15.2 Magnetic Fields
439
With the magnetic field perpendicular to the sample surface, i.e. B = μ 0 (0, 0, H ) the magneto-optic
dielectric tensor simplifies to (cf. (15.9d))
= −
ω
∗2
p
ω 2
⎛
⎝
˜
xx i ˜
xy 0
−i ˜
xy ˜
xx 0
0
0 ˜
zz
⎞
⎠
(15.30a)
˜
xx =
1 + iγ /ω
(1 + iγ /ω) 2 − (ω c /ω) 2
(15.30b)
˜
zz =
1
(1 + iγ /ω)
(15.30c)
˜
xy =
ω c /ω
(1 + iγ /ω) 2 − (ω c /ω) 2 .
(15.30d)
The in-plane component xx provides information about ω
∗
p and γ , i.e. two of the three parameters n, μ
and m
∗ are known. Additionally, the antisymmetric tensor component xy is linear in the cyclotron frequency and provides q/m
∗ . This subtle but finite birefringence depends on the strength (and orientation)
of the magnetic field and can be experimentally determined in the infrared using magneto-ellipsometry
[1450, 1451]. Such ‘optical Hall effect’ experiment allows the determination of the carrier density n,
the mobility μ, the carrier mass
4 m
∗ and the sign of the carrier charge sgn(q) with optical means. The
electrical Hall effect (Sect. 15.2.1) can reveal n, μ and sgn(q) but cannot reveal the carrier mass.
15.2.3 Energy Levels in Bulk Crystals
In a 3D electron gas (the magnetic field is along z, i.e. B = B (0, 0, 1)) the motion in the (x, y) plane is
described by Landau levels. Quantum mechanically they correspond to levels of a harmonic oscillator.
The magnetic field has no impact on the motion of electrons along z, such that in this direction a free
dispersion relation ∝ k
2
z is present. The energy levels are given as
E nk z =
n +
1
2
ω c +
2
2m
k
2
z .
(15.31)
Thus, the states are on concentric cylinders in k-space (Fig. 15.11a). The populated states of the 3D
electron gas (at 0 K) lie within the Fermi vector of length k F . For the 3D system the density of states
at the Fermi energy is a square root function of the Fermi energy (6.71). In the presence of a magnetic
field the density of states diverges every time that a new cylinder (with a one-dimensional density
of states, (6.79)) touches the Fermi surface at E F . In real systems, the divergence will be smoothed,
however, a pronounced peak or the periodic nature of the density of states is often preserved.
The period is given by the number n m of cyclotron orbits (Landau levels) within the Fermi surface.
n m +
1
2
ω c = E F .
(15.32)
If the number of carriers is constant, the density of states at the Fermi energy at varying magnetic field
varies periodically with 1/B. From the conditions (n m +
1
2
)eB 1 /m = E F and (n m + 1 +
1
2
)eB 2 /m =
E F with 1/B 2 = 1/B 1 + 1//B we find,
4 We note that mobility and effective mass defined and measured in this way may be referred to as ‘optical’. Other
definitions and approaches to the mobility or effective mass may give different results.
439
With the magnetic field perpendicular to the sample surface, i.e. B = μ 0 (0, 0, H ) the magneto-optic
dielectric tensor simplifies to (cf. (15.9d))
= −
ω
∗2
p
ω 2
⎛
⎝
˜
xx i ˜
xy 0
−i ˜
xy ˜
xx 0
0
0 ˜
zz
⎞
⎠
(15.30a)
˜
xx =
1 + iγ /ω
(1 + iγ /ω) 2 − (ω c /ω) 2
(15.30b)
˜
zz =
1
(1 + iγ /ω)
(15.30c)
˜
xy =
ω c /ω
(1 + iγ /ω) 2 − (ω c /ω) 2 .
(15.30d)
The in-plane component xx provides information about ω
∗
p and γ , i.e. two of the three parameters n, μ
and m
∗ are known. Additionally, the antisymmetric tensor component xy is linear in the cyclotron frequency and provides q/m
∗ . This subtle but finite birefringence depends on the strength (and orientation)
of the magnetic field and can be experimentally determined in the infrared using magneto-ellipsometry
[1450, 1451]. Such ‘optical Hall effect’ experiment allows the determination of the carrier density n,
the mobility μ, the carrier mass
4 m
∗ and the sign of the carrier charge sgn(q) with optical means. The
electrical Hall effect (Sect. 15.2.1) can reveal n, μ and sgn(q) but cannot reveal the carrier mass.
15.2.3 Energy Levels in Bulk Crystals
In a 3D electron gas (the magnetic field is along z, i.e. B = B (0, 0, 1)) the motion in the (x, y) plane is
described by Landau levels. Quantum mechanically they correspond to levels of a harmonic oscillator.
The magnetic field has no impact on the motion of electrons along z, such that in this direction a free
dispersion relation ∝ k
2
z is present. The energy levels are given as
E nk z =
n +
1
2
ω c +
2
2m
k
2
z .
(15.31)
Thus, the states are on concentric cylinders in k-space (Fig. 15.11a). The populated states of the 3D
electron gas (at 0 K) lie within the Fermi vector of length k F . For the 3D system the density of states
at the Fermi energy is a square root function of the Fermi energy (6.71). In the presence of a magnetic
field the density of states diverges every time that a new cylinder (with a one-dimensional density
of states, (6.79)) touches the Fermi surface at E F . In real systems, the divergence will be smoothed,
however, a pronounced peak or the periodic nature of the density of states is often preserved.
The period is given by the number n m of cyclotron orbits (Landau levels) within the Fermi surface.
n m +
1
2
ω c = E F .
(15.32)
If the number of carriers is constant, the density of states at the Fermi energy at varying magnetic field
varies periodically with 1/B. From the conditions (n m +
1
2
)eB 1 /m = E F and (n m + 1 +
1
2
)eB 2 /m =
E F with 1/B 2 = 1/B 1 + 1//B we find,
4 We note that mobility and effective mass defined and measured in this way may be referred to as ‘optical’. Other
definitions and approaches to the mobility or effective mass may give different results.