This study reports the tunable and parabolic piezoelectricity in hafnia (HfO₂) under epitaxial strain. Using first-principles calculations and experimental measurements, the researchers demonstrate that the sign of the longitudinal piezoelectric coefficient of HfO₂ can be tuned from positive to negative via epitaxial strain. Nonlinear and even parabolic piezoelectric behaviors are observed at tensile epitaxial strain. The parabolic piezoelectric behavior implies that the polarization decreases when increasing the magnitude of either compressive or tensile longitudinal strain, or equivalently, that the strain increases when increasing the magnitude of the electric field being either parallel or antiparallel to the direction of polarization. The unusual piezoelectric effects are attributed to the chemical coordination of active oxygen atoms. These striking piezoelectric features, including positive and negative signs and linear and parabolic behaviors, expand the current understanding of piezoelectricity and broaden the potential of piezoelectric applications in electro-mechanical and communications technologies. The study reveals that both the sign and magnitude of the longitudinal piezoelectricity of HfO₂-based films are tunable by strain. The piezoelectric behavior is dominated by both linear and quadratic piezoelectric coefficients. At negative and small tensile epitaxial strains, the piezoelectric coefficient is mainly from the linear coefficient, which is negative. At large tensile epitaxial strain, the coexistence of a positive linear coefficient and a negative quadratic term leads to a nonlinear dependence of polarization on strain. At the critical intermediate epitaxial strain, the piezoelectric behavior is parabolic, where polarization decreases with increasing strain magnitude. The study also shows that the piezoelectric coefficient can be tuned by epitaxial strain, which has been used to tune ferroelectricity in various compounds. The experimental results confirm the epitaxial strain-induced inversion of piezoelectricity from positive to negative. The study also demonstrates that the piezoelectric response can be used to convert the frequency of an AC field, which has not been discovered before. The findings suggest that HfO₂-based piezoelectric films have potential applications in medical devices and piezoelectric microelectromechanical systems (MEMS) due to their nanometric thickness and non-toxic nature. The study provides a deeper understanding of piezoelectricity and its potential applications in various technologies.This study reports the tunable and parabolic piezoelectricity in hafnia (HfO₂) under epitaxial strain. Using first-principles calculations and experimental measurements, the researchers demonstrate that the sign of the longitudinal piezoelectric coefficient of HfO₂ can be tuned from positive to negative via epitaxial strain. Nonlinear and even parabolic piezoelectric behaviors are observed at tensile epitaxial strain. The parabolic piezoelectric behavior implies that the polarization decreases when increasing the magnitude of either compressive or tensile longitudinal strain, or equivalently, that the strain increases when increasing the magnitude of the electric field being either parallel or antiparallel to the direction of polarization. The unusual piezoelectric effects are attributed to the chemical coordination of active oxygen atoms. These striking piezoelectric features, including positive and negative signs and linear and parabolic behaviors, expand the current understanding of piezoelectricity and broaden the potential of piezoelectric applications in electro-mechanical and communications technologies. The study reveals that both the sign and magnitude of the longitudinal piezoelectricity of HfO₂-based films are tunable by strain. The piezoelectric behavior is dominated by both linear and quadratic piezoelectric coefficients. At negative and small tensile epitaxial strains, the piezoelectric coefficient is mainly from the linear coefficient, which is negative. At large tensile epitaxial strain, the coexistence of a positive linear coefficient and a negative quadratic term leads to a nonlinear dependence of polarization on strain. At the critical intermediate epitaxial strain, the piezoelectric behavior is parabolic, where polarization decreases with increasing strain magnitude. The study also shows that the piezoelectric coefficient can be tuned by epitaxial strain, which has been used to tune ferroelectricity in various compounds. The experimental results confirm the epitaxial strain-induced inversion of piezoelectricity from positive to negative. The study also demonstrates that the piezoelectric response can be used to convert the frequency of an AC field, which has not been discovered before. The findings suggest that HfO₂-based piezoelectric films have potential applications in medical devices and piezoelectric microelectromechanical systems (MEMS) due to their nanometric thickness and non-toxic nature. The study provides a deeper understanding of piezoelectricity and its potential applications in various technologies.