Landau Level Crossings and Extended-State Mapping in Magnetic Two-dimensional Electron Gases

We present longitudinal and Hall magneto-resistance measurements of a ``magnetic'' two-dimensional electron gas (2DEG) formed in modulation-doped Zn$_{1-x-y}$Cd$_{x}$Mn$_{y}$Se quantum wells. The electron spin splitting is temperature and magnetic field dependent, resulting in striking features as Landau levels of opposite spin cross near the Fermi level. Magnetization measurements on the same sample probe the total density of states and Fermi energy, allowing us to fit the transport data using a model involving extended states centered at each Landau level and two-channel conduction for spin-up and spin-down electrons. A mapping of the extended states over the whole quantum Hall effect regime shows no floating of extended states as Landau levels cross near the Fermi level.