Artificial intelligence-driven phase stability evaluation and new dopants identification of hafnium oxide-based ferroelectric materials

Abstract

In this work, a multi-stage material design framework combining machine learning techniques with density functional theory is established to reveal the mechanism of phase stabilization in HfO₂ based ferroelectric materials. The ferroelectric phase fractions based on a more stringent relationship of phase energy differences is proposed as an evaluation criterion for the ferroelectric performance of hafnium-based materials. Based on the Boltzmann distribution theory, the abstract phase energy difference is converted into an intuitive phase fraction distribution mapping. A large-scale prediction of unknown dopants is conducted within the material design framework, and gallium (Ga) is identified as a new dopant for HfO₂. Both experiments and density functional theory calculations demonstrate that Ga is an excellent dopant for ferroelectric hafnium oxide, especially, the experimentally determined variation trends of ferroelectric phase fraction and polarization properties with Ga doping concentration are in good agreement with the predictions given by machine learning. This work provides a new perspective from machine learning to deepen the understanding of the ferroelectric properties of HfO₂ materials, offering fresh insights into the design and performance prediction of HfO₂ ferroelectric thin films.

Introduction

Since the discovery of ferroelectricity in Si-doped hafnium oxide films in 2011, there has been a great deal of interest in HfO₂-based ferroelectric thin films¹. HfO₂-based ferroelectric thin films can be annealed or doped to form a sub-stable orthorhombic phase with a non-centrosymmetric structure, resulting in excellent ferroelectric properties. Compared to traditional perovskite structure ferroelectric films, HfO₂-based ferroelectric thin films have high compatibility with complementary metal-oxide-semiconductors (CMOS) processes, good scalability, low operating voltage, moderate polarization magnitude, and high radiation resistance, making them an ideal new type of ferroelectric film material^2,3,4.

At present, there has yet to be a consensus in the academic community regarding the ferroelectric phase (f phase) transition and phase stability mechanism of HfO₂ thin films. Hafnium oxide-based ferroelectric thin films prepared experimentally often exhibit the coexistence of multiple phases. The control of ferroelectric phases in thin films and the stability of thin film properties continue to be significant challenges in the research field of hafnium oxide-based ferroelectric thin films. In recent years, researchers have explored many techniques for improving the performance of HfO₂-based ferroelectric thin films. One key focus has been on stabilizing the sub-stable orthorhombic phase, which is critical for the unique ferroelectric properties of these films. Many dopants have been reported to stabilize the non-centrosymmetric orthorhombic phase with space group Pca2₁^5,6,7, which is crucial for achieving good ferroelectric properties in these films. The proper implementation of annealing procedures is essential in forming the sub-stable orthorhombic phase in HfO₂-based ferroelectric thin films^8,9,10,11. Wei et al.¹² studied the defect properties of HfO₂, ZrO₂, and Hf_0.5Zr_0.5O₂ in their ground state monoclinic phase (m phase), ferroelectric orthorhombic phase, and transition t phase and provided insights into the mechanisms by which O vacancy and O interstitial act as significant defects. They suggested that an O-intermediate condition and low growth temperature are necessary during film preparation to effectively reduce defects and leakage current while increasing the fraction of the orthorhombic phase. Furthermore, additional techniques, including the use of proper electrode materials^13,14 and the application of mechanical stress^15,16, have also been extensively investigated for enhancing the ferroelectric properties of HfO₂-based ferroelectric thin films. The ferroelectric properties of HfO₂-based thin films are determined by the doping elements and their ratios, and researchers have extensively studied this using both experimental and first-principles calculation approaches. For example, Martin et al.¹⁷ conducted experiments to investigate the effect of Si concentration on the properties of HfO₂ films, and they observed a transition from monoclinic to orthorhombic and then to tetragonal phase (t phase) as the Si concentration increased. Yao et al.¹⁸ studied Al-doped HfO₂ films with varying doping agent concentrations ranging from 4.0 to 11.1 mol%. They observed an increase in residual polarization of the HfO₂ films, followed by a decrease, and established a correlation between the optimal annealing temperature and the concentration of the doping agent. Meanwhile, Xu et al.¹⁹ explored the impact of both cation and anion doping on HfO₂ thin films and found that increasing the doping concentration resulted in the formation of orthorhombic phases in HfO₂ crystals and a higher degree of stability for T/C phases at higher doping concentrations. Wang et al.²⁰ studied the impact of 0–25% Gd doping on phase stability using density functional theory (DFT) methods and discovered that when the doping concentration was greater than 15.5%, the tetragonal phase was more stable than the monoclinic phase. The research and development process in the traditional materials field typically involves the synthesis of samples through experiments or the study and analysis of unknown materials using DFT methods. These methods have high dependencies on experiment costs and computation time as well as theoretical models, requiring a large amount of repetitive work and resulting in a long research cycle. Despite a substantial amount of research work on doping agents and doping concentrations that have been published, the impact of other unknown doping agents on the ferroelectric properties of HfO₂ films remains a significant challenge.

Compared to the traditional trial-and-error methods and first-principles computational approaches that are characterized by long development cycles, high costs, and low efficiency, machine learning (ML) methods have been widely used for material property prediction, accelerating material design and discovery ^21,22. Machine learning has demonstrated powerful capabilities in accelerating materials optimization and design in recent years. Apart from accurately predicting performance under multiple factors, it also provides new physical perspectives for material design optimization. However, compared to other fields, material data sets are often much smaller and have weak data availability, which greatly hinders the establishment of ML models. Research has shown that the training error of ML decreases with the size of the data set. Compared to the traditional ML problems, the sure independence screening and sparsifying operator (SISSO) method based on compressed-sensing can achieve stable results from small data sets, which helps to extract effective physical descriptors from a huge and potentially highly correlated feature space. The combination of the SISSO method with traditional ML methods is a novel approach to solving existing problems. Therefore, people can establish ML models based on physical features using this method and search for more suitable doping agents in the chemical space.

In this work, we established a multistage material design framework by combining DFT and ML methods, as shown in Fig. 1. Our focus is on investigating the impact of altering the dopant and its concentration on the phase energy difference E_f-m (the difference between the energy of the orthorhombic phase and the monoclinic phase), E_f-t (the difference between the energy of the orthorhombic phase and the tetragonal phase), and P_s (spontaneous polarization) of HfO₂. Large-scale, high-throughput first-principles calculations were performed to obtain the energies of three phases for constructing the ML dataset (Fig. 1a). Based on the Boltzmann distribution theory, we calculated the ferroelectric phase fraction of HfO₂ thin films under different dopants and doping concentrations. By considering both the ferroelectric phase fraction and polarization magnitude, we propose a novel method for evaluating the ferroelectric properties of HfO₂ thin film materials, converting abstract energy differences into intuitive phase fraction distribution maps (Fig. 1b). Introducing the compressed-sensing-based SISSO method, we converged from highly correlated material property combinations to the optimal feature space, effectively addressing the issue of small datasets. We optimized and selected accurate feature descriptors for the classification problem, effectively determining the ferroelectric phase boundaries of doped HfO₂, providing a high-quality dataset for subsequent model training. By combining first-principles calculations and machine learning models, we extracted physical insights into the formation of ferroelectric phases, large-scale prediction of unknown dopants is conducted through machine learning (Fig. 1c). Finally, using the ferroelectric phase fraction and polarization as target properties, we used the developed machine learning prediction framework to reveal the stable phase properties of HfO₂ materials under doping with many unknown dopants and identify the key physical features affecting the phase energy difference. We identify Ga as a new dopant for HfO₂, and experimentally succeed in obtaining the variation patterns of ferroelectric properties and ferroelectric phase with different Ga doping concentrations (Fig. 1d), which are in good agreement with the predictions given by machine learning. The successful experimental verification confirms the feasibility of the theoretical framework for hafnium oxide ferroelectrics proposed in this study. This study bridges the research gap in machine learning for designing ferroelectric HfO₂ materials by combining first-principles calculations with machine learning methods, which is expected to provide new perspectives and breakthroughs for efficient ferroelectric HfO₂ material design.

Fig. 1: The working framework of this study.

a 225 structural models were constructed by introducing 15 dopants and considered random arrangements of dopant atoms. Large-scale, high-throughput first-principles calculations were performed to obtain the energies of three phases for constructing the machine learning dataset. b Using Boltzmann distribution theory, we calculated the proportion of ferroelectric phases in HfO₂ thin films for different dopant types and doping concentrations, converting abstract energy differences into intuitive phase fraction distribution maps. c Descriptors were formed using weighted physical features, and feature space was expanded using the SISSO method. The dataset was divided into training and validation sets for machine learning model training. Model optimization was achieved through cross-validation and grid search for hyperparameter tuning, enabling large-scale predictions of dopants via machine learning. d With ferroelectric phase fraction and polarization value as the target properties, and guided by the predictions of the machine learning model, HfGaO ferroelectric thin films were successfully prepared in experiments. The variation patterns of ferroelectric properties and ferroelectric phases at different gallium doping concentrations were obtained, validating the reliability of the machine learning model.

Results

Construction of DFT datasets and ferroelectric phase fractions

Varying the dopant and doping concentration in HfO₂ crystals can lead to changes in the crystal structure, and the energy difference between different phases is an essential basis for their stability. To investigate the stability of different phases in HfO₂ crystals, we utilized DFT calculations and selected some 2-valent to 5-valent elements as dopants for HfO₂, presenting the doping situation of HfO₂ on a large scale. The crystal structures of Ca-, Al-, Zr-, and Nb-doped monoclinic and ferroelectric phases are shown in Fig. 2a, the energy difference relative to the ground-state monoclinic phase is also shown under the condition of a doping concentration of 12.5%. This approach enabled us to gain insights into the stability of different phases in HfO₂ and investigate the effects of dopants and doping concentration on the crystal structure. The stability of different phases in this work is evaluated by the energy difference between the corresponding phase and the m phase, which is obtained by calculating the total energy of each phase and the m phase. Here, we calculated E_f-m and E_t-m, where a positive energy difference indicates that the phase has a higher energy than the m phase, while a negative energy difference indicates that the phase has a lower energy than the m phase. Although the phase formation energy can be used to compare the stability of different materials, when comparing different phases of the same material, the formation energy of a phase may only represent the transformation of absolute energy, making it difficult to directly compare relative stability. Therefore, for HfO₂, which requires the existence of a metastable ferroelectric phase, the phase energy difference can more directly reflect the likelihood of ferroelectric phase formation. The lower the energy of the ferroelectric phase, the greater the probability of its occurrence. This is because the phase energy difference directly compares the relative stability between different phases without considering the common reference state in the formation energy. If the energy of f-phase is lower than the energy of any other phase, then f-phase is the ground state phase, but it is difficult to judge whether it is the ground state phase through the formation energy.

Fig. 2: Establishing the machine learning dataset through DFT calculations.

a Crystal structures obtained from DFT calculations. The dashed line indicates the non-centrosymmetric structure of the ferroelectric phase. The energy difference ΔE relative to the ground-state monoclinic phase is also shown under the condition of a doping concentration of 12.5%. b The doping concentration varies from 0 to 12.5% according to the E_f-m displayed in different valence states. c The doping concentration varies from 0 to 12.5% according to the E_t-m displayed in different valence states.

In Fig. 2, we present the cases of undoped and doped HfO₂ with m phase as the reference state. The energy differences between f phase and m phase and t phase and m phase are 63.6 meV and 136 meV, respectively. The total energies shown here are the minimum energies obtained after structure optimization, and only doping concentrations in the range of 0–12.5% are displayed. In experiments, only Zr dopant can achieve high doping concentrations, the doping concentration should not be too high because the tetragonal t-phase formed during the crystal annealing process cannot be transformed into a ferroelectric f-phase during the cooling process²³. Generally, the dopants can have a ferroelectric phase structure only in the case of a doping concentration of 12.5%, and to easily compare the difference in energies of the different dopants, the doping concentration of less than 12.5% is therefore chosen. In Fig. 2b, c, we plotted the energy differences for 12 types of dopants collected according to their valence states. We noticed that trivalent dopants exhibit more significant advantages in terms of energy difference, and elemental doping of HfO₂ blocks can reduce the free energy difference between the f phase or t phase and the m phase. Previous studies have reported that physical factors such as dopant size and electronegativity affect the phase transition process^24,25,26, and trivalent dopants have a clear advantage due to their ionic radii being comparable to that of Hf. Additionally, in Fig. 2b, c, the energy difference effect of tetravalent silicon dopants (without charge carriers) is significantly weaker than that of divalent and trivalent dopants, which may be attributed to the carrier effect of low-valence dopants.

We have computed the total density of states (TDOS) for the f-phase of HfO₂ at a doping concentration of 12.5% (see Supplementary Fig. S1 in the Supplementary Information (SI)). It is evident that under low-valence doping (2-valent and 3-valent) and high-valence doping conditions, the Fermi level enters the valence and conduction bands, respectively. This results in the introduction of holes and electrons in hafnium oxide. In conjunction with the energy differences shown in Fig. 2, introducing these hole-doping effects significantly enhances the ferroelectricity of hafnium oxide. This observation aligns with recent research by Cao et al.²⁷, which suggests that hole doping enhances ferroelectric stability in hafnium oxide. The number of holes can be obtained by integrating the density of states near the Fermi level, and the amount of holes introduced by 2-valent doping is twice that of 3-valent doping. Consequently, at the same doping concentration shown in Fig. 2, the energy difference between the f phase and m phases is lower under 2-valent doping. Ba doping provides a better illustration of this phenomenon. Unlike Ca and Sr, the density of states indicates that Ba does not introduce holes under high-concentration doping but instead forms gap states within the bandgap. Furthermore, these states are located closer to the conduction band, resembling the behavior of Ta. As a result, the energy difference between the f phase and m phases (E_f-m) increases, which is detrimental to the stability of the ferroelectric f phase. Specifically, doping with Sr induces a lower concentration of ferroelectricity ²⁸, which is attributed to the fact that divalent Sr has more holes compared to trivalent Y, as reflected in the decreasing trend of energy difference in Fig. 2b. On the other hand, carrier injection is also one of the reasons for the fatigue or failure of ferroelectricity in ferroelectric materials. Previous studies often controlled the concentration of low-valence dopants in the low-concentration range.

In existing preparation techniques, the energy barrier differences between the various crystalline phases (including the orthorhombic phase, monoclinic phase, tetragonal phase, cubic phase, etc.) formed during annealing of ferroelectric thin films are relatively small. Therefore, regardless of whether the hafnium oxide ferroelectric films are prepared by physical or chemical vapor deposition, they exhibit the characteristics of multi phase coexistence. This is the main factor leading to the inhomogeneous ferroelectric properties and poor repeatability of hafnium oxide-based films, greatly hindering the practical device applications and commercial development of hafnium oxide-based ferroelectric films. To effectively evaluate the macroscopic ferroelectric performance of hafnium oxide-based ferroelectric films, obtaining the proportion of the ferroelectric phase in the multi-phase structure is also very important, in addition to using parameters such as polarization magnitude. The proportion of the ferroelectric phase can effectively reveal the stability and durability of the polarization of hafnium oxide-based ferroelectric films on a macroscopic scale and quickly eliminate false polarization caused by various interfering factors. In this work, we used the Boltzmann distribution to calculate the proportion of the ferroelectric phase under different doping elements and doping amounts. We evaluated the ferroelectric performance of HfO₂-based films by considering the proportion of the ferroelectric phase and the magnitude of polarization. The calculation process for the ferroelectric phase fraction is shown in the SI, which requires the energy difference of each phase under different doping elements, these abstract energies are then converted into visualized ferroelectric phase fraction distribution mapping.

The Boltzmann distribution can be used to calculate the proportions of different phases in the same system, as different phases can be regarded as different states of the system, each with its specific energy. By calculating the energy of each phase, their probabilities of occurrence, i.e., the phase fractions, can be derived using the Boltzmann distribution formula. Figure 3 shows the ferroelectric phase fractions of HfO₂ under various dopants and doping concentrations. To ensure comparability of the polarization and ferroelectric phase fraction under different doping conditions, we normalized the maximum values of the polarization magnitude. Therefore, the normalized polarization and the ferroelectric phase fraction shown in the figure represent relative magnitudes. Regarding Fig. 3, a concise observation is that the closer to the high polarization and high ferroelectric phase fraction region (i.e., the red region near the top), the more likely to achieve better ferroelectric performance. This observation underscores the robust correlation between ferroelectric phase fractions and polarization. As shown in Fig. 3a, the area of high ferroelectric phase fraction for hafnium oxide doped with Ba is tiny and ferroelectric performance only appears in a narrow, low doping region, which is difficult to achieve experimentally. Therefore, Ba is not an ideal dopant for ferroelectric hafnium oxide because its doping window is too small. For hafnium oxide doped with Al, as shown in Fig. 3d, the area of the high ferroelectric phase fraction is significantly larger. Good ferroelectric performance may occur in the range of 2% to 10% Al doping, which is in good agreement with our experimental data (see Supplementary Fig. S2 and Supplementary Table S1 the SI for details). For hafnium oxide doped with Zr, the region with a high ferroelectric phase fraction shown in Fig. 3j is more extensive and tends to occur in the high doping range of 30% to 50%, which is in good agreement with other related studies, thus providing further evidence that Zr is one of the optimal dopants for ferroelectric hafnium oxide films. Similarly, the ferroelectric phase fraction plot depicted in Fig. 3n indicates that Nb light doping is more likely to result in ferroelectric performance. It is evident from the analysis of Fig. 3 that by correlating the doping concentration with the ferroelectric phase fraction, we can assess the effectiveness of different dopants and predict the optimal doping concentration range, thus guiding optimizing the doping concentration in experiments.

Fig. 3: Ferroelectric phase fraction in the doped HfO₂ system obtained through Boltzmann distribution theory.

The ferroelectric phase fraction distributions are presented for cases where the doping elements are Ba, Ca, Sr, Al, Y, La, Gd, Si, Sn, Zr, Ti, Pb, V, Nb, and Ta, corresponding to a–o respectively. The horizontal axis represents the doping concentration, while the vertical axis represents the normalized polarization. The color gradient from purple to red indicates an increasing ferroelectric phase fraction.

Target properties of hafnium oxide based ferroelectric materials

The effective doping concentration ranges of different elements varied significantly based on DFT calculations. In the entire workflow, it is unreasonable to assume the existence of ferroelectric phase structures outside the effective doping concentration. Therefore, the first step of this work is to screen for dopants and doping concentrations that exhibit ferroelectric phase structures. In the second step of this work, to better compare the energies of the three competing phases in doped hafnium oxide, we define the relative energy of α concerning the ground state m phase with dopant A at the same doping concentration as

$$\Delta {E}_{A}^{{\rm{\alpha }}-{\rm{m}}}={E}_{A}^{{\rm{\alpha }}}-{E}_{A}^{{\rm{m}}}$$

(1)

where ${E}_{A}^{\alpha }$ and ${E}_{A}^{m}$ are the calculated total DFT energies of the doped α phase and m phase, respectively. We obtained the energy differences E_f-m and E_f-t between the ferroelectric and monoclinic phases and between the ferroelectric and tetragonal phases, respectively, based on the results from DFT calculations. The energy differences are considered to be essential foundations for the formation and stable existence of the ferroelectric phase^29,30. Polarization is a crucial criterion for assessing the ferroelectric performance of the HfO₂ system, which is highlighted in this study. The polarization magnitude of f phase HfO₂ doped samples was calculated by multiplying the Born effective charge (BEC, Z*) with the atomic displacement (d) relative to the reference centrosymmetric t phase³¹.

$${P}_{s}=\sum _{i}\frac{e}{\varOmega }{Z}_{i}^{\ast }\delta {d}_{i}$$

(2)

Where i represents the ith atom and e is denoted as the electronic charge and the considered cell volume, respectively. Density generalized perturbation theory calculates the BEC in the direction along the c-axis. Due to the complete relaxation of the doped structure, the f phase hafnium oxide crystal structure undergoes specific changes depending on the type of doped atoms and the doping concentration. Although most of these f phases maintain their ferroelectric phase structure characteristics when doped, at higher doping concentrations of certain doping elements, significant phase structure changes occur during the relaxation process, it is difficult to identify the non-contrastive ferroelectric phase structure characteristics, and in such cases, we define its polarization magnitude as zero.

Specific presentation and evaluation of the machine learning model

The multi-stage material design framework is presented in Fig. 1c. It combines classification and regression methods and integrates feature construction strategies with conventional machine learning methods. The design framework consists of two main parts: a classification task to predict whether doped HfO₂ has a ferroelectric phase structure and a regression task to predict phase energy differences and polarization magnitudes. The input data for this study, obtained from first-principles calculations, consist of relevant data for 15 types of doping elements that are divided into classification and regression datasets. As a common process in machine learning, feature engineering is an essential step. Here, the physical characteristics of the doping elements, including electronegativity (χ), ionization energy (IE), electron affinity (EA), atomic radii (R), atomic mass (m), ionic radii (r), and valence state, are used. The machine learning model can take advantage of the readily available physical features, which can be used as the most direct and accessible input for the model.

To ensure the credibility of the subsequent predictions, we first focused our efforts on the classification of ferroelectric structures. Before constructing the classification model, we used the SISSO method to build the feature space (see the SI for details). Compared to traditional machine learning methods for big data analysis, SISSO can generate effective descriptors even when using relatively sparse data. It should be noted that SISSO can be split into SIS and SO. SIS selects the subspace S_1D with the highest correlation with the target property, while the n-dimensional descriptors are selected from the subspace S_nD with a response of Δ_(n-1)D. Then, SO extracts the best n-dimensional descriptors from the union of all previously selected subspaces ${{\rm{S}}}_{{\rm{nD}}}\cup {{\rm{S}}}_{(({\rm{n}}-1){\rm{D}})}\cup \cdots \cup {{\rm{S}}}_{1{\rm{D}}}$ in the space. SISSO searches for simple and effective feature descriptors in the space, and the screened feature descriptors will be used to construct the classification model. To some extent, selecting two descriptors in the feature space simplifies the classification model and makes it more interpretable. In Fig. 4c, the density distribution of the ferroelectric phase structures and their physical feature values reveals a strong correlation between atomic radius and ferroelectric phase structures. The atomic radius associated with ferroelectric phases is primarily concentrated around R ~ 155 pm, clearly distinguishing from the atomic radius in non-ferroelectric phases. Therefore, we selected descriptors related to R from the SISSO feature space and used them to construct SVC models. To ensure simplicity and interpretability, we built each model using only two descriptors from the feature space. Further feature selection optimization was carried out, and a feature descriptor based on R and χ was preliminarily established. The classification model was constructed using the SVC method and optimized through standardization and grid search. The accuracy of the model in predicting the presence of ferroelectric structures was 96.5% (Fig. 5a). We noted that the standardized feature descriptors in the classification model are ${{(R}_{O}+{\chi }_{A})}_{S}$ and ${{(\chi }_{A}-{R}_{A})}_{S}$, which are directly related to the atomic radii and electronegativity of the doping elements.

Fig. 4: Construction of ferroelectric phase boundaries based on classification learning.

a The distribution of different valence states in the dataset was analyzed, and a pie chart was used to show the percentages of 2+, 3+, 4+, and 5+ valence states. b Confusion matrix for the classification of ferroelectric and non-ferroelectric structures based on the SVC model. c The density of physical feature values for ferroelectric and non-ferroelectric structures in the training and testing datasets, where a higher density indicates more data points.

Fig. 5: The performance of the machine learning model.

a Predictions of whether a material has a ferroelectric phase structure, where red and blue colors indicate the presence and absence of a ferroelectric phase structure, respectively. Circular symbols represent the high confidence of the model in the prediction, while triangular symbols represent predictions with a probability ranging from 0.4 to 0.6. b The effect of doping concentration variation on the volume change induced in Hf_1-xA_xO₂ (A=Gd, La, Zr, Nb, Si, Ta). c Pearson correlation coefficients among physical features. Different colors represent valence states, and the black dotted line represents the ideal prediction (predicted value equals actual value). d Predictions of E_f-m and their corresponding actual values. e Predictions of E_f-t and their corresponding actual values. f Predictions of P_s and their corresponding actual values. In d and e, the inset shows a bar chart depicting the RMSE values for various valence states.

Figure 5b shows the influence of composition on lattice constants and volume changes (see the SI for details). The doping of impurities into HfO₂ crystals can cause local deformation of the crystal structure and changes in volume³². The volume reduction or expansion caused by the doping component can be considered as a relative compressive strain change and a volume expansion change. The relative compressive strain change makes the monoclinic phase transition unstable, which also affects the orthorhombic phase to some extent. When the volume change is not significant, a phase transition occurs and the orthorhombic phase gradually stabilizes.

However, with increasing doping concentration, this relative compressive strain change can also have a negative impact, and the energy of the nonpolar phase satisfies the optimal stable condition, causing the crystal phase to shift towards the nonpolar phase, where the polar phase is at a disadvantage. For the phase transition caused by volume expansion, the strain energy caused by the size mismatch between the m phase and t phase due to volume expansion is more significant³³. The lattice distortion is relatively large, so the cubic phase dominates in this case. Next, we used Pearson correlation coefficients (PCCs) to measure the degree of correlation between features. It is worth noting that we introduced volume changes in this correlation analysis. Figure 5c shows the heatmap of the correlation coefficients between physical features, and the high correlation (PCCs = 0.946) between atomic radii and volume change links the physical model with the ML model (see the SI for details). This work not only demonstrates the excellent performance of SVC in classification problems but also provides important physical insights from the “black box” model through the new approach combining SISSO with SVC.

Next, we employed the random forest method to predict E_f-m and E_f-t, incorporating the valence states of doping elements as a feature. The valence state is the reason for the generation of holes or electrons in the HfO₂ structure. It plays an essential role in predicting energy differences, so it was a valuable addition to the relevant features. To evaluate the performance of the regression model, we used two evaluation metrics: the coefficient of determination (R²) and the mean absolute error (MAE). The test results of the random forest model are shown in Fig. 5d, e, with the R² and MAE values for the E_f-m model being 92.9% and 3.671 meV, respectively, and for the E_f-t model being 95.3% and 1.801 meV, respectively. These results demonstrate the excellent performance of our random forest model in predicting energy differences. Subsequently, we trained a polarization prediction model using the same input features but found the model performance unsatisfactory. We observed that energy difference and polarization are closely related (the energy difference reflects the stability of the ferroelectric phase). To address this, we added E_f-m and E_f-t as input features to the polarization model, and the test results are shown in Fig. 5f, with R² and MAE of 90.7% and 0.926 μC/cm², respectively. This demonstrates that the model can predict polarization magnitude within an acceptable error range (details of the machine learning model can be found in the SI).

Physical insights of the machine learning model

The SHAP method provides a unified way to explain the output of any machine learning model by connecting game theory and local interpretation, calculating the contributions of each player based on their marginal effects with or without participation^34,35. Although random forests have feature importance to explain the importance of relevant features (see Supplementary Fig. S3 in the SI), it is not easy to achieve the local contributions of each feature. Therefore, we conducted SHAP analysis on the models for energy differences (E_f-m, E_f-t) and polarization (P_s).

Figure 6a, b shows the SHAP values for individual features on the samples. Unsurprisingly, the valence state of the element contributes significantly to the energy difference since Hf is tetravalent, and the doping of other valence states can introduce holes or electrons. From Fig. 6a, b, negative SHAP values for the doping of divalent and trivalent elements can be observed, indicating that they may promote the proportion of the ferroelectric phase. This suggests that trivalent doping may be more suitable as dopants for HfO₂, which is consistent with the findings of Batra et al.³⁶. It is noted that the high electron affinity (EA) can significantly increase the energy difference, as shown in Fig. 6a. EA is defined as the energy released when an electron is added to a neutral atom. The high EA values observed under pentavalent doping exhibit a positive contribution to E_f-m and E_f-t of over 100 meV. Although other features exhibit behavior that lowers the energy difference, no reversal is observed. The high EA values may have a detrimental effect on phase stability.

Fig. 6: Performing SHAP analysis on the E_f-m and E_f-t models.

a and b show the beeswarm plots for the E_f-m and E_f-t models, respectively, with the features sorted by importance on the left. Each data point is colored according to the magnitude of the feature -value (increasing from blue to red). Positive SHAP values indicate that the energy difference increases due to the feature value, while negative SHAP values indicate that the energy difference decreases due to the feature value. c displays the SHAP analysis for a single sample (Si, Sn, Ti) in the E_f-m (left) and E_f-t (right) models, with red indicating an increase in energy difference due to the feature value, and blue indicating a decrease in energy difference due to the feature value.

Furthermore, SHAP analysis can reveal the impact of ionic radii under different coordination environments on the target property. Single-sample SHAP analysis was conducted on different ionic radii of Sn, Ti, and Si in the same valence state, as shown in Fig. 6c. The effect of different dopant ionic radii on E_f-m and E_f-t was significant (see Supplementary Figs. S4 and S5 in the SI). As the dopant ionic radii increases, the SHAP value of the ionic radii exhibits a trend of initially negative and then positive. For ionic radii alone, smaller ionic radii may have a more significant advantage than a larger ionic radius in stabilizing the ferroelectric phase. However, in terms of the overall energy difference, the energy differences for the three ions are 47.27, −11.62, and 54.12 meV, respectively. Simply using ionic radii to argue that smaller is better is not comprehensive. We note that the E_f-m value for Sn is slightly larger than that for Si. Although the negative feature contribution associated with smaller ionic radii may have an absolute advantage in E_f-m, in situations where the difference in ionic radii is not significant, other features such as electronegativity and electron affinity can have an absolute impact on the target property, and this impact is complex. The significant impact of doping ionic radii on E_f-m and E_f-t is evident in Fig. 6a, b, exhibiting opposite trends for the two properties. A smaller E_f-m value indicates a greater likelihood of forming the ferroelectric phase from an energy perspective, while a smaller E_f-t value suggests greater stability of the ferroelectric phase at the energy level. The relationship between doping ionic radii and phase energy difference does not exhibit a strict linear dependence, as shown in Fig. 7, where a linear relationship is only observed within a certain range. An opposite effect emerges with further increases in ionic radii, which could be a crucial factor in the disappearance of the ferroelectric phase structure. SHAP analysis revealed that higher dopant ionic radii within a specific range could effectively stabilize the polar phase, while high ionic radii can lead to an increase in the phase energy difference, causing instability of the ferroelectric phase structure. Starschich et al.³⁷ observed experimentally that the larger the dopant ionic radii within a certain range, the more significant the residual polarization. The results obtained from the SHAP analysis are consistent with the experimental observations, indicating that the model has extracted relevant insights at the physical level.

Fig. 7: SHAP values under different physical features.

The specific impact trends of the physical characteristics (EA, r, and R) on E_f-m (a) and E_f-t (b).

Applications of the machine learning model

To demonstrate the use of the multistage materials design framework in screening high-value HfO₂ dopants and their doping concentrations, we selected a subset of dopants as examples and computed their corresponding E_f-m, E_f-t, and calculated ferroelectric phase proportions, as shown in Fig. 8. We screened a total of 22 dopants, including 52 different valences. Following previous reports, we limited the dopant concentrations to 2%-25%. Figure 8b clearly shows that the stable ferroelectric phases are mostly found in the low doping states with lower E_f-t. Although Ag, Cd, and Sc exhibit low E_f-t at high doping concentrations, they are not ideal as HfO₂ dopants in the calculated ferroelectric phase fraction, particularly for Ag²⁺, Cd²⁺, and Sc²⁺. Interestingly, Sc³⁺ exhibited a different result. We examined the corresponding ionic radii and found that the ionic radii of Sc³⁺ in an octahedral coordination environment is 0.87 Å, which is closer to that of Hf⁴⁺ (0.83 Å). Figure 8 lists many unconfirmed doping candidates for HfO₂ in terms of first-principles calculations or experiments and some candidates that have been proven to be potentially valuable^19,36,38.

Fig. 8: Machine learning model screening results.

a The predicted results of E_f-m. b The predicted results of E_f-t. c The calculated proportion of ferroelectric phase. The horizontal axis shows different dopants and different valence states, and the vertical axis shows different doping concentrations.

In the mapping of ferroelectric phases predicted by the machine learning model, we find that Ga³⁺ doping stabilizes higher ferroelectric phase fractions, implying that there is a potentially good ferroelectric performance in HfGaO. To assess the effectiveness and accuracy of the machine learning model, we selected Ga as a new element for DFT calculations and multi-gradient doping experiments, as shown in Fig. 9. DFT calculations were performed for different doping ratios of HfGaO, and the results were compared with the predictions of machine learning model. From Fig. 9a, it can be seen that the errors of E_f-m and E_f-t are within a small range. To further validate the performance of Ga doping in experiments, we prepared Ga-doped HfO₂ films with different Hf:Ga cycle ratios by using plasma enhanced atomic layer deposition (PEALD) system (the specific experimental methods are provided in the SI). The XPS spectra are shown in Supplementary Fig. S6 in the SI, through XPS analysis we can confirm the effective doping of Ga elements and the doping concentration. After calculations we determined the doping concentrations of Ga to be 1.2%, 3.1%, 4.5%, and 9.2%, which correspond to different Hf:Ga deposition cycles, respectively (details of the Ga³⁺ valence confirmation and the XPS analysis can be found in the SI). Supplementary Fig. 9b shows a comparison between the ferroelectric phase fraction of the experimentally prepared HfGaO films and the ferroelectric phase fraction predicted by the machine learning model, and we can see that the experimentally determined variation trends of ferroelectric phase fraction and polarization properties with Ga doping concentration are in good agreement with the predictions given by machine learning. It is clear that the high ferroelectric phase fraction is accompanied by good polarization properties, appearing in the predicted high ferroelectric phase fraction and large polarization region (red region), and showing a trend of enhancement followed by weakening with increasing doping concentration (from 0% to 9.2%), which is in high consistency with the trend predicted by machine learning.

Fig. 9: Verification of reliability and usability of ML models through experiments and DFT calculations.

a The disparity between DFT computations and machine learning predictions of E_f-m and E_f-t. b Comparison between the ferroelectric phase fractions predicted by the machine learning model and those obtained from experimental samples. c Phase fractions of Ga-doped HfO₂ thin films with different doping concentrations prepared experimentally. d GIWAXS patterns and corresponding polarization of Ga-doped HfO₂ thin films with different doping concentrations prepared experimentally.

The phase composition of the HfGaO films was analyzed using grazing incidence wide angle X-ray scattering (GIWAXS) technique, and the corresponding phase fractions with different Ga doping concentrations were obtained (Fig. 9c), and the details of the calculations regarding the phase compositions are given in the SI. It is noteworthy that the ferroelectric phase fraction reaches a maximum (61%) as the doping concentration increases up to 4.5%, and the ferroelectric phase fraction decreases almost to be comparable to that of the doping concentration of 1.2% as the doping concentration increases up to 9.2%, but the composition of the phase fractions is not the same at these two doping concentrations. This is due to the fact that the fraction of the m-phase decreases and the fraction of the t-phase increases as the doping concentration increases, which is consistent with the trend of the energy difference demonstrated in Fig. 9a, where the m-phase is dominant and the ferroelectric phase increases at low doping concentrations lower than 4.5%, and the t-phase is dominant and the ferroelectric phase decreases as the Ga doping concentration continues to increase. The GIWAXS patterns and the corresponding polarization-voltage curves of HfGaO films with different Ga doping concentrations are given in Fig. 9d. We can clearly see the trends of the diffraction peak intensities corresponding to the o-phase (i.e., ferroelectric phase), t-phase, and m-phase, respectively, varying with the Ga doping concentration, and the specific phase fractions have been presented in Fig. 9c. The polarization-voltage curves we measured by the PUND (Positive­Up­Negative­Down) method so that the effect of dielectric properties on the polarization performance can be excluded. With the increase of the ferroelectric phase, the polarization magnitude of the HfGaO films increases and shows typical hysteresis characteristics, and the polarization curve reaches a maximum at 4.5% Ga doping concentration, with the remnant polarization 2P_r reaching 17.8 μC/cm², which is in perfect agreement with the process of ferroelectric phase variation. We use Ga as a brand new dopant under the prediction of the machine learning model, and experimentally succeed in obtaining the variation patterns of ferroelectric properties and ferroelectric phase with different Ga doping concentrations, which fully demonstrates that building machine learning models with excellent predictive capabilities can facilitate the search for valuable hafnium oxide doping candidates in the huge chemical space, which is orders of magnitude faster than first-principles calculations.

Discussion

Within this work, we propose a multi-stage material design framework that combines DFT and ML techniques to accelerate the discovery of candidate dopants for HfO₂ materials. We establish ML mappings between physical features and performance by selecting the DFT calculation results of 15 dopants as the dataset. Both classification and regression models demonstrate excellent performance, enabling the prediction of the ferroelectric properties of doped HfO₂ and avoiding time-consuming DFT calculations. Based on the physical interpretation of the models, we find that the ionic radius of dopants is closely related to the stability of the ferroelectric phase but not strictly linearly. Dopants with ionic radii similar to Hf may have higher effective doping concentrations, and trivalent dopants are most favorable for stabilizing the polar phase. Furthermore, we propose a new evaluation method for hafnium-based ferroelectric performance, which considers the phase energy difference and polarization effects based on a more stringent relationship in the Boltzmann distribution, thereby converting the abstract phase energies into intuitive phase fraction distribution mappings. Specifically, applying this framework to a wide range of chemical spaces, we identify Ga as a new dopant for HfO₂, and experimentally succeed in obtaining the variation patterns of ferroelectric properties and ferroelectric phase with different Ga doping concentrations. The DFT calculations and experimental results demonstrate good consistency between the predicted results of the machine learning models and the experimental performance. The material design framework proposed in this work provides a more convenient approach for exploring HfO₂ dopants and offers insights into the phase stability mechanism of hafnium-based ferroelectrics, guiding further research and material development for improving the ferroelectric performance of hafnium-based thin films.

Methods

Machine learning methods

In order to enhance the predictive accuracy of the various processes involved in this work, three ML methods were utilized. Two of the ML methods, support vector machine and random forest algorithm, were implemented in Python through the open-source scikit-learn library³⁹. The machine learning model we have chosen learns the mapping relationship between input features and output labels through the training dataset. Upon completion of training, the model is capable of making predictions or classifications on new data based on input features, which endows the machine learning model with good interpretability and makes it particularly suitable for providing new physical insights from a data-driven perspective. The third method, the sure independence screening and sparsifying operator (SISSO) method, was proposed by Ouyang et al.⁴⁰ in 2018 and aims to uncover low-dimensional material descriptors within a compressed sensing framework. This method can handle large feature spaces and highly correlated features, effectively overcoming the limitations of small and low-availability material data sets. It is worth noting that the SISSO method introduces a feature space construction function that accelerates the establishment of machine learning models through a series of mathematical operators, which are no longer limited to initial material features. This approach has been demonstrated to yield favorable outcomes in predicting material properties^22,41,42. Based on the SISSO method, effective physical descriptors that facilitate the classification of ferroelectric phase structures were uncovered.

The classification was efficiently performed using support vector machines, which was not feasible through the original feature classification. Random forest method was adopted in predicting the phase energy difference and polarization in this study, based on its excellent performance in predicting material properties^43,44. In the support vector classification (SVC), the Gaussian radial basis kernel function, which is mainly used to solve nonlinear problems, was employed to find the decision boundary more effectively by reducing the dimensionality of the data. In the machine learning process, grid search was extensively used for hyperparameter tuning of the ensemble algorithms to determine the optimal model parameters. Additionally, cross-validation was an essential step to evaluate the generalization ability of the machine learning models, where the dataset was randomly divided into 70% training and 30% testing sets. To ensure the consistency and reliability of the models, the same training and test sets were used during model training. To achieve interpretability of the model, we introduced the SHAP analysis method to explain the machine learning model and extract physical insights. This approach offers a valuable tool for interpreting the black-box machine learning models and providing physical insights for the design of materials with desired properties, which is of great importance in material science research.

DFT calculations

DFT calculations were performed using the Vienna ab initio simulation package (VASP) with a plane-wave basis set and a cutoff energy of 500 eV. The solid-state corrected Perdew−Burke−Ernzerhof exchange-correlation functional (PBEsol) was used as the DFT exchange-correlation functional. The lattice parameters obtained from the DFT calculations agree with the experimental lattice parameters of HfO₂ reported in previous works. The total energy convergence was set to 10⁻⁶eV, and the cell structure was fully relaxed using an 11 × 11 × 11 k-point mesh centered at Γ until the Hellman–Feynman forces were less than 0.01 eV/Å. For the relaxation and total energy calculations of the doped hafnium oxide supercell structures, the corresponding k-points were chosen to satisfy k_j a_j ≈ 40, where j = 1, 2, 3, in the direction of the lattice vectors. In this work, we investigate the doping behavior of hafnium oxide with various dopant elements at different concentrations. We consider three competitive phases of hafnium oxide, including the monoclinic ground state (P2₁/c, m), the polar ferroelectric phase (Pca2₁, f), and the nonpolar tetragonal phase (P4₂/nmc, t). We construct five supercell structures with dimensions of 3 × 3 × 1, 3 × 2 × 1, 2 × 2 × 1, 3 × 1 × 1, and 2 × 1 × 1, respectively, in each of the three competitive phases. We perform doping by substituting one or more dopant elements onto the Hf sites in the hafnium oxide supercell structures to achieve different doping concentrations of 2.78%, 4.17%, 6.25%, 8.33%, 12.5%, 25%, 37.5%, 50%, 62.5%, 75%, and 87.5%. The doping concentration is defined by the total atomic percentage, dop%=A/(Hf+A), in which A is the dopant atom. The dopant elements are selected based on their valence, including A²⁺ (Ca, Sr, Ba), A³⁺ (Al, Y, La, Gd), A⁴⁺ (Si, Ti, Zr, Sn, Pb), and A⁵⁺ (V, Nb, Ta).

For the construction of doping models with concentrations greater than 25%, we built doping concentrations of 25%, 37.5%, 50%, 62.5%, 75%, and 87.5% by doping 2 or more doping atoms based on the 1×1×2 supercell structure. In this case, we need to consider the random distribution of doping atoms. Taking the doping of two Ta atoms to construct a 25% doping concentration as an example, 28 doping models can be combined and arranged. The doping models for other concentrations (such as 37.5%, 50%, 62.5%, 75%, and 87.5%) follow similarly, with arrangement modes of 56, 70, 56, 28, and 1 model, respectively. We use the lowest total energy in the random structure as the reference value for the total energy of each phase, because the phase structure with the lowest energy is the most stable and most likely to form. Furthermore, by calculating the variance (σ²) of the corresponding phase under different arrangement structures, we find that the σ² of each phase energy is less than 0.5 meV/f.u., which is a very small difference compared to the energy itself. Therefore, in the case of high-concentration doping, although the random distribution of doping atoms is used, we ensure the consistency of the dataset obtained from DFT calculations by using the lowest energy phase structure as the reference.

Multi-dopant systems are typically complex, involving both isovalent and aliovalent codoping. Isovalent codoping is relatively straightforward, whereas aliovalent codoping is more intricate due to the need to consider charge compensation mechanisms and the corresponding charge carrier behavior. The machine learning model and phase fraction calculation method we have established do not impose limitations on the number of dopants. However, given the complexity of DFT modeling and constraints on computational resources, it is regrettable that we are unable to further incorporate calculations for multi-dopant systems in this study. When dealing with different dopants, doping concentrations, and phase structures, and further considering aliovalent codoping, the complexity of the DFT model significantly increases. If two atoms are doped while accounting for valence differences, and the f-phase, m-phase and t-phase are further considered, the computational demand would surge dramatically, particularly at lower doping concentrations, where the required computational time would become impractical. Current research indicates that codoping can enhance the ferroelectric properties of hafnium oxide. Specifically, in the case of isovalent codoping, experiments have shown that trivalent La-Y and La-Gd codoping improve durability and the ferroelectric phase fraction⁴⁵. For aliovalent doping, significant enhancements in ferroelectricity and the fastest switching process ever reported in HfO₂ polar devices have been observed in HfO₂ ultrathin films with III-V valent La-Ta codoping⁴⁶. In our recent study, we found that p-n type (II/III valent - V valent) codoping not only significantly enhances the ferroelectric properties of hafnium oxide but also effectively controls the defect structure, which is related to the electronic structure of the doping system⁴⁷.