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Abstract
Nanohardness and Young’s modulus of oriented monocrystalline II-IV-V2 chalcopyrite semiconductors are measured by nanoindentation for the first time. The tests are performed on (100) and (001) surfaces. Anisotropy is observed for Young’s modulus only. It is most pronounced for CdSiP2. The hardness results display linear dependence on the melting temperature (except for CdSiP2) and the values decrease with the molar mass. They can be well fitted as a function of the molar mass and the unit cell volume. Young’s modulus dependences show similar trends.
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1. Introduction
The discovery of the laser in the early 1960s stimulated new investigations targeting optical applications of ternary compounds of the I-III-VI2 and II-IV-V2 types that were first developed as semiconductor materials in the early 1950s, after the invention of the transistor. These ternary compounds are obtained by substitution from the much studied binary II-VI (→I-III-VI2) and III-V (→II-IV-V2) semiconductors. Whereas the binary structures are mostly sphalerite type (zincblende α-ZnS, space group $F\bar{4}3m$), i.e. cubic and hence optically isotropic, the ternary structures are normally of the tetragonal chalcopyrite (CuFeS2) type (space group $I\bar{4}2d$) being uniaxial and birefringent. The chalcopyrite unit cell with four formula units can be considered to be built of two sphalerite cells connected in the z-direction. The “tetragonal compression”, a reduction in length of the doubled cell along the z-axis (2-c/a) has often been considered to be the source of the birefringence.
Due to their noncentrosymmetric structure the chalcopyrites possess 2nd order nonlinear optical susceptibility and the predominantly covalent bonds lead to relatively large values of the χ(2) tensor components [1]. Thanks to their extended transparency such nonlinear and birefringent optical materials are interesting for parametric conversion in the mid-IR (3-30 µm) spectral range. Thus, chalcopyrites are indispensable above the 4-5 µm long-wave transmission limits of oxide nonlinear crystals, used at shorter wavelengths [2]. They are also attractive for generation of radiation in the far-IR (30-3000 µm), nowadays called the THz spectral range. Compared to the chalcogenides (I-III-VI2 compounds), the pnictides (II-IV-V2 compounds) are chemically stable, exhibit lower scattering losses, and are superior in terms of nonlinear coefficients, thermal conductivity, laser damage resistivity, isotropy of thermal expansion, thermo-optic coefficients and hardness. Most of the 125 possible II-IV-V2 compounds in fact do not exist, some of them crystallize in different (sphalerite, orthorhombic, or monoclinic structures) or exhibit multiple phases. 14 are known to crystallize in the chalcopyrite structure: these include all combinations of Zn, Cd (II), Si, Ge, Sn (IV) and P, As (V2), as well as MgSiP2 and ZnSnSb2 [1].
The refractive indices of four such crystals, all of them optically positive, were measured by G. D. Boyd and coworkers at Bell Labs by a prism technique in the early 1970s; these data are tabulated in [1]. Three of these compounds exhibit sufficient birefringence (>0.025) for phase-matching: ZnGeP2, ZnSiAs2, and CdGeAs2. Further efforts devoted to the growth and characterization of these chalcopyrites were focused on ZnGeP2 and CdGeAs2 which became commercially available whereas ZnSiAs2, without real advantages over ZnGeP2, was not developed. Notwithstanding its low birefringence, CdGeP2 is interesting and is applied for THz generation because optical rectification does not pose such a condition [3]. Some solid-solutions with sufficient birefringence, such as CdGe(As1-xPx)2 were also studied. The only significant step in increasing the number of practical pnictide chalcopyrite nonlinear crystals in the last 50 years has been the growth of large size, optical quality CdSiP2 by P. Schunemann et al. at BAE Systems [4,5] and its complete optical characterization and commercialization. Note that CdSiP2, together with CdSiAs2, whose optical properties remain largely unknown, are the pnictides with highest tetragonal compression and the only ones that are optically negative. Nevertheless, the interest in the growth and characterization of further pnictide chalcopyrites has never declined and very recently millimeter size monocrystalline MgSiP2 has been reported and investigated by two groups [6,7].
Apart from their good optical and/or electronic properties, the practical use and commercialization of artificial semiconductor crystals depend on their mechanical performance. The mechanical properties are related to the crystal bonding hence they can be associated with other fundamental or practical properties, such as melting temperature, optical band-gap, iconicity, etc. [8]. Thus, the mechanical response of materials is important in both – device design and fundamental physics. Hardness and Young’s modulus are simple means of characterizing the mechanical behavior of crystalline materials, which help understanding and predicting the influence of applied loads of different origin. Their knowledge is important for cutting and polishing techniques, as well as for subsequent cleaning and anti-reflection coating of the optical surfaces or estimation of thermal stress effects. The crystal anisotropy, which results from different atomic arrangements along different crystallographic directions, should influence also the mechanical properties [1]. Such anisotropy is closely related to the mechanical stability during mechanical treatment and processing [9].
While optical and electronic properties of the pnictide chalcopyrites have been widely studied since the 1970s, limited efforts have been devoted to evaluate their mechanical characteristics. Some microhardness values were compiled in [1] but the growth method, the quality of the material and its surface, the crystalline structure, orientation, etc. had not been specified. Attempts to summarize such data show large scatter in the values [10]. Comparison of experimental data on different compounds is also hard since different measuring techniques (Knoop, Vickers, etc.) exist which are also not specified, e.g. see Table 21 in [11]. Mechanical anisotropy has been addressed in more recent literature only theoretically [12–18].
In the present work we studied five pnictide chalcopyrite crystals by nanoindendation measuring simultaneously their nanohardness and Young’s modulus in a comparative manner including orientation dependence for the two major crystallographic directions.
2. Materials and methods
The oriented samples of ZnGeP2, CdSiP2, CdGeP2, CdGeAs2, and CdSnAs2 used in the present study were monocrystalline in nature. These chalcopyrites exhibit congruent melting and single crystals were grown by the horizontal gradient freeze (HGF) technique, which is a modification of the classical Bridgman method [4,19,20]. The polycrystalline charges were synthesized from stoichiometric amounts of high purity (>99.9999%) starting materials. The samples were plates with a thickness in the 0.9−2.3-mm range, double-side hand-polished on best effort basis. Both a-cut (100) and c-cut (001) plates were prepared except for CdSnAs2 for which only c-cut was available. While all the others have been applied in nonlinear optics [2,3], the bandgap of CdSnAs2 is too small for optical applications. Nevertheless, it is interesting for the present study since in terms of band-gap and melting temperature this compound represents a limiting case among the pnictide chalcopyrites.
A G200 Nanoindenter (Keysight Technologies, USA), equipped with NanoSuite 6 Software, was employed for the nanohardness and Young’s modulus measurements. Continuous stiffness mode (CSM) was chosen, since from our experience, by selecting appropriate parameters in this mode, some typical for the other modes “pop-in”, “pop-out”, kinks, and “elbow” effects [21], can be suppressed. In the CSM mode, the tip (in our case Berkovich diamond tip) penetrates gradually into the sample but in a sequence of 2-nm-amplitude steps. This is realized by applying a primary load with an oscillatory contribution superimposed on it. Each step consists of loading and unloading processes so that the nanohardness, HIT, and Young’s modulus, ES, are calculated at each depth based on the residual impression left by the tip. During unloading the material partially relaxes decreasing the probability of sample damages, this way improving the reliability of the results.
During the loading process, elastic and plastic strain fields are involved, whereas during unloading the elastic deformation recovers [22] so that elastic and plastic components of the deformation can be separated. The hardness is then defined as the resistance to plastic deformations during a constant load, whereas Young’s modulus is proportional to the elastic deformation as σ = ESε, where σ is the stress and ε is the strain.
The method of calculating HIT and ES follows the equations given in [22,23,24]. Accordingly, the nanohardness is derived as:
where P is the load and A is the contact area. The contact area is calculated from the contact area depth, hI (measured at each unloading step), taking into account the angle of the Berkovich tip: where ${h_\textrm{I}} = [{{h_\textrm{T}} - \omega ({P/S} )} ]$, ${h_\textrm{T}}$ is the total depth during the loading, S is the contact stiffness, and ω is the frequency of the oscillatory force.The composite Young’s modulus (of the system indenter - sample), Er, is calculated from:
Prior to the measurements, the nanoindenter calibration was checked by means of polished fused silica. Values of 9.5 ± 0.1 GPa for the nanohardness and 71.3 ± 0.6 GPa for Young‘s modulus were obtained, which meet the device acceptance criteria (8.5–10.5 GPa and 70–75 GPa, respectively). The “thermal drift correction” option of the equipment was used in order to suppress thermal drift effects.
A reliable comparison of the results was ensured by using identical test conditions: room temperature, oscillation frequency of 45 Hz, oscillatory amplitude of 2 nm, maximum penetration depth of 2 µm, strain rate to the target of 0.05 s−1, surface approach velocity of 10 nm/s, and surface approach distance of 1 µm. The chosen parameters enable a precise location of the sample surface, which is a mandatory prerequisite for correct determination of HIT and ES.
The samples were fixed on a specially designed cylindrical holder with a “crystal bond” adhesive (50°C flow point). Prior to the measurements, the sample surface was cleaned with acetone. At least 10 tests were performed for each sample, spaced 50 µm apart from each other, so that overlapping with plastic zones affected by previous tests was avoided. For each sample, the values of HIT and ES were averaged over all tests in the same penetration depth interval (300–1900 nm).
3. Results and discussion
Figure 1 shows representative plots of experimental nanohardness and Young’s modulus penetration depth dependences for the CdSiP2 samples. It is seen that the nanohardness dependence, Fig. 1(a), exhibits an initial increase, followed by a gradual decrease with the depth. This characteristic behavior of the curves has been attributed to various effects in the corresponding literature [25–30]. The initial increase is in fact intrinsic for the method since at the very beginning the entire load is transferred to the sample over a very small area, causing a strain-hardening effect. Another possible reason is the surface hardening during polishing, which causes some preliminary plastic deformation, i.e. increased dislocation density. The decrease with the depth can be a consequence of a material softening due to changing dislocation dynamics (dislocation nucleation enhancement and mobility), an effect known as “indentation size effect”. Quantitatively, these effects depend on the particular CSM parameters such as oscillation amplitude, frequency, etc.
Fig. 1. Representative plots of depth dependence from individual tests using different cuts of CdSiP2: (a) nanohardness, and (b) Young’s modulus.
There were some variations in the initial increase range for the different tests on the same sample but the curves showed excellent reproducibility (overlap) at values above 300 nm. This was the reason to choose an interval from 300 to 1900 nm for averaging the values. The values obtained in this way for all samples are summarized in Table 1. The melting points in the table are taken from [31] and the lattice constants from [10]. It can be seen that within our error limits, there is no anisotropy in the nanohardness. Some anisotropy is evident for Young’s modulus, most pronounced for the CdSiP2 crystal.
It is well known that the melting points similar to hardness are qualitative indicators of cohesion. In Fig. 2(a), the nanohardness dependence on the melting temperature Tm is shown. As already mentioned, there is no effect of anisotropy. The dependence follows a linear law (${H_{\textrm{IT}}} = 0.014{T_m} - 2.6$), except for CdSiP2. Our results indicate that the hardness and Young’s modulus of CdSiP2 are lower compared to ZnGeP2. Some models indeed predict such a deviation.
Fig. 2. Melting point dependence of: (a) nanohardness and (b) Young’s modulus. Solid triangles represent a-cut and open triangles – c-cut. The solid diamonds represent additional tests with randomly oriented ZnGeP2 and CdSiP2 plates polished with high optical quality [32].
Neumann [33] proposed a simple relationship between the unit cell volume V0, and the ratio H/TmK (TmK = Tm + 273), which qualitatively agrees with our results. The two fitting parameters rely on the assumption for an almost constant ionicity (averaged for the two bonds) for the entire series and experimental microhardness values. Neumann’s predictions about the behavior of the bulk modulus [33], which he related to the same parameters but with different fitting constants calibrated to experimental data available only for two compounds, also qualitatively agree with our results on Young’s modulus in Fig. 2(b), where again CdSiP2 shows a deviation from the general trend, although the latter is not linear as for the hardness in Fig. 2(a). Similar dependences as those depicted in Fig. 2 were predicted also by the plasma oscillations theory of solids [34].
The deviation for CdSiP2 can be attributed to the specific nature of the interatomic interactions in this crystal, manifested in anomalous thermal expansion - negative (contraction) along the optical axis and positive perpendicular to it [4] which apparently leads to changes in the deformation mechanisms (dislocations formation, density, flow and etc.). In fact, such a deviation has been observed also in some older microhardness tests [10].
Since the strength of the cohesive forces defines the compressibility, hardness and melting point of a crystal, i.e. stronger cohesive forces imply more resistance to compression, to indentation, and to thermal atomic motion, in [35] the hardness was related to the Debye temperature (θD = h.νm/kB, where h is Planck's constant, kB is Boltzmann's constant, and νm is the maximum lattice vibration frequency). A linear empirical relationship between hardness and Debye temperature at 0 K, H = αθD,0 – H0, was established in [36] for the II-IV-V2 chalcopyrites. Refitting their parameters using our data on the five chalcopyrites measured in the present work and their collection of data on θD,0 we obtained somewhat lower values of α = 0.038 GPa/K, and H0 = 3.34 GPa.
From a different view point, a monotonic decrease of the microhardness with the molar mass was established in [8] and Fig. 3(a) qualitatively confirms such a dependence for the nanohardness. This can be explained in terms of the chemical bond strength, where a process of metallization takes place. With different chemical bonds along different crystallographic directions, the overall strength of the crystal will be defined by the weakest of them. The ratio between ionic and covalent bond contributions in the ternary compounds crystalizing with chalcopyrite structure seems to be related to the tetragonal compression. A decrease in covalent bond energy, caused by tetragonal distortion, leads to a slight increase in the proportion of ionic bonds, which, however, does not compensate for the general decrease in bond energy. The decrease of Young’s modulus with the molar mass is seen in Fig. 3(b) but similar to the nanohardness in Fig. 3(a), the dependence is more complex and the proper description should include some functional dependence on more than just one parameter. Further investigations are required to clear this point.
Fig. 3. Molar mass dependence of: (a) nanohardness and (b) Young’s modulus. Solid triangles represent a-cut and open triangles – c-cut. The solid diamonds represent additional tests with randomly oriented ZnGeP2 and CdSiP2 plates polished with high optical quality [32].
Combining the linear dependence of H on θD,0 [36] and another empirical relationship between θD,0, the molar mass, µ, and the crystal unit cell volume, V, established in [37], the hardness can be expressed as:
With the molar mass and the cell volume calculated from Table 1, and using the experimental nanohardness values from the same table, a linear regression gives B = 70.0 GPa.(g/mol)1/2.nm5/2 and HB = 1.59 GPa for the two constants in Eq. (5), see Fig. 4. Thus the last relationship accounts better for the hardness “anomaly” observed with CdSiP2.
In Fig. (2) and Fig. (3), we have also added measurements performed by us on similar CdSiP2 and ZnGeP2 samples with unknown orientation [32]. These samples were polished with a more sophisticated technique (related to removal of more material) in order to increase the laser surface damage resistivity [38]. The comparison indicates that nanoindentation measurements are not so sensitive to the surface polishing quality as expected. This can be attributed to the fact that the equipment software eliminates automatically from the statistics tests that have been accidentally performed in the region of some existing surface defects.
As for the anisotropy experimentally observed in the Young’s modulus values in Fig. 2(b) and Fig. 3(b), similar arguments as those outlined above for the deviation of the CdSiP2 results from a general trend can be provided. In fact a clear correlation between Young’s modulus anisotropy and tetragonal compression (see Table 1) can be seen. The tetragonal compression has been initially considered to be the factor that determines the crystal birefringence, however, it turned out that this does not apply for the entire range (for all compounds belonging to the pnictide or chalcogenide chalcopyrite families). However, CdSiP2 which exhibits the largest tetragonal compression from the pnictides [31], is the only crystal from those studied in the present work that is optically negative and shows anomalous thermal expansion. The only other pnictide chalcopyrite that is optically negative, CdSiAs2, exhibits the second largest tetragonal compression [31] and at least from theoretical considerations should be the only other compound that shows also anomalous thermal expansion [39]. Note that similar predictions for the anomalous thermal expansion of CdSiP2 [39] were experimentally confirmed only later [4]. The situation with the correlation between maximum tetragonal compression, negative birefringence and anomalous thermal expansion seems similar in the chalcogenide type chalcopyrites, in particular with respect to the two most prominent representatives, the commercially available AgGaS2 and AgGaSe2 nonlinear optical crystals [31,39]. Experimental data on anisotropy of Young’s modulus could not be found in the existing literature. As mentioned in the introduction, there is a number of theoretical reports which try to predict the elasticity tensor components based on Debye model calculations [12], local density approximation method within the density functional theory [13], pseudo-potentials plane wave method within the same theory in the generalized gradient approximation [14,15,18], and first principle calculations [16,17]. Results can be found for the pnictide chalcopyrites studied in the present work except for CdSnAs2, however, the deviations for Young’s modulus are large, with the theoretical predictions being in general anisotropic (the elasticity tensor is of 4th rank) but indicating much lower values. The only occasion where the derived Young’s modulus values are close to our experimental results is found in [40], based on first principles calculations for CdSiP2.
4. Conclusion
In conclusion, we measured the nanohardness and Young’s modulus of five pnictide chalcopyrite crystals by nanoindentation using oriented samples that were grown and prepared in the same way for a reliable comparison. Two principal orientations, along the a- and c-crystallographic axes were compared. No anisotropy was established for the hardness while the anisotropy of Young’s modulus was highest for CdSiP2, negligible for ZnGeP2, and similar for GdGeP2 and CdGeAs2. The experimental findings in general do not support theoretical models from the literature concerning Young’s modulus, in particular such based on first principles calculations, with the measured values being higher.
The dependence of the measured nanohardness on melting temperature seems linear except for CdSiP2 which shows lower hardness compared to ZnGeP2 although its melting temperature is higher and its band-gap is larger. This “anomaly” seems to occur hand in hand with anomalous thermal expansion of CdSiP2 which is the pnictide chalcopyrite crystal with maximum tetragonal compression. Some empirical and semi-empirical models provide trends in the behavior of the hardness versus other parameters such as melting temperature, unit cell volume, molar mass, Debye temperature, band-gap, etc.: there is a qualitative agreement with our experimental results but a more reliable functional dependence would require data on more representatives of the same family and simultaneous involvement of more than one parameters, e.g. molar mass and unit cell volume.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. J. L. Shay and J. H. Wernick, Ternary Chalcopyrite Semi-conductors: Growth, Electronic Properties, and Applications (Pergamon Press, 1975).
2. V. Petrov, “Frequency down-conversion of solid-state laser sources to the mid-infrared spectral range using non-oxide nonlinear crystals,” Prog. Quantum Electron. 42, 1–106 (2015). [CrossRef]
3. H. P. Piyathilaka, R. Sooriyagoda, V. Dewasurendra, et al., “Terahertz generation by optical rectification in chalcopyrite crystals ZnGeP2, CdGeP2 and CdSiP2,” Opt. Express 27(12), 16958–16965 (2019). [CrossRef]
4. K. T. Zawilski, P. G. Schunemann, T. C. Pollak, et al., “Growth and characterization of large CdSiP2 single crystals,” J. Cryst. Growth 312(8), 1127–1132 (2010). [CrossRef]
5. V. Petrov, “New applications of chalcopyrite crystals in nonlinear optics,” Phys. Status Solidi C 14(6), 1600161 (2017). [CrossRef]
6. J. Chen, Q. Wu, H. Tian, et al., “Uncovering a vital band gap mechanism of pnictides,” Adv. Science 9(14), 2105787 (2022). [CrossRef]
7. J. He, Y. Guan, V. Trinquet, et al., “MgSiP2: An infrared nonlinear optical crystal with a large non-resonant phase-matchable second harmonic coefficient and high laser damage threshold,” Adv. Opt. Mater. 11(24), 2301060 (2023). [CrossRef]
8. N. A. Goryunova and Yu. A. Valov, eds., Semiconductors${A^2}{B^4}C_2^5$, [in Russian], (Sovetskoe Radio, Moscow, 1974).
9. Z. Cao, H. Yang, S. Sun, et al., “Study on the influence of anisotropy of ZnGeP2 single crystal on its surface cutting quality,” Opt. Mater. 110, 110383 (2020). [CrossRef]
10. H. Boudriot, E. Buhrig, H. Oettel, et al., “II-IV-V2-Halbleiter,” Ch. 8 in Verbindungshalbleiter (Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1986), pp. 339–391.
11. N. A. Goryunova, Complex Diamond-like Semiconductors [in Russian], (Sovetskoe Radio, Moscow, 1968).
12. L. Wei, G. Zhang, W. Fan, et al., “Anisotropic thermal anharmonicity of CdSiP2 and ZnGeP2: Ab initio calculations,” J. Appl. Phys. 114(23), 233501 (2013). [CrossRef]
13. S. K. Tripathy and V. Kumar, “Electronic, elastic and optical properties of ZnGeP2 semiconductor under hydrostatic pressures,” Mater. Sci. Eng B 182, 52–58 (2014). [CrossRef]
14. S.-R. Zhang, S.-F. Zhu, L.-H. Xie, et al., “Theoretical study of the structural, elastic, and thermodynamic properties of chalcopyrite ZnGeP2,” Mater. Sci. Semicond. Process. 38, 41–49 (2015). [CrossRef]
15. S. R. Zhang, Z. J. Zhu, H. J. Hou, et al., “Structural, elastic, optical and thermodynamic properties of CdGeP2 from theoretical prediction,” J. Ovonic Res. 18(6), 781–796 (2022). [CrossRef]
16. H. J. Hou, H. J. Zhu, S. P. Li, et al., “Structural, dynamical and thermodynamic properties of CdXP2 (X = Si, Ge) from first principles,” Indian J. Phys. 92(3), 315–323 (2018). [CrossRef]
17. Z.-L. Lv, Y. Cheng, X.-R. Chen, et al., “First principles study of electronic, bonding, elastic properties and intrinsic hardness of CdSiP2,” Comp. Mater. Sci. 77, 114–119 (2013). [CrossRef]
18. C. Zhu, C. Yang, T. Ma, et al., “Simulative calculation and analysis of structural, elastic and electronic properties of CdGeAs2,” J. Chin. Ceram. Soc. 39(2), 349–354 (2011).
19. K. T. Zawilski, P. G. Schunemann, S. D. Setzler, et al., “Large aperture single crystal ZnGeP2 for high-energy applications,” J. Cryst. Growth 310(7-9), 1891–1896 (2008). [CrossRef]
20. P. G. Schunemann and T. M. Pollak, “Method for growing crystals,” U.S. Patent No. 5,611,856 (March 18, 1997).
21. D. J. Oliver, B. R. Lawn, R. F. Cook, et al., “Giant pop-ins in nanoindented silicon and germanium caused by lateral cracking,” J. Mater. Res. 23(2), 297–301 (2008). [CrossRef]
22. W. C. Oliver and J. B. Pethica, “Method for continuous determination of the elastic stiffness of contact between two bodies,” U.S. Patent No. 4,848,141 (Jul.18, 1989).
23. W. C. Oliver and G. M. Pharr, “An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments,” J. Mater. Res. 7(6), 1564–1583 (1992). [CrossRef]
24. J. B. Pethica and W. C. Oliver, “Mechanical properties of nanometre volumes of material: use of the elastic response of small area indentations,” Mater. Res. Soc. Symp. Proc. 130, 13–23 (1988). [CrossRef]
25. T. Chudoba, “Measurement of hardness and Young’s modulus by nanoindentation,” Ch. 6 in: Nanostructured Coatings, A. Cavaleiro and J. Th. M. Hosson, eds., (Springer Science + Business Media, 2006), pp. 216–260.
26. J. J. Gilman, Chemistry and Physics of Mechanical Hardness, (Wiley, 2009).
27. G. M. Pharr, J. H. Strader, and W. C. Oliver, “Critical issues in making small-depth mechanical property measurements by nanoindentation with continuous stiffness measurement,” J. Mater. Res. 24(3), 653–666 (2009). [CrossRef]
28. K. W. Siu and A. H. W. Ngan, “The continuous stiffness measurement technique in nanoindentation intrinsically modifies the strength of the sample,” Phil. Mag. 93(5), 449–467 (2013). [CrossRef]
29. S. Shim, H. Bei, E. P. George, et al., “A different type of indentation size effect,” Scr. Mater. 59(10), 1095–1098 (2008). [CrossRef]
30. Yu. V. Milman, A. A. Golubenko, and S. N. Dub, “Indentation size effect in nanohardness,” Acta Mater. 59(20), 7480–7487 (2011). [CrossRef]
31. P. G. Schunemann and K. T. Zawilski, “New ternary chalcopyrite semiconductors for mid-IR NLO frequency conversion,” Proc. SPIE PC12405, PC124050J (2023). [CrossRef]
32. G. Exner, A. Grigorov, E. Ivanova, et al., “Nanohardness measurements of CdSiP2 and ZnGeP2 chalcopyrite-type nonlinear optical crystals,” Crystals 13(8), 1164 (2023). [CrossRef]
33. H. Neumann, “Microhardness scaling and bulk modulus-microhardness relationship in ${\textrm{A}^{\textrm{II}}}{\textrm{B}^{\textrm{IV}}}\textrm{C}_2^\textrm{V}$ chalcopyrite compounds,” Cryst. Res. Technol. 23(1), 97–102 (1988). [CrossRef]
34. V. Kumar, G. M. Prasad, and D. Chandra, “Microhardness-plasmon energy-bulk modulus relationship in ternary chalcopyrite semiconductors,” Cryst. Res. Technol. 31(4), 501–504 (1996). [CrossRef]
35. S. C. Abrahams and F. S. L. Hsu, “Debye temperatures and cohesive properties,” J. Chem. Phys. 63(3), 1162–1165 (1975). [CrossRef]
36. C. Rincón and M. L. Valeri-Gil, “Microhardness, Debye temperature and bond ionicity of ternary chalcopyrite compounds,” Mater. Lett. 28(4-6), 297–300 (1996). [CrossRef]
37. C. Rincón, “Debye temperature and melting point in ${\textrm{A}^\textrm{I}}{\textrm{B}^{\textrm{III}}}\textrm{C}_2^{\textrm{VI}}$ and ${\textrm{A}^{\textrm{II}}}{\textrm{B}^{\textrm{IV}}}\textrm{C}_2^\textrm{V}$ chalcopyrite compounds,” Phys. Stat. Sol. (a) 134(2), 383–389 (1992).
38. K. T. Zawilski, S. D. Setzler, P. G. Schunemann, et al., “Increasing the laser-induced damage threshold of single-crystal ZnGeP2,” J. Opt. Soc. Am. B 23(11), 2310–2316 (2006). [CrossRef]
39. H. Neumann, “Trends in the thermal expansion coefficients of the ${\textrm{A}^\textrm{I}}{\textrm{B}^{\textrm{III}}}\textrm{C}_2^{\textrm{VI}}$ and ${\textrm{A}^{\textrm{II}}}{\textrm{B}^{\textrm{IV}}}\textrm{C}_2^\textrm{V}$ chalcopyrite compounds,” Krist. Tech. 15(7), 849–857 (1980). [CrossRef]
40. H. J. Hou, H. J. Zhu, J. Xu, et al., “Structural, elastic, and optical properties of chalcopyrite CdSiP2 with the application in nonlinear optics from first principles calculations,” Braz. J. Phys. 46(6), 628–635 (2016). [CrossRef]
