Thermo-Mechanical Characterization of the Orthorhombic Nonlinear Optical Crystal PbGa₂GeSe₆

Abstract

Linear thermal expansion in the 70–350 K range and nanohardness and Young’s modulus at room temperature are measured for the newly developed quaternary nonlinear optical crystal PbGa₂GeSe₆ applicable in the mid-IR part of the spectrum (3–25 µm).

1. Introduction

The quaternary chalcogenide PbGa₂GeSe₆ (PGGSe) belongs to a family of orthorhombic (space group #43, Fdd2, point group mm2) crystals with a general formula AGa₂GeS(e)₆ (A = Pb, Sn) possessing non-centrosymmetric structure [1,2,3,4,5]. Most interesting from an applications perspective, they exhibit second-order nonlinear polarizability, a property of materials that allows them to convert intense light from one wavelength to another. This class of non-oxide crystals is of particular interest for frequency conversion in the mid-IR part of the spectrum due to their lower phonon energies [6]. Two kinds of nonlinear structural units are present in AGa₂GeS(e)₆ crystals: (i) tetrahedral (Ga/Ge)–S(e)₂ units and (ii) polyhedral units centered by cations with a lone electron pair (Sn²⁺ or Pb²⁺). The selenide PGGSe (i.e., A = Pb) was the first compound from this family that was discovered in 2015 [3] and it remains so far the only one that has been thoroughly optically characterized using single crystals. According to [3], its structure contains three non-centrosymmetric chromophores, [PbSe₄], [GaSe₄], and [Ga/GeSe₄], with covalent interactions between the X and Se atoms (X = Pb, Ga, Ga/Ge). Since crystals of only ~1 mm size were obtained in [3], the band-gap of PGGSe was determined from diffuse reflectance vis/near-IR spectra while the mid-IR transmission was measured with samples diluted with dry KBr and pressed into transparent sheets. The average nonlinear response of PGGSe was evaluated in [3] from relative measurements on powders using a 2.05 μm Q-switched Ho-laser. Raman spectra in back-scattering geometry and IR reflection spectra of PbGa₂GeS(e)₆ revealed that the high-frequency part is dominated by the GeS(e)₄ structural units [7], which determines the mid-IR transmission cut-off wavelength normally associated with two-phonon absorption. Fitting of the near band-gap absorption in [7] indicated an indirect nature of the band-gap in these compounds.

The melting temperature of PGGSe is 720 °C [8]. The growth of large size high optical quality single crystals by the vertical Bridgman-Stockbarger method [8] enabled the measurement of the transmission properties of PGGSe using thin plates and the dispersion and birefringence using oriented prisms, thus, in turn, leading to the fitting of the first Sellmeier equations for PGGSe, and the realization of the first phase-matched second-harmonic generation (SHG) in this material using a picosecond Ho-laser amplifier to derive the non-diagonal nonlinear coefficients: d₃₁(PGGSe) = 16.8 pm/V and d₃₂(PGGSe) = 3.6 pm/V at 2.05 µm. The band-gap energy of PGGSe was derived from transmission measurements in the visible as 2.25 eV (550 nm), larger than in [3,7], and its mid-IR transmission extends at least up to 25 µm [8]. PGGSe is an optically negative biaxial crystal with n_x < n_y < n_z and the correspondence between the principal dielectric and the crystallographic axes is xyz = cba. The angle between the two optical axes and the principal z-axis is V_z = 64°. The birefringence of PGGSe is as large as ~0.25 at 5 µm [8]. Optical damage in the SHG experiments with PGGSe was observed at 250 MW/cm² (1.25 mJ/cm²) at ~2 µm (~5 ps pulses at 1 kHz) in terms of peak on-axis values [8].

Here, we study the thermo-mechanical properties of PGGSe, including its linear anisotropic thermal expansion and the nanohardness and Young’s modulus using oriented test plates.

2. Thermal Expansion

The thermal expansion was studied using powdered single crystals with a Bruker D8 Discover Davinci X-ray diffractometer (Bruker Corporation, Madison, WI, USA) with Co Kα radiation at Kα₁ = 1.78897 and Kα₂ = 1.79285 Å, paired with an Oxford Chimera temperature control system (Oxford Cryosystems Ltd., Long Hanborough, UK).

We used FullProf (May 2021 64b) software to analyze the data in terms of a Le Bail fit. The XRD peak profiles were defined using a Thompson–Cox–Hastings formulation of the pseudo-Voigt function with axial divergence asymmetry. The background was adjusted manually and the displacement was refined only for the lowest temperature analyzed and kept fixed as temperature was increased. Figure 1 shows the PGGSe lattice parameter temperature dependence for the a, b, and c crystallographic axes. The goodness of the fits was in the χ² = 3–4 range.

Figure 1. Temperature (T)-dependent lattice constants of PGGSe (a–c): experimental data (symbols) and linear fits (lines).

The room temperature (RT = 293 K) values in Table 1 were calculated from the linear fits to the data. The values reported in [3] at the same temperature, 47.135(16) Å for a, 7.578(3) Å for b, and 12.161(4) Å for c are slightly larger, but such deviations can be attributed to different growth conditions or to different diffractometer calibrations.

Table 1. Room temperature (RT) lattice constants, relative linear thermal expansion coefficients α, and coefficient of determination R².

Direction	RT Lattice Constant, Å	α, ppm/K	R2 Factor
a	47.001(7)	7.03	0.9727
b	7.5753(6)	3.72	0.9633
c	12.161(4)	5.27	0.9707

The relative linear thermal expansion coefficients in Table 1 were derived from the slopes of the fits. They are lower compared to the widely used, commercially available AgGaSe₂ chalcopyrite (α_a = 14.1 and α_c = −11.2 ppm/K), the benchmark among the nonlinear selenides, which, in addition, shows anomalous behavior (i.e., opposite signs) [9].

Determination coefficients R² exceeding 0.95 in Table 1 are indicative of strong agreement between the experimental data and the model.

3. Nanohardness and Young’s Modulus

The nanohardness and Young’s modulus were derived from RT nanoindentation tests (Nanoindenter G200, Keysight Technologies, Santa Rosa, CA, USA). The three plates used were cut normal to the principal axes directions and had a thickness of the order of 100 µm. Thermal drift correction was applied both for the tip and the sample. Optically polished fused silica served as a calibration standard. The measured reference values were Young’s modulus of 71.3 ± 0.6 GPa and hardness of 9.5 ± 0.1 Gpa, which fit into the device acceptance criteria (70–75 GPa and 8.5–10.5 GPa, respectively). Continuous stiffness mode was chosen, with oscillatory indentation load (frequency: 45 Hz, amplitude: 2 nm) and a Berkovich tip was used. The maximum penetration depth was 2 µm. Ten consecutive indents, separated by 50 µm, were performed (see Figure 2a). The surface approach velocity and distance were 10 nm/s and 1 µm, respectively.

Figure 2. Loading/unloading curves (a), depth dependence of nanohardness (b), and Young’s modulus (c) of the b-cut PGGSe sample. The inset in (a) is a micrograph of the indents. Different colors represent the individual tests.

The elastic modulus of the sample E was estimated by the Oliver–Pharr equation [10] implemented in NanoSuite 6 Software:

$$E=\left(1-{\nu }^2\right)\sqrt{1/E_{\mathrm{r}}-(1-{\nu }_{\mathrm{i}}^2)/E_{\mathrm{i}}},$$

(1)

where ν is the material Poisson’s ratio (estimated to be 0.2), ν_i is the nanoindenter tip Poisson’s ratio, E_i is the elastic modulus of the indenter, and $E_{\mathrm{r}}=\sqrt{\pi }S/\left(2\beta \sqrt{A}\right)$, with S—the contact stiffness (measured at each oscillatory step), A—the surface contact area projected onto a plane normal to the indendation direction, and β—a geometrical factor of the indenter (for a Berkovich tip, β = 1.034).

The nanohardness was calculated as the ratio of the contact force to the contact area A. Figure 2b shows a representative plot of the nanohardness dependences on the indentation depth for the b-cut PGGSe plate. All tests, for the three directions (a, b, and c), exhibit the typical (for nanoindentation) increased values at small depths associated with surface hardening (an effect typical for relatively soft crystals as the one studied here). Beyond ~300 nm, the nanohardness of the undisturbed material monotonically decreases with the depth. Figure 2c shows the Young’s modulus depth dependence of the b-cut PGGSe plate.

The nanohardness and Young’s modulus values obtained by averaging over the interval of 300–1900 nm are summarized in Table 2. The results reveal some anisotropy for both Young’s modulus and nanohardness. Young’s modulus and nanohardness are highest for the b-cut specimen; we note that this direction exhibits the lowest relative linear expansion coefficient. For AgGaSe₂, only microhardness measurements on polycrystalline material can be found in the literature, e.g., 4.40 GPa [11].

Table 2. Young’s modulus, nanohardness, and the temperature and thermal drift correction in the nanoindentation tests of PGGSe.

Direction	Young’s Modulus, GPa	Nanohardness, GPa	Temperature, °C	Thermal Drift Correction, nm/s
a	43.5 ± 0.5	3.16 ± 0.07	26.8 ± 0.2	0.231 ± 0.077
b	56.6 ± 0.3	3.58 ± 0.04	20.1 ± 0.2	0.342 ± 0.025
c	40.2 ± 0.3	3.42 ± 0.04	20.7 ± 0.2	0.264 ± 0.015

4. Conclusions

In conclusion, we present data on the anisotropic thermal expansion of the novel nonlinear chalcogenide crystal PGGSe and its nanohardness and Young’s modulus along different directions. Such information will be useful for the crystal growth and the practical application of PGGSe in nonlinear optics, i.e., for evaluation of stresses introduced, e.g., in cutting along phase-matching directions and polishing procedures, as well as for subsequent cleaning and anti-reflection coating of the optical surfaces or evaluation of thermal stress effects in high average power laser applications.