1. Introduction

The second-order nonlinear optical (NLO) materials have attracted particular interest due to their important applications in many fields of optical parametric oscillator (OPO), difference frequency generation (DFG), laser frequency conversion, and signal communication, etc. For a NLO crystal with excellent performance, several prerequisites must be fulfilled, including high coefficient of second-harmonic generation (SHG), wide transparent region, good phase matchability, a high laser damage threshold, and the availability of corresponding large-size crystal. Several NLO crystals that are satisfied with the practical requirements in the ultraviolet and visible regions have been obtained, such as KTiOPO₄ (KTP) [1], and LiNbO₃ [2] etc. For the NLO crystals suitable in the infrared region (IR), although in recent years people have discovered many new materials (e.g., see a recent review by Chung and Kanatzidis [3]), up to now only few long-established compounds with a chalcopyrite structure, such as ZnGeP₂ [4] and AgGaS₂ [5], have practical applications. For AgGaS₂, it is the most common and representative NLO crystal in the IR region and has been widely used for DFG and OPO [6–10]. AgGaS₂ crystals exhibits relatively large SHG coefficient (d₃₆ = 18 ± 2.7 pm/V) and modestly wide transparent range in the IR spectral region (0.47-13 μm) [11], however the corresponding laser-induced damage threshold is small due to its narrow band gap (E_g = 2.64 eV) [12], and as a result AgGaS₂ is still not good enough for the application of high-energy laser. So it is necessary to find a way to increase the laser-induced damage threshold and simultaneously balance the other optical properties of AgGaS₂ crystal.

It is well known that the E_g of a semiconductor is sensitive to the external pressure, which may provide an effective way to regulate the laser-induced damage threshold of the material. For AgGaS₂, the experimental measurement showed that when the pressure is smaller than 10.2 GPa, the E_g increases linearly with pressure at the rate of about 40 meV/GPa [13,14]. Therefore, the improvement of laser-induced damage threshold of AgGaS₂ can be expected if a suitable external pressure is applied. However, it is noted that the chalcopyrite structure tends to undergo geometrical deformation under high pressure, which will lead to the change of the energy-band structure and eventually affects the NLO behaviors of the AgGaS₂. Several experimental researches have been carried out investigate the pressure-induced phase changes of AgGaS₂ crystal [15–20]. In an early work, Carlone et al. studied the pressure dependence of the Raman peaks of AgGaS₂, and they observed two pressure-induced phase transitions at 4.2 ± 0.5 GPa and 11.6 ± 0.5 GPa, respectively [15]. Werner et al. performed a powder X-ray diffraction study on AgGaS₂ at pressures up to 25 GPa, and their results showed the sequence of phase transitions is chalcopyrite → unknown structure with a possible hexagonal symmetry (at 5 GPa) → α-NaFeO₂-type lattice (at 12 GPa) → NaCl type lattice (at 15 GPa) [16]. By employing the single-crystal diffraction technique combined with a miniature diamond-anvil high-pressure cell, Kitahara et al. reported a plausible monoclinic Cc symmetry for the high-pressure phase structure above 4.2 GPa [18]. It is worth pointing out that if the AgGaS₂ crystal adopts the centrosymmetric α-NaFeO₂-type structure, the SHG effect will be quenched. In the aspect of the optical behaviors, except that the variation of the optical absorption edge of AgGaS₂ single crystal at high pressure have been reported experimentally [13], the quantitative study of the pressure-induced changes on the optical performances, especially the nonlinear optical responses of AgGaS₂ is still not clear. On the other hand, there have been many theoretical studies on the electronic structure and optical properties of AgGaS₂ [21–26], however no theoretical research has yet been undertaken to assess how the pressure affects the NLO properties of AgGaS₂ crystal. Therefore, as an important candidate of the commercial IR NLO material, it appears to us worthwhile to investigate the geometry and electronic structures of AgGaS₂ under pressure, particularly with regard to the change in the optical properties with external pressure.

Prompted by the possibility that the external pressure may provide an effective way to improve the laser-induced damage threshold of AgGaS₂, and also motivated by that it provides a suitable prototype to study the sensitive relationship between the structure and NLO performance of I-III-VI₂ ternary semiconductors with the chalcopyrite structure (e.g. AgGaSe₂, CuGaS₂, CuInS₂, etc.), in this paper density functional theory (DFT) calculations are carried out to reveal the high-pressure optical behaviors of AgGaS₂ crystal.

2. Computational details

2.1 Prediction of the stable crystal structure at different pressures

In here, a specialized genetic algorithm (GA) approach is employed to find the lowest-energy structure of AgGaS₂ crystal under different pressure. The detailed explanation of this method can be found in our previous work [27]. The main feature of the present method is that the coordination fashions of the building units are introduced to construct the structures of individuals during the GA procedure. Our results showed that such approach can obviously improve the efficiency and success rate of obtaining the stable structure of the inorganic crystals. It is well known that the building block of AgGaS₂ is [GaS₄]^5- tetrahedron at the normal pressure, which was selected as the starting point to create the initial structures of individuals. For the primitive cell of AgGaS₂, there are two Ag, two Ga, and four S atoms. The parameters used in our GA procedure, including the maximum generation number, the population size, the selective probability, the crossover probability, and the mutation probability were set to 15, 15, 0.8, 0.6, and 0.1, respectively. In the present work, the Vienna ab initio simulation package (VASP) [28–30] was employed to relax the geometry (including the cell parameters and atomic positions) of each individual. During the calculations, the projector-augmented wave (PAW) pseudopotentials and the Perdew-Burke-Ernzerhof (PBE) type exchange-correlation functionals were adopted. The kinetic energy cutoff for the plane-wave expansion and the separation of k points were set to 500 eV and 0.08 Å⁻¹, respectively.

2.2 Prediction of the band gap at different pressures

Since the results of optical calculations are directly related to the value of the E_g, an accurate estimation of the E_g is necessary if we want to obtain reliable results, especially for the system that is lacking experimental data. However, it is well known that the pure DFT method usually underestimates the E_g of semiconductors and insulators dues to insufficient cancellation of the self-interaction correction inherent in the local exchange functionals. Hence, as expected, the E_g of chalcopyrite-type AgGaS₂ at zero pressure obtained by PBE functional is about 0.91 eV, obviously lower than the experimental value of 2.64 eV [12]. To overcome above deficiency, besides utilizing the many-body GW theory, a widely used approach is to include a part of Hartree-Fock (HF) exchange in the DFT energy, which corresponds to the hybrid DFT method. In the present study, the Heyd-Scuseria-Ernzerhof (HSE06) hybrid DFT functional [31] was applied to realize a more accurate description of the E_g of AgGaS₂ under pressure, which has the form of

 

Fig. 1 (a) Variation of the calculated band gap (E_g) as a function of the percentage contribution of HF exchange, and (b) the pressure dependence of E_g of AgGaS₂ crystallized in different phases.

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2.3 Calculations of the linear and second-order nonlinear optical properties

The linear optical response of an insulator or a semiconductor is directly related to the complex dielectric function, $\epsilon (\omega )={\epsilon }_1(\omega )+i{\epsilon }_2(\omega )$, and the imaginary part ε₂(ω) is given in following equation [34],

3. Results and discussion

3.1 Structures of AgGaS₂ crystal at different pressures

In the present work, the specialized GA approach has been employed to explore the stable structures of AgGaS₂ crystal under different pressures (P) up to 10 GPa. After carefully examining the enthalpy of each structure, the most stable structure of AgGaS₂ crystal at different pressure is obtained, which is summarized in Fig. 2. Moreover, as a typical example, the detailed structural parameters of above four phases at P = 7.5 GPa including the lattice parameters, fractional atomic positions, volume per formula, lengths of Ga-S and Ag-S bonds are listed in Table 1.

 

Fig. 2 The most stable structures of AgGaS₂ crystal at different pressures predicted by the GA approaches. The Ag, Ga and S atoms are denoted by blue, brown, and yellow spheres, respectively.

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Table 1. Lattice parameters, fractional atomic positions, volume per formula, Ga-S and Ag-S bond lengths of AgGaS₂ crystallized in different structures at P = 7.5 GPa

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To determine the pressures of the phase transformations, Fig. 3(a) displays the pressure dependence of the relative enthalpies per formula unit for the different phases with respect to the chalcopyrite structure. In the early stage when the pressure is low, $I\overline{4}2d$ structure is the most stable phase, however the energy difference between $I\overline{4}2d$ and Cc structures is quit small. Actually, the monoclinic Cc phase can be seen as a “distorted chalcopyrite” structure, and the structural deformation becomes more pronounced as the pressure increases. At pressures higher than about 4.75 GPa, a phase transition occurs where the Cc structure becomes most favorable. This pressure is in good agreement with the experimental result of 4.2 GPa [18,19]. In addition, the structure of AgGaS₂ crystal at P = 5.5 GPa has been determined experimentally by the single-crystal X-ray diffraction using synchrotron radiation, and the cell dimensions predicted by the present work are a = 8.0451 Å, b = 8.0420 Å, c = 6.3993 Å, and β = 127.35°, which are also consistent with the experimental values of 8.031 Å, 8.014 Å, 6.221 Å, and 128.42° [19], respectively. Compared to the chalcopyrite structure (Fig. 2(a)), it is obvious that in the Cc phase, Fig. 2(b), AgS₄ and GaS₄ tetrahedra are all deformed to break the $\overline{4}$ symmetry in the tetragonal phase. However, we can see that the configuration of AgS₄ tetrahedra is more sensitive to the external pressure. For instance, examining the Ag-S and Ga-S bond lengths listed in Table 1 at P = 7.5 GPa, the largest deviation among four Ag-S bond lengths is about 0.15 Å, while four Ga-S bond lengths differ by only 0.02 Å, indicating that the distortion of AgS₄ tetrahedra is more remarkable than GaS₄ tetrahedra. Our results (Fig. 3(a)) show that the Cc phase is most stable up to around P = 7.18 GPa, and when the pressure is raised above this value, a new phase with Ibam symmetry is energetically preferred. A distinct feature of Ibam structure is that, under the pressure the silver atom now adopts a planar 4-fold coordinated geometry, Fig. 2(c). It seems that the Ibam phase appears only in a quite narrow range of the pressure, and at pressures higher than about 7.38 GPa, the $R\overline{3}m$ structure appears as the most stable phase. This α-NaFeO₂ type structure can be regarded as a distorted rock salt superstructure, in which Ga and Ag atoms are occupied the octahedral interstitial sites of the cubic close-packed sulfur lattice (Fig. 2(d)), and correspondingly, Ga and Ag atoms are surrounded by six sulfur atoms in a distorted octahedral environment. Compared to the above mentioned $I\overline{4}2d$, Cc, and Ibam phases, under the same pressure the $R\overline{3}m$ structure has the smallest equilibrium volume (Fig. 3(b)). Therefore, the appearance of the $R\overline{3}m$ structure leads to an obvious volume change at the transition pressure, which is consistent with the experimental observation [20]. According to the above results, under pressures up to 10 GPa, three phase transitions of AgGaS₂ are found at P = 4.75, 7.18, and 7.38 GPa, respectively. Among them, the transition at P = 4.75 GPa can be inferred to from chalcopyrite to the structure with a monoclinic Cc symmetry, the transition at P = 7.18 GPa to the orthorhombic Ibam phase, and the transition at P = 7.38 GPa to α-NaFeO₂ type structure. It is worth noting that both Ibam and α-NaFeO₂ type structures have inversion symmetry, indicating that AgGaS₂ tends to crystallize in a centrosymmetric structure at high pressures. Consequently, AgGaS₂ crystal will lose SHG response at the external pressures higher than about 7.18 GPa. So, on the following sections, we will focus on the cases that the pressure is no more than 7 GPa, and in such range of pressure only chalcopyrite and Cc structures need to be considered.

 

Fig. 3 Variation of (a) the relative enthalpy per formula with respect to the chalcopyrite structure, and (b) the volume per formula of AgGaS₂ crystal of four phases at different pressures.

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3.2 Band structure of AgGaS₂ crystal at different pressures

For the chalcopyrite AgGaS₂, the electronic structures under different pressures are explored. In Fig. 4(a), the band structure without pressure predicted by PBE functionals is provided. Basing on the results of the partial density of states (data not shown), the energy bands below −12.0 eV are dominated by S 3s states, and also contain some contributions of Ga 4s and 4p orbitals. Two energy bands distributed in −8.0 ~-5.0 eV are composed of Ga 4s and S 3p states. In the region of valence band (VB), the mixture of Ag 4d, S 3p and Ga 4p states is observed, and the top of VB is mainly derived from the hybridization of Ag 4d and S 3p orbitals. For the bottom of conduction band (CB), it primarily consists of Ga 4s states. When the pressure is applied, as an example, the band structure of P = 4 GPa is also shown in Fig. 4(a) for comparison. It is clear that AgGaS₂ crystal shows similar arrangement of the band structure and it is still a direct band gap semiconductor under the pressure. However, the compression causes the upward shift of the CB edge, which leads to the increase of the E_g (about 0.17 eV). On the contrary, the occupied energy bands exhibit downward shift, and this movement becomes more obvious for those energy bands located at lower energy. The pressure dependence of the E_g of AgGaS₂ crystallized in the chalcopyrite structure is presented in Fig. 1(b), in here the E_g is predicted by the above revHSE06 functional with 37.27% HF exchange. A linear least-square fit to the calculated data gives,

 

Fig. 4 Band structures of AgGaS₂ crystallized in (a) the chalcopyrite structure phase at P = 0 and 4 GPa, and (b) the Cc phase at P = 5 and 7 GPa. The Fermi energy is set to zero.

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The electronic structures of Cc phase are also studied, and as instances the band structures of P = 5 and 7 GPa are shown in Fig. 4(b). Compared to the chalcopyrite structure, the minimum band gap is still found to be direct at the Γ point, and the same components are also obtained for the energy bands in the VB and CB regions. Similarly, due to the upward shift of the CB edge, the enlargement of the E_g is still observed as the pressure is increased. However, it seems that for the Cc phase, the pressure has a relatively weak influence on the position of the occupied energy bands, and correspondingly, the occupied energy bands of P = 7 GPa only exhibit a slight downward movement with respect to those of P = 5 Pa. As shown in Fig. 1(b), for the Cc phase, the linear relationship between E_g and the pressure is preserved. It is noted that the value of dE_g/dP for the Cc phase (about 106 meV/GPa) is more than twice larger than that of chalcopyrite phase, implying that the pressure coefficient of the band gap is sensitive to the phase structure.

According to the E_g obtained by the revHSE06 functional, our results indicate that under pressure up to 7 GPa, the E_g of AgGaS₂ crystal increases obviously from 2.64 eV to 3.14 eV, suggesting that the laser-induced damage threshold of AgGaS₂ crystal can be improved by applying an external pressure. Furthermore, another consequence of the increasing of the band gap is that the transparency range of AgGaS₂ will be extended, in which the lower bound is reduced from 0.47 μm [11] (at normal pressure) to 0.39 μm (at P = 7 GPa).

3.3 Linear optical susceptibilities of AgGaS₂ crystal at different pressures

After obtaining the stable structure under different pressures, we then can investigate the optical properties of AgGaS₂ crystal with noncentrosymmetric arrangements. In the optical calculations, a very dense k-point mesh that results in more than 4000 k points in the first Brillouin zone was employed to guarantee the convergences of the results of the linear and second-order nonlinear optical responses, and the energy cutoff for those empty energy bands involved in the calculations was taken to be at least 30 eV above the top of VB.

The linear optical responses of AgGaS₂ crystal at the normal pressure have been extensively studied experimentally. To check the reliability of the theoretical method employed in the present work, we first focus on the calculated results of $I\overline{4}2d$ phase at P = 0 GPa. The AgGaS₂ with the chalcopyrite structure belongs to uniaxial crystal, and so there are two dielectric tensor components, corresponding to electric field perpendicular and parallel to the c-axis, namely ε^xx(ω) and ε^zz(ω), respectively. The predicted complex dielectric function without external pressure, as an example for ε^xx(ω) is displayed in Fig. 5(a). From the dielectric function, the refractive indices, n^xx(ω) and n^zz(ω) can be obtained, and then the birefringence Δn(ω) can be determined from the difference between two refraction indices (Fig. 5(b)). Additionally, the corresponding static dielectric constants (ε(0)), refractive indices (n(0)), and birefringence (Δn(0)) are listed in Table 2, and the results of other theoretical works and some available experimental values are also shown for comparison [37–41]. The calculated refractive indices in the present work agree well with the values of experimental measurements, while the results obtained by other theoretical studies are somewhat overestimated. It is noted that under the normal pressure, both theoretical and experimental magnitudes of the birefringence of AgGaS₂ is slightly small in comparison with the reasonable range of 0.06 – 0.1. From Fig. 5(b), it can be seen that the negative birefringence is predicted when the wavelength (λ) is larger than 0.502 μm (i.e. 2.47 eV). This critical wavelength is also in good agreement with the experimental observation that AgGaS₂ is a negative uniaxial crystal at λ > 0.497 μm [37]. Therefore, the optical properties of AgGaS₂ at the normal pressure can be well reproduced by our theoretical calculations.

 

Fig. 5 (a) Calculated complex dielectric function ε^xx(ω) and (b) birefringence Δn(ω) for the chalcopyrite structure, as well as (c) ε^xx(ω) and (d) Δn(ω) for the Cc phase at different pressures.

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Table 2. The calculated static dielectric constants, refractive indices, birefringence, and the magnitudes of the SHG coefficient of AgGaS₂ crystallized in the chalcopyrite structure

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Now let us discuss the linear optical susceptibilities of the chalcopyrite phase when the pressure is applied. For example, the calculated dielectric function ε^xx(ω) of AgGaS₂ crystal at P = 2 and 4 GPa are shown in Fig. 5(a). Due to the similarities in the underlying band structures, the general shapes of the dielectric function curves at different pressures are rather similar for both real and imaginary parts. However, as mentioned above the pressure will lead to the downward and upward shifts of the occupied and unoccupied energy bands, which results in the increase of the energy required for the transition from an occupied band to an unoccupied band. Consequently, the peaks in the dielectric function spectra exhibit blue shifts toward higher energy as the pressure is raised. For an example, in the case without pressure the first peak in the imaginary part of ε^xx (Fig. 5(a)) appears around 4.51 eV which shifts to 4.63 eV at P = 2 GPa, and this peak further moves to 4.73 eV at P = 4 GPa. Since the linear optical properties of a material are directly related to the dielectric function, the above blue-shift behaviors also can be observed in the spectra of the refractive indices, as well as in the birefringence curve (Fig. 5(b)) of AgGaS₂ crystal under the pressure. To more clearly show the effect of pressure on the optical performance, the calculated static dielectric constants, refractive indices, and birefringence at different pressures are summarized in Table 2, and the variations of the ε(0) are also presented in Fig. 6. It is clear that two static dielectric constants increase with the increase of the pressure, however the ε^xx(0) changes more obviously than the ε^zz(0), and when the pressure is up to 4 GPa, the increments of ε^xx(0) and ε^zz(0) are about 0.080 and 0.024, respectively. This difference between two static dielectric constants means that the optical anisotropy of AgGaS₂ is enhanced, and as shown in Fig. 6, the magnitude of static birefringence (|Δn(0)|) is increased gradually from 0.044 to 0.056 when the pressure grows up to 4 GPa. Therefore, it seems that the phase matchability of AgGaS₂ crystal can be improved by applying external pressure.

 

Fig. 6 Variations of the static dielectric functions (ε^xx(0), ε^zz(0)) and the magnitude of the static birefringence (|Δn(0)|) of AgGaS₂ crystallized in the chalcopyrite structure and Cc space group at different pressures.

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In the pressure range of 5 – 7 GPa, the chalcopyrite type structure transfers into the monoclinic Cc phase. For the Cc phase, it belongs to biaxial crystal and there are three dielectric tensor components, namely ε^xx, ε^yy and ε^zz, respectively. The dielectric tensors of AgGaS₂ crystals with the Cc structure at different pressures are calculated along the principal optical axes and for examples, the complex dielectric function ε^xx(ω) of P = 5 and 7 GPa is displayed in Fig. 5(c). Like the case of chalcopyrite phase, the peaks in the low-energy region (< 5.0 eV) also show blue shifts as the pressure increases. However, the external pressure has more obvious impact on the linear optical susceptibility of the Cc phase than that of the chalcopyrite phase, which is reflected in the fact that both the peak positions and their intensities in the dielectric function spectra (Fig. 5(c)) are sensitive to the pressure. According to the corresponding birefringence curves shown in Fig. 5(d), the magnitude of birefringence of AgGaS₂ in the IR region is increased greatly when the pressure rises. Some static linear optical properties of the monoclinic Cc phase are listed in Table 3. The changes of the ε^xx(0) and ε^zz(0) are shown in Fig. 6 (the values of ε^yy are close to ε^xx), which demonstrates that the ε^xx(0) and ε^zz(0) are increased gradually with pressure. However, the growing rate of the ε^xx(0) is apparently higher than that of ε^zz(0), indicating that the optical anisotropy of AgGaS₂ becomes more obvious with increasing of the pressure, and as a result, the Δn(0) is simultaneously increased. Moreover, from Fig. 6, it can be concluded that the linear optical responses of Cc structure are more sensitive to the pressure than those of chalcopyrite phase.

Table 3. Calculated static dielectric constants, refractive indices, birefringence, and the magnitudes of the SHG coefficient of AgGaS₂ crystallized in the Cc space group

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Therefore, our results suggest that the external pressure has deep influence upon the linear optical properties of AgGaS₂ crystal. Especially, we can significantly modify the birefringence by the pressure, and the remarkable increment of the magnitude of birefringence (from 0.044 to 0.111) can be achieved under pressure up to 7 GPa, which will help to improve the phase matching performance of AgGaS₂ crystal.

3.4 Second-order nonlinear optical properties of AgGaS₂ crystal at different pressures

As for the SHG response of AgGaS₂ in chalcopyrite phase, there are only two independent components of the SHG tensor, namely, d₁₄ and d₃₆, respectively. In the static (zero-frequency) limit, these two components are equal according to the Kleinman symmetry. For the case of P = 0 GPa, the magnitude of the static d₃₆ coefficient is predicted to be 17.64 pm/V (Table 2), and this value is very close to the experimental result of 18 ± 2.7 pm/V (at 10.63 μm) [41]. Therefore, the method used in the present work is also suitable for predicting the SHG properties of AgGaS₂ crystal. The variation of d₃₆ coefficient under pressure up to 4 GPa for chalcopyrite phase is displayed in Fig. 7(a), and the magnitude of d₃₆ is reduced gradually from 17.64 pm/V to 14.54 pm/V, implying that the SHG intensity becomes weak as the pressure is increased. According to the Eq. (5) and (6), it is apparent that the SHG coefficient is inversely proportion to the E_g, since in general the enlargement of the E_g will lead to the increasing of the energy difference between the bands m and n, namely, the ω_mn term in the denominators. If we assume that the E_g is fixed to the corresponding value of P = 0 GPa (i.e., 2.64 eV), the magnitudes of d₃₆ coefficient are changed to 17.55, 17.41, 17.24, and 17.12 pm/V for P = 1, 2, 3, and 4 GPa, respectively, which shows that now the d₃₆ coefficients at different pressures are similar. Hence, it can be concluded that above reducing of SHG intensity observed in the chalcopyrite phase can be mainly ascribed to the increasing of the E_g.

 

Fig. 7 (a) Variations of the magnitudes of the static SHG coefficients, and (b) the calculated frequency-dependent SHG coefficients of AgGaS₂ crystallized in the chalcopyrite structure and Cc space group at different pressures with an interval of 1 GPa. In figure (b), the results of P = 0, 1, 2, 3, and 4 GPa are corresponding to d₃₆ coefficient of the chalcopyrite phase, while d₃₁ for the Cc phase at P = 5, 6, and 7 GPa.

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For the monoclinic Cc phase, at static limit there are six nonvanishing independent components of the SHG tensors, and among them the d₃₁ ( = d₁₅) and d₃₂ ( = d₂₄) tensors have significant contributions to the SHG response (Table 3). Although the magnitudes of d₃₁ and d₃₂ are similar (13.55 vs. 13.48 pm/V) at P = 5 GPa, they show different variations as the pressure is further increased. From Fig. 7(a), we can see that the d₃₁ tensor decreases first with increasing of the pressure and then quickly converges to a value of about 13.0 pm/V, while the d₃₂ tensor decreases nearly linearly from 13.48 pm/V to 11.59 pm/V. Consequently, there is a somewhat large difference (about 1.5 pm/V) in the magnitudes between d₃₁ and d₃₂ tensors at P = 7 GPa. This tendency is consistent with the fact that the optical anisotropy of AgGaS₂ becomes more pronounced as the pressure is increased. Further analyses indicate that, similar to the d₃₆ of chalcopyrite phase, the variation of d₃₂ coefficient is still mainly dependent on the band gap. On the other hand, the convergent behavior of d₃₁ tensor means that the growth of energy difference between certain bands in the part of Brillouin zone may be not continued when a particular threshold of the external pressure is exceeded.

The frequency-dependent d₃₆ coefficient of chalcopyrite phase and d₃₁ coefficient of Cc phase at different pressures are also calculated by using Eq. (8) and (9), and the results are presented in Fig. 7(b). At the beginning when the energy is less than 1.0 eV, the SHG coefficients increase slowly, and with further increasing of the energy, the SHG response rapidly increases. It is clear that the position of the first peak exhibits a blue shift as the pressure is increased, and meanwhile, the peak intensity is reduced gradually. Although the SHG intensity in the IR region (< 1.6 eV) is suppressed by the external pressure, it is worth emphasizing that AgGaS₂ crystal still shows a strong SHG response because under pressure up to 7 GPa, the maximum magnitude of d₃₁ coefficient can exceed 13 pm/V in the whole IR region.

4. Conclusions

In this work, the geometries, electronic structures, and the linear and second-order nonlinear optical properties of AgGaS₂ crystal under different pressures are investigated systematically. The results obtained by a specialized GA approach show that under the pressure up to 10 GPa, the sequence of phase transitions of AgGaS₂ crystal is chalcopyrite → monoclinic Cc phase (at 4.75 GPa) → Ibam phase (at 7.18 GPa) → α-NaFeO₂-type lattice (at 7.38 GPa), which implies that the SHG response of AgGaS₂ crystal will be quenched when the pressure is raised above about 7.18 GPa. By employed the revised hybrid HSE06 functional with 37.27% HF exchange, the positive pressure coefficients of the band gap are obtained for both chalcopyrite and Cc phases, and especially the value of Cc phase is obviously larger than that of chalcopyrite phase. Under pressure up to 7 GPa, the band gap of AgGaS₂ crystal increases from 2.64 eV to 3.14 eV. This increasement of the band gap suggests that the resistance to laser-induced damage of AgGaS₂ is enhanced and the transparency range is also extended by imposing an external pressure. Meanwhile, for the linear optical susceptibilities, the blue shifts toward higher energy are observed for the peaks in the spectra of the dielectric functions, as well as in the birefringence curve of chalcopyrite and Cc phases. It is worth noting that, due to the increasing of the magnitude of birefringence, the phase matching performance of the AgGaS₂ crystal is improved as the external pressure increases. Therefore, a general improvement in the technical performances related to the linear optical properties can be expected by the pressure-induced deformation of AgGaS₂ crystal. On the other hand, for the second-order nonlinear optical susceptibilities, our calculated results reveal that both static and dynamic SHG coefficients exhibit a reduced trend with pressure, and consequently, it seems that the SHG response decreases in intensity as the pressure increases. However, the AgGaS₂ crystal still shows a strong SHG response, and the maximum magnitude of SHG coefficient can still exceed 13 pm/V in the whole IR region even under pressure up to 7 GPa. The findings of the present work can provide a straightforward way to improve the NLO performances of AgGaS₂, and also throw light on the sensitive relationship between the structure and optical properties of other I-III-VI₂ ternary semiconductors with the chalcopyrite structure.