446
15 External Fields
(a)
(b)
Fig. 15.19 a Silicon MOS (metal-oxide-semiconductor) structure of K. v. Klitzing’s et al. original experiments. b
GaAs/(Al,Ga)As heterostructure sample grown with molecular beam epitaxy for QHE measurements, chip carrier and
bond wires. Reprinted with permission from [1460]
with ν ∈ N, i.e. integer fractions of the quantized resistance ρ 0 = h/e
2
= 25812.807… , which is
also called the von-Klitzing constant. In Fig. 15.18, a spin splitting is seen for the n = 1 Landau level
(and a small one for the n = 2). We note that the topmost Hall plateau is due to the completely filled
n = 0 Landau level; the resistance is ρ 0 /2 due to the spin degeneracy of 2.
The integer quantum Hall effect, first reported in [1461, 1462], and the value for ρ 0 are found for a
wide variety of samples and conditions regarding sample temperature, electron density or mobility of
the 2DEG and the materials of the heterostructure. Besides the Hall effect in Si-based structures and
record mobility (Al,Ga)As/GaAs heterostructures (cf. Fig. 12.37), also (Mg,Zn)O/ZnO heterostructures
exhibit the effect [1463, 1464] in the same fashion.
Within the plateau, the resistivity is well defined within 10
−7 or better up to the 10
−9 regime. A
precise determination allows for a new normal for the unit Ohm [1389, 1465], being two orders of
magnitude more precise than the realization in the SI system, and an independent value for the finestructure constant α = (e
2
/ h)/(2 c 0 ). At the same time, the longitudinal resistivity, starting from the
classical value for small magnetic fields, exhibits oscillations and eventually it is close to zero for the
plateaus in ρ xy . For ρ xx values of 10
−10
have been measured, which corresponds to 10
−16
/cm
for bulk material, a value three orders of magnitude smaller than for any non-superconductor.
The interpretation of the quantum Hall effect(s) is discussed in [1466] among many other treatises.
The simplest explanation is that the conductivity is zero when a Landau level is completely filled and
the next is completely empty, i.e. the Fermi level lies between them. The temperature is small, i.e.
kT c , such that no scattering between Landau levels can occur. Thus no current, similar to a
completely filled valence band, can flow. The sheet carrier density n s is given by counting the i filled
Landau levels (degeneracy according to (15.37)) as
n s = i
e B
h
.
(15.43)
In the transverse direction energy dissipation takes place and the Hall resistivity ρ xy = B/(n s e) takes
the (scattering-free) values given in (15.42).
However, this argument is too simple as it will not explain the extension of the plateaus. As soon
as the system has one electron more or less, the Fermi energy will (for a system with δ-like density of
15 External Fields
(a)
(b)
Fig. 15.19 a Silicon MOS (metal-oxide-semiconductor) structure of K. v. Klitzing’s et al. original experiments. b
GaAs/(Al,Ga)As heterostructure sample grown with molecular beam epitaxy for QHE measurements, chip carrier and
bond wires. Reprinted with permission from [1460]
with ν ∈ N, i.e. integer fractions of the quantized resistance ρ 0 = h/e
2
= 25812.807… , which is
also called the von-Klitzing constant. In Fig. 15.18, a spin splitting is seen for the n = 1 Landau level
(and a small one for the n = 2). We note that the topmost Hall plateau is due to the completely filled
n = 0 Landau level; the resistance is ρ 0 /2 due to the spin degeneracy of 2.
The integer quantum Hall effect, first reported in [1461, 1462], and the value for ρ 0 are found for a
wide variety of samples and conditions regarding sample temperature, electron density or mobility of
the 2DEG and the materials of the heterostructure. Besides the Hall effect in Si-based structures and
record mobility (Al,Ga)As/GaAs heterostructures (cf. Fig. 12.37), also (Mg,Zn)O/ZnO heterostructures
exhibit the effect [1463, 1464] in the same fashion.
Within the plateau, the resistivity is well defined within 10
−7 or better up to the 10
−9 regime. A
precise determination allows for a new normal for the unit Ohm [1389, 1465], being two orders of
magnitude more precise than the realization in the SI system, and an independent value for the finestructure constant α = (e
2
/ h)/(2 c 0 ). At the same time, the longitudinal resistivity, starting from the
classical value for small magnetic fields, exhibits oscillations and eventually it is close to zero for the
plateaus in ρ xy . For ρ xx values of 10
−10
have been measured, which corresponds to 10
−16
/cm
for bulk material, a value three orders of magnitude smaller than for any non-superconductor.
The interpretation of the quantum Hall effect(s) is discussed in [1466] among many other treatises.
The simplest explanation is that the conductivity is zero when a Landau level is completely filled and
the next is completely empty, i.e. the Fermi level lies between them. The temperature is small, i.e.
kT c , such that no scattering between Landau levels can occur. Thus no current, similar to a
completely filled valence band, can flow. The sheet carrier density n s is given by counting the i filled
Landau levels (degeneracy according to (15.37)) as
n s = i
e B
h
.
(15.43)
In the transverse direction energy dissipation takes place and the Hall resistivity ρ xy = B/(n s e) takes
the (scattering-free) values given in (15.42).
However, this argument is too simple as it will not explain the extension of the plateaus. As soon
as the system has one electron more or less, the Fermi energy will (for a system with δ-like density of