Understanding and tuning negative longitudinal piezoelectricity in hafnia

Piezoelectric materials convert between mechanical and electrical energies and are widely used in ultrasound, sonar, sensors and power electronics. Most exhibit positive longitudinal piezoelectricity (PLPE), where polarization increases under tensile strain along the polar axis, yielding positive e33. Negative longitudinal piezoelectricity (NLPE), in which polarization decreases under tensile strain, has been rarely observed; known cases include PVDF polymers (attributed to intermixed phases), van der Waals CuInP2S6 (linked to reduced lattice dimensionality), and some theoretically predicted inorganic systems, many of which are layered with limited elastic constants that can limit electromechanical performance. Hafnia (HfO2) was discovered to be ferroelectric (polar orthorhombic Pca21) and is promising for electronic and storage devices due to CMOS compatibility, high mechanical strength, and high sound velocity. Despite its ferroelectricity, hafnia’s piezoelectric response has been less studied, with inconsistent experimental observations of longitudinal piezoelectricity (both positive and negative reported in doped films). There is a need for an intuitive, quantitative picture of hafnia’s intrinsic piezoelectric response and a systematic understanding of dopant effects. This study uses DFT to clarify the origin of NLPE in HfO2 via a bonding descriptor (WPBc) and explores how doping tunes e33 through changes in bonding asymmetry.

Prior reports of NLPE include PVDF polymers, where NLPE has been linked to intermixed crystalline phases, and van der Waals layered CuInP2S6, where reduced lattice dimensionality was implicated. Theoretical predictions identified NLPE in hexagonal ABC ferroelectrics and 2D heterobilayers. Many NLPE systems are layered and mechanically soft, potentially unsuitable for high-frequency electromechanical applications. For hafnia, ferroelectricity originates from the polar orthorhombic Pca21 phase; dopants such as Y and La have been used to enhance ferroelectric behavior, but experimental reports of the sign of e33 remain inconsistent, potentially due to texture or processing differences. Previous bonding analyses for hafnia’s NLPE emphasized the most longitudinal Hf–O bond, overlooking contributions from all coordinated bonds; a more comprehensive bonding model is warranted.

First-principles calculations were performed using VASP with PAW potentials and PBE-GGA exchange-correlation, a 520 eV plane-wave cutoff, total energy convergence of 1e-5 eV, and force tolerance of 0.01 eV/Å. I-centered k-point meshes were used: for 96-atom supercells, 3×3×3 for po and m phases; for 48-atom supercells, 3×3×6 for po and m and 4×4×3 for t phase. Piezoelectric stress coefficients were computed using DFPT workflows; elastic tensors were obtained by fitting stress–strain using six independent Green-Lagrange strains at ±0.5% and ±1%. Bonding strengths were analyzed via COHP using LOBSTER. To rationalize longitudinal piezoelectricity, the projected bond strength along c (PB) was defined for each cation–anion bond and aggregated into a weighted projected bond strength along c (WPBc) by summing positive and negative projections over all nearest neighbors around a cation. Stability screening of dopants followed a two-step process. Step 1: 49 dopants were examined at 3.125% substitution in 96-atom supercells using electronic charge compensation. The relative phase stability was characterized by E_diff = E_po(t) − E_m; ΔE_diff between doped and pure systems gauged the dopant’s stabilization of po or t relative to m. Dopants yielding negative E_diff for po advanced to Step 2. Step 2: At 12.5% substitution in 48-atom supercells, charge neutrality was enforced using oxygen vacancies (VO): +1, +2, and +3 valence dopants accompanied by removal of 3, 2, and 1 VO per two substituted Hf, respectively. Defect configurations (dopant and VO) were enumerated and ranked by electrostatic energy using pymatgen; six lowest-energy arrangements were relaxed with DFT, and the lowest-energy configuration per valence case underwent DFPT for piezoelectric and elastic properties. The distribution (average and standard deviation) of WPBc over all cations in each doped system was computed to correlate bonding asymmetry with e33.

• Undoped HfO2 exhibits a negative longitudinal piezoelectric stress coefficient e33 ≈ −1.24 C/m² (similar in magnitude but opposite in sign to AlN with e33 ≈ +1.43 C/m²). The elastic constant C33 of HfO2 exceeds that of AlN due to strong Hf–O bonding; resulting d33 is negative and smaller in magnitude than AlN due to counteracting in-plane components.
• Decomposition shows ionic contributions dominate e33 in both systems with similar magnitude (~1.85 C/m²) but opposite sign; electronic contributions are +0.61 C/m² (HfO2) and −0.42 C/m² (AlN). In HfO2, tensile strain along c causes cation (Hf) to move downward (du3/de3 < 0) and anions (O) upward (du3/de3 > 0), reducing polarization and yielding NLPE; AlN shows the opposite displacements, increasing polarization and PLPE.
• Born effective charges in HfO2 exceed nominal values (mixed ionic/covalent bonding) but this is not the origin of the sign difference; the NLPE arises from ionic displacement under strain.
• A bonding descriptor, WPBc, quantifies asymmetric bonding stiffness along c. In HfO2, the sum over seven Hf–O bonds yields negative WPBc (more stiffness below than above along c), anti-parallel to spontaneous polarization, so tensile strain expands the softer bonds more, decreasing polarization (negative e33). In AlN, WPBc is positive and aligned with polarization, giving positive e33.
• High-throughput dopant screening identified 26 dopants (after excluding Rb) that favor the polar orthorhombic phase relative to monoclinic and were used for piezoelectric analysis.
• Piezoelectric stress coefficients for selected doped HfO2 show: Sn (e33 = −2.04 C/m²), Pb (−1.72 C/m²), Rh (−1.33 C/m²) enhance the magnitude of NLPE beyond pure HfO2 and even AlN (for Sn, Pb). Eight dopants (Au, Al, Sc, Fe, Ga, Y, Ge, Ir) cause sign reversal or positive e33; La and Y doped systems have e33 near zero, making the measured sign sensitive to external factors.
• Correlation established: systems with negative average WPBc tend to have negative e33; more negative average WPBc and smaller STD(WPBc) correspond to stronger (more negative) e33. Sn-doped hafnia shows the most negative average WPBc and low STD, matching its largest NLPE.
• Elastic constants (C33) for Pb2Hf14O30, Sn2Hf14O32, and Rh2Hf14O31 decrease relative to pure HfO2 but remain >300 GPa, preserving mechanical robustness. Calculated d33 shows Sn- and Rh-doped systems have more than twofold enhancement in magnitude over pure HfO2; Sn-doped HfO2 achieves d33 ≈ 3.8 pC/N, comparable to AlN.
• Spontaneous polarization of Sn0.125Hf0.875O2 is ~0.54 C/m², similar to pure HfO2 (0.52–0.54 C/m²), and its po-phase stability improves relative to pure, supporting ferroelectricity.
• Design guidance: selecting dopants with ionic radius similar to Hf4+ can reduce local bonding fluctuations (lower STD(WPBc)); dopants less electronegative than Hf can strengthen dopant–O bonds and tune average WPBc toward more negative values.

The study addresses the origin of NLPE in ferroelectric HfO2 by showing that the longitudinal piezoelectric response is governed predominantly by ionic displacements under strain, which are dictated by asymmetric bonding stiffness along the polar axis. The WPBc descriptor provides an intuitive and quantitative link between local bonding geometry and macroscopic e33: when WPBc is anti-parallel to the spontaneous polarization, tensile strain expands softer bonds in a way that reduces polarization, resulting in NLPE. Extending this framework to doped hafnia demonstrates that both the direction (sign) and distribution (average and STD) of WPBc across cations control the sign and magnitude of e33. The correlation holds across 26 doped systems and identifies Sn, Pb, and Rh as effective dopants to enhance NLPE while maintaining high elastic stiffness suitable for devices. The findings reconcile inconsistent experimental reports by indicating that systems with e33 near zero (e.g., La, Y doped) are susceptible to external stimuli (texture, strain, size), which can flip the observed sign. Practically, the results offer materials-by-design rules: reduce local bonding fluctuations and engineer more negative average WPBc (via ionic size matching and lower dopant electronegativity) to strengthen NLPE and improve d33 without sacrificing mechanical robustness, thereby positioning Sn-doped HfO2 as a promising candidate for electromechanical components (e.g., acoustic filters) compatible with CMOS processes.

This work elucidates the negative longitudinal piezoelectricity of HfO2 through first-principles calculations and a bonding-based descriptor, WPBc, that captures asymmetric stiffness along the polar axis. The sign of e33 is set by the alignment between WPBc and spontaneous polarization, and the magnitude is strengthened by more negative average WPBc and reduced STD(WPBc). A two-step dopant screening identified 26 stabilizing dopants for the polar orthorhombic phase; among them, Sn-, Pb-, and Rh-doped HfO2 significantly enhance NLPE, with Sn achieving e33 = −2.04 C/m² and d33 ≈ 3.8 pC/N while retaining high C33 (>300 GPa). These insights provide clear design rules for tuning piezoelectricity in hafnia and potentially other ferroelectrics. Future work should incorporate additional extrinsic factors (substrate-induced strain, size effects, texture) and experimental validation to further optimize and control the piezoelectric response in device-relevant environments, as well as explore broader chemical spaces guided by WPBc-based descriptors.

The study relies on DFT/DFPT calculations with specific functionals and supercell models, which may not capture all finite-temperature, microstructural, and processing effects. Dopant screening employed fixed concentrations (3.125% for initial screening with electronic compensation; 12.5% with VO for charge neutrality) and considered a limited set of low-energy defect configurations, which may not encompass all realistic defect arrangements. Experimental reports of e33 in doped hafnia can be sensitive to external factors (substrate strain, film texture, size effects), and such multifactor influences were not fully treated here and are deferred to future work.