Doubling of Landau levels in GaAs/AlGaAs heterostructures

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Usually the Landau levels are given by (n+1/2)ℏωc with ℏωc=eB/mc. We find that there are clusters of electrons with localization so that the orbital angular momentum is not zero. There are interactions which transfer the electrons from l=0 to l≠ 0 so that there is doubling in the Landau levels with ℏωc= (1/2)g/mc where (1/2)g±=(l+(1/2)± s)/(2 l+1) For s=± 1/2 two distinct values of ωc appear whereas only one seems to be known in the literature. This is because only one value s=+1/2 is populated. When microwaves of a suitable frequency such as 13.1 GHz at a field of about 2.5 Tesla, are turned on, s=-1/2 also gets populated. For singlets s=0, (1/2)g±=1/2 so that ωc=(1/2)(eB/mc). This will double the number of oscillations in the resistivity as a function of magnetic field. This theory of doubling of the oscillations agrees with the experimental data on GaAs/AlGaAs.