Gap opening in graphene by shear strain

We exploit the concept of strain-induced band-structure engineering in graphene through the calculation of its electronic properties under uniaxial, shear, and combined uniaxial-shear deformations. We show that by combining shear deformations to uniaxial strains it is possible modulate the graphene energy-gap value from zero up to 0.9 eV. Interestingly enough, the use of a shear component allows for a gap opening at moderate absolute deformation, safely smaller than the graphene failure strain.

Figure 1

(Color online) Top view of the hexagonal graphene lattice with its lattice vectors ${\vec{a}}_{1,2}=a_0\left(\frac{3}{2},\pm \frac{\sqrt{3}}{2}\right)$, where $a_0$ is the equilibrium C-C distance. Axes $x$ and $y$ correspond to the armchair and zigzag directions, respectively. Shaded area represents the unit cell.

Figure 2

(Color online) Brillouin zone of graphene under strain. The shaded areas are the corresponding irreducible part: (a) undeformed BZ with $6/mmm$ hexagonal symmetries; (b) BZ deformed by uniaxial strain with $mmm$ rhombic symmetry; and (c) BZ deformed by shear strain with $2/m$ monoclinic symmetry.

Figure 3

(Color online) Top of the valence band (red, marked as VB) and bottom of the conduction band (green, marked as CB) of graphene under uniaxial strain. Panel a: band structure of the undeformed lattice. Panels b and c: band structure under uniaxial strain along the armchair and the zigzag directions, respectively. Symbols connect the high-symmetry points of the BZ (bottom shaded area) to the energy of the corresponding electronic states.

Figure 4

(Color online) Top of the valence band (red, marked as VB) and bottom of the conduction band (green, marked as CB) of graphene under pure shear strain with $\zeta =20\mathrm{\%}$. Symbols connect the high-symmetry points of the BZ (bottom shaded area) to the energy of the corresponding electronic states.

Figure 5

(Color online) Density of states around the Fermi level (set conventionally at 0 eV) as function of the strain parameter $\zeta$. Panel (a): graphene under pure shear deformation. Panel (b): graphene under combined shear and uniaxial deformation (along the armchair direction). The maximum value of the energy gap is observed for a strain parameter as large as $\zeta \simeq 20\mathrm{\%}$ and $\zeta \simeq 17\mathrm{\%}$, respectively.

Figure 6

(Color online) Top of the valence band (red, marked as VB) and bottom of the conduction band (green, marked as CB) of graphene under combined shear and uniaxial strain with $\zeta =15\mathrm{\%}$. The uniaxial component of the strain is applied along the zigzag (panel a) and armchair (panel b) direction. Symbols connect the high-symmetry points of the BZ (bottom shaded area) to the energy of the corresponding electronic states.