Introduction

NIR light has attracted considerable interest due to the high penetration, safety to the biological tissues, and distinct characteristic absorption by certain molecules. These advantages enable NIR light to be applied in various fields, including biomedicine, agriculture, optical communication, and anti-counterfeiting1,2,3,4. Tungsten halogen lamps, globars, and NIR pc-LEDs can be used for the NIR illumination. In particular, the pc-LEDs, which are powered by commercial blue InGaN chips, have been extensively regarded as the novel NIR light sources because of the superiority in the good integration and long service lifetime5. As is well known, the NIR phosphors almost determine the entire performance of pc-LEDs, such as the working wavelength and efficiency, the full width at half maximum (FWHM), and the quenching temperature. Among the luminescence properties, the emission wavelength is a critical reference for the practical applications6,7, as shown in Fig. 1a. For example, NIR light with a peak wavelength around 730 nm is commonly used for promoting the plant growth. While light in the 750~800 nm region is more suitable for deep tissue imaging because of the weak scattering and absorption in the biological environments8. And the strong penetration ability of NIR light in this band is also favorable for the application in optical communications. NIR light centered around 820 nm can be employed for iris recognition. The security marking as well as night version could be actualized via NIR light with EPW at 800~850 nm9,10,11. As the majority of organic molecules respond specifically to light signals beyond 850 nm, NIR sources in such wavelength regions play an important role in food analysis.

Fig. 1: Workflow for designing target NIR phosphors with the assistance of co-training ML model.
figure 1

a Different applications of NIR light according to the EPW. b The acquirement of chemical descriptors for each phosphor and the feature extraction. c The constructing of co-training ML model. The unlabeled dataset is built based on priori knowledge (I) and (II). d The utilization of proposed model for predicting EPW of hypothetical NIR phosphors in a greatly expanded chemical space. e Extraction of valuable insights from the model using the SHAP values. f The validation of model based on the experiment.

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Luminescence materials activated by transition metal (TM) and rare earth ions, such as Cr3+, Cr4+, Mn4+, Er3+, Nd3+, Yb3+, Tm3+, etc. have been widely studied in the past decades for producing the NIR phosphors12,13,14,15,16. In view of the excellent thermal stability, the relatively large internal quantum efficiency (IQE) and external quantum efficiency (EQE), great attention has been paid to Cr3+ doped garnet oxides, which can be well excited by blue light due to the strong absorption of Cr3+: 4A2g → 4T1g(4F). Since the d-orbital electrons of Cr3+ are highly susceptible to the crystal field, the NIR luminescence can be easily tuned via regulating the host matrix. It is noted that the luminescence of Cr3+ doped garnet oxides typically covers the spectral region of 650~1100 nm. Hence, for guaranteeing the output power of NIR pc-LED near the target wavelength, considerable efforts are being made to modulate the emission wavelength of Cr3+ doped garnet oxides to the desired value for certain applications, such as promoting photomorphogenesis (730 nm), vascular imaging (780 nm), ethanol content detection (930 nm), and other purposes mentioned above. Additionally, apart from the range of 950–1000 nm where light is significantly absorbed by water molecules, most of the Cr3+ emission falls within the first biological window, which is characterized by the low absorption and scattering of light by tissues, thereby allowing for deep penetration in the biological tissues and presenting potential utilization in in-vitro imaging. Garnet oxides possess cubic crystal structure with complex arrangement cations in the unit cell, as shown in Fig. 1b, thus allowing the flexible modulation of emission properties. Garnet-structured materials are usually denoted as A3B2C3O12, where A, B, and C occupy the centers of AO8 dodecahedron, BO6 octahedron, and CO4 tetrahedron, respectively. If each site is substituted by two elements with elemental coefficients that are multiples of 0.5, there would be 28,941 types of garnet oxides. Given the vast possible combinations of composition, experiment alone is apparently insufficient to achieve garnet compounds of interest under finite resources and in reasonable time scales.

Fortunately, machine learning (ML), a promising data-centric approach, is emerging as a powerful technology in the field of materials. ML can bypass the underlying physical and chemical mechanisms and establish a relationship between the targeted property of specific materials and design parameters using prevenient experimental results, thereby presenting great potential in alloys17, perovskites18, polymers19, and semiconductors20. The primary challenge for building a ML model is training on robust and diverse data, which demands compiling the database from reported literatures. And this is also the reason why the application of ML in developing TM doped NIR luminescence materials is quite limited. On the one hand, the criterion used for evaluating the luminescence properties is not uniform for different researchers. For instance, several works use T50 (the temperature corresponding to the point where the emission intensity is reduced to the half of that at room temperature) to characterize the thermal stability of phosphors, while some literatures tend to utilize I423K (the percentage between emission intensity at 423 K and that at room temperature) to discuss this issue21. In addition, the IQE and EQE are not only dependent on the element composition of host matrix, but also are determined by the synthesis condition. It has been revealed that the atmosphere, temperature, and duration for the sintering process have a significant impact on the luminescence efficiency as well as the thermal stability of Cr3+-doped garnet phosphors22. Consequently, these emission characteristics should not be employed as labels, because their robustness has been significantly diminished by different evaluation criterions and influences from multiple processing factors. Another major reason hindering the application of ML in NIR phosphors is that the reports correlated with Cr3+ activated garnets are very limited for obtaining adequate data. To our knowledge, only no more than 200 articles that report Cr3+ doped garnet oxides can be available up to now (Supplementary Table 1 in the Supplementary Material). And not all luminescence characteristics are explicitly stated in the articles. The restricted acquirement of sample data is extremely a big obstacle for training a ML to boost the NIR phosphors.

Compared with other luminescent characteristics, the EPW of Cr3+ doped garnets would be more objective and robust because they are mainly determined by the chemical composition of host. Besides, there is a close intrinsic connection between EPW and above-mentioned luminescence features through the bridge of crystal field. The EPW is known to be strongly dependent on the parameter Dq/B (Dq and B denote the parameters of crystal field and Racah), which determines the thermal stability, the luminescence efficiency, and the FWHM. Generally, according to the value of Dq/B, the crystal field strength can be classified into three regions: the weak crystal field (Dq/B < 2.3), the intermedia crystal field (Dq/B = 2.3), and the strong crystal field (Dq/B > 2.3). In the case of a strong crystal field, the luminescence originated from the spin-forbidden transition of 2E → 4A2 can be readily obtained, which presents a relatively short EPW, a narrow emission band (Supplementary Table 1 in the Supplementary Information), good thermal stability, and a large probability of obtaining high IQE and EQE. For the weak crystal field, the spin-allowed 4T2 → 4A2 transition will dominate the luminescence process and finally lead to a relatively long EPW with a broad emission, poor thermal stability, and usually low IQE and EQE. While in the intermedia crystal field, it is possible to simultaneously observe the emissions resulting from the spin-allowed and spin-forbidden transitions. The relationship between EPW and FWHM as well as IQE can be well illustrated through Fig. 1I and Supplementary Fig. 1a. In addition to EPW, the luminescence lifetime is also a robust parameter, which demonstrates close correlation with EPW through the analysis of reported literatures, as shown in Supplementary Fig. 1b. However, the lifetime data extracted from literatures are very limited (only 59, see Supplementary Table 2), which would be hard to use for constructing a reliable model. In summary, the EPW, which is easily available in the literatures, would be an effective and rational indicator for guiding the discovery of Cr3+-activated NIR phosphors. Yet, before the ML is used for directing the design of NIR phosphors, the following substantial problems should be resolved: (i) How to define an appropriate Cr3+ doped garnets dataset for training an effective ML model? (ii) Despite the potential for the ML model to accurately predict the EPW, how can we rapidly acquire NIR phosphors exhibiting high performance with the assistance of ML model?

Herein, a semi-supervised co-training ML model based on SVR and KRR (both kernel functions are radial basis function, RBF) is constructed by using EPW as the single label to boost the production of new NIR phosphors, where the output of this model is set as the mean result of two weak learners (Modelco-training = 0.5 × (ModelSVR + ModelKRR)). The detailed process of the strategy is shown in Fig. 1. Through the co-training procedure, the dataset used for training model is successfully expanded by the unlabeled (never-before-seen) samples, along with the improved accuracy and enhanced generalization of model, which is critical for further interpreting the model via the feature importance extracted by SHAP (SHapley Additive exPlanations). This enables us to further understand the key factor influencing the luminescence behavior of Cr3+ in garnets and subsequently facilitates the identification of novel NIR-emitting phosphors.

Results

Construction and analysis of two weak learners, SVR and KRR

The data of EPW utilized for training the model are acquired from 187 reported Cr3+-doped garnet phosphors, as listed in Supplementary Table 1, along with a distribution of EPW (Supplementary Fig. 2). Other limited quantities of luminescent parameters are provided in Supplementary Tables 2–4. Several features are selected based on priori knowledge. In the garnet oxides, Cr3+ ions generally occupy the octahedral sites, and the crystal field strength usually has an inverse relationship with the cation radius in the octahedral lattice sites23,24. With the increase of cation radius, the parameter Dq/B would decrease, along with the redshift and broadening of photoluminescence spectrum. Meanwhile, Dq/B could also be reduced by enlarging the cation radius of dodecahedral sites where Cr3+ can be located25. Additionally, the EPW presents strong dependence on the concentration of Cr3+ due to the comprehensive effect originated from the energy level splitting, spectral reabsorption, energy transfer, and Stokes shift26. Therefore, the ionic radii of elements in the octahedral and dodecahedral sites as well as Cr3+ doping concentration (DC) are included in the initial feature construction. Other inherent physical and chemical properties of the elements at these sites, such as valence electron concentration, electronegativity, ionization energy, and cohesive energy, are simultaneously taken into account. These features are associated with EPW in a statistically significant way, and an averaging approach is employed in cases where different elements occupy the same sites. Finally, a total of 64 features are preliminarily derived, as shown in Supplementary Table 5, where No. 1–21 features are acquired based on the mean values of elements in dodecahedral, octahedral, and tetrahedral sites. The Pearson correlation coefficients of all features with EPW are shown in Supplementary Fig. 3.

Co-training is a semi-supervised ML method based on the concept of divergence, which requires the construction of two different views, i.e. two different attribute sets. In general, co-training is actualized through the regression training of two weak learners, where a regressor is trained with labeled samples on each view. Then, a few unlabeled samples with high labeling confidence are selected by the regressors and added into the training set of the other regressor, which can be updated with these newly labeled samples. The process of mutual learning continues iteratively until the regressors no longer change or a predefined number of learning round is reached. The key of co-training lies in the generation of divergence, for which a common method is to design weak learners using two views with sufficient redundancy and conditional independence. Indeed, the conditional independence of views is usually hard to achieve. While the semi-supervised learning method with divergent views can still be effective if the regressors in each view can annotate the samples precisely27. And this strategy could ensure the difference (independence) of the learners and the accuracy (adequacy) of model prediction. Considering the fact that SVR possesses the advantage in solving problems of small dataset and nonlinear regression and KRR is particularly suitable for the cases of multicollinearity28, which exists in the EPW dataset because of the linearly correlation between the features we construct, SVR and KRR are finally chosen as the weak learners. And the rationality of the selected weak learners is further confirmed in the following discussion.

The determination coefficient (R2) and the mean absolute error (MAE), as expressed in Eqs. (1) and (2), are used to evaluate the performance of SVR and KRR based on the dataset summarized from reported articles,

$$\begin{array}{c}{{R}}^{2}=1-\frac{\mathop{\sum }\nolimits_{i=1}^{n}{({y}_{i}-{{y}}_{i})}^{2}}{\mathop{\sum }\nolimits_{i=1}^{n}{({y}_{i}-\overline{y})}^{2}}\end{array}$$
(1)
$${\rm{MAE}}=\frac{1}{n}\mathop{\sum }\nolimits_{i=1}^{n}|{y}_{i}-{{y}}_{i}|$$
(2)

where yi is the actual value; \(\overline{y}\) and \({{y}_{i}}\) denote the averages of the actual and the predicted values, respectively; n is the data number. Here, sequential feature selection (SFS) algorithm is adopted for the feature selection29. Throughout the iterative process, R2 is employed to access the model accuracy under tenfold cross-validation (CV). As shown in Fig. 2a, since R2 changes slowly after reaching the peak value, the number of features is artificially selected as 12 and 13 for SVR and KRR, respectively, to avoid the impact of redundant features on the generalization capability of model. Figure 2b shows the features obtained by the screen of SFS, accompanied by the corresponding variable importance in projection (VIP) values, which provide an explanation degree of the predicted EPW (dependent variable) by the relevant feature (independent variable). It is noteworthy that DC and atom/ion radius are all retained after the SFS screening. This result indicates that these features are closely related with EPW and simultaneously evidences the reliability of priori knowledge. It is observed that most VIP values are larger than 1, implying a substantial linear contribution of corresponding features to EPW. Consequently, both views with sufficient information are utilized to generate optimal weak learners. The datasets are randomly separated with a ratio of 0.8: 0.2 for training and testing the model. To optimize the model parameters, the grid search is implemented, during which the tenfold CV is used to increase the reliability of model. The detailed parameters for SVR and KRR models are presented in Supplementary Table 6. Finally, two original learners are acquired from two different views. Both learners display good performance within the training set, as shown in Fig. 2c, d, along with R2 values of around 0.892 and 0.870, respectively, while maintaining MAE values at 7.105 and 8.514. Besides, the learners also behave well on the test set (SVR: R2 = 0.821, MAE = 8.761; KRR: R2 = 0.850, MAE = 9.125). Obviously, these two views are able to train the weak learners effectively and would ensure the reliability of labeling samples in the following co-training procedure. We also evaluate the performances of other learners, including Random Forest, k-Nearest Neighbor (KNN), and XGBoost, as shown in Supplementary Fig. 4a–c. And the superiority of SVR and KRR would be more apparent when comparing with other learners (Supplementary Fig. 5). It should be mentioned that although the EPW is a robust parameter in comparison to other luminescence characteristics, the possible differences in calibration between different spectrometers would introduce error into our model. Meanwhile, it is also notable that the majority of spectrometers referenced in the literatures are manufactured by the same institution (Edinburgh Instruments) (Supplementary Fig. 6). This would be beneficial to the reliability of the dataset, which can be well reflected by the results shown in Fig. 2c, d. Here, we also attempt to train the models using the lifetime as label. But it is difficult to achieve a reliable model due to the insufficient data volume (Supplementary Fig. 7).

Fig. 2: Performance of weak learners.
figure 2

a The performance of two models with different number of features. b The VIP values for the features of two models after screening by the SFS algorithm. The fittings of predicted and measured values by SVR (c) and KRR (d) on the training and test sets.

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Construction of semi-supervised co-training model

Up to now, there is no database available for the Cr3+ doped garnet oxides. In this work, considering the fact that co-training has the capability of taking full advantage of the information embraced in the unlabeled samples, we will establish an unlabeled dataset based on priori knowledge. Many chemical elements, as demonstrated in Fig. 1b, are capable of forming dodecahedrons, octahedrons, and tetrahedrons within the garnet structure. However, not every combination of these elements can lead to a garnet phase. The doping formula for octahedron sites can be expressed as13

$${D}_{{r}}=|100\times \frac{{R}_{{s}}(N)-{R}_{{d}}(N)}{{R}_{{s}}(N)}|$$
(3)

where Rs and Rd are the radii of the substituted and the doped ions (Cr3+), respectively; Dr represents the radius difference between Rs and Rd, and N denotes the coordination number. The garnet structure would be readily obtained when the radius difference Dr is smaller than 30%30. The values of Dr for the ions that usually appear in octahedral sites are calculated, as shown in Supplementary Fig. 8, and the weight of these elements is modified according to the corresponding value of Dr. In addition to the radius difference, a tolerance factor τ is used to estimate the stability of garnets. The tolerance factor can be described as31

$$\tau =\frac{3\sqrt{{({R}_{{B}}+{R}_{{D}})}^{2}-\frac{4}{9}{({R}_{{A}}-{R}_{{D}})}^{2}}}{2({R}_{{C}}+{R}_{{D}})}$$
(4)

where RA, RB, and RC denote the radii of ions occupying dodecahedron, octahedron, tetrahedron, and RD represents the radius of O2−. For the garnet oxides, the distribution of tolerance factor is approximately within the range of 0.748~1.333, where the garnet structure is relatively stable32. For further guaranteeing the effectiveness of the unlabeled dataset, additional conditions are added according to the priori knowledge: (1) octahedral elements tend to be occupied by a single element or several elements with similar radii; (2) tetrahedral sites are filled with elements with high electronegativity that are conducive to structural stability, such as Ga and Ge33. Consequently, an unlabeled dataset with a size exceeding 28,941 is established, where 7000 samples are randomly selected to form the final unlabeled dataset.

Another critical aspect of the co-training is how to choose samples with the highest confidence for labeling. Previous studies on semi-supervised learning have paid much attention to the semi-supervised classification, where various classifiers can provide an estimated probability (or an approximation)34. And the labeling confidence can be determined by examining the probabilities of unlabeled examples being classified into different categories. Nevertheless, the estimated probability cannot be directly used in the regression like the case in classification due to the huge number of possible predictions from regression. According to the streamwise assumption of semi-supervised learning35, unlabeled samples with high confidence should have a high similarity with those in the labeled sample set, where the labeled values are close to each other. Thus, the MAE of regressor on the refreshed labeled sample set can be utilized as a reference to access the confidence of newly added samples. The MAE of regressor using the information provided by \(({x}_{u},{{y}_{u}})\) can be evaluated based on the labeled example set, where xu is an unlabeled instance and \({{y}_{u}}\) is the label value generated by the original regressor (\({{y}_{u}}=h({x}_{u})\)). δu, which represents the subtraction of former MAE (before adding an unlabeled sample) by latter MAE (after adding an unlabeled sample), can be expressed as

$${\delta }_{u}=\sum _{{x}_{i}\in L}|{y}_{i}-h({x}_{i})|-\sum _{{x}_{i}\in (L\cup \{({X}_{u},{{y}}_{u})\})}|{y}_{i}-{h}^{\text{'}}({x}_{i})|$$
(5)

where h and h′ denote the original regressor and the modified one that has applied the information provided by \(({x}_{u},{{y}_{u}})\), respectively. Then, in each round of marking, \(({x}_{u},{{y}_{u}})\) associated with the positive δu that possesses the maximum value can be regarded as the samples with the highest confidence.

Figure 3 depicts the schematic diagram of the regression algorithm based on co-training. L1 and L2 are obtained from the labeled dataset L by view segmenting. Us is a subset from unlabeled examples U and can be replenished after each iteration. First, we calculate the average absolute errors (former MAE1, former MAE2) originated from the prediction by h1 and h2 on the labeled samples. Then, in every round of iteration, each unlabeled sample xu from Us is separately added into Lj (j = 1, 2) to get L′j, which is used to obtain h′j. The average absolute errors (latter MAE1, latter MAE2) resulted from h′1 and h′2 on the labeled samples (L′1, L′2) are then calculated. Hence, during each round of iteration, the samples x1 and x2 that make δ1 and δ2 positive and maximal would possess the highest confidence. And the inclusion of samples with high confidence could enhance the learner generalization capability. The learning process will end when reaching the maximum number of learning iterations T or no unlabeled example is available for decreasing the MAE of regressors on the labeled example set. It should be noted that the unlabeled example chosen by h1 in each iteration will not be chosen by h2, which is beneficial to promoting the diversity of the regressors. Thus, even if h1 and h2 are similar, the examples they label for each other are still different. Finally, the predictions of these two regressors are averaged to yield the final output.

Fig. 3
figure 3

The schematic diagram of regression algorithm based on co-training.

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To fully access the regression performance of model, the semi-supervised training on D (D = U + L) is performed under different labeling rates (3%, 5%, 10%, and 15%), which are actualized by varying the data volume of U. After each round, we record the MAEs, i.e. MAE1 and MAE2, of h1 and h2 on their respective labeled sets L1 and L2, as shown in Fig. 4a–d. Under 3% labeling rate, the fitting performance of KRR model differs greatly from that of SVR especially at the end of iteration, probably due to the relatively small amount of data involved. When the labeling rate is 15%, large noises for these two models are observed at the beginning of iteration, accompanied by relatively high MAEs. Compared with the case at 10% labeling rate, the MAEs for KRR and SVR under 5% labeling rate decrease more apparently with the increment of literation, and the performance of semi-supervised model is the best among these four different situations.

Fig. 4: Performance of co-training model.
figure 4

The performance of SVR and KRR under a 3.0%, b 5.0%, c 10.0%, and d 15.0% labeling rates. e Comparison of model performance under different conditions at a 5.0% labeling rate. f The fitting performance of the ensemble model on the original label set. The fitting performance of g SVR and h KRR models on the final labeled set after co-training.

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It should be mentioned that the increase of MAE in the models has been observed at the beginning of the iteration under different labeling rates. This is mainly due to the large difference between the unlabeled samples and labeled ones at the beginning of the iteration. While such difference could lead to the accumulation of noise within the literation procedure, it also enriches the sample dataset and favors the improvement of model performance. Therefore, to understand the actual influence of unlabeled samples that differ obviously from the labeled ones, we restrict the unlabeled samples in the literation and evaluate the difference in MAE before and after the restriction. In the first 15 rounds of the iteration, the change in MAE is estimated in real time after each addition of newly unlabeled samples, and the samples resulting in a change of MAE larger than 3 are eliminated. As shown in Fig. 4e, the KRR model behaves well with the restriction of unlabeled samples in the first 30 rounds, while the performance of this model without restriction is much better in the following 170 rounds. As for the SVR model, good performance is only achieved in the first 5 rounds when restricting the unlabeled samples. Hence, it can be concluded that the samples with large differences added at the beginning of training would play a key role in improving the models.

Consequently, two models with enhanced generalization capabilities are acquired after the two weak learners label each other through 200 rounds of collaborative training at a labeling rate of 5.0%. Supplementary Fig. 9 shows the fitting behavior of co-trained SVR and KRR upon the original set. Compared with the results presented in Fig. 2c, d, R2 and MAE corresponding to SVR and KRR models have been apparently improved by the co-training procedure. Meanwhile, the final co-training model also illustrates excellent performance upon the original set, as shown in Fig. 4f. Figure 4g, h presents the fitting behaviors of these two models upon their individual datasets expanded by unlabeled samples. Both SVR and KRR demonstrate much high R2, along with small MAE. These results demonstrate that the co-training semi-supervised ML model used for predicting the EPW of Cr3+ doped garnet NIR phosphors has been successfully constructed.

Experimental validation

To verify the effectiveness of this model, 15 Cr3+ doped garnet oxides, which have never been reported before, are randomly selected from the expanded dataset (Table 1) and prepared through the high-temperature solid-state method in air atmosphere. According to the XRD characterization presented in Fig. 5a–d, all samples are well indexed to the cubic garnet phase, and the possible site occupations are shown in Fig. 5e. Since these NIR phosphors are synthesized for the first time, the XRD patterns of typical garnets are provided for references to demonstrate the crystalline of samples. It can be seen that Samples 1~ 6, 7~12, and 13~15 display high phase purity that match well with the standard cards. Slight impurity phases are observed in Samples 2, 3, and 8–11 owing to the presence of very small amounts of impurities from Sc2O3, Y2O3, In2O3, and Al2O3. The successful synthesis of garnet oxides also confirms the scientific validity of the unlabeled dataset created in this work. Figure 6a–c displays the excitation and the photoluminescence spectra for these samples. Two distinct excitation bands covering the wavelength regions of 400~520 nm and 520–720 nm are detected, which are attributed to the 4A2 (4F) → 4T1 (4F) and 4A2 (4F) → 4T2 (4F) transitions of Cr3+ ions, respectively. Excited by 450 nm light, broad NIR emission bands within 600–1100 nm are observed in all the samples. According to the definition in the reported literatures, the EPW for the as-prepared samples is also estimated by the corresponding emission peaks, and the results are shown in Table 1. Obviously, the measured EPWs (ME) are in high accordance with the predicted values (PV) via the model, accompanied by a maximum percent error (PE, PE = ∣ME − PV∣/ME) of only 2.53%. And the PE values for 12 samples are <1% (Fig. 6d). If the parameter PE smaller than 1% is regarded as the credible value, the passing rate would be 80%. The good agreement between ME and PV illustrates that the high accuracy and excellent generalization are achieved in our model.

Table 1 The luminescence performances for the prepared NIR phosphors chosen from the unlabeled samples and the comparison between the measured EPW and the predicted one
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Fig. 5: Characterization of the as-prepared phosphors.
figure 5

a–d The XRD patterns for different samples. e The crystal structures and possible occupations.

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Fig. 6: The validation of model by experiments.
figure 6

a–c The photoluminescence and excitation spectra of the as-prepared samples. d The percent error between the measured and predicted values of EPW for the samples. e The changing trends of the predicted and the measured EPWs for Samples 7–9: Lu2.5Mg0.5In2Ga2.5Ge0.5O12, Samples 10–12: Lu2CaAl2Ga2GeO12, and Samples 13, 14: Y2.5Sr0.5In2Ga2.5Si0.5O12 with different DCs. f The temperature dependence of luminescence intensities normalized according to those at room temperature.

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Samples 7~9 have the same host but different doping concentrations of Cr3+, similar case for Samples 10~12 and Samples 13, 14. It is observed from Fig. 6e that the EPWs for these three groups of samples follow the same pattern, i.e. the EPW gets large with the increase of DC in a certain host matrix. The excitation spectra nearly have no change with varying doping concentrations, indicating a relatively stable crystal field. So, the enhancement of EPW with DC could be attributed to the augment of energy level splitting originated from the interaction between the increased Cr–Cr ion pairs23,33. Meanwhile, this phenomenon agrees well with our priori knowledge and simultaneously illustrates the inherent physical mechanism involved in Cr3+ doped garnet compounds from a ML perspective. Meanwhile, the FWHM and the thermal stability for these samples are also evaluated. As shown in Table 1, the FWHM for most samples has surpassed 130 nm, with a largest value (161 nm) observed in Lu2.5Mg0.5In2Ga2.5Ge0.5O12 (Sample 9), which surpasses most of the reported literatures (see Supplementary Table 1). And the spectral-broadening with DC has been observed, which is mainly due to the enhanced electron–lattice coupling caused by the slight local structure distortion when Cr3+ ions are incorporated into the lattice sites36. In this case, the increase of FWHM with EPW is still maintained with the increment of DC. The photoluminescence spectra are recorded at different temperatures ranging from 303 to 573 K to check the thermal stabilities of these phosphors (see Supplementary Fig. 10), where I423K is used as the reference criterion. Figure 6f depicts the NIR intensities (normalized by the intensities at 303 K) for the samples at different temperatures. Nearly all the samples have thermal stabilities exceeding 60%, and the values of I423K for a third of the samples are larger than 85%. Besides, the FWHM and the thermal stability illustrate an up- and downward trend on the whole, respectively, with the increase of EPW, as summarized in Table 1, which also corresponds well with the priori knowledge. Above results further reflect the rationality of the expanded dataset by the unlabeled samples. Given the accurate model, the intrinsic factors influencing the luminescence of Cr3+ doped garnets can be objectively uncovered in the following study.

The interpretation of model for efficient orientation of target phosphors

While the co-trained model can search the chemical space efficiently and lead us to candidate materials that meet specific requirements, the ML usually behaves like a black box. As a result, it alone cannot provide an intuitive comprehension for the experimental discovery of materials, which is of great importance for promoting the exploration of innovative NIR phosphors. In order to elucidate the underlying mechanisms of our ML model, we conduct the SHAP study upon the chemical descriptors with the objective of extracting the key features that influence the EPW prediction. Moreover, considering the relationship between EPW and IQE (Supplementary Fig. 1a), the interpretation of the model would also be beneficial to understanding the factors affecting the IQE and the expeditious identification of novel materials. Figure 7 presents the SHAP analysis results for the co-training SVR and co-training KRR trained on the respective features, where Fig. 7a, c illustrates the average values of SHAP and Fig. 7b, d ranks the effect of features on the predictions of models according to the SHAP values. If the higher feature values result in more positive, the relevant feature would be positively correlated with the EPW. While, if the higher feature values cause more negative SHAP value, the feature would be inversely correlated with the EPW.

Fig. 7: Interpretable findings.
figure 7

Impact of different features on model predictions for (a, b) co-training SVR and (c, d) co-training KRR models. The left panel represents the mean impact of features, and each point in the right panel represents the impact of a particular sample.

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From Fig. 7, it is evident that the most important feature in both co-training SVR and co-training KRR is ThermCond, which represents a descriptor of the average value for elemental thermal conductivity and is positively correlated with the output of model. Among the candidate elements in garnet, Ca2+, Mg2+, Si4+, Zn2+, In3+, and Ge4+ present much high thermal conductivity. Interestingly, the non-trivalent cations (Ca2+, Mg2+, Zn2+, Si4+, and Ge4+) have been usually used to synthesize NIR phosphors with a large EPW and broad luminescence band36,37. For instance, it has been reported that the occupation of dodecahedral positions by Ca2+ and Mg2+ could lead to weak crystal fields, which is beneficial to broad NIR emissions with large EPWs38,39. Besides, as the octahedral sites are occupied by Mg2+, Zn2+, Zr4+, and Hf4+ ions, the heterovalent substitution would be caused, contributing to large EPWs as well as broad luminescence bands, which indicates relatively weak crystal strength involved in the hosts40,41,42. Furthermore, larger cations occupying the dodecahedron and octahedron would induce a weak crystal field, leading to emissions with larger peak wavelengths43,44,45. This result evidences a higher and more positive SHAP value for the IonRad descriptor (the average value of the ionic radius) in the co-training SVR. As for the co-training KRR, B_1stIonEn, which denotes the mean value of the first ionization energy of element at the octahedral position, also displays much large and many positive SHAP values. It should be mentioned that the top six ions with high first ionization energy values are Zn2+, Si4+, Ge4+, Mg2+, Hf4+, Zr4+, which correspond well with those mentioned in the discussion of ThermCond. So, it could be inferred that the elemental thermal conductivity and first ionization energy would be a pair of interrelated features.

The interpretation of the model implies that garnet oxides devoid of In3+ and other divalent and tetravalent cations, or enriched with low thermal conductivity elements, would be more conducive to the formation of phosphors with high NIR quantum yields, accompanied by relatively small EPW and a narrow emission band. Conversely, samples containing elements with high thermal conductivity are more likely to generate NIR luminescence with a large EPW and a wide band, which would be indicative of a low quantum yield. The recognition of Cr3+ activated NIR phosphors with high performance would be greatly accelerated by such rule, which could be well confirmed from the resulted phosphors demonstrated in Table 1. Clearly, most phosphors containing chemical elements that own high thermal conductivity exhibit larger EPWs (>760 nm) and much broader bands. With the purpose of efficient orientation toward phosphors possessing intense NIR emissions, the compounds containing elements with high thermal conductivity are excluded from the candidates. This exclusion is correct, which can be verified by the comparison of photoluminescence shown in Supplementary Fig. 11. As a result, the searching scope of candidates that may possess unique NIR luminescence properties is rapidly reduced to Samples 4 and 5. And because Sample 5 illustrates much strong NIR emission, the following discussion is focused on this sample.

Figure 8a shows the measured spectra used for estimating the quantum efficiency of Sample 5. Because of the limitation in the detector response, the spectrometer can only provide reasonable spectral data covering the wavelength range of 200~800 nm, through which the IQE of Lu3Y2Ga3O12: (LYGG) Cr3+ is determined to be around 72.5%. In order to acquire the actual IQE, the complete emission spectrum should be consulted46,47, as shown in the inset of Fig. 8a. In this case, the IQE for this sample is finally computed to be about 97.1%. Then, the EQE is estimated to be about 38.8% by taking into account the blue light absorption efficiency of LYGG: Cr3+ (~40.0%). Notably, the LYGG: Cr3+ presents much high IQE and EQE, which have surpassed most reported NIR-emitting phosphors with EPW around 750 nm, as shown in Fig. 8b and Supplementary Table 7. The value of Dq/B is estimated to be ~2.76 (Supplementary Fig. 12), suggesting Cr3+ ions in this sample are located in a strong crystal field. Figure 8c illustrates the corresponding photoluminescence spectra at different temperatures. With the increase of temperature from 303 to 573 K, the redshift of EPW (~14 nm) and broadened FWHM are clearly observed, which are resulted from the decreased crystal field by lattice expansion and the strengthened electron-phonon coupling effect, respectively48,49. Besides, the emission intensity of this sample maintains 86% of that at 303 K when the temperature is increased to 423 K, indicating the excellent thermal stability. The activation energy is evaluated to be about 0.34 eV, as shown in Supplementary Fig. 13. The advantage in thermal stability of LYGG: Cr3+ can also be demonstrated by comparing with other NIR phosphors, as shown in Fig. 8b.

Fig. 8: Photoluminescence properties and pc-LED applications of new NIR-emitting phosphor.
figure 8

a The calculation of IQE and EQE for LYGG: Cr3+. b The comparison of IQE and I423K for Cr3+-doped garnet phosphors with EPW around 740~770 nm (the detailed comparison can be consulted in Supplementary Table 8). c Normalized emission spectra of LYGG: Cr3+ at temperatures ranging from 303 to 573 K. d The fabrication of pc-LED and the luminescence images with and without longpass filter. e The luminescence spectra driven by different currents. f The output power and photoelectric conversion efficiency of the fabricated pc-LED under different currents. The photographs of g hand and h fish taken by the normal and NIR cameras, respectively.

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The extremely high IQE/EQE and good thermal stability of LYGG: Cr3+ suggest its potential in the application of pc-LED. Then, we fabricate a NIR-emitting pc-LED by coating LYGG: Cr3+ phosphors upon a blue chip (450 nm), as presented in Fig. 8d. The original phosphor shows green color, and red light is easily inspected when the LED is turned on without an optical filter. By using a longpass filter, bright crimson can be observed from the pc-LED. Figure 8e illustrates the emission spectra under different driving currents of blue LED. The emissions peaking around 450 nm and 750 nm, which are originated from blue LED chip and LYGG: Cr3+, respectively, are enhanced with the increment of current. As the current is elevated from 20 to 300 mA, the output of NIR light and the total output (NIR + blue) are promoted from 2.96 to 37.8 mW and 3.04 to 39.2 mW, respectively, as shown in Fig. 8f. Details are given in Supplementary Table 8. Meanwhile, both of the conversion efficiencies for blue to NIR light (ηNIR/blue) and the LED input power to NIR light (ηNIR/input) are decreased with the input current, which is mainly ascribed to the thermal issues commonly involved in the LED device. Actually, the working essence for pc-LED is an electro-optical-optical energy conversion process. And the pc-LED performance would be mainly governed by both the coated phosphors and the properties of substrate LED, including its own electro-optical conversion efficiency and electrothermal issue. Meanwhile, the manufacturing technology, for example, the type of gum and the mixing ratio between gum and phosphors, also significantly affects the behavior of pc-LED. Here, a commercial blue chip is selected as the substrate, with the inherent energy conversion efficiency in a normal level. And the manufacturing process of pc-LED has not been optimized by us. So, it can be anticipated that by using a blue LED with fine lighting properties and upgrading the fabrication technique, the pc-LED performance could be further ameliorated.

The fabricated NIR pc-LED emits NIR light in a broad wavelength range (600~1100 nm). This wavelength region is especially meaningful for the bio-imaging due to the deep penetrating capability through bio-tissues. Figure 8g shows the image of author’s hand taken with a normal camera under the lighting of fluorescence lamp. Meanwhile, the hand images are also captured by a NIR camera under the illumination of pc-LED hold within the palm. Compared with the conventional photograph, the blood vessels of hand are clearly distinguished under the lighting of pc-LED. It should be mentioned that there is a small bruise on the back of hand, which is difficult to be observed by naked eye but can be evidently identified by the NIR image. The pc-LED is also utilized to inspect the biological structure of ornamental fish, as presented in Fig. 8h. The fishbone is clearly discerned through the NIR photograph. The above results demonstrate that the derived NIR-emitting phosphors indeed exhibit outstanding properties in NIR pc-LEDs.

Discussion

For boosting the Cr3+ activated NIR garnet phosphors that are urgently demanded in the NIR-emitting pc-LEDs, we propose a semi-supervised co-training ML model in the case of small datasets. The EPW is employed as the single label, not only because it is closely relevant with the other luminescence characteristics (e.g. the quantum efficiency, the FWHM, the thermal stability) of phosphors and can be widely available in the literatures, but also, more importantly, due to the fact that the EPW is just determined by the chemical component of hosts, more objective and robust than other emission characteristics. It is the first time to our best knowledge that the co-training procedure is introduced into the training of ML models for luminescent materials. The co-training technique, which effectively expands the database of Cr3+-doped garnet compounds and solves the problem originated from the insufficient data of luminescent materials, greatly improves the generalization ability of model. The model is finally validated by the experiments. The results show that all the randomly selected garnet phosphors that have never been identified before can be successfully synthesized, and the predicted EPWs by the model are in good agreement with the measured values.

The model is then interpreted by the SHAP analysis, which can succinctly elucidate the impact of input features and avianize the inherent black box characteristic of ML. Our study on the importance of features provides an important guidance for designing NIR garnet phosphors: chemical elements with high thermal conductivity would result in large EPW, while the garnets consisting of elements with low thermal conductivity generally produce relatively short EPW. The interpretation agrees well with the conclusion condensed from previous literatures. The SHAP analysis illustrates that the balancing of chemical components is required for efficient orientation of unique NIR phosphors. This opinion finally facilitates us to develop a new Cr3+ doped garnet oxide with excellent properties, including high IQE (97.1%) and EQE (38.8%), broad emission band (600~1100 nm), and good thermal stability (luminescence intensity maintains 86% at 423 K compared with that at 303 K). The phosphors are subsequently applied to fabricate a NIR pc-LED, which exhibits great potential in bio-imaging. As the increase of reported works, more information about Cr3+ doped garnet oxides could be available. It can be anticipated that the incorporation of additional luminescence-related parameters, such as luminescence lifetime, excitation wavelength, bandgap, and structural rigidity, will further enhance the performance of the model. And the resulting model interpretation could generate additional insights that can be used to accelerate the development of unique NIR luminescent phosphors. This topic will offer a promising avenue for future research.

Methods

Materials synthesis

The phosphors discussed in this study are synthesized by traditional high-temperature solid-state method. CaCO3 (99.99%), SrCO3 (99.95%), Sc2O3 (99.9%), Y2O3 (99.9%), La2O3 (99.99%), Lu2O3 (99.9%), In2O3 (99.99%), Gd2O3 (99.99%), MgO (99.99%), Al2O3 (AR), GeO2 (99.99%), Ga2O3 (99.99%), SiO2 (AR), and Cr2O3 (99.95%) are used as raw materials. The raw materials are precisely weighed according to the stoichiometry, and 3 wt% H3BO3 is added as the flux. After thorough grinding, the raw materials are transferred into a Si-Mo rod furnace, sintered at 1550 °C for 6 h in the air atmosphere. The resulting phosphors are pressed into a thin disk for characterization.

LED fabrication

The Lu3Y2Ga3O12: Cr3+ NIR-emitting phosphor is mixed with epoxy resins A and B (mass ratio of 1:1) and then coated on a 450 nm light-emitting diode.

Characterization

X-ray diffraction patterns are measured by a Shimadzu XRD-6100 X-ray diffractometer. The diffraction angle is set at the range of 10~90° with a scanning rate of 15°/min. The excitation spectra are measured by a Shimadzu RF-6000 fluorescence spectrophotometer. The emission spectra are recorded by a fiber spectrometer (Nova, Idearoptics), where a 450 nm laser diode (Thorlabs L450P1600MM) is used as the excitation source. The temperature dependence of luminescence spectra is measured with a heating stage (Instec HCP621G). The quantum efficiency is measured by a spectrometer (FLS-1000, Edinburgh Instruments). The luminescence properties of the as-fabricated pc-LED are studied based on the OHSP-350M Multi-function Luminous Flux Tester.