Pressure-tuned many-body phases through $Γ$-K valleytronics in moiré bilayer WSe$_2$

TL;DR

Problem: understanding correlated phases in twisted bilayer WSe$_2$ where valley degrees of freedom ($Γ$ and $K$ valleys) compete. Approach: combine ab initio DFT with Wannierization and constrained RPA to build a moiré-scale Hubbard model; apply self-consistent Hartree–Fock to map the interacting phase diagram under uniaxial pressure. Key findings: interlayer distance $d$ tunes $Δ_{ΓK}$ and $w_Γ$, enabling a pressure-driven valley-transfer, yielding antiferromagnetic charge-transfer insulators, Mott-Hubbard insulators, and Kondo lattice-like metals; Γ-valley bands become flat under pressure, and no topological moiré bands are found. Significance: provides a concrete, testable route to engineer and study diverse correlated states in moiré TMDs via pressure and valley physics, linking ab initio data to many-body phenomenology and guiding experiments.

Abstract

Recent experiments in twisted bilayer transition-metal dichalcogenides have revealed a variety of strongly correlated phenomena. To theoretically explore their origin, we combine here ab initio calculations with correlated model approaches to describe and study many-body effects in twisted bilayer WSe$_2$ under pressure. We find that the interlayer distance is a key factor for the electronic structure, as it tunes the relative energetic positions between the K and the $Γ$ valleys of the valence band maximum of the untwisted bilayer. As a result, applying uniaxial pressure to a twisted bilayer induces a charge-transfer from the K valley to the flat bands in the $Γ$ valley. Upon Wannierizing moiré bands from both valleys, we establish the relevant tight-binding model parameters and calculate the effective interaction strengths using the constrained random phase approximation. With this, we approximate the interacting pressure-doping phase diagram of WSe$_2$ moiré bilayers using self-consistent mean field theory. Our results establish twisted bilayer WSe$_2$ as a platform that allows the direct pressure-tuning of different correlated phases, ranging from Mott insulators, charge-valley-transfer insulators to Kondo lattice-like systems.

Figures (5)

• Figure 1: a. Top view of the six possible stacking of untwisted bilayer WSe$_2$. Red corresponds to the W atoms, green to the Se atoms; bright colors are the top layer, faded colors the bottom layer. The three parallel (P) stackings are obtained by starting with the XX stacking and performing a lateral shift of one layer; the three antiparallel (AP) stackings are obtained from the parallel ones by rotating one layer by 180 degrees. b. The electronic structure of untwisted bilayer WSe$_2$ is characterized by the valence band maxima at $\Gamma$ and K. Here we show an illustrative band structure for the XX$_{\rm P}$ stacking, indicating the valley offset $\Delta_{\Gamma {\rm K}}$, the interlayer splittings $w_{\Gamma/K}$ and the spin-orbit coupling $\lambda_{\rm SOC}$. c. The interlayer distance $d$, defined as the W-W distance along the $z$-direction, is the dominant physical parameter that determines the electronic structure and can be tuned by applying uniaxial pressure. Here we show the interlayer distance as a function of applied pressure. d. A decrease in interlayer distance leads to an increase in interlayer hopping $w$ at the $\Gamma$-point between W $d_{z^2}$ orbitals. e. Finally, the application of pressure strongly influences the valley offset $\Delta_{\Gamma {\rm K}}$. Even though the precise value is hard to predict, here we show the tendency that for different stackings, the $\Delta_{\Gamma {\rm K}}$ increases with increasing pressure and thus with decreasing interlayer distance $d$.
• Figure 2: a. The parallel moiré structure, obtained from untwisted XX$_{\rm P}$ stacking and applying a small twist angle, consists of a hexagonal structure with locally XX$_{\rm P}$, XM$_{\rm P}$ and MX$_{\rm P}$ stacking. Note that the XM$_{\rm P}$ and MX$_{\rm P}$ regions are symmetry-related. b. The antiparallel moiré structure consists of a hexagonal structure with locally MM$_{\rm AP}$, 2H$_{\rm AP}$ and XX$_{\rm AP}$ stacking. There is no symmetry relation between the different regions of the moiré unit cell. c/d/e. By tracing the local valence band maxima at $\Gamma$ or K at the different stackings in the moiré unit cell, we can extract the moiré potential following Eqs. \ref{['eq:moire_potential']} and \ref{['eq:moire_potential2']}. Here we show the dependence of the moiré potential amplitude at ${\Gamma}$ (c) and K (d) and the moiré potential phase (e) on applied uniaxial pressure. Note that the moiré potential at $\Gamma$ is an order of magnitude larger than at K. f/g. The electronic band structure of parallel moiré structures at zero uniaxial pressure, throughout the mini-Brillouin zone, for a twist angle $\theta = 3^{\circ}$. The band structure at K (f) is barely affected by the moiré potential and does not lead to the opening up of significant moiré gaps. On the other hand, there appear isolated moiré flat bands at ${\Gamma}$ (g) with honeycomb symmetry, consistent with ARPES observations Gatti.2023. h/i. The electronic band structure of antiparallel moiré structures at zero uniaxial pressure, throughout the mini-Brillouin zone, for a twist angle $\theta = 3^{\circ}$. Similar to the case of parallel moiré structures, the band structure at K (h) shows no signs of moiré gaps. The isolated subset of moiré flat bands at ${\Gamma}$ (i) has a triangular lattice symmetry. j. The bandwidth of moiré flat bands at the top of the $\Gamma$ and K valley in parallel and antiparallel moiré structures. Most notably are the $\Gamma$-states in the antiparallel system, which attain vanishingly small bandwidth under pressure.
• Figure 3: a. Visualization of the Bloch wavefunction in the $\Gamma$ valley for antiparallel twisted bilayers, shown as the absolute value squared summed over both layers of the wavefunction. Model parameters are $\theta = 3^\circ$ and $P = 0$ GPa. The electronic states is localized in the MM$_{\rm AP}$ region of the moiré unit cell. b. Same as a, but for parallel twisted bilayers. We clearly see the emergence of a honeycomb lattice on the XM$_{\rm P}$/MX$_{\rm P}$ positions. c. Same as a, but for the K valley states in an antiparallel twisted bilayer. The states are not as localized as in the $\Gamma$ valley, and are centered around the XX$_{\rm AP}$ position. d. Same as c, but for the parallel twisted bilayers. There is less localization compared to the $\Gamma$ valley, and the states are centered at XX$_{\rm AP}$. e/f. The nearest-neighbor $t$ and next-nearest-neighbor $t'$ tight-binding hopping parameters at $\theta = 3^\circ$ as a function of pressure, for the triangular lattice models relevant for antiparallel twisted bilayers and the K valley of parallel bilayers; as well as for the honeycomb lattice model relevant for the parallel bilayer $\Gamma$ valley states.
• Figure 4: a. Qualitatively, we can already identify various many-body phases that are expected in tbWSe$_2$ under pressure. Here, we visualize six possible phases by plotting a cartoon density of states versus energy. At zero or low pressure, the charge carriers will fill up states in the weakly correlated K valley (bottom row), leading to metallic behavior. At half-filling (left column), an increase in pressure leads to a charge-transfer to the strongly correlated $\Gamma$ valley. The $\Gamma$ states will split into a lower and upper Hubbard band (LHB/UHB), yielding an insulator state. Depending on the position of the remaining K valley states, this is a charge-transfer (CT) insulator at intermediate pressures, or a Mott-Hubbard insulator at high pressures. Increasing the hole doping beyond half-filling (right column) now yields either a Kondo lattice model at intermediate pressures, or a doped Mott insulator at high pressures b/c. The qualitative picture is confirmed by numerical mean field theory calculations. Here we show the resulting phase diagram for antiparallel (b) and parallel (c) stacking at twist angle $\theta = 3^\circ$ and dielectric constant $\epsilon_0 = 30$, as a function of hole density $\nu$ and pressure $P$. We confirm that at a critical pressure the electronic charge is fully shifted from the K to the $\Gamma$ valley. At half-filling of the $\Gamma$ valley states ($\nu =1$ for antiparallel stacking, $\nu = 2$ for parallel stacking), applying pressure induces a transition from a conducting K state to an antiferromagnetic $\Gamma$ charge-transfer insulator (C-T Ins.), followed by a transition to a Mott-Hubbard insulator (M-H Ins.). Increasing the hole density $\nu$ in the charge-transfer insulator leads to a Kondo lattice regime, with localized electrons in the $\Gamma$ valley and conduction electrons in the K valley.
• Figure 5: The density of states in the Kondo lattice regime of the phase diagrams of Fig. \ref{['fig:phasediagram']}, for antiparallel ( a) and parallel ( b) stacking. In both cases the strong correlations of the $\Gamma$ states at half-filling lead to the formation of lower and upper Hubbard bands. Upon doping away from half-filling, the Fermi level ($E=0$) crosses the K valley states that are lying within the gap of the $\Gamma$ states. The result is a system with localized electrons in the $\Gamma$ valley and conduction electrons in the K valley.