next up previous contents
Next: Numerical Evaluation Methods Up: Irreversible Processes and Previous: Irreversible Processes and

The Boltzmann Equation

The Boltzmann equation is the basis for the standard models of electron transport in semiconductors in a semi-classical approximation. It consists of the classical Liouville equation (79) augmented by a master operator of precisely the form (84) to describe collisions between electrons and other particles. The Boltzmann equation is commonly written in the form [35]:


where is the force on the electron (due to the electric field, for example). is the scattering rate from to , as in (82), but now is a continuous variable. These scattering rates are conventionally obtained by evaluating the Fermi golden rule for scattering between plane-wave states with wavevectors , and . Note that these are the scattering rates between infinitely extended states, but in the Boltzmann equation, each scattering event is assumed to take place at a single point. (If we did not assume this, the continuity equation would be violated.)

The different scattering mechanisms in semiconductors and their rates has been the subject of much theoretical and experimental work. Expressions for these scattering rates can be found in the works of Conwell [36], and Ridley [37]. The evaluation of low-field transport properties of bulk semiconductors is described in detail by Rode [38]

William R. Frensley
Fri Jun 23 15:00:21 CDT 1995