Probing the anisotropy of Landau levels in phosphorene by magneto-capacitance with a parabolic potential confinement

We theoretically investigate the Landau levels (LLs) and magneto-capacitance (MC) of monolayer black phosphorus under a perpendicular magnetic field, on which a parabolic potential is applied along with the armchair and zigzag directions, respectively. By both analytically perturbative calculation and numerical diagonalization based on an effectivek⋅pHamiltonian, we find that the LLs parabolically depend on the wave vectors and show strong anisotropy as the parabolic potential is applied along with different crystal directions. Specifically, the analytical LLs obtained by perturbative calculation from a decoupled single-band Hamiltonian are in good agreement with the numerical results. Importantly, the LLs are no longer linearly dependent on the magnetic field and level index even in the low energy regime due to the confinement of parabolic potential which repaints the cyclotron orbits. Moreover, the MC spectrum clearly reflects the structure of the LLs and exhibits strong anisotropic oscillating patterns. It can be used to determine the band parameters of phosphorene, i.e., the effective masses and inter-band coupling in the absence of magnetic and electric fields.