Origin of Moiré Potentials in WS$_2$/WSe$_2$ Heterobilayers: Contributions from Lattice Reconstruction and Interlayer Charge Transfer

TL;DR

This work identifies and quantifies the multiple microscopic mechanisms generating moiré potentials in WS$_2$/WSe$_2$ heterobilayers, considering both R-type and H-type patterns. By combining lattice-reconstruction effects (local strain and piezoelectric contributions) with interlayer charge transfer, the authors determine energy modulations for conduction and valence band edges that explain miniband localization: conduction bands show strong trapping (≈200 meV from strain and ≈80 meV from stacking-induced fields in R-type; ≈173 meV and ≈83 meV in H-type), while valence bands exhibit smaller, smoother modulations. The analysis shows that interlayer charge transfer introduces a built-in field that shifts band edges differently across layers, and that the resulting excitonic moiré potential Ex = V$_c^T$ − V$_v^T$ can present double minima, consistent with moiré exciton observations. Overall, the work provides a comprehensive framework for understanding moiré potentials in these heterobilayers and their relevance to flat bands and correlated states.

Abstract

Moiré superlattices formed in WS$_2$/WSe$_2$ heterobilayers have emerged as an exciting platform to explore the quantum many-body physics. The key mechanism is the introduction of moiré potentials for the band-edge carriers induced by the lateral modulation of interlayer interactions. This trapping potential results in the formation of flat bands, which enhances the strong correlation effect. However, a full understanding of the origin of this intriguing potential remains elusive. In this paper, we present a comprehensive investigation of the origin of moiré potentials in both R-type and H-type moiré patterns formed in WS$_2$/WSe$_2$ heterobilayers. We show that both lattice reconstruction and interlayer charge transfer contribute significantly to the formation of moiré potentials. In particular, the lattice reconstruction induces a nonuniform local strain, which creates an energy modulation of 200 meV for the conduction band-edge state located at WS$_2$ layer and 20 meV for the valence band-edge state located at WSe$_2$ layer. In addition, the lattice reconstruction also introduces a piezopotential energy, whose amplitude ranges from 40 meV to 90 meV depending on the stacking and band-edge carrier. The interlayer charge transfer induces a built-in electric field, resulting in an energy modulation of 80 meV for an R-type moiré and 40 meV for an H-type moiré. Taking into account both effects from lattice reconstruction and interlayer charge transfer, the formation of moiré potential is well understood for both R-type and H-type moirés. This trapping potential localizes the wavefunctions of conduction and valence bands around the same moiré site for an R-type moiré, while around different moiré site for an H-type one.

Figures (8)

• Figure 1: (a) Relaxed atomic structure of an R-type moiré pattern formed in WS$_2$/WSe$_2$ heterobilayers. The top panel is a lateral view showing strong out-of-plane corrugation. The three high-symmetry locales along the long-diagonal of a moiré unit cell are labeled as $R^h_h$, $R^X_h$, and $R^M_h$. The moiré periodicity is 8.25 nm formed by a lattice mismatch of $4\%$ between WS$_2$ and WSe$_2$ with no relative rotation. (b) Adhesion energy as a function of interlayer stacking along the long diagonal of the relaxed moiré pattern in (a). The interlayer atomic configurations at three high-symmetry locales are given in the lower panel. (c) The extracted lattice-reconstruction induced in-plane strain distribution in the WSe$_2$ layer (upper) and WS$_2$ layer (middle). The interlayer distance $d$ is given in the lower panel, which is defined by the distance between the two W atomic planes. (d) Out-of-plane corrugation of WSe$_2$ layer (upper) and WS$_2$ layer (middle) and the interlayer distance (lower) along the long diagonal of the relaxed moiré pattern. (e)-(h) The same as (a)-(d) for the an H-type moiré pattern, where the upper WSe$_2$ layer has a rotation of 180 degree relative to the lower WS$_2$ layer.
• Figure 2: (a) Electronic band structure of a relaxed R-type WS$_2$/WSe$_2$ moiré pattern along the high-symmetry lines in the hexagonal mini-Brillouin zone (inset). The projected orbital contributions are indicated. The distribution of the wavefunctions in the moiré unit cell for the conduction (b) and valence (c) band-edge states at the energy points marked by the pentagons in (a), both of which are localized around $R^M_h$ site. The top panels show the lateral views.
• Figure 3: (a) Electronic miniband structure of a relaxed H-type WS$_2$/WSe$_2$ moiré pattern along the high-symmetry lines in the hexagonal mini-Brillouin zone. The projected orbital contributions are indicated. The distribution of the wavefunctions in the moiré unit cell for the conduction (b) and valence (c) band-edge states at the energy points marked by the pentagons in (a), which are localized around $H^h_h$ site and $H^X_h$ site respectively. The top panels show the lateral views.
• Figure 4: (a) Layer-projected band structure of a WS$_2$/WSe$_2$ heterobilayer. The type-II band edges are located at K point. A commensurate $R^X_h$ stacking is adopted. (b) Modulation of conduction band-edge energy of monolayer WS$_2$ ($V^{RS}_c$ as black line with dots) and valence band-edge energy of monolayer WSe$_2$ ($V^{RS}_v$ as red line with diamonds) as a function of local strain adopted along the long diagonal of Fig. \ref{['fig1']}(c).
• Figure 5: (a)(b) Lattice-reconstruction-induced piezocharge densities $\rho_{\text{piezo}}$ in the (a) WS$_2$ and (b) WSe$_2$ layer respectively. (c), (d) Piezopotential energy induced by piezoelectric charge shown in (a) and (b) with the screening effect considered.