Abstract
Quantum Hall effects (QHE) are observed in two-dimensional electron systems realised in semiconductors and graphene. In QHE the Hall resistance exhibits plateaus as a function of magnetic induction. In the fractional quantum Hall effects (FQHE) the values of the Hall resistance on plateaus are h/e2 divided by rational fractions, where −e is the electron charge and h is the Planck constant. The magnetic induction dependence of the Hall resistance is the strongest experimental evidence for FQHE. Nevertheless a quantitative theory of the magnetic induction and temperature dependence of the Hall resistance is still missing. Here we constructed a model for the Hall resistance as a function of magnetic induction, chemical potential and temperature. We assume phenomenological perturbation terms in the single-electron energy spectrum. The perturbation terms successively split a Landau level into sublevels, whose reduced degeneracies cause the fractional quantization of Hall resistance. The model yields all 75 odd-denominator fractional plateaus that have been experimentally found. The calculated magnetic induction dependence of the Hall resistance is consistent with experiments. This theory shows that the Fermi liquid theory is valid for FQHE.
Highlights
The basic mechanism of the integer Quantum Hall effects (QHE) (IQHE)[1,2] and FQHE3,4 is non-uniform distribution of electron density caused by the Lorentz force acting on the electrons[5]
To construct a model of the fractional quantum Hall effects (FQHE) let us first examine the theoretical mechanism of plateaus in integer QHE (IQHE), which can be quantitatively explained by adopting the Landau level εq = εNα = ωc(N + 1/2 + ζα) as the energy spectrum in equation (1)[7,8,9,10]
Considering the above analysis, let us inspect the Hall resistance data in the FQHE experiment reported in ref.[14]
Summary
The basic mechanism of the integer QHE (IQHE)[1,2] and FQHE3,4 is non-uniform distribution of electron density caused by the Lorentz force acting on the electrons[5]. To construct a model of the FQHE let us first examine the theoretical mechanism of plateaus in IQHE, which can be quantitatively explained by adopting the Landau level εq = εNα = ωc(N + 1/2 + ζα) as the energy spectrum in equation (1)[7,8,9,10]. Using the Hall resistance formula given by equation (1), we can determine the parameters λl from the experiment.
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