Negative piezoelectricity in low--dimensional materials

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.