Prediction of Li-ion conductivity in Ca and Si co-doped LiZr₂(PO₄)₃ using a denoising autoencoder for experimental data

Currently, Li-ion secondary batteries are widely used as power sources for small portable devices and laptops.^1,2 Traditional Li-ion secondary batteries use organic solutions as electrolytes, which pose fire and explosion risks owing to their flammability and volatility.^3,4 Recently, all-solid-state batteries that replace liquid electrolytes with non-flammable inorganic solid electrolytes have garnered attention.⁵ Despite the safety advantages, all-solid-state batteries still suffer from insufficient rate performance due to the relatively low Li-ion conductivity of solid electrolytes and, therefore, new highly ion-conductive materials are required. Among them, the NASICON-type solid electrolyte LiZr₂(PO₄)₃ (LZP), with high Li-ion conductivity and low reactivity with metallic Li, has gained attention.^6,7 Four polymorphs of LZP and related compositions are known: the α-phase, α′-phase, β-phase, and β′-phase.^8,9 Among these, the α-phase is reported to have the highest conductivity.^9–11 Furthermore, to enhance Li-ion conductivity, research has focused on Li-excess LZP solid electrolytes by doping lower valent cations into Zr⁴⁺ and/or P⁵⁺ sites in LZP. The combination and amount of dopants change the observed crystalline phase.^12–15 Furthermore, the crystalline phase of cation-doped LZP varies depending on the synthesis sintering conditions. Previously, we synthesized Ca and Si co-doped LZP [Li_1.45Ca_0.15Zr_1.85Si_0.15P_2.85O₁₂ (LCZSP)] under 54 sintering conditions, where the first and second heating temperatures were varied, and it was confirmed that the total Li-ion conductivity (i.e., including bulk and grain-boundary conductivities) at 30 °C varied by two orders of magnitude depending on the sintering temperature combination.¹⁴ A materials informatics approach using Bayesian optimization¹⁵ has shown promise for efficiently exploring compositions and sintering conditions to control crystal phase and optimize conductivity.^14,16–18 Such methods are used not only in LZP but also in a wide range of material optimization.^19,20 However, the black-box nature of Bayesian optimization models makes it challenging to elucidate the relationships between composition, structure, process conditions, and physical properties. Understanding and systematizing these relationships is expected to stimulate the development of new functional materials.

To understand factors affecting Li-ion conductivity, we analyzed powder X-ray diffraction (XRD) data using deep learning methods. Powder XRD is widely used for material characterization and contains a wealth of information beyond just the atomic arrangement in the crystal lattice, including lattice strain, particle size, and various other characteristics.^21,22 While most functional ceramics research utilizes XRD patterns for phase identification, only a fraction of the data is used. Therefore, we attempted to utilize all crystal information contained in XRD patterns for analysis. In particular, in LZP, where ion conductivity varies under sintering conditions, capturing subtle structural changes is crucial. However, experimental data such as XRD patterns are often challenging to use in machine learning primarily owing to difficulties in noise processing. Therefore, in addition to utilizing all information from XRD patterns, our analysis method reduces the influence of experimental data-specific noise.

Very recently, Chen et al. have successfully used Vision Transformer (ViT)^23,24 to create models that identify metal–organic frameworks and organic molecules from XRD and Fourier-transform infrared spectra.²⁴ However, the deep learning model in this study analyzes material properties in addition to identifying phases. Herein, we construct a deep learning model utilizing XRD patterns with learnable attention matrices to visually and quantitatively evaluate the relationship between crystal structure and Li-ion conductivity. Powder XRD, a powerful and frequently used analytical tool, is extensively utilized by materials researchers working on inorganic crystalline materials for phase identification and crystal-structure analysis owing to its ease of use. However, performing detailed crystal-structure analysis requires specialized knowledge of various techniques, such as the Rietveld method. The deep learning model developed in this study is expected to clarify the relationship between crystal structure and physical properties without requiring specialized knowledge, provided that several dozen XRD datasets and physical property data are available. This model is anticipated to be useful in establishing guidelines for designing functional materials.

This study focused on Li_1.45Ca_0.15Zr_1.85Si_0.15P_2.85O₁₂ (LCZSP), which is a Ca and Si-co-doped LiZr₂(PO₄)₃. Previously, we reported that the conductivity of this material varies under sintering conditions.¹⁴ A total of 54 samples were synthesized with different first and second sintering temperatures. XRD and Li-ion conductivity data were collected for each sample. The activation energy was calculated from ion conductivity data measured at 30, 60, and 90 °C. The XRD data obtained from the 2θ range of 10°–60° were normalized to a maximum intensity of 1 and used as descriptors, and the activation energy for Li-ion conduction was used as the target variable. Normalization can accommodate additional literature data; however, this step was not performed in the present study. In this study, XRD patterns of 2501 bins were used to build the model, and the differences in the number of bins due to the data format or settings of the XRD patterns were processed via numerical interpolation before feeding them into the deep learning model as inputs. However, standardizing the data based on the sample with the narrowest angle range and the smallest number of bins is necessary. The details of sample preparation, XRD, and ionic conductivity measurements are described in our previous study.¹⁶ 

XRD patterns contain noise originating not only from instrumental factors but also from human factors such as sample mounting. When capturing subtle changes in structural information, this noise can potentially influence predictive results. Therefore, in this study, we utilized a denoising autoencoder²⁵ to mitigate the impact of noise in XRD data. A denoising autoencoder is a deep learning model used for noise removal, where it takes noisy data as inputs, performs dimensionality reduction and reconstruction from compressed dimensions, and learns to reconstruct outputs that match the original data before noise addition. In this study, we created a dataset by applying four types of noise to each sample: three patterns of uniformly generated noise fluctuating within ±3.5% vertically and one pattern within ±0.24° horizontally. These noise datasets were used as inputs to train the denoising autoencoder. The inclusion of horizontal noise is expected to mitigate angular shift-related issues, such as zero-adjustment errors, and the addition of vertical noise can prevent deviations in intensity due to mixing of coarse particles and/or preferred orientation, given that these noise components are within the range of the artificial noise added to the input data of the denoising autoencoder.

The schematic illustration of the model used in this study is shown in Fig. 1. In this model, a denoising autoencoder that includes convolution layers was used to compress XRD patterns into a two-dimensional representation (the blue section in Fig. 1). By performing dimensionality reduction with the denoising autoencoder, we created representations of XRD patterns that reduce the influence of noise. From these encoded variables, the activation energy for Li-ion conduction was predicted (the yellow section in Fig. 1).

Furthermore, to enhance prediction accuracy and interpretability, we introduced an attention mechanism that learns the important regions of the XRD patterns (the light green section in Fig. 1). In particular, six-head attention matrices were introduced: four attention matrices generated using convolution layers from the features extracted by the convolution of the denoising autoencoder and two attention matrices created using linear layers. These attention matrices were then multiplied with the original features to create attention-weighted features, which were learned to improve prediction accuracy.

For training the XRD-denoising autoencoder, 90% of the data were used as training data, and the remaining 10% were used as test data to check the model performance. The batch size and learning rate were set as hyperparameters, and a grid search was performed to determine the combination with the lowest Loss_all. The batch size was set to 4, and the learning rate was set to 0.0001. Training was conducted for 20 000 epochs, and the model with the smallest Loss_all in the test data was selected as the best model. The original data before noise were added to the trained model, and the resulting attention matrix was used to examine how to interpret the XRD patterns.

Figure 3(a) is a diagnostic plot of the test data for the XRD-denoising autoencoder reconstruction displaying R² = 0.72. Figure 3(b) is a diagnostic plot of the test data for the regression prediction of Ea using w = 50. The R² score for Ea prediction exceeds 0.8, confirming the ability of the model to predict the activation energy. Figure 3(c) compares the input and output of XRD descriptors to the autoencoder. Noise types 1–3 indicate samples with vertical noise, and type 4 indicates samples with horizontal noise. Samples with vertical noise show minimal error in peak positions, but the restoration of peak heights is less accurate. While the R² score of the XRD-denoising autoencoder (0.72) indicates uncertain precision, Fig. 3(c) confirms high-precision restoration of crucial peak positions for phase differentiation. Therefore, the model effectively captures crystallographic information in the compressed dimensions. Conversely, removing the horizontal noise proved challenging.

Furthermore, a comprehensive analysis of the 2D representations compressed by the autoencoder (supplementary material Figs. S2–S4) confirms that the first compressed dimension reflects α- and β-phase formation, whereas the second compressed dimension captures features such as the first sintering temperature and the overall shape of the peaks, including the full width at half maximum, as shown in the supplementary material, Figs. S2 and S3. The correlation between the XRD profiles and the two compressed dimensions is presented in the supplementary material, Fig. S4. Because the first compressed dimension variable matched the diffraction angles of the α- and β-phases, the second heating temperature is likely related to the formation ratio of these phases. For the second compressed dimension variable, a weak correlation over a broad angular range suggests capturing information such as the peak profile.

Figure 4 shows the visualization of the attention matrices (Heads 1–6) of XRD patterns obtained from the trained model. Heads 1–4 are created by convolutional layers, while Heads 5 and 6 are created by fully connected layers. It is observed that Heads 1–4, using convolutional layers, did not capture prominent attention matrix-derived features, whereas Heads 5 and 6, using fully connected layers, successfully captured distinctive features. These results suggest that creating attention matrices using fully connected layers is superior to using convolutional layers. Notably, application of zero-filling to the outputs of Heads 1–4 degraded the autoencoder’s XRD-profile reconstruction performance and the prediction accuracy of the activation energy Ea (supplementary material Fig. S6). This result indicates that Heads 1–4, which include convolutional layers, contribute to improving the fitting accuracy of the proposed autoencoder model. One contributing factor to these results is that convolutional layers merely extract features and may not distinguish attention across similarly shaped peaks at different diffraction angles. Sample numbers in Fig. 4 are arranged in order of the first and second heating temperatures, and horizontal lines periodically appear in Heads 5 and 6, corresponding to the differences in the second heating temperature. For visualization purposes, Fig. S5 of the supplementary material displays the masks of Heads 5 and 6 for samples sorted in order of the second heating temperature. Head 5 shows a significant contribution of attention scores in samples with second heating temperatures of 1100–1150 °C, while Head 6 shows a similar trend in samples with second heating temperatures of 1100–1200 °C. Figure 5 demonstrates a strong correlation between the activation energy and the second heating temperature. These observations confirm that Heads 5 and 6 have learned attention matrix-derived features related to the second heating temperature.

The heatmap displays diffraction angles on the horizontal axis and sample numbers on the vertical axis, with the attention scores represented by color intensity.

Figure 6(a) shows the selected XRD patterns where the average attention scores of Heads 5 and 6 across all samples ≥0.8 are colored. For example, the profile indicated by the black lines in the figure represents the XRD pattern of sample No. 34, which has the lowest activation energy. The pink, blue, and green lines represent the spectra obtained from the ICSD for the α-phase LZP (No. 201935), β-phase LZP (No. 91113), and ZrO₂ (No. 100243), respectively. In addition, the star symbols indicate attribution to high attention-score areas in the ICSD data, marked with the corresponding colors. From this figure, it is observed that both Heads 5 and 6 mainly focus on specific peaks derived only from the α-phase and β-phase (represented by the single-colored star symbols). This agrees with the fact that the α-phase LZP shows superior ionic conductivity compared to the β-phase LZP, as mentioned in the introduction section. Furthermore, the significant attention score derived from ZrO₂ implies that the presence of ZrO₂ impurities may influence the activation energy of ion conductivity.

A magnified view of the XRD profiles, including highlights for high attention-score areas within the dashed box in Fig. 6(a), is presented in Fig. 6(b). A close examination reveals very subtle but distinct β-phase-specific peaks in these attention areas. Near this diffraction angle, an extremely small peak, originating from the β-phase, is expected, which likely results in a higher attention score. In the example shown in the figure, almost no peaks are visible, suggesting the formation of the α-phase. This result suggests that the model learns crystal-structure features through XRD patterns, even in the absence of information about the α- or β-phase.

In Fig. 7, we examined the differences in the attention-score distribution between samples with low and high activation energies. Note that the activation energy represents the temperature dependence of the Li-ion diffusion coefficient, where effective ion conductors typically exhibit lower activation energies. Figure 7(a) displays samples arranged in ascending order of activation energy, visually representing attention scores from Heads 5 and 6. The figure reveals that areas with higher activation energies are predominantly white, indicating numerous attention spots. Based on these results, we extracted the top five samples with low activation energies and the bottom five with high activation energies, calculated the average attention scores at each diffraction angle, and colored areas with average scores above 0.8 in Figs. 7(b) and 7(c) for Head 5 and Head 6, respectively. The black lines in Figs. 7(b) and 7(c) represent the XRD patterns of the sample with the lowest activation energy (No. 34) and the highest activation energy (No. 37). The red segments indicate high attention-score areas for samples with low activation energies, while blue segments indicate those for samples with high activation energies. From these figures, it is evident that samples with higher activation energies show more attention areas, including specific areas related to β-phase and ZrO₂, as well as areas with no clear attribution. This suggests a correlation between increased impurities and higher activation energies. Moreover, the model’s tendency to focus on entire peaks suggests its attention to factors related to XRD peak broadening, such as lattice strain or particle size. Since our previous studies indicated that the grain size and the grain-boundary structure relate to Li-ion conduction properties, the broad high attention-score areas for XRD peaks may relate to the peak profile of the XRD patterns containing crystalline size and lattice distortion derived from grain-boundaries.

Finally, we discuss the potential of the inverse analysis approach for material design. In this study, the XRD patterns of materials that achieve the user-desired property values can be predicted owing to the structure of the autoencoder. By using the trained model, a “material map”²⁶ of extrapolated space, which predicts the target properties, can be created as shown in the supplementary material, Fig. S7(a). Then, by specifying the coordinates on the map based on the predicted property values, the XRD pattern can be decoded and restored. As a demonstration, we show the decoded XRD patterns of the materials indicated by the star marks in the supplementary material, Fig. S6(a) as S6(b)–S6(d) in the supplementary material, Fig. S6(b)–S6(d). Point (b) is a coordinate on the map where Ea is expected to be minute; however, because it is currently a sparsely populated area of the data distribution, the restored XRD pattern differs significantly from those provided in this study, making it practically infeasible to prepare such a material. Conversely, at points (c) and (d), the Ea is expected to be relatively small or large within a densely populated region of the data distribution, and the restored XRD patterns have features similar to those provided. Therefore, by analyzing these restored XRD patterns in detail, guidelines for material design can be established.

In this study, we developed a deep learning model using XRD patterns and a denoising autoencoder to visually and quantitatively evaluate the relationship between XRD-derived information and activation energy for Li-ion conductivity in LZP. The model learns focus areas in XRD patterns that are crucial for predicting target variables, providing interpretability to experimental data used as descriptors. For example, in LZP XRD patterns, 2θ areas with high attention scores indicated significant information for predicting Li-ion conduction activation energy, suggesting that the material phases associated with these peaks have a substantial impact. By interpreting such learned attention regions, material researchers can gain new insights into which aspects of the crystal structure influence target properties. We believe that the indicators and insights obtained in this manner can lead to new strategies for material design, potentially accelerating the discovery of materials with excellent ionic conductivity.

Supplementary Figs. S1–S7 are provided in the supplementary material.

A part of this study is based on the results supported by a Grants-in-Aid for Scientific Research (Grant Nos. 22H02179, 24H02203, 23K21696, and 24K01157) from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan, a CREST grant from the Japan Science and Technology Agency, Japan (Grant No. JPMJCR21O6), and the Data Creation and Utilization-Type Material Research and Development Project (Grant No., JPMXP1122712807) from MEXT.

The authors have no conflicts to disclose.

Y.Y. and M.N. conceived and directed the project. Y.Y. and H.T. performed data curation descriptor building. Y.Y. coded the software used in this study with the help of S.N., K.I., M.K., and R.K. The article was mainly written by Y.Y. and M.N. through the contributions of all authors. All authors have approved the final version of the article.

Yumika Yokoyama: Data curation (equal); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Shuto Noguchi: Investigation (supporting); Methodology (equal); Software (equal); Validation (supporting); Writing – review & editing (supporting). Kazuki Ishikawa: Investigation (supporting); Methodology (equal); Software (equal); Validation (supporting); Writing – review & editing (supporting). Naoto Tanibata: Project administration (supporting); Validation (supporting); Writing – review & editing (supporting). Hayami Takeda: Data curation (lead); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (lead); Writing – review & editing (equal). Masanobu Nakayama: Conceptualization (lead); Funding acquisition (lead); Project administration (lead); Supervision (lead); Validation (lead); Writing – original draft (lead); Writing – review & editing (lead). Ryo Kobayashi: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal). Masayuki Karasuyama: Conceptualization (equal); Software (supporting); Supervision (equal); Validation (equal); Writing – review & editing (equal).

A list of heating conditions and corresponding electric resistance information and XRD data are available in Figshare: https://doi.org/10.6084/m9.figshare.26362294. The data that support the findings of this study are available from the corresponding author upon reasonable request.