Structure and Non-linear Optical Properties of b-barium Borate

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1 652 Acta Cryst. (1998). B54, 652±656 Structure and Non-linear Optical Properties of b-barium Borate D. F. Xue and S. Y. Zhang* Laboratory of Rare Earth Chemistry and Physics, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, , Jilin, People's Republic of China. (Received 1 February 1997; accepted 25 March 1998 ) Abstract The non-linear optical (NLO) properties of crystalline -BaB 2 O 4 (-barium borate, BBO) have been investigated from the chemical bond viewpoint. The contributions of each type of chemical bond to the total NLO coef cient have been quantitatively determined. The calculations indicate that the true space group of BBO is R3 rather than R3c. 1. Introduction The novel non-linear optical material -barium borate (-BaB 2 O 4 or BBO) reported by Chen et al. (1985) has found many applications. However, the space group of -BaB 2 O 4 remains controversial, as does the relationship between its crystal structure and NLO (non-linear optical) properties. The structure was rst reported by Lu et al. (1982), who assigned the space group as R3, but very close to R3c, using a hexagonal cell of dimensions a = and c = A Ê. A year later, Liebertz & Stahr (1983) reported the space group as R3c with practically the same unit-cell dimensions. FroÈ hlich (1984) reexamined the structure and concluded that the space group was R3c. Later papers have not led to a de nitive conclusion regarding the space group of BBO (Eimerl et al., 1987; Nikogosyan, 1991). Furthermore, this uncertainty has led to confusion regarding the nature of the bonding and the coordination numbers of each atom in the crystal. It nally prompted us to try to understand the origin of the non-linear optical behaviour of BBO. We believe that in a real crystal the physical properties must re ect the characters of the constituent chemical bonds. Starting from this point of view, we have studied the constituent chemical bond properties and contributions of each type of bond to the total NLO coef cient using the bond-valence theory of complex crystals (Zhang, 1991) and the modi ed Levine bondcharge model (Levine, 1973; Xue & Zhang, 1996a,b), which has been applied successfully to the calculation of the non-linearities in complex crystals. 2. Bond-valence theory According to the bond-valence theory of complex crystals (Zhang, 1991), the connection between macroscopic physical properties of a complex crystal and its constituent chemical bonds can be constructed using the bond-valence equation. This equation, applied to the chemical bond AÐB in a crystal of composition A a B b D d G g,..., involves the expression N B A a=n CA Š A N A B b=n CB Š B ; 1 where A, B, D, G,..., are different element names in the crystal formula, and a, b, d, g,..., represent the numbers of atoms of the corresponding element. N(I J) represents the coordination number of the ion I by the ion J, and N CA and N CB represent the coordination numbers of the elements A and B in the crystal. In a real crystal this equation can be applied, provided the structure is known. Then we can realize the aim of decomposing the crystal into its constituent chemical bonds. The properties of each type of chemical bond in the crystal can be derived using the following formulae. The susceptibility of any bond,, is written as ˆ 4 1 h - p =E g 2 ; where E g is the average energy gap between the bonding and the antibonding states, and p is the plasma frequency p ˆ 4 Ne e 2 =mšd A ; 3 where D and A are the correction factors of order unity (Van Vechten, 1969), and Ne is the effective valence electron density associated with the bond Ne ˆ n e = b ; 4 where b is the bond volume of bonds of type and n e is the number of effective valence electrons per bond n e ˆ ZA =N CA ZB =NCB; 5 where Z A is the number of effective valence electrons of ion A, Z A ˆ Z Aq A, q A being the effective charge of each valence electron of ion A; its value can be quantitatively determined using the present approach (Zhang, 1991). 2 # 1998 International Union of Crystallography Acta Crystallographica Section B Printed in Great Britain ± all rights reserved ISSN # 1998

2 D. F. XUE AND S. Y. ZHANG 653 The average energy gap E g can be separated into the homopolar E h and heteropolar C parts as E g 2 ˆ E h 2 C 2 E h ˆ 39:74= d 2:48 C ˆ 14:4b exp k s r 0 Z A n Z B Š=r 0 n 1 C ˆ 14:4b exp k s r 0 1=n ZA ZB Š=r 0 n<1 ; where k s ˆ 4k F=a B 1=2 9 k F ˆ 3 2 Ne Š 1=3 : 10 In (9) a B is the Bohr radius. In (7) and (8) r 0 ˆ d =2is the average radius of A and B in A Ê ; n is the ratio of numbers of the two elements B-to-A in (1) and exp k s r 0 is the Thomas±Fermi screening factor. The correction factor b has a relationship with the average coordination number N C as b ˆ NC p ; the actual value for the index p depends on the crystal structure (Levine, 1973). In our calculations of NLO crystals we have found that the value of b decreases with increasing NC. In crystals with HÐO bonds the coordination number of the H atom is always less than 3, so the average coordination number N C will be less than 3. The value of p was determined to be three (Xue & Zhang, 1996b). In the BBO crystal there are many cations with higher coordination numbers [Ba(1) 2+ and Ba(2) 2+, each participating in eight different types of BaÐO bonds]. The common expressions for b are b ˆ NC 1:48, when NC>5, and b ˆ NC 2, when N C 5. can be deduced from the above equations, provided the index of refraction for the crystal is known N C ˆ NCA= 1 n nncb= 1 n : 11 If the crystal is composed of different types of bonds (labelled ), then the total can be resolved into contributions from the various types of bonds, ˆ X F ˆ X N b b ; 12 where is the total macroscopic susceptibility which a crystal composed entirely of bonds of type would have. F is the fraction of bonds of type composing the actual crystal, N b is the number of bonds of type per cubic centimeter and b is the susceptibility of a single bond of type. The fractional ionicity f i and covalency fc of the individual bonds can be determined according to f i ˆ C 2 = E g 2 ; f c ˆ E h 2 = E g 2 : 13 The bond non-linearities can be evaluated by the following expressions and the total non-linear susceptibility d ijk is expressed as (Levine, 1973) d ijk ˆ X d ijk ˆ X F d ijk C d ijk E h Š; 14 where d ijk is the total macroscopic non-linearity which shows the contribution of bonds of type to the total non-linearity d ijk of the given complex crystal, d ijk C is the ionic fraction of the non-linear optical coef cient and d ijk E h the covalent fraction F d ijk C ˆfG ijkn b 14:4 b exp k s r 0 ZA n Z B Š b 2 C g= E g 2 d q Š 15 F d ijk E h ˆfG ijkn b s 2s 1 r 0 = r 0 r c Š 2 fc b 2 g= d q ; 16 where G ijk is the geometrical contribution of the bonds of type, which can be calculated from X G ijk ˆ 1=n b i j k ; 17 where the sum on is over all n b bonds of type in the unit cell and i is the direction cosine with respect to the ith coordinate axis of the th bond of type in the unit cell. The difference in the atomic sizes is ˆ r A r B = r A r B, where r A and r B are the covalent radii of atoms A and B (values used are taken from Dai & Shen, 1981). When the cited data for r A and r B from Dai & Shen (1981) cannot approximately satisfy the relation r A r B d s (d s represents the shortest bond length among the bonds of type ), the new value of r A used will be r A ˆ d s r B. r c ˆ 0:35r 0 is the core radius. q is the bond charge of the th bond (Levine, 1973; Xue & Zhang, 1996a,b), q ˆ n e 1= 1 Kfc Še; 18 where K is a function of the average crystal covalency F C and its best value can be determined by using the following equation (Xue & Zhang, 1996b) K ˆ 2 F c 1:1; 19 where F C is de ned as F C ˆ X N b f c : Results and discussion -BaB 2 O 4 consists of almost planar (B 3 O 6 ) 3 rings perpendicular to the polar c axis, bonded together through the barium ions (Lu et al., 1982). We have calculated the coordination numbers of all constituent ions using the atomic positions of Lu et al. (1982) and

3 654 STRUCTURE AND NON-LINEAR OPTICAL PROPERTIES Table 1. Chemical bond parameters, and linear and non-linear properties of each type of bond for BBO The crystal data used are those of Lu et al. (1982). See the text for de nitions of the quantities listed. AÐB d A E h (ev) C (ev) fc q =e G 22 d 22(10 10 e.s.u.) G 31 d 31(10 10 e.s.u.) G 33 d 33(10 10 e.s.u.) Ba1ÐO Ba1ÐO Ba1ÐO Ba1ÐO Ba1ÐO Ba1ÐO Ba1ÐO Ba1ÐO Ba2ÐO Ba2ÐO Ba2ÐO Ba2ÐO Ba2ÐO Ba2ÐO Ba2ÐO Ba2ÐO B1ÐO ² 0 B1ÐO B1ÐO B2ÐO B2ÐO B2ÐO B3ÐO B3ÐO B3ÐO B4ÐO B4ÐO B4ÐO ² The value approaches zero. Table 2. Chemical bond parameters, and linear and non-linear properties of each type of bond for BBO The crystal data used are those of FroÈhlich (1984). See the text for de nitions of the quantities listed. AÐB d A E h (ev) C (ev) fc q =e G 22 d 22(10 10 e.s.u.) G 31 d 31(10 10 e.s.u.) G 33 d 33(10 10 e.s.u.) BaÐO BaÐO BaÐO BaÐO BaÐO BaÐO BaÐO BaÐO B1ÐO B1ÐO B1ÐO B2ÐO B2ÐO B2ÐO nd the coordination number of barium by oxygen is different from the reported value of seven (Lu et al., 1982). In reality there are two types of barium ions in this crystal. Each has eightfold coordination. There are four anionic groups in each unit cell: [B(1) 3 O 6 ] 3, [B(2) 3 O 6 ] 3, [B(3) 3 O 6 ] 3 and [B(4) 3 O 6 ] 3, which are distributed over two symmetrically independent positions. The four groups are stacked on top of each other and are arranged in two pairs: [B(1) 3 O 6 ] 3 with [B(2) 3 O 6 ] 3 and [B(3) 3 O 6 ] 3 with [B(4) 3 O 6 ] 3. From the atomic positions of BBO (Lu et al., 1982), the crystal can be decomposed as follows

4 D. F. XUE AND S. Y. ZHANG 655 Table 3. Comparisons between experiment and theory applied to the non-linearities of BBO, = mm for fundamental wavelength Chen's calculated data (Li & Chen, 1985): calculated the non-linearities of the (B 3 O 6 ) 3 anionic group in BBO by neglecting the contributions of barium ions, i.e. BaÐO bonds. Chen's calculated ( 10 9 e.s.u.) Our calculated ( 10 9 e.s.u.) Experiment ( 10 9 e.s.u.) using Lu et al. (1982) Using Lu et al. (1982) Using FroÈ hlich (1984) d 22 ( ),² 5.30³ d 11 < 0.05d 22 ² 0.07 d d 31 ( )d 22 ² d 32 = d 31 ² d ² Results measured with the Maker fringe method (Chen et al., 1985; Eimerl et al., 1987; Chen et al., 1990). ³ Experimental data (Fan et al., 1989). Ba 3 (B 3 O 6 2 ˆ 3=16Ba(1)O(1) 8=3 3=16Ba(1)O(3) 2 3=16Ba(1)O( =16Ba(1)O(4) 2 3=16Ba(1)O(5) 2 3=16Ba(1)O(6) 2 3=16Ba(1)O( =16Ba(1)O(8) 8=3 3=16Ba(2)O(2) 8=3 3=16Ba(2)O(3) 2 3=16Ba(2)O(4) 2 3=16Ba(2)O( =16Ba(2)O(5) 2 3=16Ba(2)O( =16Ba(2)O(6) 2 3=16Ba(2)O(7) 8=3 1=2B(1)O(1) 1=2B(1)O(1 0 1=2B(1)O(5) 3=4 1=2B(2)O(2) 1=2B(2)O(2 0 1=2B(2)O(6) 3=4 1=2B(3)O(3) 3=4 1=2B(3)O(7) 1=2B(3)O(7 0 1=2B(4)O(4) 3=4 1=2B(4)O(8) 1=2B(4)O(8 0 : 21 By using the refractive index of BBO n 0 = at mm (Chen et al., 1985; Eimerl et al., 1987), the chemical bond parameters, and linear and non-linear optical properties of all types of constituent chemical bonds are calculated and are presented in Table 1. From Table 1 we can see the importance of the planar (B 3 O 6 ) 3 anionic group in BBO; contributions of each type of BÐO bond to the total linearity and non-linearity are much larger than those of the BaÐO bonds, especially in the calculation of tensor coef cients d 22. At the same time it should also be noted that from the chemical bond viewpoint each type of constituent chemical bond has its own contributions to the total linearity and non-linearity, although some of them cannot give dominant contributions to a certain tensor coef cient. Therefore, we believe it is not acceptable in dealing with the non-linearities of BBO to neglect the roles of the Ba(1) 2+ and Ba(2) 2+ cations (Li & Chen, 1985). In fact, barium ions (i.e. BaÐO bonds) surely play an important role in the contributions to the total NLO tensor coef cient. This becomes more obvious in the calculation of d BBO 31 and d BBO 33. In contrast, our calculations indicate that it is important to take into account all kinds of constituent chemical bonds when studying and predicting the non-linearities of such complex crystals. Owing to the uncertainty about the crystal structure of BBO, atomic positions corresponding to R3c (FroÈ h- lich, 1984) were used to give the results listed in Table 2. This structure (FroÈ hlich, 1984) has three allowed independent NLO tensor coef cients, d 22, d 31 and d 33, according to the restrictions imposed by the crystal symmetry and the Kleinman symmetry conditions (Kleinman, 1962) on non-linear optical coef cients. Accordingly, we give only the three calculated coef cients for comparison with the experimental data. Six NLO tensor coef cients, d 11, d 14, d 22, d 33, d 31 and d 32, can be calculated using the set of atomic positions (Lu et al., 1982) for space group R3. -BaB 2 O 4 should have ve independent NLO coef cients, d 22, d 11, d 14, d 31 (= d 32 ) and d 33, but as its space group is close to R3c, two NLO coef cients d 14 and d 11 are very small (Chen et al., 1990). All of our calculated results and those of Li & Chen (1985) concerning this BBO crystal are presented in Table 3. From it we can see that our results and those of Li & Chen (1985) based on the structural data of Lu et al. (1982) agree satisfactorily with the experimental data in both numerical value and sign. Therefore, Lu et al. (1982) appear to have chosen the correct structure. In Table 2 the different bond lengths d and geometrical factors G ijk calculated from the structural data of FroÈ hlich (1984) do not agree reasonably with the observed data (shown in Table 3). We conclude that this set of atomic positions does not re ect the true structure and does not explain the NLO properties. From the calculated results we nd that numerical values of d BBO 11 and d BBO 14 approach zero, con rming the

5 656 STRUCTURE AND NON-LINEAR OPTICAL PROPERTIES Table 4. Fractional atomic coordinates The re nement results of single crystals (Lu et al., 1982), space group R3, a = b = , c = A Ê, Z =6. Wyckoff position x y z Ba1 9(b) Ba2 9(b) B1 9(b) B2 9(b) B3 9(b) B4 9(b) O1 9(b) O2 9(b) O3 9(b) O4 9(b) O5 9(b) O6 9(b) O7 9(b) O8 9(b) space group reported by Lu et al. (1982) whose results lead us to believe that this crystal should be assigned to class 3. Being an excellent NLO material, BBO has an outstanding d 22 coef cient and from the calculated values listed in Table 1 we nd that there is almost no cancellation of all types of d 22 tensor coef cients corresponding to contributions from different chemical bonds. BBO has a much larger geometrical factor G BÐO 22 than any other G BÐO ij (i, j 6ˆ 2) factors and the signs of these G 22 factors are almost the same. This type of crystal structure is ideal since it leads to little cancellation of contributions from different chemical bonds. These results suggest that when searching for new highly ef cient NLO materials or designing new ones the key is to nd a good structural model. 4. Conclusion In this paper, starting from different sets of structural data for BBO, we have calculated its NLO tensor coef- cients which indicate that the atomic positions reported by Lu et al. (1982) can re ect the real situation of all constituent elements and that this structure should belong to R3 (see Table 4). From the chemical bond viewpoint, the linear and non-linear optical properties of all types of constituent chemical bonds are analysed. The contributions of barium ions cannot be neglected in the calculation. Our calculations suggest that the coordination number of the barium ions should be eight, rather than seven, as reported by Lu et al. (1982). This work was supported by the State Key Program of Basic Research of the People's Republic of China. References Chen, C. T., Wu, B. C., Jiang, A. D. & You, G. M. (1985). Sci. Sin. B, 28, 235±243. Chen, C. T., Wu, Y. C. & Li, R. K. (1990). J. Cryst. Growth, 99, 790±798. Dai, A. B. & Shen, M. C. (1981). Periodic Table of the Element. Shanghai: Scienti c Technology Press (in Chinese). Eimerl, D., Davis, L., Velsko, S., Graham, E. K. & Zalkin, A. (1987). J. Appl. Phys. 62, 1968±1983. Fan, Y. X., Eckardt, R. C., Byer, R. L., Chen, C. T. & Jiang, A. D. (1989). IEEE J. Quantum Electron. QE-25, 1196±1199. FroÈ hlich, R. (1984). Z. Kristallogr. 168, 109±112. Kleinman, D. A. (1962). Phys. Rev. 126, 1977±1979. Levine, B. F. (1973). Phys. Rev. B, 7, 2600±2626. Li, R. K. & Chen, C. T. (1985). Acta Phys. Sin. 34, 823±829. Liebertz, J. & Stahr, S. (1983). Z. Kristallogr. 165, 91±93. Lu, S. F., Ho, M. Y. & Huang, J. L. (1982). Acta Phys. Sin. 31, 948±955. Nikogosyan, D. N. (1991). Appl. Phys. A, 52, 359±368. Van Vechten, J. A. (1969). Phys. Rev. 182, 891±905. Xue, D. F. & Zhang, S. Y. (1996a). J. Phys. Condens. Matter, 8, 1949±1956. Xue, D. F. & Zhang, S. Y. (1996b). J. Phys. Chem. Solids, 57, 1321±1328. Zhang, S. Y. (1991). Chin. J. Chem. Phys. 4, 109±115.

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