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 effective k · p Hamiltonian, 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.

中文翻译：

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通过具有抛物线势限制的磁电容探测磷烯中朗道能级的各向异性


我们从理论上研究了垂直磁场下单层黑磷的朗道能级（LL）和磁电容（MC），分别沿着扶手椅和锯齿形方向施加抛物线势。通过解析微扰计算和基于有效k·p哈密顿量的数值对角化，我们发现LLs抛物线依赖于波矢，并且随着抛物线势沿着不同的晶体方向施加，表现出很强的各向异性。具体来说，通过解耦单带哈密顿量的微扰计算获得的解析 LL 与数值结果非常一致。重要的是，即使在低能量状态下，由于重新绘制回旋加速器轨道的抛物线势的限制，LL 也不再线性依赖于磁场和能级指数。此外，MC 谱清楚地反映了 LL 的结构，并表现出强烈的各向异性振荡模式。它可用于确定磷烯的能带参数，即在没有磁场和电场的情况下的有效质量和带间耦合。