15.2 Magnetic Fields
433
B
V H
w
I
x
y
z
v
F
B
Fig. 15.5 Scheme of the Hall-effect geometry. The movement of one electron in the longitudinal electric current I is
shown schematically. The coordinate system (x, y, z) and the directions of the magnetic field B, the drift velocity of an
electron v and the resulting Lorentz force F are given. The transverse field E y is given by V H /w
Fig. 15.6 Hallbar
geometry. The current I is
driven by a source from
contact 1 to 2. The Hall
voltage is measured
between contacts 3 and 4,
the longitudinal voltage
between 3 and 5
1
2
3
4
5
6
I
I
V =V
xy
H
V xx
B z
lation is due to the Lorentz force. The related transverse voltage is called the Hall voltage and the
resistivity ρ xy = E y /j x the Hall resistivity [28, 31, 32]. Many aspects of the Hall effect are discussed
in [1427]. For thin-film samples typically Hall bars [1428] (see Fig.15.6 and also Fig. 15.19, for a
reasonable measurement of the Hall voltage the ratio of length and width of the Hall bar should be at
least 3) or the van-der-Pauw geometry (Fig. 15.7) and method are used [1429–1431].
For band transport in the relaxation time approximation (Chap. 8.2), the steady-state equation of
motion is (in the vicinity of an isotropic extremum)
m
∗ v
τ
= q (E + v × B) .
(15.8)
We note that this equation of motion is also valid for holes, given the convention of Sect. 6.10.1, i.e.
positive effective mass and charge. With the cyclotron frequency ω c = q B/m
∗ the conductivity tensor
is (j = qnv = σ E)
σ =
⎛
⎝
σ xx σ xy 0
σ yx σ yy 0
0 0 σ zz
⎞
⎠
(15.9a)
σ xx = σ yy = σ 0
1
1 + ω 2
c τ 2 = σ 0
1
1 + μ 2 B 2
(15.9b)
σ xy = −σ yx = σ 0
ω c τ
1 + ω 2
c τ 2 = σ 0
μ B
1 + μ 2 B 2
(15.9c)
433
B
V H
w
I
x
y
z
v
F
B
Fig. 15.5 Scheme of the Hall-effect geometry. The movement of one electron in the longitudinal electric current I is
shown schematically. The coordinate system (x, y, z) and the directions of the magnetic field B, the drift velocity of an
electron v and the resulting Lorentz force F are given. The transverse field E y is given by V H /w
Fig. 15.6 Hallbar
geometry. The current I is
driven by a source from
contact 1 to 2. The Hall
voltage is measured
between contacts 3 and 4,
the longitudinal voltage
between 3 and 5
1
2
3
4
5
6
I
I
V =V
xy
H
V xx
B z
lation is due to the Lorentz force. The related transverse voltage is called the Hall voltage and the
resistivity ρ xy = E y /j x the Hall resistivity [28, 31, 32]. Many aspects of the Hall effect are discussed
in [1427]. For thin-film samples typically Hall bars [1428] (see Fig.15.6 and also Fig. 15.19, for a
reasonable measurement of the Hall voltage the ratio of length and width of the Hall bar should be at
least 3) or the van-der-Pauw geometry (Fig. 15.7) and method are used [1429–1431].
For band transport in the relaxation time approximation (Chap. 8.2), the steady-state equation of
motion is (in the vicinity of an isotropic extremum)
m
∗ v
τ
= q (E + v × B) .
(15.8)
We note that this equation of motion is also valid for holes, given the convention of Sect. 6.10.1, i.e.
positive effective mass and charge. With the cyclotron frequency ω c = q B/m
∗ the conductivity tensor
is (j = qnv = σ E)
σ =
⎛
⎝
σ xx σ xy 0
σ yx σ yy 0
0 0 σ zz
⎞
⎠
(15.9a)
σ xx = σ yy = σ 0
1
1 + ω 2
c τ 2 = σ 0
1
1 + μ 2 B 2
(15.9b)
σ xy = −σ yx = σ 0
ω c τ
1 + ω 2
c τ 2 = σ 0
μ B
1 + μ 2 B 2
(15.9c)