Transport in superlattices on single-layer graphene

We study transport in undoped graphene in the presence of a superlattice potential both within a simple continuum model and using numerical tight-binding calculations. The continuum model demonstrates that the conductivity of the system is primarily impacted by the velocity anisotropy that the Dirac points of graphene develop due to the potential. For one-dimensional superlattice potentials, new Dirac points may be generated, and the resulting conductivities can be approximately described by the anisotropic conductivities associated with each Dirac point. Tight-binding calculations demonstrate that this simple model is quantitatively correct for a single Dirac point, and that it works qualitatively when there are multiple Dirac points. Remarkably, for a two-dimensional potential which may be very strong but introduces no anisotropy in the Dirac point, the conductivity of the system remains essentially the same as when no external potential is present.

Figure 1

Schematic representation of the superlattice potentials used showing the axis selection. The system is infinite along the $x$ direction and has a finite length along the $y$ direction. The dashed patterns on each side of the $y$ direction indicate the leads for the Landauer conductance calculations. The superlattice barriers can be parallel (a) or perpendicular (b) to the direction of transport. We also consider a chessboard-like two-dimensional superlattice potential in which $d_y\simeq d_x$ (c).

Figure 2

(a) Schematic representation of the superlattice used in the tight-binding calculations showing the axis selection. The system is infinite along the $x$ direction and has a finite length $L=\sqrt{3}{\mathit{Na}}_0$ along the $y$ direction. The superlattice has a vertical period $d_{\text{SC}}=3M{a}_0$ in which periodic boundary conditions are imposed. We include a sketch of the Brillouin zone with the same axis selection for a potential applied along the $x$ direction (b) and the $y$ direction (c). The length of the reduced Brillouin zone ($2\pi /d_{\text{SC}}$) is indicated. The splitting of the Dirac points is also sketched. Note that the new Dirac points always move perpendicularly to the direction of the potential.

Figure 3

For a potential along the $x$ direction as depicted in Fig. 1a, in the left panels we plot the transmission $T(q)$ per spin channel as a function of $q{d}_{\text{SC}}$ for a superlattice of period $d=42a_0$ and amplitudes $\mathrm{Ṽ}=\frac{V_{0}d}{2\hslash v_F}<\pi$ (top left panel) and for a period $d=54a_0$ and amplitudes $\mathrm{Ṽ}>\pi$ (bottom left panel). Note that for this orientation of the potential, $d=d_{\text{SC}}$. In the right panels, we plot the distribution of the eigenvalues of the transmission matrix for the different values of $\mathrm{Ṽ}$. The length of the stripe is $L=100\sqrt{3}{a}_0$.

Figure 4

Transport parallel to the superlattice barriers. Top (bottom) panel shows the conductivity (Fano factor) for a graphene sheet with $L=500\sqrt{3}{a}_0$ and superlattice period $d=24a_0$ (solid blue line) and $L=200\sqrt{3}{a}_0$ and superlattice period $d=36a_0$ (dashed red line) as a function of the normalized barrier height $V_{0}d$. Dotted line corresponds to the conductivity obtained in the continuum model assuming independent transport channels, Eq. (8), in the top panel and to the pseudodiffusive value $F=1/3$ in the bottom panel.

Figure 5

For a potential along the $y$ direction as depicted in Fig. 1b, in the left panels we plot the transmission $T(q)$ per spin channel as a function of $q{d}_{\text{SC}}$ for a superlattice potential of period $d=38.1a_0$ and normalized amplitudes $\mathrm{Ṽ}=\frac{V_{0}d}{2\hslash v_F}<\pi$ (top left panel) and $\mathrm{Ṽ}>\pi$ (bottom left panel). In the right panels, we plot the distribution of the eigenvalues of the transmission matrix for the different values of $\mathrm{Ṽ}$. The length of the stripe is $L=500\sqrt{3}{a}_0$.

Figure 6

Transport perpendicular to the superlattice barriers. Top (bottom) panel shows the conductivity (Fano factor) for a graphene sheet with $L=500\sqrt{3}{a}_0$ and superlattice periods $d=34.6a_0$ (solid blue line) and $d=74.2a_0$ (dashed red line) as a function of the normalized barrier height $V_{0}d$. Dotted line corresponds to the conductivity obtained in the continuous model assuming independent transport channels, Eq. (8), in the top panel and to the pseudodiffusive value $F=1/3$ in the bottom panel.

Figure 7

In the top (bottom) panel, we plot the conductivity (Fano factor) as a function of the normalized barrier height $V_0{d}_y$ for a graphene sheet in the presence of a two-dimensional superlattice with a fixed vertical period of $d_x=48a_0$ and different horizontal periods $d_y=45a_0$ (blue solid line) and $d_y=48.5a_0$ (red dashed line). The dotted line corresponds to the conductivity of pristine graphene (${\sigma }_0$) in the top panel and to the pseudodiffusive limit ($1/3$) in the bottom panel. The length of the graphene sheet is $L=200\sqrt{3}{a}_0$. Inset: the conductivity as a function of the normalized barrier height in the proximity of the critical potential $V_c$ in which a new pair of Dirac points is created.