML MoTe$_2$ along the $\Gamma$-M direction. The calculated dispersions are over-plotted for comparison. (f),(g) Dispersion images measured along the $\Gamma$-K direction on natural 2 ML and t-BL MoTe$_2$ (sample S2, $\theta$ = 5.0$^\circ$). (h) Comparison of EDCs at the $\Gamma$ point for 2 ML and t-BL MoTe$_2$ to extract the deep energy splitting, marked by arrows in (f),(g).
• Figure 2: (a) Schematic of the t-BL MoTe$_2$ with a twist angle of $\theta=4.0^{\circ}$ (sample S3). (b) Dispersion image measured on t-BL MoTe$_2$ along the $\Gamma$-K direction. (c),(d) Dispersion images acquired under right/left circular polarization (RCP/LCP). (e),(f) Dispersion images acquired under horizontal/vertical linear polarization ($p$-pol./$s$-pol.).
• Figure 3: (a) Schematic of t-DBL MoTe$_2$ with $\theta = 4.0^{\circ}$ (sample S4). (b) Optical image of sample S4. (c)-(e) Intensity maps at energies of -1.4, -1.9 and -2.3 eV, showing the twisted region with Brillouin zones marked by red and blue hexagons. (f)-(h) Intensity maps measured on the bottom 2 ML region at the same energies as (f)-(h), with its Brillouin zone marked by blue hexagon. (i) Dispersion image measured along the $\Gamma$-K direction on the t-DBL MoTe$_2$ region. (j) Dispersion image measured on bottom 2 ML along the $\Gamma$-K direction.
• Figure 4: (a)-(d) Dispersion images measured on t-DBL MoTe$_2$ with $\theta = 0.0^{\circ}$, $4.0^{\circ}$, $10.0^{\circ}$, $63.0^{\circ}$ along the $\Gamma$-K direction. (e)-(h) Schematic summary of band structures at deep energies for different twist angles. (i) EDCs at the $\Gamma$ point to extract energy splitting of the lower $p_z$ band at different twist angles. (j) Extracted energy splitting of the deep valence bands as a function of twist angle.
• Figure 5: (a)-(d) Calculated band structure for t-DBL MoTe$_2$ with twist angles of $\theta = 0^{\circ}$, $3.89^{\circ}$, $9.43^{\circ}$, $60^{\circ}$. (e) Atomic structure of AB-AB stacked MoTe$_2$. (f)-(i) Lattice relaxation with modulated effective interlayer spacing for different twist angles. (j) Atomic structure of AB-BA MoTe$_2$, which corresponds to 60$^\circ$ t-DBL MoTe$_2$ with rhombohedral stacking.