Complete quantitative nonlinear-optical characterization of SnGa4S7 and SnGa4Se7: orientation-resolved determination of the full second-order susceptibility tensor and high-level computational benchmarking

The ternary chalcogenides SnGa4S7 and SnGa4Se7 were reported in 2014 as promising mid-infrared second-harmonic-generation (SHG) materials with large powder conversion efficiencies, high laser-damage thresholds, and wide transparency, but their nonlinear optical response was quantified only by the relative powder method, which cannot resolve individual tensor coefficients, and their electronic structure was characterized only by density-functional calculations of the kind that systematically underestimate semiconductor band gaps. More than a decade of follow-up literature contains theoretical studies of these compounds yet no oriented single-crystal measurement. The open problem is the quantitative completion of their characterization: absolute, refractive-index-corrected measurement of all symmetry-allowed second-order susceptibility coefficients $d_{ij}$ (with signs and uncertainties) and of refractive-index dispersion on oriented single crystals of both compounds, together with first-principles benchmarking of band gaps and SHG tensors against these measured values at a level of theory that remedies the density-functional band-gap underestimate.

Background

Second-harmonic generation (SHG) is the nonlinear optical (NLO) process in which a non-centrosymmetric crystal converts two photons of frequency $\omega$ into one photon at $2\omega$; its strength is governed by the second-order susceptibility tensor, usually tabulated as coefficients $d_{ij}$ in pm/V that link the polarization induced in crystal direction $i$ to products of the electric-field components along axes $j$ (in the standard convention $d_{il}=\tfrac{1}{2}\chi^{(2)}_{ijk}(-2\omega;\omega,\omega)$, the pair of field indices $jk$ being contracted to a single index $l=1,\dots,6$, so that $d_{il}$ is dispersive and is meaningful only together with the optical frequencies at which it is quoted). Only crystals lacking inversion symmetry exhibit a nonzero $\chi^{(2)}$, and only certain tensor elements are allowed by the crystal point group. Mid-infrared (mid-IR, roughly 2.5–25 $\mu$m wavelength) NLO crystals with a wide transparency window, large $d_{ij}$, and high laser-induced damage threshold (LIDT) are key enabling components for tunable mid-IR coherent sources, with commercial benchmarks such as AgGaS$_2$. A central experimental difficulty is that a newly discovered compound is normally obtained only as microcrystalline powder, so its SHG response is screened by the Kurtz–Perry powder method: the second-harmonic intensity of a sieved powder is compared with that of a reference powder under the same laser firing. This relative method yields only an effective coefficient $d_{\mathrm{eff}}$, averages over all crystal orientations, omits the refractive-index difference between sample and reference, and cannot give the signs or individual magnitudes of the tensor elements. Quantitative use of a material — predicting phase matching (the condition that fundamental and harmonic waves propagate with matched refractive indices so the generated light adds coherently), designing frequency-conversion devices, and comparing with theory — instead requires the full $d_{ij}$ tensor measured on oriented single crystals with absolute calibration, together with refractive-index dispersion, commonly fitted with Sellmeier equations (a standard empirical formula giving the refractive index as a function of wavelength). On the theory side, electronic-structure calculations based on density-functional theory (DFT) with the local-density approximation (LDA) or generalized-gradient approximation systematically underestimate band gaps of semiconductors; remedies include the HSE06 hybrid functional (which mixes a fraction of exact exchange into the exchange–correlation treatment) and the modified Becke–Johnson (mBJ) potential, both standard tools for obtaining reliable gaps and, in turn, frequency-dependent NLO coefficients. In this setting, Luo, Lin, Cui, Zhang, Zhang, He, and Cheng (Chem. Mater. 2014) reported the non-centrosymmetric chalcogenides SnGa$_4$S$_7$ and SnGa$_4$Se$_7$ (tin gallium sulfide and selenide), obtained by high-temperature solid-state reaction, as mid-IR SHG materials with large powder conversion efficiencies, high damage thresholds, and wide transparency. The two compounds are isostructural and crystallize in the monoclinic polar space group $Pc$ (No. 7), that is, point group $m$ with the glide plane perpendicular to the unique axis $b$; in the standard contracted notation with $x\parallel a$, $y\parallel b$, $z\parallel c$, this point group allows exactly ten independent second-order coefficients, $d_{11}, d_{12}, d_{13}, d_{15}, d_{24}, d_{26}, d_{31}, d_{32}, d_{33}, d_{35}$, which reduce to six if Kleinman symmetry may be invoked at the working wavelengths ($d_{15}=d_{31}$, $d_{24}=d_{32}$, $d_{26}=d_{12}$, $d_{35}=d_{13}$). Being monoclinic, both are optically biaxial: only the dielectric axis along $b$ is fixed by symmetry, while the other two rotate within the $ac$ plane as functions of wavelength. The SHG response was quantified only by powder relative-intensity measurements — the selenide was reported to reach a second-harmonic intensity of about $3.8\times$ that of AgGaS$_2$ at 2.05 $\mu$m, with a laser-damage threshold of about $4.6\times$ AgGaS$_2$ — using the standard powder–reference approximation that neglects refractive-index differences, while the large NLO response was analysed theoretically and attributed to the stereochemically active lone pair of Sn$^{2+}$ and to the cooperative polarities of the [SnQ$_4$] and [GaQ$_4$] units. What therefore remains to be done is the quantitative part of the characterization: the fully quantitative, orientation-resolved determination of the complete second-order susceptibility tensor (for example from oriented single-crystal measurements with refractive-index correction), the measurement of refractive-index dispersion, and further high-level computational benchmarking of the optical/NLO properties. The follow-up literature has not closed this gap: an independent DFT+mBJ optoelectronic study of SnGa$_4$Q$_7$ appeared in 2015 (Khan et al.), and the original group extended theory to terahertz difference-frequency-generation sources based on calculated susceptibilities in 2017 (Cheng et al., who also noted that conversion efficiency grows with crystal size), but no report of bulk single-crystal growth or orientation-resolved tensor measurement of either compound has appeared; work on the Sn–Ga–Q system shifted to other stoichiometries (SnGa$_2$GeS$_6$, SnGa$_2$GeSe$_6$, Sn$_2$Ga$_2$S$_5$), while the structural analogues PbGa$_4$S$_7$ and PbGa$_4$Se$_7$ were likewise studied mainly at the powder level.

Problem Statement

For the mid-infrared SHG materials SnGa$_4$S$_7$ and SnGa$_4$Se$_7$: replace the powder-derived relative SHG figures of the original 2014 report (a Kurtz–Perry effective response referenced to AgGaS$_2$) with a complete quantitative nonlinear-optical characterization. Specifically, (i) obtain phase-pure, oriented single crystals of each compound, determine their orientation (e.g., by Laue back-reflection or single-crystal X-ray diffraction), and measure all symmetry-allowed independent second-order susceptibility coefficients $d_{ij}$ — the ten allowed by point group $m$, or the six that remain if Kleinman symmetry is established at the working wavelengths — with their signs, magnitudes, and uncertainties, using an absolutely calibrated, orientation-resolved technique (such as Maker-fringe, wedge, or calibrated phase-matched SHG measurements at stated wavelengths within the compounds' transparency window) with the refractive-index (Fresnel/boundary) corrections that the powder-reference approximation omits; (ii) measure the refractive-index dispersion of the three principal dielectric axes and, because both compounds are biaxial, the wavelength dependence of the orientation of the two axes that symmetry does not fix within the $ac$ plane (e.g., by minimum-deviation or ellipsometric methods, customarily reported as Sellmeier fits) both to support those corrections and to establish phase-matching behavior; and (iii) benchmark first-principles predictions of the electronic gaps and of every independent SHG tensor element, computed at a level of theory that remedies the known LDA/GGA band-gap underestimate (for example a hybrid functional such as HSE06, the modified Becke–Johnson potential, or a many-body treatment), against these measured values for both compounds, quoting calculated and measured coefficients at the same optical frequencies and with their uncertainties so that the comparison can be made element by element. The objective is one coherent completion of the quantitative NLO characterization of this compound pair: the tensor, dispersion, and computational components jointly convert the 2014 screening result into reliably usable, theory-cross-checked material data.

References

1. Zhong-Zhen Luo; Chen-Sheng Lin; Hong-Hua Cui; Wei-Long Zhang; Hao Zhang; Zhang-Zhen He; Wen-Dan Cheng, 'SHG Materials SnGa4Q7 (Q = S, Se) Appearing with Large Conversion Efficiencies, High Damage Thresholds, and Wide Transparencies in the Mid-Infrared Region', Chemistry of Materials 26(8), 2743-2749 (2014). DOI 10.1021/cm5006955, https://doi.org/10.1021/cm5006955
2. Wilayat Khan; Sikander Azam; Fahad Ali Shah; Souraya Goumri-Said, 'DFT and modified Becke Johnson (mBJ) potential investigations of the optoelectronic properties of SnGa4Q7 (Q = S, Se) compounds: Transparent materials for large energy conversion', Solid State Sciences 48, 244-250 (2015). DOI 10.1016/j.solidstatesciences.2015.08.015, https://doi.org/10.1016/j.solidstatesciences.2015.08.015
3. Wen-Dan Cheng; Chen-Sheng Lin; Hao Zhang; Yi-Zhi Huang; Guo-Liang Chai, 'Theoretical Evaluation of Terahertz Sources Generated From SnGa4Q7 (Q=S, Se) as Infrared Nonlinear Optical Materials', ChemPhysChem 18, 519-525 (2017). DOI 10.1002/cphc.201601128, https://doi.org/10.1002/cphc.201601128
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5. Zhe-Xiong Zheng; Zhi-Xin Qiu; Chao-Hong Xie; Yang-Ping Zhang; Xiao-Ming Jiang; Bin-Wen Liu; Guo-Cong Guo, 'Remarkable phase-matchable second-harmonic generation realized by strong polarities of [PbSe3] and [GaSe4] functional motifs in PbGa4Se7', Science China Materials 66, 2795-2802 (2023). DOI 10.1007/s40843-022-2417-2, https://doi.org/10.1007/s40843-022-2417-2
6. Zhi-Hui Shi; Yang Chi; Zong-Dong Sun; Wenlong Liu; Sheng-Ping Guo, 'Sn2Ga2S5: A Type of IR Nonlinear-Optical Material', Inorganic Chemistry 58, 12002-12006 (2019). DOI 10.1021/acs.inorgchem.9b02021, https://doi.org/10.1021/acs.inorgchem.9b02021