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10 Recombination
10.2.5 Recombination Dynamics
The carrier densities n and p, are decomposed into the densities n 0 and p 0 in thermodynamic equilibrium and the excess-carrier densities δn and δ p, respectively
n = n 0 + δn
(10.13a)
p = p 0 + δ p .
(10.13b)
Here, only neutral excitations are considered, i.e. δn = δ p. Obviously the time derivative fulfills
∂n
∂t
=
∂ δn
∂t
, and correspondingly for the hole density. The equation for the dynamics
˙
n = ˙
p = −Bnp + G th = −B (n p − n 0 p 0 ) = −B (n p − n
2
i )
(10.14)
can be written as
∂ δ p
∂t
= −B (n 0 δ p + p 0 δn + δn δ p) .
(10.15)
The general solution of (10.15) is given by
δ p(t) =
(n 0 + p 0 ) δ p(0)
[n 0 + p 0 + δ p(0)] exp [B t (n 0 + p 0 )] − δ p(0)
.
(10.16)
In the following, we discuss some approximate solutions of (10.15). First, we treat the case of a small
(neutral) excitation, i.e. δn = δ p n 0 , p 0 . The dynamic equation is in this case
∂ δ p
∂t
= −B (n 0 + p 0 ) δ p .
(10.17)
Then the decay of the excess-carrier density is exponential with a time constant (lifetime) τ given by
τ =
1
B (n 0 + p 0 )
.
(10.18)
In an n-type semiconductor additionally n 0 p 0 , and thus the minority carrier lifetime τ p is
τ p =
1
B n 0
.
(10.19)
If the nonequilibrium carrier densities are large, i.e. n ≈ p n 0 , p 0 , e.g. for strong injection, the
kinetics obeys
∂ δ p
∂t
= −B (δ p)
2
,
(10.20)
and the transient has the form
δ p(t) =
δ p(0)
1 + B t δ p(0)
,
(10.21)
where δ p(0) is the excess hole density at time t = 0. Such a decay is called hyperbolic and the
recombination is bimolecular. The exponential decay time is formally τ
−1
= Bδ p(t) and is thus time
and density dependent. A detailed discussion of minority carrier lifetime is given in [949].