Moiré Ferroelectricity-Driven Band Engineering in Twisted Square Bilayers

Abstract

We develop the moiré band theory for M-valley twisted square homobilayers with layer groups $P$-$42m$ and $P$-$4m2$, and propose candidate material realizations. We show that moiré ferroelectricity-originating from sliding ferroelectricity in the untwisted bilayers-provides an independent control knob for miniband engineering in addition to interlayer tunneling. The competition between these two effects enables controlled switching between layer-resolved bilayer minibands and an effective single isolated miniband. Remarkably, these systems exhibit an emergent momentum-space nonsymmorphic symmetry in the absence of external magnetic fields. Large-scale \emph{ab initio} calculations identify Cu$_2$WS$_4$ and GeCl$_2$ as representative materials realizing the ferroelectricity- and tunneling-dominated regimes, respectively. Our results establish twisted square homobilayers as a promising platform for correlated band engineering beyond moiré hexagonal systems.

Figures (4)

• Figure 1: (a) Brillouin zones (BZ) of the top and bottom layers, shown as red and blue squares, respectively. The dashed (solid) gray square denotes the moiré BZ used in the continuum model (DFT calculations), where the high-symmetry points are marked by Greek letter without (with) prime. (b,c) OP and corresponding bilayer point groups for $\{E|\tau\}$ stacking in the layer group $P$-$42m$ and $P$-$4m2$, respectively. Orange lines indicate the presence of OP. Translation vectors G(0,0), C(0.5,0.5), A(0.5,0), and B(0,0.5) are labeled. (d) Spatial distribution of $|\Delta_{T}(\mathbf{r})|$ with $w_0=20$ meV. Spatial patterns of the lowest-harmonic FE potential for (e) $P$-$42m$ with $w^{(2)}_1=-w^{(2)}_2=30$ meV, and (f) $P$-$4m2$ with $w^{(1)}_1=-w^{(1)}_2=-20$ meV.
• Figure 2: Band structure calculated from the continuum model for (a) $P$-$42m$ and (b) $P$-$4m2$, where FE potential dominates over interlayer tunneling. Solid (dashed) lines denote bands without (with) interlayer tunneling. The parameters are $m=0.6m_0$ ($m_0$ the free electron mass), $\theta=6^\circ$, $a=5$ Å. For (a): $w^{(1)}_1=-w^{(1)}_2=20$ meV, $w^{(2)}_1=-w^{(2)}_2=30$ meV, $w_0=10$ meV. For (b): $w^{(1)}_1=-w^{(1)}_2=-20$ meV, $w_0=6$ meV, $w'_0=5$ meV. The low-energy bands of the top (bottom) layer in the absence of interlayer tunneling are highlighted in brown (jasper). The corresponding tight-binding models are schematically illustrated in (c) and (d) using the same color scheme, where $t_i$ denote the dominant hopping amplitudes for the top layer.
• Figure 3: (a) Atomic structure of monolayer Cu$_2$MX$_4$ from top and side views. (b) Differential charge-density distributions and OP directions for $\{E|\tau\}$-stacked bilayers with $\bm{\tau}=(1/4,1/4)$ and $(3/4,1/4)$. Yellow and blue isosurfaces denote electron accumulation and depletion upon stacking, respectively. (c) OP distribution for different stacking configurations; the arrow direction indicates the in-plane polarization component, while the color encodes the out-of-plane component. (d) Band splitting at the M point as a function of stacking. (e) Band structures of twisted bilayer Cu$_2$WS$_4$ at $\theta=7.6^\circ$. Gray dots show DFT results, and red lines denote the fitting from the continuum Hamiltonian.
• Figure 4: (a) Atomic structure of monolayer GeCl$_2$.(b) Differential charge-density distributions and OP directions for $\{E|\tau\}$-stacked bilayers with $\bm{\tau}=(0,1/2)$ and $(1/2,0)$. (c) OP distributions for different stacking configurations. (d) Band splitting at the M point as a function of stacking. (e) Band structures of twisted bilayer GeCl$_2$ at $\theta=6.7^\circ$ by DFT calculations and continuum model. The style follows that of Fig. \ref{['fig3']}.