Unique Landau-level structure of monolayer black phosphorus under an exponentially decaying magnetic field

We study the Landau-level spectrum of a monolayer black phosphorus under an exponentially decaying magnetic field along one spatial dimension. The results show that unlike the case in a constant magnetic field, the number of Landau levels in the inhomogeneous magnetic field is finite, and the Landau-level structure of the system is strongly dependent on the inhomogeneity of the magnetic field. In particular, the crossing of some Landau levels apparently occurs, and the accidental degeneracy points between the levels for the conduction and valence bands are highly anisotropic due to the anisotropic effective masses in monolayer black phosphorus. The above unique characteristics of the Landau-level structure in an exponentially decaying magnetic field could be directly confirmed by the magneto-absorption and transport measurements.