Riccati equation for simulation of leads in quantum transport

We present a theoretical procedure with numerical demonstration of a workable and efficient method to evaluate the surface Green's function of semi-infinite leads connected to a device. Such a problem always occurs in quantum transport calculations but also in the study of surfaces and heterojunctions. We show here that these semi-infinite leads can be properly described by real-energy Green's functions obtained analytically by a smart solution of the Riccati matrix equation. The performance of our method is demonstrated in the case of a multichain two-dimensional electron-gas system, composed of a central ribbon connected to two semi-infinite leads, pierced by two opposite magnetic fields.

Figure 1

Schematics of a device configuration for two-probe transport calculations: the central scattering region S is embraced by left and right leads periodic along the charge transport direction.

Figure 2

Schematic representation of the semi-infinite multiple chain right lead. The dimension of the intracolumn matrix $h_R$ and nearest-neighbor intercolumn matrices $C$ and $C^{†}$ equals the number of rows of the device.

Figure 3

Currents map for right injected carriers in a square topology ribbon composed of 240 chains; the scattering region is 330 lattice constants wide, and the bias ${\mu }_R-{\mu }_L$ is infinitesimal. In the system two magnetic fields are present. Both magnetic fields have a value of 5 T, but the one in the upper part of the system points toward the reader, while the other points in the opposite direction. In the system we have introduced a hardwall potential mimicking the $h/e^2$ shape.

Figure 4

Schematic representation of the semi-infinite two-chain right lead. (a) The intracolumn matrix is indicated by $h$, and the nearest-neighbor intercolumn matrices are indicated by $C$ and $C^{†}$. (b) System of two independent chains equivalent to the system represented in (a).