Layerwise Stratification and Band Reordering in Twisted Multilayer MoTe$_2$

Abstract

We introduce a generalizable, physics informed strategy for generating training data that enables a machine learning force field accurate over a broad range of twist angles and stacking layer numbers in moire systems. Applying this to multilayer twisted MoTe2 (tMoTe2), we identify a structural and electronic stratification: the two moire interface (MI) layers retain substantial lattice reconstruction even in thick multilayers, while outer bulk like layers show rapidly attenuated distortions.Surprisingly, this stratification becomes strongest not in the ultra-small twist angle regime (<~1°), where in plane domain formation is well known, but rather at intermediate angles (2-5°). Simultaneously, interlayer hybridization across the MI-bulk boundary is strongly suppressed, leading to electronic isolation. In twisted double bilayer MoTe2, this stratification gives rise to coexisting honeycomb and triangular lattice motifs in the frontier valence bands. We further demonstrate that twist angle and weak gating can create energy shift of bands belonging to the two motifs, producing Chern band reordering and nonlinear electric polarization with modest hole doping. Our approach allows efficient simulation of multilayer moire systems and reveals structural-electronic separation phenomena absent in bilayer systems.

Figures (4)

• Figure 1: (a) Schematic structure of $n+m$ tMoTe$_2$ with twist angle $\theta$ near R stacking. Layers are indexed as MI$-m$ to MI$+n$. Inset: (Right) layer-resolved bands color-coded by layer polarization with red at MI and blue at bulk-like layers. $\ket{K_{\text{MI}+j}, \uparrow/\downarrow}$ shows the monolayer state at Brillouin zone corner K of the layer $\text{MI}+j$ with spin $\uparrow/\downarrow$. (Left) $1+2$ tMoTe$_2$ for training data generation. (b) Force parity plot for MLFF tested on $4.41^\circ$$1+2$ tMoTe$_2$, comparing the MLFF-model predicted and DFT calculated ionic forces on each atom (green dots). Red line of $y=x$ shows conditions of exact match. (c) Force parity plot for $6.01^\circ$$2+2$ tMoTe$_2$. (d, e) Differential information entropy distribution of local atomic environment for (d) $4.41^\circ$$1+2$ (red), $6.01^\circ$$2+2$ (blue), and (e) $5+5$ tMoTe$_2$ under $\theta=5.09$° (blue), $3.15$° (orange), and $1.41$° (green) relative to $1+2$ training set. Frequency density is the normalized distribution of $\delta H$ datapoints over the full range.
• Figure 2: (a) The average magnitude of the Mo in-plane displacement $\bar{u}_{xy}$ across twist angles and layer indices in $5+5$ tMoTe$_2$. (b) $\bar{u}_{xy}$ at MI layers across twist angles and thickness in $n+n$ ($n=1, 2, ..., 5$) tMoTe$_2$. Lines are guides to the eye.
• Figure 3: (a) Band structure of $2.88$° $2+2$ tMoTe$_{2}$ with color coded layer polarization $S(\alpha, \mathbf{k})$ (defined in main text). $C_{K, \alpha}$ labels the valley Chern number. (b, c) Electron density $\rho_e$ of states at $\gamma$, integrated along z and projected onto 2D. (b) $|\psi_{(\text{MI}, 1),\gamma}|^2+|\psi_{(\text{MI}, 2),\gamma}|^2$ and (c)$|\psi_{(\text{B}, 1),\gamma}|^2+|\psi_{(\text{B}, 2),\gamma}|^2$ for $2.88$° $2+2$ tMoTe$_{2}$. The density maxima in each layer are marked by black dots. The moiré cell is marked by black dashed boxes. (d, e) Valence bands of (d) $2.65$° and (e) $2.28$° $2+2$ tMoTe$_2$. Dashed box in (d) marks overlapping bands $(\text{MI}, 1)$, $(\text{B}, 1)$, and $(\text{B}, 2)$.
• Figure 4: (a) Valence bands of $2.88$° $2+2$ tMoTe$_2$ under $E_z=-3.3$ mV/nm. Red and blue bands are layer polarized at $\text{MI}\pm1$ and $\text{MI}\pm2$, respectively. A chemical potential corresponding to filling factor $\nu=-0.4$ is marked by black dashed line. (b) The response of out-of-plane electric polarization per hole $p_z$ to out-of-plane electric field $E_z$ at hole filling $\nu=-0.4$. The total value, contributions from $\text{MI}\pm1$ and $\text{MI}\pm2$ are marked with black solid, red dashed, and blue dashed lines, respectively. (c) Color map of the total electric polarization per hole $p_z$ as a function of hole filling factors and out-of-plane electric fields.