Composite Fermions in a Long-Range Random Magnetic Field: Quantum Hall Effect versus Shubnikov–de Haas Oscillations

We study transport in a smooth random magnetic field, with emphasis on composite fermions (CFs) near half-filling of the Landau level. When either the amplitude of the magnetic field fluctuations or its mean value $\overline{B}$ is large enough, the transport is percolating in nature. While at $\overline{B}=0\mathrm{}$ the percolation enhances the conductivity ${\sigma }_{\mathrm{xx}}$, increasing $\overline{B}$ leads to a sharp falloff of ${\sigma }_{\mathrm{xx}}$ and, consequently, to the quantum localization of CFs. We show that the localization is a crucial factor in the interplay between the Shubnikov–de Haas and quantum Hall oscillations and that the latter are dominant in the CF metal.