Spin-polarized Second Harmonic Generation from the Antiferromagnetic CaCoSO Single Crystal

The spin-polarized second harmonic generation (SHG) of the recently synthesized CaCoSO single crystal is performed based on the calculated electronic band structure. The calculation reveals that the spin-up (↑) channel of CaCoSO possesses a direct energy gap (Γv-Γc) of about 2.187 eV, 1.187 eV (Kv-Kc) for the spin-down (↓) channel and an indirect gap (Γv-Kc) of about 0.4 eV for the spin-polarized CaCoSO single crystal. The linear optical properties obtained reveal that the recently synthesized crystal exhibits considerable anisotropy with negative uniaxial anisotropy and birefringence favor to enhance the SHG. We have calculated the three non-zero tensor components of the SHG and found the is the dominat component, one with a large SHG of about (d33 = 6.936 pm/V at λ = 1064 nm), the half value of KTiOPO4 (KTP). As the values of (↑) < (↓) < spin-polarized are related to the values of the energy gap of (↑) 2.187 eV> (↓) 1.187 eV> spin-polarized gap 0.4 eV; therefore, a smaller energy gap gives better SHG performance. Furthermore, the microscopic first hyperpolarizability, βijk, is calculated.

A great effort has been made to synthesize novel nonlinear optical (NLO) crystals which are able to produce visible, UV and IR laser radiation at wavelengths that are presently inaccessible via conventional sources which have been in demand for several applications [1][2][3][4] . The NLO materials form a large group of semiconducting and dielectric materials with diverse optical, electrical, and structural properties. Some of these NLO materials appear to be promising candidates for optical frequency conversion applications in solid-state-based tunable laser systems [5][6][7] . These have potentially significant advantages over dye lasers because of their easier operation and the potential for more compact devices. Tunable frequency conversion in the mid-IR is based on optical parametric oscillators (OPO) using pump lasers in the near-IR. On the other hand, frequency doubling devices allow one to expand the range of powerful lasers in the far-IR. The transformation of frequencies within the wide spectral range of electromagnetic radiation (between 1-15 μ m) for ultrafast lasers (from picoseconds up to femtoseconds) and large power densities can lead to significant progress in laser engineering. The method used for such frequency transformations is usually the generation of harmonic frequencies by employing high quality NLO crystals possessing large second optical susceptibilities 8,9 . In particular, the main method employed for frequency transformation is the second harmonic generation (SHG), where organic and inorganic crystals are now among the best 10,11 . Werake and Zhao 12 have reported the observation of the SHG in GaAs induced by pure spin current. Takas and Aitken 13 have presented the thermal analysis and SHG efficiency of α -Na 3 PO 3 S oxythiophosphate compound. Upon exposure to 1064 nm incident radiation, noncentrosymmetric α -Na3PO3S produces an SHG efficiency 200 times that of α -quartz and is nonphase-matchable (type 1). Wu et al. 14 have synthesized acentric BaHgS 2 by a conventional solid-state reaction method. It has been found that the powder of BaHgS 2 exhibits a large SHG response of about 6.5 times compared with that of reference AgGaS 2 at a fundamental wavelength (2.09 μ m). Liu et al. 15 have synthesized a new oxychalcogenide, BaGeOSe 2 which contains acentric mixed-anion basic building units of GeO 2 Se 2 tetrahedrons and BaOSe 6 polyhedrons. These GeO 2 Se 2 tetrahedrons and BaOSe 6 polyhedrons are responsible for the large NLO response. Li et al. 16 reported that the combination of the highly electropositive alkali metal (Na) and Zn with a d 10 electronic configuration into a crystal structure affords one new IR NLO material, Na 2 ZnGe 2 S 6 , which exhibits excellent properties including a wide transparent region, a large band gap, and especially a balance between the strong NLO coefficient (30 × KDP) and high LDT (6 × AgGaS2), which indicate a promising application in the IR region. A few years ago it was reported that AgGaX 2 (X = S, Se) [17][18][19] and ZnGeP 2 20,21 exhibit good SHG coefficients for application in the IR region. In the UV and visible regions, great achievements have been made by providing many excellent NLO materials like β-BaB2O4 (BBO), LiB3O5 (LBO) and KBe2BO3F2 (KBBF) [22][23][24][25][26] . Kang et al. 27 have designed two novel NLO carbonates KBeCO 3 F and RbAlCO 3 F 2 using the first-principles theory. The investigated crystals are structurally stable and possesses very large energy band gaps, and considerable optical anisotropy. They reported that KBeCO 3 F and RbAlCO 3 F 2 are very promising deep-UV NLO crystals alternative to KBBF. Liang et al. 28 have investigated the usage of metal sulfides with diamond-like (DL) for NLO applications in the mid-IR spectral region. The linear and NLO properties of the DL-metal sulfides are analyzed on the basis of first-principles calculations. It has been found that it is relatively easy to achieve good balance between the band gap and the NLO performance. Moreover, moderate birefringence Δ n (~0.03-0.10) is crucial for practical mid-IR NLO applications. They reported that several metal sulfides with normal DL and defect DL structures, show excellent mid-IR NLO properties. These provide an useful information for the design and discovery of novel materials possesses good mid-IR NLO performance. Lin et al. 29 have reported that the ab initio approaches have the ability to accurately predict the optical properties in NLO crystals, and the developed analysis tools are very important to explore their intrinsic mechanism. This microscopic understanding is crucial for designing novel crystals with large NLO properties. It is expected that the first-principle approaches will deeply enhance the search efficiency and help the researchers to save resources in the exploration of novel NLO crystals with good performance.
Recently, the oxide chalcogenide compounds have become considerable candidates for several applications, for instance, as p-type transparent conductors [30][31][32][33][34][35] . Among these oxide chalcogenide compounds, the CaCoSO compound has been recently synthesized by Salter et al. 36 . It has been reported that the CaCoSO is isostructural with CaZnSO, BaCoSO and CaFeSO [36][37][38][39][40] . Since the newly synthesized CaCoSO crystallizes in non-centro-symmetry, this results in the loss of inversion symmetry, which in turn gives a considerable SHG. As the crystal structure of the CaCoSO single crystal has just been reported 36 and there is no further information regarding its linear and nonlinear optical properties, we think it would be worthwhile to perform a comprehensive theoretical calculation to calculate the spin-polarized linear, and nonlinear optical properties and the microscopic first hyperpolarizability. Moreover, it is important to highlight that there is no previous report on the spin-polarized nonlinear optical properties and the microscopic first hyperpolarizability for the CaCoSO. In the current work, we aim to find new materials without inversion symmetry and possessing considerable SHG, which have been in demand for many industrial, medical, biological and entertainment applications.

Density Functional Calculations
Very recently, Salter et al. 36 have synthesized CaCoSO in the form of a single crystal with a P6 3 mc space group, cell parameters a = b = 3.7415(8) Å, c = 11.106(2) Å, V = 134.64(6) Å 3 and Z = 2, at room temperature 36 . Utilizing the reported x-ray diffraction data of CaCoSO 36 , we performed comprehensive ab-initio calculations based on the full-potential method within the generalized gradient approximation (PBE-GGA) plus Hubbard Hamiltonian (U). In a step forward to gain accurate results, the experimental geometrical parameters 36 are optimized using PBE-GGA 41 . The optimized crystal structure of CaCoSO single crystal is shown in Fig. 1(a-c). The resulting geometrical parameters are used to perform the calculations employing the ab-initio LAPW+ lo full-potential method utilizing the Wien2k code [42][43][44] . It is well known that for highly localized electrons, the Coulomb repulsion between the electrons in open shells should be taken into account; therefore, the Hubbard-like on-site repulsion should be added to the Kohn-Sham Hamiltonian. In the present work, we used the method of Anisimov et al. 45 and Liechtenstein et al. 46 , where the Coulomb (U) and exchange (J) parameters are used. We applied the U on the 3d orbitals of the Co atom. Several U values are used to obtain a qualitative agreement with experimental data; we found 0.20 Ry is the optimal U value for Co-3d. In the current calculation, the self-consistency is obtained using 900 k  points in the irreducible Brillouin zone (IBZ). The self-consistent calculations are converged since the total energy of the system is stable within 0.00001 Ry. The spin-polarized linear and nonlinear optical properties and the microscopic first hyperpolarizability are calculated using 25000 k  points in the IBZ, as accurate calculations require a dense sampling of the BZ.

Results and Discussion
The spin-polarized linear optical properties of the antiferromagnetic CaCoSO are calculated based on the spin-polarized electronic band structure. The complex dielectric function consists of real and imaginary parts 47,48 . For the hexagonal symmetry, the imaginary part consists of two tensor components: these are ε ω ⊥ ( ) 2 and ε ω ( ) 2 . We have performed calculations for ε ω ⊥ ( ) 2 and ε ω ( ) 2 using the expression given elsewhere 47,48 . The calculated imaginary part for the spin-up (↑ ) and spin-down (↓ ) channels are shown in Fig. 2(a). Due to the fact that the spin-up channel has a different structure than that of the spin-down channel (see it is located at around 1.187 eV, which confirms that the spin-up channel possesses an energy band gap larger that that of the spin-down channel (see Fig. S1 Supplementary Materials). We should emphasize that the broadening is taken to be 0.1 eV, which is traditional for oxide crystals and is typical of the experimental accuracy. In the spin-up channel the fundamental absorption edge occurs due to the optical transition from Co-3p/3d, S-3p, and O-2p bands of the VBs to Ca-s/p, Co-4s/3p and O-2s bands of the CBs, while in the spin-down channel it occurs between Co-3p/3d, O-2p, S-3p VBs and Co-3p/3d, Ca-3p, O-2p, S-3p CBs. The main structure which is confined between 5.0 eV and 10.0 eV occurs due to the optical transitions between Ca-4s/3p, Co-4s/3p/3d, O-2s/2p, S-3s/3p VBs and Ca-4s/3p, Co-4s/3p, O-2s CBs in the spin-up channel. Whereas in the spin-down channel it is formed due to the transitions from Ca-4s/3p, Co-4s/3p, O-2s/2p, S-3s/3p VBs to Ca-4s/3p, Co-4s/3p/3d, O-2s CBs.
From the calculated , as shown in Fig. 2(c,d). The optical conductivity is directly related to the energy band structure of solids 50 ; therefore, deep insight into the electronic structure of the materials can be further obtained from the optical conductivity. Furthermore, from the imaginary part of the optical conductivity, the values can be obtained (Table 1). Figure 2(e) illustrates the absorption coefficients of CaCoSO for the spin-up and spin-down channels. The fundamental absorption edge of the spin-up channel occurs at 2.187 eV, whereas it is 1.187 eV for the spin-down channel, which reveals that the absorption edge of the spin-down is smaller than that of the spin-up, confirming our previous observation at Fig. 2(a).
The reflectivity spectra (Fig. 2f) reveal that the CaCoSO possesses low reflectivity at the low energy range and the highest reflectivity at around 13.0 eV, which represents the lossless region. To support this statement the loss function is calculated and presented in Fig. 2(g). The refractive indices are very important quantities to determine the phase matching conditions which are necessary for the second harmonic generation. The spin-up/ down refractive indices of the CaCoSO single crystal are calculated and presented in Fig. 2(h). From the parallel and Δn(0). and perpendicular tensor components of the refractive indices, the birefringence can be obtained. The calculated spin-up/down and spin-polarized birefringence are presented in Fig. 2(i). The calculated values of the birefringence at the static limit and at the wavelength 1064 nm are listed at Table 1. It is clear that the CaCoSO single crystal exhibits relatively large birefringence. The electron clouds of the CaO 3 S 4 and CoS 3 O (Fig. 1b,c) exhibit a planar shape with conjugated electron orbitals, which make the CoS 3 O groups the main source of the large birefringence in CaCoSO. It is well known that the birefringence determines partly whether an NLO material has the value of study 51 . It is clear that there is a considerable anisotropy between the two tensor components of the optical properties. The high electron density configuration and strong anisotropy for Co-S and Co-O indicate the main contribution of CoS 3 O to the optical anisotropy. The optical anisotropy is a favor to enhance the SHG. Furthermore, we have calculated the spin-up (↑ ) and spin-down (↓ ) and the spin-polarized second harmonic generation for the CaCoSO single crystal. The formalisms which are used to calculate the complex nonlinear optical are presented elsewhere [52][53][54] . The investigated single crystal possesses three non-zero tensor components. These are 113, 311 and 333; only one of these tensor components is the most intensive one and acts as the dominant one. From the calculated (↑ )(↓ ) (Fig. 3a), we can notice that the (↑ )(↓ ) χ ω ( ) 333 (2) is the most intensive among the others; therefore, χ ω ( ) 333 (2) is the dominant due to the crystal's symmetry. Figure 3(b) illustrates the (light long dashed curve-red color); -lower panel-Calculated (↑ )ε 2 (ω) (dark solid curve-black color); Calculated (↑ ) ε 2 (ω/2) (dark dashed curve-red color); Calculated (↓ )ε 2 (ω/2) (light dotted dashed curve -green color); Calculated (↓ )ε 2 (ω/2) (dark solid curve-blue color). dominant tensor component for the spin-up, spin-down and the spin-polarized χ ω ( ) 333 (2) for the CaCoSO single crystal. The SHG values of the three tensor components are listed in Table 2, which shows that the investigated crystal possesses a large SHG of about 1.628 pm/V at the static limit and 13.872 pm/V (d 33 = 6.936 pm/V) at λ = 1064 nm. The obtained value of the SHG at λ = 1064 nm is about the half value of the well-known NLO crystal KTiOPO 4 (KTP) 55 . Following Table 2 (2) is related to the value of the energy gap (↑ ) 2.187 eV> (↓ ) 1.187 eV> spin-polarized gap 0.4 eV; therefore, we can see that our finding is supported by the statement: a smaller energy gap gives better SHG performance.
The dominant tensor component χ ω ( ) 333 (2) consists of imaginary and real parts as shown in Fig. 3(c) for the spin-up, spin-down channels. The imaginary and real parts further consist of 2ω/ω inter-/intra-band contributions as shown in Fig. 3(d,e). In order to understand the origin of the spectral features of the dominant tensor component χ ω ( ) 333 (2) , the imaginary part of the dielectric function as a function of ω/2 and ω is associated with χ ω ( ) 333 (2) , as shown in Fig. 3(f). This clearly shows that the structure below the half energy gap (1.093 eV(↑ ), 0.593 eV(↓ )) is associated with 2ω reson a n c e , w h e r e a s t h e s t r u c t u r e a b o v e t h e f u n d a m e n t a l e n e r g y g a p ( 2 . 1 8 7 e V (↑ ), 1.187 eV(↓ )) is associated with interference between 2ω and ω resonances and the tail is mainly due to ω resonance. The values of the microscopic first hyperpolarizability, β ijk 56,57 , vector component along the dipole moment direction for the dominate tensor component are calculated at the static limit and at λ = 1064 nm, as shown in Table 2.

Conclusions
Based on the experimental crystallographic data of the recently synthesized CaCoSO single crystal, the spin-polarized SHG is calculated and the origin of the large SHG is discussed in detail. It has been found that the spin-up channel of the CaCoSO single crystal possesses direct energy (Γ v -Γ c ) of about 2.187 eV, 1.187 eV (K v -K c ) for the spin-down channel and an indirect gap (Γ v -K c ) of about 0.4 eV for the spin-polarized CaCoSO single crystal. The linear optical properties reveal that the recently synthesized crystal exhibits considerable anisotropy with negative uniaxial anisotropy and birefringence favor to enhance the SHG. It is clear that CaCoSO single crystal exhibits relatively large birefringence. The electron clouds of the CaO 3 S 4 and CoS 3 O groups exhibit a planar shape with conjugated electron orbitals, which make the CaO 3 S 4 and CoS 3 O groups the main source of the large birefringence in CaCoSO. It is well known that the birefringence determines partly whether an NLO material has the value of study. The high electron density configuration and strong anisotropy for Co-S and Co-O indicate the main contribution of CaO 3 S 4 and CoS 3 O to the optical anisotropy. The optical anisotropy is a favor to enhance the SHG. We have calculated the three non-zero tensor components of the SHG and found the χ ω ( ) 333 (2) is the dominant one, with a large SHG of about (d 33 = 6.936 pm/V at λ = 1064 nm), the half value of the well-known NLO crystal KTiOPO 4 (KTP). The (↑ ) χ ω ( ) 333 (2) < (↓ ) χ ω ( ) 333 (2) < spin-polarized χ ω ( ) 333 (2) is related to the value of the energy gap of the (↑ ) 2.187 eV> (↓ ) 1.187 eV> spin-polarized gap 0.4 eV; therefore, we can see that our finding is supported by the statement: a smaller energy gap gives better SHG performance. Furthermore, the values of the microscopic first hyperpolarizability, β ijk , for the dominant tensor component is calculated at the static limit and at λ = 1064 nm.   and β ijk of CaCoSO, in pm/V at static limit and at λ = 1064 nm. Where 1 pm/V = 2.387 × 10 −9 esu.