Plotting 2D Landau levels There are these constants: Electron charge = q = 1.602×10-19 Coulomb Electron effective mass (GaAs) = mass = 0.067×9.11×10-31 kg Hbar = 1.5457×10-34 J-s μB = Bohr magneton = 9.274×10-24 J/tesla g-factor (GaAs)=0.44 electro spin = ½ The LL does not consider electron spin. Now, adding electron spins into the picture, there is Zeeman splitting: ΔE(B)=spin×g×μB×B. So, the LL energy is now labeled by ν, the filling factor, and 0 < ν < infinity. If ν = 1, this means that the lowest “spin-up” LL is 100% filled, while the lowest “spin-down” LL is totally empty. Each LL has a degeneracy of eB/h. First of all, the energy versus B is obtained by: Energy = (n+1/2)hbar×ω± spin×g×μB×B. For this plot below, the filling factor is ranged from zero to 20, and the magnetic field B is from zero to 10 tesla. There are apparently 10 lines, and this is because the Zeeman splitting energy is too small compared with the LL quantization energy. 01234567891000.0330.0670.10.130.170.2Landau levels vs. Magnetic FieldMagnetic field in teslaEnergy in eV0.164250EeV nu B,()100 BNext, we want to calculate the magnetic fields at which the filling factors are integers. Because the LL degeneracy is eB/h, and we know the n2D, so, the magnetic field values are obtained --- B=n2D×h/(q×ν). The result is plotted below. The filling factor ν is shown as a function of the corresponding magnetic fields. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.502468101214161820nu (FillingFactor) vs. B201nu4.13620.20681 Bnu nu() Now, knowing the magnetic field values, we can label these magnetic fields with the LL energy plot, to obtain the Fermi level oscillation picture.0123456789100.0330.0670.10.130.170.2Landau levels vs. Magnetic FieldMagnetic field in teslaEnergy in eV0.164250EeV nu B,()EeV nu Bnu nu(),()100 B Bnu nu(), This plot below is identical to the one above. We merely changed the scale in magnetic field and in the energy scale. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 56.66667.1040.001330.0020.002670.003330.004Landau levels vs. Magnetic FieldMagnetic field in teslaEnergy in eV0.0040EeV nu B,()EeV nu Bnu nu(),()50B Bnu nu(), Now, based on this plot, we can finally follow the solution and trace the Fermi level as a function of the magnetic field.Finally, we notice that the oscillatory Fermi level versus B has a pattern --- the oscillatory feature is periodic in 1/B. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.00670.01330.020.02670.03330.04Landau levels vs. Magnetic FieldMagnetic field in teslaEnergy in eV0.048.51046 105−×EeV nu B,()EeV nu Bnu nu(),()50.11B1Bnu nu(), This periodic behavior is often used in the analysis when the data does not show the text-book like 2D Landaul
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