8.4 High-Field Transport
237
(a)
(b)
(c)
Fig. 8.12 Distribution of electrons in silicon in momentum space (cmp. Fig. 6.35c) for electric fields of a 10 kV/cm, b
10 2 kV/cm and c 10 3 kV/cm. Adapted from [756]
already at moderate fields. Then, the electron temperature [755] becomes larger than the lattice temperature. With increasing electrical field the carriers can gain more and more energy and will on average
populate higher states, assuming a non-Boltzmann (and non-Fermi) statistical distribution [756]. The
electron distribution in k-space is depicted for silicon for three different electric fields in Fig. 8.12b,c.
Hot carriers suffer additional scattering processes that are discussed in the following, namely optical
phonon emission, intervalley scattering and impact ionization.
8.4.1 Drift-Saturation Velocity
If the carrier energy is large enough it can transfer energy to the lattice by the emission of an optical
phonon. This mechanism is very efficient and limits the maximum drift velocity. Such behavior is
non-ohmic. The limiting value for the drift velocity is termed the drift-saturation velocity. It is given
by [757]
v s =
8
3π
ω LO
m ∗ .
(8.29)
This relation can be obtained from an energy-balance consideration. The energy gain per unit time in
the electric field is equal to the energy loss by the emission of an optical phonon.
q v · E =
LO
τ
,
(8.30)
where τ is the typical relaxation time constant for LO phonon emission. Together with (8.3) we find
(8.30) except for the pre-factor, which is close to 1. The exact pre-factor results from a quantummechanical treatment. For Ge the drift-saturation velocity at room temperature is 6 × 10
6 cm/s, for Si it
is 1 × 10
7 cm/s (Fig. 8.13a). The carrier velocity also depends on the crystallographic direction [758].
8.4.2 Negative Differential Resistivity
In GaAs, the initially linear regime (constant mobility) saturates at a maximum drift velocity of about
2 × 10
7 cm/s for about 3 kV/cm; for higher fields, a reduction in drift velocity (with increasing field!)
is present (1.2 × 10
7 cm/s at 10 kV/cm, 0.6 × 10
7 cm/s at 200 kV/cm), as shown in Fig. 8.13a. This
237
(a)
(b)
(c)
Fig. 8.12 Distribution of electrons in silicon in momentum space (cmp. Fig. 6.35c) for electric fields of a 10 kV/cm, b
10 2 kV/cm and c 10 3 kV/cm. Adapted from [756]
already at moderate fields. Then, the electron temperature [755] becomes larger than the lattice temperature. With increasing electrical field the carriers can gain more and more energy and will on average
populate higher states, assuming a non-Boltzmann (and non-Fermi) statistical distribution [756]. The
electron distribution in k-space is depicted for silicon for three different electric fields in Fig. 8.12b,c.
Hot carriers suffer additional scattering processes that are discussed in the following, namely optical
phonon emission, intervalley scattering and impact ionization.
8.4.1 Drift-Saturation Velocity
If the carrier energy is large enough it can transfer energy to the lattice by the emission of an optical
phonon. This mechanism is very efficient and limits the maximum drift velocity. Such behavior is
non-ohmic. The limiting value for the drift velocity is termed the drift-saturation velocity. It is given
by [757]
v s =
8
3π
ω LO
m ∗ .
(8.29)
This relation can be obtained from an energy-balance consideration. The energy gain per unit time in
the electric field is equal to the energy loss by the emission of an optical phonon.
q v · E =
LO
τ
,
(8.30)
where τ is the typical relaxation time constant for LO phonon emission. Together with (8.3) we find
(8.30) except for the pre-factor, which is close to 1. The exact pre-factor results from a quantummechanical treatment. For Ge the drift-saturation velocity at room temperature is 6 × 10
6 cm/s, for Si it
is 1 × 10
7 cm/s (Fig. 8.13a). The carrier velocity also depends on the crystallographic direction [758].
8.4.2 Negative Differential Resistivity
In GaAs, the initially linear regime (constant mobility) saturates at a maximum drift velocity of about
2 × 10
7 cm/s for about 3 kV/cm; for higher fields, a reduction in drift velocity (with increasing field!)
is present (1.2 × 10
7 cm/s at 10 kV/cm, 0.6 × 10
7 cm/s at 200 kV/cm), as shown in Fig. 8.13a. This