Abstract
We investigate theoretically the Landau levels (LLs) and magneto-transport properties of phosphorene under a perpendicular magnetic field within the framework of the effective k·p Hamiltonian and tight-binding (TB) model. At low field regime, we find that the LLs linearly depend both on the LL index n and magnetic field B, which is similar with that of conventional semiconductor two-dimensional electron gas. The Landau splittings of conduction and valence band are different and the wavefunctions corresponding to the LLs are strongly anisotropic due to the different anisotropic effective masses. An analytical expression for the LLs in low energy regime is obtained via solving the decoupled Hamiltonian, which agrees well with the numerical calculations. At high magnetic regime, a self-similar Hofstadter butterfly (HB) spectrum is obtained by using the TB model. The HB spectrum is consistent with the LL fan calculated from the effective k·p theory in a wide regime of magnetic fields. We find the LLs of phosphorene nanoribbon depend strongly on the ribbon orientation due to the anisotropic hopping parameters. The Hall and the longitudinal conductances (resistances) clearly reveal the structure of LLs.
Highlights
Index n and magnetic field B at low-field regime, which means the LLs in phosphorene are similar with that in conventional semiconductor two dimensional gases (2DEGs)
Owing to the anisotropic energy dispersions, i.e., the effective masses, the Landau splittings of conduction and valence band are different for a fixed magnetic field, and the wavefunctions corresponding to the LLs show strong anisotropic behavior
In the low field regime, we found that the LLs linearly depend both on the LL index n and magnetic field B, which is similar with that of conventional semiconductor two-dimensional electron gas
Summary
Index n and magnetic field B at low-field regime, which means the LLs in phosphorene are similar with that in conventional semiconductor two dimensional gases (2DEGs). Owing to the anisotropic energy dispersions, i.e., the effective masses, the Landau splittings of conduction and valence band are different for a fixed magnetic field, and the wavefunctions corresponding to the LLs show strong anisotropic behavior. We obtain an analytical expression for the LLs in low energy regime via solving a decoupled Hamiltonian, which agrees well with the numerical data in low energy regime. At high-field regime, magneto-level spectrum, i.e., the Hofstader butterfly (HB) spectrum, is obtained by using a tight binding (TB) model. We find that the results obtained by the effective k·p Hamiltonian and TB model agree with each other in weak magnetic field cases. By using Kubo formula, we find the Hall and the longitudinal conductances (resistances) clearly reveal the structure of LLs
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