Nonlinear Optical Properties of Mixed Oxides Crystals CsNbMoO6 and

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C: Plasmonics; Optical, Magnetic, and Hybrid Materials 6

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Nonlinear Optical Properties of Mixed Oxides Crystals CsNbMoO and CsTaMoO. A Periodic CPHF/KS Study Liubov A. Varlamova, Stanislav Konstantinovich Ignatov, Diana G. Fukina, Anton A. Konakov, Artem E. Masunov, and Evgeny V. Suleimanov J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b07117 • Publication Date (Web): 29 Sep 2018 Downloaded from http://pubs.acs.org on October 4, 2018

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KH2PO4 LiNbO3 CH4N2O SiO2

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KH2PO4 LiNbO3 CH4N2O SiO2

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dmax deff η ρ

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U KDP/LiNbO3

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Nonlinear Optical Properties Of Mixed Oxides Crystals CsNbMoO6 and CsTaMoO6. A Periodic CPHF/KS Study L. A. Varlamova1 *, S. K. Ignatov1, D. G. Fukina1, A. A. Konakov1, A. E. Masunov1,2,3,4, E. V. Suleymanov1 1

2

3

Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia

NanoScienece Technology Center, University of Central Florida, Orlando, FL 32826, USA

Photochemistry Center RAS, Federal research center Crystallography and Photonics Russian Academy of Science, ul. Novatorov 7a, Moscow 119421, Russia 4

South Ural State University, Lenin pr. 76, Chelyabinsk 454080, Russia

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National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoye shosse 31, Moscow, 115409, Russia * e-mail: [email protected]

Abstract. Inorganic crystals are advantageous for use as second harmonic generation (SHG) materials because of their large laser damage threshold and mechanical strength. Here we focus on the defect pyrochlores based on mixed oxides of MoO3, Ta2O5, Cs2O, Nb2O5 which were synthesized recently. We predict the linear optical properties (refractive indices and linear polarizabilities) and nonlinear ones (first hyperpolarizabilities and nonlinear optical tensors) for the crystals of the defect pyrochlores CsNbMoO6 and CsTaMoO6 using the Coupled-Perturbed Hartree-Fock/Cohn-Sham field method (CPHF/KS) in both full-electron and pseudopotential basis sets. To validate the more accurate computational protocol, the crystals KH2PO4, LiNbO3, (NH2)2CO and α-SiO2 were used as benchmarks. The best agreement of the predicted nonlinear optical parameters with experimental ones is achieved in calculations at the DFT/PBE0 level in the POB-TZVP basis. This comparison becomes quantitative when the dimensionless relative value of the nonlinear optical activity ρ is used. The calculated relative SHG characteristics of the pyrochlore crystals are ρ(CsNbMoO6/LiNbO3) = 0.016 and ρ(CsTaMoO6/LiNbO3) = 0.022, ρ(CsNbMoO6/KDP) = 0.39 and ρ(CsTaMoO6/KDP) = 0.52. The symmetry of the sublattices of disordered atoms significantly affects the SHG parameters of the crystals and the values of the calculated parameters for sublattices of different symmetry can differ by an order of magnitude. Keywords: Non-linear optical properties, second harmonic generation, defect pyrochlores crystals, CPHF/KS calculations ACS Paragon Plus Environment

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1. Introduction Materials with non-linear optical (NLO) properties enable many modern technologies, including those for creating tunable lasers, optoelectronic devices, optical communication, information processing and storage systems1. Some of these technologies require materials capable of second harmonic generation (SHG), combined with large laser damage threshold and mechanical strength. Inorganic NLO materials, which satisfy these and other requirements have been used for years for these purposes. Organic NLO materials, on the other hand, may be advantageous in that their properties can be easily tuned by molecular modifications. To a certain extent, properties for some of the inorganic materials can be tuned by variable composition. Pyrochlores represent a family of phases, which allow for this tunability. When certain structural features are present, such defect pyrochlores may display NLO properties. Pyrochlores are a broad class of mixed metal oxides, isostructural to the mineral pyrochlore (Ca,Na)2Nb2O6F. They form a large family of compounds which demonstrate many useful physical properties such as ionic or electronic conductivity, magnetic, catalytic2-6, ferroelectric, and nonlinear optical properties.7-9 Ideal alpha-pyrochlore structure represents cubic phase with centrosymmetric space group Fd-3m and stoichiometry A2B2X6X', where A is a large low valence cation, B is a small more highly charged cation capable of octahedral coordination, X is oxygen anions, and X' is hydroxide, fluoride anions, as well as water molecules, weakly interacting with B10. In many cases alpha-pyrochlore structures lose weakly bonded X' ion and create a series A2-xB2X6X'1-y of defect pyrochlore compounds with ionvacancies11. The end member of this series is the beta-pyrochlore with general formula AB2X6. Losing all X' ions and half A-ions simultaneously leads to conservation of the cubic centrosymmetric structure with space group Fd-3m (ideal beta-pyrochlore structure12). When B position is occupied by the elements in v and vi oxidation state simultaneously (e.g. Nb5+, Ta5+, ACS Paragon Plus Environment


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Sb5+, W6+, Mo6+, Te6+), the general symmetry of structure may decrease because of intraoctahedral distortions of B metal coordination sphere. This leads to form defect pyrochlores family which retain the cubic structure, reduces the Fd-3m space group symmetry to F23, F4132, Fd3, F-43m sub-groups13 without inversion center8,9,14-16. This work is focused on defect pyrochlores based on mixed oxides of Cs2O, MoO3, Ta2O5, and Nb2O5. These compounds were synthesized earlier17 and were shown that for some compositions and crystalline modifications they possess nonlinear optical activity.18 This class of materials allows for a wide variation in the composition of the crystals, which makes the synthesis and optimization of whose NLO properties labor intensive. In addition, for each given composition, several crystallographic positions can be fractionally occupied by metal ions. This leads to the emergence of many structural polytypes, some of which may have high NLO characteristics, while others do not have NLO properties. From this point of view, it is of interest to carry out a rapid virtual screening of the properties of possible crystal structures and composition variations, choosing the most promising candidates for subsequent synthesis and experimental study. Here we aim to establish computational protocol for rapid screening of nonlinear optical properties over a wide range of compositions, polytypic modifications and ionic ordering. To predict optical properties of interest, the method should allow for rapid evaluation of the refractive index, polarizability, first and second hyperpolarizability. This task can be accomplished by quantum chemical calculations within the framework of various approximations, among which the Hartree-Fock (HF) method or the density functional theory (DFT) is often used, followed by the application of an electric field to estimate linear and hyperpolarizability by numerical differentiation.19,20 Alternative application of analytical differentiation by means of the coupled-perturbed Hartree-Fock CPHF method21-23 (or KohnSham CPKS in the case of DFT) has for a long time been limited to isolated molecules, but recently it became available for crystals.

21-23

Unfortunately, for the compounds of interest,


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additional difficulties arise due to the lack of well-tested basic sets and pseudopotentials for heavy atoms, as well as DFT functionals. Previously, a limited number of papers appeared in the literature, where the validity and accuracy of the NLO properties prediction was studied. For fluorooxoborates, NLO predictions were systematically considered, depending on the chosen functional, pseudopotential, and type of basis functions.24 In another study, full-electron calculations of 3D periodic systems using Gaussian type basis set and B3LYP functional were performed to predict elastoelectric and piezoelectric tensors using the CPKS method as implemented in CRYSTAL program.25 In addition, calculations of NLO properties based on optical spectra calculated in the LDA/PAW approximation in the VASP code were carried out.26 Calculation of polarization effects based on the LDA and the norm-conserving Trowler-Martin pseudopotentials in the ABINIT27 code. The authors of Ref.11 concluded that both methods considered give similar results and comparison with the experimental data demonstrates satisfactory agreement. For one of the compounds, the estimates of the maximum components of hypersusceptibility are consistent with experiment and with each other. For another compound, the predicted second harmonic generation (SHG) activity was not observed, in the authors' opinion, due to the experimental difficulties. Unfortunately, in this paper, the effect of heavy metal atoms could not be analyzed and the issue of various crystalline modifications within the same composition and its effect on the calculated properties was not considered. Here we study the NLO properties of new representatives of the defect pyrochlore class CsNbMoO6 and CsTaMoO6. For the benchmarking purposes, we also consider typical organic and inorganic NLO materials of several classes. On this basis, we evaluate the ability of our methods to predict linear and nonlinear optical properties and select the most accurate one, which will be applied to various crystalline modifications of new prospective NLO materials such as CsNbMoO6 and CsTaMoO6. Since the structure of these crystals allows various



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sublattices of transition metals, we further investigate the effects of particular ion ordering on calculated NLO properties. A separate issue considered in this paper is the analysis of the choice of selection criteria for candidate structures exhibiting SHG activity that would allow comparison of crystals of different types of symmetry, which cannot be done by direct comparison of the maximum components of the hyperpolarizability tensor due to the different symmetry of this tensor for crystals of different syngonies.

2. Computational Details Calculations of nonlinear optical properties of crystals were carried out using CoupledPerturbed Hartree-Fock (or Kohn-Sham in case of density functional theory) CPHF/K21-23,28 method, implemented in CRYSTAL14 program.29 This implementation is based on formulas:

α tu = −

β tuv

4 nk

2 =− nk

 occ virt k * k ,t k k ,u  ℜ ∑k ∑ ∑a ∑p Caµ Ωµν Cν pU pa  µν   BZ

k ,v  k ,i*     ∂U pa k ,v k ,u k * k , v k  k* k ,u k U C F C U U P ℜ × − C F C s +      ∑k ∑a ∑p t ,u ,v pa ∑ ∑ pµ µν ν q qa ∑b pb bµ µν ν a ∂ku    µν  q  

1.a

1.b

Here nk is the number of k-points in the first Brillouin zone; summation indices a (b, c ...) and p (q, r ...) run over occupied and virtual crystalline orbitals respectively; Ukv is the unknown antiHermitian matrix of the nondiagonal block; Сk are unperturbed eigenvectors; Ωk is the matrix representing the perturbed electric field; P is an operator permuting the differentiation indices; δUk,vpa/δku is determined by the derivative of Сkµi with respect to k. Coordinates of all atoms in a unit cell were optimized. Calculations of energy and optical characteristics were performed with two theoretical methods: Hartree-Fock and density functional theory (DFT) with PBE030 functional. It is known from the literature that the HartreeFock method usually underestimates hyperpolarizability, while DFT overestimates it.31 Two ACS Paragon Plus Environment

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types of basic sets were used: the full-electronic minimal basis STO-3G32-34 (basis A) and the basic POB-TZVP33,35 with polarization functions, supplemented by the effective core potentials for the Cs, Nb, Mo, Ta atoms,36 and full-electron for the remaining atoms (basis B). The absence of polarization functions in the basis A excludes the effects of atomic polarizability from the model but does not greatly impair the quality of NLO predictions of properties in the case of molecular materials.37 It would be interesting to see how this affects the properties of ionic crystals. The combination of two methods and two basis sets produces four levels of theory, considered in this paper. In the following text we refer to them as the levels of theory 1 (HF/A), 2 (HF/B), 3 (PBE0/A) and 4 (PBE0/B). The convergence of the SCF procedure at these levels of the theory with the use of default parameters was not achieved or was very slow. To improve the convergence, non-standard values of the Anderson mixing parameter (keywords ANDERSON FMIXING 30 and ANDERSON 60) were chosen. The points of the Brillouin zone for the calculation of crystalline orbitals were specified on the basis of shrink factors 6 and 8 with increased accuracy of integrals calculation (keyword TOLINT 8 8 8 8 16). The well-studied birefringent crystals of potassium dihydrogen phosphate (KDP) KH2PO4, lithium niobate LiNbO3, urea (NH2)2CO and α-quartz α-SiO2 were chosen as the benchmark set. The structural parameters of all the studied systems are presented in Table 1.

Table 1. Experimental structural characteristics38-41 of test and investigated crystals17,18 Crystal Test crystals KH2PO4 LiNbO3 (NH2)2CO SiO2

Space groupa

a, Å

b, Å

c, Å

α, º

β, º

γ, º

I42d (122) R3c (161) P421m (116) P312 (154)

7.5256

7.5256

7.0593

90

90

90

5.1483

5.1483

13.8631

90

90

120

5.565

5.565

4.684

90

90

90

4.9134

4.9134

5.4051

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CsNbMoO6

F43m 10.4000 10.4000 10.4000 (216) CsTaMoO6 F43m 10.4104 10.4104 10.4104 (216) a – symmetry group, number on international crystallographic tables42

90

90

90

90

90

90

Based on the calculated components of the nonlinearly optical tensor, the effective magnitude of the hyperpolarizability tensor (deff) was calculated as the convolution of the tensor over three orthogonal vectors corresponding to the standard orientation of the crystal:43

d eff = pexdpez pey = peydp ezp ex = p ez dp exp ey

2

Here pex,y,z are unit vectors oriented along the crystal axes; d is the hyperpolarizability tensor; deff is the scalar product of the first vector by the tensor-vector product. In this formula, the product is the convolution of the tensor with respect to three vectors. The value of deff is of particular interest, since it is used in practical calculations of NLO materials. The values of deff were calculated according to the formulae43 for different space groups. Since the values of deff depend on the direction of the orientation of the incident beam relative to the crystal axes, in this paper we averaged these values for all angles of orientation of the incident beam relative to the crystal axes in the zero-point orientation. Since for low space group symmetry the formulae28 are not applicable, in order to calculate the deff of these crystals, we used the convolution of the tensors obtained by two vectors in a spherical coordinate system in accordance with the formulae:

a i = ∑ ∑ dijk n j nk

3.a

n j = nk = {sin θ cos ϕ ,sin θ sin ϕ ,cosθ }

3.b

j

d

2 eff

k

1 = 4π

π 2π

3

∫ ∫ ∑ a Jdθ dϕ 2 i

o o i =1


3.c

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The SHG efficiency of a crystal converter can be characterized by the dimensionless coefficient η proportional to the ratio of the powers of the single wave and the double frequency wave generated by this crystal under the fixed experimental conditions:44 2 2 2 P2ω 2π deff L Pω  Vk ⋅ L  = sinc 2   2 2 Pω ε 0cn1 n2λ2 A  2 

4.a

deff2 P2ω :η = 3 Pω n

4.b

Here Pω,2ω are the powers of the rays at single and double frequency; L is the thickness of the crystal; ξ0 is the dielectric constant; c is speed of light; ni are the refractive indices at wavelengths λi taking into account the polarizations in the synchronism direction; λi is wavelength; A is the cross-sectional area of the laser beam with frequency ω0; ∆k is the wave detuning vector; sinc is a parametric sine function. In this paper we consider both existing types of SHG, i.e. besides changing the convolution type,43 the reducible refractive indices no·no·ne for the first type and no·ne·ne for the second type SHG generation. Note that since deff is averaged over different angles of the incident beam directions, the value of η is also an averaged characteristic and static coefficients of refraction are used in calculations.

3. Results and Discussion 3.1 Benchmark calculations. The calculated values of the refractive indices for ordinary and extraordinary beams, the nonzero components of the hyperpolarizability tensor, the effective values of the hyperpolarizability tensor, and the coefficient η for four benchmark set crystals in comparison with the experimental data45 are presented in Table 2. Table 2. Optical parameters of test crystals calculated at different theory levels in comparison with the experimental data ACS Paragon Plus Environment


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Test crystals

Level of theory

Experimental data

1 2 3 4 Refractive indices for ordinary ray: no KH2PO4 1.34960 –a 1.44872 1.48892 1.49535 LiNbO3 1.89970 1.84106 2.18512 2.13093 2.2351 (NH2)2CO 1.15442 1.37253 1.18659 1.42610 1.47197 SiO2 1.29586 1.40450 1.40028 1.48564 1.53483 Refractive indices for extraordinary ray: ne KH2PO4 1.29776 –a 1.41263 1.43451 1.46041 LiNbO3 1.71543 1.72974 1.98919 1.98493 2.1581 (NH2)2CO 1.33267 1.47374 1.37843 1.55324 1.5856b SiO2 1.30879 1.41201 1.41708 1.49594 1.54360 Nonzero components of the hyperpolarizability tensor: d KH2PO4 d36 = 0.389 – a d36 = 0.963 d36 = 0.407 d36 (1014)=0.389 LiNbO3 d22= 0.725 d22=1.264 d22= 0.858 d22=0.516 d22 (1058)=2.46 d31 = –2.146 d31 = –0.745 d31 = –6.330 d31 = –5.421 d31 (1058)= –4.64 d33 = –7.887 d33 = –5.967 d33 =–20.131 d33=–17.778 d33 (1058)= –41.70 (NH2)2CO d36 = 0.378 d36 = 0.649 d36 = 0.104 d36 = 0.757 d36 (600) = 1.17 SiO2 d11=0.3211 d11=0.175 d11=0.728 d11=0.379 d11 (1064) = 0.30 Effective values of the hyperpolarizability tensor for I and II type SGH and their mean value: deff(I), deff(II), deff(ave) KH2PO4 0.158 –a 0.390 0.165 0.158 0.105 –a 0.260 0.110 0.105 a 0.129 – 0.319 0.135 0.129 LiNbO3 1.073 0.372 3.165 2,710 2,320 0.098 0.171 0,116 0,070 0,332 0.324 0.252 0,606 0,435 0,878 (NH2)2CO 0.120 0.207 0.033 0.241 0.241 0.102 0.175 0.028 0.205 0.205 0.111 0.190 0.030 0.222 0.222 SiO2 0.065 0.035 0.147 0.077 0.061 0.043 0.024 0.098 0.051 0.041 0.053 0.029 0.120 0.063 0.050 coefficient of efficiency for I and II type SGH and their mean value: η eff(I), η eff(II), η eff(ave) KH2PO4 0.016 –a 0.078 0.013 0.012 0.010 0.048 0.008 0.007 0.013 0.061 0.010 0.009 LiNbO3 0.325 0.069 1.799 1.316 0.060 0.006 0.018 0.005 0.002 0.037 0.044 0.036 0.099 0.051 0.047 (NH2)2CO 0.017 0.032 0.001 0.038 0.038 0.019 0.024 0.001 0.028 0.028 0.014 0.028 0.001 0.032 0.033 SiO2 0.005 0.004 0.020 0.005 0.007 0.003 0.001 0.001 0.003 0.002 0.004 0.002 0.016 0.004 0.002 a - it was not possible to achieve the SCF convergence at given level of theory


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As follows from the data in Table 2, the calculated values of refractive indices are lower than the experimental values for all calculation methods and all compounds. Figure 1 shows the deviations from the experimental values of the refractive index calculated by the CPHF/KS method.

KH2PO4 CH4N2O SiO2

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LiNbO3

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Figure 1. Error in estimating the refractive index for test crystals at four levels of the theory,%.

As one can see from the Figure 1, the best agreement with experiment is achieved in the case of quantum chemical calculation at level 4, the relative deviation from the experimental values, averaged for the four test compounds is 3.3%. The worst results are obtained at level 1, with an average relative error of about 15.6%. The data given do not allow to conclude whether an improvement in the basis or a transition from HF to DFT leads to unambiguous improvement of the results for all compounds. However, in general, the calculation at level 4 is much better than in all other cases. At this level, the worst results are obtained for LiNbO3, which can be explained by the presence of a heavy transition element (Nb) in the system. Note that for the urea crystal, the values of the estimation errors depend significantly on the completeness of the basic set: in the case the STO-3G basis, the error is much larger than in the case of POB-TZVP. As will be shown below, this fact is also manifested in subsequent calculations, and is especially strong in the case of calculating DFT.



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∆dmax,%

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KH2PO4 LiNbO3

120

CH4N2O SiO2

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Method Figure 2. Errors for the maximal components of the hyperpolarizability tensor. The relative unsigned errors of the maximal components of the hyperpolarizability tensor are shown in Figure 2. As one can see from these data, the error in calculating the components of the hyperpolarizability tensor is minimal in the case of the theory level 4 in this case it amounts to 55% for compounds of various types, with the maximum values observed for lithium niobate. Incidentally, this compound has the largest maximum component among test compounds, and also contains heavy atoms. In the case of level 3 (a combination of a narrow basis and a higher method), the results deviate most strongly from the experimental values. The deviations of the calculated results from the experimental values for the maximum components of the hyperpolarizability tensor dij, the effective values of the tensor deff, and the power factor of the SHG η, averaged over the four test compounds are presented in Figure 3. In addition, the figure shows the ratios of the power factors ρ = ηi /ηj averaged over all pairs of test compounds i and j, which will be discussed below. As follows from these data, the errors in estimating the values of deff and η slightly differ from the errors dij and, therefore, are not a reliable parameter for comparing the SHG activity of crystals. In contrast, the value of ρ has a


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noticeably smaller deviation, which makes it a preferable parameter for comparing the activity of different materials.

dmax deff 90

η ρ

60

∆, %

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Method Figure 3. Averaged over the test compounds relative deviations from experimental values for the calculated maximum components of the hyperpolarizability tensor dij, the effective value of the tensor deff, the power factor of the SHG η, and the ratio ρ= ηi/ηj. One can see from the Figure 3, that the absolute calculated values of η do not agree with experimental values well. Thus, this parameter cannot serve as a quantitative characteristic in the theoretical predictions of NLO activity. However, if we consider the values of η relative to some reference compound, for example KDP or LiNbO3, the correlation is significantly improved. Relative values of the predicted NLO activity were successfully used earlier for comparison with measurements and in molecular systems.46 The correlation between the theoretical and calculated values of ρKDP= ηi/ηKDP and ρLiNbO3 = ηi/ηLiNbO3 (i is the index of test compound) relative to two standards: KDP and LiNbO3 are shown in Figure 4.



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Metod 1 Metod 2 Metod 3 Metod 4

0,9

ηcalc

4

ηcalc

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0,6

2 0,3

0 0

2

4

6

0,0 0,0

0,3

0,6

0,9

1,2

ηexp

ηexp

Figure 4. Correlation between the experimental and theoretical values of ρKDP= ηi/ηKDP (left) and ρLiNbO3 = ηi/ηLiNbO3 (right).

One can see from this figure, that calculations at levels 1-3 demonstrate a weak correlation between theoretical and experimental ρ. In contrast, calculation at level 4 produced a very good correlation (the correlation coefficient is over 0.99, with the slope very close to 1, and the cut-off line on the y-axis is close to zero). When constructing similar dependencies for I and II types of SHG, the correlation coefficient is about 65%, which may be explained by mixed generation mechanism. Thus, the calculation at level 4 can be used to reliably estimate the relative SHG activity coefficient for various compounds. The data shown in Figuge 4 refer to the mixed type of generation, i.e. represent a linear combination of expressions for I and II type SHG, with coefficients 0.5. This type of generation is closest to the conditions of experimental measurements of powder samples. However, a similar picture holds also for other types of SHG, including type I and type II generation. Thus, the ratio of η to one of the test compounds bith calculated at level 4 gives a universal dimensionless value ρ, by means of which it is possible to characterize an unknown nonlinear converter with respect to the studied reference structures. We will use this value as a criterion for preliminary selection of compounds for further study. Earlier it was shown that when the wavelength is increased above λ = 975 nm, the preferential type of second harmonic generation changes from type II to type I.47 These ACS Paragon Plus Environment

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conditions were used to obtain the data considered here (λ≥1064 nm)3. However, for monocrystals, it is also possible to observe pure generation of type I or II. The effect of the generation type on the calculated values of ρ is shown in Figure 5. As follows from these data, the values of ρ obtained for the type I generation, as well as for the mixed type of generation, are much more consistent with the experimental values. Thus, to estimate the SHG activity of unknown compounds at wavelengths ≥1064 nm, it is preferable to use the values of ρ, calculated for the type I generation.

10

ρ KDP/LiNbO3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Exp Exp mix calc I type calc II type calc Ave type calc

5

0

1

2

3

4

Metod

Figure 5. A comparison of the radiation power ratios ρ = ηKDP/ηLiNbO3 calculated at the levels of the theory 1, 3, 4 for the generation types I (empty), II (colored) and mixed type (half-colored) with experimental values of 3 (blue) and 30 calculated from the experimental parameters crystal (red) line.

3.2 NLO properties of CsNbMoO6 and CsTaMoO6. The crystals CsNbMoO6 and CsTaMoO6 were synthesized by methods described in Ref.17 The structure of the synthesized crystals was investigated by the X-ray diffraction method and corresponds to the structural type of pyrochlore, with the symmetry group F43m (216), and the lattice parameters given in Table 1. The structure of the unit cell is shown in Figure 6a. According to the results of X-ray diffraction analysis, the compounds CsNbMoO6 and ACS Paragon Plus Environment


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CsTaMoO6 have 16 crystallographic positions in the unit cell (Fig. 6b), which can be equally occupied by BV atoms (BV = Nb or Ta) or Mo, with equal probability of filling each of the 16 positions by either one or another atom. With different choice of the mutual arrangement of the BV and Mo atoms, different unit cells with different symmetry groups are obtained, and depending on the symmetry, the optical properties of the crystals, including the number of optic axes, the symmetry of the hyperpolarizability tensor, and the possible phase-matching conditions. To evaluate the effect of different ion ordering on the NLO properties of the compounds studied, three possible variants of the mutual arrangement of the Mo and BV atoms in the cell were considered. The mutual arrangement of these atoms was chosen to achieve a significant difference in the structure of the cells. The selected variants of Mo/BV sublattices and their corresponding reduced-symmetry cells are shown in Fig.7. The symmetry of the selected structures was determined using the program FINDSYM48 and corresponded to the space groups P4m2 (115), Cmm2 (35) and Cm (8).

(a)

(b)

Figure 6. The structure of the unit cell of the investigated compounds Cs Mo BV O6 (BV = Ta, Nb) (a) and the sublattice of the transition element atoms Mo/ BV (b). The blue atoms are Cs, gray is O, green is Mo/ BV.


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16

P4m2 (115)

Cmm2 (35)

Cm (8)

Figure 7. Selected variants of the Mo / BV sublattices (upper row) and the corresponding elementary cells of the reduced symmetry of the Cs Mo BV O6 compounds (lower row). Blue atoms – Cs, gray – O, green – Mo, olive – BV. The calculations of the selected structures were carried out in the CRYSTAL14 program at levels 2 and 4. The arrangement of the atoms in the unit cell corresponded to the structures shown in Figure 7 (the symmetry of the cells corresponded to space groups 115, 35 and 8, regardless of the type of BV atom.) For the structures studied, refractive indices, the nonzero components of the hyperpolarizability tensor, its effective values and the ratio of the SHG power ratio η were calculated.



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Table 3. The calculated refractive indices, the nonzero components of the hyperpolarizability tensor, the effective values of the hyperpolarizability tensor, and the power ratio η of the CsMo BVO6 compounds with the elementary cells of the symmetry groups 115, 35, 8. Calculation at the 2-level theory Space 115 group nx 1.87420 (ne) ny 1.87885 (no) nz 1.87885 (no)

Calculation at the 4-level theory CsNbMoO6

35

8

1.86376 1.87529 1.89030

1.86575 1.87420 (ne) 1.87430 1.87885 (no) 1.88265 1.87885 (no) d11 = 0.167 d15 = 0.263 d16 = 0.053 d36 = 0.272 d13 = 0.065 d24 = –0.349 d33 = –0.051 0.013 0.001 0.272 0.136 0.066 0.014 0.087 0.000 0.0001 0.0024 0.0187 0.0019 0.0022 0.0001 0.0039 0.0000 CsTaMoO6 1.80901 2.24838 (ne) 1.82216 2.26016 (no) 1.83653 2.26016 (no) d11 = –0. 235 d15 = –0. 342 d16 = –0. 075 d36 = 0.984 d13 = –0. 078 d24 = 0. 459 d33 = 0. 072 0.016 0.019 0.492 0.418 0.086 0.016 0.115 0.000 0.0002 0.0044 0.2773 0.0281 0.0041 0.0002 0.0074 0.0000

d15= 0.342 d31 = 0.342 d24= –0.552 d32 = –0.552 d33 = 0.040

dij

d36 = 0.415

deff

0.208

0.176

η

0.0856

0.0488

nx ny nz

1.81922 (ne) 1.83106 (no) 1.83106 (no)

2.30937 2.33942 2.37937

dij

d36 = 0.446

d15= 0.270 d31 = 0.270 d24= –0.508 d32 = –0.508 d33 = 0.056

deff

0.223

0.189

η

0.1073

0.0109

0.000 0.086 0.138 0.0000 0.0038 0.0098

0.000 0.066 0.127 0.0000 0.0025 0.0090

0.033 0.000 0.000 0.0053 0.0000 0.0000

0.038 0.000 0.000 0.0047 0.0000 0.0000

115

35

8

2.30937 2.33942 2.37937

2.31083 2.33843 2.36099 d11 = –0.489 d15 = 1.797 d16 = 0.465 d13 = –0.205 d24 = –1.657 d33 = 0.218 0.042 0.022 0.449 0.044 0.414 0.000 0.001 0.037 0.054 0.001 0.046 0.000

d15= 1.069 d31 = 1.069 d24= –3.412 d32 = –3.412 d33 = 1.989 0.000 0.267 0.853 0.0000 0.0189 0.1922

0.373 0.000 0.000 0.0961 0.0000 0.0000

2.19181 2.23769 2.30167 d15= 0.211 d31 = 0. 211 d24= –3.141 d32 = –3.141 d33 = 2.434 0.000 0.053 0.785 0.0000 0.0008 0.1855

0.466 0.000 0.000 0.1001 0.0000 0.0000

2.19939 2.23796 2.27288 d11 = –0. 050 d15 = –2.094 d16 = –0.732 d13 = –0.147 d24 = 2.163 d33 = 0.229 0.030 0.011 0.524 0.031 0.541 0.000 0.0003 0.0645 0.0832 0.0003 0.0887 0.0000

The difference between the refractive indices for structures of different symmetry does not exceed 9·10–5. A small difference between these values leads to the fact that phase matching conditions in these structures are most likely not fulfilled. Structures 2 and 3 (space groups 35 and 8) are biaxial and are characterized by three refraction coefficients. This fact also indicates that achieving phase-matching conditions in such systems is even less likely. The maximum difference between the indices nx. ny. nz is observed for symmetry structures Cmm2 (group 35).


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The difference between the refractive indices nx. ny. nz is greater for the structures with Ta atoms. This trend takes place irrespective of the method of calculation and the space group. If we consider that the intensity of SHG for tantalum structures is lower, it can be assumed that the difference in refractive index is determined by the basis set used. As the comparison of the calculated characteristics of the structures 115, 35 and 8 shows, the values of the NLO characteristics change noticeably when passing from uniaxial structures to biaxial ones. The coefficients of activity for uniaxial structures are almost an order of magnitude higher than those for biaxial structures. At the same time, NLO parameters of biaxial structures vary much less, usually within a few tens of percent. The deff calculation from the Eq. (3.c) shows that the structure with the space group Cmm2 has the maximum SHG activity. The results of calculations of the optical parameters of compounds for which reliable experimental data exist, show that the performance of the CPHF/KS calculations for the NLO activity parameters differ significantly depending on the quantum chemical theory and basis set used. The best agreement between the calculated parameters and the experimental data takes place in the case of the theory level 4 (DFT/PBE0 in the POB-TZVP basis). A somewhat worse agreement holds in the case of level 2 (HF in the basis of POB-TZVP) in calculating the maximum components of the hyperpolarizability tensor and the refractive indices of urea and quartz, as well as level 3 (DFT / PBE0 in the basis of STO-3G) in calculating the refractive index of KDP and LiNbO3. Thus, when choosing the method for calculating compounds with heavy atoms, the choice of basis and pseudopotential is of paramount importance in comparison with the choice between HF and DFT methods. For the calculation of NLO parameters, an urgent need is the elaboration of improved the basic sets, in particular, their calibration taking into account optical characteristics. At the same time, the influence of the atomic basis in calculating the refractive indices obtained directly from the CPHF/KS calculation is less significant than the influence of the quantum chemical theory. When calculating the components of the



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hyperpolarizability tensor, the theory level 2 is better than 3, which demonstrates a greater importance of the basis. The mean error is greater for the tensor components than that for the refractive indices, which explains the fact that for the values of deff, η, ρ, the level of theory 2 yields better results than level 3. Probably, the significant deviations from the experimental values are a consequence of the fact that the basis sets used in the calculations were calibrated by piezoelectric properties.14 Thus, in order to calculate the optical characteristics, it is essential to improve the basis sets and, possibly, the DFT functionals. Their calibration from experimental data for optical characteristics was proposed in the literature.49 This is evidenced by the fact that the worst agreement with experiment in the test structures is observed for lithium niobate, in the calculations of which a pseudopotential, also calibrated for piezoelectric properties, was used. The present study shows that the determination of the parameters of NLO activity of crystals can be significantly improved by using the relative values of ρ, apparently, due to the compensation of errors in the quantities included in this ratio. The use of this criterion has the additional advantage that it becomes possible to compare crystals of different symmetries, which is difficult when using the dmax values, usually used in practice. The contribution to the calculation error in deff, η, ρ also introduces uncertainty of the phase matching conditions. Usually it is assumed in theoretical calculations that the value of sinc in Eq. (4a) is equal to or close to 1. However, this condition is not always satisfied in practice, especially in the case of powder samples or biaxial crystals. Consequently, theoretical results tend to overestimate the values of ρ, which is demonstrated by Figure 5. When calculating the structures of the test compounds corresponding to the different mutual arrangement of Mo and BV atoms, the calculated NLO characteristics of uniaxial and biaxial crystals were substantially different. The influence of the sublattice symmetry on the NLO parameters strongly depends on the symmetry of the unit cell, but in the case of biaxial


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crystals their SHG activity indices differ by no more than factor of 2, whereas between similar values for uniaxial and biaxial crystals such quantities differ by an order of magnitude. This fact indicates that the effect of statistical disorder in such crystals is very noticeable and, apparently, primarily affects the feasibility of phase matching conditions. Out investigation of the influence of the atom distribution in the sublattice of heavy metals on optical characteristics shows that the values of the refractive indices, at least for the sublattices considered, vary insignificantly, as do the general components of the hyperpolarizability tensors. However, this difference is somewhat more pronounced for the values of deff, η, and ρ. Calculation of the NLO characteristics of representatives of a new class of defect pyrochlores CsNbMoO6 and CsTaMoO6 shows that their SHG activity with respect to LiNbO3 is of 0.0161 and 0.0219, with respect to KDP 0.3869 and 0.5249, for Nb and Ta structures, respectively. The phase matching conditions are not satisfied in the powder state and depend strongly on the structural disorder of the sample. Thus, the production of single-crystal SHG converters requires selection of syntheic conditions that assures stoichiometry and allows control over structural disorder.

4. Conclusions We predict SHG activity of defect pyrochlore crystals CsBVMoO6 (BV = Nb or Ta) by CPHF/KS method using HF and DFT/PBE0 formalism combined with two Gaussian type basis sets. To estimate the accuracy of these predictions, we benchmark our methods on the crystals of KH2PO4 (KDP), LiNbO3, (NH2)2CO (urea), and α-SiO2 where reliable experimental data are available. Next, we study the influence of the ion ordering on the NLO properties, arising due to the different distributions of ions Mo or BV over d-metal sublattice positions. We calculate both linear optical properties (refractive indices and linear polarizabilities) and nonlinear ones (first ACS Paragon Plus Environment


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hyperpolarizabilities and nonlinear optical tensors) for different polytypic unit cells of CsNbMoO6 and CsTaMoO6 at different levels of theory. The results obtained allow us to make the following conclusions: (1) the best agreement of the calculated nonlinear optical parameters with experiment is achieved in the case of calculations at the DFT/PBE0 level with the POBTZVP basis; (2) agreement becomes quantitative when the dimensionless relative NLO activity ρ = ηi/ηX is used for this comparison (ηi is the ratio of powers of the single and double frequency rays for the unknown compound, ηX is the same ratio for the reference compound); (3) the calculated NLO characteristics ρ(CsNbMoO6/LiNbO3)=0.016 and ρ(CsTaMoO6/LiNbO3)=0.022, ρ(CsNbMoO6/KDP)=0.39 and ρ(CsTaMoO6/KDP)=0.52; (4) the symmetry of the sublattices of disordered ions significantly affects the SHG parameters of the crystal and the values of the calculated parameters for sublattices of different symmetry can differ by an order of magnitude.

Acknowledgments This work was supported by the Ministry of Education and Science of the Russian Federation (assignments 4.5510.2017/8.9 and 4.8337.2017/BCh), and the Russian Foundation for Basic Research (project 17-03-00912). A.E.M acknowledges support by the Act 211 by Government of the Russian Federation (Contract No. 02.A03.21.0011) in part of benchmarking calculations, and by the Russian Science Foundation (Contract No. 14-43-00052) in part of hyperpolarizability predictions. He also greatful to the Ministry of Education and Science of the Russian Federation (award No. 4.1157.2017/4.6) to the “Improving of the Competitiveness” program of the National Research Nuclear University MEPhI. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) resource Stampede2 at UT Austin through allocation TGDMR180004. Supporting Information


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The calculated electronic band gaps of the reference compounds and the compounds under investigation are given in Table S1 of Supporting Information along with the available experimental data and short discussion on this property.

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TOC graphics


Nonlinear Optical Properties of Mixed Oxides Crystals CsNbMoO6 and

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