  Bulk black phosphorus (BP) consists of puckered layers of phosphorus atoms.
Few-layer BP, obtained from bulk BP by exfoliation, is an emerging candidate as
a channel material in post-silicon electronics. A deep understanding of its
physical properties and its full range of applications are still being
uncovered. In this paper, we present a theoretical and experimental
investigation of phonon properties in few-layer BP, focusing on the
low-frequency regime corresponding to interlayer vibrational modes. We show
that the interlayer breathing mode A3g shows a large redshift with increasing
thickness; the experimental and theoretical results agreeing well. This
thickness dependence is two times larger than that in the chalcogenide
materials such as few-layer MoS2 and WSe2, because of the significantly larger
interlayer force constant and smaller atomic mass in BP. The derived interlayer
out-of-plane force constant is about 50% larger than that in graphene and MoS2.
We show that this large interlayer force constant arises from the sizable
covalent interaction between phosphorus atoms in adjacent layers, and that
interlayer interactions are not merely of the weak van der Waals type. These
significant interlayer interactions are consistent with the known surface
reactivity of BP, and have been shown to be important for electric-field
induced formation of Dirac cones in thin film BP.
