INflUENCE OF UNIAXIAL PRESSURE ON THE CHARACTERISTICS OF LAMB AND SH-WAVE PROPAGATION IN LINBO3 CRYSTALLINE PLATES Текст научной статьи по специальности «Физика»

DOI: 10.17516/1997-1397-2021-14-1-105-116 УДК 548.0:534+004.942

Influence of Uniaxial Pressure on the Characteristics of Lamb and SH-wave Propagation in LiNbO3 Crystalline Plates

Sergey I. Burkov* Oleg N. Pletnev

Siberian Federal University Krasnoyarsk, Russian Federation

Pavel P. Turchin^

Siberian Federal University Krasnoyarsk, Russian Federation Kirensky Institute of Physics, Federal Research Center KSC SB RAS

Krasnoyarsk, Russian Federation

Olga P. Zolotova*

Reshetnev Siberian State University of Science and Technology

Krasnoyarsk, Russian Federation

Boris P. Sorokin§

Technological Institute for Superhard and Novel Carbon Materials

Moscow, Troitsk, Russian Federation

Received 12.09.2020, received in revised form 22.10.2020, accepted 15.11.2020 Abstract. Theoretical study of uniaxial pressure influence on the propagation characteristics of Lamb and SH-waves in lithium niobate plate is carried out. Electromechanical coupling coefficients and controlling coefficients of the pressure influence on phase velocity are calculated in various directions. Transformation and hybridization of acoustic modes upon a pressure influence have been derived in details. PACS: 43.25.Fe; 43.35.Cg; 77.65.-j

Keywords: piezoelectric plate, Lamb wave, SH-wave, uniform pressure influence, computer simulation. Citation: S.I. Burkov, O.N. Pletnev, P.P. Turchin, O.P. Zolotova, B.P. Sorokin, Influence of Uniaxial Pressure on the Characteristics of Lamb and SH Wave Propagation in LiNbO3 Crystalline Plates, J. Sib. Fed. Univ. Math. Phys., 2021, 14(1), 105-116. DOI: 10.17516/1997-1397-2021-14-1-105-116.

Introduction

Studies of the propagation of the normal elastic waves in piezoelectric plates known as Lamb wave (LW) modes have been carried out for a quite some time [1,2]. Note, that in practice the Lamb waves have some advantage in comparison with Rayleigh surface acoustic waves (SAW) owing to a smaller acoustic attenuation at microwave band and keeping a convenient possibility

* sburkov@sfu-kras.ru

tpturchin@sfu-kras.ru

izolotova@sibsau.ru https://orcid.org/0000-0003-2927-4278

§ bpsorokin1953@yandex.ru © Siberian Federal University. All rights reserved

of excitation by an interdigital transducer (IDT). The influence of external static impacts on LW characteristics in the piezoelectric plates of PLZT ceramics and Y-cut of the lithium niobate crystal (LNC) was studied in [3,4], where the prospect of LW application in such devices as sensors of an electric field E as well as the E-controlled delay lines was studied. It was noted, that the sensitivity of LW based devices was the better than that based on Rayleigh wave propagation.

The E-influence on the characteristics of symmetric and antisymmetric Lamb waves in piezoelectric plates was theoretically studied in [5,6]. However, this study was carried out in the framework of a perturbation theory without taking into account the nonlinear material tensors of a crystal. An experimental and theoretical study of the E-influence on the quasishear horizontal (QSH) wave propagated on the thin LNC plate of X, Y, and Z-cuts was performed in [7,8], where promising directions of an elastic wave propagation in creating signal processing devices were obtained.

A detailed study of the E-influence on the characteristics of the fundamental (zero) LW mode in thin LNC plates was performed in [9,10], where a strong dispersion dependence of LW phase velocity vs an electric field was marked. It was noted, that in this case an E-influence on the phase velocity has a substantial dependence vs the frequency, thus such dependence for some directions of the wave propagation could be changed from the linear to the quadratic one by varying the frequency [11].

Based on the LNC plate the high-quality acoustic resonator tuned by an electric field was demonstrated in [12]. General theory of the SAW propagation in a piezoelectric crystal as well as the reflection and refraction of elastic waves on the interface between two elastic media subjected to the pressure, was considered in [13,14]. The E-influence on LW propagation in piezoelectric plates taking as an example of silicosillenite piezoelectric crystals was considered in detail in [15,16].

Changes in the material properties of a crystal under an influence of uniform mechanical pressure studied in [17,18], were used in stabilizing the resonance frequency and, in particular, compensating the frequency drift of the acoustic resonators based on langasite and quartz crystals [19,20]. The influence of an initial stress on Love wave propagation in the piezoelectric layered structures (PLS) as "piezoelectric layer/isotropic substrate" and "isotropic layer/piezoelectric substrate" has been studied in a number of works [21,22].

The aim of the present paper is concerned with a numerical analysis of an influence of uniaxial mechanical pressure on the characteristics of symmetric and antisymmetric Lamb waves as well as SH-waves taking the LNC plates as an example and including into consideration the complete set of the LNC non-linear electromechanical constants.

1. Theory of elastic wave propagation in piezoelectric plates under uniform pressure influence

Starting equation of motion for small amplitude waves, as well as the equations of electrostatics and the state equations of the piezoelectric medium subjected the influence of uniform mechanical pressure have obtained earlier [23] as

pqua = tab,b + ua,pqtpq; d m,m = (1)

TAB = CA BCD VCD - e*MAB EM; D M = £*MN EN + eMAB VAB ■

In Eq. (1) the following notations are introduced: p0 is the density of a crystal in the unde-

formed (initial) state, UA is the vector of dynamical elastic displacements, tab is the tensor of a thermodynamic stress, DM is the electric displacement field vector, Tpq = —TPPPq is the static uniaxial stress tensor, Pq is the unit vector of a pressure force, and ncD is the strain tensor. It was assumed that the compression stress should have a negative sign. Here and after the time-dependent variables are marked by "tilde" symbol. The comma after the subscript denotes a spatial derivative, and coordinate Latin indices vary from 1 to 3. Here and further, the rule of summation over repeated indices will be used.

Effective elastic, piezoelectric and dielectric constants in the approximation of a linear dependence on the magnitude of static mechanical stress T were defined in [24] as:

CABKL = CABKL — CABKLQRSQRMN PM PN t;

eNAB = eNAB — GnabklSklmn Pm Pn T; (2)

£mn = £mn — HNMAB SABKLPKPLT ■

Here Cabkl, eNAB, and £nMN are the second-order elastic, piezoelectric and dielectric constants, respectively; SMBKL is the tensor of elastic compliances; Cmbklqr, eNABKL, and HNMAB are the third-order elastic, nonlinear piezoelectric and electrostrictive material tensors, respectively.

Substituting into Eq. (1) the solutions for the elastic displacements and electric potential in a conventional form of the plane monochromatic waves, one can obtain the linearized Green-Christoffel equations as:

|Tbc (T) — P0U25bc] ac = 0;

rBC = \p*ABCD + (2CMBFN Sadcf + 5bc 5am 5dn) Pm Pn t] kAkD;

(3)

rC4 = epAC kP kA; r4C = rC4 + 2eAFD SMNC F PM PN TkAkD;

T44 = —£pQkp kQ■

Here kA is an acoustic wave vector.

Let the X3 axis of an operational coordinate system is directed along the outer normal to the surface of the plate, occupying the 0 < X3 < h space, and the Xi axis coincides with the wave propagation direction. Here h is the plate thickness.

The propagation of an elastic wave in a piezoelectric layered structure under a uniform pressure must satisfy to the corresponding boundary conditions. The boundary condition for the normal components of a stress tensor is that their equality to zero of the layer should be fulfilled on the free surface. The continuity of the tangential components of an electric field is ensured by the conditions on the continuity of the electric potential p as well as the normal component of an electrical displacement vector at a crystal-vacuum interface. All such conditions can be written as:

T3A = 0\x3=h ; D3 = Dvac\x3=h; = v\x3=h ;

3 3 3 (4)

T3A = 0\x3=0 ; D3 = Dvac\X3 = 0 ; P3 = p|x3 = 0 ■

In the case of mechanical stress application orthogonally the free surface (P || X3) the elastic properties of a loading medium must be taken into account. Assuming that a uniaxial stress in such geometry occurs without a hard contact with a free surface specimen, for example, by means of gaseous static equipment, the mechanical boundary conditions can be written in the form [24]:

T3J + UjkT3K =0 (X3 = h). (5)

Substituting the solutions in a conventional form of plane homogeneous waves into the Eq. (4), one can obtain a system of equations for calculating the propagation parameters of acoustic waves in a piezoelectric plate as:

(6)

[an(e*AB +2sabkpe3ABpkPpt) k^aA + (e*kkR - ieo) a^ exp(ik3n)h) = 0;

n=1 8

E [an CB3KL + 2sEpmncAAbklpmpnt) k^apn - ep3Bk^a^] exp(ik3n)h) = 0;

n=1 8

E( fi * I ocA f~<A U V 1 ( n) ( n) * 1 ( n) ( n) _ „

an \CB3KL + 2SKPMNc3BKLPMPNt) kL aP - eP3BkP a4 = 0;

n=1 8

e an (e*AB + 2sabkp e3AB pk pp t) k^a^ + (e*N kN + i^o^ a4n) = 0.

n=1

Here aRl), an, and k^ designate the amplitudes, weight coefficients and wave numbers of the nth partial wave (n = 1, ..., 8) in the crystalline plate, respectively.

In Eqs. (6), all the changes in the configuration of an anisotropic continuous medium associated with its static deformation and, in particular, with changes in the shape of a specimen (so-called geometric non-linearity), as well as the changes in the material constants (see Eq. (2)) of a piezoelectric crystal (so-called physical non-linearity) under the influence of hard mechanical stress have been taken into account. Final system of boundary conditions has 8 homogeneous equations relatively the unknown weight coefficients an. The calculation of elastic wave parameters was carried out using a conventional method of partial waves [23]. The equality to zero of the determinant of boundary condition matrix was used to obtain a required Lamb wave phase velocity.

2. Dispersion dependences of Lamb and SH-wave propagation parameters

Based on Eqs. (1)-(6), a computer calculation of such parameters of Lamb and SH-waves as phase velocities, electromechanical coupling coefficients (EMCCs), controlling coefficients of phase velocities for LNC plate under a lot of uniform mechanical stress was carried out. Electromechanical coupling coefficients were calculated by a conventional relation:

K2 = 2(v - Vm), (7) v

where the phase velocities v and vm should be defined at free or shorted surface boundary conditions, respectively. Controlling coefficients of the phase velocity upon application of an uniaxial pressure were taken in the form:

1 ( Av \

ap=vMUpJ Ap ^ (8)

Here the value P is the pressure magnitude. Data on the linear and nonlinear electromechanical properties for the LiNbO3 crystal were taken from [25].

Fig. 1 represents the dispersion dependences of the phase velocities, EMCCs, and ap controlling coefficients in the [100] direction of a wave propagation for the (001) plate orientation and 3 kinds of an uniaxial pressure application along the [100], [010], and [001] directions. The

range of the hf product considered was varied from 0 up to 16000 m/s. Analyizing Fig. 1 (a) one can see that the phase velocity curves of the symmetric S0 and antisymmetric A0 modes tend to Rayleigh wave phase velocity at an elevated frequency as well as a large-scale plate thickness. At the same conditions, the dispersive phase velocity of the SH0 mode tends to an appropriate value of the bulk acoustic wave (BAW) of quasishear type which has a phase velocity with comparatively slow value (so-called QSS wave). Fig. 1 (b) shows that all the considered modes have non-zero electromechanical coupling, but the maximum value for the fundamental (zero) modes is achieved for the SH0 mode as K2 = 0.15 for comparatively thin plates at hf = 250 m/s, while for the symmetric mode So, the EMCC values have a much lower magnitude as K2 = 0.014, which is consistent with work [26]. It should be noted that in the LNC plates the hybridization of elastic modes [27] occurs mainly at the intersection points of dispersion curves. However, when one of free surface has a metallization, the dispersion curves break apart and the mode types will change [15,28]. But a similar effect doesn't occur upon a metallization of both free surfaces of a plate. The first modes of Lamb and SH1 waves have large EMCC values (Fig. 1 (b)) in comparison with the zero modes.

The maximum value of the controlling coefficient aP = 2.13 • 10-11 Pa-1 at hf = 2200 m/s for LW fundamental modes is achieved for the S0 mode when P || [100], i.e. along the propagation direction (Fig. 1 (c)). But the maximal aP value of the A0 mode is somewhat higher. So, for hf = 6050 m/s values it becomes equal to 2.37 • 10-11 Pa-1. But at hf >8000 m/s the aP coefficient becomes equal to 2.25 • 10-11 Pa-1 coinciding with the similar value calculated for non-dispersive Rayleigh wave. This can be explained by the transformation of the S0 mode into Rayleigh type at a higher frequency. The aP dispersion dependence of the SH0 mode increases monotonously when hf > 6500 m/s, and tends to the appropriate value observed for the QSS BAW as 3.01 • 10-11 Pa-1. The range of the changes in the aP values observed for higher-order elastic Lamb modes is significantly larger. For example, this fact can be quite seen for the S1 symmetrical mode where the maximal 3.02 • 10-11 and minimal 6.9 • 10-12 Pa-1 aP values are realized at hf = 3450 and 15450 m/s, respectively.

In the case of P || [010], i.e. orthogonally to the sagittal plane and propagation direction, the maximal value of aP = 1.56 • 10-11 Pa-1 at hf = 2800 m/s is achieved for the S0 mode in the vicinity of the S0 and SH1 hybridization (Fig. 1 (d)). Hybridization effect which consists in the existence of coupled modes and energy exchange under the conditions of space-time synchronism, is displayed on the behavior of the aP coefficients, which in this case can serve as a qualitative parameter of a hybridization level. In particular, the P application along the direction [010] increases the interaction area of the S0 and SH1 modes as in the case of a slight change in the propagation direction of the elastic wave as shown in [27]. For the SH0 mode, the maximal value of aP = 2.66 • 10-11 Pa-1 at hf = 5600 m/s. The maximal values of aP = 2.27 • 10-11 and 4.51 • 10-11 Pa-1 for the S1 and S2 modes are realized at hf = 15900 and 4100 m/s, respectively. The dispersive dependences of the aP coefficients of SH1 and SH2 modes are distinguished sufficiently, because when the hf product increases, the aP for the SH1 similarly the SH0 mode tends to appropriate value of QSS BAW mode, but the aP coefficient for the SH2 tends to appropriate value of the QFS BAW mode as 2.96 • 10-11 Pa-1. Note that for some modes the aP coefficients are equal to zero, i.e. an application of uniaxial mechanical pressure does not affect on their phase velocities. For example, this is fulfilled for the SH0 mode at hf = 10250 m/s as well as for the S0 mode at hf = 4300 m/s.

Fig. 1 (e) represents the dispersive dependencies of the aP coefficient when uniaxial mechanical pressure is applied along the normal to free surface of a plate, i.e. along the threefold axis

Fig. 1. Dispersion dependences of the parameters of Lamb and SH-waves in Z-cut of lithium niobate crystal plate. (a) Phase velocities; (b) EMCCs when one side of a plate was metalized; (c) aP controlling coefficients at P || [100]; (d) aP controlling coefficients at P || [010]; (e) aP controlling coefficients at P || [001]. Red curves should be associated with the An modes, dashed green ones with the Sn modes, and dashed blue ones with the SHn modes

Fig. 2. Normalized components of the displacement vectors of the A0 and S0 modes in lithium niobate plate at hf = 10250 m/s. (a) S0 mode at P = 0; (b) A0 mode at P || [001]; (c) S0 mode at P ||[001]

[001]. A significant increasing in the aP coefficients of the A0 mode is observed at small values of hf product, in particular aP = 6.45 • 10"11 Pa"1 at hf = 50 m/s. For the SH0 mode at hf > 5500 m/s there is a monotonous growth of the aP coefficient tending the appropriate value obtained under the same conditions for the QSS BAW as aP = 3.6 • 10"12 Pa"1. The behavior of the aP dispersion dependence of the higher order modes as A2, and SH2 is basically the same as in the case of P application orthogonally to the sagittal plane, considered above. The values of the aP coefficients of the A2 and SH2 modes at hf = 12000 m/s are 1.63 • 10"11 and 4.4 • 10"13 Pa"1, respectively.

Fig. 3. Dispersive dependencies of the parameters of Lamb and SH-waves propagating [100] direction in the LNC Y-cut under the uniaxial pressure application. (a) EMCCs; (b) aP coefficients at P ||[100]; (c) aP coefficients at P ||[001]; (d) aP coefficients at P ||[010]. Red curves should be associated with the An modes, dashed green ones with the Sn modes, and dashed blue ones with the SHn modes

When the uniaxial pressure P ||[001] is applied along the normal to a free plate surface, one can observe a peculiarity in the behavior of Lamb fundamental modes. Fig. 2 represents the normalized components of displacement vector for A0 and S0 modes at hf = 10250 m/s. In the case without the pressure influence, the A0 and S0 modes at hf > 9000 m/s degenerate into

Rayleigh surface wave with the phase velocity as 3823.66 m/s. Elastic displacements in the A0 and S0 fundamental modes have a maximal value on the free surfaces and are practically equal to zero in the middle plane of a crystalline plate (Fig. 2 (a)) [1]. Application of the uniaxial pressure at P || [001] leads to removing an above mentioned degeneration and splitting Rayleigh SAW mode back into the A0 and S0 modes. The displacements of Lamb modes A0 and S0 are concentrated at the lower and upper free surface of the plate, respectively (Figs. 2 (b), 2 (c)). If the S0 mode is transformed into the SAW mode with the phase velocity vR = 3822.39 m/s at P = 108 Pa and aP = 1.93 • 10"12 Pa"1, then the mode A0 has the phase velocity vAo = 3821.38 m/s and aP = 4.54 • 10"12 Pa"1 (Fig. 1 (e)).

On Fig. 3 the values of EMCCs and aP coefficients of Lamb and S H modes propagating in the [100] direction of Y-cut upon an application of uniaxial mechanical pressure are represented. In this case all the modes have a piezoelectric activity (Fig. 3 (a)). The maximal EMCC values are achieved for the S0 mode as K2 = 0.35 at hf = 7060 m/s and for SH0 mode as K2 = 0.39 at hf = 450 m/s. In the case when uniaxial pressure is applied along the wave propagation direction P ||[100], a series of interactions between the modes also arise, in particular, between S0 and SH1, S1 and SH2, as well as between the A2 and SH2 modes. This reflects in a behavior of aP coefficients (Fig. 3 (b)). For example, in the interaction region between the S0 and SH1 modes at hf = 3200 m/s the changes in aP coefficients realize within the range of 2.6 • 10"11 up to 7.04 • 10"12 Pa"1, and of 3.41 • 10"12 up to 3.11 • 10"11 Pa"1, respectively.

When uniaxial pressure P ||[001] is applied orthogonally to the sagittal plane of Y-cut, the aP values of almost all modes are sufficently smaller than those in the case P ||[100] considered above. Additionally there are the zero aP coefficients for some modes (Fig. 3 (c)).

Fig. 4. Dispersive dependences of the controlling coefficients in Lamb and S H modes propagating [001] direction in LNC X-cut under an application of uniaxial pressure. (a) aP coefficients at P ||[001]; (b) aP coefficients at P ||[010]; (c) aP coefficients at P ||[100]. Red curves should be associated with the An modes, dashed green ones with the Sn modes, and dashed blue ones with the SHn modes

When the pressure is applied along the normal to free surface P ||[010], a significant increase in aP coefficient of mode A0 at small hf values takes place. In particular, aP = 6.7 • 10"11 Pa"1 at hf = 50 m/s (Fig. 3 (c)). Similarly, a "splitting" of the phase velocities for the A0 and S0 modes occurs. If P = 0, at hf > 9000 m/s the fundamental modes A0 and S0 degenerate into Rayleigh wave with the phase velocity vSaw = 3798.8 m/s. Upon the P application the A0 mode

degenerates into Rayleigh wave with the vSAW = 3796.7 m/s and aP = 5.63 • 10-12 Pa-1, then the S0 mode has the phase velocity vSo = 3795.44 m/s and aP = 8.54 • 10-12 Pa-1.

It should be noted that the controlling coefficients grow significantly in the region of interaction between the S0 and SH1 modes, aP = 3.99 • 10-11 and -1.08 • 10-11 Pa-1, respectively (Fig. 3 (d)).

On Fig. 4 the dispersive dependences of the aP coefficients for Lamb and S H modes which propagate along the [001] direction in LNC X-cut upon the P application are presented. In this case, in particular, the A0 and S0 fundamental Lamb modes don't degenerate now into the SAW mode in all the considered range of the hf variation at all the variants of P application. When uniaxial pressure is applied along the propagation direction as P || [001], the aP coefficients of the A0 and SH0 modes tend to the appropriate value aP = 1.57- 10-11 Pa-1 observed in the SAW mode, but aP coefficient of the S0 mode tends to 1.8 • 10-11 Pa-1 value as the QSS BAW mode (Fig. 4 (a)). Note, that only at P ||[001] there are zero values of the aP coefficients. Application of an uniaxial pressure P || [100] significantly increases the hybridization effect between the S0 and SH1 modes. In particular, for the S0 and SH1 modes the value of the aP coefficients are 3.7 • 10-11, and 2.4 • 10-11 Pa-1, respectively (Fig. 4 (c)).

The extremal values of aP coefficients are summarized in the Tab. 1.

Table 1. Extremal values of aP coefficients for Lamb and SH modes in lithium niobate

Cut Mode Pressure force direction hf, m/s Phase velocity, m/s ap, -i1 Pa-1

Z-cut Ai [100] 2800 10271.4 3.02

S2 11600 4838.07 0

SH2 [010] 6050 8483.2 3.65

So 4300 4022.26 0

SHo 5600 4492.4 2.66

SHo 10300 4229.1 0

So [001] 1950 6031.8 0

Ai 4500 6376.1 1.75

S3 9200 6276.1 1.74

SHo 7650 4401.06 0

Y-cut SH2 [100] 12250 4860.7 0

Ao 5600 3778.7 2.36

So 1250 6663.9 2.67

So [010] 3150 5079.5 3.99

SHo 3450 4536.5 0

X-cut Ao [010] 4600 3434.9 0

Si 5050 6708.7 4.81

So [100] 2950 5257.1 3.68

Ai 5000 6710.8 4.5

3. Conclusion

Using the general results obtained in the Section 3, one can analyze in detail the dispersion of the main parameters for different acoustic modes in a piezoelectric plate under an application of uniaxial pressure. A prerequisite for this analysis is the knowledge of all the linear and nonlinear electromechanical constants of a crystal.

Dispersive dependences of the phase velocities, EMCCs, aP controlling coefficients vs the hf

product for the X-, Y- and Z-cuts of lithium niobate single crystals under the homogeneous uniaxial pressure applied in a lot of directions have been calculated. It should note the highest values of aP coefficients for the mode A0 in the thin plates, for example, aP = 5.13 • 10-11, 6.7 • 10-11, and 6.4 • 10-11 Pa-1 at hf = 50 m/s in the X-, Y- and Z-cuts, respectively. It has been demonstrated that for various applications of uniaxial mechanical pressure, the interaction of elastic wave modes can occur. Such kind of study can be usefull for the physical acoustics as well as in point of view in the practical applications, for example in acoustic sensors based on Lamb or SH-waves.

This work was supported by the grant of the Russian Science Foundation (project # 16-1210293).

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Влияние одноосного механического давления на характеристики волн Лэмба и бН-волн в пластинах кристалла Ь^ЪС3

Сергей И. Бурков Олег Н. Плетнев

Сибирский федеральный университет Красноярск, Российская Федерация

Павел П. Турчин

Сибирский федеральный университет Красноярск, Российская Федерация Институт физики им. Л.В.Киренского ФИЦ КНЦ СО РАН Красноярск, Российская Федерация

Ольга П. Золотова

Сибирский государственный университет науки и технологий им. Решетнева

Красноярск, Российская Федерация

Борис П. Сорокин

Технологический институт сверхтвердых и новых углеродных материалов

Москва, Троицк, Российская Федерация

Аннотация. Проведено теоретическое исследование влияния одноосного механического давления на характеристики распространения акустических волн в пластине ниобата лития. Рассчитаны коэффициенты электромеханической связи и коэффициенты управляемости при различных вариантах приложения внешнего механического одноосного давления.

Ключевые слова: пьезоэлектрическая пластина, волна Лэмба, £Л-волна, влияние однородного давления, компьютерное моделирование.