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 distributionf k of a system of charged particles obeying classical statistics in a uniform fieldF, $$\frac{{\partial f_k }}{{\partial t}} + F\frac{{\partial f_k }}{{\partial k}} = \smallint d^3 k'(W_{kk'} f_{k'} - W_{k'k} f_k ),$$ will be solved analytically for a special class of transition ratesW kk′=const·h k ·ν k ·ν k′ for any initial distribution.h k is the Maxwell distribution andν k >0 can be interpreted as ak-dependent relaxation frequency. The constant relaxation approximation (ν k =ν) will be used to discuss the drift velocitiesu for all the fields and temperaturesT for certain types of band structuresE(k). Bands with lineark-dependence for largek 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 $$\mathop {\lim }\limits_{F \to \infty } u (F,T) = \mathop {\lim }\limits_{T \to \infty } u (F,T) = 0$$ for all the periodic band structures.