Effective K valley Hamiltonian for TMD bilayers under pressure and application to twisted bilayers with pressure-induced topological phase transitions

TL;DR

This work addresses how perpendicular pressure tunes the topological properties of twisted MoTe$_2$ bilayers. It builds a symmetry-based, low-energy Hamiltonian near the $K$ valley and fits pressure-dependent couplings to ab initio DFT data, then applies the framework to twisted bilayers by treating twist as a position-dependent interlayer shift. The key contributions are explicit pressure-dependent expressions for intra- and interlayer couplings, demonstration of pressure-induced valley Chern-number transitions, and identification of pressure ranges that flatten moiré bands. The findings offer a practical, non-interacting tool to study and tune topological phases in TMD moiré systems under pressure, with potential for in situ control of electronic topology and band structure.

Abstract

Motivated by recent studies on topologically non-trivial moiré bands in twisted bilayer transition metal dichalcogenides (TMDs), we study MoTe$_2$ bilayer systems subject to pressure, which is applied perpendicular to the material surface. We start our investigation by first considering an untwisted bilayer system with an arbitrary relative shift between layers; a symmetry analysis for this case permits us to obtain a simplified effective low-energy Hamiltonian valid near the important $\mathbf{K}$ valley region of the Brillouin zone. Ab initio density functional theory (DFT) was then employed to obtain relaxed geometric structures for pressures within the range of 0.0 - 3.5 GPa and corresponding band structures. The DFT data were then fitted to the low-energy Hamiltonian to obtain a pressure-dependent Hamiltonian. We then apply our model to a twisted system by treating the twist as a position-dependent shift between layers - here, we assume rigid layers, which is a crucial simplification. In summary, this approach allowed us to obtain the explicit analytical expressions for a Hamiltonian that describes a twisted MoTe\textsubscript{2} bilayer under pressure. Our Hamiltonian then permitted us to study the impact of pressure on the band topology of the twisted system. As a result, we identified many pressure-induced topological phase transitions as indicated by changes in valley Chern numbers. Moreover, we found that pressure could be employed to flatten bands in some of the cases we considered.

Figures (17)

• Figure 1: Illustration of the MoTe2 homobilayer in (a) AA and (b) AB/BA stacking configurations
• Figure 2: (Right) DFT band structures of MoTe2 bilayer for $\mathbf{d}_{0}$ displacement and $\mathbf{d}_{1}$ displacement. The two topmost valence bands (the red lines) close to the $\mathbf{K}$ valley have a parabolic shape as indicated by a circle we used to mark them. The energy has been shifted to place zero energy between the two topmost valence bands. (Left) The Brillouin zone and high-symmetry paths for MoTe2 homobilayer.
• Figure 3: The reciprocal lattice vectors $\mathbf{G}_j$ connect point $\mathbf{K}$ to the nearest equivalent $\mathbf{K}$ points.
• Figure 4: Illustration of the moiré Brillouin zone (mBZ), which was constructed by rotating the top and bottom layer Brillouin zone by $+\theta/2$ and $-\theta/2$, respectively, and setting the $\mathbf{K}$ valley of each BZ as valleys of the newly constructed mBZ named as $\boldsymbol{\kappa}_-$ and $\boldsymbol{\kappa}_+$. The black dot indicates the original position of $\mathbf{K}$ for the untwisted BZ.
• Figure 5: Illustration of point $p$ in the moiré Brillouin zone as measured from $\boldsymbol{\gamma}$ and $\boldsymbol{\kappa}_-$.