Effects of strain on electronic properties of graphene

We present first-principles calculations of electronic properties of graphene under uniaxial and isotropic strains, respectively. The semimetallic nature is shown to persist up to a very large uniaxial strain of 30% except a very narrow strain range where a tiny energy gap opens. As the uniaxial strain increases along a certain direction, the Fermi velocity parallel to it decreases quickly and vanishes eventually, whereas the Fermi velocity perpendicular to it increases by as much as 25%. Thus, the low energy properties with small uniaxial strains can be described by the generalized Weyl’s equation while massless and massive electrons coexist with large ones. The work function is also predicted to increase substantially as both the uniaxial and isotropic strain increases. Hence, the homogeneous strain in graphene can be regarded as the effective electronic scalar potential.

Figure 1

(Color online) (a) Hexagonal lattice of graphene. ${\mathit{a}}_1$ and ${\mathit{a}}_2$ are the lattice vectors. With $Z\left(A\right)$ strain, ${\mathit{a}}_1=\left(a_x,a_y\right)$ and ${\mathit{a}}_2=\left(-a_x,a_y\right)$. ${\mathit{\delta }}_i\left(i=1,2,3\right)$ connects three nearest neighbors. (b) The first BZ with high symmetric points. Energy contours for graphene (c) without strain, (d) $A$ strain of 20%, and (e) $Z$ strain of 20%. The scale bar for contours is in unit of eV. The $\pi$ and ${\pi }^{\ast }$ bands with various (f) $A$ strains and (g) $Z$ strains along the line of $k_y=0$ in (b).

Figure 2

(Color online) Calculated band structures of the strained graphene around the $S$ point with a $Z$ strain of 24.0%, 26.2%, 26.5%, and 27.0% (from left to right panels), respectively.

Figure 3

(Color online) The group velocities $\left(v_{Ai}\right)$ of $\pi$ and ${\pi }^{\ast }$ electrons with the $A$ strain (a) and $v_{Zi}$ with the $Z$ strain along the direction $i$ ($=1,2,3,4$ in insets) in an unit of isotropic group velocity $\left(v_0\right)$ without strain. The angle between the direction A2 and A3 in inset of (a) is $52°$ and one between Z2 and Z3 in (b) is $38°$.

Figure 4

(Color online) Calculated density of states of graphene with (a) the $A$ strain and (b) $Z$ strain.

Figure 5

(Color online) Calculated work functions $\left(\Phi \right)$ of graphene with the $A$ and $Z$ strains.

Figure 6

(Color online) (a) Calculated Fermi velocity variation under the $I$ strain in an unit of the Fermi velocity $\left(v_0\right)$ without strain. (b) Calculated work functions $\left(\Phi \right)$ the $I$ strain. We set the work function without strain to zero here.