Strain effect on the optical conductivity of graphene

Within the tight-binding approximation, we study the dependence of the electronic band structure and of the optical conductivity of a graphene single layer on the modulus and direction of applied uniaxial strain. While the Dirac-cone approximation, albeit with a deformed cone, is robust for sufficiently small strain, band dispersion linearity breaks down along a given direction, corresponding to the development of anisotropic massive low-energy excitations. We recover a linear behavior of the low-energy density of states, as long as the cone approximation holds, while a band gap opens for sufficiently intense strain, for almost all, generic strain directions. This may be interpreted in terms of an electronic topological transition, corresponding to a change in topology of the Fermi line, and to the merging of two inequivalent Dirac points as a function of strain. We propose that these features may be observed in the frequency dependence of the longitudinal-optical conductivity in the visible range, as a function of strain modulus and direction, as well as of field orientation.

Figure 1

(Color online) Schematic representation of the strained (red) vs unstrained (blue) honeycomb lattice for $\theta =\pi /4$. Open (closed) symbols refer to the A (B) sublattices, respectively.

Figure 2

(Color online) Contour plots of the dispersion relations within the 1BZ for the valence band, $E_{\mathbf{k}1}$ (left panel), and conduction band, $E_{\mathbf{k}2}$ (right band), Eq. (9). Here, we are depicting the situation corresponding to a strain modulus of $\epsilon =0.18$ along the generic direction $\theta =\pi /4$. Solid blue lines are separatrix lines and occur at an electronic topological transition, dividing groups of contours belonging to different topologies. Either line passes through one of the critical points $M_{\ell }$ $\left(\ell =1,2,3\right)$, defined as the middle points of the 1BZ edge (solid black hexagon).

Figure 3

(Color online) Showing the DOS prefactor ${\rho }_1$, Eq. (19), normalized with respect to its value ${\rho }_1^{\left(0\right)}$ in the absence of strain, as a function of the strain modulus $\epsilon$, for various strain angles. The strain direction $\theta$ increases from $\theta =0$ (armchair direction, corresponding to the lowest curve) to $\theta =\pi /6$ (topmost curve). All other cases can be reduced to one of these exploiting the symmetry properties of the lattice.

Figure 4

(Color online) Energy dependence of the DOS over the whole bandwidth for increasing strain modulus $\epsilon =0–0.25$ and fixed strain direction $\theta =0$ (top panel) and $\theta =\pi /6$ (bottom panel). In both cases, the DOS slope close to the Fermi energy increases as a function of strain. However, while the DOS remains gapless for $\theta =0$, a nonzero gap opens around $E=0$ at a critical strain for $\theta =\pi /6$.

Figure 5

(Color online) Energy dependence of the DOS over the full bandwidth for fixed strain modulus $\epsilon =0.25$ and varying strain direction.

Figure 6

(Color online) Polar plot of $v_{\lambda }\left(\varphi \right)$ (with $\lambda =2$, i.e., for the conduction band) around ${\mathbf{k}}_D$, $v_{\lambda }\left(\varphi \right)$, normalized with respect to its value in the absence of strain, $v_f^{\left(0\right)}$. Strain is here applied at a generic fixed angle $\theta =\pi /4$. The anisotropy of the Fermi velocity increases with increasing strain, until the shape of $v_{\lambda }\left(\varphi \right)$ breaks at $\epsilon =0.28$. This corresponds to the existence of a direction (solid blue line), Eq. (21), along which the dispersion relation $E_{\mathbf{q}\lambda }$ displays a nonlinear character.

Figure 7

(Color online) Polar plots of longitudinal-optical conductivity ${\sigma }_{ll}/{\sigma }_0$, Eq. (29), as a function of frequency $\omega >0$ (polar axis) and electric field orientation $\varphi$ (azimuthal direction). Here, we set $\mu =0$ and $k_{\text{B}}T=0.025\text{eV}$. Strain is applied along the $\theta =0$ (armchair) direction, and the strain modulus increases from left to right, and from top to bottom $\left(\epsilon =0,0.075,0.175,0.275\right)$.

Figure 8

(Color online) Longitudinal-optical conductivity ${\sigma }_{ll}/{\sigma }_0$, Eq. (29), as a function of frequency $\omega >0$, for fixed strain modulus $\epsilon =0.1$ and strain direction $\theta =0$ (armchair). Different lines refer to various orientations of the electric field $\left(\varphi =0,\pi /4,\pi /2\right)$.

Figure 9

(Color online) Same as Fig. 7 but for strain applied along the $\theta =\pi /6$ direction.

Figure 10

(Color online) Same as Fig. 8 but for strain applied along the $\theta =\pi /6$ direction.

Figure 11

(Color online) Same as Fig. 7 but for strain applied along the $\theta =\pi /4$ direction.

Figure 12

(Color online) Same as Fig. 8 but for strain applied along the $\theta =\pi /4$ direction.

Figure 13

(Color online) Showing the scaled position of the center $\left(q_{x0}/E,q_{y0}/E\right)$ of an elliptical section of the Dirac cone around ${\mathbf{k}}_D$ at constant energy $E$, Eq. (A1). Each line refers to a given strain angle $\theta$ with respect to the lattice $x$ axis, and varying strain modulus $\epsilon =0–0.2$. E.g., the directions $\theta =0$ and $\theta =\pi /6$ correspond to the vertical bottom and horizontal left line, respectively.