  In this study, we investigate the underlying mechanisms of the negative
piezoelectricity in low--dimensional materials by carrying out
first--principles calculations. Two--dimensional ferroelectric CuInP$_2$S$_6$
is analyzed in detail as a typical example, but the theory can be applied to
all other low--dimensional piezoelectrics. Similar to three--dimensional
piezoelectrics with negative piezoelectric responses, the anomalous negative
piezoelectricity in CuInP$_2$S$_6$ results from its negative clamped--ion term,
which cannot be compensated by the positive internal strain part. Here, we
propose a more general rule that having a negative clamped--ion term should be
universal among piezoelectric materials, which is attributed to the "lag of
Wannier center" effect. The internal--strain term, which is the change in
polarization due to structural relaxation in response to strain, is mostly
determined by the spatial structure and chemical bonding of the material. In a
low--dimensional piezoelectric material as CuInP$_2$S$_6$, the internal--strain
term is approximately zero. This is because the internal structure of the
molecular layers, which are bonded by the weak van der Waals interaction,
responds little to the strain. As a result, the magnitude of the dipole, which
depends strongly on the dimension and structure of the molecular layer, also
has a small response with respect to strain. An equation bridging the internal
strain responses in low--dimensional and three--dimensional piezoelectrics is
also derived to analytically express this point. This work aims to deepen our
understanding about this anomalous piezoelectric effect, especially in
low--dimensional materials, and provide strategies for discovering materials
with novel electromechanical properties.
