Effective contact model for geometry-independent conductance calculations in graphene

A geometry-independent effective model for the contact self-energies is proposed to calculate the quantum conductance of patterned graphene devices using Green's functions. A Corbino disk, being the simplest device where the contacts cannot be modeled as semi-infinite ribbons, is chosen to illustrate this approach. This system's symmetry allows an analytical solution against which numerical calculations on the lattice can be benchmarked. The effective model perfectly describes the conductance of Corbino disks at low-to-moderate energies, and is robust against the size of the annular device region, the number of atoms on the edge, external magnetic fields, or electronic disorder. The contact model considered here affords an expedient, flexible, and geometry-agnostic approach that easily allows the consideration of device dimensions encompassing several million atoms, and realistic radial dimensions of a few hundreds of nanometers.

Figure 1

Left panel: lattice representation of a graphene Corbino disk of inner radius $R_i$ and outer radius $R_o$ contacted to an effective contact. Right panel: a closeup of an edge region showing a schematic of graphene atoms bound to the effective contact.

Figure 2

(a) Energy dependence of the conductance normalized by $R_i$ for different values of $R_i$ and $R_o$, calculated using Eq. (6) (labeled “D”) and the lattice Green's function approach (labeled “G”). (b) Conductance of a Corbino Disk with $R_i=66$ nm and $R_o=76$ nm using the Green's function approach with the effective self-energy term ${\Sigma }_q=-i\Delta$, for different values of $\Delta$. The line labeled by “D” was calculated using Eq. (6)

Figure 3

(a) Conductance of a Corbino disk of fixed $R_i=1.5$ nm and varying width $W=R_o-R_i$. The inset shows the conductance at the Dirac point as function of the width. (b) Conductance of a Corbino disk of fixed width $W=10$ nm and different values of $R_i$ and $R_o$.

Figure 4

(a) Radial conductivity of a Corbino disk with fixed width $W=10$ nm and different values of $R_i$ and $R_o$. The black dot in the vertical axis marks the value of the universal minimum of conductivity, ${\sigma }_0=4e^2/\pi h\approx 0.6(2e^2/h)$. (b) Conductance of a Corbino disk having $R_i=66$ nm and $R_o=76$ nm for different values of an external, perpendicular, and homogeneous magnetic field, $B$. The panels in the second row show the conductance of a Corbino disk having $R_i=28$ nm and $R_o=38$ nm with on-site disorder (c), and a finite density of vacancies (d). In (e) and (f) we present a closeup of the low-energy region for the curves in (b) and (d), respectively.

Figure 5

Conductance of a ballistic Corbino disk ($R_i=0.1$ $\mu$m and $R_o=0.2$ $\mu$m) in GaAs as a function of the Fermi energy, $E_F$. The curve was obtained using the effective contact Green's function Eq. (A4). Inset: schematic of the polar grid used in the finite difference decomposition of the Schrödinger equation.