de Haas-van Alphen Effect, Exchange Splitting, Spin-Orbit Interaction, and Magnetic Breakdown in Ferromagnetic Nickel

Abstract

The detailed band structure of ferromagnetic nickel is investigated in the neighborhood of the symmetry point $X$ in the Brillouin zone. The Hamiltonian is constructed in the k·p representation, taking into account exchange and spin-orbit interactions as well as the coupling of spins to the magnetic induction vector B. Group-theoretical arguments, suitably modified in the presence of ferromagnetism, are employed to determine the form of this Hamiltonian. When exchange and spin-orbit interactions are taken into account, very few accidental degeneracies are allowed. Thus the spin-orbit coupling, in general, removes accidental degeneracies between bands of opposite spin and causes a hybridization of spin-up and spin-down energy bands. In the case of ferromagnetic nickel, the resultant spin-dependent energy gaps exhibit a strong dependence on the axis of spin quantization, and consequently they permit magnetic breakdown to occur between bands of opposite spin for a select range of applied-magnetic-field directions. These effects provide an explanation for the anomalous behavior of the de Haas-van Alphen amplitude which is observed experimentally.