In this article, we explore the anisotropic electron energy loss spectrum (EELS) in monolayer phosphorene based on ab-initio time dependent density functional theory calculations. Similar to black phosphorous, the EELS of undoped monolayer phosphorene is characterized by anisotropic excitonic peaks for energies in vicinity of the bandgap, and by interband plasmon peaks for higher energies. On doping, an additional intraband plasmon peak also appears for energies within the bandgap. Similar to other two dimensional systems, the intraband plasmon peak disperses as ω pl ∝ √ q in both the zigzag and armchair directions in the long wavelength limit, and deviates for larger wavevectors. The anisotropy of the long wavelength plasmon intraband dispersion is found to be inversely proportional to the square root of the ratio of the effective masses: ω pl (qŷ)/ω pl (qx) = mx/my.

Based on extensive first principle calculations, we explore the thickness dependent effective dielectric constant and slab polarizability of few layer black phosphorene. We find that the dielectric constant in ultra-thin phosphorene is thickness dependent and it can be further tuned by applying an out of plane electric field. The decreasing dielectric constant with reducing number of layers of phosphorene, is a direct consequence of the lower permittivity of the surface layers and the increasing surface to volume ratio. We also show that the slab polarizability depends linearly on the number of layers, implying a nearly constant polarizability per phosphorus atom. Our calculation of the thickness and electric field dependent dielectric properties will be useful for designing and interpreting transport experiments in gated phosphorene devices, wherever electrostatic effects such as capacitance, charge screening etc. are important.

The giant carrier mobility of graphene is significantly reduced due to external perturbations, such as substrate based charge impurities, and their impact can be minimized by encapsulating graphene between hexagonal boron nitride (hBN) layers. Using density functional theory (DFT) based ab initio calculations, we study the static response of such a composite by placing it in a vertical electric field. We find that at relatively low electric field (∼0.1 V/Å), although the relative permittivity (εr) of a composite stack increases with the number of layers, εr for a fixed stack thickness is independent of the field strength. However, at higher electric field strength, εr increases monotonically with the applied field strength even for a fixed stack thickness, signifying nonlinear response. The relative permittivity changes more readily for graphene rich stacks as compared to hBN rich stacks, which is consistent with the property of the pristine phases. We also present an empirical formulation to calculate the thickness and stacking dependent effective dielectric constant of any arbitrary stack of graphene–hBN layers, which fits very well with the ab initio calculations. Our empirical formulation will also be applicable for van der Waals stacks of other two-dimensional materials and will be useful for designing and interpreting transport experiments, where electrostatic effects such as capacitance and charge screening are important.

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