10.2 Band–Band Recombination
307
r sp (E) + r st (E) = r abs (E) .
(10.7)
Since for absorption and stimulated emission the same quantum-mechanical matrix element is responsible, B 1 = B 2 . If the population functions are Fermi-Dirac distributions with quasi-Fermi levels F n
and F p (Sect. 7.6), the detailed balance (10.7) yields
C(E 1 , E 2 ) = B 1 (E 1 , E 2 ) N ph
exp
E − (F n − F p )
kT
− 1
.
(10.8)
In thermodynamic equilibrium, i.e. F n = F p ,
C(E 1 , E 2 ) = N 0 B 1 (E 1 , E 2 ) = B .
(10.9)
If the constant B, the bimolecular recombination coefficient, is independent of the energy E, the
integration for the net bimolecular recombination rate r B can be executed analytically and we find
r B =
∞
E g
r sp (E) + r st (E) − r abs (E)
dE
(10.10)
= B n p
1 − exp
−
F n − F p
kT
.
In thermodynamic equilibrium, of course, r B = 0. The recombination rate Bnp is then equal to the
thermal generation rate G th
G th = B n 0 p 0 .
(10.11)
The bimolecular recombination rate typically used in Shockley–Read–Hall (SRH) [942, 943] kinetics is
r B = B (n p − n 0 p 0 ) .
(10.12)
Values for the coefficient B are given in Table 10.1. In the case of carrier injection, np is larger than
in thermodynamical equilibrium, i.e. n p > n 0 p 0 , and the recombination rate is positive, i.e. light
is emitted. If the carrier density is smaller than in thermodynamical equilibrium, e.g.. in a depletion
region, absorption is larger than emission. This effect is also known as ‘negative luminescence’ [944]
and plays a role particularly at elevated temperatures and in the infrared spectral region.
Table 10.1 Bimolecular recombination coefficient at room temperature for a number of semiconductors. Data for GaN
from [945], Si from [946], SiC from [947], other values from [948]
Material
B (cm 3 /s)
GaN
1.1 ×10 −8
GaAs
1.0 ×10 −10
AlAs
7.5 ×10 −11
InP
6.0 ×10 −11
InAs
2.1 ×10 −11
4H-SiC
1.5 ×10 −12
Si
1.1 ×10 −14
GaP
3.0 ×10 −15
307
r sp (E) + r st (E) = r abs (E) .
(10.7)
Since for absorption and stimulated emission the same quantum-mechanical matrix element is responsible, B 1 = B 2 . If the population functions are Fermi-Dirac distributions with quasi-Fermi levels F n
and F p (Sect. 7.6), the detailed balance (10.7) yields
C(E 1 , E 2 ) = B 1 (E 1 , E 2 ) N ph
exp
E − (F n − F p )
kT
− 1
.
(10.8)
In thermodynamic equilibrium, i.e. F n = F p ,
C(E 1 , E 2 ) = N 0 B 1 (E 1 , E 2 ) = B .
(10.9)
If the constant B, the bimolecular recombination coefficient, is independent of the energy E, the
integration for the net bimolecular recombination rate r B can be executed analytically and we find
r B =
∞
E g
r sp (E) + r st (E) − r abs (E)
dE
(10.10)
= B n p
1 − exp
−
F n − F p
kT
.
In thermodynamic equilibrium, of course, r B = 0. The recombination rate Bnp is then equal to the
thermal generation rate G th
G th = B n 0 p 0 .
(10.11)
The bimolecular recombination rate typically used in Shockley–Read–Hall (SRH) [942, 943] kinetics is
r B = B (n p − n 0 p 0 ) .
(10.12)
Values for the coefficient B are given in Table 10.1. In the case of carrier injection, np is larger than
in thermodynamical equilibrium, i.e. n p > n 0 p 0 , and the recombination rate is positive, i.e. light
is emitted. If the carrier density is smaller than in thermodynamical equilibrium, e.g.. in a depletion
region, absorption is larger than emission. This effect is also known as ‘negative luminescence’ [944]
and plays a role particularly at elevated temperatures and in the infrared spectral region.
Table 10.1 Bimolecular recombination coefficient at room temperature for a number of semiconductors. Data for GaN
from [945], Si from [946], SiC from [947], other values from [948]
Material
B (cm 3 /s)
GaN
1.1 ×10 −8
GaAs
1.0 ×10 −10
AlAs
7.5 ×10 −11
InP
6.0 ×10 −11
InAs
2.1 ×10 −11
4H-SiC
1.5 ×10 −12
Si
1.1 ×10 −14
GaP
3.0 ×10 −15