Page 198 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Semi-Classical Boltzmann Transport Equation
shape from the unit circle to an ellipse. With applied constant external
force the initial circular density is shifted and distorted. Both cases con-
serve the total initial area, i.e., the particle density.
6.1.2 The Scattering Term
,,
In the case where Cf kx t)) ≠ 0 there are particles removed from a
(
(
position in the three dimensional wave vector subspace of the phase
space and put to a new position. This process is nothing else than a
momentum transfer in a scattering process. Suppose the scattering pro-
cess itself takes place on a time-scale much shorter than that defined by
the streaming motion and the time between two scattering processes.
This leads to the assumption that on the latter time-scale all scattering
processes happen instantaneously, i.e., it takes no time for the particle to
switch from one velocity to another one due to the scattering process.
This situation is shown in Figure 6.2 (a-c). Momentum is transferred in a

−
+
kp k k
x x
x x
k
− p
+

Figure 6.2. a) Momentum transfer in a scattering process, b) single particle trajectory
without applied external force field, and c) trajectory in the presence of scattering and
applied external field.

very short time from an external reservoir to the particle. Imagine small
liquid molecules hitting an object immersed in the liquid or electrons
being hit by phonons. In the case of absent external driving force this
causes the single particle to follow a trajectory as sketched in Figure 6.2
(b) where the particle follows straight lines in space with constant veloc-
ity until it suffers the next scattering event. The situation changes with
applied external field as depicted in Figure 6.2 (c). The trajectories

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