8.2 Conductivity
225
In a real semiconductor, at finite temperatures, impurities, phonons and defects (finally also the
surface) will contribute to scattering. In the relaxation-time approximation it is assumed that the
probability for a scattering event, similar to friction, is proportional to the (average) carrier velocity.
The average relaxation time τ is introduced via an additional term ˙
v = −v/τ that sums up all scattering
events.
1 Thus, the maximum velocity that can be reached in a static electric field is given by (steady-state
velocity)
v = −
e E τ
m ∗ .
(8.3)
The current density per unit area is then linear in the field, i.e. fulfills Ohm’s law
j = n q v =
n e
2 E τ
m ∗ = σ E .
(8.4)
The conductivity σ in the relaxation-time approximation is given by
σ =
1
ρ
=
n e
2
τ
m ∗ .
(8.5)
In the case of a cylindrically symmetric mass such as for electrons in silicon or germanium, for the
effective mass in (8.5) the effective conductivity mass must be used,
1
m ∗
σ
=
1
3
2
m t
+
1
m l
.
(8.6)
The specific resistivity is the inverse of the conductivity. Metals have a high conductivity (see Table 8.1),
e.g. for Cu at room temperature σ = 5.8 × 10
5
−1 cm
−1 . At low temperatures (4 K) the conductivity
is even a factor of 10
5 higher. The mean free path d = τ v F is
d =
σ m
∗ v F
n e 2 ,
(8.7)
v F being the Fermi velocity (E F = m
∗ v
2
F /2). For copper, d = 3 mm at low temperature (and thus
susceptible to the sample geometry) while at room temperature the mean free path is only about 40 nm.
However, this becomes an issue when the metal line width and height of interconnects in integrated
circuits approaches this length scale [715] (see Sect. 24.5.5).
In semiconductors, the carrier concentration depends strongly on the temperature. At zero temperature the conductivity is zero. Also, the scattering processes and thus the relaxation time constant exhibit
a temperature dependence. The conductivity spans a large range from insulating to almost metallic
conduction (see Table 8.1).
8.3 Low-Field Transport
First we consider only small electric fields. The real meaning of this will only become clear in Sect. 8.4
on high-field transport. In the low-field regime the velocity is proportional to the electric field.
1 Going beyond the relaxation time approximation is discussed in Appendix J.
225
In a real semiconductor, at finite temperatures, impurities, phonons and defects (finally also the
surface) will contribute to scattering. In the relaxation-time approximation it is assumed that the
probability for a scattering event, similar to friction, is proportional to the (average) carrier velocity.
The average relaxation time τ is introduced via an additional term ˙
v = −v/τ that sums up all scattering
events.
1 Thus, the maximum velocity that can be reached in a static electric field is given by (steady-state
velocity)
v = −
e E τ
m ∗ .
(8.3)
The current density per unit area is then linear in the field, i.e. fulfills Ohm’s law
j = n q v =
n e
2 E τ
m ∗ = σ E .
(8.4)
The conductivity σ in the relaxation-time approximation is given by
σ =
1
ρ
=
n e
2
τ
m ∗ .
(8.5)
In the case of a cylindrically symmetric mass such as for electrons in silicon or germanium, for the
effective mass in (8.5) the effective conductivity mass must be used,
1
m ∗
σ
=
1
3
2
m t
+
1
m l
.
(8.6)
The specific resistivity is the inverse of the conductivity. Metals have a high conductivity (see Table 8.1),
e.g. for Cu at room temperature σ = 5.8 × 10
5
−1 cm
−1 . At low temperatures (4 K) the conductivity
is even a factor of 10
5 higher. The mean free path d = τ v F is
d =
σ m
∗ v F
n e 2 ,
(8.7)
v F being the Fermi velocity (E F = m
∗ v
2
F /2). For copper, d = 3 mm at low temperature (and thus
susceptible to the sample geometry) while at room temperature the mean free path is only about 40 nm.
However, this becomes an issue when the metal line width and height of interconnects in integrated
circuits approaches this length scale [715] (see Sect. 24.5.5).
In semiconductors, the carrier concentration depends strongly on the temperature. At zero temperature the conductivity is zero. Also, the scattering processes and thus the relaxation time constant exhibit
a temperature dependence. The conductivity spans a large range from insulating to almost metallic
conduction (see Table 8.1).
8.3 Low-Field Transport
First we consider only small electric fields. The real meaning of this will only become clear in Sect. 8.4
on high-field transport. In the low-field regime the velocity is proportional to the electric field.
1 Going beyond the relaxation time approximation is discussed in Appendix J.