Machine learning for predicting Debye temperature and screening inorganic phosphor hosts

The machine-learning model to predict Debye temperature first requires training using a large, diverse set of data. In this case, the 2610 DFT-based bulk (B DFT ) and shear (G DFT ) moduli extracted from the Materials Project database can be converted into an approximate Debye temperature (Θ D,DFT ) based on the average sound velocity (ν m ) following Eq. 1, where h is Planck constant, k B is Boltzmann constant, n is the number of atoms per formula unit, N A is Avogadro constant, ρ is the crystal structure’s density, M is the molar mass, and ν m is the average sound velocity33:

$$\Theta _{{\mathrm{D,DFT}}} = \frac{h}{{k_{\mathrm{B}}}}\left[ {\frac{{3n}}{{4\pi }}\left( {\frac{{N_{\mathrm{A}}\rho }}{M}} \right)} \right]^{\frac{1}{3}}v_{\mathrm{m}}.$$ (1)

In a polycrystalline material, ν m can be approximated following Eq. 2:

$$v_{{\mathrm{m}},{\mathrm{DFT}}} = \left[ {\frac{1}{3}\left( {\frac{2}{{v_{{\mathrm{T}},{\mathrm{DFT}}}^3}} + \frac{1}{{v_{{\mathrm{L}},{\mathrm{DFT}}}^3}}} \right)} \right]^{ - \frac{1}{3}},$$ (2)

where ν L and ν T are the longitudinal and transverse sound velocity, respectively, which can be obtained from the DFT calculated B DFT and G DFT using Eq. 3:

$$v_{\mathrm{L}} = \left( {\frac{{B_{{\mathrm{DFT}}} + \frac{{4G_{{\mathrm{DFT}}}}}{3}}}{\rho }} \right)^{\frac{1}{2}}\;{\mathrm{and}}\;v_{\mathrm{T}} = \left( {\frac{{G_{{\mathrm{DFT}}}}}{\rho }} \right)^{\frac{1}{2}}.$$ (3)

Once Θ D,DFT was obtained for all 2610 compounds, a machine-learning algorithm was developed to predict the Debye temperature using support vector machine regression (SVR) (Θ D,SVR ). The 10-fold cross-validation regression analysis for Θ D,SVR is plotted in Fig. 1a. The statistics of the cross-validated fit show excellent agreement between the Θ D,DFT and the Θ D,SVR with a root mean-squared error of cross-validation of 59.9 K and a mean absolute error of 37.9 K. The coefficient of determination (r2) is 0.89 indicating that the descriptor set does an excellent job expressing the materials Debye temperatures. Nearly all of the data (97%) falls between 50 and 750 K, as highlighted by the darker regions of the plot. The cross-validated Θ D,SVR tends to be slightly underestimated, particularly for materials with extremely high Debye temperatures (>1500 K), which is potentially due to the lack of data with extremely high Debye temperature. In fact, there are only four compounds, BN, Be 2 C, B 6 O, and LiB 6 C, that fall notably below the expected Θ D,SVR , suggesting that our machine-learning model does not capture the chemistry of these very light compounds. Although there are a few incorrect predictions, Fig. 1b shows that more than 75% of the compounds are predicted within ≈15% of the DFT calculated value.

Fig. 1 Cross-validation plots for the machine-learning model predicting Debye temperatures. a Ten-fold cross-validation (CV) predicted Debye temperature (Θ D,SVR ) versus calculated Debye temperature (Θ D,DFT ). The ideal line is shown as dashed line and the fit line is shown as solid line. b Fraction of compounds according to their percent error between CV predicted Θ D,SVR and Θ D,DFT . The red curve shows the trend Full size image

Once the training set has been constructed, the machine-learning algorithm can be used to estimate Θ D,SVR for compounds compiled in crystallographic databases. The extraction of the crystal structures from PCD followed the same criteria as listed above, supplying 118,287 unique compounds for prediction. This list includes 15,770 binary compounds, 56,266 ternary compounds, and 46,251 quaternary compounds, of which 27,698 (23%) contain rare-earth elements that cannot be readily calculated using conventional DFT and 19,384 (16%) have either disordered positions or site mixing, which is also generally inaccessible with DFT. From this extended list, the search for potential phosphor hosts can be immediately reduced by adhering to a few simple phosphor design criteria. First, knowledge of the band gap is essential for screening phosphor hosts using the sorting diagram, and each compound was further cross-referenced against the Materials Project to ensure that the (Perdew, Burke, and Ernzerhof (PBE)-level) E g,DFT is available. Additionally, a majority of rare-earth substituted phosphor hosts contain at least three elements, that is, are ternary phases or higher; therefore, in this study binary compounds are excluded. These two basic requirements reduced the number of possible compounds by 94% to 7504. The list of potential rare-earth substituted inorganic phosphor hosts could be further honed considering rare-earth substituted phosphor hosts are customarily composed of only the elements highlighted by the blue boxes shown in Fig. 2; any compound that does not follow this set of composition requirements was therefore also removed from the list of potential phosphors10. Finally, any compound following all of these criteria but still calculated to be a metal, that is, E g = 0, was also removed from the list narrowing the final number of possible phosphor hosts to 2071. The Θ D,SVR of these 2071 compositions was subsequently predicted with the machine-learning model and associated with the DFT calculated band gap (E g,DFT ). This entire dataset was used to screen potential inorganic phosphor hosts and is provided as Supporting Information.

Fig. 2 The elements that compose rare-earth substituted phosphor hosts. Potential phosphor hosts were restricted to the highlighted elements. Any compound containing any other element was excluded from the materials screening diagram Full size image

Plotting the 2071 compounds’ E g,DFT as a function of their predicted Θ D,SVR , illustrated in Fig. 3a, creates an expanded sorting diagram that is convenient for balancing these two properties. The darker regions on this plot represent a higher density of potential phosphor hosts. Some (8%) of the compounds on the sorting diagram have very narrow (0 eV < E g,DFT ≤ 0.4 eV) PBE-level band gaps, suggesting that these phases are not likely viable options as rare-earth substituted phosphor hosts. There are also a considerable number (22%) with a Θ D,SVR <250 K, indicating that these materials may not have the structural rigidity necessary for a high Φ.

Fig. 3 Machine learning predicted Debye temperature against calculated bandgap. a Machine learning predicted Debye temperature (Θ D,SVR ) against the density functional theory calculated bandgap (E g,DFT ) for 2071 compounds predicted. The darker regions occur where there are overlapping data. Classes of common phosphor hosts are highlighted, including b borates and sulfides, c nitrides and (oxy)halides, d silicates and fluorides, and e aluminates and phosphates Full size image

Decomposing the sorting diagram into the compositional information indicates several trends related to the potential of different phosphor hosts. For example, Fig. 3b shows that borates tend to have wide band gaps and surprisingly high Debye temperatures owing to boron’s small size that allows dense polyhedral packing. Sulfides have a Θ D,SVR < 500 K and an E g,DFT only in the range of 0–4 eV, which suggests that any emission will likely have a rather low Φ when substituted with a rare-earth ion. Conversely, nitrides (Fig. 3c) have moderate band gaps (<4.5 eV), but a generally high Θ D,SVR due to the covalently bonded, corner-sharing [SiN 4 ] tetrahedra. This allows the predicted Debye temperature to surpass 600 K for many of the compounds and supports the numerous efficient nitride based rare-earth inorganic phosphors. Silicates (Fig. 3d) and aluminates (Fig. 3e), which are perhaps the most common classes of phosphor hosts, tend to have wide band gaps, but they also have a range of prospective Θ D,SVR values. Experimentally there are many silicates and aluminates with a high luminescence quantum efficiency, most notably Y 3 Al 5 O 12 :Ce3+ or (Ba/Sr) 2 SiO 4 :Eu2+, as well as numerous reports of materials that have low luminescence quantum efficiency reinforcing the range of Debye temperatures and band gaps14,17. In contrast, fluorides (Fig. 3d) all show very wide band gaps ranging between 4 and 8 eV but low Debye temperatures (<500 K) due to the ionic bonding in these phases. Finally, phosphates (Fig. 3e) tend to have wide bandgaps >4 eV) and moderate Debye temperatures (≈500 K) signifying there is potential for continued development in this space. In fact, a significant number of rare-earth substituted phosphates have been recently reported such as Na 3 Sc 2 (PO 4 ) 3 :Eu2+ 15, BaRbPO 4 :Eu2+34, and LiSrPO 4 :Eu2+35, supporting the sorting diagram’s suggestion.

Analyzing metadata-like composition is clearly indispensable for directing the search for new materials, with the potential to shift the search toward remarkable composition space, such as focusing on borates. Indeed, a few high-efficiency borate phosphors have already been reported. For example, Ce3+-substituted Ba 2 Y 5 B 5 O 17 has a blue emission with an Φ of 70% at room temperature that can be increased to nearly 90% by substituting the harder cation Lu3+ for the comparatively softer Y3+36,37. Similarly, NaSrBO 3 :Ce3+ and NaCaBO 3 :Ce3+ both have a reported Φ of ≈75%38,39. Many of the borate phosphors including K 2 Al 2 B 2 O 7 :Eu2+ are also thermally robust showing minimal loss of luminescence at high temperature40. Although the search for borate phosphors is directed based on the machine-learning algorithm, the discovery of these materials was still achieved through traditional means including isovalent substitution of known materials, rationally searching phase space, or through serendipitous discovery.

The development of an enhanced sorting diagram also makes it possible to target specific compounds as prospective high Φ phosphors. For example, examining the sorting diagram shows that CaAlB 3 O 7 has a very high Θ D,SVR of 732 K and E g,DFT of 5.3 eV due to the presence of alternating layers of [AlO 6 ] octahedral and [BO 4 ] tetrahedra connected through edge and corner sharing, with the Ca2+ decorated among the very dense anionic network41. Li 5 Rb 2 B 7 O 14 is also a phase of interest given that Θ D,SVR is 703 K and E g,DFT is 4.5 eV. This phase is a non-centrosymmetric crystal structure that contains a three-dimensional network of polyborate chains producing the expected rigid crystal structure42. Both of these phases are viable phosphor hosts and worth consideration. Additionally, perhaps the most unique crystal structure suggested by the sorting diagram is NaBaB 9 O 15 (highlighted in Fig. 3b). This compound stands out among the 2071 potential phosphors due to its high Θ D,SVR of 729 K and E g,DFT of 5.5 eV. NaBaB 9 O 15 is a particularly remarkable compound because the composition suggests that the bonding should be primarily ionic and not a rigid crystal structure as implied by the Θ D,SVR . Therefore, experimentally investigating this unanticipated potential phosphor host will lend support to the sorting diagram as a tool for identifying new high Φ rare-earth substituted inorganic phosphors.

Phosphor synthesis and crystal structure refinement

The synthesis of NaBaB 9 O 15 yielded a phase pure product according to laboratory (Cu Kα) powder X-ray diffraction. The structure of the unsubstituted, that is, x = 0, sample was analyzed using high-intensity, high-resolution synchrotron X-ray powder diffraction data, shown in Fig. 4. The crystal structure was obtained from the Rietveld refinement with details provided in Table 1 and the refined atomic coordinates and isotropic displacement parameters listed in Table 2.

Fig. 4 Rietveld refinement of NaBaB 9 O 15 synchrotron X-ray powder diffraction data. The observed data are colored black, the refinement is colored red, and the difference is colored blue Full size image

Table 1 Rietveld refinement data of NaBaB 9 O 15 Full size table

Table 2 Unit-cell parameters determined using Rietveld refinement Full size table

NaBaB 9 O 15 adopts the non-centrosymmetric trigonal space group R3c (no. 161). As illustrated in Fig. 5, the crystal structure contains a three-dimensional framework of [B 3 O 7 ]5– subunits in which two [BO 3 ]3– trigonal planar units and one [BO 4 ]5– tetrahedron are linked by the vertices. The refined B–O bond lengths within the trigonal planes range between 1.33 and 1.38 Å, while the bond lengths within the [BO 4 ]5– tetrahedron are slightly longer (1.45–1.49 Å). The arrangement of the [B 3 O 7 ]5– units generates large tunnels along the [001] direction that are occupied by Ba2+ and Na+. The Ba2+ ions are coordinated in a nine-vertex distorted tri-capped trigonal prism, whereas six oxygen atoms form a highly distorted trigonal antiprism surrounding the Na+ ions. This combination of corner sharing [B 3 O 7 ]5– polyhedra is remarkable because it constructs a network that is not especially dense. In fact, the calculated density of NaBaB 9 O 15 is only 2.669 g cm–3 which is half the density of the other reported Na-Ba-B-O phases43,44. Such a porous anionic network would not be predicted to form a rigid crystal structure. However, the high Θ D likely arises due to the presence of high-frequency vibrational modes from the unique [B 3 O 7 ]5− structural unit, which forms an extended and intertwined polyanionic backbone. Moreover, the high concentration of very light boron atoms also gives rise to the presence of high-frequency phonon modes. This combination of composition and crystal structure both contribute to the anomalously high Debye temperature for NaBaB 9 O 15 . Because machine learning has minimal bias, the prediction is undiscriminating in which crystal structures are suggested, highlighting the importance of using this approach to identify phases that may otherwise be disregarded.

Fig. 5 Crystal structure of NaBaB 9 O 15 . a Crystal structure of NaBaB 9 O 15 viewed in [010] direction with the associated [BaO 9 ], [NaO 6 ], and [B 3 O 7 ] polyhedral subunits highlighted. b The arrangement of the [B 3 O 7 ]5– units generates large tunnels along the [001] direction, which are alternately filled by Ba2+ and Na+ Full size image

Once the crystal structure is refined from high-resolution scattering data, it is possible to experimentally estimate Θ D based on the Debye–Waller theory as a confirmation of the machine-learning prediction45,46. Although this is a considerable approximation that assumes a single acoustic branch and ignores the optical modes, it is still valuable for comparing the machine-learning predicted Debye temperature to an experimental value. Using the mean-square atomic displacement parameters, <\(u_{\mathrm{eq}}^2\)>, the Θ D,i can be calculated for each crystallographically independent atom, i, following Eq. 4, where m is the atomic mass and u eq,i is the atomic displacement parameter. The overall Debye temperature (Θ D ) can then be approximated by taking the stoichiometrically weighted average of Θ D,i according to the formula NaBaB 9 O 15 :

$$\Theta _{{\mathrm{D}},i} = \sqrt {\frac{{3\hbar ^2TN_{\mathrm{A}}}}{{m_ik_{\mathrm{B}}u_{{\mathrm{eq}},i}}}}.$$ (4)

Evaluating for Θ D using the refined <\(u_{\mathrm{eq}}^2\)> (Table 2) suggests that the Debye temperature should be ≈635 K, which is only 12% lower than the machine learning predicted Θ D,SVR . This is in an excellent agreement considering that the B and G values used in the training set are derived from DFT using the Voigt–Reuss–Hill approximation and neglects the temperature dependence of the elastic moduli. Moreover, the scattering refinement data are based on synchrotron X-ray diffraction, which is not particularly adept at evaluating atomic displacement parameters. Nevertheless, the small difference indicates that machine learning is indeed a viable technique for quickly estimating Θ D,SVR .

Photoluminescent properties of the identified phosphor

The high Θ D,SVR and wide PBE-level calculated band gap suggests that NaBaB 9 O 15 has strong potential as an inorganic phosphor host. Considering that the PBE-level calculations significantly underestimate E g , a HSE06 hybrid functional calculation was conducted using the Vienna ab initio Simulation Package47,48,49 to ensure that this compound has a sufficient band gap to tolerate the rare-earth’s electronic transitions. The result confirms NaBaB 9 O 15 indeed has a very wide calculated E g,HSE of 7.3 eV. The phosphor was then prepared by substituting Eu2+ as the luminescent center for Ba2+ following NaBa 1–x Eu x B 9 O 15 (x = 0, 0.005, 0.01, 0.02, 0.03, 0.04, and 0.05). The compound maintains phase purity across the entire substitution range examined, as plotted in Supplementary Figure 1. Measuring the photoluminescence excitation spectrum indicates that Eu2+ has a broad ultra-violet excitation band ranging from 230 to 385 nm, as shown in Fig. 6a. The spectrum contains a main excitation peak centered at λ ex = 275 along with two shoulders at λ ex = 315 and λ ex = 365 nm. The emission spectra collected by exciting at these three wavelengths produces an identical emission peak with a maximum emission wavelength at 416 nm, indicating that the multiple excitation peaks arise from the electronic transitions between the Eu2+ 4f orbitals to multiple Eu2+ 5d orbital excited states. The emission spectra can be fit by a single Gaussian peak, depicted in Fig. 6a (and Supplementary Figure 2b), corresponding to the emission from a single crystallographically independent Eu2+. This is in agreement with the refined crystal structure that only contains one substitution site, Ba(1). The single Eu2+ site was further supported by measuring the photoluminescence lifetime (Supplementary Figure 3). The data could be fit with a single exponential function described in Eq. 5:

$$I = I_0 + Ae^{\frac{{ - t}}{\tau }},$$ (5)

where I is the luminescent intensity, t is time, A is parameter, and τ is the decay time, to reveal that the luminescence decay occurs with a 0.842 μs decay time. The emission is a bright violet with Commission Internationale de l'Eclairage (CIE) 1931 coordinate of (0.1620, 0.01522) and has a very narrow full-width at half-maximum (FWHM) of only 1962 cm–1 (34.5 nm). These data show that this material eclipses the industry standard blue-emitting phosphor BaMgAl 10 O 17 :Eu2+, which has a FWHM of ≈2200 cm−1 (≈50 nm)50, but NaBaB 9 O 15 :Eu2+ is still broader than the recently reported blue-emitting AELi 2 [Be 4 N 6 ]:Eu2+ (AE = Sr, Ba) phosphors that have a FWHM of only ≈1200 cm−1 (≈25 nm)51.

Fig. 6 Photoluminescent properties of NaBa 0.97 Eu 0.03 B 9 O 15 . a Excitation (dashed line) and emission (black solid line) spectra measured at room temperature. The Gaussian fit of the emission spectrum is solid gray. b Measured photoluminescent quantum yield (PLQY) of NaBaB 9 O 15 substituted with varying concentrations of the rare-earth Eu2+ under 315 nm excitation. PLQY for each concentration was measured three times and the error bar represents the standard deviation Full size image

Given that the crystal structure of NaBaB 9 O 15 host is likely a highly rigid phosphor host based on the predicted Θ D,SVR , the Φ of NaBaB 9 O 15 :Eu2+ system was extensively analyzed. First, the rare-earth concentration was optimized by varying the loading of Eu2+. As shown in Fig. 6b, the 3 mol% Eu2+ concentration in NaBaB 9 O 15 :Eu2+ has the highest Φ = 95(1)%, and increasing the Eu2+ concentration causes an abrupt drop in Φ. Exciting the phosphor at 340 nm shows the same optimized concentration of Eu2+ with a slight decrease in Φ to ≈75% with an increase in the rare-earth content beyond 3 mol% again causing a drop in Φ. The origin of the emission loss stems from an energy transfer process between adjacent luminescence centers. The critical distance between luminescent centers for this concentration quenching process (R C ) can be calculated following Eq. 6, where V is the volume of the unit cell, x c is the optimized rare-earth concentration, and N is the number of crystallographically independent Eu2+ positions in a unit cell52. In NaBaB 9 O 15 :Eu2+, using a volume of 1857.28 Å, an optimized concentration of x = 0.03, and six Eu2+ crystallographic sites, the critical distance is approximately 27 Å. Such a large R C means that self-quenching is not a very prominent mechanism for non-radiative relaxation in this crystal structure, partly explaining the excellent Φ:

$$R_{\mathrm{C}} \approx 2\left( {\frac{{3V}}{{4\pi x_{\mathrm{c}}N}}} \right)^{1/3}.$$ (6)

Beyond concentration quenching, inorganic phosphors can also undergo thermal quenching, where the luminescence intensity is lost due to an increase in non-radiative relaxation. Probing the temperature dependence of the emission spectrum from 80 to 500 K (Supplementary Figure 4) reveals that NaBa 0.97 Eu 0.03 EuB 9 O 15 possesses an outstanding thermal response. As shown in Fig. 7a, the emission spectrum broadens slightly with increasing temperature going from FWHM = 32.5 nm (1849.5 cm–1) at 80 K to FWHM = 40.1 nm (2286.1 cm–1) at 500 K. This change can be attributed to increased electron–phonon line broadening at elevated temperatures associated with the excited state 5d orbitals of Eu2+5. More importantly, the emission spectrum at high temperature does not show any thermal quenching, which is the loss of luminescence with increasing temperature, even as the temperature approaches 500 K (Fig. 7b). There is a slight loss of emission intensity once the temperature is >360 K; however, this decrease is only by ≈15%. The integrated area of the emission peak shows no change across the entire temperature range examined because the loss of emission intensity is accompanied by a widening of the FWHM of the emission spectrum, countering the loss of intensity and leading to minimal thermal quenching up to 500 K.

Fig. 7 Temperature-dependent photoluminescent properties of NaBa 0.97 Eu 0.03 B 9 O 15 phosphor. a Contour plot of the normalized emission spectra excited at 340 nm as a function of temperature. b Relative integrated intensity of the emission spectra (rel. intger. int.) and the relative intensity of the emission peak (rel. peak int.)—as a function of temperature Full size image

Temperature-dependent powder X-ray diffraction data (Supplementary Figure 5a) indicate that there is no structural modification at elevated temperatures based on refining the crystal structure’s unit cell volume as a function of temperature (Supplementary Figure 5b). Thus, the origin of the outstanding thermal properties in this phosphor can be explained by evaluating the two classical non-radiative pathways that dominate thermal quenching53. One process is through thermally induced photoionization, where increasing the temperature can lead to the excited-state electrons occupying the Eu2+ 5d orbitals being thermally promoted into the phosphor host’s conduction band preventing re-emission. The best method to restrict photoionization is by ensuring that the phosphor host has strong crystal field splitting and a very wide E g . Separating the rare-earth orbitals from the host conduction band mitigates the probability of thermal excitation at any reasonable temperature, and, therefore, prevents this quenching pathway. Although the crystal field splitting is not considerable, bearing in mind that the borate polyanion is a weak-field ligand, the E g,HSE for NaBaB 9 O 15 :Eu2+ is still 7.3 eV. This sizeable band gap is likely sufficient to separate rare-earth orbitals from the host conduction band, thereby minimizing this quenching pathway.

The second non-radiative relaxation process occurs via electron–phonon coupling of the ground state and excited state potential energy surfaces53. In rare-earth substituted phosphors, the excitation of an electron from the ground state to the excited state causes a simultaneous relative displacement (Δr) of the excited state nuclear coordinates with respect to the ground state nuclear coordinates. If the Δr separating the potential energy surfaces is large, then the two potential energy surfaces can cross (interact), which supplies a mechanism for non-radiative relaxation via electron–phonon coupling13. Conversely, in systems where Δr is small, the potential energy surfaces do not interact and the relaxation process can proceed uninhibited, ideally by luminescence. Empirically the best approach for preventing a large Δr, as it relates to changes in bond lengths, might be to target rigid crystal structures13,54. However, recent research has showed that Θ D , as a proxy for rigidity, is not always predictive of thermal quenching55,56.

Alternatively, a method to probe Δr in a solid state material is to estimate the electron–phonon coupling by ascertaining a material’s Huang–Rhys factor S57,58. Experimentally, S can be extracted by measuring the FWHM of the rare-earth substituted phosphor’s emission peak as a function of temperature and fitting these data with Eq. 7:

$${\mathrm{FWHM}}(T) = \sqrt {8\;{\mathrm{ln}}2} \;\hbar {\mathrm{\omega }}\;\sqrt {S\;{\mathrm{coth}}\frac{{\hbar \omega }}{{2k_{\mathrm{B}}T}}},$$ (7)

where ℏω is the mean phonon energy and k B is the Boltzmann constant59,60. A system that has strong electron–phonon coupling (large Δr) tends to have S > 5, whereas weak electron–phonon coupling (small Δr) occurs when S < 1. Fitting the temperature-dependent luminescence data for NaBaB 9 O 15 :Eu2+ (Supplementary Figure 6) reveals the Huang–Rhys factor is only 2.098, indicating that this material is approaching the weak electron–phonon coupling regime, and, therefore, should have a relatively small Δr. The consequence is that the two potential energy surfaces in this phosphor are not expected to intersect and the electron–phonon coupling is predicted to be minimal. Thus, this mechanism is also not anticipated to be a primary contributor to non-radiative relaxation in this phosphor. The combination of limited concentration quenching, the absence of photoionization as a quenching mechanism, and the lack of overlap between the ground and excited state potential energy surfaces leads to the robust thermal behavior for NaBaB 9 O 15 :Eu2+.

Conclusions