Transport in Quantum Wires

Abstract

In various circumstances, the electric field in an active device region may not only be inhomogeneous, but also too large for applying linear response theory. In such cases the difference between the initial state of the circuit, that is often an equilibrium state, and the final steady state cannot be regarded as being caused by a small perturbation. However, as was explained in Chap. 10, if only the steady state is addressed, one may construct a heuristic initial density matrix ϱ ₀ corresponding to a non-equilibrium state that is ‘close’ to the real steady state. By ‘close’ we mean that, by construction, both the heuristic density matrix and the true density matrix are sharing a finite set of global observables. The underlying idea is that in such a framework the time evolution leading from the heuristic state to the true steady state can be treated perturbatively as far as the finite set of common observables is concerned. In the case of homogeneous and static fields [48, 49, 50, 123, 124, 68] the most natural candidate for ϱ ₀ turned out to be based on a boosted Fermi-Dirac distribution ϱ ^B₀ describing a non-interacting electron gas at an elevated electron temperature Te and carrying a uniform drift velocity v _D, i.e.

$$F(E(k) - \hbar k \cdot {{\upsilon }_{D}} - \mu ,{{T}_{e}}) = {{\left[ {1 + \exp \left( {\frac{{E(k) - \hbar k \cdot {{\upsilon }_{D}} - \mu }}{{{{k}_{B}}{{T}_{e}}}}} \right)} \right]}^{{ - 1}}},$$

(1)

where E(k) and μ respectively denote the one-electron dispersion relation and the equilibrium chemical potential or Fermi level.