In a real crystal, in which defects and impurities are inevitable, things may
change. Let suppose that only the lowest Landau level is entirely occupied and
start to reduce B z . Accordingly, the degeneration decreases, and a growing number
of electrons is forced to move to the adjacent level that may be too high. From the
energetic point of view, for them it becomes more convenient to be entrapped into
localized impurity sites. Therefore, such electrons are not movable and, consequently, they cannot participate to the conductivity. A further decrease of B z reduces
the Landau energy separation and for trapped electrons becomes convenient to fill an
available lower vacant level. It comes out that the Landau levels in a real crystal are
entirely occupied or entirely unfilled over a broad range of B z . The plateau in every
step of ρ xy is described in the framework that there are always occupied Landau
levels with the exclusion of those B z amplitudes, falling in the narrow regions in
which a transition from the last filled Landau level to the next unoccupied one
occurs. Therefore, the magnetoresistance ρ xx is equal to zero because when some
Landau levels are entirely full and others completely void, there is no chance of
electron scattering. In the narrow transition range in which s takes the contiguous
value there will be non-completely filled levels, scattering becomes possible, consequently ρ xx is not more null (Persano-Adorno et al. 2018b).
Students should observe that in an ideal structure, without defects, the IQHE
would not be possible.
16.3.4 Extension Phase to the Fractional Quantum Hall
Effect
The fractional quantum Hall effect is a particular case of quantum Hall effect which
arises in a small number of low-dimensionality systems, (for example in graphene)
where electrons assume a collective behavior, exhibiting unexpected features. The
FQHE is observable in these 2D materials at very high magnetic fields (>150 T) and
very low temperatures (<1 K). The principal difference between the IQHE and the
FQHE lies on the basic electron feature: they behave as free (noninteracting)
particles in IQHE and correlated interacting electrons in the FQHE. In FQHE the
equal spacing of the Landau levels is substituted by a collective behavior of
composite quasiparticles whose charge is smaller than e and the level separation is
proportional to B z
½ . The unexpected circumstance that the charge of these composite
quasiparticles is smaller than e has been ascribed to bits of magnetic field, that
attached to each electron, give rise to a new object whose properties are different
from those of a free or a bound electron. These quasiparticles are affected by an
effective magnetic field B z *, different from the external applied field B z . Their
movement appears do not depend strictly on the magnetic field amplitude and they
may behave as bosons or fermions depending on the magnetic field (Laughlin 1983).
16 Inquiry-Based Approach and Numerical Simulations: A Powerful Integration in. . .
211
change. Let suppose that only the lowest Landau level is entirely occupied and
start to reduce B z . Accordingly, the degeneration decreases, and a growing number
of electrons is forced to move to the adjacent level that may be too high. From the
energetic point of view, for them it becomes more convenient to be entrapped into
localized impurity sites. Therefore, such electrons are not movable and, consequently, they cannot participate to the conductivity. A further decrease of B z reduces
the Landau energy separation and for trapped electrons becomes convenient to fill an
available lower vacant level. It comes out that the Landau levels in a real crystal are
entirely occupied or entirely unfilled over a broad range of B z . The plateau in every
step of ρ xy is described in the framework that there are always occupied Landau
levels with the exclusion of those B z amplitudes, falling in the narrow regions in
which a transition from the last filled Landau level to the next unoccupied one
occurs. Therefore, the magnetoresistance ρ xx is equal to zero because when some
Landau levels are entirely full and others completely void, there is no chance of
electron scattering. In the narrow transition range in which s takes the contiguous
value there will be non-completely filled levels, scattering becomes possible, consequently ρ xx is not more null (Persano-Adorno et al. 2018b).
Students should observe that in an ideal structure, without defects, the IQHE
would not be possible.
16.3.4 Extension Phase to the Fractional Quantum Hall
Effect
The fractional quantum Hall effect is a particular case of quantum Hall effect which
arises in a small number of low-dimensionality systems, (for example in graphene)
where electrons assume a collective behavior, exhibiting unexpected features. The
FQHE is observable in these 2D materials at very high magnetic fields (>150 T) and
very low temperatures (<1 K). The principal difference between the IQHE and the
FQHE lies on the basic electron feature: they behave as free (noninteracting)
particles in IQHE and correlated interacting electrons in the FQHE. In FQHE the
equal spacing of the Landau levels is substituted by a collective behavior of
composite quasiparticles whose charge is smaller than e and the level separation is
proportional to B z
½ . The unexpected circumstance that the charge of these composite
quasiparticles is smaller than e has been ascribed to bits of magnetic field, that
attached to each electron, give rise to a new object whose properties are different
from those of a free or a bound electron. These quasiparticles are affected by an
effective magnetic field B z *, different from the external applied field B z . Their
movement appears do not depend strictly on the magnetic field amplitude and they
may behave as bosons or fermions depending on the magnetic field (Laughlin 1983).
16 Inquiry-Based Approach and Numerical Simulations: A Powerful Integration in. . .
211