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Physical review. B, Condensed matter

Steady-state quantum kinetic equation

Supriyo Datta

Published Sep 15, 1989• 96 citations
Publisher

Abstract

Starting from the Dyson equation in the Keldysh formulation, we derive a kinetic equation for steady-state quantum transport under the simplifying assumption that the inelastic scattering is caused by uncorrelated point scatterers, such as impurities with internal degrees of freedom. This assumption allows us to write a transport equation that involves only the electron density and not the spatial correlations of the wave function. Assuming local thermodynamic equilibrium we then simplify the transport equation to a form which resembles the B\"uttiker-Landauer formula extended to include a continuous distribution of probes.

DOI
10.1103/physrevb.40.5830

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Starting from the Dyson equation in the Keldysh formulation, we derive a kinetic equation for steady-state quantum transport under the simplifying assumption that the inelastic scattering is caused by uncorrelated point scatterers, such as impurities with internal degrees of freedom. This assumption allows us to write a transport equation that involves only the electron density and not the spatial correlations of the wave function. Assuming local thermodynamic equilibrium we then simplify the transport equation to a form which resembles the B"uttiker-Landauer formula extended to include a continuous distribution of probes.
A description of quantum transport for electrons in the presence of a random impurity potential is provided in terms of a kinetic equation, which is obtained from the diagrammatic structure for the linear response functions. When the electron-electron interaction is neglected, the kinetic equation is of the Boltzmann type, with an effective scattering kernel that signals localization effects. With the further inclusion of the interaction, a suitable distribution function for the energy of the quasiparticles can be introduced even in the presence of strong disorder. In this case the kinetic equation has the diffusive form, with diffusion coefficient and Landau interaction parameters that get renormalized by the interplay of disorder and interaction. The quasiparticles kinetic equation is also shown to contain all known results for the hydrodynamic correlation functions.
Collective orders and photoinduced phase transitions in quantum matter can evolve on timescales which are orders of magnitude slower than the femtosecond processes related to electronic motion in the solid. Quantum Boltzmann equations can potentially resolve this separation of timescales, but are often constructed by assuming the existence of quasiparticles. Here we derive a quantum Boltzmann equation which only assumes a separation of timescales (taken into account through the gradient approximation for convolutions in time), but is based on a nonperturbative scattering integral, and makes no assumption on the spectral function such as the quasiparticle approximation. In particular, a scattering integral corresponding to nonequilibrium dynamical mean-field theory is evaluated in terms of an Anderson impurity model in a nonequilibrium steady state with prescribed distribution functions. This opens the possibility to investigate dynamical processes in correlated solids with quantum impurity solvers designed for the study of nonequilibrium steady states.
A formulation of the Kadanoff-Baym-Keldysh theory of nonequilibrium quantum statistical mechanics is developed in order to describe nonperturbatively the effects of the electric field on electron-phonon scattering in nondegenerate semiconductors. We derive an analytic, gauge-invariant model for the spectral density of energy states that accounts for both intracollisional field effect and collisional broadening simultaneously. A kinetic equation for the quantum distribution function is derived and solved numerically. The nonlinear drift velocity versus applied field characteristics is also evaluated numerically. Many features of our nonlinear theory bear formal resemblance to linear-response theory.
A single-time quantum transport equation, which includes effects beyond the quasiparticle approximation, is derived for Fermi systems in the framework of nonequilibrium real-time Green's functions theory. Ternary correlations are incorporated in the kinetic description via a cluster expansion for the self-energies (e.g., the transport vertex and the width) truncated at the level of three-body scattering amplitudes. A finite temperature/density formulation of the three-body problem is given. Corresponding three-body equations reduce to the well-known Faddeev equations in the vacuum limit. In equilibrium the equation of state contains virial corrections proportional to the third quantum virial coefficient.
We investigate the potential of a quantum Boltzmann equation without momentum conservation, i.e., a kinetic equation for the local distribution function, for describing correlated electron systems out of equilibrium. The equation can be derived either for the dynamics of well-defined quasi- particles, neglecting the momentum conservation of the scattering in the spirit of dynamical mean field theory (DMFT), or, equivalently, as the quantum Boltzmann equation for the local occupied density of states in non-equilibrium DMFT. The approach may thus be applied even in cases where well-defined quasiparticles do not exist, with the main assumption that the spectral function remains rigid under the non-equilibrium evolution. We compare the result of the Boltzmann equation to non-equilibrium DMFT for the case of photo-carrier relaxation in Mott insulators, where processes on very different timescales emerge, i.e., impact ionization, intra-Hubbard-band thermalization, and full thermalization. Since the Boltzmann without momentum conservation is much simpler, it may allow us to simulate more complicated systems, whole devices, and much longer times.
We develop the theory of quantum transport of electrons through systems with strong correlations between fermionic and internal spin degrees of freedom. The atomic representation for the Hamiltonian of a device and nonequilibrium Green’s functions constructed using the Hubbard operators allow overcoming difficulties in the perturbation theory encountered in the traditional approach because of a larger number of bare scattering amplitudes. Representing the matrix elements of effective interactions as a superposition of terms each of which is split in matrix indices, we obtain a simple method for solving systems of very many equations for nonequilibrium Green’s functions in the atomic representation. As a result, we obtain an expression describing the electron currents in a device one of whose sites is in tunnel coupling with the left contact and the other, with the right contact. We derive closed kinetic equations for the occupation numbers under conditions where the electron flow leads to significant renormalization of them.