  Two-dimensional crystals are emerging materials for nanoelectronics.
Development of the field requires candidate systems with both a high carrier
mobility and, in contrast to graphene, a sufficiently large electronic bandgap.
Here we present a detailed theoretical investigation of the atomic and
electronic structure of few-layer black phosphorus (BP) in order to predict its
electrical and optical properties. This system has a direct bandgap, tunable
from 1.51 eV for a monolayer to 0.59 eV for a 5-layer sample. We predict that
the mobilities are hole-dominated, rather high and highly anisotropic. The
monolayer is exceptional in having an extremely high hole-mobility (of order
10000 cm$^{2}$ V$^{-1}$ s$^{-1}$) and anomalous elastic properties which
reverse the anisotropy. Light absorption spectra indicate linear dichroism
between perpendicular in-plane directions, which allows optical determination
of the crystalline orientation and optical activation of the anisotropic
transport properties. These results make few-layer BP a promising candidate for
future electronics.
