Geometrical properties of strained and twisted moiré heterostructures

TL;DR

This work reviews how linear elasticity in two-dimensional materials sets the foundation for understanding how twist and various strains reshape moiré geometries in bilayers and heterobilayers. By formalizing the deformation through displacement, strain, and the transformation tensor that combines rotation and strain, the authors show how uniaxial, shear, and biaxial strains generate a wide spectrum of moiré patterns, including quasi-1D, square, and hexagonal geometries, and how strain modifies the moiré Brillouin zone. The review also surveys experimental strategies to apply and tune strain—such as substrate bending, process-induced stress, and sliding methods—and highlights remarkable phenomena like 1D channels, rectangular patterns, and giant atomic swirls. Overall, strain emerges as a powerful, tunable knob that complements twist in tailoring moiré lattices and potentially enabling new correlated electronic phases in 2D materials.

Abstract

The experimental observations of many interaction-driven electronic phases in moiré superlattices have stimulated intense theoretical and experimental efforts to understand and engineer these correlated physics. Strain is a powerful tool for manipulating and controlling the geometrical and electronic structures of moiré superlattices. This review provides a comprehensive introduction to the geometry of strained moiré superlattices. First, starting from the linear elasticity theory, we briefly introduce the general formalism of small deformations in two-dimensional materials, and discuss the particular cases of uniaxial, shear and biaxial strain. Then, we apply the theory to twisted and strained moiré materials, mainly focusing on the hexagonal homobilayers, hexagonal heterobilayers and monoclinic lattices. Special moiré geometries, like the quasi-unidimensional patterns, square patterns and hexagonal, are theoretically predicted by manipulating the strain and twist. Finally, we review recently developed strain techniques and the special moiré geometries realized via these approaches. This review aims at equipping the reader with a robust understanding on the description and implementation of strain in moiré materials, as well as highlight some major breakthroughs in this active field.

Figures (16)

• Figure 1: Schematic effect of three common types of strain in moiré systems: uniaxial, shear and biaxial. The red dashed-line shows the undeformed hexagon and the blue solid-line shows the deformation of each strain type (uniaxial and shear, both with direction along the x-axis).
• Figure 2: Strain-dependent construction of moiré vectors. The left panels shows the deformed reciprocal vectors $\mathbf{b}_{i,\pm}$ in two honeycomb lattices subject to a twist $\theta=5^{\circ}$ and uniaxial strain with magnitude $\epsilon=5\%$ and direction $\phi=60^{\circ}$. Taking the difference between the deformed reciprocal vectors does not yield, in this case, the smallest moiré vectors $\mathbf{G}_{i}$. A smaller moiré vector $\mathbf{G}'_{2}=\mathbf{G}_{1}+\mathbf{G}_{2}$ is obtained by taking the difference between the deformed reciprocal vectors $\mathbf{b}'_{2,\pm}=\mathbf{b}_{1,\pm}+\mathbf{b}_{2,\pm}$. The right panel shows the superlattice spanned by the moiré vectors. Adapted under the terms of the CC BY license from Ref. escudero2024designing. Copyright (2024) by the American Physical Society.
• Figure 3: Examples of moiré patterns in two equal hexagonal layers with relative twist $\theta=2^{\circ}$ and uniaxial heterostrain. All cases shown correspond to equal length moiré vectors (Section \ref{['subsec:uniaxial']}). Panel (a) shows three moiré pattern corresponding to (from left to right): $\epsilon=0$ (no strain), $\epsilon\approx1.64\%$, $\phi\approx-9.4^{\circ}$, $\epsilon\approx4.44\%$, $\phi\approx-0.9^{\circ}$ and $\epsilon\approx-2.28\%$, $\phi\approx-22.7^{\circ}$. The relative angles between the real space moiré vectors are, respectively, $\beta_{R}=60^{\circ},90^{\circ},140^{\circ},30^{\circ}$. In each case the Wiger-Seitz cell of the superlattice is shown in white. The small inset below each panel show the strain magnitude (in a scale of $5\%$) and the strain direction relative to the $x$ axis. Panel (b) shows the evolution of the Wigner-Seitz cell and the repeated moiré pattern within it, for moiré vector angles (from left to right) $\beta_{R}=40^{\circ},60^{\circ},80^{\circ},90^{\circ},100^{\circ},120^{\circ},140^{\circ}$. For each case, the bar underneath indicates the strain magnitude (thick line) in a scale of $5\%$ (thin line). Adapted under the terms of the CC BY license from Ref. escudero2024designing. Copyright (2024) by the American Physical Society.
• Figure 4: Variation of the real space moiré vectors lengths $A'_{1}=\left|\mathbf{G}_{1}^{R}\right|$ and $A'_{2}=\left|\mathbf{G}_{2}^{R}\right|$, as a function of the twist angle and the strain magnitude, for the cases of two homobilayer ($\mathrm{WSe_{2}-WSe_{2}}$) and heterobilayer ($\mathrm{WSe_{2}}-\mathrm{MoSe_{2}}$) hexagonal lattices. The configuration is such that the top layers is only twisted and the bottom layer is only strained. Panel (a) shows the moiré lengths $A'_{1},A'_{2}$ (equal) under biaxial strain magnitude $\epsilon_{b}$; dotted ellipses indicate the regions of moiré lenghts $A'_{1},A'_{2}=10\,\mathrm{nm}.$ Panel (b) shows the moiré lengths $A'_{1}$ (top) and $A'_{2}$ (bottom) under uniaxial strain magnitude $\epsilon_{u}$ at fixed direction $\phi=0^{\circ}$. The spots labeled as I, II and III indicate the unidimensional, square and hexagonal moiré geometries, respectively (cf. Section \ref{['sec:Moire_special']}). The points $\nu_{1}$ and $\nu_{2}$ in the heterobilayer case indicate the vertexes of the divergence curves (unidimensional patterns). Adapted under the terms of the CC BY license from Ref. kogl2023moire. Copyright (2023) The author(s).
• Figure 5: Variation of the real-space angle $\beta_{R}$ between the moiré vectors (top) and the moiré area $M=\left|\mathbf{G}_{1}^{R}\times\mathbf{G}_{2}^{R}\right|$ (bottom), as a function of the twist angle and the strain magnitude, for the cases of two homobilayer ($\mathrm{WSe_{2}-WSe_{2}}$) and heterobilayer ($\mathrm{WSe_{2}}-\mathrm{MoSe_{2}}$) hexagonal lattices (same configuration as in Figure \ref{['fig:Strain_Homobilayer_Heterobilayer']}). Panel (a) shows the case of uniaxial strain with magnitude $\epsilon_{u}$ at fixed direction $\phi=0^{\circ}$. Panel (b) shows the case of shear strain with magnitude $\epsilon_{s}$ at fixed direction $\phi=0^{\circ}$. In both cases, the spots labeled as I, II and III indicate the unidimensional, square and hexagonal moiré geometries (see also Section \ref{['sec:Moire_special']}). Adapted under the terms of the CC BY license from Ref. kogl2023moire. Copyright (2023) The author(s).