Energy-Level Structure of Zinc in Magnetic Fields Using a Single-Band Effective Hamiltonian

The zero-magnetic-field energy-band structure of the midplane of the second band of zinc is Fourier analyzed and the coefficients E(→R) are used to construct the magnetic effective Hamiltonian $H({{→}}{{π}})={Σ}{R}^{}E({{→}}{R})exp(i{{→}}{R}·{}{{{→}}{{π}}}{{}})({{→}}{{π}}={{→}}{p}+{e{{→}}{A}}{c})$. This effective Hamiltonian is cast into the form of a matrix operator, and the eigenvalues are computed by numerical means for a range of magnetic fields. These exact solutions of the effective Hamiltonian display characteristics which can be compared with semiclassical results. The energy levels of the effective Hamiltonian form bands in a "magnetic zone" which is a zone in reciprocal space much smaller than the Brillouin zone. Within a magnetic zone the wave function and energy are functions of a vector →q which is in many respects similar to the zero-field wave vector →k. These magnetic bands can be associated with Landau levels corresponding to circular, lens, or triangle shaped contours, and as the magnetic field is varied the energies of the magnetic bands evolve in the manner that Landau levels are predicted to evolve from semiclassical arguments. The spacing of lens and triangle bands confirms the Onsager rule of equal areas between Landau contours; however, a systematic deviation is noted, the maximum deviation being about 5%. The Roth correction to the Onsager rule roughly predicts this deviation. The bands associated with lens and triangle contours display considerable broadening indicating strong coupling between these orbits.