New Generation of Massless Dirac Fermions in Graphene under External Periodic Potentials

We show that new massless Dirac fermions are generated when a slowly varying periodic potential is applied to graphene. These quasiparticles, generated near the supercell Brillouin zone boundaries with anisotropic group velocity, are different from the original massless Dirac fermions. The quasiparticle wave vector (measured from the new Dirac point), the generalized pseudospin vector, and the group velocity are not collinear. We further show that with an appropriate periodic potential of triangular symmetry, there exists an energy window over which the only available states are these quasiparticles, thus providing a good system to probe experimentally the new massless Dirac fermions. The required parameters of external potentials are within the realm of laboratory conditions.

Figure 1

Schematic diagram showing an equienergy contour (ellipse) with $E=\hslash v_0{k}_0+\hslash v_{0}m{G}_0/2$ of the newly generated massless Dirac fermions. The quasiparticle wave vector $\mathbf{k}$, the generalized pseudospin vector (see text) $\mathbf{c}$, and the group velocity vector ${\mathbf{v}}_g$ are represented by solid arrows, dashed arrows, and dash-dotted arrows, respectively, for graphene in an even periodic potential.

Figure 2

(a) A TGS with muffin-tin type of periodic potential with a spatial period $L$. The potential is $U_0$ inside the gray disks with diameter $d$ and zero outside. (b) The SBZ of a TGS. (c) The energy separation $\Delta E$ between states in the first and the second band above the original Dirac point energy versus the wave vector $k$ along the path $\tilde{K}\tilde{M}{\tilde{K}}^{\prime }$ in a TGS given by $U_0=0.5\mathrm{eV}$, $L=10\mathrm{nm}$, and $d=5\mathrm{nm}$.

Figure 3

(a) Energy dispersion relation of a TGS with external potential with $U_0=0.5\mathrm{eV}$, $L=10\mathrm{nm}$, and $d=5\mathrm{nm}$ for the first and the second band above the original Dirac point energy as a function of wave vector $\mathbf{k}$ from the original Dirac point. Arrows indicate the $\tilde{M}$ points of the SBZ around which new massless Dirac fermions are generated. (b) The DOS of charge carriers in electron orbits (bright and red) and hole orbits (dark and blue) in the TGS characterized in (a). The original Dirac point energy is set at zero. Dashed black line shows the DOS of pristine graphene. The arrow indicates the new Dirac point energy.