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Time‐Dependent Momentum Distribution of Polarons at Arbitrary Temperature and Electric Field
Author(s) -
Brosens F.,
Devreese J. T.
Publication year - 1982
Publication title -
physica status solidi (b)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.51
H-Index - 109
eISSN - 1521-3951
pISSN - 0370-1972
DOI - 10.1002/pssb.2221110222
Subject(s) - legendre polynomials , polaron , electric field , hamiltonian (control theory) , physics , boltzmann equation , momentum (technical analysis) , distribution function , condensed matter physics , phonon , scattering , field (mathematics) , boundary value problem , electron , charge carrier , mathematical physics , quantum mechanics , mathematics , mathematical optimization , finance , pure mathematics , economics
The charge carriers in a polar semiconductor are assumed to be described by the Fröhlich polaron Hamiltonian, i.e. the interaction with the LO phonons is considered to be the dominant scattering mechanism. In the presence of an electric field, and at sufficiently weak electron–phonon coupling, the dynamics of the charge carriers are studied from the Boltzmann equation. A rigorous solution for the time‐dependent momentum distribution function is found at arbitrary temperature and electric field. This solution consists of a recurrence relation between the expansion coefficients α n ( p, t ), if the momentum distribution function is expanded in Legendre polynomials: \documentclass{article}\pagestyle{empty}\begin{document}$ f(p,t) = {\textstyle{1 \over 2}}\sum\limits_{n = 0}^\infty {(2n + 1)} {\rm a}_{\rm n} (p,t)P_n (\cos \theta ) $\end{document} where θ is the angle between the electric field and the momentum p under consideration. It is shown that from Boltzmann's equation one of these expansion coefficients is left completely unspecified, and has to be determined from physical boundary conditions.