10.10 ABC Model
329
η int =
B n
2
A n + B n 2 + C n 3 =
B n
A + B n + C n 2 .
(10.59)
10.11 Field Effect
The emission of electrons from a trap is thermally activated with an ionization energy E i = E C − E t . If
the trap is in a strong electric field E, the emission probability can change. An acceptor-like trap after
removal of the electron is neutral and its potential is short range. A donor has a long-range Coulomb
potential after ionization. In an electric field, these potentials are modified as visualized in Fig. 10.30.
Various additional processes can now occur.
10.11.1 Thermally Activated Emission
For the δ-like potential the ionization energy remains unchanged. For the Coulomb potential the barrier
in the field direction is lowered by
i = e
e
π π 0 r
√
E .
(10.60)
The emission rate e n is increased in the field by exp((E i /kT ). This effect is called the Poole–Frenkel
effect [1028] and can be quite important. For silicon and E = 2 × 10
5 V/cm and i = 100 meV a
50-fold increase of the emission rate at room temperature is expected. As an example the Poole–Frenkel
effect for the electron emission from (neutral) interstitial boron in silicon (B
0
i → B
+
i + e
− ) is shown
in Fig. 10.31, following the enhancement of e n ∝ exp(
√
E). The extrapolation to E = 0 agrees with
the EPR result [275, 1029].
5
10.11.2 Direct Tunneling
Carriers can tunnel from the trap level through the barrier in the field direction into the conduction band.
This process is temperature independent. The transmission factor of a barrier is (in WKB approximation) proportional to exp[−(2/
√
2m [V (x) − E] dE]. The emission probability for a triangular
Fig. 10.30 Field effect at
a a δ-like potential and b a
Coulomb potential
E i
E i
(a)
(b)
5 The slope of the line in Fig. 10.31 is slightly smaller than expected from (10.60).
329
η int =
B n
2
A n + B n 2 + C n 3 =
B n
A + B n + C n 2 .
(10.59)
10.11 Field Effect
The emission of electrons from a trap is thermally activated with an ionization energy E i = E C − E t . If
the trap is in a strong electric field E, the emission probability can change. An acceptor-like trap after
removal of the electron is neutral and its potential is short range. A donor has a long-range Coulomb
potential after ionization. In an electric field, these potentials are modified as visualized in Fig. 10.30.
Various additional processes can now occur.
10.11.1 Thermally Activated Emission
For the δ-like potential the ionization energy remains unchanged. For the Coulomb potential the barrier
in the field direction is lowered by
i = e
e
π π 0 r
√
E .
(10.60)
The emission rate e n is increased in the field by exp((E i /kT ). This effect is called the Poole–Frenkel
effect [1028] and can be quite important. For silicon and E = 2 × 10
5 V/cm and i = 100 meV a
50-fold increase of the emission rate at room temperature is expected. As an example the Poole–Frenkel
effect for the electron emission from (neutral) interstitial boron in silicon (B
0
i → B
+
i + e
− ) is shown
in Fig. 10.31, following the enhancement of e n ∝ exp(
√
E). The extrapolation to E = 0 agrees with
the EPR result [275, 1029].
5
10.11.2 Direct Tunneling
Carriers can tunnel from the trap level through the barrier in the field direction into the conduction band.
This process is temperature independent. The transmission factor of a barrier is (in WKB approximation) proportional to exp[−(2/
√
2m [V (x) − E] dE]. The emission probability for a triangular
Fig. 10.30 Field effect at
a a δ-like potential and b a
Coulomb potential
E i
E i
(a)
(b)
5 The slope of the line in Fig. 10.31 is slightly smaller than expected from (10.60).