Introduction

The optical static refractive index, delineating light’s deflection and refraction in materials under invariant conditions (i.e., free from dynamic effects such as changing electric fields or frequencies), serves as a pivotal and fundamental metric for elucidating the static light-matter interactions. The optical static refractive index usually remains constant over a specific light spectrum regime, typically in the visible to near-infrared range. This important physical quantity not only informs the design of optical components, but also elucidates the underlying electronic properties of materials, which facilitates various applications including optoelectronics, photonic crystals, lenses, and optical coatings1,2,3,4,5,6,7,8,9,10. While experimental methods such as ellipsometry and spectrophotometry, being widely employed in many laboratories, are able to offer reliable refractive indices of materials, these approaches frequently need complex parameter fitting. Furthermore, these methods are also limited by the time-consuming process in sample preparation, environmental constraints, and high costs. These restrictions underscore the urgent need for advanced predictive models that can efficiently navigate this complexity and realize the fast and accurate prediction of optical properties.

So far, great endeavors have been devoted to the prediction of optical static refractive index in various materials using both quantum calculations and semi-empirical models. Quantum calculations, such as density functional perturbation theory (DFPT)11,12, Bethe–Salpeter equation (BSE)13, and Time-dependent density functional theory (TD-DFT)14, have proven to be powerful tools in predicting and analyzing the optical properties. These methods excel in capturing the intricate interactions between electrons and the electromagnetic field, providing high-precision insights into phenomena such as excitonic effects and electronic transitions. By accurately calculating static refractive indices and other optical constants, these approaches significantly enhance the process of discovering and optimizing materials for advanced optical applications. For example, DFPT can be used to calculate phonon dispersion relations and related optical properties, while BSE excels in modeling exciton binding energies, both of which are crucial for understanding a material’s optical response. Despite these advantages, these quantum calculations still suffer from significant computational cost, which make fast material screening over the enormous material space a formidable task. Meanwhile, there have also been great efforts over the past decade to explore the semi-empirical models for predicting the optical static refractive index. These models including Moss15,16,17, Ravindra18,19,20, and Naccarato21 models, are typically represented by the simple mathematical equations that coupled with limited physical or fitting parameters (bandgaps, electronegativity, etc.). These models exhibit the decent capability of predicting the optical static refractive index, providing a much more efficient approach than quantum calculations in material screening. While being efficient and attractive, the existing semi-empirical models still struggle in properly accounting for the complex interplay of electronic transitions and material-specific factors (crystal structural complexity, impurities, etc.). In particular, these complexities distort the local electronic environments and result in the discrepancies between model prediction and actual value. Furthermore, this shortcoming restricts their utility in high-throughput screening for the discovery of new optical materials with reliable optical properties22,23,24. Therefore, there is a critical need for more advanced models that combine computational efficiency with high predictive accuracy, so as to reduce the dependence on costly experiments and time-consuming quantum calculations.

In this study, we overcome the above-mentioned challenges by proposing a novel predictive model for optical refractive indices of materials. Inspired by the chemical bond theory that correlates dielectric properties with the chemical bonds in crystal structure, we integrate crystal structure, intrinsic physical quantities, temperature and elemental doping effects into an enhanced framework for optical static refractive index prediction. Our established model not only extends the applicability of chemical bond theory to more complex structures and materials but also incorporates detailed considerations of temperature and elemental doping effects. We demonstrate the model’s exceptional accuracy using a series of complex oxides such as spinel, pyrochlore, and perovskite, where the predicted optical static refractive indices align remarkably well with experimental data. The broad applicability, coupled with robust efficiency and high accuracy in the established model, significantly propels the development of new materials and spectral designs. Moreover, our model offers valuable theoretical insights for adjusting refractive indices in solid solutions and doping and provides new theoretical perspectives to understand the physical mechanisms driving dielectric and optical properties.

Results

Establishment of the predictive model for optical static refractive index

Our predictive model commences with the Phillips, Van Vechten, and Levine (PVL) chemical bond theory25,26. It is worth noting that the PVL theory establishes the quantitative correlation between electrical susceptibility χ and chemical bond. In particular, the electrical susceptibility χ of any chemical bond in materials can be described by \({\varepsilon }^{\mu }=1+4\pi {\chi }^{\mu }.\) Here \({\chi }^{\mu }=\frac{1}{4\pi }\frac{{\left({\rm{\hslash }}{{\rm{\omega }}}_{p}^{\mu }\right)}^{2}}{{\left({E}_{g}^{\mu }\right)}^{2}}\) where \({\rm{\hslash }}\) is Planck constant, \({E}_{g}^{\mu }\) denotes the average energy gap, being expressed as

$${\left({E}_{g}^{\mu }\right)}^{2}={\left({E}_{h}^{\mu }\right)}^{2}+{\left({E}_{c}^{\mu }\right)}^{2}$$
(1)

where \({E}_{h}^{\mu }\) and \({E}_{C}^{\mu }\) represent the average covalent energy gap caused by homopolar potential energy and the average ionic energy gap caused by ionic potential energy, respectively (see 3 “The complete derivation of the optical static refractive index prediction model” in Supplementary Material for more detail).

\({{\rm{\omega }}}_{p}^{\mu }\) is the free electron plasma frequency, representing the energy difference between bonding and antibonding molecular orbitals for the \(\mu\) th bond type. μ represents the presence of the μ-type chemical bonds in the material. These parameters are crucial for determining electronic transitions and bond stability. In our work, we utilize microscopic dielectric theory to quantitively calculate the term \({\left({{\hslash }}{{\rm{\omega }}}_{p}^{\mu }\right)}^{2}\) by \({\left({{\hslash }}{{\rm{\omega }}}_{p}^{\mu }\right)}^{2}=\frac{4\pi {N}_{e}^{\mu }}{{m}_{e}}{D}_{\mu }{A}_{\mu }\) where \({N}_{e}^{\mu }\), \({m}_{e}\), \({D}_{\mu }\), and \({A}_{\mu }\) are the effective valence electron density of any type of bond in a crystal, electronic mass, d-electron correction factor, and Penn correction factor (The definition, physical meaning, and calculation process for these parameters are presented in Supplementary Section 2 “The development of predictive model for optical static refractive index” for more details). From this formula, we clearly see the \({D}_{\mu }\) is the key factor that significantly affects \({{\rm{\omega }}}_{p}^{\mu }\) and corresponding electrical susceptibility \({\chi }^{\mu }\). However, rational correction for d-electron is still non-trivial. To fully take the influence of d-electrons into consideration, especially for transition and noble metals, we utilize \({D}_{\mu }=1+{\Gamma }_{\mu }\) where \({\Gamma }_{\mu }\) is the ratio between the number of d empty holes and the number of valence electrons. In this work, \({\Gamma }_{\mu }\) can be calculated by27

$${\Gamma }_{\mu }=\frac{{\rm{number}}\; {\rm{of}}d\;{\rm{holes}}\; {\rm{in}}\; \mu \; {\rm{th}}\; {\rm{bond}}}{{\rm{number}}\; {\rm{of}}\; {\rm{valence}}\; {\rm{electrons}}\; {\rm{in}}\; \mu \; {\rm{th}}\; {\rm{bond}}}=\frac{{N}_{{holes}}^{\mu }{m}_{\mu }}{{m}_{\mu }{z}_{A}^{\mu }+6{n}_{\mu }}$$
(2)

For transition metals, \({\varGamma }_{\mu }=\frac{\left(10-\left({z}_{A}^{\mu }-2\right)\right){m}_{\mu }}{{m}_{\mu }{z}_{A}^{\mu }+6{n}_{\mu }}=\frac{{m}_{\mu }(12-{z}_{A}^{\mu })}{{m}_{\mu }{z}_{A}^{\mu }+6{n}_{\mu }}\). So \({{\rm{D}}}_{\mu }=1+{\varGamma }_{\mu }=\frac{{12m}_{\mu }+6{n}_{\mu }}{{m}_{\mu }{z}_{A}^{\mu }+6{n}_{\mu }}\) where, \({N}_{{holes}}^{\mu }\), \({z}_{A}^{\mu }\), \({m}_{\mu }\) and \({n}_{\mu }\) are the number of d holes of cation in \(\mu\) th bond, valence electron of cation of \(\mu\) th bond and valence electron of cation of \(\mu\) th bond, respectively. It is worthy to note that for material that no d electrons, the number of d holes of cation in \(\mu\)-th bond \({N}_{{holes}}^{\mu }\) becomes 0. Thus, \({\varGamma }_{\mu }=0\) and \({D}_{\mu }=1\) for the materials that no d electrons (see Section 2—d-electron correction factor \({D}_{\mu }\) and an example in Section 8-II “Materials with no d-electrons” in Supplementary Material for more details).

Such d correction considers the effect from both unfilled d shells and empty conduction d-level bands. Based on this d-electron correction, the electrical susceptibility can be calculated by27

$${\chi }^{\mu }=\frac{{\left({{\hslash }}{{\rm{\omega }}}_{p}^{\mu }\right)}^{2}}{4\pi {{E}_{g}^{\mu }}^{2}}\left(1+{\Gamma }_{\mu }\right){A}_{\mu }$$
(3)

Consequently, the optical static refractive index of materials with diverse chemical bonds can be achieved via

$${n}^{2}={\rm{\varepsilon }}=1+4\pi \sum _{\mu }{F}^{\mu }{\chi }^{\mu }$$
(4)

where \({F}^{\mu }\) is the scaling factor for the number of μ-type bonds, being expressed as \(\frac{{a}_{\mu }{N}_{{BA}}^{\mu }}{\sum _{\mu }{a}_{\mu }{N}_{{BA}}^{\mu }}\).\({a}_{\mu }\) is number of elements corresponding to the cations per formula unit in the \(\mu\) th bond. \({N}_{{BA}}^{\mu }\) is number of nearest anions of cations in \(\mu\) th bonds. In this work, \({F}^{\mu }\) represents the relative contribution of the \(\mu\) th type of chemical bond to the material’s total refractive index. It reflects the bond’s weight in terms of its abundance and significance in the crystal structure (see Section 2-VII “The proportion of \(\mu\) th chemical bond” \({F}^{\mu }\) in Supplementary Material for more details). To efficiently accelerate the high-throughput prediction of the static dielectric and optical properties of materials and to further determine the parameters affecting the optical static refractive index, such as crystallographic parameters and elemental bonding parameters, we rigorously derive that the optical refractive index can be expressed as

$${n({d}_{\mu },V,{z}_{A}^{\mu })}^{2}=1+\frac{4\pi \theta }{V}\sum _{\mu }\frac{{F}^{\mu }{D}_{\mu }{\gamma }_{\mu }\zeta {A}_{\mu }}{{d}_{\mu }{{\rm{\sigma }}}_{\mu }}$$
(5)

where \({\gamma }_{\mu }\), \({\zeta }_{\mu }\), \({d}_{\mu }\), \({{\rm{\sigma }}}_{\mu }\), \(V\) and \({z}_{A}^{\mu }\) are effective valence electron number, total chemical bond volume, bond length, bond energy factor, volume per formula unit and valence electron number. Here \(\theta ={\hslash }^{2}{e}^{2}{{m}_{e}}^{-1}\) (see Section 2 “The development of predictive model for optical static refractive index” and Section 3 “The complete derivation of the optical static refractive index prediction model” in Supplementary Material for more details).

Figure 1a schematically illustrates the workflow of the predictive model for the static optical refractive index. The model takes several physical quantities derived from material crystallography as inputs, including inter-atomic bonding, geometric arrangement, and metrics related to local chemical environments (i.e., binary bonding unit and valence electron configuration), as shown in Fig. 1b. For example, the binary bonding formula of a complex oxide with chemical formula can be written as \({\sum }_{\mu }{{A}_{{m}_{\mu }}B}_{{n}_{\mu }}\), where \(A\) and \(B\) are cations and anions, respectively. \({a}_{\mu }\) and \({b}_{\mu }\) represent the number of elements corresponding to the cations and anions per formula unit in the \(\mu\) th bond. Also, \({m}_{\mu }={N}_{{BA}}^{\mu }{a}_{\mu }/{N}_{{cA}}^{\mu }\) and \({n}_{\mu }={N}_{{AB}}^{\mu }{b}_{\mu }/{N}_{{cB}}^{\mu }\) are the stoichiometric ratios of binary bonding formulas for each \(\mu\)-type of chemical bond. \({N}_{{BA}}^{\mu }\)(\({N}_{{AB}}^{\mu }\)) is the number of nearest B (A) atoms around a specific A (B) atom. \({N}_{{cA}}^{\mu }\) and \({N}_{{cB}}^{\mu }\) are the coordination numbers for A and B, respectively (see Supplementary Section 2 for more details).

Fig. 1: The schematic of the predictive model for static optical refractive index.
figure 1

a Schematic workflow for predicting the static optical refractive index. b The input intrinsic physical quantities are derived from material crystallography. c The connection of various parameters in the predictive model.

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Once we obtain the elemental bonding parameters based on intrinsic physical quantities, crystallographic prior parameters and key parameters, the optical static refractive index of the material can be finally obtained. For new materials, we can conduct the model prediction of optical static refractive index with X-ray diffraction (XRD) or density functional theory (DFT) optimization results (see Fig. 1c and Section 4 “Prediction of the optical static refractive index for new materials” in Supplementary Material for more details).

It is also important to note that our model uses physical parameters as inputs, rather than artificially fitted parameters. In particular, the optical refractive index can be determined by the crystallographic, electron configuration, and elemental composition. Such model couples bonding information with other physical quantities such as bond length, volume, etc. to predict the optical static refractive index, being highly efficient for screening potential materials with target dielectric/optical properties.

In reality, the finite temperature and impurity (i.e., doping effect) would also play a vital role in optical static refractive index of materials. However, existing models do not seem to rigorously consider temperature and impurity effects, which sacrifices the capability of model prediction. Therefore, we carefully examine temperature and impurity effects on optical static refractive index prediction and integrate them into our model. To tackle the predictive uncertainty from temperature and impurity for materials, the chemical bond change and critical physical quantities affected by temperature and impurity are fully scrutinized, and the temperature dependence and impurity dependence optical static refractive index with the degree of substitution (x) at temperature T can be consequently calculated by

$$\begin{array}{l}{n\left({\rm{V}}\left(x,T\right),{d}_{{\mu }^{{\prime} }}\left(x,T\right),{z}_{A}^{{\mu }^{{\prime} }}\left(x\right)\right)}^{2}=1+\frac{4\pi \theta }{V\left(x,T\right)}\\\qquad\mathop{\sum}\nolimits _{{\mu }^{{\prime} }}\frac{{F}^{{\mu }^{{\prime} }}\left(x\right){\gamma }_{{\mu }^{{\prime} }}\left(x\right){\zeta }^{{\prime} }\left(x,T\right){D}_{{\mu }^{{\prime} }}\left(x\right){A}_{{\mu }^{{\prime} }}\left(x,T\right)}{{d}_{{\mu }^{{\prime} }}\left(x,T\right){{\rm{\sigma }}}_{{\mu }^{{\prime} }}\left(x\right)}\end{array}$$
(6)

This formula clearly indicates that the change in the optical static refractive index due to temperature effects is solely attributed to variations in bond length and volume per formula unit at different temperatures. For instance, the increase in total chemical bond volume (\(\zeta \left(T\right)\)) with rising temperature can be attributed to the elongation of bond length (\({d}_{\mu }\left(T\right)\)) due to thermal expansion, as described by \(\zeta \left(T\right)=\sum _{\mu }{\left({{a}_{\mu }{N}_{{cA}}^{\mu }d}_{\mu }\left(T\right)\right)}^{3}\). Moreover, the degree of substitution (\(x\)) is crucial in determining the resultant properties of the doped material, as it denotes the number of substituted atoms in the material. When the material is doped, the bond type \(\mu\) becomes \({\mu }^{{\prime} }\), and its parameters become functions of the x, enabling the determination of the substitution-dependent optical static refractive index. Unlike temperature corrections for lattice thermal expansion, doping corrections must consider the stoichiometric ratio, number of valence electrons, ion valence, and atomic coordination number. For instance, the total chemical bond volume (\({\zeta }^{{\prime} }\left(x\right)\)) changes with variations in the degree of substitution (x), depending on the type (\({\mu }^{{\prime} }\)) and length of chemical bonds (\({d}_{{\mu }^{{\prime} }}\left(x\right)\)) introduced by impurities, which is described by \({\zeta }^{{\prime} }\left(x\right)=\sum _{{\mu }^{{\prime} }}{\left({{{a}_{\mu }}^{{\prime} }{N}_{{cA}}^{{\mu }^{{\prime} }}d}_{{\mu }^{{\prime} }}\left(x\right)\right)}^{3}\) (see Section 6 “Temperature effect on the prediction of optical static refractive index” in Supplementary Material for more details).

In Eq. \(\left(6\right)\), we have consolidated complex crystal structures, chemical bonds, temperature, and impurities to predict the optical static refractive index. Meanwhile, our approach can also facilitate high-throughput screening of the optical static refractive index across various materials, as depicted in Fig. 2.

Fig. 2: The validation of predictive model for a variety of complex oxides with different crystal structures.
figure 2

The predicted optical static refractive indices and experimental45,46,47,48,49,50,51,52,53,54,55,56,57,58,59 data are compared. a 13 cubic perovskite oxides, b 3 cubic spinel oxides, c 3 cubic perovskite oxides, d 3 orthorhombic perovskite oxides, and e 3 rhombohedral perovskite oxides. Error bars in predictive data indicate the predictive refractive index from 0 K to 273 K while the error bars in experimental data are raised from various experimental values reported in different literature.

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Model validation for pristine complex oxides

To validate our predictive framework, we selected 41 complexes but representative oxides with well-documented experimental optical static refractive indexes where Fig. 2 shows 13 pyrochlores (R2M2O7 cubic type in Fig. 2a), 3 spinels (RM2O4 cubic type in Fig. 2b, and 9 perovskites which include 3 RMO3 cubic type in Figs. 2c and 3RMO3 orthorhombic type in Fig. 2d and 3RMO3 rhombohedral RMO3 type in Fig. 2e). All other verified results are shown in Supplementary Fig. S8-1. The reliability of the input parameters in this model is demonstrated in Figs. S8-2 and Table S8-1. Additionally, the calculated parameters and corresponding crystal structures of these materials are presented in Supplementary Tables S8-2 to S8-6 and Figs. S2–1, Figs. S8–3 to S8-6. We compared the data calculated by current model against other predictive models including Moss15,16,17,28,29, Ravindra18,30,31, Herve and Vandamme5, Reddy32, Kumar and Singh33, and S.K. Tripathy20, etc. By analyzing the defined errors such as relative error (\({\rm{RE}}( \% )=\left|{n}_{\rm{model}}-{n}_{\exp .}\right|/\left|{n}_{\exp .}\right|\), where \({n}_{\rm{model}}\) and \({n}_{\exp .}\) represent the static optical refractive index predicted by various theoretical models and determined experimentally, respectively), we clearly note our model exhibits superior predictability in the static refractive index across various examined complex oxides. In particular, the results detailed in Figs. S8-1f underscore our model’s high accuracy and excellent consistency with experimental values. For instance, the root mean squared error (RMSE) for cubic R2M2O7, RM2O4, RMO3, orthorhombic RMO3, and rhombohedral RMO3 are also much lower than other examined models.

Effect of intrinsic physical properties on the prediction of optical static refractive index

To further clarify the critical factors affecting the optical static refractive, we systematically assess the influence of a variety of physical parameters including d-electron number, bond length, lattice constant as well as the coordination number on optical static refractive index. Taking cubic pyrochlore (R2M2O7) as representative crystal structure, we observe a linear descent in the optical static refractive index with the increase of d-electron in either R or M crystal site. It is worth noting that the d-empty holes can result in highly localized d-electrons. This localization effect enhances the electron density and polarization ability of the material, thereby increasing the refractive index. The established correlation between the reduced radii of R-site and/or M-site cations in R2M2O7 and the enhancement of ionic bond strength is evident from the increased polarization, as shown in Figure S8-7a. This correlation underpins the observed reduction in the optical static refractive index from zero to ten d electrons. Conversely, larger ion radii, associated with less charges, lead to weaker ionic bonds and reduced lattice energy. The refractive index of M-site induced by d-electrons exhibits a higher sensitivity compared to that of R-site due to the presence of M-site d-orbitals at the bottom of the pyrochlore conduction band (Fig. S8-7b). Unlike the linear correlation between d-electron and optical static refractive index, the relationships among bond length, lattice constant, coordination number, and optical static refractive index are apparently nonlinear (Fig. 3b–d). In particular, the optical static refractive index decreases with the drop of lattice constant, volume, and coordination number until it reaches a threshold, after which it stabilizes (Fig. 3c). In contrast, the optical static refractive index increases progressively with bond length across all three chemical bond types. This variance in bond length trends results from the combined effects of bond length on plasma frequency and average energy gap, which in turn affect the optical static refractive index. Notably, Supplementary Fig. S8-8 to S8-12 investigate the effects of d-electrons number, chemical bond length, volume, coordination number, and charge number of cubic spinel (CoFe2O4), cubic perovskite (BaTiO3), rhombohedral perovskite (CaTiO3), rhombohedral perovskite (ScBO3 (R3̅c)) and MgTiO3 (\(R\bar{3}c\)). A higher coordination number indicates that there are more coordinated atoms or ions surrounding the central atom or ion, leading to the dispersion and weakening of the electron cloud. Consequently, the directionality and strength of the high-coordination chemical bonds may be diminished, resulting in an overall reduction of the material’s polarization, thereby lowering the refractive index. Moreover, Fig. 3e depicts the optical static refractive indices of several common pyrochlore oxides with different temperatures. It can be seen that the optical static refractive index increases further with the increase of temperature, but the gradient of the optical static refractive index with temperature is small, which is consistent with the fact that the electronic refractive index is insensitive to temperature34,35 (Supplementary Fig. S6-1).

Fig. 3: The influence of different intrinsic parameters on optical static refractive index.
figure 3

a Number of d-electrons and unoccupied orbitals for R and M-site atoms, b three types of different chemical bond lengths, c Lattice constants and crystal volume of the material, d coordination number for R and M-site atoms for representative cubic pyrochlore Gd2Zr2O7. e Temperature-dependent refractive index for various examined pyrochlore oxide materials.

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Effect of doping on the prediction of optical static refractive index

After clarifying the role of intrinsic parameters and temperature effect in optical static refractive indices, we further elucidate the mechanisms and underlying physics through which the dopant elements affect four representative doping systems by employing the established model in this work. The model validation workflow of the optical static refractive index for doped material system is shown in Figs. S8–19. Figure 4a demonstrates the predictive optical static refractive indices for various doped systems, the scrutinized doping systems encompass binary oxide and perovskite systems with diverse space groups including Gd2−xMnxO3(0 ≦ x ≦ 0.6), Ba1−xSrxO3(0 ≦ x ≦ 0.3), La1−xScxFeO3(0.025 ≦ x ≦ 0.125) and La1−xCaxMnO3(0.3 ≦ x ≦ 1). The calculated parameters and corresponding crystal structures of these materials are presented in Supplementary Tables S8–23, S8–26, S8–28, S8–29 and Fig. S8–21, S8–24, S8–26, S8–29, respectively. In particular, the trend and magnitude of the optical static refractive index with varying doping can be easily captured by current model, as seen in Fig. 4a. Such integration of theoretical and experimental validation provides additional support for our model, offering novel theoretical foundations for optimizing optical static refractive index through doping engineering. The experimental XRD and static refractive index data of these materials are available in Supplementary Figs. S8–20, S8–23 and Figs. S8–24, S8–2736,37,38,39,40.

Fig. 4: The validation of the doping effect in the model.
figure 4

a Verification of the doping effects in various material systems, including \({{\rm{Gd}}}_{2-x}{{\rm{Mn}}}_{x}{{\rm{O}}}_{3}(0\le x\le 0.6)\), \({{\rm{Ba}}}_{1-x}{{\rm{Sr}}}_{x}{\rm{Ti}}{{\rm{O}}}_{3}(0\le x\le 0.3)\),\({{{La}}_{1-x}{\rm{Sc}}}_{x}{\rm{Fe}}{{\rm{O}}}_{3}(0.025\le x\le 0.125)\) and \({{\rm{La}}}_{1-x}{{\rm{Ca}}}_{x}{Mn}{{\rm{O}}}_{3}(0.3\le x\le 1)\). b–e Investigation of the crystal volume \({V}_{c}\), bond length \({d}_{{\mu }^{{\prime} }}\), stoichiometric ratio \({m}_{{\mu }^{{\prime} }}\), \({n}_{{\mu }^{{\prime} }}\) and chemical bond ratio \({F}_{{\mu }^{{\prime} }}\) of Gd2−xMnxO3 (0 ≤ x ≤ 1) at different Mn-substitution. Solid scatter represents XRD experimental (exp.)36,37,38,39,40 results from XRD refinements. O, O’ and O” in (d) and (e) means three different positions of the O28e atom. f–i The model’s predicted optical static refractive indices for a range of material systems, including \({{\rm{Gd}}}_{2-x}{{\rm{Mn}}}_{x}{{\rm{O}}}_{3}\) (x = 0.03, 0.05, 0.1), \({{\rm{Ba}}}_{1-x}{{\rm{Sr}}}_{x}{\rm{Ti}}{{\rm{O}}}_{3}\) (x = 0, 0.2, 0.25, 0.3), \({{{La}}_{1-x}{\rm{Sc}}}_{x}{\rm{Fe}}{{\rm{O}}}_{3}\) (x = 0.05, 0.075, 0.1, 0.125), and \({{\rm{La}}}_{0.5}{{\rm{Ca}}}_{0.5}{Mn}{{\rm{O}}}_{3}\), were compared with various semi-empirical models and experimental data to evaluate their accuracy.

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Figure 4b–e shows the variation of intrinsic quantities (volume \({V}_{c}\), bond length \({d}_{{\mu }^{{\prime} }}\), stoichiometric ratio \({m}_{{\mu }^{{\prime} }}\), \({n}_{{\mu }^{{\prime} }}\) and chemical bond ratio \({F}_{{\mu }^{{\prime} }}\)) of representative Gd2−xMnxO3(0 ≦ x ≦ 0.6) for different Mn-substitution. In Gd2-xMnxO3(0 ≦ x ≦ 0.6) system, \({Mn}^{3+}\) ions (radii 64.5 pm) are incorporated into \({{Gd}}^{3+}\) ions (radii 93.8 pm), the decrease in volume (Fig. 4b) due to the lattice contraction makes the atomic interactions stronger, further increasing the refractive index. Simultaneously, the number of 3d electron decreases from 10 to 5 without changing the coordination number, resulting in an increase in the optical static refractive index. This reduction weakens the mutual repulsion between electrons, allowing the remaining d electrons to participate more effectively in the polarization process. As the degree of substitution (\(x\)) increases in Gd2−xMnxO3(0 ≦ x ≦ 0.6), the number \({{a}_{\mu }}^{{\prime} }\) and \({{b}_{\mu }}^{{\prime} }\) of Mn atoms and O bonded to Mn rises, leading to a gradual increase in the stoichiometric ratios \({m}_{{\mu }^{{\prime} }}={N}_{{OM}n}^{\mu }{{a}_{\mu }}^{{\prime} }/{N}_{{cM}n}^{\mu }\) and \({n}_{{\mu }^{{\prime} }}={N}_{{MnO}}^{\mu }{{b}_{\mu }}^{{\prime} }/{N}_{{cO}}^{\mu }\) for the Mn–O bond. Concurrently, the chemical bonding ratio \({F}_{{\mu }^{{\prime} }}\) for the Mn–O bond linearly escalates, while the opposite trend is observed for Gd atoms, as shown in Fig. 4d, e. This doping mechanism is also applicable to the La1−xScxFeO3(0.025 ≦ x ≦ 0.125) and La1−xCaxMnO3(0.3 ≦ x ≦ 1) systems, as illustrated in Fig. 4a. Furthermore, literature indicates that the band gaps of \({\rm{CaMn}}{{\rm{O}}}_{3}\) and \({\rm{L}}{\rm{aMn}}{{\rm{O}}}_{3}\) are 1.55 eV and 0.4 eV41 respectively, suggesting that Ca doping can enhance the band gap of \({\rm{L}}{\rm{aMn}}{{\rm{O}}}_{3}\). Therefore, Ca doping reduces the optical static refractive index of the \({\rm{L}}{\rm{aMn}}{{\rm{O}}}_{3}\) system, aligning with the trend observed in Fig. 4. In addition, we also compared our model with existing semi-empirical ones in Fig. 4f-i. Notably, the model established in this work accurately predicts the impurity-dependent optical static refractive indices for various doping systems, while other models fail. This robustness likely arises from its ability to capture subtle changes in intrinsic parameters induced by impurities. Unlike conventional models, which often rely on simple parameters like band gaps to estimate optical properties, our model primarily uses crystallographic information, making it highly sensitive to impurities. As a result, it can more precisely predict impurity-dependent optical static refractive indices.

Experimental validation of predictive optical static refractive index for La doped Gd1.894Zr1.894O6.629

The above-mentioned analysis has successfully demonstrated the predictive capability of established framework for optical static refractive index in complex crystal structures. Furthermore, our model also accurately captures the role of temperature as well as doping effects on the optical static refractive index. To further verify the validation of established model, we now show direct experimentally measured optical static refractive index using the common and representative pyrochlore Gd1.894Zr1.894O6.629 with La element dopant. We initially utilize solid-state reaction to synthesize a series of La doped Gd1.894Zr1.894O6.629 with La element being Gd1.874La0.02Zr1.894O6.629, Gd1.834La0.06Zr1.894O6.629, Gd1.794La0.1Zr1.894O6.629. Figure 5a shows the Energy-dispersive X-ray spectroscopy (EDS) from a high-angle annular dark-field scanning transmission electron microscope (HAADF-STEM) of La-doped samples where we can observe that each element (i.e., Gd, La, Zr, and O) is homogeneously distributed and no extra phases form in the sample. The X-ray diffraction (XRD) pattern of the Gd1.834La0.06Zr1.894O6.629 sample (the XRD Rietveld refinement for other La doped samples are shown in Supplementary Figs. S8–28 to S8–31 and Tables S8–30) suggests the single-phased cubic pyrochlore structure is formed. The Rietveld refinement using XRD pattern indicates Gd1.834La0.06Zr1.894O6.629 has the lattice parameters of a = b = c = 10.544\(08\pm 0.01896\) Å and α = β = γ = 90°, further confirming the Fd\(\bar{3}\) m space group. Such crystallographic information is also consistent with identified structure in Transmission Electron Microscope (TEM) images. Furthermore, we find Gd and La occupy the (Gd/La)O8 dodecahedral site while Zr is distributed on ZrO6 octahedral site (Fig. 5c). Figure 5e depicts the changes in bond length and crystal volume for three types of chemical bond in Gd1.894−xLaxZr1.894O6.629 over different La-substitution (i.e., x is 0.2, 0.6, and 0.1). The high-precision fitting of the model data to the experimental data validates the extrapolated bond lengths and volume across all degree of substitution (\(x\)) in this material system (Supplementary Tables S8–24). Notably, Fig. 5f effectively demonstrates the high predictive performance of the model, that the refractive index values and trends calculated by our model align closely with the experimental values, with minor discrepancies potentially arising from experimental uncertainties such as light scattering or porosity. This congruence demonstrates the model’s accuracy in quantitatively predicting static refractive indices and its utility in guiding doping modifications.

Fig. 5: Experimental characterization and prediction of optical static refractive index for Gd1.894Zr1.894O6.629 with various La-substitution.
figure 5

a EDS spectrum of Gd1.834La0.06Zr1.894O6.629 material. b The X-ray diffraction pattern of Gd1.834La0.06Zr1.894O6.629 after refinement by Rietveld method. c The schematic illustration of Gd1.834La0.06Zr1.894O6.629 crystal structure with multiple elements shown on the structural units. d The high-resolution TEM image of Gd1.834La0.06Zr1.894O6.629. e The crystal volume and bond length of Gd1.834−xLaxZr1.894O6.629 (x = 0, 0.02, 0.06, 0.1). f The refractive index of Gd1.834−xLaxZr1.894O6.629 (x = 0, 0.02, 0.06, 0.1). The error bars reflect the data obtained from multiple high-precision XRD refinements.

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Discussion

In this work, a series of complex oxides including 41 kinds of crystal structures (i.e., cubic R2M2O7, cubic RM2O4, cubic RMO3, orthorhombic RMO3, and rhombohedral RMO3) and 5 doped systems are fully scrutinized where the optical static refractive indices predicted via our proposed model agree very well with experimentally gauged ones. This consistency further demonstrates that the developed model can successfully predict the optical static refractive index of new materials based on the crystal structure obtained from experimental X-ray diffraction (XRD) or theoretical density functional theory (DFT), using only physical quantities uniquely determined by the crystal structure as input. Such simple physical parameters consolidated within the model would not only enable quantitative refinement of optical static refractive index for a variety of materials, but also ensure high throughput searches for new materials with targeted static optical properties. In particular, the developed model accounts for at various degrees of substitution, offering theoretical guidance for optimizing the optical static refractive index through doping engineering. For instance, we find that doped systems with low d-electron counts, long bond lengths, small lattice constants, and/or low anion coordination tend to yield higher refractive indices. Notably, our established model can be seamlessly integrated with state-of-the-art machine learning, providing efficient guidance for spectral design in complex materials and bypassing the hurdle of conventional trial-and-error approaches. It is also worth noting that our model cannot fully address the impact of macroscopic defects such as porosity, surface morphology on optical static refractive index of material. Meanwhile, our model is specifically designed to address the optical static refractive index in complex dielectric oxides, not metal-like materials and small-gap materials. For complex dielectric oxides with a large band gap, the refractive index exhibits weak frequency dependence between 0 eV and the band gap (the optical range). On one hand, this focus arises from the distinct electronic properties of dielectrics, where the absence of free electrons that dominate the optical response in metals allows for reliable prediction of the optical static refractive index. In dielectrics, the optical behavior is primarily determined by bound electrons and their interactions with the crystal lattice, making precise modeling and prediction of their refractive index feasible. Conversely, metal-like materials exhibit plasmonic behavior and significant free electron contributions, resulting in complex optical phenomena that are beyond the scope of our current model. On the other hand, for small-gap materials, improvements are required to capture inter-band transitions, excitonic effects, and dispersion due to electron-photon interactions. Future developments of the model could incorporate advanced methods, such as Bethe–Salpeter equation + GW Approximation (BSE + GW), to address these limitations. Therefore, while our model provides valuable insights and predictions for dielectric oxides, it does not extend to materials where metallicity and small band gaps significantly influence optical properties. Moreover, for non-dielectric materials, the model’s predictability can be improved by incorporating electron-phonon interactions. Tackling these challenges remains a critical objective for the development of more comprehensive refractive index prediction models.

Overall, we have successfully established one robust model that shows strong capability of predicting optical static refractive index of complex materials with high accuracy and efficiency. Through rigorous experimental validation, we demonstrate the generality, wide applicability across diverse crystal structures and space-group material systems. Furthermore, our results advance the understanding of the role of crystallography, d-electrons in optical static refractive index and provide physical insights that may enable doping engineering against the tunability and design of optical static refractive index of materials.

Methods

Density functional theory (DFT) calculations

All structural optimizations were performed using density functional theory (DFT) as implemented in the Vienna Ab initio Simulation Package (VASP)42. The electron–ion interactions were described by the projector-augmented wave (PAW)43 method, and the exchange-correlation functional was approximated using the generalized gradient approximation with the Perdew–Burke–Ernzerhof functional for solids (GGA-PBEsol)44. A plane-wave cutoff energy of 520 eV was employed for all calculations to ensure the convergence of the total energy. The Brillouin zone was sampled using a Monkhorst–Pack k-point grid with a spacing of 0.04 Å−1. For materials with different space groups, the k-point densities were scaled appropriately according to the reciprocal lattice sizes to maintain consistency across various unit cells. The energy convergence criterion for electronic self-consistency was set to 10−8 eV, while the force threshold for ionic relaxation was set to 0.02 eV/Å. To enhance convergence near the Fermi level, Gaussian smearing with a width of 0.05 eV was applied. Spin-polarized calculations were conducted to account for magnetic effects. For ionic relaxations, the conjugate gradient algorithm was utilized, and both atomic positions and cell shapes were optimized to achieve fully relaxed structures.

Synthesis and characterization of \({\rm{Gd}}_{1.894-x}{\rm{La}}_{x}{\rm{Zr}}_{1.894}{\rm{O}}_{6.629}\) (\(x\) = 0, 0.02, 0.06, 0.1)

Synthesis of \({\rm{Gd}}_{1.894-x}{\rm{La}}_{x}{\rm{Zr}}_{1.894}{\rm{O}}_{6.629}\) (\(x\) = 0, 0.02, 0.06, 0.1)

The pyrochlore‐type \({\rm{Gd}}_{1.894-x}{\rm{La}}_{x}{\rm{Zr}}_{1.894}{\rm{O}}_{6.629}\) materials were fabricated via the solid‐state reaction method. Initially, the powders of La2O3 (99.9%, Shanghai Aladdin Bio-Chem Technology Co., Ltd.), Gd2O3 (99.9%, Shanghai Aladdin Bio-Chem Technology Co., Ltd.) and ZrO2 (99.99%, Shanghai Aladdin Bio-Chem Technology Co., Ltd.) were stoichiometrically weighted. Next, the mixed reactive materials were ball‐milled for 48 h in the absolute ethanol and dried at 80 °C in an electric oven for 24 h. Then, the reactants were calcined at 1400 °C for 12 h to obtain the single‐phase \({\rm{Gd}}_{1.894-x}{\rm{La}}_{x}{\rm{Zr}}_{1.894}{\rm{O}}_{6.629}\) powders. Finally, bulk \({\rm{Gd}}_{1.894-x}{\rm{La}}_{x}{\rm{Zr}}_{1.894}{\rm{O}}_{6.629}\) samples were sintered at 1430 °C for 10 h.

Characterization

X‐ray diffractometer (XRD) measurements at a scan rate of 0.5°/step with Cu-\({K}_{\alpha }\) radiation (DX2700, China) was performed to determine the phase composition and Rietveld refinement of the synthesized samples. The microstructure morphologies and compositions of the samples were investigated by a high-angle annular dark-field scanning transmission electron microscope (HAADF-STEM, Thermo Scientific, Spectra 300) with an Energy-dispersive X-ray spectroscopy (EDS Thermo Scientific, Talos F200X G2). The solar reflectivity and the optical band gap were characterized by a UV–VIS–NIR spectrophotometer (Lambda 950, PerkinElmer, USA) in the range of 0.25–2.5 µm. The infrared reflectivity spectra were characterized by a Fourier-transform infrared spectrometer (FTIR, Nicolet 6700, USA) in the range of 2.5–16.7 µm.