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arXiv:0810.5635 · Physical Review E

Two-time quantum transport and quantum diffusion

P. Kleinert

Published May 8, 2009• 5 citations
arXivHTMLPDF

Abstract

Based on the nonequilibrium Green's function technique, a unified theory is developed that covers quantum transport and quantum diffusion in bulk semiconductors on the same footing. This approach, which is applicable to transport via extended and localized states, extends previous semiphenomenological studies and puts them on a firm microscopic basis. The approach is sufficiently general and applies not only to well-studied quantum-transport problems, but also to models, in which the Hamiltonian does not commute with the dipole operator. It is shown that even for the unified treatment of quantum transport and quantum diffusion in homogeneous systems, all quasimomenta of the carrier distribution function are present and fulfill their specific function. Particular emphasis is put on the double-time nature of quantum kinetics. To demonstrate the existence of robust macroscopic transport effects that have a true double-time character, a phononless steady-state current is identified that appears only beyond the generalized Kadanoff-Baym ansatz.

DOI
10.1103/physreve.79.051107

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Based on the nonequilibrium Green's function technique, a unified theory is developed that covers quantum transport and quantum diffusion in bulk semiconductors on the same footing. This approach, which is applicable to transport via extended and localized states, extends previous semiphenomenological studies and puts them on a firm microscopic basis. The approach is sufficiently general and applies not only to well-studied quantum-transport problems, but also to models, in which the Hamiltonian does not commute with the dipole operator. It is shown that even for the unified treatment of quantum transport and quantum diffusion in homogeneous systems, all quasimomenta of the carrier distribution function are present and fulfill their specific function. Particular emphasis is put on the double-time nature of quantum kinetics. To demonstrate the existence of robust macroscopic transport effects that have a true double-time character, a phononless steady-state current is identified that appears only beyond the generalized Kadanoff-Baym ansatz.
A systematic and unambiguous method of deriving generalized transport equations, i.e., equations for distribution functions having a single-time structure, on the basis of the nonequilibrium Green function is obtained, if the common Kadanoff-Baym (KB) ansatz is replaced by a modification which we call the generalized KB ansatz. This new ansatz is fully consistent with the dynamical structure of the theory and is independent of any specific representation. The resulting equations appear to be the zeroth-order approximation of a systematic expansion in terms of the collision duration. In the case of the electron transport in a strong homogeneous electric field, the generalized ansatz is shown to be in agreement with the superoperator methods, whereas the KB ansatz is known to fail in this case.
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.
We study time-dependent quantum transport through a correlated double quantum dot (DQD) model system by means of time propagation of the nonequilibrium many-body Green's function. The theory is an extension of the Kadanoff-Baym approach for finite inhomogeneous systems ( Phys. Rev. Lett. , 98 (2007) 153004) to open inhomogeneous systems and generalizes the Meir-Wingreen formula to include initial correlations and memory effects. Important features of the theory are 1) the possibility to study the ultrafast dynamics of transients and other time-dependent regimes and 2) the inclusion of exchange and correlation effects in a conserving approximation scheme. We calculate time-dependent local currents and densities for different many-body approximations and highlight the role of initial correlations and memory effects on the transient dynamics. Furthermore we show that coherent charge oscillations on the DQD are strongly affected by the confined Coulomb interaction and can be directly related to the local equilibrium spectral density.
Full understanding of the relaxation mechanisms and far-from-equilibrium transport in modern mesoscopic structures requires that such systems be treated as open. We therefore generalize some of the core elements of the Kadanoff-Baym-Keldysh nonequilibrium Green's function formalism, inherently formulated for closed systems, to treatment of an open system, coupled with its environment. We define the two-time correlation functions and analyze the influence of the memory effects on the open-system transport. In the transient regime, the two-time correlation functions clearly show four distinct terms: a closed-system-like term, an entanglement term, and two memory terms that depend explicitly on the initial state of the environment. We show that it is not possible to completely eliminate the influence of the environment by a fortunate choice of the initial state, and approximating the system as closed is valid only in the limit of negligible system-environment coupling, which is never the case in the transient regime. We derive the transport equations for transients that properly account for the system-environment coupling. On the other hand, we address the important issue of transport in a far-from-equilibrium steady state. We show that, once a steady state is reached, the balance between the driving and relaxation forces implies that the two-time correlation functions regain a closed-system-like form, but with an effective, modified system Hamiltonian, and with the system statistical operator unrelated to that of the initial state. We emphasize that the difference between the transient and the far-from-equilibrium steady-state regimes, crucial for theoretical investigation of nonequilibrium quasiparticle transport, effectively lies within the different relative magnitude of the combined entanglement and memory terms with respect to the closed-system-like term in two-time correlation functions.
Construction of the electron quantum transport equations from Non-Equilibrium Green's Functions is presently based on Reconstruction Theorems, reducing full description of a non-equilibrium system to a dynamic theory in terms of one-particle quantities. This approach, stemming from the original Kadanoff-Baym Ansatz, and based on a suitably renormalized NGF formalism, can be brought to a close parallel with the Time Dependent Density Functional Theory. This concerns, in particular, the role of correlations at a finite initial time. A universal treatment of these initial conditions is possible for the switch-on states, while more general initial states have so far to be treated individually.
The quantum Boltzmann equation (QBE), originally formulated as the quantum extension of Boltzmann’s classical transport theory by Uehling and Uhlenbeck and subsequently developed within the nonequilibrium Green-function formalisms of Kadanoff–Baym and Keldysh, provides the central kinetic framework for describing relaxation and transport in weakly interacting quantum many-body systems. Although its formal structure is well established, conventional derivations often obscure the microscopic equivalence between operator and nonequilibrium Green-function approaches. Wick factorization is typically applied without explicitly enumerating real-time contractions, while Green-function treatments encode the same physics in Born-level self-energies that conceal their operator origin. In this paper, we present a pedagogical microscopic derivation of the quantum Boltzmann equation that makes this equivalence fully explicit. Starting from the ladder-operator formulation, we enumerate all fifteen Wick pairings of the second-order six-field correlator, identifying three Hartree–Fock contributions and twelve real-time Born scattering channels generated by a quartic, number-conserving interaction Hamiltonian. We then establish a one-to-one correspondence between these twelve operator contractions and the Keldysh components of the second-order Born self-energy, demonstrating explicitly how the Uehling–Uhlenbeck collision integral emerges after Wigner transformation and quasiparticle projection. Under controlled Born–Markov and quasiparticle approximations, this work offers a transparent synthesis of operator and Green-function formulations, and clarifies how kinetic fluctuations can be understood as emergent under coarse-graining.