Possible States for a Three-Dimensional Electron Gas in a Strong Magnetic Field

If a three-dimensional semimetal or doped semiconductor is placed in a sufficiently strong magnetic field, then a change in its transport properties will occur. If the electron-impurity interaction is dominant, then the magnetic field will produce localization of the electron wavefunctions, sometimes described as magnetic freezeout. If the electron-electron interaction is more important, then some type of collective transition may occur. Spin-density waves, charge-density waves, valley-density waves, excitonic insulators, and Wigner crystallization have been proposed to occur under various circumstances. As a generalization to three-dimensions of the integral quantized Hall effect, we show that for electrons in periodic or quasiperiodic potential, when the Fermi level lies in an energy gap, the zero temperature conductivity tensor is given by σ i j = (e 2 /2π h) ∑ k ε i j k G k, where G is a reciprocal-lattice-vector of the potential. We discuss the effect of impurities and dislocations on this result.