The Tight-Binding and the Nearly-Free-Electron Approach to Lattice Electrons in External Magnetic Fields

Abstract

The energy structure for lattice electrons in an external magnetic field is treated by extension of either the tight-binding or the nearly-free-electron method. The secular problems resulting from both methods are identical, if the coupling integrals and the real lattice characterizing the tight-binding method are interchanged with the Fourier integrals and the orbit lattice characterizing the nearly-free-electron method. A Bloch-type transformation reducing these essentially two-dimensional secular problems to one dimension is performed, following the procedure introduced in a preceding paper. For rational fluxes $\frac{2\pi N}{M}$ per lattice cell, an even finite secular determinant is left, yielding a splitting of each zero-field band into $M$ subbands. Total agreement between the level structures resulting from the two approaches is found. The relation of these structures to the de Haas-van Alphen periods in the case of magnetic breakdown is discussed.