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DOI: 10.17516/1997-1397-2022-15-1-80-87 УДК 538.9
Anisotropy of the Electromechanical Characteristics of SH-Waves and Lamb Waves in Yttrium Aluminum Borate Single Crystals
Pavel P. Turchin*
Siberian Federal University Krasnoyarsk, Russian Federation Kirensky Institute of Physics Federal Research Center KSC SB RAS Krasnoyarsk, Russian Federation
Sergey I. Burkov* Vladimir I. Turchin* Oleg N. Pletnev Marina Yu. Chulkova Anastasia G. Nechepuryshina
Siberian Federal University Krasnoyarsk, Russian Federation
Received 10.04.2021, received in revised form 10.06.2021, accepted 20.08.2021
Abstract. The anisotropy of the electromechanical properties of SH-waves and Lamb waves in yttrium aluminum borates, which are nonmagnetic representatives of the RMe3(BO3)4 single crystals family (where R=Y, La-Lu; M=Fe, Al, Cr, Ga, Sc) with unique properties of magnetoelectrics and multiferroics, has been studied. In the process of the numerical simulation of the acoustic waves characteristics, the values of linear electromechanical constants of YAl3(BO3)4 single crystals, previously measured by ultrasonic pulse echo and quasi-static methods, have been used.
Keywords: surface acoustic, SH and Lamb waves, piezoelectrics, multiferroics, yttrium aluminum borates.
Citation: P.P. Turchin, S.I. Burkov, V.I. Turchin, O.N. Pletnev, M.Yu. Chulkova, A.G. Nechepuryshina, Anisotropy of the Electromechanical Characteristics of SH-Waves and Lamb Waves in Yttrium Aluminum Borate Single Crystals, J. Sib. Fed. Univ. Math. Phys., 2022, 15(1), 80-87. DOI: 10.17516/1997-1397-2022-15-1-80-87.
Introduction
Single crystals of the RMe3(BO3)4 trigonal rare-earth oxyborates family (where R=Y, La-Lu; M=Fe, Al, Cr, Ga, Sc), depending on the composition and thermodynamic conditions, have piezoelectric, magnetoelectric and multiferroic properties [1-3]. Since giant magnetoelectric [4] and magnetodielectric [5] effects were discovered in (RFe3(BO3)4) ferroborates, aluminum borates RAl3(BO3)4 are promising for applications in laser technology [6-9]. Recently, there has been a growing interest in the study of the macroscopic physical properties of these crystals, primarily
* pturchin@sfu-kras.ru
t https://orcid.org/0000-0001-5198-2145
* https://orcid.org/0000-0001-5584-4794
© Siberian Federal University. All rights reserved
for the study of microscopic magneto-elastic-electric interactions in them, as well as for expanding their applications in functional electronics [4,5,10-15].
Yttrium aluminum borate YAl3(BO3)4 (point symmetry 32) in the series of oxyborates is a nonmagnetic single crystal and characterises the anisotropy of the elastic-electric interaction in them. Previously, we obtained experimental values of the electromechanical constants of this single crystal by echo-pulse ultrasonic [16] and quasi-static [16,17] methods. This also gave an opportunity to study the anisotropy of the main characteristics of bulk (BAW) and surface (SAW) acoustic waves in yttrium aluminum borates. The present article studies the dispersion of the electromechanical characteristics of SH-waves and Lamb waves and assesses the surface effect on the magnitude of the electromechanical interaction in yttrium aluminum borates. Numerical simulation is based on the experimental values of electromechanical constants [16,17].
1. Theory and values of material constants of a single crystal
SH-waves and Lamb waves propagation in a piezoelectric crystal plate is considered in the operating orthogonal coordinate system, in which the X3 axis is directed along the outer normal to the surface of the medium occupying the space X3 < h and X3 > 0, and the Xi axis coincides with the direction of wave propagation. For the waves of small amplitude, the wave equation, electrostatic equations, and equations of state of the piezoelectric medium have the following form [18]:
PoUi = Tik,k, Dm,m = 0.
Tik Cikpq epq enik En: (1)
Dn enik eik + enm Em.
Equation (1) contains the following notations: p0 is the crystal density, Ui is elastic displacements vector, ej and Tj are tensors of infinitesimal mechanical deformations and stresses, Ei and Di are vectors of electric field strength and induction, Cijki, eijk and ej are tensors of elastic, piezoelectric and dielectric constants. Hereinafter, summation over a twice repeating index is meant.
Acoustic waves propagation in a piezoelectric plate of h thickness must correspond to the boundary conditions [19] of equality to zero of the normal stress tensor components at the crystal-vacuum interface. The continuity of the tangential to the interface components of the electric field strength vector is provided by the condition of continuity of the electric potential y, as well as by the condition of continuity of normal components of the induction vector [20].
The detailed description of experimental studies of the values of elastic, piezoelectric, and dielectric constants is given in [16]. The elastic constants were found by solving the inverse problem of crystal acoustics [21] from the measured values of the BAW velocities in the basic and rotated crystallographic directions. To determine BAW velocities, an echo-pulse ultrasonic acoustic method, with an accuracy of 10~4 for absolute measurements, has been used. The measurements were performed at a frequency of 28 MHz. The absolute values of the piezoelectric constants have also been found from the measured values of the BAW velocities, but for piezoactive acoustic modes [16]. To clarify the values of the piezoelectric constants, quasi-static measurements of the dijk piezomodules, which are related with eijk by the equation [22] have been carried out
eijk = dilmClmjk . (2)
In this method, a DMA 242 C device was used to create a precision variable dynamic loading. The signs of the piezoelectric constants were found by the direct measurement of the piezoelectric effect in the crystallophysical coordinate system, where C14 < 0. The high-frequency dielectric
permittivity is determined from the low-frequency one, taking into account the piezoelectric contribution.
Table 1. Values of electromechanical constants of YAl3 (BO3)4 single crystals at room temperature
Elastic constants C\M, 1010 N/m2
C11 C12 C13 C14 C33 C44 C66
40.47±0.05 21.14±0.05 9.75±0.05 -2.35±0.1 27.09±0.05 7.49±0.01 9.67±0.05
Piezoelectric constants e^?, C/m2, dj?, 10 12C/N Dielectric constants e^/e0
eii e14 du d14 FT> e11 e33
-1.06±0.07 -0.27±0.04 -6.0±0.3 -7.2±0.4 11.7±0.1 11.1±0.1
The experimental values of the linear electromechanical constants for YAl3(BO3)4 single crystals are given in the Tab. 1.
2. Anisotropy of velocities and electromechanical coupling coefficients of SH-waves and Lamb waves
Lamb and SH-waves velocity values were found by solving equations (1) subject to boundary conditions [19,20], taking into consideration the material constants' values given in the Table. The values of the electromechanical coupling constant K2 was calculated according to the formula:
K2 = 2(v - Vm), (3)
v
where vm is the phase velocity on the metallized surface, and in the case of Lamb waves of both metallized surfaces.
Fig. 1 shows the dispersion dependences of the phase velocities and K2 of the Lamb and SH-waves in Z and Y-cuts in the direction [100] and X-cuts in the direction [001] of the elastic wave propagation. The range of the considered values of hxf (thicknessxfrequency) is from 0 to 18000 m/s. The range of the phase velocities variation of Lamb and SH-waves is from the value of the phase velocity of the quasi-longitudinal BAW 10573 m/s to 3959 m/s SAW in Z and Y-cuts. But in X-cut, the range of Lamb and SH-waves phase velocity variations is from 8533 m/s (QL) to 4228 m/s (SAW). In this case, all elastic wave modes are piezoactive. The maximum K2 value for the fundamental mode is achieved for the So mode of the Lamb wave in Z and Y-cuts in hxf range from 250 m/s to 3000 m/s and K2 = 0.09 for the Y-cut when hxf = 2250 m/s, but in Z-cut K2=0.026 (Fig. 1d,e). It should be noted that in Y-cut for all elastic wave modes K2 values are almost three times higher than the corresponding values in Z-cut. In X-cut, the maximum value for the S0 mode is K2= 0.003 when hxf = 4450 m/s. The situation with the antisymmetric mode A0 is similar. The maximum K2 = 0.042 (Fig. 1f) in X-cut is achieved for the SH0 mode when hxf = 1650 m/s. It is necessary to note that the obtained K2 values are significantly higher than those obtained in langasite single crystals [23].
(a) (b) (c)
0.040i
Fig. 1. Dispersion dependences of phase velocities and K2 of Lamb and SH-waves: a, d) Z-cut; b, e) Y-cut; c, f) X-cut
One of the features of Lamb waves propagation in these cuts is the presence of interaction (hybridization) regions [24] between the Lamb wave modes, in which a significant change in the K2 values of the interacting modes of the elastic wave takes place. For example, in the Z-cut between Si and SH2 modes when hxf = 6050 m/s or S2 and SH3 when hxf = 9650 m/s. The peculiarity of the phase velocities in the interaction region of elastic wave modes is reflected in the tab in Fig. 1a. Hybridization is manifested more significantly in X-cut, where interaction occurs only between the So and A1 modes when hxf = 7050 m/s and 9750 m/s. Moreover, K2 changes of the S0 mode in the hybridization region occur from K2 = 0.03 to K2 = 0.006. However, it should be noted that in Y-cut there is an interaction only between the S0 and SH0 modes, where the K2 change is insignificant.
Fig. 2 shows the anisotropy of phase velocities and K2 in Z-cut of the YAl3(BO3)4 crystal of Lamb and SH-waves at values hxf = 1050, 6050, and 10650 m/s. Along with the fundamental modes of the Lamb wave, Fig. 2a, d) demonstrates phase velocities and the K2 of nondispersive acoustic BAW and SAW. The maximum values K2 = 0.104 are achieved for the fast shear wave QFS at an angle of 30° with the direction [100]. The maximum value K2 = 0.07 is also achieved for the SH0 mode in the same direction. For quasi-longitudinal BAW, as well as for the S0 mode, the maximum K2 value in the direction of the elastic wave propagation [100] is K2=0.03 and K2 = 0.024, correspondingly. It should be noted that the minimum K2 values in the YAl3(BO3)4 plate are for SAW, which does not exceed the value K2 = 0.0013.
With an increase in the hx f values, the K2 value of the Lamb wave modes decreases
>
So
QFS
" - — _
SH„
QSS^. - • —• -
SAW
Ao
>
1QL A,
'—
—- — — SH,
S QFS 1 s, ---
s„ ..................-T ~ SH
—
A« QSS
20 (p° 40
60
20 (po 40
60
(a)
(b)
(c)
Fig. 2. Anisotropy of phase velocities and K2 of Lamb waves in Z-cut 0°, 0°) of the YAB crystal plate at different values of thicknessxfrequency a, d) hxf= 1050 m/s; b,e) hxf= 6050 m/s; c, f) hxf = 10650 m/s
(Fig. 2e, f). Moreover, the maximum K2 values are for SH-waves and antisymmetric Ai. For example, the K2 values of the elastic wave modes A1, SH0 and SH1 at an angle of 30° with the direction [100] are 0.008, 0.018, 0.02 (Fig. 2e) when hxf = 6050 m/s, correspondingly. However, when hxf = 10650 m/s K2 = 0.014 for the SH0 mode (Fig. 2f).
Fig. 3 shows the anisotropy of phase velocities and K2 of Lamb and SH-waves in Y-cut of the YAl3(BO3)4 crystal when hxf =1050, 6050, and 10650 m/s. In Y-cut, all BAWs are also piezoactive, but the maximum ECC value is achieved for a longitudinal wave QL in the direction [100] K2=0.03. For non-dispersive SAW, the maximum ECC value K2=0.011 is achieved in the direction of an elastic wave propagation at an angle of 28° with the axis [100]. For the fundamental modes of the Lamb wave, the maximum ECC values when hxf = 1050 m/s for the elastic wave modes S0, A0 and SH0 are 0.074, 0.026, and 0.004, correspondingly (Fig. 3d). It should be noted that the maximum K2 values are achieved in Y-cut.
As hxf increases, the ECC values decrease for all elastic wave modes. However, in Y-cut, an interaction between the elastic wave modes also arises, and leads to a sharp change in the K2 values of interacting modes in the hybridization region (Fig. 3e, f). For example, between the elastic wave modes SH0 and S1 when hxf = 6050 m/s or SH0 and A1, SH2 and S1 when hxf = 10650 m/s (Fig. 3e, f). It should also be noted that in the direction [001] of acoustic wave propagation, the maximum K2 value when hx f =6050 m/s is achieved for the elastic wave mode SH1 K2=0.034, while the other modes have close to zero K2 values.
So
QFS
" - —
SH0
QSS^ . ^ •
SAW
A„
11 10
I/3 £
A,
"SH.
SH! QFS 1 s, ~ - -
~0 ..................~ — * SH
---— * — -__
A0 QSS
20 <po 40
60
20 9o 40
60
(a)
(b)
(c)
Fig. 3. Anisotropy of phase velocities and K2 of Lamb waves in Y-cut (0°, 90°, of the YAB crystal plate at different values of thicknessxfrequency a, d) hxf = 1050 m/s; b, e) hxf = 6050 m/s; c, f) hxf = 10650 m/s
Conclusion
In this study, the values of elastic and piezoelectric material constants obtained by acoustic and quasi-static measurements are used to study the electromechanical characteristics of acoustic waves: bulk, surface, SH-waves, and Lamb waves in the plates of yttrium aluminum borate single crystals. The range of changes in phase velocities (from 800 m/s to 11800 m/s) and the maximum value of the electromechanical coupling coefficient (K2 = 0.14 for the So mode at Euler angles of 0°, 90°, 0°) have been established. The propagation directions and hxf values for which hybridization of acoustic waves is observed have been found in the cuts under study. The phase velocities of Lamb and SH-waves, as well as K2 waves significantly exceed the similar ones in langasite single crystals [23].
The study was carried out within the framework of the state assignment of the Ministry of Science and Higher Education of the Russian Federation (research project code FSRZ-2020-0011).
References
[1] D.Khomskii, Trend: Classifying multiferroics: Mechanisms and effects, Physics, 2(2009), no. 20. DOI: 10.1103/Physics.2.20
[2] R.Ramesh, N.A.Spaldin, Multiferroics: progress and prospects in thin films, Nanoscience and Technology: A Collection of Reviews from Nature Journals, 2010, 20-28.
DOI: 10.1142/9789814287005_0003
[3] N.A.Spaldin, M.Fiebig, The renaissance of magnetoelectric multiferroics, Science, 309(2005), no. 5733, 391-392. DOI: 10.1126/science.1113357
[4] K.C.Liang et al, Giant magnetoelectric effect in HoAl3(BO3)4, Physical Review B, 83(2011), no. 18, 180417. DOI: 10.1103/PhysRevB.83.180417
[5] A.A.Mukhin et al,. The giant magnetodielectric effect in the multiferroic SmFe3(BO3)4, Letters to the Journal of Experimental and Theoretical Physics, 93(2011), no. 5, 305-311. DOI: 10.1134/S0021364011050079
[6] K.N.Girbachenya et al., High-frequency Er3+, Yb3+: YAl3(BO3)4 microchip laser with longitudinal diode pumping, Devices and methods of measurement, 2(2012), no. 5, 79-81.
[7] A.S.Aleksandrovsky et al., Upconversion luminescence of YAl3(BO3)4:(Yb3+, Tm3+) crystals, Journal of Alloys and Compounds, 496(2010), no. 1-2, 18-21.
DOI: 10.1016/j.jallcom.2010.02.089
[8] G.Wang et al, Cr3-doped borates-potential tunable laser crystals? Radiation Effects and Defects in Solids, 136(1995), no. 1-4, 43-46. DOI: 10.1080/10420159508218789
[9] L.Zheng et al., >1 MW Peak Power at 266 nm in Nonlinear YAl3(BO3)4 (YAB) Single Crystal, In Lasers and Electro-Optics (CLEO), 2015 Conference on. IEEE, 2015.
DOI: 10.1364/CLEO_AT.2015.JTu5A.33
10] N.V.Volkov et al., Magnetization, magnetoelectric polarization and specific heat of HoGa3(BO3)4, JETP Letters, 99(2014), no. 2, 67-75. DOI: 10.1134/S0021364014020106
11] A.I.Popov, D.I.Plokhov, A.K.Zvezdin, Quantum theory of magnetoelectricity in rare-earth multiferroics: Nd, Sm, and Eu ferroborates, Physical Review B., 87(2013), no. 2, 024413. DOI: 10.1103/PhysRevB.87.024413
12] A.K.Zvezdin et al., On magnetoelectric effects in gadolinium ferroborat GdFe3(BO3)4, JETP Letters, 81(2005), no. 6, 272-276. DOI: 10.1134/1.1931014
13] T.N.Gaydamak et al., Elastic and piezoelectric moduli of Nd and Sm ferroborates, Low Temperature Physics, 41(2015), no. 8, 614-618. DOI: 10.1063/1.4929719
14] V.I.Zinenko et al., Vibrational spectra, elastic, piezoelectric and magnetoelectric properties of HoFe3(BO3)4 and HoA13(BO3)4 crystals, Journal of Experimental and Theoretical Physics, 117(2013), no. 6, 1032-1041. DOI: 10.1134/S1063776113140203
15] G.A.Vziagina et al., Magnetoelastic effects in terbium ferroborate, Low Temperature Physics, 34(2008), no. 11, 901-908. DOI: 10.1063/1.3009584
16] P.P.Turchin et al., Electromechanical Properties and Anisotropy of Acoustic Waves Characteristics in Single Crystals YAl3(BO3)4, Journal of Siberian Federal University. Mathematics & Physics, 12(2019), no. 6, 756-771. DOI: 10.17516/1997-1397-2019-12-6-756-771
17] P.P.Turchin et al., Application of DMA 242 C for Quasi-Static Measurements of Piezoelectric Properties of Solids, Journal of Siberian Federal University. Mathematics & Physics, 13(2020), no. 1, 97-103. DOI: 10.17516/1997-1397-2020-13-1-97-103
[18] D.Royer, E.Dieulesaint, Elastic waves in solids II: generation, acousto-optic interaction, applications, Springer Science & Business Medial, 1999.
[19] I.A.Viktorov, Rayleigh andLamb Waves, Springer US, 1967.
[20] J.Rajagopalan, K.Balasubramaniam, C.V.Krishnamurthy, A phase reconstruction algorithm for Lamb wave based structural health monitoring of anisotropic multilayered composite plates, The Journal of the Acoustical Society of America, 119(2006), no. 2, 872-878. DOI: 10.1121/1.2149775
[21] K.S.Aleksandrov, G.T.Prodaivoda, Anisotropy of the elastic properties of minerals and rocks, Novosibirsk, Publishing house of the SB RAS, 2000.
[22] J.F.Nye et al., Physical properties of crystals: their representation by tensors and matrices, Oxford university press, 1985.
[23] S.I.Burkov, O.P.Zolotova, B.P.Sorokin, P P.Turchin, Calculation of Thermostable Directions and the Effect of External Electric Field on the Propagation of Lamb and SH Waves in a Langasite_Crystal Plate, Acoustical Physics, 58(2012), no. 6, 650-657.
DOI: 10.1134/S1063771012050065
[24] I.E.Kuznetsova, B.D.Zaytsev, A.A.Teplykh, I.A.Borodina, Hybridization of acoustic waves in piezoelectric plates, Acoustical Physics, 53(2007), no. 1, 64-69.
DOI: 10.1134/S1063771007010071
Анизотропия и электромеханические характеристики SH-волн и волн Лэмба в монокристаллах иттриевого алюмобората
Павел П. Турчин
Сибирский федеральный университет Красноярск, Российская Федерация
Сергей И. Бурков Владимир И. Турчин Олег Н. Плетнев Марина Ю. Чулкова Анастасия Г. Нечепурышина
Сибирский федеральный университет Красноярск, Российская Федерация
Аннотация. Исследована анизотропия электромеханичеких характеристик ЯИ-волн и волн Лэмба в иттриевых алюмоборатах, которые являются немагнитным представителем семейства монокристаллов И,Ме3(БО3)4 (где И,=У, Ьа-Ьи; М=Ре, А1, Сг, Оа, Яс) с уникальными свойствами магнито-электриков и мультиферроиков. При численном моделировании характеристик акустических волн использованы значения линейных электромеханических постоянных монокристаллов УА13(БО3)4, измеренных ранее ультразвуковым эхо-импульсным и квазистатическим методами.
Ключевые слова: поверхностные акустические, ЯИ- и Лэмба волны, пьезоэлектрики, мультифер-роики, алюмоборат иттрия.
