Tight-binding approach to uniaxial strain in graphene

We analyze the effect of tensional strain in the electronic structure of graphene. In the absence of electron-electron interactions, within linear elasticity theory, and a tight-binding approach, we observe that strain can generate a bulk spectral gap. However, this gap is critical, requiring threshold deformations in excess of 20% and only along preferred directions with respect to the underlying lattice. The gapless Dirac spectrum is robust for small and moderate deformations and the gap appears as a consequence of the merging of the two inequivalent Dirac points only under considerable deformations of the lattice. We discuss how strain-induced anisotropy and local deformations can be used as a means to affect transport characteristics and pinch off current flow in graphene devices.

Figure 1

(Color online) Tension geometry considered in the text. The zigzag direction of the honeycomb lattice is always parallel to the axis $Ox$.

Figure 2

(Color online) (a) Honeycomb lattice geometry. The vectors ${\mathit{\delta }}_1=a\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$, ${\mathit{\delta }}_2=a\left(0,1\right)$ ${\mathit{\delta }}_3=a\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$ connect $A$ sites (red/dark) to their $B$ site (blue/light) neighbors. (b) The first Brillouin zone of undeformed graphene with its points of high symmetry.

Figure 3

(Color online) (a) Plot of $t_1/t_2$ vs $t_3/t_2$ as a function of strain, $\epsilon$ and $\theta$. Closed lines are isostrain curves and arrowed lines correspond to the trajectory of the point $\left(t_1/t_2,t_3/t_2\right)$ as $\epsilon$ increases calculated at constant angle. The graph is symmetric under reflection on both axes. In the shaded area the spectrum is gapless. The blue isostrain line $\left(\epsilon \approx 0.23\right)$ corresponds to the gap threshold. In panel (b) we show the angular dependence of $t_{1,2,3}$ for $\epsilon =0.05$ and $\epsilon =0.23$.

Figure 4

(Color online) Top row shows density plots of the energy dispersion, $E\left(k_x,k_y\right)$, for $\left\{\epsilon =0,\theta =0\right\}$ (a), $\left\{\epsilon =0.2,\theta =\pi /2\right\}$ (b), and $\left\{\epsilon =0.2,\theta =0\right\}$ (c). In (d) we have a cut of (c) along $k_y=0$, showing the merging of the Dirac cones as strain increases and the ultimate appearance of the gap. In panel (e) we compare the gap given by Eq. (18) (line) with the result obtained from direct minimization of the energy in the full BZ (dots).

Figure 5

(Color online) (a) A close up of the energy dispersion close to its minimum for tension along $\theta =0$ and $\epsilon =0.05$. The solid white lines show the intersection of the Bragg planes that define the boundary of the first BZ in the deformed lattice, while the dashed white lines represent the same boundary lines in equilibrium. It is clear that the Dirac point lies neither at $\mathit{K}$ nor at its deformed counterpart. The energy contours are labeled in eV. (b) A contour plot of the absolute value of the Fermi velocity, $\hslash v_F={\nabla }_{\mathit{k}}E\left(\mathit{k}\right)$ for the same region shown in (a).

Figure 6

(Color online) Anisotropy in the Fermi velocity as a function of strain for a Fermi energy of 50 meV. The vertical axis shows the ratio of the maximum to the minimum values of $v_F$ in the entire Fermi surface.