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# Ba3Mg3(BO3)3F3 polymorphs with reversible phase transition and high performances as ultraviolet nonlinear optical materials

## Abstract

Nonlinear optical (NLO) materials are the vital components of future photoelectric technologies as they can broaden the tunable wavelength range supplied by common laser sources. However, the necessary prerequisites for a practical NLO material are rather strict. Accordingly, considerable efforts have been focused on finding potential NLO materials. Here we report two asymmetric beryllium-free borates Pna21- and P$$\bar 6$$2m-Ba3Mg3(BO3)3F3 featuring NLO-favorable 2[Mg3O2F3(BO3)2] layered structures. The reversible phase transition among two polymorphs was demonstrated by multiple experimental tests. The optical measurements reveal that Pna21-Ba3Mg3(BO3)3F3 possesses the optical properties required for ultraviolet NLO applications. Remarkably, Pna21-Ba3Mg3(BO3)3F3 has a large laser damage threshold, a deep-ultraviolet cutoff edge, a favorable anisotropic thermal expansion as well as the capacity of insolubility in water. These optical properties can be comparable or superior to that of commercial NLO material β-BaB2O4, which make Pna21-Ba3Mg3(BO3)3F3 a promising ultraviolet NLO material.

## Introduction

Nonlinear optical (NLO) materials applied from ultraviolet (UV) to infrared spectral ranges are the vital components of future photoelectric technologies1,2,3,4. Accordingly, considerable and sustained efforts have been focused on designing and synthesizing new potential NLO materials5,6,7,8,9. Generally speaking, a practical UV NLO material should simultaneously satisfy the following criteria: a noncentrosymmetric structure; a broad transparent spectral range down to the UV region (λ < 400 nm); a relatively large second-order nonlinear coefficient (dij > 0.39 pm/V); a moderate birefringence to satisfy the phase-matching condition, chemical stability (non-deliquescent), and resistance to laser (large laser damage threshold, large LDT); and ease of growth of large single crystals with high optical quality10. Targeting above conditions, until now, the borate system plays a dominant role in UV NLO materials because borates possess wide optical transparency windows, varied acentric structure types, high LDTs, and large polarizabilities11,12,13,14,15,16,17,18. Therefore, a variety of commercial borate-based UV NLO materials have been widely investigated and developed. Among them, borates with planar [BO3]3− anionic groups have always been at the center of the search for NLO materials with large second-harmonic generation (SHG) response and birefringence, which mainly benefits from the π-conjugated molecular orbitals of [BO3]3− units19,20,21,22. Based on this, several borates with [BO3]3− NLO-active units in coplanar and aligned configuration were continuously reported with good optical properties, including KBe2BO3F2 (KBBF)19, BaAlBO3F220, NaSr3Be3B3O9F421, Li Sr(BO)22, etc.

For the sake of finding new novel UV or even deep-UV (λ < 200 nm) NLO crystals, an effective design tactic is to introduce fluorine into the structural framing of borate system based on the following considerations. First, the fluorine with the largest electronegativity (χ = 3.98) can widen the transparency area and make the cutoff edge of borates blue-shift into UV or even deep-UV spectral region19,20,21,23,24,25. Second, substituting oxygen with fluorine in [BO4]5− maternal blocks to form [BOxF4−x](x+1)− (x = 0, 1, 2, 3) fundamental building blocks can make the anionic groups more flexible, and the negative effective charges at the terminal oxygen will be partly reduced, resulting in a suitable SHG coefficient and birefringence26,27,28,29. Third, the introduction of fluoride-containing flux can effectively decrease the viscosity of the borate systems during the growth of single crystals, which is a desirable choose for growing the large single crystals20,24,25,30.

For us, besides selecting the appropriate system (borate) and introducing anion (fluorine), how to effectively design UV NLO materials with presupposed structures and properties is also critically important. The following ideas were considered for designing the target molecules: first, in order to manipulate the B–O fundamental building blocks into isolated planar [BO3]3− configuration, we adjusted the ratio of cations and boron to >1.0 based on the rules proposed by Becker31. Second, we use the strategy of chemical cosubstitution to make structural modifications based on the classic NLO materials with the goal of obtaining new molecular structures contain NLO-favorable layered structures32,33.

Motivated by these, considering that s-block elements Be and Mg, Sr and Ba appear tightly clustered in the periodic table with the similar electronic configurations, sharing many common physicochemical properties. Therefore, we attempted to substitute the Be and Sr atoms of Sr2Be2B2O7 (SBBO)1 with Mg and Ba atoms through chemical cosubstitution idea to create new SBBO-like borates, which leads us to discover two new beryllium-free borates, Pna21- and P$$\bar 6$$2m-Ba3Mg3(BO3)3F3 (Pna21- and P$$\bar 6$$2m-BMBF), featuring SBBO-like layered structures without structural instability. The 2[Mg3O2F3(BO3)2] layers in both of them not only preserve the NLO-favorable structural habits of SBBO, but also introduce the Mg–F bonds served between layers as bridges to enhance the interlayer interaction. Based on the top designing molecule structures, large-size single crystals of Pna21-BMBF were also grown through finding the suitable flux system. The optical measurements based on the crystal reveal that Pna21-BMBF possesses the NLO properties required for the UV NLO applications, which indicates that Pna21-BMBF is a potential UV NLO material.

## Results

### The reversible phase transition between two polymorphs

Single crystal X-ray diffraction reveals that two new borates crystallize into the asymmetric space group of Pna21 (No. 33) and P$$\bar 6$$2m (No. 189), respectively (Table 1 and Supplementary Table 1). Interestingly, the molecular formula of both phases was determined as Ba3Mg3(BO3)3F3 through structure solutions, which have the same simplest molecular formula with BaMgBO3F (monoclinic, Cc) reported by Li. et al.34. And the relatively low temperature factors and residual factors verify the correctness of those two crystal models, which also indicates the existence of two polymorphs. Furthermore, the BMBF polymorphs and phase transitions are further confirmed by high-temperature in situ powder X-ray diffraction and thermal expansion coefficients experiment: based on the results of thermal gravimetric and differential scanning calorimetry analysis (Fig. 1a), we make circular heating and cooling the Pna21-BMBF samples in the region of 100–650 °C. When the temperature rises to 500 °C, the Pna21-BMBF phase start to transfer into P$$\bar 6$$2m-BMBF phase, and P$$\bar 6$$2m-BMBF is stable up to 650 °C. When the temperature begins to decrease, P$$\bar 6$$2m-BMBF phase can also transfer into Pna21-BMBF, indicating the reversibility of this phase transition behavior. More intuitively, the disappearance and appearance of X-ray diffraction peaks at about 2θ = 23.690, 26.994, and 27.686° in BMBF series can help us to judge this phase transition (Fig. 1b, c). The measured average thermal expansion coefficients based on the three as-polished (100), (010), and (001) wafers occur sudden change at about 450 °C, which is caused by the phase transitions of BMBF series (Fig. 1d).

### Crystal structures of Pna21- and P$$\bar 6$$2m-BMBF

The structural evolution among SBBO, Pna21-, and P$$\bar 6$$2m-BMBF are shown in Fig. 2, which verifies the validity of structure-oriented design strategy. As shown in Fig. 2a, b, e, BMBF series possess a similar layered crystal structures, their crystal structures are both composed of 2[Mg3O2F3(BO3)2] layers (Supplementary Figure 1) along the a-axis for Pna21-BMBF and c-axis for P$$\bar 6$$2m-BMBF, and those single layers are further connected by the Mg–F bonds to construct a three-dimensional framework with tunnels running along the c (or a)-axis. In the asymmetric unit of Pna21-BMBF, the Ba, Mg, B, and F atoms occupy three crystallographically unique positions, for the O atoms there are nine unique positions. While for P$$\bar 6$$2m-BMBF with higher symmetry, the Ba, Mg, B, O, and F atoms occupy one, one, two, two, and one unique positions (Supplementary Tables 2, 3, 5, and 6). In both BMBF polymorphs, the B atoms possess only one coordination type, the [BO3]3− triangles (Fig. 2c, d). The B–O bond distances and O–B–O bond angles locate in the range from 1.341 to 1.395 Å, 117.6 to 120.9 ° for Pna21-BMBF, and 1.349 to 1.381 Å, 117.0 to 122.8 ° for P$$\bar 6$$2m-BMBF, respectively (Supplementary Tables 4 and 7). The Mg atoms in both polymorphs are six-coordinated into the MgO4F2 octahedra with four O and two F atoms locating in the equatorial and axial positions, respectively. Three MgO4F2 units share three O atoms to generate a six-membered ring cluster Mg3O9F6 (Supplementary Figure 2), then those clusters link together via the axial Mg–F bonds to build an isolated 3[Mg3O9F3] triangular prism (Supplementary Figure 3). Interestingly, two [BO3]3− units locate in and out of the six-membered ring Mg3O9F6 to form polymer and further polymerize into 2[Mg3O2F3(BO3)2] infinite layers, and the connection is by bridging Mg–F bonds to form a three-dimensional multilayered structure. All the Ba atoms are coordinated into BaOnFm (m + n = 10 and 11) polyhedra, located between the adjacent 2[Mg3O2F3(BO3)2] layers, which is quite like the arrangement of the Sr atoms in SBBO1.

Structurally, the triangular [BO3]3− units in both Pna21- and P$$\bar 6$$2m-BMBF adopt nearly coplanar configuration. This is propitious to generate large SHG response and birefringence because of the strong anisotropy resulting from their different polarizabilities in different directions, which is similar to the NLO materials containing π-planar triangular [CO3]2−35,36. The F–Mg–F bonds act as bridge-linking role to connect the neighboring layers, which is similar to the [B2O5]4− dimers in LiNa5Be12B12O3337 and the [B3O6]3− groups in Cs3Zn6B9O2138,39. Further, the electrostatics force of the interaction between layers in Pna21- and P$$\bar 6$$2m-BMBF was evaluated and |FMg–F| is stronger than that of |FK–F| in KBBF (Supplementary Table 8), indicating that the electrostatic interactions between Mg–F bonds in Pna21- and P$$\bar 6$$2m-BMBF provide better linkage of neighboring layers, which is favorable to improve the layering tendency and the same results can be concluded from the ease of the growth of block-like crystals of BMBF series than that of KBBF.

### Crystal growth and optical quality of Pna21-BMBF

After various attempts, Pna21-BMBF crystal with dimensions up to 16 × 14 × 8 mm3 has been grown at the optimal conditions (Fig. 3a). Pna21-BMBF crystal has never cracked during cutting and polishing into (100), (010), and (001) wafers (Fig. 3b). The crystallization quality of the as-grown crystal was checked by the X-ray rocking curve. As shown in Fig. 3c, the full-width at half-maximum on (100) face of Pna21-BMBF crystal was measured to be 18″, indicating high crystalline quality. The conoscopic interference pattern (Fig. 3d) shows that the as-grown Pna21-BMBF crystal is biaxial, optical homogeneous, and does not have any strain inside. When taken together, Pna21-BMBF crystal shows high optical quality, and the optical measurements based on this crystal are reliable.

### High performances as UV NLO materials

Pna21-BMBF was subjected to physicochemical property characterizations based on the single-crystal level.

Thermal expansion coefficients are vital parameters for crystal growth and application in devices. As can be seen in Fig. 1d, the average thermal expansion coefficients in the region from 100 to 450 °C are evaluated to be αa = α(100) = 2.17 × 10−5 K−1, αb = α(010) = 1.63 × 10−5 K−1, and αc = α(001) = 1.53 × 10−5 K−1, respectively. The value of αa/αc is about 1.43 for Pna21-BMBF crystal, which is smaller than that of β-BaB2O4 (β-BBO) (αc/αa = 9)40. Therefore, Pna21-BMBF crystal exhibits a more favorable anisotropic thermal expansion, which will effectively protect the crystals from cracking caused by thermal expansion during crystal growth and optical devices fabrication.

To study the water solubility of Pna21-BMBF, we dipped the target Pna21-BMBF (0.113 g) crystal as well as the references α- and β-BBO crystals (0.159 and 0.105 g) into 20 mL de-ionized water at room temperature. As plotted in Fig. 1e, the weight of Pna21-BMBF did not change in the measured time ranges, while for α- and β-BBO crystals, they were dissolved in water in 90 h. These indicate that Pna21-BMBF is chemically stable and not soluble in the water.

We carried out the SHG measurements by the Kurtz–Perry method41 with incident lasers at λ= 1064 nm. Meanwhile, the well-known NLO material KH2PO4 (KDP) was used as the reference. The comparison of SHG signals produced by Pna21- and P$$\bar 6$$2m-BMBF polycrystalline samples in the same particle sizes ranging from 200 to 250 μm reveals that Pna21- and P$$\bar 6$$2m-BMBF exhibit suitable SHG responses of 1.8 and 2.0 × KDP, respectively. Such SHG responses are large enough for the applications of UV NLO materials and it is also comparable to those reported borates with deep-UV cutoff edges, such as KBBF (1.26 × KDP)19, LiNa5Be12B12O3337 (1.4 × KDP), and Li4Sr(BO3)2 (2 × KDP)22. According to the results, the SHG efficiency increases with the raising particle sizes, indicating that both BMBF polymorphs exhibit type I phase-matching behavior in 1064 nm incident lasers. In addition, the effective NLO coefficients (deff) of Pna21- and P$$\bar 6$$2m-BMBF crystal are estimated to be about 1.3 and 1.4 × KDP according to the square roots of the ratios of their corresponding SHG signal intensities. In order to verify the results, large sizes of single crystals should be grown and cut to evaluate the SHG coefficients based on the crystal level, which is discussed in detail in the following parts.

The as-grown crystals were cut and optically polished to 1 mm thickness for the spectrum measurement. As presented in Fig. 4b, the Pna21-BMBF crystal exhibits a broad transmission of 184–3780 nm, indicating that the application of Pna21-BMBF crystal can cover a spectral range from UV to near-infrared. Obviously, the UV cutoff edge of Pna21-BMBF is about 184 nm, which is 5 nm shorter than that of β-BBO (189 nm) in the same measured condition. And also the high transmittance of Pna21-BMBF in the UV spectral range makes it possible to be applied as a UV NLO material. The short cutoff edge is beneficial to obtain high LDT, therefore, LDT measurements were carried out using a pulsed nanosecond laser (1064 nm, 10 ns, and 10 Hz). A well-polished high-quality (001) wafer of Pna21-BMBF has a LDT of 6.2 GW/cm2, which is comparable to β-BBO crystal (6.0 GW/cm2) at the same experimental conditions.

The refractive indices of Pna21-BMBF were measured using prism coupling method. As it follows the inequality nz − ny < ny − nx, which depicts that Pna21-BMBF is a negative biaxial crystal (consistent with results of conoscopic interference patterns), with the birefringence (Δn = nz − nx) ranging from 0.044@1064 nm to 0.045@532 nm and the values are comparable with those of the commercial UV NLO materials LiB3O5 and CsLiB6O10n = 0.045 and 0.049@1064 nm)10. A least-square method was used to fit the dispersion parameters of the refractive indices, ni, using the Sellmeier Eq. (1):

$$n_i^2 = {{A}} + \frac{{{B}}}{{\lambda ^2 - {{C}}}} - {{D}}\lambda ^2$$
(1)

where λ represents the wavelength in units of µm, and A, B, C, and D are the Sellmeier parameters. The Sellmeier Eq. (1) deduced by the least-square fitting of all the measured refractive indices have been fitted as:

$$\begin{array}{ccccc}\\ n_x^2 = & 2.58654 + \frac{{0.01672}}{{\lambda ^2 - 0.00904}} - 0.00987\lambda ^2\\ \\ n_y^2 = & 2.71284 + \frac{{0.02264}}{{\lambda ^2 + 0.01829}} - 0.00486\lambda ^2\\ \\ n_z^2 = & 2.72604 + \frac{{0.01908}}{{\lambda ^2 - 0.01344}} - 0.00819\lambda ^2\\ \end{array}$$
(2)

The calculated and experimental values of the refractive indices at five different wavelengths are summarized in Supplementary Table 9. The calculated values are consistent with experimental ones to the fourth decimal place, which indicates that the fitted Sellmeier equations based on the current data are reliable (Fig. 4c). Based on above fitted Sellmeier equations, when considering the type I phase-matching condition, n(ω) = n(2ω), the shortest type I phase-matching wavelengths in the xz and xy plane are evaluated to be 310 and 322 nm (Fig. 4d), respectively. Therefore, the Pna21-BMBF crystal can generate 532 and 355 nm light by direct second and third harmonic generation from a 1064 nm laser.

In order to determine the individual NLO coefficients of Pna21-BMBF, Maker fringe measurements were carried out. The schematic of the Maker fringe measurement system is shown in Fig. 5. For the SHG coefficient measurements, the orientation of the crystals and the relationships of Pω and P2ω are shown in Fig. 6a, c. The (001)-cut and (100)-cut plane-parallel uncoated plates of Pna21-BMBF crystals (4 mm × 4 mm × 1 mm) were used to measure the related NLO coefficients. And also a (110)-cut plate (5 mm × 5 mm × 2 mm) of the KDP crystal was used to measure the d36 (KDP) NLO coefficient as the reference. By fitting the calculated Maker fringes based on the current measurements and calculations, the NLO coefficient of the Pna21-BMBF crystal relative to d36 (KDP) has been determined as d33 = 1.3 × d36 (KDP) (Fig. 6b, d). However, the Maker fringe for d31 is too weak to be observed. As d36 coefficient of KDP is 0.39 pm/V, the absolute NLO coefficients of Pna21-BMBF crystal based on the measurements and calculations are evaluated to be d33 = 0.51 pm/V.

## Discussion

In order to achieve a deep investigation about the structure-property relationship, theoretical calculations based on density functional theory methods were performed42,43,44. The total and partial density of states projected on the constitutional atoms of Pna21-BMBF crystal are given in Fig. 7a. Clearly, the energy bands can be divided into several regions. The uppermost part of valence bands from −7.5 to 0 eV is essentially dominated by O 2p states and an appreciable contribution of F 2p orbitals with a small amount of B 2p states. The obvious narrow band with sharp peak locates at about −10.7 eV entirely dominated by Ba 5p orbitals and also with a small contribution of O 2p states. The last part from −19.4 to −15.9 eV is mainly generated from O 2s, O 2p, and B 2s, B 2p, indicating strong sp hybridization of the B and O orbitals in the [BO3]3− groups. While the bottom of the conduction band is essentially composed of Ba 5d, Ba 5s, B 2p, and O 2p orbitals. In principle, the optical properties are mainly determined by the electronic transitions among the states near Fermi level, which are mainly occupied by the O 2p nonbonding orbitals, B–O hybridization states and Ba 5d states in Pna21-BMBF. Besides, the interatomic interactions within the BO3 units and MgO4F2 octahedra can be clearly visualized by the electron localization function diagrams. Results show the ionic character of Mg–O and Mg–F bonds in the MgO4F2 octahedra as well as the covalent nature of B–O bonds in the [BO3]3− units (Fig. 7b). Through the Mulliken population analysis, the same conclusions can also be substantiated by the calculated overlap populations for Mg–F, Mg–O and B–O, which are 0.12–0.21, 0.21–0.55, and 0.85–0.97, respectively (higher populations reflect stronger covalence).

The NLO properties of Pna21-BMBF were also calculated using a theoretical technique under the static limit within the length gauge. The calculated SHG tensors are evaluated to be d33 = 0.47, d32 = −0.39, and d31 = 0.09 pm/V, respectively. These values are well consistent with experimental values presented above (d33 = 0.51 pm/V). The SHG density analysis can be used to intuitively and accurately display the SHG-contributed orbitals for an NLO crystal. Furthermore, we investigated the SHG origin of Pna21-BMBF from the perspective of orbital analysis using SHG density analysis. And the SHG process is denoted by two virtual transition processes, namely virtual electron (VE) and virtual hole processes. The contributions of VE process to d31, d32, and d33 SHG tensors are up to 83.7%, 81.6%, and 86.5%, respectively, indicating that the SHG effects of three tensors mainly originate from the VE processes. Thus, only the largest d33 SHG tensor of VE process is analyzed and the results are drawn in Fig. 8. In the occupied states, the nonbonding 2p orbitals of O (1, 2, 3, 5, 8, 9) atoms in [B(2)O3]3− and [B(3)O3]3− groups give more dominant contributions to the SHG effects, while for the occupied states, π orbitals of the [B(1, 2, 3)O3]3− units as well as 2p orbitals of F (1, 2, 3) atoms in the MgO4F2 octahedra give more dominant contributions for the SHG effects, revealing the p–(p, π*) charge-transfer excitation mechanism in Pna21-BMBF. Both occupied and unoccupied states of the Ba cations, in analogy to alkali metal and alkali earth metal ions in other NLO crystals, have very little contribution on the SHG effect, which agrees well with the anionic group theory10. To sum up, the large SHG response of Pna21-BMBF can be regarded as the synergistic effects of the planar [BO3]3− and distorted MgO4F2 octahedra.

In summary, by starting from the SBBO structure, we designed and synthesized two beryllium-free SBBO-like borates through chemical cosubstitution strategy. A high-quality Pna21-BMBF crystal with dimensions up to 16 × 14 × 8 mm3 has been grown by the top-seeded solution growth method with a BaF2-H3BO3-LiF flux system. Linear and nonlinear optical measurements reveal a sufficiently large SHG coefficient (d33 = 0.51 pm/V), suitable birefringence (0.045@532 nm), type I phase-matchability, and chemical stability. Remarkably, Pna21-BMBF has a large LDT (~6.2 GW/cm2), a deep-UV cutoff edge (λ ~ 184 nm), a weak anisotropic thermal expansion as well as the capacity of insolubility in water, these optical properties could be comparable or superior to that of commercial NLO materials β-BBO (Supplementary Table 10). Thus, these merits on optical properties make Pna21-BMBF a promising UV NLO material.

## Methods

### Synthesis and crystal growth

Polycrystalline samples of targeted Pna21-BMBF were synthesized through solid-state reaction method by mixing the raw materials BaF2, BaCO3, MgO, and H3BO3 according to the molar ratio of 3:3:6:6. The mixture was preheated at 400 °C for 48 h. After that, the temperature was gradually raised to 600 °C for Pna21-BMBF, with several intermediate mixings and grindings. Single crystals of Pna21-BMBF were grown by top-seeded solution growth method with the flux BaF2-H3BO3-LiF at a molar ratio of Pna21-BMBF/H3BO3/BaF2/LiF = 1:1:0.5:3. A mixture of Pna21-BMBF polycrystalline samples, BaF2, H3BO3, and LiF were placed into a Φ 30 × 30 mm platinum crucible and melted at 750 °C in a temperature-programmable electric furnace. The solution was held at 750 °C for 24 h to form a homogeneous solution. After that, cooled to 715 °C and dipped the Pna21-BMBF seed crystal into the solution and held for 2 h that allowed the seed crystal surface to melt. The solution was slowly cooled to the saturation temperature about 712 °C. Then the Pna21-BMBF crystal was grown by cooling the solution at a rate of 0.2 °C/d to 706 °C. When the growth procedure finished, the Pna21-BMBF crystal was drawn out of the solution and cooled down to room temperature at a rate of 20 °C/h. While for P$$\bar 6$$2m-BMBF single crystals, the loaded compositions are Ba(NO3)2 (0.5227 g, 2 mmol), BaF2 (0.3507 g, 2 mmol), MgO (0.0403 g, 1 mmol) and H3BO3 (0.4328 g, 7 mmol). All the reagents were mixed homogeneously and transferred to a platinum crucible. The samples were heated to 880 °C in 24 h and held at this temperature for 72 h and then cooled to 700 °C at a rate of 1.0 °C/h, after that, cooled to room temperature at a rate of 15 °C/h. During the process of crystallization, colorless and transparent P$$\bar 6$$2m-BMBF crystals were formed in the platinum crucible.

### Characterization

The selected single crystals with dimensions up to 0.12 × 0.10 × 0.01 mm3 for Pna21-BMBF and 0.16 × 0.07 × 0.06 mm3 for P$$\bar 6$$2m-BMBF were glued on glass fibers for structure determination by single-crystal X-ray diffractometer. An APEX II CCD diffractometer equipped with monochromatic Mo Kα radiation was used for the single-crystal data collection. The Bruker Suite software package was used to reduce the collected data. The numerical absorption corrections were carried out with the SADABS program and integrated with the SAINT program45. The original structures were established by the direct method and refined by the full-matrix least-squares program on SHELXL46. The PLATON program was used for checking the possible missing symmetry elements, but no higher symmetries were found47. Investigation of the thermal behavior Pna21-BMBF crystal was performed using a NETZSCH STA 449 C simultaneous thermal analyzer. The sample of 6.5 mg was enclosed in a platinum crucible and heated from 40 to 1200 °C at a rate of 5 °C/min. The measurements were carried out in an atmosphere of flowing N2. The powder X-ray diffraction data of Pna21-BMBF were collected at room temperature using an automated Bruker D2 X-ray diffractometer (Supplementary Figure 4). The transmittance spectra of Pna21-BMBF water were collected by a Solid Spec-3700DUV spectrophotometer for the range of 180–2500 nm in an atmosphere of flowing N2 and by a Shimadzu IRAffinity-1 Fourier spectrometer for the range extended from 2500 to 4000 nm (Supplementary Figure 5). The powder SHG measurements for targeted Pna21- and P$$\bar 6$$2m-BMBF were carried out on the basis of the Kurtz–Perry method41. Polycrystalline samples of Pna21- and P$$\bar 6$$2m-BMBF were ground and sieved into the following particle size ranges: 20–38; 38–55; 55–88; 88–105; 105–150; 150–200; and 200–250 μm. During the measurement, Pna21- and P$$\bar 6$$2m-BMBF samples were irradiated with a Q-switched Nd:YAG laser with light wavelength of 1064 nm. Then the intensity of the frequency-doubled of 532 nm output emitted from Pna21- and P$$\bar 6$$2m-BMBF were recorded by a digital oscillometer equipped with a photomultiplier tube. The microcrystalline KDP samples with the same particle sizes were used as references. For LDT test, the crystals were cut plane parallel and polished to optical quality using a Unipol-300 grinding/polishing machine. A Q-switched Nd:YAG laser with a pulse width of 10 ns and repetition frequency of 10 kHz was employed as the fundamental light source (1064 nm). The second-harmonic signal generated from the sample wafer was detected by a side window photomultiplier tube, averaged by a fast-gated integrator and boxcar average, and then automatically recorded by a computer. A cut-KDP (110) wafer was used as a reference. The refractive index measurements of Pna21-BMBF were carried out using (100), (010), and (001) crystal plates on the Metricon model 2010/M prism coupler (Metricon Co.) at five wavelengths (406.9, 514.0, 636, 964.8, and 1546.7 nm), and the accuracy of the measurements is estimated to be 2 × 10−4. Maker fringe measurements of Pna21-BMBF were performed on two crystal wafers cut perpendicular to the a- and b-crystallographic axes in order to measure d31, d32, and d33 NLO coefficients. Then, the data were fitted according to the Maker fringe theory. The detected second-harmonic power P2ω is expressed as the following Eq.(3):

$$P_{2\omega }\left( \theta \right) = \frac{{512\pi ^2}}{{cw^2}}d^2P_\omega ^2f\left( {n,\theta } \right)\sin ^2\Psi$$
(3)
$$f\left( {n,\theta } \right) = \frac{1}{{\left( {n_\omega ^2 - n_{2\omega }^2} \right)^2}}t_\omega ^4T_{2\omega }p^2\left( \theta \right)R\left( \theta \right)\beta \left( \theta \right)$$
(4)

where c is the speed of light in vacuum, w is the radius of light beam, nω and n2ω are the refractive indices under the fundamental and harmonic wavelength, d is the NLO coefficient, Pω is the fundamental beam power, Ψ = 2πL(nωcosθω − n2ωcosθ2ω)/λω, and sin2Ψ determines the minima oscillating position. tω and t2ω are the transmission coefficients of fundamental wave and harmonic wave, p(θ) is the projection factor, R(θ) is the incident multiple-reflection correction, and β(θ) is the beam size correction. The f (n, θ) part depicts the envelope of the Maker fringe. By fitting the calculated and measured Maker fringes, a constant, C= 512πd2P2ω/(2) can be obtained. The magnitude of NLO coefficients of Pna21-Ba3Mg3(BO3)3F3 can be determined relatively to d36 of KDP crystal, and the coefficient equation can be expressed as:

$$d_{{\mathrm{sample}}} = \sqrt {\frac{{C_{{\mathrm{sample}}}}}{{C_{\mathrm{KDP}}}}} d_{36}\left( {{\mathrm{KDP}}} \right)$$
(5)

### Computational methods

First-principles calculations of Pna21-BMBF based on density functional theory were performed by a planewave pseudopotential calculation package CASTEP48. The exchange and correlation effects were treated by Perdew–Burke–Ernzerhof in the generalized gradient approximation49. The interactions between the ionic cores and electrons were described by norm-conserving pseudopotentials50. The cutoff energy for the planewave basis was 990 eV and the Brillouin zone was sampled by 4 × 2 × 3 Monkhorst-Pack k-point.

### Data availability

The authors declare that the data supporting the findings of this study are available within the article and Supplementary Information files, or from the corresponding authors upon reasonable request.

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## Acknowledgements

This work was financially supported by National Natural Science Foundation of China (Grant Nos. 21501194, 51425206, 51602341, and 91622107), the National Key Research Project (Grant Nos. 2016YFB1102302 and 2016YFB0402104), Xinjiang Key Research and Development Program (Grant No. 2016B02021), National Basic Research Program of China (Grant No. 2014CB648400), and West Light Foundation of the Chinese Academy of Sciences (Grant No. 2016-YJRC-2).

## Author information

M.M. and M.Z. grew the single crystals and performed the experiments, data analysis, and paper writing. H.W. supervised the optical experiments. Z.Y. developed the theoretical calculations. Y.S. performed the crystal data collection. J.S. and S.P. designed and supervised the experiments. All the authors discussed the results and commented on the manuscript.

### Competing interests

The authors declare no competing interests.

Correspondence to Junliang Sun or Shilie Pan.

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