  The Poisson's ratio is a fundamental mechanical property that relates the
resulting lateral strain to applied axial strain. While this value can
theoretically be negative, it is positive for nearly all materials, though
negative values have been observed in so-called auxetic structures. However,
nearly all auxetic materials are bulk materials whose microstructure has been
specifically engineered to generate a negative Poisson's ratio. In the present
work, we report using first principles calculations the existence of a negative
Poisson's ratio in a single-layer, two-dimensional material, black phosphorus.
In contrast to engineered bulk auxetics, this behavior is intrinsic for single
layer black phosphorus, and originates from its unique, puckered structure,
where the pucker can be regarded as a re-entrant structure that is comprised of
two coupled orthogonal hinges. As a result of this atomic structure, a negative
Poisson's ratio is observed in the out-of-plane direction under uniaxial
deformation in the direction parallel to the pucker, with the Poisson's ratio
becoming increasingly negative with both increased tension and compression. The
puckered structure also results in highly anisotropic in-plane Poisson's
ratios, which are found to be 0.4 in the direction perpendicular and 1.28 in
the direction parallel to the pucker.
