Moir\'e Band Engineering in Twisted Trilayer WSe2

We present a systematic theoretical study on the structural and electronic properties of twisted trilayer transition metal dichalcogenide (TMD) WSe$_2$, where two independent moir\'e 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 $\alpha\beta$ and $\beta\alpha$ domains, where the two moir\'e patterns shift to minimize overlap, while in alternating trilayers, $\alpha\alpha'$ domains emerge with aligned moir\'e patterns. A key feature of trilayer TMDs is the summation of moir\'e 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 $\alpha\beta$ 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 moir\'e potential summation in trilayer systems, offering a broader platform for designing moir\'e-based electronic and excitonic phenomena beyond those achievable in bilayer TMDs.