330
10 Recombination
Fig. 10.31 Field effect for
electron emission from
interstitial boron in silicon
(T = 65 K). The filled
circles represent
experimental data from
DLTS, the data point
marked with an empty
circle is from EPR (zero
field). The line is a linear fit
and extrapolation. Adapted
from [1029]
10
3
10
2
1
10
0
10
Electron emission rate e (s )
n
-1
0
50
100
150
200
E ( (V/cm) )
1/2
1/2
Si:B
E (V/cm)
10
4
2 10
3
2 10
4
0
4 10
4
barrier is then
e n =
e E
4
√
2m ∗ E i
exp
−
4
√
2m ∗ E
3/2
i
3 e E
.
(10.61)
In the case of a Coulomb-like potential the argument of the exponent in (10.61) needs to be multiplied
by a factor 1 − ((E i /E i )
5/3 with i from (10.60).
10.11.3 Assisted Tunneling
In a thermally assisted tunneling process the electron on the trap level is first excited to a virtual level
E t + E ph by phonon absorption and then tunnels out of the trap (photon-assisted tunneling). From the
energetically higher level the tunneling rate is higher. The probability is proportional to exp(E ph /kT ).
The additional energy can also be supplied by a photon (photon-assisted tunneling).
10.12 Recombination at Extended Defects
10.12.1 Surfaces
A surface (cmp. Chap. 11) is typically a source of recombination, e.g. through midgap levels induced
by the break of crystal symmetry. The recombination at surfaces is modeled as a recombination current
j s = −e S (n s − n 0 ) ,
(10.62)
where n s is the carrier density at the surface and S is the so-called surface recombination velocity.
The surface recombination velocity for GaAs is shown in Fig. 10.32. For InP, if the surface Fermi
level is pinned close to midgap, the surface recombination velocity increases from ∼ 5 × 10
3 cm/s for
a doping level of n ∼ 3 × 10
15 cm
−3 to ∼ 10
6 cm/s for a doping level of n ∼ 3 × 10
18 cm
−3 [1030].
For Si, the surface recombination rate depends on the treatment of the surface and lies in the range
10 Recombination
Fig. 10.31 Field effect for
electron emission from
interstitial boron in silicon
(T = 65 K). The filled
circles represent
experimental data from
DLTS, the data point
marked with an empty
circle is from EPR (zero
field). The line is a linear fit
and extrapolation. Adapted
from [1029]
10
3
10
2
1
10
0
10
Electron emission rate e (s )
n
-1
0
50
100
150
200
E ( (V/cm) )
1/2
1/2
Si:B
E (V/cm)
10
4
2 10
3
2 10
4
0
4 10
4
barrier is then
e n =
e E
4
√
2m ∗ E i
exp
−
4
√
2m ∗ E
3/2
i
3 e E
.
(10.61)
In the case of a Coulomb-like potential the argument of the exponent in (10.61) needs to be multiplied
by a factor 1 − ((E i /E i )
5/3 with i from (10.60).
10.11.3 Assisted Tunneling
In a thermally assisted tunneling process the electron on the trap level is first excited to a virtual level
E t + E ph by phonon absorption and then tunnels out of the trap (photon-assisted tunneling). From the
energetically higher level the tunneling rate is higher. The probability is proportional to exp(E ph /kT ).
The additional energy can also be supplied by a photon (photon-assisted tunneling).
10.12 Recombination at Extended Defects
10.12.1 Surfaces
A surface (cmp. Chap. 11) is typically a source of recombination, e.g. through midgap levels induced
by the break of crystal symmetry. The recombination at surfaces is modeled as a recombination current
j s = −e S (n s − n 0 ) ,
(10.62)
where n s is the carrier density at the surface and S is the so-called surface recombination velocity.
The surface recombination velocity for GaAs is shown in Fig. 10.32. For InP, if the surface Fermi
level is pinned close to midgap, the surface recombination velocity increases from ∼ 5 × 10
3 cm/s for
a doping level of n ∼ 3 × 10
15 cm
−3 to ∼ 10
6 cm/s for a doping level of n ∼ 3 × 10
18 cm
−3 [1030].
For Si, the surface recombination rate depends on the treatment of the surface and lies in the range