15.2 Magnetic Fields
447
states) be located in the upper or lower Landau level, respectively. Then, the longitudinal conductivity
should no longer be zero and the Hall resistivity deviates from the integer fraction of ρ 0 . In general, a
single particle picture is insufficient to model the IQHE.
The accepted model for the explanation of the QHE, supported now with plenty of experimental
evidence, is the edge state model where quantized one-dimensional conductivity of edge channels, i.e.
the presence of conductive channels along the sample boundaries, is evoked [1475].
The most fundamental arguments for the origin of the IQHE come from gauge invariance and the
presence of a macroscopic quantum state of electrons and magnetic flux quanta [1468]. This model
holds as long as there are any extended states at all in the inhomogeneously broadened density of
states (Landau levels) (Fig. 15.14). These edge states arise from the fact that the quantum Hall state
has another topology than the (topologically trivial) surrounding [463, 1469, 1470]. Each plateau
corresponds to a distinct topological phase characterized by a Chern number (cmp. (6.29)) related to
the ‘Hofstadter butterfly’ [1471, 1472] (cf. Sect. G.3.1).
It is remarkable that the classical Hall effect is based on homogeneous conduction for a proper
evaluation (cf. Fig. 15.7e), while the quantum Hall effect involves electrical transport only along the
edges. A detailed microscopic picture of the edge channels is also of interest. Due to depletion at the
boundary of the sample, the density of the 2DEG varies at the edge of the sample and ‘incompressible’
stripes develop for which ∂μ/∂n s → ∞. When the filling factor is far from an integer, the Hall voltage
is found to vary linearly across the conductive channel and the current is thus homogeneous over the
sample (Fig. 15.20d). In the Hall plateau, the Hall voltage is flat in the center of the channel and exhibits
drops at the edges, indicating that the current flows along the boundary of the sample (edge current)
[1473] in agreement with predictions from [1476]. Although the current pattern changes with varying
magnetic field, the Hall resistivity remains at its quantized value.
In graphene, the IQHE plateaus take on the resistance values [1249, 1477],
1
ρ xy
= ±4
n +
1
2
e
2
h
,
(15.44)
with n ∈ N 0 . This new ‘half integer’ quantization condition can be translated into the quantized filling
factor ν = ±4 (n + 1/2) in the usual QHE formalism of (15.42). The condition (15.44) is due to
the different nature of Landau levels for a linear dispersion (Dirac spectrum) [1478]. The +/−-sign
denotes the QHE for electrons and holes, respectively. The factor g = 4 stems from the degeneracy of
the Landau levels, accounting for spin degeneracy (when the Landau level separation is much larger
than the Zeeman spin splitting) and for the sublattice degeneracy. The two n = 0 Hall plateaus have
even been observed (for B = 45 T) at room temperature [1479].
15.2.8.2 Fractional QHE
For very low temperatures and in the extreme quantum limit, novel effects are observed when the
kinetic energy of the electrons is smaller than their Coulomb interaction. New quantum Hall plateaus
are observed at various fractional filling factors ν = p/q. We note that the effects of the fractional
quantum Hall effect (FQHE) in Fig. 15.21 mostly arise for magnetic fields beyond the n = 1 IQHE
plateau. The filling factor ν = n/(eB/ h) is now interpreted as the number of electrons per magnetic
flux quantum φ 0 = h/e.
The effects of the FQHE cannot be explained by single-electron physics. The plateaus at fractional
fillings ν occur when the Fermi energy lies within a highly degenerate Landau (or spin) level and imply
the presence of energy gaps due to many-particle interaction and the result of correlated 2D electron
motion in the magnetic field.
447
states) be located in the upper or lower Landau level, respectively. Then, the longitudinal conductivity
should no longer be zero and the Hall resistivity deviates from the integer fraction of ρ 0 . In general, a
single particle picture is insufficient to model the IQHE.
The accepted model for the explanation of the QHE, supported now with plenty of experimental
evidence, is the edge state model where quantized one-dimensional conductivity of edge channels, i.e.
the presence of conductive channels along the sample boundaries, is evoked [1475].
The most fundamental arguments for the origin of the IQHE come from gauge invariance and the
presence of a macroscopic quantum state of electrons and magnetic flux quanta [1468]. This model
holds as long as there are any extended states at all in the inhomogeneously broadened density of
states (Landau levels) (Fig. 15.14). These edge states arise from the fact that the quantum Hall state
has another topology than the (topologically trivial) surrounding [463, 1469, 1470]. Each plateau
corresponds to a distinct topological phase characterized by a Chern number (cmp. (6.29)) related to
the ‘Hofstadter butterfly’ [1471, 1472] (cf. Sect. G.3.1).
It is remarkable that the classical Hall effect is based on homogeneous conduction for a proper
evaluation (cf. Fig. 15.7e), while the quantum Hall effect involves electrical transport only along the
edges. A detailed microscopic picture of the edge channels is also of interest. Due to depletion at the
boundary of the sample, the density of the 2DEG varies at the edge of the sample and ‘incompressible’
stripes develop for which ∂μ/∂n s → ∞. When the filling factor is far from an integer, the Hall voltage
is found to vary linearly across the conductive channel and the current is thus homogeneous over the
sample (Fig. 15.20d). In the Hall plateau, the Hall voltage is flat in the center of the channel and exhibits
drops at the edges, indicating that the current flows along the boundary of the sample (edge current)
[1473] in agreement with predictions from [1476]. Although the current pattern changes with varying
magnetic field, the Hall resistivity remains at its quantized value.
In graphene, the IQHE plateaus take on the resistance values [1249, 1477],
1
ρ xy
= ±4
n +
1
2
e
2
h
,
(15.44)
with n ∈ N 0 . This new ‘half integer’ quantization condition can be translated into the quantized filling
factor ν = ±4 (n + 1/2) in the usual QHE formalism of (15.42). The condition (15.44) is due to
the different nature of Landau levels for a linear dispersion (Dirac spectrum) [1478]. The +/−-sign
denotes the QHE for electrons and holes, respectively. The factor g = 4 stems from the degeneracy of
the Landau levels, accounting for spin degeneracy (when the Landau level separation is much larger
than the Zeeman spin splitting) and for the sublattice degeneracy. The two n = 0 Hall plateaus have
even been observed (for B = 45 T) at room temperature [1479].
15.2.8.2 Fractional QHE
For very low temperatures and in the extreme quantum limit, novel effects are observed when the
kinetic energy of the electrons is smaller than their Coulomb interaction. New quantum Hall plateaus
are observed at various fractional filling factors ν = p/q. We note that the effects of the fractional
quantum Hall effect (FQHE) in Fig. 15.21 mostly arise for magnetic fields beyond the n = 1 IQHE
plateau. The filling factor ν = n/(eB/ h) is now interpreted as the number of electrons per magnetic
flux quantum φ 0 = h/e.
The effects of the FQHE cannot be explained by single-electron physics. The plateaus at fractional
fillings ν occur when the Fermi energy lies within a highly degenerate Landau (or spin) level and imply
the presence of energy gaps due to many-particle interaction and the result of correlated 2D electron
motion in the magnetic field.