Tunable band gap in graphene with a noncentrosymmetric superlattice potential

We show that, if graphene is subjected to the potential from an external superlattice, a band gap develops at the Dirac point provided the superlattice potential has broken inversion symmetry. As numerical example, we calculate the band structure of graphene in the presence of an external potential due to periodically patterned gates arranged in a triangular graphene superlattice (TGS) or a square graphene superlattice with broken inversion symmetry, and find that a band gap is created at the original and, in the case of a TGS, the “second generation” Dirac point. This gap, which extends throughout the superlattice Brillouin zone, can be controlled, in principle, by changing the external potential and the lattice constant of the superlattice. For a square superlattice of lattice-constant 10 nm, we have obtained a gap as large as 65 meV, for gate voltages no larger than 1.5 V.

Figure 1

(Color online) Schematic of graphene plane subjected to a periodic external potential $V\left(x,y\right)$ consisting of a periodic arrangement of circular contacts, within each of which $V$ is a constant. The large circles are arranged on a triangular lattice of edge $b$. The small circles are not midway between the large circles; for the case shown, they are centered at $0.4b\widehat{x}$ relative to the centers of the large circles.

Figure 2

(Color online) Same as Fig. 1, except that the large and small circular contacts are arranged on a square lattice. Each small circle is now centered at $0.375b\widehat{x}$ relative to the center of the nearest large circle.

Figure 3

(Color online) SBS for graphene subjected to an external potential produced by a triangular superlattice of contacts, as shown in Fig. 1. The calculation is carried out using $b=10\text{nm}$, $V_1=0.5\text{V}$, $V_2=-1.65\text{V}$, $r_1/b=0.25$, $r_2/b=0.1$, and the small circles are located at $0.4\widehat{x}$. The $\Gamma$ point of the superlattice corresponds to one of the $K$ points of the original graphene lattice. If only the large circular contacts were present, the SBS would have Dirac points at both $\Gamma$ and $M$ of the SBS Brillouin zone, at both of which the density of states would go linearly to zero. Inset: first Brillouin zone of a triangular lattice showing the three high-symmetry points $\Gamma$, $M$, and $K$.

Figure 4

(Color online) SBS for graphene subjected to an external potential produced by a square superlattice of contacts, as in Fig. 2. The $\Gamma$ point of the superlattice corresponds to one of the $K$ points of the original graphene lattice. In this case, we use $b=10\text{nm}$, $V_1=0.5\text{V}$, $V_2=-1.4\text{V}$, and $r_1/b=0.25$, $r_2/b=0.125$, and the small circles are located at $0.375\widehat{x}$. Inset: first Brillouin zone of a square lattice showing the three high-symmetry points $\Gamma$, $X$, and $M$.

Figure 5

(Color online) Plot of the gap at the Dirac point $\Gamma$ as a function of the voltage $V_2$ (in $V$) on the smaller circles. Full curve: square lattice; dashed curve: triangular lattice. For the square lattice, we use $b=10\text{nm}$, $V_1=0.5\text{V}$, $r_1/b=0.25$, and $r_2/b=0.125$, and small circles are located at $0.375\widehat{x}$. For the triangular lattice, $b=10\text{nm}$, $V_1=0.5\text{V}$, $r_1/b=0.25$, and $r_2/b=0.1$, and the small circles are located at $0.4\widehat{x}$.

Figure 6

(Color online) Plot of the gap at the Dirac point as function of $r_2/b$. Full curve: square lattice; dashed curve: triangular lattice. For the square lattice, $b=10\text{nm}$, $V_1=0.5\text{V}$, $V_2=-1.4\text{V}$, and $r_1/b=0.25$, and small circles are located at $0.375\widehat{x}$. For the triangular lattice, $b=10\text{nm}$, $V_1=0.5\text{V}$, and $V_2=-1.65\text{V}$, $r_1/b=0.25$, and small circles are located at $0.4\widehat{x}$.