Algorithm for efficient elastic transport calculations for arbitrary device geometries

With the growth in interest in graphene, controlled nanoscale device geometries with complex form factors are now being studied and characterized. There is a growing need to understand new techniques to handle efficient electronic transport calculations for these systems. We present an algorithm that dramatically reduces the computational time required to find the local density of states and transmission matrix for open systems regardless of their topology or boundary conditions. We argue that the algorithm, which generalizes the recursive Green's function method by incorporating the reverse Cuthill-McKee algorithm for connected graphs, is ideal for calculating transmission through devices with multiple leads of unknown orientation and becomes a computational necessity when the input and output leads overlap in real space. This last scenario takes the Landauer-Buttiker formalism to general scattering theory in a computational framework that makes it tractable to perform full-spectrum calculations of the quantum scattering matrix in mesoscopic systems. We demonstrate the efficacy of these approaches on graphene stadiums, a system of recent scientific interest, and contribute to a physical understanding of Fano resonances which appear in these systems.

Figure 1

The six sites of a model nearest-neighbor tight-binding system are shown with index labels. Lines between sites indicate off-diagonal entries in the Hamiltonian.

Figure 2

Recursive Green's function methods demonstrated on a nearest-neighbor single-orbital tight-binding Hamiltonian for a graphene stadium. Each block of the Hamiltonian is represented by different colors of atoms. In (a) and (b), we have a common two-terminal left-to-right system. In (a) the recursive algorithm is determined by LRGF and in (b) by the outward wave method. In (c) a single lead enters from the left for studying the full scattering matrix of reflected wave functions, as described in Sec. 3e. In (d), the outward wave method is applied to an ensemble situation, in which the entire boundary of the device region is treated as a potential location for attaching a lead, as described in Sec. 3d.

Figure 3

Transmission coefficient for the graphene stadium (Ref. 9) using the outward wave method (black) and the original data (gray dashed). Differences between the two data are shown at the bottom. Deviations arise from disparities in sampling points and the infinitesimal $\eta$ parameter. We publish results for a very small $\eta$ parameter of $2.7\times {10}^{-5}\text{eV}$.

Figure 4

(Top) Time of calculation for a single energy point for relativistic stadiums using full inversion (blue stars), LRGF (green crosses), and the outward wave method (red pluses). Standard deviations above and below are indicated by whisker bars. (Middle) Estimated time for transmission calculations, in arbitrary units, computed from the optimization function discussed in Sec. III A. (Bottom) Estimated memory requirements for each system. The clusters we used had a memory limit of 16 GB, which is indicated by the magenta dotted line. Our simulations suggest that the memory estimation for LRGF and the outward wave method are undervalued. For the transmission calculation, all three methods returned the same transmission coefficient at the energy point within the precision of the machine. The number of basis functions in each calculation is a linear function of the system size parameter.

Figure 5

The size of each submatrix, in order of submatrix index, for outward wave (solid red line) and LRGF (dotted green line) for stadiums of system size parameters 40 and 10 (inset).

Figure 6

Geometry for the Helmholtz resonator configuration.

Figure 7

Transmission function for the relativistic stadium (dashed line), its Helmholtz-resonator equivalent (solid line), and mode mixing in the single-lead reflection matrix (dotted line).