Confining and repulsive potentials from effective non-Abelian gauge fields in graphene bilayers

We investigate the effect of shear and strain in graphene bilayers, under conditions where the distortion of the lattice gives rise to a smooth one-dimensional modulation in the stacking sequence of the bilayer. We show that strain and shear produce characteristic Moiré patterns which can have the same visual appearance on a large scale, but representing graphene bilayers with quite different electronic properties. The different features in the low-energy electronic bands can be ascribed to the effect of a fictitious non-Abelian gauge field mimicking the smooth modulation of the stacking order. Strained and sheared bilayers show a complementary behavior, which can be understood from the fact that the non-Abelian gauge field acts as a repulsive interaction in the former, expelling the electron density away from the stacking domain walls, while behaving as a confining interaction leading to localization of the electronic states in the sheared bilayers. In this latter case, the presence of the effective gauge field explains the development of almost flat low-energy bands, resembling the form of the zeroth Landau level characteristic of a Dirac fermion field. The estimate of the gauge field strength in those systems gives a magnitude of the order of several tens of tesla, implying a robust phenomenology that should be susceptible of being observed in suitably distorted bilayer samples.

Figure 1

Different types of Moiré patterns in graphene bilayers, which can be obtained (a) by means of shear, pulling laterally the layers in opposite directions, and (b) by applying tensile strain, pulling the layers in the direction perpendicular to the hexagon rows in the honeycomb lattice.

Figure 2

(a), (b) Low-energy bands around the Fermi level of an undoped bilayer of the type shown in Fig. 1, obtained by means of a tight-binding approximation for a geometry of infinite length along the vertical direction and length $L=210\sqrt{3}a$ ($a$ being the C-C distance) along the horizontal direction, with periodic (a) and open (b) boundary conditions. (c), (d) Plots obtained in similar fashion as (a), (b), representing the low-energy bands (from a given Dirac valley) of a graphene bilayer of the type shown in Fig. 1, for a geometry of infinite length along the vertical direction and length $L=633a$ along the horizontal direction with periodic (c) and open (d) boundary conditions. The energy $\varepsilon$ is given in eV and the momentum in units of the inverse of the C-C distance.

Figure 3

Local density from the states in the four low-energy bands closer to $\varepsilon =0$ in Fig. 2, at respective momenta [from (a) to (d)] $k_y=0.01\pi /a,0.014\pi /a,0.019\pi /a$, and $0.023\pi /a$ ($a$ being the C-C distance).

Figure 4

(a), (b) Local density from the states with $k_x=0$ in the linear branch close to zero energy (a) and at the bottom of the quadratic band right above zero energy (b) in the band structure shown in Fig. 2. (c), (d) Local density from the states with $k_x=0.014\pi /a$ ($a$ being the C-C distance) at the top of the quadratic band right below zero energy (c) and at the top of the quadratic band right above zero energy (d) appearing to the right in the band structure of Fig. 2.

Figure 5

(a) Zoom view of the low-energy part of the band structure shown in Fig. 2. The energy $\varepsilon$ is given in eV and the momentum in units of the inverse of the C-C distance. (b) Plot of the local density from the states at the crossing of the linear branches within the approximately flat bands in (a).

Figure 6

Zoom view of the low-energy bands for a sheared bilayer in the effective gauge field model, obtained with a single-harmonic approximation to the interlayer potentials for $\lambda =0.1$ eV and $\lambda L/v_F=6.2\pi$. The energy $\varepsilon$ is given in eV and the momentum in units of the inverse of the C-C distance.