ISSN 0020-1685, Inorganic Materials, 2008, Vol. 44, No. 5, pp. 524–530. © Pleiades Publishing, Ltd., 2008. Original Russian Text © A.B. Dubovskii, E.A. Tyunina, E.N. Domoroshchina, G.M. Kuz’micheva, and V.B. Rybakov, 2008, published in Neorganicheskie Materialy, 2008, Vol. 44, No. 5, pp. 601–607.
Composition Effect on the Elastic Properties of Langasite A. B. Dubovskiia, E. A. Tyuninab, E. N. Domoroshchinab, G. M. Kuz’michevab, and V. B. Rybakovc a
All-Russia Research Institute of Mineral Materials Synthesis, Institutskaya ul. 1, Aleksandrov, Vladimir oblast, 601650 Russia b Lomonosov State Academy of Fine Chemical Technology, pr. Vernadskogo 86, Moscow, 119571 Russia c Moscow State University, Vorob’evy gory 1, Moscow, 119899 Russia e-mail:
[email protected] Received July 11, 2007
Abstract—The frequency coefficient, an important performance parameter of piezoelectric materials, which is used in determining vibrational frequencies of crystalline piezoelectric elements, is shown to vary along the length of a crystal. The dependences of the frequency coefficient and elastic stiffness coefficient on unit-cell parameters have the form of a parabolic function with a maximum, while the dependence of the frequency coefficient on 1 ⁄ ( ρ ) (ρ is the density of the crystal) is linear. Polarity changes within a sample are revealed, which may be due to the stress arising from variations in melt composition during crystal growth. DOI: 10.1134/S0020168508050178
INTRODUCTION Single-crystal La3Ga5SiO14 (langasite, LGS) is an attractive piezoelectric material, which is used in the fabrication of bulk- and surface-acoustic-wave bandpass filters and diverse piezoelectric and piezoresonance sensors. One important performance parameter of a piezoelectric material is its frequency coefficient å (kHz mm), which is used in determining vibrational frequencies of piezoelectric elements. The frequency coefficient depends on the density and elastic constants of the material and, therefore, would be expected to vary insignificantly in a particular direction. In studies of langasite, however, we observed a strong scatter of M. In this paper, we address the origin of this effect. EXPERIMENTAL From an 86-mm-diameter langasite crystal of nominal composition La3Ga5SiO14, grown by the Czochralski technique in a Kristall-3M growth unit (0001 growth direction, vrotation = 1–15 rpm, vgrowth = 1.2 mm/h, 98–99% Ar + 1–2% é2 atmosphere), we prepared a series of wafers 8 mm in diameter and 0.097 mm in thickness, normal to the [10 10 ] direction (Y-cut). Powder samples prepared by grinding portions of the wafers were examined by x-ray diffraction (XRD) on a DRON-3M diffractometer (CuKα radiation, flat graphite monochromator, sample rotation, starch additions to prevent preferential orientation, counting time of 10 s per data point, scan step of 0.02°). Diffraction line profiles were fitted with the convolution of a Gaussian and Lorentzian.
Qualitative XRD phase analysis, with the use of ICDD PDF-2 data, revealed no impurity phases (detection limit, 2 wt %). Lattice parameters were determined by least squares refinement among reflections in the range 2θ = 10°– 100°, using α-Al2O3 powder with a = 4.759(1) Å and c = 12.993(2) Å (National Institute of Standards and Technology, USA) as an external standard. In refining the composition and structure of the powder samples, all computations were performed with DBWS-9411 [1]. The intensity data set was reduced using the PROFILE FITTING V 4.0 program [2]. The peaks were fitted by a Voigt function at full width at half maximum = 8.0. After refining the scale factor, zero counter position, sample displacement, and lattice parameters, we refined the line profile and structural parameters. The structural parameters were refined in several steps: first, only atomic position coordinates; next, isotropic thermal parameters at fixed positional parameters; then, site occupancies at fixed positional and thermal parameters; thermal parameters together with site occupancies; and, finally, anisotropic thermal parameters. Microregions (0.2 × 0.2 × 0.2 mm) in the same wafers were examined by XRD on a CAD-4 four-circle diffractometer [3] at room temperature (MoKα radiation, graphite monochromator, ω scan mode). Lattice parameters were refined among 20–25 reflections with interplanar spacings d > 0.874 Å. In preliminary data processing, we used WinGX software [4], with an empirical absorption correction [5]. Full-matrix least squares refinements with SHELX-97 [6] and scattering factors for neutral atoms included anisotropic temperature factors for all atoms. The intensity data collection
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Table 1. Unit-cell parameters and counting statistics for the crystals studied Sample
1
a*, Å c*, Å V*, Å3 a**, Å c**, Å V**, Å3 Absorption coefficient, mm–1 (MoKα) Measured reflections Observed reflections R1 [I > 4σ(I)], % Flack parameter, x
2
8.168(3) 5.094(1) 294.32(27) 8.1639(6) 5.0955(5) 294.11(7) 22.07 588 585 4.77 –0.04(9)
3
4
5
6
7
8
8.162(3) 8.166(2) 8.167(2) 8.161(2) 5.091(2) 5.097(2) 5.0984(7) 5.096(2) 293.72(33) 294.35(26) 294.50(19) 293.93(26) 8.1662(3) 8.1667(2) 8.1672(6) 8.1673(5) 8.1694(5) 8.1682(2) 8.1655(5) 5.0970(2) 5.0968(2) 5.0978(6) 5.0974(6) 5.0984(5) 5.0973(2) 5.0960(5) 294.36(4) 294.39(3) 294.48(8) 294.47(7) 294.68(6) 294.53(3) 294.26(6) 22.12 22.07 22.06 22.10 588 585 8.67 –0.03(17)
588 583 4.84 0.04(8)
869 850 2.28 0.06(4)
588 586 4.31 –0.09(8)
* Unit-cell parameters of a microregion. ** Unit-cell parameters of ground crystals.
conditions and refinement procedure were the same for all of the samples. The frequencies of fundamental transverse shear modes were determined using polished 8-mm-diameter wafers (Y-cut). The amplitude response of the wafers was measured with a Kh1-54 meter. From the frequency and thickness, we determined the frequency coefficient M (kHz mm):
Since we measured the fundamental frequency, we have n = 1, 2å = ví (ví is the slow transverse acoustic wave velocity), and C 66 = v T ρ = (2M)2ρ. In this way, from the frequency coefficient å we determined the D elastic stiffness coefficient C 66 (GPa) since the measurements were performed on Y-cuts. This is a resonance method for determining stiffness coefficients. D
2
D
n C 66 M = tf = --- -------, 2 ρ where t is the thickness of the piezoelectric element, f is its vibrational frequency, n = 1, 2, 3 … is the harmonic number, ρ is the density of the element, and D
E
C 66 = C 66 +
2 e 26 ----S e 22
2
S
( e 26 is the piezoelectric coefficient, and e 22 is the dielectric permittivity).
RESULTS AND DISCUSSION The structure of langasite (La3Ga(1)Ga3(2)(GaSi)(3)O14, sp. gr. P321, Z = 1) has three gallium sites (Ga(1), octahedral; Ga(2), tetrahedral; (GaSi)(3), trigonal-pyramidal), one lanthanum site (dodecahedral), and three oxygen sites. The composition of langasite crystals can in general be represented as
(La3 – uu)(Ga1 – vv)(1)(Ga1 – ww)3(2)(Ga1 − xSix)2(3)(O14 – yy), indicating that vacancies () are possible in the La [7–9], Ga(1), Ga(2) [8–10], and O [7–10] sites. In the (GaSi)(3) site, any relationship between Ga and Si is possible: Ga = Si, Ga > Si, or Ga < Si [7–10]. Clearly, all this may cause variations in lattice parameters. Tables 1–4 summarize XRD results for microregions 0.2 × 0.2 × 0.2 mm in dimensions. The smallest R-factor was obtained for plate 6 . Its refined composition is almost stoichiometric: La3Ga4(Ga0.52Si0.48(1))2(O13.98(2)0.02). That sample had the largest lattice parameters. Close to it in composi-
tion was a microsample from plate 1: La3Ga4(Ga0.509Si0.491(1))2O13.99(1). Samples 1 and 6 were identical in a parameter (Table 1) but differed in Ò: 5.094(1) and 5.0984(7) Å, respectively. It cannot be ruled out that the c parameter responds to compositional changes (in this case, oxygen vacancies and/or Ga/Si ratio in the (GaSi)(3) site) to a greater extent than does a. The refined compositions of the microregions studied in plates 2 and 4 are La3(Ga0.9840.016(3))Ga3(Ga1.060Si0.940(6))(O13.970.03(1)) and
La3(Ga0.980.02(2))(Ga2.95(1)0.05)(Ga1.004Si0.996(7))(O13.89(1) 0.11(4)),
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Table 2. Positional parameters, site occupancies (µ), and equivalent isotropic thermal parameters (Ueq × 102, Å2) in the refined structures of single crystals Sample La x µ Ueq Ga(1) µ Ueq Ga(2) x µ Ueq Uiso Ga(3) x y z µ Ueq Si(3) µ O(1) x y z µ Ueq O(2) x y z µ Ueq Uiso O(3) x y z µ Ueq‚
1
2
4
6
8
0.41866(1) 1.0 0.391(2)
0.41848(2) 1.0 0.567(4)
0.41894(3) 1.0 0.741(5)
0.41876(4) 1.0 0.702(7)
0.58132(1) 1.0 0.229(2)
1.0 0.709(7)
0.984(3) 0.98(2)
0.98(2) 1.13(2)
1.0 0.94(2)
1.0 0.468(6)
0.76514(2) 1.0 0.419(4)
0.76504(5) 1.0 0.779(8)
0.76526(7) 0.983(3) 0.76(1)
0.76532(8) 1.0 0.77(1)
0.23475(3) 1.0 0.283(3)
1/3 2/3 0.53391(7) 0.509(1) 0.270(6)
1/3 2/3 0.5327(1) 0.530(3) 0.60(1)
1/3 2/3 0.5325(2) 0.502(3) 0.56(2)
1/3 2/3 0.5325(2) 0.518(8) 0.60(3)
2/3 1/3 0.53413(6) 0.508(1) 0.120(5)
0.491(1)
0.470(3)
0.498(3)
0.482(8)
0.492(1)
1/3 2/3 0.1873(4) 1.0 0.85(3)
1/3 2/3 0.1849(11) 1.0 2.8(1)
1/3 2/3 0.1933(9) 1.0 1.20(8)
1/3 2/3 0.1933(11) 1.0 1.23(10)
2/3 1/3 0.1880(3) 1.0 0.74(3)
0.4678(1) 0.3104(2) 0.3136(2) 1.0 1.13(2)
0.4661(3) 0.3137(3) 0.3120(4) 1.0 1.20(5)
0.4664(4) 0.3115(4) 0.3171(6) 0.983(7) 1.65(6)
0.4652(6) 0.3113(6) 0.3179(7) 1.0 1.74(7)
0.5324(1) 0.6901(1) 0.3167(2) 0.977(3) 0.87(2)
0.2235(2) 0.0813(1) 0.7633(2) 1.0 1.44(3)
0.2248(3) 0.0815(3) 0.7620(3) 1.0 1.21(5)
0.2210(4) 0.0807(4) 0.7651(5) 1.0 1.91(7)
0.2219(6) 0.0807(7) 0.7630(6) 1.0 1.96(7)
0.7768(2) 0.9183(2) 0.7640(2) 1.0 1.33(3)
Note: Samples 1, 2, 4, and 6 are right-handed: La in position 3e (x 0 0), Ga(1) in 1a (0 0 0), Ga(2) in 3f (x 0 1/2), Ga(3) in 2d, O(1) in 2d, and O(2) and O(3) in 6g; sample 8 is left-handed. INORGANIC MATERIALS
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Table 3. Principal bond distances (dij, Å) in the structures studied Crystal La–2O(3) –2O(2) –2O(2) –2O(1) 〈La–O〉 Ga(1)–6O(3) Ga(2)–2O(2) –2O(3) 〈Ga(2)–O〉 (Ga,Si)(3)–O(1) –3O(2) 〈(Ga,Si)(3)–O〉
1 2.345(2) 2.502(1) 2.850(1) 2.629(1) 2.582 2.004(1) 1.931(1) 1.806(1) 1.869 1.766(2) 1.724(1) 1.735
2 2.337(3) 2.476(2) 2.869(2) 2.623(2) 2.576 2.014(2) 1.917(2) 1.803(2) 1.860 1.771(6) 1.751(2) 1.756
4 2.355(3) 2.504(3) 2.872(3) 2.639(2) 2.593 1.984(3) 1.912(3) 1.811(3) 1.862 1.729(5) 1.733(3) 1.732
6 2.354(4) 2.509(4) 2.876(4) 2.640(2) 2.595 1.996(4) 1.903(4) 1.805(3) 1.854 1.729(6) 1.739(4) 1.737
8 2.345(1) 2.513(1) 2.855(1) 2.6284(7) 2.585 1.999(1) 1.922(1) 1.806(1) 1.864 1.764(2) 1.716(1) 1.728
Table 4. Compositions refined by single-crystal XRD (microregions) and profile analysis (ground single crystals) Composition
Sample no.
microregions
ground single crystals
1
La3Ga4(Ga1.018Si0.982(2))(O13.990.01(1)) R = 4.8%
2
La3(Ga0.9840.016(3))Ga3(Ga1.060Si0.940(6))(O13.970.03(1)) R = 8.7%
La3Ga4(Ga1.07Si0.93(1))(O13.970.03) Rp = 10.15, Rwp = 13.08 La3Ga4(Ga1.05Si0.95(1))(O13.980.02) Rp = 9.82, Rwp = 13.02 La3(Ga0.990.01(1))Ga3(Ga1.05Si0.95(1))O13.960.04 Rp = 8.83, Rwp = 11.08 La3Ga4(Ga1.030Si0.970(8))(O13.990.01) Rp = 8.07, Rwp = 11.07
3 4
La3 (Ga0.980.02(2))(Ga2.95(1)0.05)(Ga1.004Si0.996(7))(O13.890.11(4)) R = 4.8%
5 6
La3Ga4(Ga1.04Si0.96(2))O13.980.02(2) R = 2.3%
7 8
La3Ga4(Ga1.017Si0.983(3))O13.990.01 R = 4.3%
respectively, indicating that these microregions had the lowest structural perfection. Analysis of the bond distances in the structures of the microregions studied (Table 3) indicates that plate 2, which was found to be the most Ga-rich, has the largest (Ga,Si)(3)–O(1) and (Ga,Si)(3)–O(2) bond distances in the trigonal-pyramidal sites. The average cation–anion distances in the trigonal-pyramidal sites correlate with those in the dodecahedra: as the former increase, the latter drop, and vice versa. Moreover, the Ga–O bond distance in the Ga(1)O6 octahedra correlates with the Ga(1) site occupancy: the presence of vacancies in this site reduces the bond distance. An interesting feature of our samples is that their polarity changes along the length of the crystal: plates INORGANIC MATERIALS
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La3(Ga0.970.03(1))Ga3(Ga1.09Si0.891(2))(O13.910.09) Rp = 8.72, Rwp = 12.06 La3(Ga0.99(2)0.02) Ga3(Ga1.05Si0.95(3))(O13.960.04) Rp = 7.76, Rwp = 10.82 La3(Ga0.98(2)0.02)(1)Ga3(Ga1.05Si0.95(6))(O13.940.06) Rp = 8.91, Rwp = 11.94 La3Ga(Ga2.97(1)0.03)(Ga1.05Si0.95(2))(O13.930.07) Rp = 10.63, Rwp = 13.55
1, 2, 4, and 6 are right-handed, while plate 8 is lefthanded. It cannot be ruled out that this is due to the stress arising from variations in melt composition during crystal growth. Table 4 lists the refined compositions of ground samples (profile analysis). Comparison of the powder samples and microregions in the same plates indicates the plates are inhomogeneous. The same is evidenced by the fact that, in some instances, the powder samples and microregions differ in unit-cell parameters (Table 1). Plate 6 was found to be the most homogeneous, while plates 1, 2, and 8 were the least homogeneous. The elastic properties of an anisotropic crystal can be fully described using elastic stiffness coefficients, Òij, or elastic compliance coefficients, sij. They are fourth-
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The stiffness coefficients Òij can be determined from ultrasound velocities. The bulk- and surface-acousticwave velocities were measured in [11, 12]. The stiffness coefficients Òij can be determined from sound velocities using a set of relations, which is in part a complex for off-diagonal (i j) Òij Ò coefficients. For some waves (in this case, only [X/X] and [–45°/+45°] waves), the piezoelectric effect was considered earlier. The following relations were used [11]:
a, Å (‡)
8.1700
6 7 8.1675
5
4 3 2
8
8.1650
c 44 = (2f2b)2ρ, where f2 is the resonance frequency of shear vibrations along the Y-cut, and b is the sample width; 1 E E -, s 11 = s 22 = --------------2 2 4 f 1l ρ where f1 is the resonance frequency of longitudinal vibrations of the ï-cut, and l is the sample length. E
1 8.1625 c, Å 5.0990
(b)
5.0985
6
5.0980
E
4
5.0975
5
5.0970
7
2 3
5.0965
8
5.0960 5.0955
1
5.0950 1360
1370
1380
1390
1400 å, kHz mm
s 14 can be found from the difference between the tension–compression frequencies along the length of ï-cuts rotated about the ï axis through –30° and +30°. s 11 – s 12 E E Since c 44 = --------------------------------------------and c 66 = 2 s 44 ( s 11 – s 12 ) – 2s 14 s 44 E E ------------------------------------------------ , one determines s44, s 12 , and s 13 . 2 2s 44 ( s 11 – s 12 ) – 2s 14 Using the following relations, one can determine the piezoelectric moduli d11 and d14 and the piezoelectric coefficients e11 and e14:
Fig. 1. Unit-cell parameters (a) a and (b) c vs. frequency coefficient.
E
T
rank tensors inversely related to one another. In crystals with the langasite structure (class 32), only six Òij coefficients are independent. They are commonly designated Ò11, Ò12, Ò13, Ò14, Ò33, and Ò44. The stiffness tensor has 18 independent components, which can be represented in the form of a symmetrical 6 × 6 matrix [11]: 0
c 12 c 11 c 13 – c 14 0
0
c 13 c 13 c 33 0
0
0
c 14 – c 14 0 c 44 0
0
0
0
0
0 c 44 c 14
0
0
0
0 c 14 c 66
c66 = (Ò11 – Ò12)/2. ∆ E The inverse elastic matrix is given by s ij = (–1)i + j -----ijc- , ∆ c where ∆Ò and ∆ ij are, respectively, the determinant and minor Òij of the elastic stiffness matrix. c
1/2
d 14 = 2k 23 ( s 33 ε 11 ) , E
c 11 c 12 c 13 c 14 0
1/2
d 11 = k 12 ( s 11 ε 11 ) , T
D 1/2
e 11 = d 11 ( c 11 – c 12 ) + d 14 c 14 = k 26 ( c 66 ε 11 ) , E
E
E
E
D
E
e 14 = 2d 11 c 14 + d 14 c 44 , where ε11 is dielectric permittivity. Figures 1 and 2 plot the frequency coefficient å D (kHz mm) and elastic stiffness coefficient C 66 , (GPa), respectively, versus the ‡ and Ò parameters of the powder samples. All of the plots are seen to be parabolic. Kaminskii et al. [12] reported å = 1382 kHz mm, and Sakharov et al. [13] obtained 1380 kHz mm, which demonstrates that this parameter depends on growth conditions. Figure 3 demonstrates that the frequency coefficient is a linear function of 1 ⁄ ( ρ ) . Consequently, M increases with decreasing density. Thus, the present results on the dependences of the frequency coefficient and elastic stiffness coefficient on the unit-cell parameters and, accordingly, on composition are consistent with earlier data on the dependences of the physical properties (piezoelectric moduli [14–16] INORGANIC MATERIALS
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8.1650
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and dielectric permittivity [14–16]) of langasite-type compounds on their composition (and, hence, on their unit-cell parameters) and the influence of growth defects (central core, striations, inclusions, cracks, and twins) on the propagation of surface acoustic waves [11, 17], structural perfection (optical homogeneity, lack of cracks, twins, and inclusions) on their acoustic properties [11, 18–20], and point defects (oxygen vacancies) on their electrical conductivity and dielectric properties (dielectric permittivity and dielectric loss tangent) [8, 21].
1
REFERENCES
8.1625 c, Å 5.0990
(b)
5.0985
6
5.0980
4
5.0975
5 2 3
5.0970 5.0965
7 8
5.0960 1
5.0955 5.0950 5.0945 42.050
42.775
43.500
44.225
44.950 C D66, GPa
Fig. 2. Unit-cell parameters (a) a and (b) c vs. elastic stiffness coefficient.
M, kHz mm 1405 8 56 7
1400 1395 1390
4
1385 1380
2
3
1375 1370 1365 1 1360 0.4166 0.4168 0.4170 0.4172 0.4174 0.4176 1 ⁄ ( ρ calc ) Fig. 3. Frequency coefficient vs. 1 ⁄ ( ρ calc ) . INORGANIC MATERIALS
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