432
15 External Fields
1.42
1.50
1.46
(ii)
(i)
(iv)
(iii)
(v)
AlGaAs/
GaAs
(b)
1.45
1.43
1.47
AlGaAs/
GaAs
(ii)
(i)
(iii)
(a)
Fig. 15.4 Impact of electric fields on the absorption spectrum of n × (9.5 nm GaAs/9.8 nm Al 0.32 Ga 0.68 As) multiple
quantum well structures. a Electric field along the [001] growth direction (n = 50), (i)–(v) E = 0, 0.6, 1.1, 1.5, and
2 × 10 5 V/cm. b Electric field within the interface plane (n = 60), (i, ii, iii) E = 0, 1.1, and 2×10 5 V/cm. Adapted
from [1422]
15.2 Magnetic Fields
In magnetic fields, electrons (or holes) perform a cyclotron motion with frequency ω c = eB/m
∗ , i.e.
a motion perpendicular to the magnetic field on a line of constant energy in k-space. This line is the
intersection of a plane perpendicular to the magnetic field and the respective isoenergy surface in kspace. For semiconductors with anisotropic mass, such as Si and Ge, the quantum theory of cyclotron
resonance has been given in [1423]. The physics of semiconductors in magnetic fields is covered in
detail in [1424].
The ballistic cyclotron motion can only occur between two scattering events. Thus, a significantly
long path along the cyclotron trajectory (classically speaking) and the connected magnetotransport
properties are only possible when
– ω c τ 1, i.e. when the average scattering time τ is sufficiently large. This requires high mobility.
– the magnetic field is sufficiently strong and the temperature sufficiently low, i.e. c kT , such
that thermal excitations do not scatter electrons between different Landau levels.
– the cyclotron path is free of geometric obstructions.
An external magnetic field also produces a Zeeman-like splitting of the spin states. For the electron,
the energy splitting is given by
= g
∗
e μ B B ,
(15.7)
where B is the magnetic-field amplitude and g
∗
e the (effective) electron g-factor. This value differs
from the free-electron value in vacuum of g e = 2.0023 due to the presence of spin-orbit interaction
(see Sect. 15.2.3). Values for g
∗
e at low carrier density and low temperatures are 2 for Si, 1.2 for InP and
ZnSe, −1.65 for CdTe, −0.44 for GaAs, −15 in InAs, and −50 for InSb. In [1425] the temperature
dependence of g
∗
e in GaAs, InP and CdTe is also measured and discussed. The electron g-factor
increases in thin GaAs/(Al,Ga)As quantum wells [1426].
15.2.1 Classical Hall Effect
An electrical current along the x (longitudinal) direction in a perpendicular magnetic field B = (0, 0, B)
along z causes an electric field E y along the transverse (y) direction (Fig. 15.5). The charge accumu-
15 External Fields
1.42
1.50
1.46
(ii)
(i)
(iv)
(iii)
(v)
AlGaAs/
GaAs
(b)
1.45
1.43
1.47
AlGaAs/
GaAs
(ii)
(i)
(iii)
(a)
Fig. 15.4 Impact of electric fields on the absorption spectrum of n × (9.5 nm GaAs/9.8 nm Al 0.32 Ga 0.68 As) multiple
quantum well structures. a Electric field along the [001] growth direction (n = 50), (i)–(v) E = 0, 0.6, 1.1, 1.5, and
2 × 10 5 V/cm. b Electric field within the interface plane (n = 60), (i, ii, iii) E = 0, 1.1, and 2×10 5 V/cm. Adapted
from [1422]
15.2 Magnetic Fields
In magnetic fields, electrons (or holes) perform a cyclotron motion with frequency ω c = eB/m
∗ , i.e.
a motion perpendicular to the magnetic field on a line of constant energy in k-space. This line is the
intersection of a plane perpendicular to the magnetic field and the respective isoenergy surface in kspace. For semiconductors with anisotropic mass, such as Si and Ge, the quantum theory of cyclotron
resonance has been given in [1423]. The physics of semiconductors in magnetic fields is covered in
detail in [1424].
The ballistic cyclotron motion can only occur between two scattering events. Thus, a significantly
long path along the cyclotron trajectory (classically speaking) and the connected magnetotransport
properties are only possible when
– ω c τ 1, i.e. when the average scattering time τ is sufficiently large. This requires high mobility.
– the magnetic field is sufficiently strong and the temperature sufficiently low, i.e. c kT , such
that thermal excitations do not scatter electrons between different Landau levels.
– the cyclotron path is free of geometric obstructions.
An external magnetic field also produces a Zeeman-like splitting of the spin states. For the electron,
the energy splitting is given by
= g
∗
e μ B B ,
(15.7)
where B is the magnetic-field amplitude and g
∗
e the (effective) electron g-factor. This value differs
from the free-electron value in vacuum of g e = 2.0023 due to the presence of spin-orbit interaction
(see Sect. 15.2.3). Values for g
∗
e at low carrier density and low temperatures are 2 for Si, 1.2 for InP and
ZnSe, −1.65 for CdTe, −0.44 for GaAs, −15 in InAs, and −50 for InSb. In [1425] the temperature
dependence of g
∗
e in GaAs, InP and CdTe is also measured and discussed. The electron g-factor
increases in thin GaAs/(Al,Ga)As quantum wells [1426].
15.2.1 Classical Hall Effect
An electrical current along the x (longitudinal) direction in a perpendicular magnetic field B = (0, 0, B)
along z causes an electric field E y along the transverse (y) direction (Fig. 15.5). The charge accumu-