Page 225 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
1 m
4π 2 ∫
3
(
2
,,
------------- (
Q =
–
a
3
Ω k vv ) vv–( a ) f kx t)d k (6.83)
Equations (6.78), (6.80), and (6.82) are referred to as the hydrodynamic
model for the motion of electrons in a semiconductor crystal. Together
with the same set of equations that can be derived for the defect electron
or hole density they form the basis for bipolar charge carrier transport
p
in semiconductors. However, note that (6.78), (6.80) and (6.82) are a
finite set of equations. It contains more unknowns than equations, i.e., it
is underdetermined. Therefore, simplifying assumptions are needed to
give a closed form. For specific forms of simplifying assumptions regard-
ing the temperature tensor T we refer the reader to the literature [6.4,
6.5].
6.3.3 Solving the Drift-Diffusion Equations

There is only one important equation missing to complete the set of
hydrodynamic equations, the Poisson equation (4.35). This equation
relates the moving charge carriers to the formation of the internal electro-
static potential in a crystal and is written as

(
∇• ( εψ) = qn – p + N – N ) = – ρ (6.84)
∇
A
D
where N and N D stand for the ionized acceptor and donor concentra-
A
tion respectively. So far there is nothing special about these equations
and one could think that it is straight forward to solve them via a simple
finite difference scheme. This approach is far from having any chance to
result in a solution of realistic problems in semiconductor transport.

The subset of the hydrodynamic equations that contains only the density
balance for electrons and holes together with (6.84) is referred to as the
drift-diffusion model. Let us focus on the stationary problem (n˙ = , 0
p˙ = 0 ), that results in the following three partial differential equations

222 Semiconductors for Micro and Nanosystem Technology