Semiconductor Bloch equations

The semiconductor Bloch equations (abbreviated as SBEs) describe the optical response of semiconductors excited by coherent classical light sources, such as lasers. They are based on a full quantum theory, and form a closed set of integro-differential equations for the quantum dynamics of microscopic polarization and charge carrier distribution. The SBEs are named after the structural analogy to the optical Bloch equations that describe the excitation dynamics in a two-level atom interacting with a classical electromagnetic field. As the major complication beyond the atomic approach, the SBEs must address the many-body interactions resulting from Coulomb force among charges and the coupling among lattice vibrations and electrons. The SBEs are one of the most sophisticated and successful approaches to describe optical properties of semiconductors originating from the classical light–matter interaction, once the many-body theory are systematically included.

The optical response of a semiconductor follows if one can determine its macroscopic polarization ${\displaystyle \mathbf {P} }$ as a function of the electric field ${\displaystyle \mathbf {E} }$ that excites it. The connection between ${\displaystyle \mathbf {P} }$ and the microscopic polarization ${\displaystyle P_{\mathbf {k} }}$ is given by

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