250
8 Transport
In the case of nondegeneracy, i.e. when the Fermi level is within the band gap and not closer than
about 4kT to the band edges, η = ln(n/N C ). Then ∂η/∂n = 1/n, and the equation simplifies to
D = (kT /q)μ, i.e. the ‘regular’ Einstein-relations,
D n = −β
−1
μ n
(8.59a)
D p = β
−1
μ p .
(8.59b)
In this case, (8.54a,b) read
j n = −eμ n n E − kT μ n ∇n
(8.60a)
j p = eμ p p E − kT μ p ∇ p .
(8.60b)
We recall that both diffusion coefficients are positive numbers, since μ n is negative. Generally, the
diffusion coefficient depends on the density. A Taylor series of the Fermi integral yields
D n = −β
−1
μ n
1 + 0.35355
n
N C
− 9.9 × 10
−3
n
N C
2
+ · · ·
.
(8.61)
8.12 Continuity Equation
The balance equation for the charge is called the continuity equation. The temporal change of the charge
in a volume element is given by the divergence of the current and any source (generation rate G), e.g.
an external excitation, or drain (recombination rate U ). Details about recombination mechanisms are
discussed in Chap. 10. Thus, we have
∂n
∂t
= G n − U n −
1
q
∇ · j n = G n − U n +
1
e
∇ · j n
(8.62a)
∂ p
∂t
= G p − U p −
1
e
∇ · j p .
(8.62b)
In the case of nondegeneracy we find, using (8.54ab)
∂n
∂t
= G n − U n − μ n n ∇ · E − μ n E ∇n + D n n
(8.63a)
∂ p
∂t
= G p − U p − μ p p ∇ · E − μ p E ∇ p + D p p .
(8.63b)
In the case of zero electric field these read
∂n
∂t
= G n − U n + D n n
(8.64a)
∂ p
∂t
= G p − U p + D p p ,
(8.64b)
and if the stationary case also applies: