materials. The experimental evidences show that ρ xy varies by steps regulated by the
law ρ xy ¼ h/(se
2 ), where s is a positive integer dependent on B z (Tsui 1999). The
coefficient h/e
2 is a universal constant independent on the used sample. Experimental results acquired in a low-mobility two-dimensional electron gas in GaAs/
AlxGa1-xAs at a lattice temperature T ¼ 66 mK and a carrier concentration
n ¼ 1.93 Â 10
11 /cm
3 , shows the concomitant quantization of the Hall resistivity
ρ xy characterized by the step behavior and the fall of the magnetoresistance ρ xx to
very low values (for more details, see Fig. 7 and its description in Persano-Adorno
et al. 2018b).
16.3.3 Explanation Phase
In the EXPLAIN phase, the learners stimulated by the educator should understand
that the IQHE is due to the circumstance that the closed circular orbits covered by the
electrons in a gas of free (independent) electrons only assume quantized energies,
called Landau Levels, whose values are solutions of the Schrödinger equation:
E n ¼ (n + 1/2)ħω c , with n ¼ 1,2,3 . . . . The mechanism originating the quantized
Hall resistivity depends on how many of these energy levels are occupied by
electrons (Kittel 2005). The number of permitted orbitals on each orbit is constant
for a given amplitude of the applied magnetic field. The electrons cannot dwell in the
energy gaps in between or in the quasi-continuous k z (unaffected by B z ) states that do
not occur in a 2D system. Figure 16.3 shows how the transition from the quasicontinuous Fermi levels to the highly degenerate Landau levels arises. Since the
degeneration rises with the B z amplitude, the population of the filled Landau levels
will depend on B z and on the size of the conductor. The electrons fill these levels up
to the last one that may be entirely filled or not.
We point out that these properties pertain to an ideal crystal structure.
Fig. 16.3 Left: quasicontinuous states of a Fermi
electron gas at T ¼ 0 in
absence of perturbations;
center: rearrangement of the
electron distribution in
presence of a magnetic field
without impurity/phonon
scattering (Landau levels);
right: rearrangement of the
electron states when
impurities cause scattering
and electron trapping
(localized states). (Adapted
with permission from Longo
2011)
210
D. Persano-Adorno
law ρ xy ¼ h/(se
2 ), where s is a positive integer dependent on B z (Tsui 1999). The
coefficient h/e
2 is a universal constant independent on the used sample. Experimental results acquired in a low-mobility two-dimensional electron gas in GaAs/
AlxGa1-xAs at a lattice temperature T ¼ 66 mK and a carrier concentration
n ¼ 1.93 Â 10
11 /cm
3 , shows the concomitant quantization of the Hall resistivity
ρ xy characterized by the step behavior and the fall of the magnetoresistance ρ xx to
very low values (for more details, see Fig. 7 and its description in Persano-Adorno
et al. 2018b).
16.3.3 Explanation Phase
In the EXPLAIN phase, the learners stimulated by the educator should understand
that the IQHE is due to the circumstance that the closed circular orbits covered by the
electrons in a gas of free (independent) electrons only assume quantized energies,
called Landau Levels, whose values are solutions of the Schrödinger equation:
E n ¼ (n + 1/2)ħω c , with n ¼ 1,2,3 . . . . The mechanism originating the quantized
Hall resistivity depends on how many of these energy levels are occupied by
electrons (Kittel 2005). The number of permitted orbitals on each orbit is constant
for a given amplitude of the applied magnetic field. The electrons cannot dwell in the
energy gaps in between or in the quasi-continuous k z (unaffected by B z ) states that do
not occur in a 2D system. Figure 16.3 shows how the transition from the quasicontinuous Fermi levels to the highly degenerate Landau levels arises. Since the
degeneration rises with the B z amplitude, the population of the filled Landau levels
will depend on B z and on the size of the conductor. The electrons fill these levels up
to the last one that may be entirely filled or not.
We point out that these properties pertain to an ideal crystal structure.
Fig. 16.3 Left: quasicontinuous states of a Fermi
electron gas at T ¼ 0 in
absence of perturbations;
center: rearrangement of the
electron distribution in
presence of a magnetic field
without impurity/phonon
scattering (Landau levels);
right: rearrangement of the
electron states when
impurities cause scattering
and electron trapping
(localized states). (Adapted
with permission from Longo
2011)
210
D. Persano-Adorno