Moiré Band Engineering in Twisted Trilayer WSe2

TL;DR

This paper develops a comprehensive continuum framework to study twisted trilayer WSe$_2$ with two independent moiré patterns, focusing on lattice relaxation and the resulting electronic structure near the valence-band edge. In helical stacks, relaxation induces $\,\alpha\beta\,$ and $\beta\alpha$ domains, and the middle-layer moiré potential doubles in depth, producing a Kagome lattice and flat bands localized on layer 2; in alternating stacks, deeper triangular wells yield strongly localized bound states. The authors show that a moderate perpendicular electric field can polarize layers and hybridize orbitals across layers, enabling graphene-like Dirac bands, flat bands, and $s$/$p$/$d$-orbital hybrid states, expanding the moiré engineering space beyond bilayer TMDs. Overall, the work highlights moiré potential summation as a design principle for tunable, multi-orbital electronic landscapes in trilayer moiré systems, with implications for correlated and topological phenomena.

Abstract

We present a systematic theoretical study on the structural and electronic properties of twisted trilayer transition metal dichalcogenide (TMD) WSe$_2$, where two independent moiré patterns form between adjacent layers. Using a continuum approach, we investigate the optimized lattice structure and the resulting energy band structure, revealing fundamentally different electronic behaviors between helical and alternating twist configurations. In helical trilayers, lattice relaxation induces $αβ$ and $βα$ domains, where the two moiré patterns shift to minimize overlap, while in alternating trilayers, $αα'$ domains emerge with aligned moiré patterns. A key feature of trilayer TMDs is the summation of moiré potentials from the top and bottom layers onto the middle layer, effectively doubling the potential depth. In helical trilayers, this mechanism generates a Kagome lattice potential in the $αβ$ domains, giving rise to flat bands characteristic of Kagome physics. In alternating trilayers, the enhanced potential confinement forms deep triangular quantum wells, distinct from those found in bilayer systems. Furthermore, we demonstrate that a moderate perpendicular electric field can switch the layer polarization near the valence band edge, providing an additional degree of tunability. In particular, it enables tuning of the hybridization between orbitals on different layers, allowing for the engineering of diverse and controllable electronic band structures. Our findings highlight the unique role of moiré potential summation in trilayer systems, offering a broader platform for designing moiré-based electronic and excitonic phenomena beyond those achievable in bilayer TMDs.

Figures (17)

• Figure 1: Schematic figure of the real-space distribution of the potential on the middle layer of twisted trilayer TMD with (a) helical and (b) alternate twist. On each row, the left two panels show the contributions from the moiré pattern between layers 1 and 2 (moiré 12) and between layers 2 and 3 (moiré 23), where darker regions indicate higher potential energy. The right panel displays the total potential in the commensurate domain, obtained by summing the moiré 12 and 23 potentials with the corresponding lateral shift.
• Figure 2: (a) Stacking geometry of a twisted homo-trilayer TMD with a helical twist. Green, black, and orange sheets represent layers 1, 2, and 3, respectively. (b) Moiré pattern arrangement in the helical twist trilayer. Blue and red dots indicate the MM points of moiré 12 (formed between layers 1 and 2) and moiré 23 (formed between layers 2 and 3), respectively. The gray rhombus marks a moiré-of-moiré unit cell. (c) Representative local moiré structures in (b), where empty and filled triangles denote MX and XM domains, respectively. (d) Side view of the local atomic configurations at specific points in (c). (e–h) Corresponding structures and atomic configurations for the alternate twist case.
• Figure 3: A double-moiré system with a common periodicity, which is given by $(n,m,n',m')=(2,3,2,2)$, corresponding to $(\theta^{12},\theta^{23})=(7.3^\circ,5.8^\circ)$. (a) Real space structure of the non-relaxed moiré lattices 12 (blue) and 23 (red), where the vertexes represent MM stacking point. A gray rhombus is a common period (the moiré-of-moiré period). (b) Momentum -space sturcture. Green, black, and orange hexagons represent the monolayer BZ of layer 1, 2, and 3 respectively, and blue and red hexagons are the BZs for the moiré 12 and 23. (c) A magnified view of the region around the $K_{+}$-valley, where gray hexagons represent the BZ of the moiré-f-moiré pattern.
• Figure 4: (a) Band structure and the layer-projected DOS near the valence band edge for a relaxed $\alpha\beta$-stacked twist trilayer WSe$_{2}$ with $(\theta^{12}, \theta^{23})=(1.0^{\circ}, 1.0^{\circ})$. Blue numbers by the bands indicate the Chern number of the $K_+$ valley. (b) Contour maps of the intralayer potential $V_1, V_2, V_3$. Blue and red lattices represent moiré lattices of 12 and 23, respectively. (c) Real-space distribution of wavefunctions for representative bands, averaged over the Brillouin zone. The colors indicate the layer distribution, as shown in the triangular diagram in the inset. (d-f) Corresponding figures for a commensurate $\beta\alpha$-stacked trilayer with $(\theta^{12}, \theta^{23})=(1.0^{\circ}, 1.0^{\circ})$.
• Figure 5: (a) Structural relaxation in a helical twist trilayer WSe$_2$ with $(\theta^{12},\theta^{23})=(1.07^\circ,0.93^\circ)$. The left two panels present schematic illustrations of the double moiré lattice for non-relaxed and relaxed cases, where blue and red dots represents MM stacking points of moiré 12 and moiré 23. A gray rhombus represents a moiré-of-moiré unit cell. The right two panels plot the interlayer binding energy $U_{B}^{12}$ and $U_{B}^{23}$ in the relaxed structure, where bright and dark regions correspond to MM and MX/XM stacking, respectively. Small magenta dots indicate the MM stacking without lattice relaxation. (b) Band structure and layer-resolved density of states (DOS) of the same system. (c) Contour maps of the intralayer potentials $V_1, V_2, V_3$. (d) Real-space distribution of representative wavefunctions, averaged over the shaded energy regions in the DOS plot of (b). The colors indicate the layer distribution.