Strain-induced band gaps in bilayer graphene

We present a tight-binding investigation of strained bilayer graphene within linear elasticity theory, focusing on the different environments experienced by the $A$ and $B$ carbon atoms of the different sublattices. We find that the inequivalence of the $A$ and $B$ atoms is enhanced by the application of perpendicular strain ${\varepsilon }_{zz}$, which provides a physical mechanism for opening a band gap, most effectively obtained when pulling the two graphene layers apart. In addition, perpendicular strain introduces electron-hole asymmetry and can result in linear electronic dispersion near the $K$ point. Our findings suggest experimental means for strain-engineered band gaps in bilayer graphene.

Figure 1

Graphene bilayer, with $A$ carbon atoms in black and $B$ carbon atoms in gray (top). Electronic spectrum near the $K$ point in absence (bottom left) and presence (bottom right) of a bias voltage $W$ between the layers.

Figure 2

Low-$k$ band structure $E\left(k\right)$ for perpendicularly uniformly strained bilayer graphene for various values of $|c\frac{\partial V}{\partial c}{\varepsilon }_{zz}|=f(t_{\perp }+c\frac{\partial t_{\perp }}{\partial c}{\varepsilon }_{zz})$.

Figure 3

Plots of the variation of $\frac{t_{\perp }}{c}(\frac{1}{{\varepsilon }_{zz}}-2)$ (blue, full line, expansion) and $\frac{t_{\perp }}{c}(\frac{1}{{\varepsilon }_{zz}}+2)$ (violet, dashed line, contraction) with ${\varepsilon }_{zz}$. The two horizontal lines represent the values of $|\frac{\partial V}{\partial c}|$ resulting from using the two extreme values for $V$ considered in the present work—$V=-2$ eV (olive, dashed-dotted line) and $V=-0.25$ eV (green, dashed line).

Figure 4

Plots of the variation of $V\left(c^{\prime }\right)$ with $c^{\prime }=c(1+{\varepsilon }_{zz})$ for $V\left(c\right)=-0.25$ eV (blue, full line) and $V\left(c\right)=-2$ eV (violet, dashed line).