15.2 Magnetic Fields
437
R H =
1
q n
r H ,
(15.20)
with the so-called Hall factor r H given by
r H =
γ
α 2 + ω 2
c γ 2 ,
α =
τ
1 + ω 2
c τ 2
, γ =
τ
2
1 + ω 2
c τ 2
.
(15.21)
The Hall factor depends on the scattering mechanisms and is of the order of 1. For large magnetic
fields the Hall factor approaches 1. For small magnetic fields we have
R H =
1
q n
τ
2
τ 2 .
(15.22)
The mobility calculated from (cf. (15.9d)) σ 0 R H is called the Hall mobility μ H and is related to the
mobility via
μ H = r H μ .
(15.23)
It is assumed so far that the free carrier density and mobility are homogeneous within the volume
of current transport. Multi-layer models can be fitted to experimental Hall data in order to account
for different conduction channels in different layered parts of the sample [1437]. E.g., in a two-layer
model, contributions from bulk and surface/interface conduction can be separated [1438–1440].
The magnetic field dependence of σ can be used in a general case to separate contributions of
carriers with different density and mobility (including its sign) without assumptions and obtain the
mobility spectrum s(μ) (MSA, mobility spectral analysis),
σ xx =
∞
−∞
s(μ)
1
1 + μ 2 B 2 dμ
(15.24a)
σ xy =
∞
−∞
s(μ)
μB
1 + μ 2 B 2 dμ ,
(15.24b)
as a generalization of (15.13), (15.9b) and (15.9c) [1441–1443]. Examples are the separation of electron conductivity in (GaAs-) - and (InAs-) X-Minima in a GaAs/(Al,Ga)As/InAs double quantum
well structures [1441] (Fig. 15.10a), substrate and 2DEG electron conductivity in (Al,Ga)N/GaN heterostructures [1444], and electrons and holes in an InAs/GaSb quantum well [1442] (Fig. 15.10b).
In the case of hopping conduction (Chap. 8.8), the theory of Hall effect is more involved [1445,
1446]. If carriers are transported by hopping, generally, they may not be free to move as expected by
(15.8) in response to the applied magnetic field and the Lorentz force. A Hall effect occurs only at
the junction of three (or more) hopping sites [1445]. The sign of the experimentally determined Hall
coefficient is often opposite to the one expected from the carrier type according to (15.12), e.g., as
studied in a-Si [1447, 1448]. The sign anomaly depends on intricacies such as the local site geometry
and interference processes taking place among bonding and antibonding orbitals of various numbers
as summarized in [1449].
437
R H =
1
q n
r H ,
(15.20)
with the so-called Hall factor r H given by
r H =
γ
α 2 + ω 2
c γ 2 ,
α =
τ
1 + ω 2
c τ 2
, γ =
τ
2
1 + ω 2
c τ 2
.
(15.21)
The Hall factor depends on the scattering mechanisms and is of the order of 1. For large magnetic
fields the Hall factor approaches 1. For small magnetic fields we have
R H =
1
q n
τ
2
τ 2 .
(15.22)
The mobility calculated from (cf. (15.9d)) σ 0 R H is called the Hall mobility μ H and is related to the
mobility via
μ H = r H μ .
(15.23)
It is assumed so far that the free carrier density and mobility are homogeneous within the volume
of current transport. Multi-layer models can be fitted to experimental Hall data in order to account
for different conduction channels in different layered parts of the sample [1437]. E.g., in a two-layer
model, contributions from bulk and surface/interface conduction can be separated [1438–1440].
The magnetic field dependence of σ can be used in a general case to separate contributions of
carriers with different density and mobility (including its sign) without assumptions and obtain the
mobility spectrum s(μ) (MSA, mobility spectral analysis),
σ xx =
∞
−∞
s(μ)
1
1 + μ 2 B 2 dμ
(15.24a)
σ xy =
∞
−∞
s(μ)
μB
1 + μ 2 B 2 dμ ,
(15.24b)
as a generalization of (15.13), (15.9b) and (15.9c) [1441–1443]. Examples are the separation of electron conductivity in (GaAs-) - and (InAs-) X-Minima in a GaAs/(Al,Ga)As/InAs double quantum
well structures [1441] (Fig. 15.10a), substrate and 2DEG electron conductivity in (Al,Ga)N/GaN heterostructures [1444], and electrons and holes in an InAs/GaSb quantum well [1442] (Fig. 15.10b).
In the case of hopping conduction (Chap. 8.8), the theory of Hall effect is more involved [1445,
1446]. If carriers are transported by hopping, generally, they may not be free to move as expected by
(15.8) in response to the applied magnetic field and the Lorentz force. A Hall effect occurs only at
the junction of three (or more) hopping sites [1445]. The sign of the experimentally determined Hall
coefficient is often opposite to the one expected from the carrier type according to (15.12), e.g., as
studied in a-Si [1447, 1448]. The sign anomaly depends on intricacies such as the local site geometry
and interference processes taking place among bonding and antibonding orbitals of various numbers
as summarized in [1449].