Straintronics and twistronics in bilayer graphene

TL;DR

This work develops a general framework to study twist- and strain-engineered bilayer graphene by building commensurate moiré supercells for arbitrary twist and strain, combining atomistic tight-binding with a strain-extended continuum model. It shows that strain, especially shear, broadens the narrow bands, splits van Hove features, and shifts Dirac points in the moiré Brillouin zone, with the bandwidth minimum depending on strain magnitude and direction. The study demonstrates that electrostatic (Hartree) interactions compete with strain-induced band broadening, leading to bandwidths that can rival those of unstrained twist configurations, and reveals strain-driven topological transitions in the narrow bands, characterized by valley Chern numbers that can become asymmetric under interactions. Overall, strain provides a tunable knob to access flat-band and topological phases in twisted bilayer graphene, and the combined TB and continuum approaches yield a consistent, predictive picture for designing strain-twistronic devices.

Abstract

The interplay of twist and strain in bilayer graphene enables the formation of moiré patterns and narrow bands that host correlated and topological phases. While magic-angle twisted bilayer graphene has been widely studied, strain provides an additional and realistic control knob for band engineering. In this work, we first generate a global method to construct commensurate supercells for arbitrary twist and strain. Then, using atomistic tight-binding and strain-extended continuum models to study the commensurate structures, we identify configurations that minimize the bandwidth beyond the magic angle. The results reveal a strong dependence of band narrowing and topology on strain type, magnitude, direction and lattice relaxation. Particularly, shear strain produces a stronger distortion than uniaxial strain. Including electron-electron interactions through a self-consistent Hartree potential shows that strain broadens the bare bands while reducing electrostatic renormalization. Strain also drives topological transitions as the narrow and remote bands hybridize, establishing twisted and strained bilayer graphene as a tunable platform for flat-band and topological phenomena.

Figures (12)

• Figure 1: (a) Moiré pattern of bilayer graphene with a twist angle $\theta=3.89{}^{\circ}$ and uniaxial strain $\epsilon_{u}=3.7\%$. The atoms in the top and bottom layers are plotted with blue and red dots, respectively. The AA, AB and DW stacking regions are labeled. Due to the strain the AA regions are elliptical. (b) Illustration of the moiré Brillouin zone, with the reciprocal lattice vectors labeled by $\mathbf{G}_1$ and $\mathbf{G}_2$. We label six corner $K$ points. The red dashed line is the momentum path for the band structure plots. (c), (d), (e) Schematically show the uniaxial, biaxial and shear strains, respectively. Only the top layer (blue line) is deformed. The undeformed bottom layer is plot with red lines. (f) Schematic construction of the moiré cell with the lattice vectors of the two graphene layers. (g) Schematic construction of the moiré cell in three steps by following the Eq. (\ref{['strainvec']}) with $\mathcal{E} \to \mathcal{E}_{b}+\mathcal{E}_{u/s}$: (1) an isotropic rescaling corresponding to $\epsilon_{b}$; (2) an anisotropic rescaling corresponding to $\epsilon_{u}$ in a direction given by $\phi$; (3) a rotation by an angle $\theta$. We only plot the first two steps in (g).
• Figure 2: (a) Position of the Dirac points projected within the mBZ, for the commensurate solutions of magic angle $\theta\sim1.05^{\circ}$ without strain. (b) The same case as (a) for uniaxial strain $\epsilon_{u}\sim0.1\%$. (c) The same case as (a) for shear strain $\epsilon_{s}\sim0.1\%$. In each case, plots on the left and middle sides show the Dirac points in the top (red) and bottom (blue) layers, and the mBZ periodically translated from the origin for the $K$ and $K^{\prime}$ valleys, respectively. Plots on the right side show the position of the Dirac points within the mBZ, when translated by the moiré vectors. In (b) and (c), the positions of Dirac points from the top strained layer are slightly shifted from the five possible positions in the mBZ.
• Figure 3: Band structure and DOS for the commensurate structures of TBG with $\theta=1.05{}^{\circ}$, $\phi=0{}^{\circ}$ and (a) no strain, (b) uniaxial strain $\epsilon_{u}=0.1\%$, (c) shear strain $\epsilon_{s}=0.1\%$, calculated by the TB (top panel) and continuum (bottom panel) models. In the TB band structures of (b) and (c), the color represents the expectation of the valley operator, with $\langle \hat{V}_z \rangle \approx 1$ if a state belongs to valley $K$ (red line) and $\langle \hat{V}_z \rangle \approx-1$ if a state belongs to valley $K'$ (black line). The momentum path is illustrated in Fig. \ref{['fig_structure']}(b).
• Figure 4: Evolution of DOS with uniaxial strain direction $\phi$ in TSBG with $\theta=1.05{}^{\circ}$ and $\epsilon_u=0.3\%$, calculated by the TB (left side) and continuum (right side) models. We label $E=0\ \text{meV}$ with black vertical dashed lines. The curves are relatively shifted to make the plot clear.
• Figure 5: Band structure and DOS for relaxed TBG with $\theta=1.05{}^{\circ}$ and (a) no strain, (b) uniaxial strain $\epsilon_{u}=0.1\%$, (c) shear strain $\epsilon_{s}=0.1\%$. The band structure and DOS of rigid cases are plotted with gray dots. The colors in the band structure are the same as in Figure \ref{['fig_uni_band']}. Note that, in the plots, the energy range in the relaxed case is almost two times larger than the energy range of the rigid case.