  Stacking two-dimensional (2D) materials into multi-layers or
heterostructures, known as van der Waals (vdW) epitaxy, is an essential degree
of freedom for tuning their properties on demand. Few-layer black phosphorus
(FLBP), a material with high potential for nano- and optoelectronics
applications, appears to have interlayer couplings much stronger than graphene
and other 2D systems. Indeed, these couplings call into question whether the
stacking of FLBP can be governed only by vdW interactions, which is of crucial
importance for epitaxy and property refinement. Here, we perform a theoretical
investigation of the vibrational properties of FLBP, which reflect directly its
interlayer coupling, by discussing six Raman-observable phonons, including
three optical, one breathing, and two shear modes. With increasing sample
thickness, we find anomalous redshifts of the frequencies for each optical mode
but a blueshift for the armchair shear mode. Our calculations also show
splitting of the phonon branches, due to anomalous surface phenomena, and
strong phonon-phonon coupling. By computing uniaxial stress effects,
inter-atomic force constants, and electron densities, we provide a compelling
demonstration that these properties are the consequence of strong and highly
directional interlayer interactions arising from electronic hybridization of
the lone electron-pairs of FLBP, rather than from vdW interactions. This
exceptional interlayer coupling mechanism controls the stacking stability of BP
layers and thus opens a new avenue beyond vdW epitaxy for understanding the
design of 2D heterostructures.
