Kinetic theory of quantum transport at the nanoscale

We present a quantum-kinetic scheme for the calculation of non-equilibrium transport properties in nanoscale systems. The approach is based on a Liouville-master equation for a reduced density operator and represents a generalization of the well-known Boltzmann kinetic equation. The system, subject to an external electromotive force, is described using periodic boundary conditions. We demonstrate the feasibility of the approach by applying it to a double-barrier resonant tunneling structure.

Figure 1

Upper panel: the DBRTS in the absence of an electric field. Middle panel: total potential in the presence of external electric field for low ${\gamma }_0$ (see text). Lower panel: spatial distribution of the current. Short dashed line: Hamiltonian current $j\left(x\right)$. Long dashed line: current due to inelastic collisions $j_L\left(x\right)$. Solid line: total current $j_T\left(x\right)=j\left(x\right)+j_{\mathcal{L}}\left(x\right)$. Within the numerical precision of the calculation $j_T$ is constant as required by continuity (5). All plots use a nonlinear scale for the position to magnify the barrier region.

Figure 2

Two calculated $I\text{−}V$ characteristics of a DBRTS. The curve with the highest peak current (solid line, circles) is the one corresponding to the lower value of ${\gamma }_0$. The energy of the dashed line on the left corresponds to twice the energy difference between the resonant level and the Fermi level. The second dashed vertical line is displaced to the right by twice the energy bandwidth. According to a simple model of resonance, transmission should occur in the region between the two lines.

Figure 3

The diagonal elements $f_{nn}$ of the density matrix in the steady state. Stronger bias leads to a stronger deviation from the unperturbed Fermi-Dirac (FD) distribution.