# Exact solution of the Boltzmann equation in the relaxation approximation, existence of negative differential conductivity for certain types of energy bands and its implication for the fluctuation spectrum and the noise temperature

## Abstract

The Boltzmann equation for the distribution fk of a system of charged particles obeying classical statistics in a uniform field F, {Mathematical expression} will be solved analytically for a special class of transition rates Wkk′=const·hk·νk·νk′ for any initial distribution. hk is the Maxwell distribution and νk>0 can be interpreted as a k-dependent relaxation frequency. The constant relaxation approximation (νk=ν) will be used to discuss the drift velocities u for all the fields and temperatures T for certain types of band structures E(k). Bands with linear k-dependence for large k give rise to drift velocities saturating for large fields. For bands with the periodicity of the reciprocal lattice, the zero drift-theorem has been proved. It states that {Mathematical expression} for all the periodic band structures. This theorem is even correct for a general Wkk′ if certain restrictions are made. Finally, making use of the Markov character of the conditional probability (Green's function) solution of the Boltzmann equation, the velocity fluctuation spectrum S is calculated for E(k)=A(1-cos a k). It will be shown that S(F, T, 0) remains positive for the critical field and all temperatures, and therefore the noise temperature diverges on approaching the critical field. © 1969 Springer-Verlag.