Page 195 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
duce the concept of scattering. This term will obtain more meaning in
Chapter 7, where we look at the interactions between the various sub-
systems.
Chapter Goal The goal of this chapter is to show how to describe transport phenomena
and how to compute the transport of electrons through a semiconductor
device.

Chapter We start with the Boltzmann transport equation, describing its terms, and
Roadmap show how it should be modified for particular particle types. Zooming in,
we first describe local equilibrium. Next, we see how local equilibrium
and the global balance equations of thermodynamics are related. This
leads us to a number of models for classical and semi-classical transport
of charge carriers in silicon, including the very successful drift-diffusion
equations. Finally, we look at numerical methods used to solve the trans-
port equations.

6.1 The Semi-Classical Boltzmann Transport
Equation
The semi-classical Boltzmann transport equation (BTE) gives a descrip-
tion of the electronic system in terms of single particle distribution func-
,,
(
tions f px t) , i.e., a time-dependent density in a six-dimensional
single particle phase space. A single particle is identified by its three spa-
tial coordinates and three momentum values. For the momentum of a free
particle we know that p = —k = mv . In this case we could write the
(
,,
distribution function in terms of velocities f vx t) . From the discus-
sion in Section 3.3.3 we know that this simple relation does not hold in a
semiconductor in general. Though we shall restrict our discussion to
energies near the band extrema, where p = m∗ v holds, with the effec-
tive mass m∗ , we prefer to take the wave vector as the independent
k
variable. This will remind us that in the case of electron we are dealing
with wave phenomena as well as with particle properties. A more formal

192 Semiconductors for Micro and Nanosystem Technology