Page 226 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

From Global Balance to Local Non-Equilibrium
⋅
Drift–
Diffusion ∇ D = ρ (6.85)
Equations where D is the displacement vector with the constitutive equation
D = – ε ε ∇ψ (6.86)
r 0
where the dielectric constants ε and ε are defined in Chapter 4.
r 0
⋅
∇ j = qR (6.87a)
n
∇ j ⋅ p = – qR (6.87b)
The net recombination rate describes a destruction of holes and elec-
R
trons in pairs, i.e., the events per unit time an electrons goes from the
conduction band into the valence band. Therefore, the same number
occurs in both equations (6.87a) and (6.87b). The respective constitutive
equations for the currents are

j = – qµ n∇ψ + qD ∇n (6.88a)
n
n
n
–
j = – qµ p∇ψ qD ∇p (6.88b)
p
p
p
where µ n and µ p are the electron and hole mobility, respectively, and
D n and D p denote the respective diffusion constants. The Einstein rela-
tions relate these two quantities by
k T 0
B
D = ------------µ (6.89a)
n n
q
and

k T 0
B
D = ------------µ (6.89b)
p p
q
In (6.89a) and (6.89b) we encounter the temperature T 0 . This means that
our model discards any information about the temperature distribution in
positional space, i.e., we have assumed a global uniform temperature of
electrons and holes that coincides with the lattice temperature.

Semiconductors for Micro and Nanosystem Technology 223