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International Journal of Quantum Chemistry

Density matrix methods for semiconductor Coulomb dynamics

James W. Dufty, Chang Sub Kim, M. Bönitz, Rolf H. Binder

Published Jan 1, 1997• 5 citations
Publisher

Abstract

Current experiments on semiconductor devices using femtosecond lasers provide new theoretical challenges for the description of charge carrier dynamics. Among the new features of such experiments are states driven very far from equilibrium and probes on time scales short compared to scattering and other characteristic material relaxation times. Standard many-body methods must be modified and extended to accommodate these features. We propose that the quantum hierarchy for reduced density operators is an ideal formulation of such initial value problems and describe how the dominant effects of exchange and charge correlations can be accounted for in a simple and physically transparent closure of the hierarchy of equations. The transformations, approximations, and interpretation can be accomplished independent of any particular matrix representation. Decomposition into kinetic equations for band occupation densities and polarization densities follows in a straightforward way after the many-body problem has been brought under control. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 929–940, 1997

DOI
10.1002/(sici)1097-461x(1997)65:5<929::aid-qua56>3.0.co;2-q

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Current experiments on semiconductor devices using femtosecond lasers provide new theoretical challenges for the description of charge carrier dynamics. Among the new features of such experiments are states driven very far from equilibrium and probes on time scales short compared to scattering and other characteristic material relaxation times. Standard many-body methods must be modified and extended to accommodate these features. We propose that the quantum hierarchy for reduced density operators is an ideal formulation of such initial value problems and describe how the dominant effects of exchange and charge correlations can be accounted for in a simple and physically transparent closure of the hierarchy of equations. The transformations, approximations, and interpretation can be accomplished independent of any particular matrix representation. Decomposition into kinetic equations for band occupation densities and polarization densities follows in a straightforward way after the many-body problem has been brought under control. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 929–940, 1997
The problem of quantum dynamics in open systems has gained attention in recent decades and not the least due to the advances made in quantum transport in molecular systems. The main motivation behind quantum transport and molecular electronics is the futuristic goal to be able at some point to replace, or to complement, the silicon-based technology and to make the electronic devices faster. On a fundamental level, one has to deal with time-dependent processes where electron-electron or electron- phonon interactions are of great importance, and they can cause profound quantitative and qualitative changes on the physical and dynamical properties of electronic systems compared to the non-interacting case. Most of the studies of quantum transport have been focused on the steady-state description while neglecting the short-time dynamics. However, the dynamical effects are of great importance since fast- switching processes play a pivotal role in the operation of future devices. We studied the problem of time-dependent electron transport through the Anderson impurity model by using many-body perturbation theory (MBPT) together with Keldysh Green’s functions as well as with time-dependent density functional theory (TDDFT). These methods were compared with numerically exact time-dependent density-matrix renormalization group (tDMRG) method. We found that the many-body perturbation theory results beyond Hartree-Fock approximation were in close agreement with tDMRG results. In addition we studied the possibility of multistablity in the density and current of an interacting nanoscale junction as well as how to reversibly switch between the multiple solutions in time domain. An accurate theoretical treatment of electron correlation even in as simple model as an interacting electron gas at metallic densities still continues to be a challenge; especially description of features in the photoemission spectra due to electron correlations provides a theoretical challenge. The many-body perturbation theory yields a systematic way to study electron-electron (electron-phonon) correlations in various systems. One of the widely used approximations in MBPT is the GW approximation in which the bare interaction line is replaced with screened interaction line in the first order exchange diagram. The GW approximation gives good estimates for the band gap values close to experimental ones but especially the self-consistent GW approximation has a number of deficiencies like washing out of plasmon features and overestimation of bandwidths compared to experiment. One way to improve GW calculations is to include vertex corrections. Unfortunately, the straightforward inclusion of vertex corrections yields negative spectra in some frequency regions. We developed a diagrammatic approach to construct self-energy approximations with positive spectral properties. Our approach consists of expressing a self-energy of response diagram as a product of half-diagrams after which a minimal set of additional diagrams is identified to construct a perfect square. We applied this method to study vertex corrections in a homogeneous electron gas. In addition we analyzed the diagrammatic content of photocurrent with density functional theory. The expression for the photocurrent was obtained as an integral over the Kohn-Sham spectral function renormalized by effective potentials that depend on the exchange correlation kernel of current density functional theory. The expression for the photocurrent gives us the angular dependence of the photocurrent but it does not provide a direct access to the kinetic energy distribution of the photoelectrons.
The theoretical description of materials' properties driven out of equilibrium has important consequences in various fields such as semiconductor spintronics, nonlinear optics, and continuous and discrete quantum information science and technology. The coupling of a quantum many-body system to an external bath can dramatically modify its dynamics compared to that of closed systems; new phenomena like relaxation and decoherence appear as a consequence of the non-unitary evolution of the quantum system. In addition, electron-electron correlations must be properly accounted for in order to go beyond a simple one-electron or mean-field description of the electronic system. Here, we discuss a first-principles methodology based on the evolution of the electronic density matrix capable of treating electron-environment interactions and electron-electron correlations at the same level of description. The effect of the environment is separated into a coherent contribution, like the coupling to applied external electromagnetic fields, and an incoherent contribution, like the interaction with lattice vibrations or the thermal background of radiation. Electron-electron interactions are included using the nonequilibrium Green's function plus the generalized Kadanoff-Baym ansatz. The obtained non-Markovian coupled set of equations reduces to the ordinary Lindblad quantum master equation form in the Markovian limit.
Summary form only given. Effects of exciton-exciton correlations (e.g. bound biexciton, exciton-exciton scatterings) have been amply demonstrated in the /spl chi//sup (3)/ contributions to experiments measuring the interband polarization in semiconductor quantum wells. Increasing attention is currently being focussed on higher order excitonic correlation effects. One higher-order quantity is the incoherent exciton density (defined as total exciton density minus the interband polarization squared) which is created by excitons scattering each other out of the coherent component. Other than its role in the dephasing and relaxation of the system, the incoherent exciton density has not received much attention. In this contribution, we present results of our theoretical study on the behavior of this quantity and related correlation functions under conditions that inhibit relaxation. Our theory follows the general formulation of bosonic theories for excitons: the basic degrees of freedom are exact bosons, and the fermionic exchange effects are incorporated into the boson-boson and boson-photon effective Hamiltonians. We truncate the effective interaction at the two-boson level. No formal bosonization procedure is used: the matrix elements of our effective Hamiltonian are chosen in such a way that in the coherent /spl chi//sup (3)/ limit, our boson theory yields formally identical results as the standard fermionic dynamics controlled truncation formalism. Standard many-body techniques are used to derive equations of motion for densities, correlation functions, and measurable quantities such as the interband polarization.
A reduced-density-matrix description is developed for coherent linear and non-linear electromagnetic interactions of quantized electronic systems in the presence of environmental decoherence and relaxation phenomena. Applications of interest include many-electron atomic systems and semiconductor materials. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations of the reduced-density-matrix approach are developed in a unified manner. The general non-Markovian formulations provide a rigorous foundation for the systematic introduction of the commonly used Born (lowest-order perturbation-theory) and Markov (short-memory-time) approximations. A preliminary semiclassical perturbation-theory treatment of the electromagnetic interaction is introduced, based on a coupled hierarchy of relationships for the field components of the reduced density operator.
Abstract A reduced-density-matrix description is developed for coherent linear and non-linear electromagnetic interactions of quantized electronic systems in the presence of environmental decoherence and relaxation phenomena. Applications of interest include many-electron atomic systems and semiconductor materials. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations of the reduced-density-matrix approach are developed in a unified manner. The general non-Markovian formulations provide a rigorous foundation for the systematic introduction of the commonly used Born (lowest-order perturbation-theory) and Markov (short-memory-time) approximations. A preliminary semiclassical perturbation-theory treatment of the electromagnetic interaction is introduced, based on a coupled hierarchy of relationships for the field components of the reduced density operator.
A generalized Bloch-Maxwell formulation for laser-field-coupled semiconductors is derived from a two-band model which includes direct Coulomb interactions. The momentum-dependent, microscopic, electron-hole equations of motion in the time-dependent Hartree-Fock approximation and neglecting interband exchange interactions form the starting point for the formulation. A self-consistent set of coupled equations in four dynamical variables for the medium, together with the electric-field amplitude coupled through the Maxwell wave equation in a semiclassical approximation, are obtained to lowest order in the coherent Coulomb exchange interaction and the density-of-states distribution. Intrinsic optical bistability is predicted in a steady state due to a carrier density-dependent redshift of the band edge due explicitly to coherent Coulomb exchange. Integration of the dynamical equations for conditions which correspond to the ultrafast time regime exhibit intrinsic adiabatic inversion, adiabatic following, anomalous Rabi cycling, and unique, as well as fast, optical switching, all of which depend upon the coherent Coulomb exchange interaction.