Twistraintronics in Square Moire Superlattices of Stacked Graphene Layers

TL;DR

This work reports the first controlled observation of strain-induced square moiré patterns in bilayer graphene by selectively displacing graphene wrinkles to apply shear-like heterostrain. Using STM/STS, the authors show elliptically elongated AA domains and two Van Hove singularities near the Fermi level, whose splitting is modulated by the applied strain. A continuum model incorporating twist, shear strain, and Hartree electrostatics accurately reproduces the observed electronic features, identifying a minimum-elastic-energy shear configuration as the origin of the square moiré geometry. The results establish twistraintronics as a viable route to access highly correlated electronic states in moiré lattices with square symmetry, expanding the design space beyond conventional trigonal twist patterns. The study highlights the synergy between local strain control and spectroscopic probes as a path toward anisotropic superconductivity and other correlated phenomena in engineered 2D heterostructures.

Abstract

We report the first observation of controlled, strain-induced square moire patterns in stacked graphene. By selectively displacing native wrinkles, we drive a reversible transition from the usual trigonal to square moire order. Scanning tunneling microscopy reveals elliptically shaped AA domains, while spectroscopy shows strong electronic correlation in the form of narrow bands with split Van Hove singularities near the Fermi level. A continuum model with electrostatic interactions reproduces these features under the specific twist-strain combination that minimizes elastic energy. This work demonstrates that the combination of twist and strain, or twistraintronics, enables the realization of highly correlated electronic states in moire heterostructures with geometries that were previously inaccessible.

Figures (13)

• Figure 1: Local strain control via STM-based manipulation of graphene wrinkles. (a–d) STM sequence showing reversible moiré switching (trigonal → square → trigonal) by laterally shifting a nearby wrinkle: (a) The region of interest (black square) exhibits a strain-free trigonal moiré geometry; (b) approaching the wrinkle yields a square geometry; (c–d) sweeping the wrinkle across the area and then retracting it releases strain and restores the trigonal geometry. (e–g) Zoomed-in views of the same area after each manipulation step; dashed parallelograms mark the moiré unit cells. (h) Schematic of the manipulation mechanism (see Supplemental Material, animation wrinkle_manipulation.mp4). STM parameters: $I_T$ = $50\,\mathrm{pA}$, $V_{bias}$ = $600\,\mathrm{mV}$.
• Figure 2: STM topography and spatially-resolved spectroscopy of trigonal and square moiré superlattices in TBG. (a-c) Atomic-resolution STM images of trigonal (a) and square moiré patterns (b,c) , both with a periodicity of $\approx 12\,\mathrm{nm}$. In each panel, the black shapes outline the corresponding moiré unit cell. (c) Atomic-resolution STM image of a single square moiré unit cell. Colored arrows trace the closed-loop path along which differential conductance ($dI/dV$) spectra lines were acquired. (d) STM image of a large strained moiré. The non-uniform periodicity is caused by an inhomogeneous strain profile. (All STM data available with atomic resolution in the Supplementary Material). (e) Two-dimensional map of dI/dV intensity as a function of energy (horizontal axis) and spatial position along the four segments of the path (vertical axis; color-coded to match the arrows in the left inset). Spectra show consistent features along vertical lines, indicating spatial homogeneity. Similar sharp features can be seen in the horizontal lines of spectra. (f) Average of all $dI/dV$ curves in (e), showing LDOS vs. Energy. The red curve shows the average; the area shaded in light orange between the black dashed curves is the one-standard deviation interval. STM parameters: $I_T$ = $340\,\mathrm{pA}$, $V_{bias}$ = $50\,\mathrm{mV}$(a); $I_T$ = $50\,\mathrm{pA}$, $V_{bias}$ = $16\,\mathrm{mV}$(b); $I_T$ = $230\,\mathrm{pA}$, $V_{bias}$ = $10\,\mathrm{mV}$(c); $I_T$ = $50\,\mathrm{pA}$, $V_{bias}$ = $25\,\mathrm{mV}$(d).
• Figure 3: (a) Square moiré pattern formed in a bilayer graphene configuration with a twist angle $\theta\approx1.125{}^{\circ}$ and shear strain with magnitude $\epsilon_s\approx-0.526\%$ and direction $\phi=30{}^{\circ}$ (see Ref. SM). The top and bottom layers are rotated by $\pm\theta/2$ and strained with equal magnitude but opposite direction. (b) Colormap of atomic positions in the square pattern, indicating the stacking regime of each atom relative to the closest atom in the other layer, raging from directly on top (AA), to in between (DW), to bernal stacking (AB/BA). The gold stars point the DW that indicates the transition from AA to AB/BA stacking. (c) LDOS along the four directions $A,B,C,D$ shown in (b), for energies $E=E_{F}\pm50\,\mathrm{meV}$ around the Fermi energy $E_{F}$. (d) 3D plot of the band structure. (e) Total density of states with Hartree (red line) and without Hartree (gray dashed-line). The electronic properties are obtained from the continuum model with strain, including the electrostatic interactions (Hartree potential) with a filling of $\nu=+1$ electron per moiré unit cell (see Ref. SM for details).
• Figure S1: (a) Plot of the function $f\left(\epsilon\right)$ given by Eq. \ref{['eq:fe']}, which determines the moiré length $L_{M}$ through Eq. \ref{['eq:Lmoire']}. The green vertical line at $\epsilon=2\sqrt{3}-3$ corresponds to the minimum (traceless) shear strain case, at which $f\simeq0.966$ takes its maximum value. (b) Evolution of the moiré length $L_{M}\left(\theta,\epsilon\right)$ as a function of the twist angle $\theta$ and $\epsilon$. The contour black dashed curves correspond to constant moiré lengths $L_{M}$ from 8 to 14 nm. The dashed gray region represent the range of moiré lengths extracted from the experiment results shown in Figure \ref{['fig:exp']}(c). The magenta dot-dashed curve corresponds to the average moiré length $\tilde{L}_{M}=12.1205\,\mathrm{nm}$. The horizontal green line corresponds to the value $\epsilon=2\sqrt{3}-3\simeq0.42641$ of minimum shear strain configuration, and the gold star indicates its intercept with the average moiré length $\tilde{L}_{M}$ at a twist angle $\theta\simeq1.125^{\circ}$.
• Figure S2: Square patterns for the same moiré length $\tilde{L}_{M}=12.1025\,\mathrm{nm}$, and different parameters $\epsilon=-1,-0.5,-0.1,0.1,2\sqrt{3}-3,0.9127$. The case $\epsilon=2\sqrt{3}-3\approx0.42641$ of minimum strain (traceless, shear strain) is highlighted in a green box. The last case $\epsilon=0.9127$ corresponds to the uniaxial heterostrain. Note that in order to preserve the same moiré length, for each parameter $\epsilon$ there is a different twist angle $\theta$; cf. Eq. \ref{['eq:Lmoire']}. As $L_{M}$ is the same in all cases, up to an overall rotation the moiré patterns look practically the same at the moiré scale. However, they have markedly different electronic properties, cf. Figure \ref{['fig:DOS_all']}.