8.4 High-Field Transport
241
Fig. 8.18 Impact
ionization rate as a function
of primary carrier energy
for electrons (solid line)
and holes (dashed line) in
silicon at room
temperature. The curves
are fit to results from a
Monte-Carlo simulation.
Adapted from [772, 773]
0
1
3
2
4
5
Primary energy (eV)
Ionization rate (s )
-1
10
15
10
14
13
10
12
10
11
10
10
10
9
10
8
10
7
10
Si
e
h
Thus so-holes have typically the smaller threshold.
3 At energies where impact ionization occurs, nonparabolicities are typically important, thus (8.31)–(8.33) are only indicative. The threshold behavior
and the dependence of the scattering rate as a function of the primary carrier energy in Si, calculated
considering the detailed band structure, is shown in Fig. 8.18.
The generation rate G of electron–hole pairs during impact ionization is given by
G = α n n v n + α p p v p ,
(8.34)
where α n is the electron ionization coefficient. It describes the generation of electron–hole pairs per
incoming electron per unit length. α p denotes the hole ionization coefficient. The coefficients depend
strongly on the applied electric field. They are shown in Fig. 8.19. They also depend on the crystallographic direction.
The impact ionization initiated by electrons and holes in silicon has been calculated considering the
full band structure using a Monte Carlo technique in [772] and [773], respectively. In both cases the
impact ionization rate is anisotropic for excess energies smaller than 3 eV and become isotropic above.
The average energies at the moment of generation of secondary generated carriers depends linearly on
the primary electron or hole energy.
The energy dependence of the electron initiated impact ionization rate has been calculated for GaAs,
GaN and ZnS considering details and anisotropy of the band structure in [774]. The rates averaged over
the Brillouin zone are compared in Fig. 8.20. Because of the large band gap of GaN, impact ionization
can only be generated by electrons in higher conduction bands. The sharp increase of ionization rate for
GaN around 5.75 eV correlates with a large valence band DOS from hole states at the zone boundary.
8.5 High-Frequency Transport
The above consideration pertained to dc (or slowly varying) fields. Now, we consider an ac field. It
accelerates the carriers but at the same time the dissipative force in the relaxation-time approximation
is present, i.e. (for electrons)
m
∗
˙
v = −e E − m
∗ v
τ
.
(8.35)
3 Assuming m so = m e , m e m hh and 0 E g , E thr
so /E thr
e ≈ 1 − (m e /m hh )(1 + g )/2 < 1.
241
Fig. 8.18 Impact
ionization rate as a function
of primary carrier energy
for electrons (solid line)
and holes (dashed line) in
silicon at room
temperature. The curves
are fit to results from a
Monte-Carlo simulation.
Adapted from [772, 773]
0
1
3
2
4
5
Primary energy (eV)
Ionization rate (s )
-1
10
15
10
14
13
10
12
10
11
10
10
10
9
10
8
10
7
10
Si
e
h
Thus so-holes have typically the smaller threshold.
3 At energies where impact ionization occurs, nonparabolicities are typically important, thus (8.31)–(8.33) are only indicative. The threshold behavior
and the dependence of the scattering rate as a function of the primary carrier energy in Si, calculated
considering the detailed band structure, is shown in Fig. 8.18.
The generation rate G of electron–hole pairs during impact ionization is given by
G = α n n v n + α p p v p ,
(8.34)
where α n is the electron ionization coefficient. It describes the generation of electron–hole pairs per
incoming electron per unit length. α p denotes the hole ionization coefficient. The coefficients depend
strongly on the applied electric field. They are shown in Fig. 8.19. They also depend on the crystallographic direction.
The impact ionization initiated by electrons and holes in silicon has been calculated considering the
full band structure using a Monte Carlo technique in [772] and [773], respectively. In both cases the
impact ionization rate is anisotropic for excess energies smaller than 3 eV and become isotropic above.
The average energies at the moment of generation of secondary generated carriers depends linearly on
the primary electron or hole energy.
The energy dependence of the electron initiated impact ionization rate has been calculated for GaAs,
GaN and ZnS considering details and anisotropy of the band structure in [774]. The rates averaged over
the Brillouin zone are compared in Fig. 8.20. Because of the large band gap of GaN, impact ionization
can only be generated by electrons in higher conduction bands. The sharp increase of ionization rate for
GaN around 5.75 eV correlates with a large valence band DOS from hole states at the zone boundary.
8.5 High-Frequency Transport
The above consideration pertained to dc (or slowly varying) fields. Now, we consider an ac field. It
accelerates the carriers but at the same time the dissipative force in the relaxation-time approximation
is present, i.e. (for electrons)
m
∗
˙
v = −e E − m
∗ v
τ
.
(8.35)
3 Assuming m so = m e , m e m hh and 0 E g , E thr
so /E thr
e ≈ 1 − (m e /m hh )(1 + g )/2 < 1.