We study the influence of strong spin-orbit interaction on the formation of flat bands in relaxed twisted bilayer WSe$_2$. Flat bands, well separated in energy, emerge at the band edges for twist angles ($theta$) close to 0$^o$ and 60$^o$. For $theta
$ close to 0$^o$, the interlayer hybridization together with a moir{e} potential determines the electronic structure. The bands near the valence band edge have non-trivial topology, with Chern numbers equal to +1 or $-$1. We propose that this can be probed experimentally for twist angles less than a critical angle of 3.5$^o$. For $theta$ near 60$^o$, the flattening of the bands arising from the K point of the unit cell Brillouin zone is a result of atomic rearrangements in the individual layers. Our findings on the flat bands and the localization of their wavefunctions for both ranges of $theta$ match well with recent experimental observations [1,2].
Transition metal dichalcogenide (TMD) moire heterostructures provide an ideal platform to explore the extended Hubbard model1 where long-range Coulomb interactions play a critical role in determining strongly correlated electron states. This has led
to experimental observations of Mott insulator states at half filling2-4 as well as a variety of extended Wigner crystal states at different fractional fillings5-9. Microscopic understanding of these emerging quantum phases, however, is still lacking. Here we describe a novel scanning tunneling microscopy (STM) technique for local sensing and manipulation of correlated electrons in a gated WS2/WSe2 moire superlattice that enables experimental extraction of fundamental extended Hubbard model parameters. We demonstrate that the charge state of local moire sites can be imaged by their influence on STM tunneling current, analogous to the charge-sensing mechanism in a single-electron transistor. In addition to imaging, we are also able to manipulate the charge state of correlated electrons. Discharge cascades of correlated electrons in the moire superlattice are locally induced by ramping the STM bias, thus enabling the nearest-neighbor Coulomb interaction (UNN) to be estimated. 2D mapping of the moire electron charge states also enables us to determine onsite energy fluctuations at different moire sites. Our technique should be broadly applicable to many semiconductor moire systems, offering a powerful new tool for microscopic characterization and control of strongly correlated states in moire superlattices.
Introduction of a twist between layers of two-dimensional materials which leads to the formation of a moire pattern is an emerging pathway to tune the electronic, vibrational and optical properties. The fascinating properties of these systems is ofte
n linked to large-scale structural reconstruction of the moire pattern. Hence, an essential first step in the theoretical study of these systems is the construction and structural relaxation of the atoms in the moire superlattice. We present the Twister package, a collection of tools that constructs commensurate superlattices for any combination of 2D materials and also helps perform structural relaxations of the moire superlattice. Twister constructs commensurate moire superlattices using the coincidence lattice method and provides an interface to perform structural relaxations using classical forcefields.
Moire superlattices in transition metal dichalcogenide (TMD) heterostructures can host novel correlated quantum phenomena due to the interplay of narrow moire flat bands and strong, long-range Coulomb interactions1-5. However, microscopic knowledge o
f the atomically-reconstructed moire superlattice and resulting flat bands is still lacking, which is critical for fundamental understanding and control of the correlated moire phenomena. Here we quantitatively study the moire flat bands in three-dimensional (3D) reconstructed WSe2/WS2 moire superlattices by comparing scanning tunneling spectroscopy (STS) of high quality exfoliated TMD heterostructure devices with ab initio simulations of TMD moire superlattices. A strong 3D buckling reconstruction accompanied by large in-plane strain redistribution is identified in our WSe2/WS2 moire heterostructures. STS imaging demonstrates that this results in a remarkably narrow and highly localized K-point moire flat band at the valence band edge of the heterostructure. A series of moire flat bands are observed at different energies that exhibit varying degrees of localization. Our observations contradict previous simplified theoretical models but agree quantitatively with ab initio simulations that fully capture the 3D structural reconstruction. Here the strain redistribution and 3D buckling dominate the effective moire potential and result in moire flat bands at the Brillouin zone K points.
Using a multiscale computational approach, we probe the origin and evolution of ultraflatbands in moire superlattices of twisted bilayer MoS$_2$, a prototypical transition metal dichalcogenide. Unlike twisted bilayer graphene, we find no unique magic
angles in twisted bilayer MoS$_2$ for flatband formation. Ultraflatbands form at the valence band edge for twist angles ($theta$) close to 0$^circ$ and at both the valence and conduction band edges for $theta$ close to 60$^circ$, and have distinct origins. For$ theta$ close to 0$^circ$, inhomogeneous hybridization in the reconstructed moire superlattice is sufficient to explain the formation of flatbands. For $theta$ close to 60$^circ$, additionally, local strains cause the formation of modulating triangular potential wells such that electrons and holes are spatially separated. This leads to multiple energy-separated ultraflatbands at the band edges closely resembling eigenfunctions of a quantum particle in an equilateral triangle well. Twisted bilayer transition metal dichalcogenides are thus suitable candidates for the realisation of ordered quantum dot array.
The tunability of the interlayer coupling by twisting one layer with respect to another layer of two-dimensional materials provides a unique way to manipulate the phonons and related properties. We refer to this engineering of phononic properties as
Twistnonics. We study the effects of twisting on low-frequency shear (SM) and layer breathing (LBM) modes in transition metal dichalcogenide (TMD) bilayer using atomistic classical simulations. We show that these low-frequency modes are extremely sensitive to twist and can be used to infer the twist angle. We find unique ultra-soft phason modes (frequency $lesssim 1 mathrm{cm^{-1}}$, comparable to acoustic modes) for any non-zero twist, corresponding to an textit{effective} translation of the moir{e} lattice by relative displacement of the constituent layers in a non-trivial way. Unlike the acoustic modes, the velocity of the phason modes is quite sensitive to twist angle. As twist angle decreases, ($theta lesssim 3^{circ}, gtrsim 57^{circ}$) the ultra-soft modes represent the acoustic modes of the emergent soft moir{e} scale lattice. Also, new high-frequency SMs appear, identical to those in stable bilayer TMD ($theta = 0degree/60degree$), due to the overwhelming growth of stable stacking regions in relaxed twisted structures. Furthermore, we find remarkably different structural relaxation as $theta to 0^{circ}$, $to 60^{circ}$ due to sub-lattice symmetry breaking. Our study reveals the possibility of an intriguing $theta$ dependent superlubric to pinning behavior and of the existence of ultra-soft modes in textit{all} two-dimensional (2D) materials.
We develop parameters for the interlayer Kolmogorov-Crespi (KC) potential to study structural features of four transition metal dichalcogenides (TMDs): MoS$_2$, WS$_2$, MoSe$_2$ and WSe$_2$. We also propose a mixing rule to extend the parameters to t
heir heterostructures. Moire superlattices of twisted bilayer TMDs have been recently shown to host shear solitons, topological point defects and ultraflatbands close to the valence band edge. Performing structural relaxations at the DFT level is a major bottleneck in the study of these systems. We show that the parametrized KC potential can be used to obtain atomic relaxations in good agreement with DFT relaxations. Furthermore, the moire superlattices relaxed using DFT and the proposed forcefield yield very similar electronic band structures.
