Landau levels and magneto-transport property of monolayer phosphorene

The group V element phosphorus can exist in several allotropes and black phosphorus (BP) is the most stable phase under normal conditions1. Recently, layered BP has attracted intensive attention because of its unique electronic properties and potential applications in nanoelectronics2,3,4,5,6,7,8. In the bulk form, BP is a van der Waals-bonded layered material where each layer forms a puckered surface due to sp3 hybridization2,3. BP possesses a direct band gap 0.3 eV located at Z point3,4. This direct gap increases to 1.5–2 eV when the thickness decreases from bulk to few layers and eventually monolayer via mechanical exfoliation3,5,9. Hence, BP is an appealing candidate for tunable photodetection from the visible to the infrared part of the spectrum10. Further, the field-effect-transistor (FET) based on few layer BP is found to have an on/off ratio of 105 and a carrier mobility at room temperature as high as 103 cm2/V·s3,5, which make BP a favorable material for next generation electronics. The low energy physics of monolayer BP (phosphorene) around Γ point can be well described by an anisotropic two band k·p model2, which agrees well with a tight binding (TB) model11. To date, various interesting properties for phosphorene have been predicted theoretically and verified experimentally, including those related to strain induced gap modification2, tunable optical properties12, layer controlled anisotropic excitons13, quantum oscillations in few layers BP14,15,16 etc. However, the Landau levels (LLs) and magneto-transport (MT) properties of this unique anisotropic system remain unexplored. In this work, we study the LL spectra and MT properties of phosphorene under a perpendicular magnetic field. By using an effective k·p Hamiltonian, we find that the LLs linearly depend both on energy 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). Interestingly, 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. Further, we find the LLs of phosphorene nanoribbon depend strongly on the ribbon orientation due to the anisotropic hopping parameters. In order to detect those interesting magneto energy spectra, we calculate MT properties of phosphorene within the framework of the linear response theory. By using Kubo formula, we find the Hall and the longitudinal conductances (resistances) clearly reveal the structure of LLs.