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AIP conference proceedings

Extension of the Semiconductor Bloch Equations for Spin Dynamics

C. Lechner

Published Jan 1, 2005• 1 citation
Publisher

Abstract

We present a theoretical approach to spin dynamics in semiconductors and semiconductor heterostructures based on the density matrix formalism. It consists in an extension of the coherent Semiconductor Bloch equations for the two‐level system to a six‐level system which accounts for the spin and polarization degrees‐of‐freedom and includes carrier‐carrier scattering beyond the Hartree‐Fock truncation. This extension provides the concept to formulate spin dynamics including spin relaxation and dephasing times on a microscopic level.

DOI
10.1063/1.1994607

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We present a theoretical approach to spin dynamics in semiconductors and semiconductor heterostructures based on the density matrix formalism. It consists in an extension of the coherent Semiconductor Bloch equations for the two‐level system to a six‐level system which accounts for the spin and polarization degrees‐of‐freedom and includes carrier‐carrier scattering beyond the Hartree‐Fock truncation. This extension provides the concept to formulate spin dynamics including spin relaxation and dephasing times on a microscopic level.
We present an approach to spin dynamics by extending the optical Bloch equations for the driven two-level system to derive microscopic expressions for the transverse and longitudinal spin-relaxation times. This is done for the six-level system of electron and hole subband states in a semiconductor or a semiconductor quantum structure to account for the degrees of freedom of the carrier spin and the polarization of the exciting light and includes the scattering between carriers and lattice vibrations on a microscopic level. For the subsystem of the spin-split electron subbands we treat the electron-phonon interaction in second order and derive a set of equations of motion for the $2\ifmmode\times\else\texttimes\fi{}2$ spin-density matrix, which describes the electron spin dynamics and contains microscopic expressions for the longitudinal (T1)({T}_{1})(T1​) and transverse (T2)({T}_{2})(T2​) spin-relaxation times. Their meaning will be discussed in relation to experimental investigations of these quantities.
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
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.
The density matrix is a useful concept which will allow one to deal with a large collection of spinors with a statistical distribution at time t = 0 and follow their time evolution by tracking the time evolution of a new operator (the density matrix). This will require the extension of the Bloch sphere to the Bloch ball concept, in which the vector representing the collection of spinors will be able to reach points in the interior of the Bloch sphere as a function of time. This requires the inclusion of damping and scattering mechanisms representing interactions among spinors, and also their interactions with the environment. The phenomenological equations allowing a description of these relaxation mechanisms are known as Bloch’s equations. They are introduced here through a connection to the Bloch sphere and Bloch ball concepts. The longitudinal and transverse relaxation times, T1 and T2, respectively, are also introduced and some numerical examples (results of Monte-Carlo simulations) are presented to illustrate their physical meaning.
The density matrix is a useful concept which will allow one to deal with a large collection of spinors with a statistical distribution at time t = 0 and follow their time evolution by tracking the time evolution of a new operator (the density matrix). This will require the extension of the Bloch sphere to the Bloch ball concept, in which the vector representing the collection of spinors will be able to reach points in the interior of the Bloch sphere as a function of time. This requires the inclusion of damping and scattering mechanisms representing interactions among spinors, and also their interactions with the environment. The phenomenological equations allowing a description of these relaxation mechanisms are known as Bloch’s equations. They are introduced here through a connection to the Bloch sphere and Bloch ball concepts. The longitudinal and transverse relaxation times, T1 and T2, respectively, are also introduced and some numerical examples (results of Monte-Carlo simulations) are presented to illustrate their physical meaning.
The dynamics of electrons and holes in semiconductors and semiconductor heterostructures have been extensively discussed in the literature, and are governed by their mutual Coulomb interaction, by their interaction with lattice vibrations (phonons) and by their radiative recombination (for an overview see e.g. Ref. [1]). A powerful technique to study the carrier dynamics in semiconductors utilizes short optical pulses to photoexcite electrons from the valence to the conduction band (i.e. to create electron-hole pairs) that subsequently relax until the photon energy stored in the medium is dissipated and the system has recovered the thermal equilibrium. The dynamics of the carriers photoexcited by the short optical pulse can be followed by monitoring two main optical observables: the interband population difference and the induced polarization. Both quantities decay with time after the photoexcitation due to the carrier dynamics. While the induced polarization (which is an ensemble average over many photoexcited dipole moments) is sensitive to scattering events that break the phase relationship among the individual dipoles and/or with the light field creating them (dephasing processes), the population difference is sensitive only to events that change the energy state of the carriers. Transient coherent optical spectroscopy addresses the time-evolution of the induced polarization in the time-regime during and immediately following the photoexcitation by a short laser pulse, in which the photoexcited dipoles have retained a defined phase relationship among each other and with the exciting coherent radiation.