Orbital magnetization of three-dimensional Dirac electrons in the quantum limit

In this study, we evaluated the dependence of magnetization of three-dimensional Dirac electrons in the quantum limit on the magnetic field and temperature. The magnetization was calculated by differentiating the free energy with respect to the magnetic field. The field and temperature dependence of the chemical potential were entirely considered under the canonical ensemble condition. The total magnetizationMconsisted of two contributions from the conductionMcand valenceMvbands.Mvwas insensitive to temperature and exhibited sub-linear field dependence, which is consistent with the previous research on Dirac electrons. By contrast,Mcwas sensitive to both temperature and magnetic field, yielding a non-trivial contribution to the totalM. As a result, the properties of totalMchanged at approximatelykBT≃EF, whereEFis the Fermi energy measured from the band bottom andkBis the Boltzmann constant. At low temperatureskBT≲EF,Mexhibited sub-linear field dependence, whereasMexhibited super-linear field dependence at high temperatureskBT≳EF. This qualitative change in the field dependence ofMwill play a significant role in the magnetization of Dirac electrons with smallEF.