The Kubo Formula and Linear Response
In much of semiconductor transport theory, it is our main aim to calculate the response of the distribution of electrons, within our device or bulk semiconductor sample, to an applied perturbation. While this perturbation is usually an electric field, it may also be a magnetic field, temperature gradient, density gradient, pressure, or any combination of these generalized forces. Energy from these forces is coupled to the electrons and subsequently decays to the lattice via interaction with the phonons. In small semiconductor devices, we would like to know the entire time dependence of the appropriate interactions, and usually we use some form of kinetic theory based upon the Boltzmann transport equation, which itself is not valid on the short time scales. The response is directly related to the nature of the scattering of the electrons by the lattice, and is often characterized in terms of relaxation times, such as the momentum relaxation time and the energy relaxation time. In turn, these averaged quantities are the macroscopic effects of the microscopic fluctuations introduced by the scattering processes themselves. As a consequence, it is possible to relate them directly to the averaged response to the spectrum of the fluctuations themselves—the traditional fluctuation-dissipation theorem.
KeywordsDensity Matrix Linear Response Boltzmann Transport Equation Kubo Formula Velocity Operator
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