10.9 Band–Impurity Recombination
325
(a)
(b)
Fig. 10.27 a Experimental values of the electron lifetime in heavily p-doped (In,Ga)As on InP at room temperature.
The dashed lines show dependencies of Auger (∝ N
−2
A , C p = 8.1 × 10 −29 cm −6 s −1 ) and band-band recombination
(∝ N
−1
A , B = 1.43 × 10 −10 cm −3 s −1 ). Adapted from [1014]. (b) Experimental Auger lifetimes in p-type (squares) and
n-type (circles) silicon at 300 K. The dashed (solid) line is theory for p-type (n-type) material. Adapted from [1015]
10.9 Band–Impurity Recombination
A very important recombination process is the capture of carriers by impurities. This process is in
competition with all other recombination processes, e.g. the radiative recombination and the Auger
mechanism. The band–impurity recombination is the inverse process to the carrier release from impurities and intimately related to carriers statistics (Chap. 7). It is particularly important at low carrier
densities, for high dopant concentration and in indirect semiconductors since for these the bimolecular
recombination is slow. This process is generally considered to be non-radiative since no photons close
to the band edge are emitted.
4
10.9.1 Shockley–Read–Hall Kinetics
The theory of capture on and recombination involving impurities is called Shockley–Read–Hall (SRH)
kinetics [942]. An example of radiative band–impurity recombination (of the type shown in Fig. 10.28a)
is shown in Fig. 10.8 for the (e,A
0 ) recombination at the carbon acceptor in GaAs.
We consider electron traps [1016] (see Fig. 10.28) with a concentration N t with an energy level E t .
In thermodynamic equilibrium they have an electron population
f
0
t =
1
exp
E t −E F
kT
+ 1
,
(10.41)
where f t is the nonequilibrium population of the trap. Then the capture rate r c is proportional to the
unoccupied traps and the electron concentration, r c ∝ n N t (1 − f t ). The proportionality factor has the
form v th σ n , where v th is the thermal velocity v th =
√
3kT /m ∗ ≈ 10
7 cm/s and σ n is the capture cross
section that is of atomic scale, typically ∼ 10
−15 cm
2 . The capture cross section can be related to the
optical absorption cross section [588, 589].
In order to make the following calculation more transparent, we put the effective-mass ratio
√
m 0 /m ∗
into σ in the following and thus have the same thermal velocity v th =
√
3kT /m 0 for electrons and
4 Depending on the energetic depth of the trap, mid or far infrared photons can be emitted.
325
(a)
(b)
Fig. 10.27 a Experimental values of the electron lifetime in heavily p-doped (In,Ga)As on InP at room temperature.
The dashed lines show dependencies of Auger (∝ N
−2
A , C p = 8.1 × 10 −29 cm −6 s −1 ) and band-band recombination
(∝ N
−1
A , B = 1.43 × 10 −10 cm −3 s −1 ). Adapted from [1014]. (b) Experimental Auger lifetimes in p-type (squares) and
n-type (circles) silicon at 300 K. The dashed (solid) line is theory for p-type (n-type) material. Adapted from [1015]
10.9 Band–Impurity Recombination
A very important recombination process is the capture of carriers by impurities. This process is in
competition with all other recombination processes, e.g. the radiative recombination and the Auger
mechanism. The band–impurity recombination is the inverse process to the carrier release from impurities and intimately related to carriers statistics (Chap. 7). It is particularly important at low carrier
densities, for high dopant concentration and in indirect semiconductors since for these the bimolecular
recombination is slow. This process is generally considered to be non-radiative since no photons close
to the band edge are emitted.
4
10.9.1 Shockley–Read–Hall Kinetics
The theory of capture on and recombination involving impurities is called Shockley–Read–Hall (SRH)
kinetics [942]. An example of radiative band–impurity recombination (of the type shown in Fig. 10.28a)
is shown in Fig. 10.8 for the (e,A
0 ) recombination at the carbon acceptor in GaAs.
We consider electron traps [1016] (see Fig. 10.28) with a concentration N t with an energy level E t .
In thermodynamic equilibrium they have an electron population
f
0
t =
1
exp
E t −E F
kT
+ 1
,
(10.41)
where f t is the nonequilibrium population of the trap. Then the capture rate r c is proportional to the
unoccupied traps and the electron concentration, r c ∝ n N t (1 − f t ). The proportionality factor has the
form v th σ n , where v th is the thermal velocity v th =
√
3kT /m ∗ ≈ 10
7 cm/s and σ n is the capture cross
section that is of atomic scale, typically ∼ 10
−15 cm
2 . The capture cross section can be related to the
optical absorption cross section [588, 589].
In order to make the following calculation more transparent, we put the effective-mass ratio
√
m 0 /m ∗
into σ in the following and thus have the same thermal velocity v th =
√
3kT /m 0 for electrons and
4 Depending on the energetic depth of the trap, mid or far infrared photons can be emitted.