15.2 Magnetic Fields
435
allows the determination of the carrier type and the carrier density.
1
If both types of carriers are present simultaneously, the conductivity (two-band conduction) is given
by the sum of electron and hole conductivity (8.11),
σ = σ e + σ h .
(15.13)
The Hall constant (15.10) is then
R H =
1
e
−n μ
2
e (1 + μ
2
h B
2
) + p μ
2
h (1 + μ
2
e B
2
)
n 2 μ 2
e (1 + μ
2
h B 2 ) − 2 n p μ e μ h (1 + μ e μ h B 2 ) + p 2 μ
2
h (1 + μ 2
e B 2 )
.
(15.14)
Under the assumption of small magnetic fields,
2 i.e. μB 1, the Hall coefficient is
R H =
1
e
−n μ
2
e + p μ
2
h
(−n μ e + p μ h ) 2 +
n p (−n + p) μ
2
e μ
2
h (μ e − μ h )
2
(−n μ e + p μ h ) 4
B
2
+ . . .
.
(15.15)
For small magnetic field this can be written as
R H =
1
e
p − n β
2
( p − n β) 2 ,
(15.16)
with β = μ e /μ h < 0. For large magnetic fields, i.e. μB 1, the Hall coefficient is given by
R H =
1
e
1
p − n
.
(15.17)
In Fig. 15.8, the absolute value of the Hall coefficient for InSb samples with different doping concentrations is shown. The p-doped samples exhibit a reverse of the sign of the Hall coefficient upon
increase of temperature when intrinsic electrons contribute to the conductivity. The zero in R H occurs
for n = p μ
2
h /μ
2
e = n i /|β|. For high temperatures, the Hall coefficient for n- and p-doped samples is
dominated by the electrons that have much higher mobility (Table 8.2).
The simultaneous conduction in a band and an impurity band (cmp. Sect. 8.6) has been separated
with a suitable model assuming two conduction channels for holes [1434] (Fig. 15.9).
In (15.8) an isotropic mass was considered. For semiconductors with multiple valleys and anisotropic
extrema, in particular the conduction bands of Si and Ge (cf. Sect. 6.9.2), the Hall coefficient has been
derived in [1435, 1436]. With K = m l /m t being the mass anisotropy (cmp. Table 6.5), the Hall
coefficient (15.12) changes to,
R H =
1
q n
3 K (K + 2)
(2K + 1) 2 .
(15.18)
In the derivation of the (unipolar) Hall coefficient we had assumed that all carriers involved in the
transport have the same properties, in particular that they are subject to the same scattering time. This
assumption is generally not the case (cmp. Chap. J) and we need to operate with the ensemble average
of the discussed quantities. The ensemble average of an energy-dependent quantity ζ(E) over the
1 Using the Hall effect, the net free charge carrier concentration is determined. We note that the concentration of fixed
charges in semiconductors can be investigated by depletion layer spectroscopy (Sect. 21.2.4).
2 We note that for a mobility of 10 4 cm 2 /Vs, μ −1 is a field of B = 1 T.
435
allows the determination of the carrier type and the carrier density.
1
If both types of carriers are present simultaneously, the conductivity (two-band conduction) is given
by the sum of electron and hole conductivity (8.11),
σ = σ e + σ h .
(15.13)
The Hall constant (15.10) is then
R H =
1
e
−n μ
2
e (1 + μ
2
h B
2
) + p μ
2
h (1 + μ
2
e B
2
)
n 2 μ 2
e (1 + μ
2
h B 2 ) − 2 n p μ e μ h (1 + μ e μ h B 2 ) + p 2 μ
2
h (1 + μ 2
e B 2 )
.
(15.14)
Under the assumption of small magnetic fields,
2 i.e. μB 1, the Hall coefficient is
R H =
1
e
−n μ
2
e + p μ
2
h
(−n μ e + p μ h ) 2 +
n p (−n + p) μ
2
e μ
2
h (μ e − μ h )
2
(−n μ e + p μ h ) 4
B
2
+ . . .
.
(15.15)
For small magnetic field this can be written as
R H =
1
e
p − n β
2
( p − n β) 2 ,
(15.16)
with β = μ e /μ h < 0. For large magnetic fields, i.e. μB 1, the Hall coefficient is given by
R H =
1
e
1
p − n
.
(15.17)
In Fig. 15.8, the absolute value of the Hall coefficient for InSb samples with different doping concentrations is shown. The p-doped samples exhibit a reverse of the sign of the Hall coefficient upon
increase of temperature when intrinsic electrons contribute to the conductivity. The zero in R H occurs
for n = p μ
2
h /μ
2
e = n i /|β|. For high temperatures, the Hall coefficient for n- and p-doped samples is
dominated by the electrons that have much higher mobility (Table 8.2).
The simultaneous conduction in a band and an impurity band (cmp. Sect. 8.6) has been separated
with a suitable model assuming two conduction channels for holes [1434] (Fig. 15.9).
In (15.8) an isotropic mass was considered. For semiconductors with multiple valleys and anisotropic
extrema, in particular the conduction bands of Si and Ge (cf. Sect. 6.9.2), the Hall coefficient has been
derived in [1435, 1436]. With K = m l /m t being the mass anisotropy (cmp. Table 6.5), the Hall
coefficient (15.12) changes to,
R H =
1
q n
3 K (K + 2)
(2K + 1) 2 .
(15.18)
In the derivation of the (unipolar) Hall coefficient we had assumed that all carriers involved in the
transport have the same properties, in particular that they are subject to the same scattering time. This
assumption is generally not the case (cmp. Chap. J) and we need to operate with the ensemble average
of the discussed quantities. The ensemble average of an energy-dependent quantity ζ(E) over the
1 Using the Hall effect, the net free charge carrier concentration is determined. We note that the concentration of fixed
charges in semiconductors can be investigated by depletion layer spectroscopy (Sect. 21.2.4).
2 We note that for a mobility of 10 4 cm 2 /Vs, μ −1 is a field of B = 1 T.