Page 231 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
With (6.100) we can immediately solve the homogeneous equation
(6.99), to give
t
 
,
f ( kx t) = f k t() x t() 0)exp –  ∫ Γ kt'()) t' (6.101)
,,
,
(
(
d
h
 
0
This equation tells us that with Γ = 0 all the phase space density
,
(
f k 0() x 0() 0) would move according to (6.100) to the new position at
,
k t() and x t() to give f k t() x t() 0,( , ) . The fact that Γ ≠ 0 lowers the
t
arriving density at time . This is because while moving some of the den-
sity is scattered out from the deterministic path. There is a particular
solution of the differential equation found that reads
t t
 
,,
,
(
(
f ( kx t) = ∫ exp –  ∫ Γ kt'()) t' ∑ W ( f k' τ() x τ() τ)) τ (6.102)
,
d
d
p k'k
 
0 τ k'
The total solution given by f = f + f is obtained by iteratively insert-
h p
f
ing into (6.101) and (6.102). This procedure is very interesting for sta-
tionary problems in spatially homogeneous materials.
Direct The direct integration requires a discretization of the partial differential
Integration equation, the BTE. Remember, however, that this requires a discretiza-
tion in a six-dimensional phase space. The problem then grows as N 6 ,
where N is the number of discretization points for each of the phase
space directions. Therefore, this method is not suitable for problems that
require the full dimensionality because no sufficient accuracy may be
achieved.
Monte Carlo This is a stochastic method that works by creating stochastic free flight
Simulation between the collisions (see Box 6.1). Since all the scattering probabilities
are known, one can generate a random series of collision–free determin-
istic motions of a particle, e.g., according to (6.100). At the and of each
deterministic motion stands a scattering process, which in turn is also
generated stochastically. The distribution function is generated either
f

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