Microwave Hall conductivity of the two-dimensional electron gas in GaAs-AlxGa1−xAs

For the first time the quantum Hall effect was observed in ${\sigma }_{\mathrm{xy}}$ at microwave frequencies in the two-dimensional electron gas in GaAs-${\mathrm{Al}}_{0.35}$ ${\mathrm{Ga}}_{0.65}$As using a crossed waveguide arrangement. These frequencies were ${10}^3$–${10}^4$ higher than those used in previous experiments where a so-called low-frequency breakdown of the integer quantum Hall effect was observed. Our microwave results show that this effect does not occur in GaAs-${\mathrm{Al}}_{\mathrm{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$As heterostructures. Therefore, we have to rule out the mechanism of an effective delocalization at low frequencies and the existence of long semiclassical orbits in the sample.