Quantum electron transport beyond linear response

After a brief historical survey and summary of previous work involving the linear response theory (LRT) Hamiltonian, H=H(system)−AF(t), where the latter part refers to the effect of an applied field F(t) as in Kuboʹs theory, we deal with the nonlinear problem in which the applied electric field of arbitrary strength, as well as a possible magnetic field, are included in the system Hamiltonian, H=H[system(E,B)]. In Part A of this study we deal with the general formalism on the many-body level. Projection operators are applied to the Von Neumann equation in the interaction picture, H(system)=H0+λV, which after the Van Hove limit, λ→0, t→∞, λ2t finite, leads to the master equation for ∂ρ/∂t, containing both the Pauli–Van Hove diagonal part involving the transitions Wγγ′, and a nondiagonal quantum-interference part, as obtained by us previously. Likewise, the full many-body current operator JA is obtained by manipulation of the Heisenberg equation of motion for dA/dt in the interaction picture. The second division, Part B, concerns the quantum Boltzmann theory. Employing the formalism of second quantization (c†,c fermion operators, a†,a boson operators) we obtain a general quantum Boltzmann equation for an electron gas in interaction with phonons or impurities in the adiabatic approximation (no phonon drag). On the ‘one-body’-level the current operator leads to a generalized Calecki form for arbitrary electric and magnetic fields. For extended states this transport equation yields the Boltzmann transport equation (BTE) for the Wigner occupancy function f(k,r,t) of a Fermi gas, thus furnishing a definitive proof, fully rooted in the quantum description, for Boltzmannʹs much criticized irreversible BTE of 1871. For localized states Titeicaʹs foreseen ‘collisional current’ (1935) is established in terms of the matrix elements of the state-centers and the hopping probabilities wζ ζ′. Also, for the limiting case of small fields, qEΔRζ ζ′ kT, our previous LRT results as well as the well-known results for the magneto-conductivity by Adams and Holstein and by Argyres and Roth, are recovered.