Atomic displacements drive flat band formation and lateral electron and hole separation in near-60 degree twisted MoSe2/WSe2 bilayers

Abstract

Transition metal dichalcogenide (TMD) bilayers with an interlayer twist exhibit a moire super-period, whose effects can manifest in both structural and electronic properties. Atomic displacements can lead to reconstruction into domains of aligned stacking, and flat bands can form that may host correlated electron states. In heterobilayers angular mismatch is nearly unavoidable, so understanding the consequences of an interlayer twist is essential. Using ab initio density functional theory, we find that in near-60 degree twisted MoSe2/WSe2 bilayers valence and conduction band flat bands emerge at ~3 degree twist. Despite relatively limited reconstruction at these angles, atomic displacement creates a polarization gradient that forms a confining potential, localizing and laterally separating electrons and holes within the moire supercell. Excitons formed from flat band electrons and holes should therefore have not only the out-of-plane dipole moment familiar from MoSe2/WSe2 interlayer excitons, but an in-plane dipole moment as well.

Figures (5)

• Figure 1: Structural relaxation of 62.88$^\circ$ twisted MoSe$_2$/WSe$_2$ bilayer. (a) Unrelaxed and (b) relaxed structure of 62.88$^\circ$ twisted MoSe$_2$/WSe$_2$. Top inset shows the monolayer TMD components of the bilayer, and circular insets identify the regions of high symmetry stacking in the moiré supercell. The moiré supercell length, $\lambda_M$, is 6.56 nm. Panels (c, d) show the in-plane displacement fields of (c) the Mo atoms and (d) the W atoms. The displacement field is defined as $\mathbf{u}_k = \mathbf{r}^f_k - \mathbf{r}^0_k$, where $\mathbf{r}^0_k$ is the position of atom $k$ in the unrelaxed lattice, and $\mathbf{r}^f_k$ is the relaxed position of atom $k$. Selenium displacement fields are qualitatively the same and are shown in the SI SI.) Atoms in the Mo and W layers relax with an opposing twist about the ABBA and BB high symmetry points, while the atoms near the AA high symmetry point relax relatively little.
• Figure 2: Emergence of flat bands with decreasing twist angle. Band structures of MoSe$_2$/WSe$_2$ bilayers at (a) 64.41$^\circ$, (b) 63.89$^\circ$, (c) 63.48$^\circ$, and (d) 63.15$^\circ$ plotted with respect to momentum in the mini-Brillouin zone. The valence band flat band emerges at 63.89$^\circ$, and the conduction band flat band emerges at 63.48$^\circ$. As the twist angle decreases, the bandwidth of each flat band narrows. Valence band flat bands are two-fold degenerate, while conduction band flat bands are six-fold degenerate. Bands are considered "flat" if they are separated from the continuum of valence or conduction bands by a gap that spans the entire mini-Brillouin zone.
• Figure 3: Electronic bands and localization of flat band states for $62.88^\circ$ MoSe$_2$/WSe$_2$. (a) Electronic bands as a function of mini Brillouin zone momentum. Both conduction band flat bands are six-fold degenerate. The highest valence band flat band is two-fold degenerate. Panels (b-e) show the modulus squared of the wavefunctions corresponding to the circled states in (a). Each panel shows four moiré supercells: the top row shows the weight of the conduction band flat band state on the Mo (left) and W (right) atoms, and the bottom row shows the weight of the valence band flat band state on the Mo (left) and W (right) atoms. Conduction band weights are the sum of the six states that make up the conduction band flat band. Valence band weights are the sum of the two states that make up the valence band flat band. The conduction band states are localized around the AA-stacked region in the moiré supercell, while the valence band states are localized largely around the ABBA regions of the moiré supercell. Conduction band states have weight in both Mo and W layers, while valence band states have weight only in the W layer. The weight of the state on the Se atoms is an order of magnitude lower than the metal atom weights SI.
• Figure 4: Electronic structure of 60$^\circ$-aligned MoSe$_2$/WSe$_2$ bilayers. (a) Band structure of aligned AA-stacked MoSe$_2$/WSe$_2$. The conduction band minimum is at Q (between K and $\Gamma$ points), and the valence band maximum is at K. Inset: There are two K points in the system, K and K', and six Q points. In the twisted system, conduction and valence band flat band degeneracies and layer localizations are consistent with orgins in the band edges of the aligned system. All high-symmetry stacking band structures (AA, ABBA, BB) have band edges at the same Brillouin zone points. (b) Band alignments of the valence band at K and the conduction band at Q for the three high-symmetry 60$^\circ$-aligned MoSe$_2$/WSe$_2$ bilayers. If the twisted bilayer could be modeled as a patchwork of high-symmetry stacked regions, then electrons would be expected to reside in the ABBA region and holes in the BB region, but this is not what is observed in full twist calculations. The conduction band minimum (CBM) at Q in ABBA is 52 meV lower than the CBM at AA. The valence band maximum (VBM) at K in BB is 9 meV higher than the VBM at AA and 18 meV higher than the VBM at ABBA.
• Figure 5: In-plane atomic relaxation leads to induced charge density and moiré potential. (a) Polarization derived from the in-plane displacement field, ${\bf u}$, in 62.88$^\circ$ twisted MoSe$_2$/WSe$_2$ for the MoSe$_2$ layer (top) and the WSe$_2$ layer (bottom). $\lambda_M = 6.56$ nm in all panels. (b) Induced charge in (top) MoSe$_2$ and (bottom) WSe$_2$, computed from the negative divergence of the polarization. Induced charge is at a maximum at the AA-stacked regions and a minimum in the ABBA-stacked regions. (c) Potential landscape derived from the charge density induced by in-plane displacements, showing a maximum at the AA-stacked regions and a minimum at the ABBA-stacked regions, with a local minimum at the BB-stacked regions. This is consistent with the localization of the electrons and holes plotted in Figure 3, where the electrons are drawn to the areas of positive potential and the holes drawn to the areas of negative potential. Electron and hole lateral localization can thus be attributed mainly to piezoelectric effects in this heterobilayer.