Understanding electron behavior in strained graphene as a reciprocal space distortion

The behavior of electrons in strained graphene is usually described using effective pseudomagnetic fields in a Dirac equation. Here we consider the particular case of a spatially constant strain. Our results indicate that lattice corrections are easily understood using a strained reciprocal space, in which the whole energy dispersion is simply shifted and deformed. This leads to a directional-dependent Fermi velocity without producing pseudomagnetic fields. The corrections due to atomic wave function overlap changes tend to compensate such effects. Also, the analytical expressions for the shift of the Dirac points, which do not coincide with the $K$ points of the renormalized reciprocal lattice, as well as the corresponding Dirac equation are found. In view of the former results, we discuss the range of applicability of the usual approach of considering pseudomagnetic fields in a Dirac equation derived from the old Dirac points of the unstrained lattice or around the $K$ points of the renormalized reciprocal lattice. Such considerations are important if a comparison is desired with experiments or numerical simulations.

Figure 1

(a) Unstrained graphene lattice showing the vectors ${\delta }_i$ that point to the neighbors of type $A$ sites, (b) the same lattice under a uniform stress, and (c) the first Brillouin zone of the reciprocal lattice for unstrained (dashed lines) and strained (solid lines) graphene. Note how the reciprocal lattice is contracted in the direction where the lattice is stretched, and the change of the ${\mathit{K}}_0$ symmetry point into $\mathit{K}$. (d) How the distortion of the reciprocal lattice transforms the original Dirac cone (left) into a distorted one (right) with a directional-dependent Fermi velocity.

Figure 2

Isoenergetic curves obtained from the energy dispersion given by Eq. (9). A blowup is presented around the Dirac points ${\mathit{K}}_D$ for each surface. (a) Unstrained graphene; (b) strained graphene with $\beta =0,{\epsilon }_{xx}=0.05,{\epsilon }_{xy}=0,{\epsilon }_{yy}=-\nu {\epsilon }_{xx}$; and (c) strained graphene with $\beta \approx 3$ and the same strain tensor as in (b). Note how although the strain tensor is the same in (b) and (c), the ellipses are rotated by $\pi /2$, since the reciprocal space deformation and hooping effects tend to compensate.

Figure 3

The Dirac cone in unstrained (dashed line) and strained (solid line) graphene and the experimental observation of electron behavior for a probe that shifts the chemical potential ($\mu$) with respect to the Fermi energy. The shaded (green) rectangle indicates the width of the thermal selector due to the Fermi-Dirac distribution. The effective Dirac equation with pseudomagnetic fields can be obtained by developing around the original ${\mathit{K}}_0$ points or in the Dirac points ${\mathit{K}}_D$ of the strained lattice. For $\mu =0$, only the latter approach will work for low temperatures.