Integer quantum Hall effect in a spin-orbital coupling system

Abstract

Electron transport mechanism of a two-dimensional infinite slab subjected to Rashba spin-orbital coupling is studied in this paper. We calculate the Hall conductance and the longitudinal resistance of the integer quantum Hall effect (IQHE). In a strong magnetic field, the Landau levels of electrons increase rapidly at large wave vectors due to the constraint of the two edges of the sample while they remain flat at small wave vectors. Although the Zeeman effect can split the energy levels of spin degeneracy under a strong magnetic field, the spacing between the Landau levels is exactly equal to the spin splitting, thus the spin degeneracies have not been fully resolved. The spin-orbital coupling fully resolves the spin degeneracies of the energy levels. This is the key to reproducing the IQHE. Electrons with rapid increasing energies are localized at the two edges of the sample and transport along the edges to form separated currents with opposite directions. In this case, back scattering of electrons is prohibited due to the localization of these two branches. Since the electrons on the upper and lower edges originate respectively from the left and right electrode, they also have the chemical potentials of the electrons in those electrodes, respectively. The computation result shows that the Hall conductance appears as plateaus at integer times of <i>e</i><sup>2</sup>/<i>h</i>. Temperature influences the accuracy of the Hall plateaus. As an international resistance standard, exceeding a critical temperature can produce significant errors to the Hall plateaus. Below the critical temperature, the accuracy can reach 10<sup>–9</sup>. Finally the mechanism of the longitudinal resistance of the IQHE is discussed and computed numerically. It is shown that only the wave-functions with opposite and small wave vectors have a significant overlap in the bulk of the sample and thus contribute to the longitudinal resistance. Due to the separation of currents in different directions in space, the longitudinal resistance does vanish at the Hall plateaus but it appears when the Hall conductance jumps from one plateau to another one.