Conductance and polarization in quantum junctions

We revisit the expression for the conductance of a general nanostructure—such as a quantum point contact—as obtained from the linear-response theory. We show that the conductance represents the strength of the Drude singularity in the conductivity $\sigma \left(k,k^{\prime };i\omega \to 0\right)$. Using the equation of continuity for electric charge we obtain a formula for conductance in terms of polarization of the system. This identification can be used for direct calculation of the conductance for systems of interest even at the ab initio level. In particular, we show that one can evaluate the conductance from calculations for a finite system without the need for special “transport” boundary conditions.

Figure 1

Semiclassical interpretation of a finite-field potential $V\left(x\right)$ applied to a wire. The local Fermi spheres are depicted as for 2D gas for clarity. The filled area on top of the equilibrium spheres indicates accelerated electrons, while the empty area inside the equilibrium spheres represents states emptied by the deccelerating effect of the uphill field.

Figure 2

The dependence of conductance of free-electron 1D gas on imaginary frequency. Black line corresponds to the analytical result (20) and the numerical data labels “$L\text{−}N$” represent system of length $L\text{a.u.}$ with $N$ discretization points in real space. Apart from the low-energy drop to zero, which is characteristic of a finite system, the functions converge to the exact result for the infinite system. This opens the possibility of calculating the dc conductance by extrapolation from moderate values of $\alpha$.

Figure 3

The dependence of conductance, given by Landauer formula, on the Fermi energy ($k_F=\pi ∕3$ in our calculations) for a square barrier of height $e_F$ and width ${\lambda }_F∕3$ (upper graph) and the dependence of the numerically calculated conductance on the imaginary energy (lower graph). $G\left(\alpha \right)$ approaches the exact result of infinite system before it turns rapidly towards zero.