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Physical review. B, Condensed matter

Effective Bloch equations for semiconductors

Markus Lindberg, Stephan W. Koch

Published Aug 15, 1988• 640 citations
Publisher

Abstract

Generalized Bloch equations for laser-excited semiconductors are derived applying quantum-mechanical projection-operator techniques. The equations include phase-space filling and the many-body Coulomb effects. The coherent part of the equations is evaluated for the regime of ultrafast-pump-probe excitation and shown to reduce to the inhomogeneously broadened two-level Bloch equations for the different momentum states if the proper Coulomb enhancement in the density of states is taken into account. For the high-excitation quasithermal regime a generalization of the Elliott formula for the absorption spectrum is derived which is valid not only for bulk semiconductors, but also for quantum-well structures and other systems with reduced spatial dimensions.

DOI
10.1103/physrevb.38.3342

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Physical Review Letters•Aug 13, 1990•127 citations
Generalized Bloch equations for laser-excited semiconductors are derived applying quantum-mechanical projection-operator techniques. The equations include phase-space filling and the many-body Coulomb effects. The coherent part of the equations is evaluated for the regime of ultrafast-pump-probe excitation and shown to reduce to the inhomogeneously broadened two-level Bloch equations for the different momentum states if the proper Coulomb enhancement in the density of states is taken into account. For the high-excitation quasithermal regime a generalization of the Elliott formula for the absorption spectrum is derived which is valid not only for bulk semiconductors, but also for quantum-well structures and other systems with reduced spatial dimensions.
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.
A comprehensive numerical study of the optical Stark effect and associated phenomena in semiconductors is presented. The results are obtained from numerical solutions of the full semiconductor Bloch equations for bulk and quantum-well structures. Coherent oscillations, the optical Stark effect, and adiabatic following, i.e., ultrafast bleaching and recovery of the exciton, are discussed. The relative importance of the different contributions in the semiconductor Bloch equations is identified. Optical-absorption spectra are computed for different exciton dephasing models and a large variety of pump-probe excitation parameters.
Ultrafast nonlinear optical spectroscopy is used to study the coherent dynamics of optically excited electron-hole pairs in semiconductors. Coulomb interaction implies that the optical inter-band transitions are dominated, at least at low temperatures, by excitonic effects. They are further enhanced in quantum confined lower-dimensional systems, where exciton and biexciton effects dominate the spectra even at room temperature. The coherent dynamics of excitons are at modest densities well described by the optical Bloch equations and a number of the dynamical effects known from atomic and molecular systems are found and studied in the exciton-biexciton system of semiconductors. At densities where strong exciton interactions, or many-body effects, become dominant, the semiconductor Bloch equations present a more rigorous treatment of the phenomena. Ultrafast degenerate four-wave mixing is used as a tool to study the coherent exciton dynamics, and the importance of performing transform limited spectroscopy is demonstrated throughout.
The set of phenomenological model equations which we proposed to determine the scaling behavior of the low-density Rabi oscillations [Phys. Rev. B 48, 17 811 (1993)] is justified from the coherent part of the semiconductor Bloch equations in lowest order in the light-matter coupling. In contrast to similar short-time studies, it is shown that for resonant excitation of an excitonic state the Rabi oscillations can be described by a Ginzburg-Landau type equation which includes only the exciton-exciton interaction of the resonantly excited state but no state-filling terms or contributions from detuned excitonic or continuum resonances. Explicit calculations are performed for the bulk two-band jellium model and for a one-dimensional two-band Hubbard model.
A many-body analysis of spectral hole burning and light-induced band renormalization in the gain region of an initially inverted semiconductor is presented. The results are based on the numerical solution of the semiconductor Bloch equations, specialized to the case of noncopropagating pump-and-probe light fields. These equations account for coherent pump-probe scattering and include the Coulomb correlation between charge carriers. They also contain optical dephasing and carrier-carrier scattering rates that are computed within the random-phase approximation.