Introduction

Piezoelectric materials constitute a class of functional materials capable of converting energy between mechanical energy and electrical energy. Since the initial discovery of the piezoelectric effect in quartz by the Curie brothers in 18801, research on piezoelectric materials has thrived and found extensive applications in various fields, including medical ultrasound, sonar, sensors, vibration-powered electronics and more2,3,4,5. Most piezoelectric materials exhibit positive longitudinal piezoelectricity, in which the polarization (P) increases with external tensile strain (\(\varepsilon\)) along the polar direction (Fig. 1a) and results in a positive piezoelectric coefficient e33 (\(\partial P/\partial \varepsilon\)). In contrast, materials with negative longitudinal piezoelectricity are rarely reported. One well-known exception is the ferroelectric poly(vinylidene fluoride) (PVDF), where the polarization reduces with tensile strain in the polar direction (Fig. 1b), exhibiting negative longitudinal piezoelectricity6,7. However, the origin of the negative longitudinal piezoelectric effect (NLPE) in PVDF has been attributed to the intermixed crystalline phase and no NLPE has been reported for a single phase. Besides, You et al. recently discovered the NLPE in the van der Waals layered ferroelectric CuInP2S6 in the experiment, which was attributed to the reduced lattice dimensionality8. In addition, several inorganic compounds have been theoretically predicted to possess NLPE9,10,11. For example, Liu et al. demonstrate NLPE in several hexagonal ABC ferroelectrics, and Arora et al. report NLPE in 2D heterobilayers. It is worth noting that almost all of the above structures are layered stacks with limited elastic constants12, which can lead to significant mechanical losses and degraded performance for some electromechanical applications, such as high-frequency resonators, filters and ultrasonic motors.

Fig. 1: Negative piezoelectric response in HfO2.
figure 1

Schematic diagrams of the a positive and b negative longitudinal piezoelectric effect, where the polarization increases and decreases with the tensile strain along the polar direction. In addition, the crystal structures of AlN and HfO2 are provided with the numbers representing different nearest neighboring anions of the cation. c The piezoelectric coefficient e33, \({e}_{33}^{\rm{electron}}\), \({e}_{33}^{\rm{ion}}\), d33, and elastic constant C33 of HfO2 and AlN. The data from literatures25,29,30 are marked with stars or triangles. Factors of the ionic contribution for piezoelectric coefficient, d the atomic strain sensitivity \({du}/d\varepsilon\) and e the Born effective charge \({Z}_{33}^{* }\). Schematic diagrams describe the definition of f PBc, g WPBc, and schematic draws for the mechanism responsible for the h positive piezoelectric effect and i negative piezoelectric effect.

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Hafnia, a well-known high-k gate insulator, was first recognized for its ferroelectricity in silicon-doped thin films in 201113. Since then, numerous researchers have dedicated their efforts to understanding the origin of ferroelectric behaviors in hafnia and exploring methods to control its ferroelectricity13,14,15,16,17. Previous investigations have identified the ferroelectricity of hafnia originates from the polar orthorhombic phase (space group: Pca21)18,19,20,21. Ferroelectric hafnia has now emerged as one of the most promising materials in the fields of electronic devices and storage applications13,22,23. Interestingly, despite being a ferroelectric material, hafnia’s electromechanical response, especially its piezoelectric properties, has received relatively little attention. However, due to its high mechanical strength, high speed of sound, and compatibility with CMOS24, hafnia has the potential to serve as a candidate material for the piezoelectric bulk acoustic filter, offering an alternative to the widely used aluminum nitride (AlN). Recent first-principles studies predicted that ferroelectric hafnia presents the NLPE25,26. However, the experimental measurement results of the longitudinal piezoelectricity of hafnia are still inconsistent. For example, the positive piezoelectric response was measured in La- and Y-doped HfO2, while Dutta and his collaborators observed a negative piezoelectric response in La-doped HfO225,27,28. They consider the difference may be caused by different textures of the sample caused by dopant. However, the effect of dopants on the piezoelectricity of hafnia has been little studied to date. Therefore, it is crucial to provide an intuitive and reasonable physical image to understand the intrinsic piezoelectric response of hafnia, while systematically exploring the influence of dopants on the longitudinal piezoelectric response.

In this study, a comprehensive exploration of the negative longitudinal piezoelectricity of hafnia and doped hafnia was undertaken based on density functional theory (DFT) calculations. An intuitive understanding of NLPE was first provided by introducing the asymmetric bonding stiffness along the strain direction, which is characterized quantitatively by weighted projected bond strength around cation in the c direction (WPBc). Our finding indicates that the anti-parallel between WPBc and bulk spontaneous polarization can result in the decrease of polarization with respect to tensile strain in the c direction, and thus a negative piezoelectricity. The influence of WPBc on the piezoelectric effect of hafnia is further examined in 26 stability-screened doped hafnia. Our investigation confirms that the sign of the piezoelectric coefficient is strongly dependent on the direction of the WPBc with respect to the spontaneous polarization and suggests that the piezoelectric coefficient e33 could be evaluated by the average WPBc and standard deviation (STD) of WPBc.

Results and discussion

Understanding the NLPE from a quantitative DFT calculation and qualitative bonding environment analysis

To unlock the origin of NLPE in ferroelectric hafnia (po-HfO2), the piezoelectric properties of AlN with positive longitudinal piezoelectric effect (PLPE) were calculated as a reference. As shown in Fig. 1b, the spontaneous polarization (Ps) of po-HfO2 is caused by the off-centering of half of the oxygen anions within the unit cell with respect to the centrosymmetric cation sublattice, leading to a positive Ps in this unit cell. Additionally, each Hf atom is surrounded by seven nearest neighboring O atoms, and the bond lengths range from 2.04 to 2.27 Å. For AlN (Fig. 1a), the Ps is also positive and is attributed to the off-centering of the nitrogen anions concerning the centrosymmetric cation sublattice. Each Al atom is surrounded by four N atoms with similar bond lengths. The precise values of the bond length are provided in Supplementary Tables 2 and 3. As shown in Fig. 1c, the calculated e33 of HfO2 is −1.24 C/m2, which is similar to that of AlN (1.43 C/m2) in magnitude but with an opposite sign. Moreover, the elastic constant C33 of HfO2 is larger than that of AlN, induced by the strong ionic bond between Hf and O. Combining eij and Cij, the piezoelectric strain coefficient (d33) of HfO2 has been determined to be a negative value but lower than that of AlN in magnitude. Our calculated results of AlN are in agreement with the previous DFT predictions29,30. According to the definition of \({d}_{33}={\sum }_{i=1}^{6}{e}_{3i}({{C}^{-1})}_{i3}\), contributions from the in-plane components of e3i and (C−1)i3 need to be taken into account. Due to the larger and negative in-plane e31 and e32 of HfO2 relative to that of AlN (Supplementary Table 13), the contribution of e33 is partially counteracted, which results in a relatively lower d33 of HfO2.

To illustrate the dominant factors of NLPE, different components contributing to e33 are discussed. The expression of e33 is given by

$${e}_{33}={e}_{33}^{\rm{electron}}+{e}_{33}^{\rm{ion}}={e}_{33}^{\rm{electron}}+\sum _{k}\frac{e}{{ab}}{Z}_{33}^{* }(k)\frac{{{du}}_{3}(k)}{d{\varepsilon }_{3}}$$
(1)

The first term evaluates the contribution of electrons to the piezoelectric coefficient and is calculated with the ions fixed at their equilibrium fractional coordinates. The second term arises from the ionic contribution and measures the microscopic atomic displacements with respect to the strain. Here, \(a\) and \(b\) stand for the equilibrium lattice parameters, k represents each atom in the unit cell, \({Z}_{33}^{* }\) is the Born effective charge and u3 is the atomic coordinate along the c direction. It is clearly seen from Fig. 1c that in both systems, the ionic contribution dominates the total value of e33, and the absolute value is almost the same (1.85 C/m2), but with the opposite sign. The electronic contribution of e33 in HfO2 (0.61 C/m2) is greater than that in AlN (−0.42 C/m2), which leads to the magnitude of e33 for HfO2 being slightly smaller than that of AlN. The calculated \({{du}}_{3}/d{\varepsilon }_{3}\) and \({Z}_{33}^{* }\), as displayed in Fig. 1d, e, unambiguously demonstrate that the NLPE in hafnia is accompanied by the negative displacement of cations (\(d{u}_{3}/d{\varepsilon }_{3}\)(Hf) < 0) as well as the positive displacement of anions (\(d{u}_{3}/d{\varepsilon }_{3}\)(O) > 0) with respect to the tensile strain along c axis. This means that, in response to the macroscopic tensile strain along the c-axis, the cation center will move downward, while the anion center move upward. Due to the upward direction of Ps, this movement will yield a decrease of polarization and, thus, negative e33. Conversely, for AlN with PLPE, the strain-induced atomic displacement for cations and anions are positive (\(d{u}_{3}/d{\varepsilon }_{3}\)(Al) > 0) and negative (\(d{u}_{3}/d{\varepsilon }_{3}\)(N) < 0), respectively, resulting in increased polarization and, thus, positive e33. The Born effective charge \({Z}_{33}^{* }\) for each atom is, in general, consistent with the nominal charge, but the value is greater than the nominal charge in HfO2, suggesting a blend of ionic and covalent bonds. Whereas in AlN, \({Z}_{33}^{* }\) for each atom is smaller than its nominal charge, indicating a more pronounced covalent character31. Importantly, this nuance is not responsible for the opposite sign of the longitudinal piezoelectric effect in AlN and HfO2. This finding differs from the prior NLPE mechanism reported in van der Waals layered materials and hexagonal ABC ferroelectrics, where electronic contribution typically dominated due to their special stacking nature and interlayer bonding8,9,32, and a detailed analysis can be found in Supplementary Note 1. In HfO2, due to the asymmetric but similar ionic/covalent bond properties on both sides of hafnium, atoms are more sensitive to the strain, resulting in a large du/dε. Combined with the large effective charges of hafnium (>5e), the ionic contribution is dominated in HfO2.

The preceding discussion yields a quantitative conclusion that the NLPE is caused by the anomalous displacement of ions under strain. To offer a more intuitive understanding of the negative piezoelectricity of HfO2, its bonding environment is compared with AlN. Although previous studies have discussed the NLPE of HfO2 from a bonding perspective, the correlation between the most longitudinal Hf-O bond and e33 has been discussed by overlooking other Hf-O bond contributions25,26. Therefore, a rational and intuitive model needs to be established. Given our focus on the longitudinal piezoelectricity, we characterize the projected bond strength between cation and anion along the c direction (PBc), as schematically shown in Fig. 1f. Centered around the cation, a positive PBc represents the projection of its bonding with the anion is along the +c direction, while a negative value indicates that the projected bond strength is in the −c direction. Mapping in po-hafnia (Fig. 1g), each Hf is bonded with seven O anions with similar bond lengths, four bonds above the Hf cation showing positive PBc and three bonds below Hf cation showing negative PBc. We define the weighted PBc (WPBc) as the sum of all positive PBc (WPBc+) plus the sum of all the negative PBc (WPBc−) around a central cation. Due to the similar Hf-O bond lengths, the Hf–O1 bond with the smallest longitudinal angle gives a stronger contribution to the WPBc−, finally resulting in a negative WPBc for Hf. The PBc value for each Hf–O bond is listed in Supplementary Table 7. Therefore, in the structure with negative WPBc, we can assume that a stiff bond is located below a soft bond along the c direction, as shown in Fig. 1i. When a tensile strain is applied along the c direction, it is evident that the soft bond will expansion easier than the stiff bond. Due to the anti-parallel direction between Ps and WPBc, this expansion yields to a decrease of bulk polarization and a negative e33.

In comparison, each Al in the AlN crystal structure is surrounded by four nearest neighbor N anions (Fig. 1a). Among them, the Al–N1 bond, which is parallel to the c direction, gives a stronger contribution to the WPBc+ and leads to an overall positive WPBc (see Supplementary Table 8 for detailed values). For the structure with positive WPBc, it can be considered that a stiff bond is located above a soft bond along the c direction (Fig. 1h). Differing from HfO2, the WPBc and Ps share the same direction in this case. As a consequence, with respect to the tensile strain in the c direction, the larger expansion of the soft bond will cause an increase in the bulk polarization, resulting in a positive e33. The validation in other materials (PbTiO3, GeTe, and ZrO2) can be found in Supplementary Fig. 2 and Tables 4–6, 9–12). In line with the above conclusion, when the direction of WPBc is parallel to the spontaneous polarization (positive WPBc), the e33 is positive. When the direction of WPBc is antiparallel to the spontaneous polarization (negative WPBc), the e33 is negative.

Stability screening in doped hafnia

In the above discussion, quantitative and qualitative explanations were provided to elucidate the origin of NLPE in hafnia. We first propose that the sign of the piezoelectric coefficient is intricately linked to the direction of the WPBc with respect to the Ps. The primary contributor to the NLPE is identified as the opposing direction of these two vectors. Nevertheless, whether this conclusion is universal and how the value of WPBc impacts the piezoelectric coefficient remains unclear. We address these questions through a systematic investigation of doped hafnia systems. Prior to calculating the piezoelectric coefficient, it is imperative to identify dopants that facilitate the formation of the po-HfO2 phase. Earlier investigations have indicated that the po-HfO2 is metastable23,33. However, more recent studies have demonstrated that doping can effectively stabilize its ferroelectricity34,35,36,37. To address the high computational cost associated with accurately modeling the effect of 49 dopants on the energetics of hafnia, a two-step screening process is performed36. as illustrated in Fig. 2a. To ascertain the energy ordering of different phases of doped hafnia, the energy difference (\({E}_{\rm{diff}}\)) of the po phase and the t phase (the most stable phase at high temperature) with respect to the m phase (the most stable phase at room temperature) is defined as

$${E}_{\rm{diff}}={E}_{{po}(t)}-{E}_{m}$$
(2)

where \({E}_{{po}(t)}\) and \({E}_{m}\) are the DFT energies of po(t) and m phases, respectively. To emphasize the direct role of a dopant in the stabilization of the po and t phases, we further calculate the relative energy difference (\({\triangle E}_{\rm{diff}}\)) between the doped phases and pure phases by

$${\triangle E}_{\rm{diff}}={E}_{\rm{diff}}^{D}-{E}_{\rm{diff}}^{\rm{pure}}$$
(3)

Here, a negative \({\triangle E}_{\rm{diff}}\) represents that the dopant favors (or stabilizes) the po or t phase over the m phase more than the dopant-free pure case. Furthermore, if the energy of the po phase is lower than the energy of the t phase, one can anticipate that such dopants will enhance the ferroelectric behavior in hafnia.

Fig. 2: Phase stability screening of hafnia in the presence of various dopants.
figure 2

a Overall screening process and modeling conditions in each step. b Energy difference of the po phase and the t phase with respect to the m phase at 12.5% doping concentration, as computed using Eq. 2. c Relative energy difference between the doped phases and pure phases at 12.5% doping concentration, as computed using Eq. 3. Dashed lines are employed to distinguish doping elements with different valence states. Hollow symbols in c represent that the energy of hafnia with these dopants is lower in the t phase compared to the po phase.

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In the first step, we investigated the influence of 49 dopants on the phase stability of po phase and m phase of hafnia. Based on previous findings36,38,39, the energy difference between the po phase and the m phase has consistently exhibited a monotonic trend with doping concentration increase. Thus, to expedite the screening process while ensuring computational reliability, a 96-atom supercell with a 3.125% doping concentration is constructed for the po phase and the m phase through electron compensation (see details in method), we calculate the \({\triangle E}_{\rm{diff}}\) using Eq. 3. Among them, 34 dopants with negative \({\triangle E}_{\rm{diff}}\) for po phase are identified as favoring the ferroelectric phase and were further studied in step 2. In this step, we increase the sophistication of the model by introducing an appropriate oxygen vacancy to achieve charge neutrality and constraining the doping concentration at 12.5% for practical reasons. The number and arrangement of defects in the po, m and t phases are introduced in the method section. We found that the oxygen vacancies are energetically preferred in the vicinity of the dopants, especially for divalent element doping, and oxygen vacancy tends to occupy the O site with 3 Hf–O bonds, which is consistent with the findings of Batra et al.23. The \({E}_{\rm{diff}}\) of the doped systems in Fig. 2b indicates that in most systems the m phase is still the most stable phase, consistent with the previous findings36. However, compared to the undoped phase, the relative energy change of the po and t phases with respect to the m phase is significant. We plot the \({\triangle E}_{\rm{diff}}\) in Fig. 2c to identify the dopants that can be expected as the promoters of the po phase36. Among them, 27 out of 34 dopants favor the po phase. However, in the case of Rb, the relaxed structure exhibits significant distortion, and we exclude it for further analysis. The remaining 26 elements doped po-HfO2 are selected for the study of piezoelectric properties. It is noteworthy that certain dopants predicted in our study, such as Y and La, have been experimentally identified to favor the ferroelectric phase40.

Piezoelectric properties in doped hafnia

To the best of our knowledge, there has been little research into the piezoelectric coefficient of doped hafnia in previous works and even less research into how to control its piezoelectric coefficient. In our work, the piezoelectric coefficients of 26 doped po-HfO2 are illustrated in Fig. 3a. Among them, Pb, Rh, and Sn demonstrate an active alloying effect on the enhancement of the piezoelectric coefficient e33 of hafnia, which equals to −1.72 C/m2, −1.33 C/m2, and −2.04 C/m2, respectively. Especially, the e33 value for Pb- and Sn-doped hafnia surpasses that of wurtzite AlN29,30, implying their substantial potential for electromechanical devices. Conversely, the remaining elements exhibit a weakening effect on the piezoelectric coefficient of hafnia, with eight elements (Au, Al, Sc, Fe, Ga, Y, Ge, Ir) doped hafnia even exhibiting sign reversal of piezoelectric coefficient or a positive piezoelectric response. As can be seen, the e33 of La- and Y-doped HfO2 is almost zero. Therefore, in experiments, the sign of e33 in both systems is more susceptible to external stimuli, such as substrate strain and size effect, resulting in opposite signs. For example, the previous calculation has shown that epitaxial strain can change the sign of e33 of HfO225. Here, we focus on the effect of elemental doping on piezoelectricity, while the influence of muti-factor will be investigated in future work.

Fig. 3: Piezoelectric response in doped hafnia.
figure 3

a Calculated e33 based on DFPT in selected doped systems. b The distribution of WPBc of cations in the doped hafnia, a positive (negative) value represents the WPBc is in the c (−c) direction. All error bars shown in this figure represent ±standard deviation and are based on counting statistics for each point. The wider the error bars, the larger the local bonding fluctuations of cations.

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Furthermore, to examine whether the sign of the piezoelectric coefficient is determined by the direction of WPBc and to understand how the value of WPBc influences the piezoelectric coefficient in doped hafnia, we conduct a meticulous study of the chemical bonding environment of cations. In contrast to undoped hafnia, the bonding environment around each cation differs in the doped systems, the distribution of WPBc around all cations in each given system is depicted in Fig. 3b. It is evident that the WPBc values vary significantly among different doping systems, with the systems exhibiting substantial fluctuations displaying relatively low e33, particularly noticeable in the trivalent doped hafnia. As with undoped hafnia, to quantitively characterize the direction and value of WPBc of the doped systems, we employ the average WPBc as a reference. From Fig. 3b, it is clear that there is a correlation between the sign of WPBc and e33, and most systems associated with negative WPBc exhibit negative piezoelectric values. Additionally, there appears to be a discernible positive correlation between the average WPBc value and the piezoelectric value. For instance, in Sn-doped hafnia with the most negative e33, the average WPBc is the lowest.

Here, we attempt to establish an intuitive correlation between WPBc and e33. The standard deviation (STD) of WPBc is introduced to quantitatively measure local bonding fluctuations of cations, while the average WPBc is employed to characterize the overall bonding direction and strength. The correlation between e33 and the bonding environment descriptors, STD-WPBc and average WPBc, are illustrated in Fig. 4a. The results clearly indicate that the more negative the average WPBc and the smaller the standard deviation (STD) of WPBc, the stronger the piezoelectric response. These observations align with the conclusions drawn from our analysis of pure hafnia. The introduction of the STD-WPBc descriptor in the doping system reflects the structural distortion caused by defects, a largely distorted structure is also less responsive to changes in internal polarization under strain41. These findings offer valuable insights for selecting suitable dopants when designing hafnia with enhanced piezoelectric or ferroelectric performance. For instance, choosing dopants with an ionic radius similar to Hf4+ appears to reduce local bonding fluctuations. Additionally, selecting dopants with lower electronegativity than hafnia can enhance bond strength between dopants and oxygen42, potentially resulting in a larger average WPBc.

Fig. 4: Impact of bonding environment on e33 and promising candidates.
figure 4

a Piezoelectric coefficient e33 distribution versus the average WPBc and the standard deviation of WPBc. Among them, three doped hafnia systems (Pb-, Sn-, Rh-) with e33 greater than pure hafnia are selected for further simulations for b the elastic constant C33 and piezoelectric coefficient d33.

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Finally, to provide experimenters with more intuitive information, we calculate the elastic constant C33 and piezoelectric strain coefficient d33 of Pb2Hf14O30, Sn2Hf14O32, and Rh2Hf14O31, as shown in Fig. 4b. A complete result of elastic tensors and piezoelectric tensors are presented in the Supplementary Tables 14-15. In comparison to pure HfO2, the elastic constants of all three systems have decreased. However, these values remain higher than 300 GPa, which indicates reliable mechanical strength. Combining eij and Cij, we calculate the d33. Among them, the d33 of the Pb-doped system slightly decreases compared to that in pure hafnia, while the Sn- and Rh-doped systems show more than two times enhancement. Especially, the d33 of Sn-doped HfO2 is about 3.8 pC/N, comparable to that of AlN. In addition, the calculated spontaneous polarization of Sn0.125Hf0.875O2 is 0.54 C/m2, which is similar to that previously reported spontaneous polarization of pure HfO2 (0.52–0.54 C/m2)25,43. Considering that the stability of the po phase of Sn0.125Hf0.875O2 is better than that of the pure phase, the possibility of polarization switching could make Sn0.125Hf0.875O2 a promising candidate for ferroelectricity. According to the result, we encourage more experimental trials to further explore the piezoelectric properties of similar compounds.

In this work, we carried out an extensive investigation on the negative longitudinal piezoelectricity from both quantitative DFT calculation and qualitative bonding environment analysis. From the computational results, this counterintuitive piezoelectric effect stemming from the negative ionic contribution of e33, facilitated by the reverse strain displacement of cations and anions compared to the positive piezoelectric materials. Furthermore, the weighted projected bond strength along the c direction (WPBc) is first proposed to quantitatively characterize the asymmetric bonding stiffness along the strain direction. Our finding indicates that the direction of WPBc with respect to spontaneous polarization is the essential reason for the sign of the longitudinal piezoelectric coefficient. Further meticulous study on the chemical bonding environment of cations in doped hafnia confirms the relationship between WPBc and e33. It is found that the sign of the piezoelectric coefficient is strongly dependent on the direction of the WPBc with respect to the spontaneous polarization, a more negative average WPBc and a lower standard deviation (STD) of WPBc can lead to a stronger piezoelectric response. Importantly, the Sn-doped hafnia with the lowest value of average WPBc and STD-WPBc is identified to have the highest piezoelectric coefficient (−2.04 C/m2) compared to other dopants. We hope this work could inspire future computational and experimental studies in the understanding of the piezoelectric effect. In addition, these insights can be used to discover new choices for next-generation electromechanical devices.

Methods

DFT calculations

In this work, all calculations based on density functional theory (DFT) were performed within the Vienna Ab initio Simulation Package (VASP), using the projector augmented-wave (PAW) method44,45. The wave functions were expanded in a plane-wave basis set with an energy cutoff of 520 eV. The exchange-correlation interaction was treated within the generalized gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE) format46. The convergence criteria were set to 10-6 eV for total energy and 0.01 eV/Å for the force on each atom. In the DFT calculations, the Γ-centered k-point meshes were adopted. For the supercell with 96 atoms, a 3 \(\times\) 3 \(\times\) 3 k mesh was chosen for polar orthorhombic (po) and monoclinic (m) phases. For supercell with 48 atoms, 3 \(\times\) 3 \(\times\) 6 k mesh was chosen for po and m phases, and 4 \(\times\) 4 \(\times\) 3 k mesh for the tetragonal (t) phase, respectively. The piezoelectric stress coefficients were calculated by using the preset workflow based on density functional perturbation theory (DFPT), while the elastic tensors were computed by the workflow that fits the strain-stress relationship by performing six independent components of Green–Lagrange strain tensor with four magnitudes (±0.5%, ±1%) for the original structure47,48. Bonding strengths were obtained by the crystal orbital Hamilton population (COHP) method, which is implemented in the LOBSTER code49.

Construction of doped hafnia models

For isovalent dopants replacing Hf, the systems maintain charge neutrality. For aliovalent dopants, at 3.125% doping concentration, electron compensation is applied to the 96-atom supercell to accelerate the screening process, where a single dopant substituting an Hf atom and the charge is electronically compensated. At 12.5% doping concentration, various numbers of oxygen vacancies (Vo) were introduced for charge compensation. Specifically, substituting every two Hf with +1 valence state dopants is accompanied by the removal of three Vo, similarly, with +2 and +3 valence state dopants require the removal of two and one Vo, respectively, this approach has been widely utilized to construct the ferroelectric HfO2 defect structure36,38,39. The defects, such as doping element and Vo were arranged in supercells with 48 atoms based on low electrostatic energy criteria, as implemented in the Python Material Genomics (pymatgen) code50. Considering the large number of possible arrangements of defects (>104, for dopants with +1 valence state), the configurations were sorted in sequence with the electrostatic energy, and then six configurations with the lowest electrostatic energy were selected and relaxed with DFT calculations51. The configuration with the lowest energy was then chosen as a representative of each valence state and proceeded with DFPT calculations. The position arrangement of the defects in doped hafnia systems is shown in Supplementary Fig. 3.