We investigate the electronic properties of N -layer black phosphorus by means of an analytical method based on a recently proposed tight-binding Hamiltonian involving 14 hopping parameters. The method provides simple and accurate general expressions for the Hamiltonian of N -layer phosphorene, which are suitable for the study of electronic transport and optical properties of such systems, and the results show the features that emerge as the number of layers increases. In addition, we show that the N -layer problem can be translated into N effective monolayer problems in the long wavelength approximation and, within this analytical picture, we obtain expressions for the energy gap and the effective masses for electrons and holes along the N -layer black phosphorus plane directions as function of the number of layers, as well as for the Landau levels as function of perpendicular magnetic field.

Hydrodynamic flow in two-dimensional electron systems has so far been probed only by dc transport and scanning gate microscopy measurements. In this work we discuss theoretically signatures of the hydrodynamic regime in near-field optical microscopy. We analyze the dispersion of acoustic plasmon modes in two-dimensional electron liquids using a non-local conductivity that takes into account the effects of (momentum-conserving) electron-electron collisions, (momentum-relaxing) electron-phonon and electron-impurity collisions, and many-body interactions beyond the celebrated Random Phase Approximation. We derive the dispersion and, most importantly, the damping of acoustic plasmon modes and their coupling to a near-field probe, identifying key experimental signatures of the crossover between collisionless and hydrodynamic regimes. arXiv:1812.09889v1 [cond-mat.mes-hall]

We investigate the energy spectrum of single layer black phosphorene nanoribbons (BPN) by means of a low-energy expansion of a recently proposed tight-binding model that describes electron and hole bands close to the Fermi energy level. Using the continuum approach, we propose boundary conditions based on sublattice symmetries for BPN with zigzag and armchair edges and show that our results for the energy spectra exhibit good agreement with those obtained by using the five-parameter tight-binding model. We also explore the behaviour of the energy gap versus the nanoribbon width W . Our findings demonstrate that band gap of armchair BPNs scale as 1/W 2 , while zigzag BPNs exhibit a 1/W tendency. We analyse the different possible combinations of the zigzag edges that result two-fold degenerate and non-degenerate edge states. Furthermore, we obtain expressions for the wave functions and discuss the limit of validity of such analytical model.