Page 233 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
by time average or by an ensemble average, depending on whether the
problem is ergodic or not. This method is very convenient since it follows
the microscopic motion of a particle. Also, depending on how accurate
the result must be, it is more or less time consuming. Modern Monte
Carlo techniques are refined with respect to the questions asked. For high
accuracy needed in only specific parts of the phase space there are
already highly optimized algorithms and even commercially available
computer programs.

Spherical The idea behind this method is to expand the distribution function in
Harmonics terms of spherical harmonic functions using the coordinate reference sys-
Expansion
(
tem k θϕ) in -space.
,,
k
(
,
,
,
m
n
n
m
f kx) = Y f k x,( ) + ∑ Y f ( k x) + ∑ Y f ( k x) + ... (6.103)
0 0 1 1 2 2
n m
The information about the angular dependencies is now put into the
,,,
spherical harmonics Y m with l = 012... given by
l

 P ( cos θ)cos ( mϕ), m = 012 …
,,,
m
Y m =  l (6.104)
l
,
,
,
 P ( cos θ)sin ( mϕ), m = – 0 – 1 – 2 …
m
l

These are orthogonal functions in the θϕ,( ) space. Therefore, inserting
(6.103) into the BTE and multiplying by a given Y o k gives after integra-
tion an equation for f ( k x) . In this way we obtain a whole hierarchy of
,
k
o
equations without the assumption of a local equilibrium like for the
moment equations. The disadvantage is that it is still a four-dimensional
problem and thus is best suited for two-dimensional problems in posi-
tional space. An additional drawback is that not the whole hierarchy can
be solved for and higher order objects have to be neglected at some point.
k
Nevertheless, this method gives good results if a resolution in -space is
needed.
230 Semiconductors for Micro and Nanosystem Technology