Searching for silicate nonlinear optical materials by combining calculation and experiment

Abstract

Nonlinear optical (NLO) materials have been applied in many high technology fields. In this work, the first-principles high-throughput screening pipeline (FHSP) has been used for searching NLO materials from the silicate system. 253 silicates collected in the Inorganic Crystal Structural Database (ICSD) have been successfully calculated. Among them, 8 crystals were screened out after two-step evaluation. Finally, 3 compounds were selected for synthesis after comprehensive investigation. The SHG intensities tested using the Kurtz–Perry method of KNbSi₂O₇, PbZnSiO₄, and Na₂TiSiO₅ are 6.7, 2.9, and 2.8 times that of KDP, respectively. Their UV absorption edges are 243 nm, 276 nm, and 280 nm, respectively. This work provides a full view of silicates as NLO materials and obtains excellent candidates for the next step of development.

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Introduction

Nonlinear optical (NLO) crystals are key materials in laser systems owing to their functions of frequency conversion, generation of entangled photon pairs, attosecond pulse generation, etc.^1–5 Frequency conversion through NLO materials is a convenient and efficient approach to generate laser beams in a wider range.^6–8 β-BaB₂O₄ (BBO),⁹ LiB₃O₅ (LBO),¹⁰ AgGaSe₂ (AGSe),¹¹ AgGaS₂ (AGS),¹² and ZnGeP₂ (ZGP)¹³ are commercialized materials that are widely used in the ultraviolet (UV) and mid-far-infrared (M-F-IR) wavelength range.¹⁴ With the development of laser technology, NLO crystals with optimal performance are urgently required.

Silicates have wide transmission ranges from ultraviolet to mid-IR such as those of Bi₁₂SiO₂₀ (0.37–6.0 μm),^15,16 Ba₂TiOSi₂O₇ (0.3–5.0 μm),¹⁷ Y₂SiO₅ (0.2–5.0 μm), Gd₂SiO₅ (0.2–5.0 μm), and Lu₂SiO₅ (0.2–5.0 μm).¹⁸ In addition, the silicate system has potential to generate large SHG coefficients, such as those of Li₂K₄[(TiO)Si₄O₁₂] (4.5 × KDP) and Li₂Rb₄[(TiO)Si₄O₁₂] (4.5 × KDP),¹⁹ Ba₂TiOSi₂O₇ (5.5 × KDP),¹⁷ PbSrSiO₄ (5.8 × KDP),²⁰ Sr₂TiSi₂O₈ (8.0 × KDP),²¹ and Cs₂B₄SiO₉ (4.6 × KDP).²² Besides, silicates usually exhibit large band gaps and correspondingly high laser-induced damage thresholds (LDTs), as well as stability in open air.

In this work, non-centrosymmetric silicate crystals collected in the Inorganic Crystal Structural Database (ICSD) are comprehensively investigated using the first-principles high-throughput screening pipeline for NLO materials (FHSP-NLO) and experimental confirmation.²³ Finally, three silicates, KNbSi₂O₇, PbZnSiO₄, and Na₂TiSiO₅, meeting strict multiple criteria are screened out and synthesized using a high-temperature solid-state method. They possess excellent properties, including large SHG coefficients, suitable birefringence, and short cut-off edges. In addition, statistical analysis based on the different classification of cations shows that d⁰-transition metal (TM) and lone pair (LP) cations play a vital role in birefringence and SHG response.

Methods

Synthesis

KNbSi₂O₇, PbZnSiO₄, and Na₂TiSiO₅ were synthesized by the conventional solid-state method. The initial reagents including SiO₂ (Aladdin, 99%), K₂CO₃ (Aladdin, 99.5%), Nb₂O₅ (Aladdin, 99.9%), PbO (Aladdin, 99.9%), ZnO (Aladdin, 99%), NaOH (Aladdin, 95%) and TiO₂ (Aladdin, 99%) were used without further purification. Stoichiometric amounts of reactants were mixed thoroughly and heated in the corundum crucibles. The reactants, reaction time, and temperature for each compound are listed below.

KNbSi₂O₇. KNbSi₂O₇ was synthesized using K₂CO₃ (0.086 g), Nb₂O₅ (0.166 g), and SiO₂ (0.150 g). After weighing the raw materials, they were ground evenly, and the mixtures were transferred into an alumina crucible. The mixtures were heated from room temperature to 1170 °C in 5 h, held at that temperature for 24 h, cooled to 1100 °C in 70 h, then cooled to 1000 °C in 72 h, and finally cooled to room temperature in 24 h.

PbZnSiO₄. PbZnSiO₄ was obtained using PbO (1.114 g), ZnO (0.406 g), and SiO₂ (0.300 g). The mixtures were heated from room temperature to 750 °C in 3 h, held at that temperature for 72 h, and cooled to room temperature in 3 h. The mixtures were then removed and thoroughly ground. The same method was used for refiring twice to improve the purity of the material.

Na₂TiSiO₅. Na₂TiSiO₅ was synthesized using NaOH (0.200 g), TiO₂ (0.200 g), and SiO₂ (0.150 g) as raw materials. The mixtures were heated from room temperature to 850 °C in 6 hours and kept at this temperature for 10 hours, then finally cooled to room temperature in 3 h.

Powder XRD

KNbSi₂O₇, PbZnSiO₄, and Na₂TiSiO₅ were tested using a Bruker-D8 X-ray diffractometer. The experimental XRD spectra of the title compounds were measured using X-rays with Cu Kα radiation (λ = 1.541 Å) at room temperature, and then compared with the theoretical XRD pattern.

Infrared spectroscopy

The IR spectra of KNbSi₂O₇, PbZnSiO₄, and Na₂TiSiO₅ were recorded on an IR Affinity-1 Fourier transform infrared spectrometer (Nicolet iS10, America) in the range of 400–3000 cm⁻¹. The silicates were ground and mixed with KBr.

UV-vis-NIR diffuse-reflectance spectra

The UV-vis-NIR diffuse reflectance spectra of KNbSi₂O₇, PbZnSiO₄, and Na₂TiSiO₅ were measured at room temperature with a SolidSpec-3700DUV spectrophotometer. The wavelengths of the spectra were in the range 190–1100 nm.

SHG measurement

Through the Kurtz-Perry method,²⁴ the powder SHG (PSHG) responses of KNbSi₂O₇, PbZnSiO₄, and Na₂TiSiO₅ were investigated using a Q-switch laser at 1064 nm. Silicates and KDP were ground and sieved into different particle size ranges: 20–38, 38–55, 55–88, 88–125, 125–160, 160–200, and 200–250 μm.

Computational method

Theoretical calculations were performed using the CASTEP package based on density functional theory (DFT).²⁵ The CASTEP package has been used to perform self-consistent field (SCF) and optical calculations with normal conserved pseudopotentials (NCPs). The exchange correlation energy is described by the Perdew–Burke–Ernzerhof (PBE) functional within the generalized gradient approximation (GGA).^26,27 The energy cutoff of the plane-wave base was set to 750 eV. The SCF calculations were performed with a convergence criterion of 1 × 10⁻⁶ eV per atom on total energy. Numerical integration of the Brillouin zone was performed using the Monkhorst–Pack scheme with k-point separation set as 0.07 Å⁻¹ in the Brillouin zone for SCF and 0.04 Å⁻¹ for optics calculation. The empty band was set to 3 times the valence band in the optical calculation to ensure the convergence of the refractive index and SHG coefficients.

Results and discussion

First, 279 noncentrosymmetric (NCS) structures containing Si and O elements were obtained from the Inorganic Crystal Structural Database (ICSD). Then, the band gap, birefringence, and second-order susceptibility χ⁽²⁾ of these silicates were calculated using FHSP-NLO. Finally, 253 structures were calculated successfully. The chemical formula, ICSD collection-code, space group, band gap (E_g), birefringence (Δn), second-order susceptibility (χ⁽²⁾), and the maximum χ⁽²⁾ tensor of all 253 structures are listed in the ESI.† As shown in Fig. 1, the maximum χ⁽²⁾ tensors of these silicate crystals are plotted against their band gaps E_g-GGA. One can easily find that the maximum χ⁽²⁾ obviously decreases with the increase in the band gap, which is consistent with the theory.

Fig. 1 The statistical chart of calculated results for 253 silicates.

As we mentioned above, silicates have a wide transmission window from UV to mid-IR and accordingly they can be used as NLO materials working in a wide wavelength range. In order to screen the materials that meet the conditions from UV to mid-IR, we adopted a new screening method. Firstly, we screened out the structures that meet the criteria for UV NLO crystals, i.e. E_g > 6.0 eV and χ⁽²⁾ > 0.78 pm V⁻¹. Secondly, the structures that meet the criteria for IR NLO crystals (E_g > 2.5 eV and χ⁽²⁾ > 7.8 pm V⁻¹) were screened out. Thirdly, we joined the two points (marked as G (2.5, 7.8) and H (6, 0.78)) with a straight line. Then, the materials located to the top-right of this line were screened out. Finally, 17 structures (located in the green shading of Fig. 1) with excellent properties were selected.

The calculated properties of the selected 17 structures are listed in Table 1. There are 14 materials because some structures are duplicated. It is well known that the PBE functional tends to underestimate the band gap. To obtain accurate predictions of the band gap, a HSE (hybrid screened exchange) functional was used to perform the band structure calculation.^28–30 Both the values, E_g-GGA and E_g-HSE, are listed in Table 1. Based on our previous calculations, it is found that the sum-over-states method cannot precisely predict the birefringence of materials that contain Ti or Nb element, such as Ba₂TiGe₂O₈.³¹ To improve the accuracy of the birefringence calculation, the density-functional perturbation theory (DFPT) method was adopted for the selected structures. Then we screened the materials with birefringence calculated using the DFPT method greater than 0.04. Subsequently, 8 materials were selected from the above 14 materials, namely PbZn(SiO₄),³² Na₂TiSiO₅,³³ Ca₂(SiO₄),³⁴ KNbSi₂O₇,³⁵ CaZr(Si₂O₇),³⁶ BaY₆(Si₃B₆O₂₄)F₂,³⁷ Ba₂Y₃(SiO₄)₂F₅,³⁸ and CsSiB₃O₇.³⁹ After investigation and experimental verification, we chose three materials for further synthesis and characterization, i.e. KNbSi₂O₇, PbZnSiO₄, and Na₂TiSiO₅. For the other materials, Ca₂(SiO₄) shows a virtual vibration frequency that demonstrates the dynamic instability of its structure. Indeed, a centrosymmetric structure that is more reasonable than this NCS structure had been reported.⁴⁰ Actually, when we synthesized this material the obtained phase was a centrosymmetric structure. The synthesis temperature of CaZr(Si₂O₇) is too high, and our laboratory conditions cannot meet the requirements. BaY₆(Si₃B₆O₂₄)F₂ and Ba₂Y₃(SiO₄)₂F₅ are multiphase structures, and the pure phase products are difficult to obtain. The NLO properties of CsSiB₃O₇ have been reported by Sun and Lin et al.,³⁹ and its SHG effect is 0.8 times that of KDP.

Table 1 Chemical formulas, ICSD collection numbers, space groups (SG), calculated band gaps both using GGA and HSE06 (E_g-GGA and E_g-HSE, units: eV), birefringences calculated using the SOS and DFPT method (Δn-SOS and Δn-DFPT) at 1064 nm, and SHG coefficients (χ⁽²⁾) of 17 selected materials. The scissor operators that are set as the difference between E_g-GGA and E_g-HSE are used in the calculation of χ⁽²⁾

No.	Formula	ICSD	SG	E g-GGA (eV)	E g-HSE (eV)	Δn-SOS	Δn-DFPT	χ ^(2) (+scissor) (pm V^−1)
1	Ba2TiSi2O8	201 844	P4bm	4.05	5.82	0.060	0.034	χ 113 = χ223 = −3.064, χ333 = 10.654
2	Ba2TiSi2O8	201 845	P4bm	4.05	5.83	0.061	0.035	χ 113 = χ223 = −3.039, χ333 = 10.631
3	Ba2TiSi2O8	4451	P4bm	4.06	5.78	0.060	0.033	χ 113 = χ223 = 3.077, χ333 = −10.835
4	PbZn(SiO4)	26 840	Pna21	3.26	4.21	0.033	0.045	χ 113 = −1.811, χ223 = −1.134, χ333 = 8.959
5	Pb4Zn2[(SO4)(SiO4)(Si2O7)]	100 296	P21	3.57	4.69	0.011	0.009	χ 112 = 0.513, χ123 = 1.074
								χ 222 = −7.164, χ233 = 0.766
6	Na2TiSiO5	259 048	Pmc21	3.48	5.30	0.026	0.046	χ 113 = −1.787, χ223 = −0.194, χ333 = 5.454
7	Ca2(SiO4)	182 053	P63mc	3.55	4.07	0.078	0.057	χ 113 = χ223 = −1.017, χ333 = −7.814
8	Na2TiSiO5	82 153	Pmc21	3.48	5.24	0.026	0.049	χ 113 = −1.872, χ223 = −0.178, χ333 = 5.582
9	KNbSi2O7	72 111	P4bm	2.93	5.32	0.125	0.071	χ 113 = χ223 = 2.971, χ333 = −0.81
10	Ba(Li2SiO4)	260 259	P63mc	4.44	5.72	0	0.001	χ 113 = χ223 = 0.085, χ333 = −4.369
11	CaZr(Si2O7)	203 131	C2	4.5	6.59	0.062	0.052	χ 112 = −0.799, χ123 = −1.858
								χ 222 = −1.075, χ233 = 1.562
12	BaBe(SiO4)	86 792	Cm	4.74	5.66	0.020	0.019	χ 111 = −0.027, χ113 = −2.262, χ122 = 1.720
								χ 133 = −2.666, χ223 = −0.980, χ333 = 3.162
13	BaY6(Si3B6O24)F2	30 674	P3	5.40	6.73	0.033	0.045	χ 111 = −χ122 = −0.494, χ113 = χ223 = 2.517
								χ 333 = −3.105
14	Ba2Y3(SiO4)2F5	290 289	Pba2	5.09	6.58	0.035	0.046	χ 113 = −0.371, χ223 = 0.401, χ333 = −2.173
15	BaBe2(Si2O7)	263 133	Pmn21	5.83	7.58	0.017	0.006	χ 113 = 0.384, χ223 = 0.643, χ333 = −1.620
16	CsSiB3O7	255 774	Pna21	5.43	7.27	0.044	0.056	χ 113 = −0.400, χ223 = −0.962, χ333 = 1.150
17	LiB(SiO4)	67 536	I4̄	6.58	8.51	0.004	0.008	χ 113 = −0.158, χ123 = 0.705, χ223 = 0.158

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The powder XRD spectra of KNbSi₂O₇, PbZnSiO₄, and Na₂TiSiO₅ match well with the simulated ones derived from the CIF data (see Fig. 2(a)–(c)). The Rietveld refinement of PXRD data of pure polycrystalline samples was performed (Fig. S1–S3†). The details of the Rietveld refinement of PXRD are provided in Table S1.† It verified the phase purity of the title compounds. In addition, the existence of the fundamental building blocks (FBBs) was confirmed using the experimental IR spectra.⁴¹ The IR spectra of KNbSi₂O₇, PbZnSiO₄, and Na₂TiSiO₅ samples in the wavenumber range 400–3000 cm⁻¹ are shown in Fig. 2(d)–(f). According to calculated IR spectrum and the experimental results of IR spectroscopy, the IR absorption band near 900 cm⁻¹ belongs to the stretching vibration of SiO₄ tetrahedra. The IR absorption band from 600 to 400 cm⁻¹ belongs to the bending vibration of SiO₄ tetrahedra.⁴² In particular, in the compound KNbSi₂O₇, the IR spectral characteristic peak at 1234 cm⁻¹ belongs to the Si–O–Si stretching vibration in the Si₂O₇ dimer. The peak at 706 cm⁻¹ is due to the Nb–O stretching modes of NbO₆ octahedra. The above infrared results can confirm the existence of fundamental building blocks (FBBs) in the title compounds.

Fig. 2 Powder XRD of (a) KNbSi₂O₇, (b) PbZnSiO₄, and (c) Na₂TiSiO₅. Infrared spectroscopy of (d) KNbSi₂O₇, (e) PbZnSiO₄, and (f) Na₂TiSiO₅.

The crystal structure of KNbSi₂O₇ belongs to NCS polar space group P4bm with a = b = 8.7404(8) Å, c = 8.136(1) Å, α = β = γ = 90°, Z = 2. In its structure, NbO₆ octahedrons are connected to each other by sharing para-position O atoms to form chains. The chains are further connected by a Si₂O₇ dimer to form a three-dimensional (3D) skeleton with channels where K atoms are located (see Fig. 3(a)). The UV-vis-NIR diffuse reflectance spectra of KNbSi₂O₇, PbZnSiO₄, and Na₂TiSiO₅ are shown in Fig. 4(a)–(c). The spectra indicate that KNbSi₂O₇ has a short UV cutoff edge of 243 nm, and the absorption edge of KNbSi₂O₇ is 4.31 eV, which is larger than those of Li₂K₄[(TiO)Si₄O₁₂] and Li₂Rb₄[(TiO)Si₄O₁₂].¹⁹ KNbSi₂O₇ exhibits a strong SHG intensity of 6.7 times that of KDP under 1064 nm fundamental wave laser radiation (Fig. 4(d)). It is larger than those of recently reported silicate NLO materials, such as Li₂K₄[(TiO)Si₄O₁₂] (4.5 × KDP) and Li₂Rb₄[(TiO)Si₄O₁₂] (4.5 × KDP),¹⁹ Ba₂TiOSi₂O₇ (5.5 × KDP),¹⁷ PbSrSiO₄ (5.8 × KDP),²⁰ and Sr₂ZnSi₂O₇ (35 × α-SiO₂).⁴³ It is worth noting that Crosnier et al. has tested the SHG effect of KNbSi₂O₇ to get an estimate of the phase transition temperature and identify it as a good candidate for a nonlinear optical material.³⁵ Subsequently, Gopalakrishnan et al. measured the SHG intensity of KNbSi₂O₇ as 200 times that of α-SiO₂.⁴⁴ The above results support our calculations and experimental results. Afterwards, single crystals of KNbSi₂O₇ with a size of 10 × 10 × 1 mm³ were fabricated.⁴⁵ In order to investigate the structure–property relationship, the NLO-related properties were calculated by using DFT method. The partial density of states (PDOS) (see Fig. 5(a)) reveals that the valence band maximum (VBM) is predominantly contributed by the orbitals of O nonbonding 2p. The main components of the conduction band minimum (CBM) are the 4d orbitals of Nb. The SHG-weighted densities presented in Fig. 5(b) and (c) illustrate that the electron densities that mainly contribute to SHG arise from O and Nb atoms in occupied and unoccupied states, respectively.

Fig. 3 (a) Crystal structures of KNbSi₂O₇. (b) Crystal structures of PbZnSiO₄. (c) Crystal structures of Na₂TiSiO₅.

Fig. 4 UV–vis–NIR diffuse reflectance spectra of (a) KNbSi₂O₇, (b) PbZnSiO₄, and (c) Na₂TiSiO₅. (d) Phase-matching measurement results for KNbSi₂O₇, PbZnSiO₄, Na₂TiSiO₅ and KDP.

Fig. 5 (a) Partial DOS of KNbSi₂O₇, SHG density in the (b) occupied state and (c) unoccupied state of KNbSi₂O₇.

PbZnSiO₄ compound crystallizes in a NCS polar space group Pna2₁ and the cell parameters are a = 8.2441(1) Å, b = 18.96259(20) Å, c = 5.06(1) Å, α = β = γ = 90°, Z = 8. In PbZnSiO₄, the 3D network is constructed by the ZnO₄ and the SiO₄ tetrahedron through shared corner O atoms. The Pb atoms are located in the vacancies of the network (see Fig. 3(b)). PbZnSiO₄ exhibits a UV cutoff edge of 276 nm and an absorption edge of 3.27 eV. The measured SHG response of PbZnSiO₄ is about 2.9 times that of KDP (Fig. 4(d)). The PDOS (see Fig. S4(a)†) reveals that the VBM is predominantly contributed by the nonbonding 2p orbitals of O. The main components of the CBM are predominantly contributed by the Pb 6p orbitals. One can easily find that the 6s orbitals of Pb appear at the VBM. This is the obvious character of the stereochemical active lone pair (SCALP) effect of the Pb atom. The SHG weighted density shows that the main source of electron density is the O atom in the occupied state (see Fig. S5(a)†) and Pb atoms and O atoms in the unoccupied state (see Fig. S5(b)†).

Na₂TiSiO₅ crystallizes in the crystal system and belongs to the Pmc2₁ space group with the cell parameters a = 9.1222(5) Å, b = 4.8040(3) Å, c = 9.8293(5) Å, α = β = γ = 90°, Z = 4. The structural unit of Na₂TiSiO₅ is composed of two independent Na atoms, one Ti atom, two Si atoms and five O atoms. Each Si atom is coordinated by four O atoms to form a SiO₄ tetrahedron. Each Ti atom is coordinated by five O atoms to form a TiO₅ quadrangular pyramid. The compressed TiO₅ square pyramid connects with the SiO₄ group by sharing four bottom O atoms of TiO₅ to form a 3D network. The top O atoms are only coordinated with the Ti atom. The sodium ions are located in the voids of the structure to balance the charge of this anionic framework (see Fig. 3(c)). The short UV cutoff edge of Na₂TiSiO₅ is determined to be about 280 nm and its absorption edge is 3.66 eV. Na₂TiSiO₅ shows a SHG intensity 2.8 times that of KDP at 1064 nm laser irradiation (Fig. 4(d)). The PDOS (see Fig. S4(b)†) reveals that the VBM is predominantly contributed by the 2p orbitals of O. The main components of the CBM are predominantly contributed by the 3d orbitals of Ti. The SHG weighted density shown in Fig. S6 (a) and (b)† indicates that O atoms, especially the top O atoms in the TiO₅ quadrangular pyramid, are the main source in the occupied state. In the unoccupied state, the main sources are Ti atoms.

As shown in Table 1, one can find that cations have a significant effect on the band gap, birefringence, and SHG coefficients. In order to clarify the influence of different cation types on NLO-related properties, the 253 structures are divided into four categories according to their cation types: (1) d⁰-TM cations, (2) lone pair (LP) cations, (3) d¹⁰-TM cations, and (4) alkalis and alkaline-earth metal cations as well as trivalent rare-earth (TRE) cations. If the crystal contains more than one kind of cation, the crystal will be classified as the former categories as listed above. As show in Fig. 6, the χ⁽²⁾ and Δn dispersions of different cations are obviously different. We count the average performance of 253 silicates according to the cation kinds. The duplicate structures are excluded from the statistic. As shown in Fig. 7, one can find that cation types have an obvious influence on the E_g, Δn, and χ⁽²⁾ values. The d⁰-TM, LP, and d¹⁰-TM cations reduce the band gap compared to the alkali/alkaline-earth/TRE metal cations. In contrast, they can induce larger χ⁽²⁾ and Δn. In particular, d⁰-TM and LP cations can significantly improve the χ⁽²⁾ values.

Fig. 6 Using GGA functionals the maximum χ⁽²⁾ and birefringence (Δn) vs. band gap calculated of 253 selected silicates divided into four categories: structures containing (1) d⁰-TM cations, (2) lone pair (LP) cations, (3) d¹⁰-TM cations, and (4) alkali metal and alkaline-earth metal cations.

Fig. 7 Statistical average values of the band gap (E_g-GGA), maximum SHG tensor (|χ⁽²⁾|_max), and birefringence (Δn) of silicates that contain d⁰-TM cations, LP cations, d¹⁰-TM cations, alkalis, alkaline-earth metal cations, and trivalent rare-earth cations (IA, IIA, TRE).

Conclusion

In summary, an effective first-principles high-throughput screening pipeline (FHSP) based on the DFT method is adopted for the silicate system for searching for excellent NLO materials. Subsequently, 8 compounds are screened out by two-step calculation. Finally, KNbSi₂O₇, PbZnSiO₄, and Na₂TiSiO₅ are selected and synthesized successfully using the conventional solid-state method. KNbSi₂O₇ exhibits a large birefringence of 0.071 and a strong phase-matchable SHG intensity of 6.7 times that of KDP. PbZnSiO₄ and Na₂TiSiO₅ exhibit SHG intensities of 2.9 and 2.8 × KDP at 1064 nm, respectively. The UV absorption edges of the three title compounds are 243 nm, 276 nm, and 280 nm, respectively. Based on the statistic results of all the 253 silicates we calculated, one can find that the d⁰-TM and LP cations are significant sources for SHG and birefringence and are the preferred choice for exploring new NLO materials in silicates. This work provides a full view of silicates to search for new NLO materials and obtain excellent candidates for the next step of development.

Conflicts of interest

The authors declare no competing financial interest.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (grant No. 52072109, 21975062, and 22101069) and the Science and Technology Project of Hebei Education Department (grant No. BJK2023029 and QN2023234).

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Footnote

† Electronic supplementary information (ESI) available: Calculated band structure and partial density of states (PDOS) of PbZnSiO₄ and Na₂TiSiO₅. The chemical formula, ICSD collection-code, space group, band gap (E_g), birefringence (Δn), second-order susceptibility (χ⁽²⁾), and maximum χ⁽²⁾ tensor (|χ⁽²⁾|_max) of all 253 silicates. Database is accessed by visiting https://nlo.hbu.cn. See DOI: https://doi.org/10.1039/d3qi00979c

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