Prediction of metastable energy level distribution of D³⁺ (D = Cr and Fe) doped phosphors based on machine learning

Abstract

The energy level transitions in phosphor materials critically determine their emission characteristics, and accurately predicting the energy level distribution of ions in these materials is critical for determining their luminescence behavior. However, reliance on multiple experimental methods to determine energy level distributions is inefficient, consuming both time and resources. There is an urgent need for a rapid and accurate method to predict the energy level distribution of ions in crystals. This paper employs regression models based on machine learning to propose a method for predicting the energy level distribution rules of Cr³⁺ and Fe³⁺ in various doped crystals, and identifies the position and distribution patterns of these levels in different doped crystals, as well as their impact on luminescence characteristics. Furthermore, a dataset detailing the energy level distributions of Cr³⁺ and Fe³⁺ doped into different phosphor materials was established. Eight machine learning regression algorithms were selected for model construction, and a comprehensive evaluation and comparison of these algorithms were conducted. The results demonstrate that robust regression delivers the best overall performance. Using trained models, predictions were made for the ²E and ⁴T₁ energy levels in new Cr³⁺ and Fe³⁺ doped phosphor materials. The prediction errors of the optimal algorithms for these materials were all in the range of about 1%, with the best prediction error at just 0.0056%. This study introduces an innovative approach for predicting and optimizing the energy level structures and luminescence properties of phosphor materials.

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1. Introduction

Transition metal ions, particularly Cr³⁺ and Fe³⁺, are crucial for the formation of optical properties in doped phosphor materials.¹ Cr³⁺-activated near-infrared phosphor materials can achieve both narrow-band and broad-band emissions, finding applications across various scenarios.² The d–d transitions of Cr³⁺ in crystals can generate broad photoluminescence from far-red (spin-forbidden ²E–⁴A₂) to near-infrared (spin-allowed ⁴T₂–⁴A₂) regions, making Cr³⁺ one of the ideal activators for near-infrared light emission.³ Currently, there are numerous phosphor materials activated by Cr³⁺ that exhibit excellent properties, such as Mg₄Ta₂O₉:Cr³⁺ with its long peak wavelength and broad-band near-infrared emission,⁴ and GaTaO₄:Cr³⁺ which produces broad-band near-infrared emission under 460 nm blue light excitation.⁵ Besides Cr³⁺, Fe³⁺ also holds potential as a dopant. Based on its half-filled electronic configuration, Fe³⁺ ions can emit near-infrared light through ⁴T₁–⁶A₁ transitions.⁶ The abundance and non-toxic nature of Fe³⁺ ions make them an environmentally friendly doping option.⁷ The energy states of Cr³⁺ and Fe³⁺ ions in doped crystals, which involve specific electronic configurations and transitions, determine the material's optical properties, such as absorption and emission spectra.⁸ Therefore, studying the closely related energy levels of ions and the luminescence behavior of phosphor materials, such as the ²E energy level of Cr³⁺ and the ⁴T₁ energy level of Fe³⁺, is essential, which not only elucidates the luminescence behavior of these materials, but also plays a key role in optimizing their functional performance in various applications. However, determining these specific energy levels through traditional experimental methods is challenging, often requiring significant time, resources, and complex procedures, thus presenting challenges in the research and optimization of Cr³⁺ and Fe³⁺-doped phosphor materials.⁹

Benefiting from advancements in computer science, machine learning, as a data-driven approach, possesses the advantage of balancing development efficiency and cost. It is emerging as a novel paradigm in materials science research.¹⁰ The motivation for integrating machine learning algorithms into the study of doped phosphor materials stems from the vast amount of data generated over the past few decades related to these materials. Despite this abundance of data, certain fundamental aspects of the underlying physics of phosphors remain unclear. The data-driven nature of machine learning is particularly advantageous for addressing problems where the fundamental physics is not fully understood yet rich data are available. Consequently, this approach can accelerate the research and design of phosphor materials.¹¹ Li et al.¹² utilized a data-driven approach combining high-throughput DFT calculations and data mining of experimentally validated data to discover a monophase white-light phosphor (Sr₂AlSi₂O₆N:Eu²⁺), which exhibited an exceptional emission bandwidth. Jiang et al.¹³ constructed wavelength and thermal stability machine learning models by extracting relevant features, and through an active learning-based multi-objective optimization approach, identified garnet-type phosphors Lu_1.5Sr_1.5Al_3.5Si_1.5O₁₂:Ce that demonstrated excellent thermal stability at targeted wavelengths. Jang et al.¹⁴ created a high-quality dataset of inorganic phosphors and their optical properties using machine learning methods to predict optical characteristics such as the maximum photoluminescence (PL) emission wavelength and PL decay time, achieving significant predictive success. Kim et al.¹⁵ integrated machine learning models with an experimental dataset of synthesized Ba_0.9−xSr_xMgAl₁₀O₁₇:Eu²⁺, establishing a relationship between photoluminescence (PL) and crystal properties. However, there are currently few targeted studies on the metastable energy levels ²E and ⁴T₁, which are closely related to the luminescence properties of Cr³⁺ and Fe³⁺ ions in doped crystals, and there is a lack of a method that can effectively predict the energy levels of these ions in doped crystals without the need for experimental measurements, and at very low time and resource costs.

Therefore, this paper aims to use machine learning methods to predict the ²E and ⁴T₁ energy levels of Cr³⁺ and Fe³⁺ doped in various phosphor materials, which are closely related to the luminescence properties of the materials. Based on a detailed study of the luminescence behavior of Cr³⁺ and Fe³⁺ doped phosphor materials and factors influencing the transitions of these ions, this work has established a dataset necessary for the model through an in-depth literature review and rigorous data extraction methods. The dataset includes crystal field parameters D_q, Racah parameters B and C, and the ²E and ⁴T₁ energy levels of Cr³⁺ and Fe³⁺. Based on this, eight machine learning regression algorithms (linear regression, robust regression, lasso regression, ridge regression, elastic net, decision tree, random forest, and gradient boosting) were selected to build models to predict the metastable energy levels ²E and ⁴T₁ of Cr³⁺ and Fe³⁺ in doped crystals. A comprehensive evaluation of these algorithms’ predictions on known data was conducted. Subsequently, some new phosphor materials doped with Cr³⁺ and Fe³⁺ were selected, and the trained models were used to predict the ²E and ⁴T₁ energy levels of Cr³⁺ and Fe³⁺ in these materials. Comparisons with actual values were made to further verify the effectiveness of the proposed prediction method.

2. Method

2.1. Machine learning in materials science

Machine learning aims to identify patterns in given data, subsequently making predictions or decisions. If we define a model function F(x,ω), where x is the input variable and ω represents the model parameters, the goal is to approximate an unknown true function g(x) that directly links input to output. During training, the model's performance is measured using a loss function L(y;F(x,ω)), with a mean squared error being a common choice, i.e., L(y;F(x,ω)) = ||y − F(x,ω)||². The core of machine learning is to optimize the parameters ω to minimize the loss, thereby enhancing the model's predictive ability on unknown data. Based on the learning method used during training, machine learning is primarily divided into supervised learning, unsupervised learning, and reinforcement learning.¹⁶ Under this classification, typical machine learning models are shown in Fig. 1. The training data for supervised learning contain input samples and their corresponding output labels, and the goal is to use the training data and data feedback to learn the relationship between a given input and a given output. Unsupervised learning deals with unlabeled data and the goal is to discover hidden structures or patterns in the data. Reinforcement learning is a learning model of the decision-making process in which an agent learns how to act to maximize some cumulative reward by interacting with the environment. This research predominantly utilizes supervised learning, where the training data include input samples and their corresponding output labels, specifically the phosphor material parameters D_q, B, C, and the specific energy levels of ions in a doped environment; for Cr³⁺, this is the ²E level, and for Fe³⁺, the ⁴T₁ level. The objective is to use the training data and feedback to learn the relationships between given inputs and outputs, thereby enabling the prediction of the ²E and ⁴T₁ energy levels of doped ions.

Fig. 1 Common algorithms in machine learning classification and categories.

This study evaluates various machine learning tools that phosphor material researchers can utilize. The tools are categorized into several types: machine learning/deep learning (ML/DL) frameworks, comprehensive ML libraries, cloud computing platforms, data science and ML platforms, and scientific computing software. Additionally, for each category, several popular options commonly used by researchers are listed, and the characteristics or main functions of each software or tool are succinctly and clearly described, as shown in Table 1. This section enables researchers to more conveniently select the tools that best suit their specific research needs, thereby effectively advancing scientific research in the field of phosphor materials.

Table 1 Classification and characteristic comparison of commonly used machine learning software or tools for phosphor materials research

Categories	Software/tools	Characteristics
ML/DL frameworks	PyTorch^17	Deeply integrated with Python
		Dynamic computation graph and powerful GPU support
	TensorFlow^18	Distributed computing and GPU acceleration
		Rich API multi-platform
	Keras^19	The architecture is modular and scalable
		Can run on multiple underlying frameworks
Comprehensive ML libraries	Scikit-learn^20	Open source ML library, simple, and effective
		A wide range of classic ML algorithms
	Weka^21	Open source ML and data mining library
		Provide user-friendly GUI
Cloud computing platforms	Microsoft Azure ML^22	Full-process services from data preparation to model deployment
	Google Cloud AI Platform^23	One-stop model training and tuning
		Integrate Google AI technology
Data science and ML platforms	RapidMiner^24	Integrated data science workflow designer
		Supports a wide range of ML/DL algorithms
	H2O^25	Open source ML platform that supports a wide range of statistical and ML algorithms
	Kaggle^26	Data set access and sharing, and open community
Scientific computing software	MATLAB^27	Provide ML/DL toolbox
		Strong mathematics and visualization skills

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2.2. Analysis of the luminescence behaviour of Cr³⁺ and Fe³⁺ doped phosphor materials and the main factors affecting ion transitions

In compounds doped with Cr³⁺ ions, these ions typically manifest a 3d³ electronic configuration, giving rise to doublet states such as ²E and quartet states such as ⁴A₂—the ground state of energy.²⁸ Under the influence of a strong crystal field, the ²E level ascends to the position of the lowest excited state, undergoing transitions to ⁴A₂via spin-forbidden processes, thereby emitting far-red light. The emission spectrum is characterized by a narrow-band peak, observable in materials such as La₂ZnTiO₆:Cr³⁺ and GdAlO₃:Cr³⁺.²⁹ Conversely, in a weak crystal field, the ⁴T₂ state emerges as the lowest excited state, facilitating spin-allowed transitions to the ground state and resulting in a broadband near-infrared emission spectrum, as seen in LiInSi₂O₆:Cr³⁺ and ScF₃:Cr³⁺. In environments of medium crystal field strength, both ²E–⁴A₂ and ⁴T₂–⁴A₂ transitions occur concurrently in the Cr³⁺ electronic structure, with the emission spectrum displaying both peak and broadband emissions, exemplified by Sc₄Zr₃O₁₂:Cr³⁺ and CaSc₂O₄:Cr³⁺.³⁰ This investigation predominantly explores the ²E–⁴A₂ transition of Cr³⁺ ions, known for producing distinct line emissions (R line). In contrast, Fe³⁺ ions exhibit a 3d⁵ electronic configuration with a solitary sextet state ⁶A₁, serving as the ground energy state. The excited states predominantly comprise quartet states (⁴T₁, ⁴T₂, etc.), where transitions are also spin-forbidden. Upon doping into various matrices, Fe³⁺ may exhibit near-infrared photoluminescence due to ⁴T₁–⁶A₁ transition, underscoring its significant role in the absorption and emission processes of light in Fe³⁺.³¹Fig. 2 illustrates the schematic diagram of the ²E and ⁴T₁ energy level transitions in Cr³⁺ and Fe³⁺ doped crystals, highlighting the ²E–⁴A₂ and ⁴T₁–⁶A₁ transitions that are instrumental to the luminescence properties of these doped phosphors. Since the transitions of Cr ions are d–d transitions occurring in the d orbitals, the effect of electron cloud rearrangement on energy level transitions is minimal, and the influence of the crystal field dominates the transitions of Cr³⁺ ions.³² The Tanabe–Sugano diagram³³ takes into account both the crystal field splitting D_q and the interactions between electrons (Racah parameters) affecting the transitions of Cr³⁺ energy levels. The crystal parameter D_q describes the splitting effect of the crystal field on electron states, while Racah parameters B and C represent the Coulomb interactions between electrons, playing a critical role in determining energy levels and transitions. The transitions of Fe³⁺ ions are also d–d transitions, with D_q, B, and C as the main factors influencing their energy level transitions.³⁴ Based on this analysis, this study aims to explore, through machine learning methods, the complex relationships between parameters D_q, B, and C and the ²E energy levels of Cr³⁺ and the ⁴T₁ energy levels of Fe³⁺, thereby predicting the energy levels of ²E and ⁴T₁.

Fig. 2 Schematic diagram of ²E and ⁴T₁ energy level transitions doped with Cr³⁺ and Fe³⁺ in crystals.

2.3 Analysis of algorithms

This paper aims to use machine learning algorithms to predict the ²E and ⁴T₁ energy levels of Cr³⁺ and Fe³⁺ doped in various phosphor materials. Specifically, this paper attempts to explore the complex relationship between the crystal parameter D_q, Racah parameters B and C, and the distribution of ion energy levels, so as to realize the prediction of ion energy levels in doped crystals. Regression models are supervised learning methods used to predict continuous value outputs in machine learning, mainly by analyzing the relationship between input features and the target variable. The goal of this model is to establish a mathematical equation that can describe the mapping relationship between inputs and outputs as accurately as possible. Therefore, in this paper, the regression models are chosen for the prediction of energy levels. This study selected eight typical regression algorithms for exploration, including linear regression, robust regression, lasso regression, ridge regression, elastic net, decision tree, random forest, and gradient boosting. In the field of machine learning, choosing the right algorithm is critical to the performance and accuracy of the model. These eight algorithms are all more classical and representative of regression algorithms, covering a wide range of techniques from simple to complex, and have the ability to handle different types of data, thus providing a comprehensive and effective analysis tool for research. The article conducts in-depth research and discussion on the basic principles, mathematical models, and applicability of these algorithms, and further analyzes and summarizes these models in this research context based on the final experimental results, with specific analysis content as follows.

The primary objective of the linear regression algorithm³⁵ is to determine the optimal weights of the independent variables to minimize the error between the model's predicted values and the actual observed values. The linear regression model is based on the fundamental assumption of a linear relationship, meaning that the dependent variable can be accurately predicted by a linear combination of the independent variables. This can be initially assessed through a scatter plot to judge whether a linear relationship exists between the variables. It is suitable for basic regression that does not focus on outliers, as shown in the following form:

where y_i represents the observed values, β₀ is the intercept, β_j are the coefficients, and x_ij are the input features.

Robust regression³⁶ is a method specifically designed to handle outliers in regression analysis. It employs robust loss functions to minimize the impact of outliers, thereby enhancing the robustness of the model and making it more effective in the presence of irregularities in the dataset. The mathematical model includes a robust loss function L(u), such as the Huber loss, which is formulated as follows:

Lasso regression³⁷ incorporates an L1 regularization term into traditional regression models. While fitting a generalized linear model and minimizing the sum of squared residuals, it constrains the complexity of the model through a penalty term. This approach facilitates variable selection and regularization, enhancing the model's predictive accuracy and interpretability. The formula is as follows:

where λ is the regularization parameter.

Ridge regression³⁸ includes an L2 regularization term, which helps to suppress the size of the model coefficients and thus prevent overfitting. It minimizes the sum of squared residuals and imposes a penalty on large coefficients to balance bias and variance. The formula is as follows:

Elastic net³⁹ combines the features of ridge regression (L2 regularization) and lasso regression (L1 regularization). It is particularly suited for scenarios with multiple correlated independent variables. Adjusting the balance between the two types of regularization effectively controls the complexity and sparsity of the model, thereby optimizing the model's predictive performance and interpretability. The formula is as follows:

where α is the mixing parameter between L1 and L2 regularization.

In decision tree regression,⁴⁰ each internal node represents a decision rule on a feature, which is based on maximizing the homogeneity (or minimizing the heterogeneity) of the data after each node split. Specifically, each split is chosen based on the feature that most reduces the variance of the target variable. The criterion typically used to select the best split is the mean squared error (MSE), which is calculated as follows:

where y_i is the actual value of the i-th observation in the node, ŷ is the average value of all observations in the same node, and N is the number of observations in the node. During the construction of the decision tree, the algorithm progressively splits the data until the stopping criteria are satisfied.

Random forest⁴¹ is an ensemble of decision trees, each built on a bootstrap sample of the original dataset. At each split point, a subset of features is randomly selected to determine the best split. The final prediction of the model is obtained by averaging the predictions from all the trees, which is expressed as follows:

where B represents the number of trees, and f̂_b(x) is the prediction result from the b-th tree. Random forest reduces the model's variance by increasing the independence between decision trees, effectively preventing overfitting. It is suitable for handling regression tasks with high-dimensional features and complex data structures.

Gradient boosting regression⁴² optimizes the overall prediction accuracy by iteratively adding base models (typically weak learners, such as simple linear regression models or decision trees) to minimize the loss function. Specifically, the model update formula at each step can be expressed as:

where F_m(x) is the model after the m-th iteration, F_m−1(x) is the model from the previous iteration, and h(x_i) is a base model selected from the function space H that maximally reduces the loss function. This method effectively fits complex data structures by progressively reducing prediction errors. In this study, decision trees are chosen as the weak learners for gradient boosting.

2.4. Experimental datasets and details

This article conducts a series of precise search strategies, providing a comprehensive literature review of relevant research fields. Specifically, the study focuses on the crystal field parameter D_q and the Racah parameters B and C, which play crucial roles in understanding and predicting the electronic energy levels of ions in doped phosphors. During data collection, we systematically extracted data from renowned academic databases such as Scopus, Web of Science, ScienceDirect, and IEEE Xplore, which is fundamental for constructing machine learning models. The dataset for Cr-doped samples is shown in Table 2,^43–71 and the dataset for Fe-doped samples is presented in Table 3.^72–96 These data, after preprocessing, serve as input features for the regression models listed in Section 2.3, enabling the prediction of Cr³⁺ and Fe³⁺ behavior in different environments.

Table 2 Spectral parameters of Cr³⁺ doping in different environments (unit: cm⁻¹)

Crystal	D q	B	C	^2E	Crystal	D q	B	C	^2E
KAl(MoO4)2	1494.8	585.5	3049	13 517	CdO·P2O5	1545	700	3230	14 815
Y3Ga5O12	1626	645	2950	14 472	PbO–Ga2O3–P2O5	1557	660	3061	14 045
LiAl5O8	1779	741	2875	14 085	Pb3O4–ZnO–P2O5	1523	709	3232	14 864
Ba2Mg(BO3)2	1967	718	2756	14 327	MgO	1615	586	3249	14 164
NaAl(WO4)2	1548	615.6	3083	13 822	KZnF3	1323	731	3437	15 523
YAl3(BO3)4	1680	672	3218	14 663	LiNbO3	1355	644	3026	13 772
NaMg3Al(MoO4)5	1440	676	2945	13 755	MgSrAl10O17	1549	677	3179	14 506
LiNbO3: ZnO	1342	646	3022	13 755	MgAl2O4	1814.9	544.7	3224.9	14 577
LiBP	1577	707	3146	14 600	SiO2:B2O3:Na2O:Al2O3:CaO:ZrO2	1574	792.2	3005	14 684
BaAl2O4	1811	533	2862	14 180	GdScO3	1553	574	3211	13 947
Sc2(MoO4)3	1408	608	3054	13 643	LiBaF3	1546	702	3300	15 039
LaSc3(BO3)4	1529	675	3448	14 620	LiBaF3	1509	701	3292	15 006
YAl(BO3)4	1680	672	3225	14 599	NaNH4SO4·2H2O	1750	735	3220	15 015
GdAl(BO3)4	1695	673	3380	14 599	(NH4)2Mg(SO4)2·6H2O	2043	676	3371	13 923
Silicate glass	1562	853	2898	14 810	(CH3)4NCdCl3	2043	722	2845	13 602
CdO	1540	619	3327	14 595	YCrO3	1656.6	542.9	2962.3	13 643
SbPO	1560	629	3171	14 185	YAl3(BO3)4	1662	684	3194	14 636

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Table 3 Spectral parameters of Fe³⁺ doping in different environments (unit: cm⁻¹)

Crystal	D q	B	C	^4T1	Crystal	D q	B	C	^4T1
KTaO3	−640	640	2500	14 569	ZnCdO	920	840	2500	14 595
Mg2Al4Si5O18–Al2O3–MgAl2O4–SiO2	816	538	2944	14 104	Al2O3	1527	650	3160	9450
SrB4O7	790	700	3000	17 852	ZnGa2O4	1560	600	3100	8400
PVA capped ZnSe nanoparticles	720	720	2500	16 389	Li0.5Ga2.5O4	1550	560	3280	9200
CdO nanopowders	920	690	2750	14 594	ZnS	900	720	2700	14 788
K2SnCl4·H2O	820	831	2198	13 986	LiGa5O8	906	594	3737	18 349
Ca3Ga2Sn3O12	941	670	3047	14 749	LiAl5O8	952	638	3868	19 084
Ca8Mg(SiO4)4Cl2	667	517	3267	17 331	[NH4]2[Mg(H2O)6](SO4)2	740	757	2381	15 267
LiGa5O8	700	565	3000	15 790	Humite	760	690	2769	15 240
LiAl5O8 (ordered)	800	644	2960	15 255	NH4Cl	675	645	2838	16 291
β-LiAlO2	883	630	3000	14 750	Sr(NO3)2	1450	934	2059	12 024
β-LiGaO2	961	713	2995	14 970	K2SO4-ZnSO4	1040	700	2800	14 385
KAlO2	771	593	3074	16 030	Cadmium borosulphate glass	1010	707	2778	14 081
CsAlO2	680	523	3152	16 360	Y3Fe5O12	1336	783	2928	11 357
β-BaB2O4	915	680	2800	14 702	Jadeites	1290	566	3470	12 563

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After collecting and integrating data from relevant databases, a dataset suitable for machine learning models was constructed through a series of data preprocessing steps. In the data preprocessing phase, this study employed feature standardization techniques, specifically Z-score normalization. This method standardizes data features by adjusting their mean to 0 and standard deviation to 1. The core formula is as follows:

Z = (X − μ)/σ (9)

where X represents the original data, μ is the sample mean, and σ is the sample standard deviation. The purpose of this standardization is that many machine learning algorithms, such as linear regression, typically operate on the premise that all features should be centered around zero and have similar variances. Additionally, standardization helps reduce bias caused by differences in feature scales, ensuring that each feature is treated equally in terms of importance during model training.

This article predicts the ²E and ⁴T₁ energy levels of Cr³⁺ and Fe³⁺ doped phosphor materials using machine learning methods, with the experimental steps detailed in Fig. 3. It is particularly emphasized that appropriate division of the dataset is crucial for assessing the model's performance on unknown data, which directly relates to the model's generalization ability. This study employed a 7 : 3 ratio to split the dataset into training and testing sets. Specifically, the training set comprises the majority of the data (70%), where the model learns to recognize or predict patterns, thus adjusting the model parameters. For the Cr³⁺-doped phosphor dataset, 23 samples were allocated to the training set, and for the Fe³⁺-doped dataset, 21 samples were designated for training. The remaining 30% of the data makes up the testing set, and is used to evaluate the model's performance on unknown data. The testing set contains 11 samples for the Cr³⁺-doped dataset and 9 samples for the Fe³⁺-doped dataset.

Fig. 3 Flow chart and step analysis of machine learning to predict the metastable energy level model of phosphor materials.

3. Results and discussion

Building on the methodologies described, this paper employs machine learning regression algorithms to predict the ²E and ⁴T₁ energy levels of Cr³⁺ and Fe³⁺ doped in phosphor materials. For this purpose, the study developed an ensemble model featuring eight different machine learning regression algorithms, allowing for the selection of the most suitable algorithm based on specific requirements. After prediction, the model outputs a predicted vs. actual plot based on the training and testing sets, and integrates an error distribution histogram into this plot. These visualizations facilitate a comprehensive assessment of model performance and aid in diagnosing potential issues. Additionally, the model provides the calculated parameter coefficients and performance evaluation metrics for the selected regression algorithms. Overall, the model developed in this paper effectively evaluates the performance of each algorithm in the scope of this study.

3.1 Comparative analysis

In this article, the established model is used to predict the doping of Cr³⁺ and Fe³⁺, specifically focusing on the prediction of the ²E energy level for Cr³⁺ doping. The predicted vs. actual plot for the ²E energy level output is shown in Fig. 4.

Fig. 4 Different regression models based on training sets and test sets predicted vs. actual values of Cr³⁺ doped ²E energy (cm⁻¹) scatter plots.

Fig. 4 presents a comparative analysis of the actual and predicted values of the ²E energy level (cm⁻¹) for Cr³⁺ doping, based on the training and testing datasets. The analysis utilizes eight different regression models: linear regression, robust regression, lasso regression, ridge regression, elastic net, decision tree, random forest, and gradient boosting. Additionally, a small error distribution histogram is embedded in Fig. 4. In the main chart, the x-axis represents the actual values of the ²E energy, while the y-axis displays the predicted values generated by the models, with different colors distinguishing the data points from the training set (purple squares) and the testing set (red squares). The black diagonal dashed line in the chart indicates the ideal prediction outcome, where the model's predicted values equal the actual values. The inset on the right side of the figure shows the distribution of relative errors for the testing set, with the horizontal axis representing the percentage of relative error and the vertical axis showing the probability density of errors.

Table 4 provides the coefficients of the parameters D_q, B, and C for Cr³⁺ doping, calculated using the training dataset across eight regression models. The larger the magnitude of the coefficients, the more significant the variable's impact on the predictions. Notably, the decision tree, random forest, and gradient boosting models do not provide traditional coefficients but offer feature importance scores instead. These scores reflect the contribution of each feature to the model's predictive outcomes.

Table 4 Coefficients calculated using 8 regression models in the case of Cr³⁺ doping

Parameter	Linear regression	Robust regression	Lasso regression	Ridge regression	Elastic net	Decision tree	Random forest	Gradient boosting
D q	−52.48	89.52	−55.89	−25.21	−2.72	0.40	0.32	0.26
B	152.13	299.86	242.71	232.86	143.18	0.15	0.35	0.35
C	269.00	322.10	203.99	192.39	156.62	0.45	0.33	0.39

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Additionally, the model provides common metrics for evaluating regression model performance, including mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), and the coefficient of determination (R²). The specific formulas for these coefficients are shown in Table 5. These metrics are essential for assessing the accuracy and reliability of the regression models used in the study.

Table 5 Calculation formula for model evaluation indicators

Metrics	Calculation formula
MAE
MSE
RMSE
R ^2

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where y_i represents the actual values, and ŷ_i are the predicted values. For the first three metrics—MAE, MSE, and RMSE—the lower the values, the better the performance of the model. Conversely, for R², values closer to 1 indicate a stronger explanatory power of the model. The specific values of these performance metrics for the Cr³⁺ doped model are presented in Table 6. These metrics quantify the model's performance in terms of accuracy and reliability.

Table 6 Eight regression model evaluation index values under Cr³⁺ doping

Model	MAE	MSE	RMSE	R ^2
Linear regressing	306.93	109 164.08	330.40	0.57
Robust regression	149.80	33 603.59	183.31	0.82
Lasso regression	298.28	110 731.08	332.76	0.59
Ridge regression	286.90	114 974.23	339.08	0.58
Elastic net	359.05	164 626.27	405.74	0.41
Decision tree	207.25	93 820.50	306.30	0.56
Random forest	295.30	115 887.01	340.42	0.64
Gradient boosting	244.91	104 747.35	323.65	0.57

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Table 6 clearly shows that robust regression significantly outperforms other algorithms in predicting Cr³⁺ doping. It achieves the lowest mean absolute error (MAE) at 149.80, indicating small prediction errors and excellent predictive performance. Its R² value of 0.82 demonstrates that the model captures the primary trends in the data, reflecting its robustness and good fit. This may be due to robust regression's use of methods that minimize the impact of outliers, allowing the model to perform better with noisy data. In this task, linear regression has an MAE of 306.91 and an R² of 0.57, indicating a moderate fit to the data. Linear regression assumes a linear relationship between features and the target variable, which can be limiting if the data contain non-linear relationships or outliers. Lasso regression does not stand out, with an MAE of 298.28 and an R² value of 0.59. Lasso employs L1 regularization for feature selection, but the results suggest that this has limited effectiveness in this task. Ridge regression performs similarly to linear and lasso regression, with an MAE of 286.90 and an R² of 0.58. It uses L2 regularization to reduce multicollinearity issues, smoothing weights to prevent overfitting, yet the performance improvement in this task is limited. Elastic net exhibits the largest error with an MAE of 359.05 and the lowest R² at 0.41, indicating that the model struggles to handle complex relationships in the data effectively. Elastic net combines L1 and L2 regularization but performs less effectively than lasso or ridge regression separately, possibly due to an inadequate balance between the L1 and L2 regularization terms. The decision tree has a relatively low MAE of 207.25 and an R² value of 0.56. Decision trees capture nonlinear relationships in data through a tree-like structure and, while they can achieve low errors, they are prone to overfitting. Random forest, which integrates multiple decision trees to reduce overfitting and enhance model robustness, achieves an MAE of 295.30 and a relatively higher R² of 0.64, reflecting a better explanatory capability of the model. Gradient boosting, which incrementally adds weak learners to minimize errors, has an MAE of 244.91 and an R² value of 0.57. This suggests that while the model reduces errors, its explanatory power could be further improved.

Similarly, Fig. 5 displays a comparative analysis between the actual and predicted values of the ⁴T₁ energy level (cm⁻¹) for Fe³⁺ doping, based on the training and testing datasets. This visualization aids in assessing the accuracy and performance of the predictive models used in this context.

Fig. 5 Different regression models based on training sets and test sets predicted vs. actual values of Fe³⁺ doped ⁴T₁ energy (cm⁻¹) scatter plots.

Table 7 provides the coefficients (D_q, B, and C) calculated based on the training dataset for Fe³⁺ doping using eight different regression models. Table 8 displays the specific values of the performance evaluation metrics for models applied to Fe³⁺ doping. These tables offer critical insights into the effectiveness and predictive capabilities of the regression models under different doping conditions.

Table 7 Coefficients calculated by 8 regression models in the case of Fe³⁺ doping

Parameter	Linear regression	Robust regression	Lasso regression	Ridge regression	Elastic net	Decision tree	Random forest	Gradient boosting
Dq	−1845.00	−2710.05	−2174.64	−1685.22	−904.81	0.79	0.83	0.66
B	1015.04	1171.94	1492.22	845.08	333.58	0.04	0.05	0.03
C	1573.03	1748.95	1871.93	1273.70	462.92	0.17	0.12	0.31

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Table 8 Eight regression model evaluation index values under Fe³⁺ doping

Model	MAE	MSE	RMSE	R ^2
Linear regression	1044.85	1839 904.42	1356.43	0.67
Robust regression	771.99	1220 312.12	1104.68	0.81
Lasso regression	906.70	1552 565.42	1246.02	0.58
Ridge regression	994.45	1973 838.90	1404.93	0.69
Elastic net	1760.25	4318 256.00	2078.04	0.39
Decision tree	1083.89	1815 872.78	1347.54	0.71
Random forest	737.16	1100 334.25	1048.97	0.72
Gradient boosting	1098.88	1696 438.16	1302.47	0.82

---

The data from Table 8 indicate that compared to the Cr³⁺ doping scenario, the MAE significantly increased across all algorithms for Fe³⁺ doping, but some models were better at capturing the overall trends and complex relationships in the data, thus improving the R² values. This reflects that model selection and optimization strategies need to vary based on different data characteristics.

For linear regression in the Fe³⁺ doping context, the MAE was notably high at 1044.85, suggesting a large error, yet the R² value improved to 0.67, indicating a better fit than in some other models. Robust regression continued to perform well with a lower MAE of 771.99 and an excellent R² of 0.81, demonstrating its robustness. Lasso regression still showed poor performance with an MAE of 906.7 and an R² value of 0.58. Similar to linear regression, ridge regression saw some improvement in R² to 0.69, but still had a high MAE of 994.45. Elastic net remained the least effective, with the highest MAE of 1760.25 and the lowest R² value of 0.39, indicating poor predictive performance and failure to capture data trends. Both decision tree and random forest showed increases in R² to 0.71 and 0.72, respectively. However, the decision tree had a higher MAE of 1083.89 compared to the random forest, which had an MAE of 737.16, demonstrating good predictive capabilities. Gradient boosting achieved the highest R² value at 0.82, suggesting that it was better at explaining overall data trends, though its MAE of 1098.88 indicates that accuracy still needs to be improved.

3.2. Prediction of several new Cr³⁺ and Fe³⁺ doped phosphor materials

To validate the efficacy of the proposed models, additional new phosphor materials doped with Cr³⁺ and Fe³⁺ were selected for further testing, using trained models to predict their ²E or ⁴T₁ energy levels. The newly selected Cr³⁺-doped phosphor materials include [C(NH₂)₃]M(HCOO)₃(M = Mg²⁺),⁹⁷ YScO₃,⁹⁸ and Zn₂TiO₄.⁹⁹ The new Fe³⁺-doped phosphor materials are Sr₉Ga(PO₄)₇¹⁰⁰ and CaCdPH.¹⁰¹ The predictions for the ²E energy levels of these Cr³⁺-doped materials and the ⁴T₁ energy levels of the Fe³⁺-doped materials were performed using the trained models. The comparison of the model predictions with the actual values of the ²E or ⁴T₁ energy levels is shown in Table 9. This step provides a crucial test of the model's predictive capabilities on new and diverse materials.

Table 9 Comparison between the predicted and actual values of different algorithm models

	Material	Actual value	Linear regression	Robust regression	Lasso regression	Ridge regression	Elastic net	Decision tree	Random forest	Gradient boosting
Cr^3+ (^2E)	[C(NH2)3] M(HCOO)3	14 552	14 492.51	14 560.59	14 404.86	14 442.04	14 472.21	14 600	14 524.13	14 604.73
	YScO3	14 124	14 143.37	14 081.43	14 250.16	14 224.95	14 259.16	13 947	14 132.19	14 367.70
	Zn2TiO4	13 966	13 911.97	13 993.46	14 210.27	14 165.20	14 185.79	13 923	14 201.56	14 342.69
Fe^3+ (^4T1)	Sr9Ga(PO4)7	13 333	13 675.77	13 332.26	13 769.65	13 918.46	14 305.07	14 385	15 497.43	14 634.82
	CaCdPH	17 726	16 263.50	17 242.89	15 797.87	15 867.27	15 185.38	16 389	16 368.63	17 546.85

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Table 9 indicates that the models proposed in the article are effective in predicting the ²E or ⁴T₁ energy levels of phosphor materials doped with Cr³⁺ and Fe³⁺. The prediction results from various algorithms are closely aligned with the actual energy levels.

To provide a clearer comparison of the predictive performance of each algorithm, a bar chart depicting the percentage of absolute error in the predictions for these materials has been constructed, as shown in Fig. 6. This visualization helps to assess the precision of each model in practical applications, highlighting their strengths and potential areas for improvement.

Fig. 6 Histogram of the percentage error of different algorithms predicting the results of the materials.

From Fig. 6, it is evident that, overall, the model's prediction errors for the energy levels under Fe³⁺ doping are higher than those for Cr³⁺ doping, which is consistent with the characteristics of the model predictions on known data discussed in Section 3.1. For [C(NH₂)₃]M(HCOO)₃, robust regression achieved the best results, with an error of only 0.059%. For YScO₃, random forest produced the best results, with an error of only 0.058%. For Zn₂TiO₄, robust regression was the most accurate, with an error of 0.1966%. For Sr₉Ga(PO₄)₇, robust regression significantly outperformed other algorithms, with an error of only 0.0056%. For CaCdPH, gradient boosting performed the best, with an error of 1.01%. These results demonstrate that the machine learning models used in this study can predict the energy levels of ions in phosphor materials with a minimum error as low as 0.0056%, achieving highly precise predictions. Additionally, robust regression consistently showed high performance across all five materials. Linear regression, lasso regression, and ridge regression had similar levels of accuracy, while random forest and gradient boosting showed more variability and fluctuation in their prediction accuracy. Overall, the machine learning prediction models developed in this study achieved excellent results in predicting the ²E or ⁴T₁ energy levels in phosphor materials doped with Cr³⁺ and Fe³⁺. Moreover, it can be found from the existing research results in this paper that robust regression performs outstandingly. Robust regression reduces the influence of outliers by adopting robust loss functions such as Huber's loss to improve the stability of the model, which makes the model less susceptible to the influence of a small portion of atypical sample points in the data, and thus has excellent prediction performance.

3.3. Challenge and limitations

Despite the considerable potential demonstrated by machine learning technologies in the research of doped phosphor materials, their application is still constrained by several challenges and limitations. These challenges primarily include issues concerning the availability and quality of data, the complexity and interpretability of models, difficulties in feature engineering, and the need for interdisciplinary knowledge.

Firstly, regarding data availability and quality, the effectiveness of data-driven machine learning models heavily depends on having access to large, diverse, and accurate datasets. Thus, securing high-quality and diverse training datasets present a significant challenge. In the field of phosphor materials research, the complexity of material compositions, physical processes, and performance measurements makes it particularly difficult to obtain comprehensive and high-quality datasets. Secondly, the complexity and interpretability of models are major concerns. The doping processes in phosphor materials involve complex chemical and physical changes, often requiring sophisticated models for accurate simulation. Additionally, many efficient models, such as deep learning models, although precise in their predictions, lack sufficient interpretability. This makes understanding the underlying mechanisms behind model predictions challenging. Furthermore, feature engineering is crucial for enhancing model prediction accuracy. Effective feature extraction is vital for improving model accuracy, yet identifying which features best describe the intrinsic properties or external behaviours of materials remains a significant challenge in practical applications. Lastly, the demand for interdisciplinary knowledge cannot be overlooked. Research in this field not only requires deep knowledge of chemistry, physics, and materials science but also support from computer science and a thorough understanding of the chemical and physical characteristics of doped phosphor materials, which is itself a challenge. Facing these challenges, enhancing the quality of data collection, developing efficient yet more transparent and interpretable machine learning models, conducting in-depth feature engineering, and deeply integrating knowledge across these disciplines are key pathways to improving research outcomes.

5. Conclusions

This paper utilizes machine learning methods to predict the ²E and ⁴T₁ energy levels of Cr³⁺ and Fe³⁺ ions doped in crystals, which are closely related to the luminescence properties of the phosphor materials doped with these ions. According to the task requirements, eight machine learning regression algorithms (linear regression, robust regression, ridge regression, elastic net, decision tree, random forest, and gradient boosting) were selected to build models for prediction. A comprehensive evaluation of each algorithm's performance on the dataset was conducted, and the results indicated that robust regression achieved excellent outcomes in predicting both Cr³⁺ and Fe³⁺. Subsequently, some new phosphor materials doped with Cr³⁺ and Fe³⁺, such as [C(NH₂)₃]M(HCOO)₃:Cr³⁺, YScO₃:Cr³⁺, Zn₂TiO₄:Cr³⁺, Sr₉Ga(PO₄)₇:Fe³⁺, and CaCdPH:Fe³⁺ were also selected. The trained models were used to predict their ²E and ⁴T₁ energy levels. The predictions show that for these materials, the best-performing algorithm in the ensemble model achieved a prediction error of only 0.0056%, and the optimal prediction errors for all materials were controlled to be approximately 1%, fully reflecting the effectiveness of the machine learning prediction models established in this study. This study provides a novel approach to predicting and optimizing the energy level structures and luminescence properties of phosphor materials. In the future, we will explore more possibilities of machine learning methods in phosphor material characterization, and we will try to use more complex and high-performance models or algorithms to accelerate the discovery of new materials and deeper study of complex properties in materials.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of interest

The authors declare that they have no conflicts of interest.

Acknowledgements

This work was supported by the Natural Science Foundation of Jiangsu Higher Education Institutions of China (grant number 23KJA510005) and the Open Foundation of State Key Laboratory of Luminescent Materials and Devices, South China University of Technology (grant number 2023-skllmd-15).

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Footnote

† These authors contributed equally to this work.

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