Engineering 2D square lattice Hubbard models in 90$^\circ$ twisted Ge/SnX (X=S, Se) moiré supperlattices

TL;DR

The paper addresses realizing square-lattice Hubbard physics in 2D moiré materials by proposing 90° twists of rectangular-lattice Ge/Sn chalcogenides to generate square moiré lattices with flat conduction minibands. Through first-principles DFT, tight-binding fits, and a Gamma-valley continuum model, the authors show the lowest conduction miniband can be mapped to a square-lattice Hubbard model with tunable nearest-neighbor and next-nearest-neighbor hoppings $t$ and $t'$, and on-site interaction $U$, with $U/t \\approx 7.8$ and $t'/t$ in the few-tenths range depending on the stack. They extend the design to multilayers and near-90° twists, demonstrating both isotropic and anisotropic square-lattice realizations, and they use DFT$+U+V$ and functional renormalization group (FRG) to predict correlation-driven Néel order at half filling and gate-tunable competition between stripe antiferromagnetism and $d$-wave superconductivity upon doping, tunable via displacement fields. The work provides a versatile, tunable solid-state platform for exploring the square-lattice Hubbard model, including magnetism, charge order, and unconventional superconductivity, with moiré wavelengths below 10 nm and a broad set of candidate rectangular-lattice 2D materials for realization.

Abstract

Due to the large-period superlattices emerging in moire two-dimensional (2D) materials, electronic states in such systems exhibit low energy flat bands that can be used to simulate strongly correlated physics in a highly tunable setup. While many investigations have thus far focused on moire flat bands and emergent correlated electron physics in triangular, honeycomb and quasi-one-dimensional lattices, tunable moire realizations of square lattices subject to strong correlations remain elusive. Here we propose a feasible scheme to construct moire square lattice systems by twisting two or more layers of 2D materials in a rectangular lattice by 90 degrees. We demonstrate the concept with twisted GeX/SnX (X=S, Se) moire superlattices and calculate their electronic structures from first principles. We show that the lowest conduction flat band in these systems can be described by a square lattice Hubbard model with parameters which can be controlled by varying the choice of host materials, number of layers, and external electric fields. In particular, twisted double bilayer GeSe realizes a square lattice Hubbard model with strong frustration due to the next nearest neighbour hopping that could host unconventional superconductivity{, in close analogy to the Hubbard model for copper-oxygen planes of cuprate high-temperature superconductors}. The presented scheme uses 90-degree twisted 2D materials with rectangular unit cells as a promising platform for realizing the physical phenomena of square lattice Hubbard models, establishing a new route for studying its rich phase diagram of magnetism, charge order, and unconventional superconductivity in a highly tunable setting.

Figures (12)

• Figure 1: General strategy of constructing moiré square lattices using 2D rectangular lattices. (a) Schematic illustration of the 90$^\circ$ twisted 2D layers. (b) Flow chart of the construction of the moiré square superlattice using 2D layers of rectangular lattices with different lattice constants a and b (a $\neq$ b). (c) Brillouin zone of the 90$^\circ$ twisted rectangular lattices.
• Figure 2: Moiré structure of 90$^\circ$ twisted bilayer GeSe. The top and the side views of the atomic structure are shown in upper left and lower left panels, respectively. A$^t$ in the side view is used to indicate the layer twisted by 90$^\circ$ with respect to the layer labeled as A. The moiré unit cell is indicated by black solid lines. The black dashed line represents the in-plane $C_{2d}$ rotational axis. The top and the side views of the atomic structures of the local stacking configurations of AA$^\alpha$, AA$^\beta$, AB$^\alpha$, and AB$^\beta$ are shown in right panels. The Ge and the Se atoms are represented by green/blue and grey/pink balls, respectively.
• Figure 3: Moiré flat bands of 90$^\circ$ twisted bilayer GeSe. (a) Low-energy band structure near the band edge. The dashed line indicates the Fermi level. (b) The enlarged band structure corresponding to the shaded region in (a), which shows the dispersion of the moiré flat band at the CBE. The black solid line and the cyan dots denote the results calculated with DFT and the effective TB model, respectively. (c) Schematic diagram of the square lattice effective TB model. (d) Top view (upper panel) and side view (lower panel) of partial charge density distribution in real space for the flat-band state at the CBE.
• Figure 4: Moiré flat bands of 90$^\circ$ twisted bilayer GeS and SnS. The atomic and the electronic structures of 90$^\circ$ twisted bilayer GeS (a, b) and SnS (c, d). The low energy flat bands at the CBE are fitted well by the effective TB model.
• Figure 5: $\Gamma$-valley continuum model for 90$^\circ$ twisted bilayer GeX/SnX (X=S, Se). Continuum model and ab-initio bandstructure for the square lattice moiré candidates (a) GeS, (b) SnS and (c) GeSe.