Effective Hamiltonian approach for magnetic band structure and additional oscillations in the magnetization originating from the Fermi surface at finite magnetic fields

Abstract

The one-electron Schr\"odinger equation in a two-dimensional periodic potential and a homogeneous magnetic field $B$ perpendicular to the plane is solved exactly for rational flux quantum numbers per unit cell ${\ensuremath{\Phi}}_{c}∕{\ensuremath{\Phi}}_{0}=p∕q$. For comparison, the spectrum around a certain flux quantum number ${p}_{0}∕{q}_{0}$ has also been obtained by semiclassical quantization of the exact magnetic band structure (MBS) at ${p}_{0}∕{q}_{0}$. To implement and justify this procedure, a generalized effective Hamiltonian theory based on the MBS at finite magnetic fields has been established. The total energy as a function of ${\ensuremath{\Phi}}_{c}∕{\ensuremath{\Phi}}_{0}$ shows a series of kinks, where each kink indicates an insulating state. The kinks of each series converge to a metallic state. The magnetization contains information not only about the band structure (at zero-magnetic field), but also about the magnetic band structures (for finite fields). The period of the oscillations in $M[1∕(B\ensuremath{-}{B}_{0})]$ is determined by the Fermi surface cross sections for the MBS at ${B}_{0}$. The height of the steps in $M(B)$ provides the energy gap in the MBS at $B$. Unlike the standard Lifshitz-Kosevich type approaches, our theoretical de Haas-van Alphen spectra contain the effects of magnetic breakdown, forbidden orbits, and interband coupling implicitly.