In this section, Boltzmann's equation is introduced as an important
foundation for semi-classical transport description in semiconductor devices.
In order to describe the complete device behavior, theoretically every single
carrier within the device would have to be described by solving Newton's
classical equations of motion
The Hamilton function
represents the total carrier energy and can be
split into two parts: The potential energy
and the kinetic energy
. The potential energy incorporates the conduction and valence band
edge energies
for electrons and holes, respectively,
considering
band gap narrowing, and the electrostatic potential
. Thus,
can be expanded as
Instead of keeping the focus on every single carrier, a statistical description
is introduced. Neglecting Heisenberg's uncertainty principle, each carrier
is exactly described by its momentum and position, so it takes a certain place
within 6 dimensional
,
-space. The normalization of all present
carriers within the volume under investigation results in the distribution
function, which is the solution variable of Boltzmann's equation. It
incorporates the carrier density within 6-dimensional
-space.
In the sequel, the Boltzmann transport equation is derived from phenomenological considerations and
its range of validity is discussed. Formally, Boltzmann's equation
represents a seven-dimensional integro-differential equation within the phase
space
. However, it is accessible using a book-keeping
background, which is presented in Section 3.2.2.1. Originally
formulated for the description of statistical mechanics of gases, it is the
cornerstone for the classical description of transport in semiconductors as
well. The transport models derived in this thesis later on are based on the
Boltzmann transport equation.