10.9 Band–Impurity Recombination
327
rate e n is (apart from the exponential term) ∝ T
2 if σ is temperature independent. Charge conservation
requires in nonequilibrium (and of course in equilibrium) r c − r v = g c − g v . From this we obtain the
population of the trap in nonequilibrium:
f t =
σ n n + σ p p t
σ n (n + n t ) + σ p ( p + p t )
.
(10.51)
The recombination rate r b−i of the band–impurity recombination is then
r b−i = −
∂δn
∂t
= r c − g c
(10.52)
=
σ n σ p v th N t
σ n (n + n t ) + σ p ( p + p t )
(n p − n 0 p 0 ) .
Using the ‘lifetimes’
τ n 0 = (σ n v th N t )
−1
(10.53)
τ p 0 = (σ p v th N t )
−1
,
(10.54)
this is typically written as
r b−i =
1
τ p 0 (n + n t ) + τ n 0 ( p + p t )
(n p − n 0 p 0 ) .
(10.55)
For an n-type semiconductor the Fermi level is above E t and the traps are mostly full. Thus hole capture
is the dominating process. The equation for the dynamics simplifies to
∂δ p
∂t
= −
p − p 0
τ p 0
.
(10.56)
Thus, an exponential decay with minority-carrier lifetime τ p 0 (or τ n 0 for p-type material) occurs.
A recombination center is most effective when it is close to the middle of the band gap (midgap
level). The condition ∂r b−i /∂ E t = 0 leads to the trap energy E
max
t
with the maximum recombination
rate being located at
E
max
t
=
E C + E V
2
− kT ln
σ n N C
σ p N V
.
(10.57)
The curvature ∂
2 r b−i /∂ E
2
t at E
max
t
is proportional to −(np − n 0 p 0 ) and thus indeed is negative in the
presence of excess carriers. However, the maximum can be fairly broad.
The SRH kinetic presented here is valid for low densities of recombination centers. A more detailed
discussion and a more general model can be found in [1018].
A typical example for a recombination center is gold in silicon. The minority carrier lifetime
decreases from 2 × 10
−7 s to 2× 10
−10 s upon increase of the Au concentration from 10
14 to 10
17 cm
−3 .
The incorporation of recombination centers is an important measure for the design of high-frequency
devices [1019]. Due to importance in silicon technology the recombination properties of many metals
in silicon have been investigated, in particular Fe-contamination and the role of FeB-complexes [1020–
1022].
A reduction in minority-carrier lifetime can also be achieved by irradiation with high-energy particles
and the subsequent generation of point defects with energy levels at midgap.
327
rate e n is (apart from the exponential term) ∝ T
2 if σ is temperature independent. Charge conservation
requires in nonequilibrium (and of course in equilibrium) r c − r v = g c − g v . From this we obtain the
population of the trap in nonequilibrium:
f t =
σ n n + σ p p t
σ n (n + n t ) + σ p ( p + p t )
.
(10.51)
The recombination rate r b−i of the band–impurity recombination is then
r b−i = −
∂δn
∂t
= r c − g c
(10.52)
=
σ n σ p v th N t
σ n (n + n t ) + σ p ( p + p t )
(n p − n 0 p 0 ) .
Using the ‘lifetimes’
τ n 0 = (σ n v th N t )
−1
(10.53)
τ p 0 = (σ p v th N t )
−1
,
(10.54)
this is typically written as
r b−i =
1
τ p 0 (n + n t ) + τ n 0 ( p + p t )
(n p − n 0 p 0 ) .
(10.55)
For an n-type semiconductor the Fermi level is above E t and the traps are mostly full. Thus hole capture
is the dominating process. The equation for the dynamics simplifies to
∂δ p
∂t
= −
p − p 0
τ p 0
.
(10.56)
Thus, an exponential decay with minority-carrier lifetime τ p 0 (or τ n 0 for p-type material) occurs.
A recombination center is most effective when it is close to the middle of the band gap (midgap
level). The condition ∂r b−i /∂ E t = 0 leads to the trap energy E
max
t
with the maximum recombination
rate being located at
E
max
t
=
E C + E V
2
− kT ln
σ n N C
σ p N V
.
(10.57)
The curvature ∂
2 r b−i /∂ E
2
t at E
max
t
is proportional to −(np − n 0 p 0 ) and thus indeed is negative in the
presence of excess carriers. However, the maximum can be fairly broad.
The SRH kinetic presented here is valid for low densities of recombination centers. A more detailed
discussion and a more general model can be found in [1018].
A typical example for a recombination center is gold in silicon. The minority carrier lifetime
decreases from 2 × 10
−7 s to 2× 10
−10 s upon increase of the Au concentration from 10
14 to 10
17 cm
−3 .
The incorporation of recombination centers is an important measure for the design of high-frequency
devices [1019]. Due to importance in silicon technology the recombination properties of many metals
in silicon have been investigated, in particular Fe-contamination and the role of FeB-complexes [1020–
1022].
A reduction in minority-carrier lifetime can also be achieved by irradiation with high-energy particles
and the subsequent generation of point defects with energy levels at midgap.