Anisotropy of dc electric field influence on acoustic wave propagation in piezoelectric plate Текст научной статьи по специальности «Физика»

УДК 548.534

Anisotropy of DC Electric Field Influence on Acoustic Wave Propagation in Piezoelectric Plate

Sergey I. Burkov* Olga P. Zolotova

Institute of Engineering Physics and Radio Electronics,

Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041,

Russia

Boris P. Sorokin^

Technological Institute for Superhard and Novel Carbon Materials, Centralnaya, 7a, Troitsk, Moscow region, 142190,

Russia

Kirill S. Aleksandrov

Received 10.12.2010, received in revised form 10.02.2011, accepted 20.04.2011

Anisotropy of dc electric field influence on the different types of acoustic waves in the piezoelectric plate has been investigated by means of computer simulation. Detail calculations have made for bismuth germanium oxide crystals.

Keywords: dc electric field, Lamb wave.

Introduction

Investigation of acoustic wave propagation in the piezoelectric plates under the bias electric field leads to the possibility of the controlling of acoustoelectonic devices parameters. Detail theory of the dc electric field or mechanical stress influences on the bulk acoustic (BAW) and surface acoustic (SAW) waves propagation in piezoelectric crystals had been derived in [1-6]. The studying of the anisotropic propagation of the zero order Lamb wave in the lithium niobate piezoelectric plates has fulfilled by the authors [7, 8]. Peculiarities of Lamb waves and surface waves with the horizontal polarization (SH) propagating along the high symmetry directions of the cubic piezoelectric crystals under the dc electric field have been investigated earlier [9].

To optimize the acoustoelectronic device it is necessary to find both an appropriate crystal direction and a value of the (h x f) product for a given frequency range (h — the crystalline plate thickness, f — the frequency). In the present paper the anisotropy of Lamb and SH—waves parameters in the bismuth germanium oxide (Bii2GeO20) piezoelectric crystal under the influence of dc electric field has been investigated by means of computer simulation.

* sburkov@sfu-kras.ru t bpsorokin2@rambler.ru © Siberian Federal University. All rights reserved

1. Propagation Theory of Lamb and SH-waves in

Piezoelectric Plate under the Influence of Homogeneous DC Electric Field

Influence of homogeneous dc electric field E on Lamb and SH-wave propagation conditions in piezoelectric crystalline plate has been considered on the basis of the theory of bulk acoustic waves propagation in piezoelectric crystals subjected to the action of a bias electric field [1].

Wave equations and electrostatics equation written in the natural state for homogeneously deformed crystals without center of symmetry have the form [2]:

poUi = Tik,k, (1)

D =0 (

Here po is the density of crystal taken in non-deformed (initial) state, Ui is the vector of dynamic elastic displacements, rik is the tensor of thermo-dynamical stresses and Dm is the vector of the induction of electricity. Here and further the tilde sign is marked the time dependent variables. Comma after the lower index denotes a spatial derivative and Latin coordinate indexes are changed from 1 to 3. Here and further the summation on twice recurring lower index is understood.

State equations can be written as:

~ __~ * TTl

Tik = Cikpq 'Ipq — enikEni (2)

Dn = enik Vik + £nmEm,

where пав is the deformation tensor and effective elastic, piezoelectric, dielectric constants are defined by:

Ciklm = Ciklm + ( Ciklmpqdjpq — ejiklm) MjE,

enik enik + (enikpqdjpq + Hnjpq') MjE, (3)

£nm = £rim + (Hnmikdjik + ^mj) Mj E■

In (3) dc electric field is the value of dc electric field applied to the crystal, Mj is the unit vector of E—direction, CEklmpq, enikpq, e^imj, Hnmik are nonlinear elastic, piezoelectric, dielectric and electrostrictive constants (material tensors), djpq and enik are the piezoelectric tensors, CEklm and enm are elastic and dielectric tensors. Substituting (2) into (1) we can obtain Green-Christoffel equation in a general form which can be used for the analysis of bulk acoustic waves propagation in the case of E—influence.

Let's use coordinate system X3 axis directs along the external normal to the surface of a media occupying the space h > X3 > 0, and the wave propagation direction coincides with Xi axis. Plane waves propagating in the piezoelectric plate are taken in the form:

Ui = ai exp t [i (kjXj — ut)\, ,

<p = a.4 expt [i (kjXj — ut)],

where ai and a4 are amplitudes of elastic wave and electric potential ip concerned closely with the wave, and kj are components of wave propagation vector. Taking into account (2) and (3) the substitution (4) into (1) gives us equation in a specific form. So if the electric field is applied to piezoelectric crystal, Green-Christoffel equation can be written as

[rpq(E) — po^2Spq] Uq = 0, (5)

where Green-Christoffel tensor has the form:

rpq (Cip<

q 4

eimq kikm,

+ 2djkq CEpkmMj E)kikn

r4q — Tq4 + 2eikmdjkq Mj kikmEi

(6)

r

44

— —F* k k

Propagation of acoustic waves in the piezoelectric plate under the E—influence should satisfy to boundary conditions of zero normal components of the stress tensor on the boundaries "crystal-vacuum". Continuity of the electric field components which are tangent to the boundary surface is guaranteed by the condition of the continuity of the electrical potential and normal components of the electric displacement vector:

T3k f — f — D = D =

= 0, f[I], flII\ d[i ,

Dl11],

x3 — 0 x3 < 0

X3 > h

x3 < 0

X3 > h.

x3

h;

(7)

Here the upper index "I" is concerned to the half-space X3 > h and index "II" — to the halfspace X3 < 0. Substituting the solutions (4) into equations (7) and neglecting of the terms which are proportional E2 (and higher order ones), finally we have obtained the system of equations useful to analyze the change of the wave's structure arising as a consequence of crystal symmetry variation and new effective constants appearance:

^n=1 Cn

En=1 Cn

En=1 Cn

n=1 Cn

C* k(n) a(n) + P* k(n)a(n) C3jklkl ak + ek3j kk a4

expt (ik3n)h^ — 0;

e3kik;(n)akn) - (e3kkhl) - ieo)a4n) exPt ^k3n)h

C3jkiki

k

n) n)

0;

k

e3kik(n)akn) - (e*3k+ ieo)a^ — 0

+ e*k.3j k

(n) (n)

k3jkk a4

(n) k

0;

v(n)l _

(8)

Index n = 1,... 8 corresponds to the number of one of eight partial waves; Cn and a^ are its weight coefficients and amplitudes, respectively.

It can remember that the equations (8) were obtained at the assumption of homogeneity of applied dc electric field without taking into account the edge effects. But these equations take into account all changes of the crystal density and the form of crystal sample arising as a consequence of finite deformation of piezoelectric media under the action of strong dc electric field [2].

2. Anisotropy of DC Electric Field Influence on the

Acoustic Wave Parameters in the Bismuth Germanium Oxide Piezoelectric Plate

As a model media the bismuth germanium oxide (23 point symmetry) has been used. Taken into account equations (8) and linear and nonlinear material properties from [2], the computer calculation of the main parameters such as phase velocity, electromechanical coupling coefficient (EMCC), and controlling coefficient of phase velocity

av = VM(^) aE.0 (9)

has been carried out.

Analyses of the variation of waves parameters was fulfilled for (001) and (110) crystalline planes when dc electric field was directed along some kind of X1, X2 and X3 axes.

2.1. Acoustic Wave Propagation in the (001) Crystalline Plane

If acoustic wave is propagating in the (001) crystal plane, the application of dc electric field along Xi axis, i.e. along the acoustic wave propagation direction, or along X2 axis, i.e. orthogonal to the sagittal plane, leads to decreasing of the crystal symmetry to triclinic one in general case except the field directions coinciding with basic axes of the crystal. Last variants have been considered in details earlier [9,10]. Under the action of dc electric field along X3 basic axis (E || [001]), i.e. orthogonal to the free surface of the crystal plate, the crystal symmetry decreases to monoclinic one (2 point symmetry). Twofold axis coincides with the [001] direction and there are induced some effective elastic constants:

Ci6 = (Ci66 di4 — ei24 )E ; C36 = (C144d14 — e114) E; e15 = (e156di4 + H55) E; e32 = (e134d14 + H32) E;

Phase velocities of the zero and first order modes of Lamb wave propagating in the (001) crystalline plane with (h x f ) values up 500 to 3000 m/s are shown on Fig. 1. If (h x f ) values are

C26 = (C155d14 — e134) E; C45 = (C456 d14 — e156) E ; e31 = (e124d14 + H31 ) E; £33 = (e114d14 + H33) E.

a

v, 103 m/s

- Ai SHi

-—^---OT ^

HT

-

QFS Ao

1 1

[100] [110] (p/ [010] b

c

Fig. 1. Phase velocities of acoustic waves propagating in the (001) plane of Bi12GeO20 crystal at E = 0 and various values of h x f (m/s): a) h x f = 500; b) h x f = 1000; c) h x f = 1500; d) h x f = 2000; e) h x f = 2500; f) h x f = 3000. Curves for the quasi-longitudinal, fast and slow quasi-shear bulk acoustic waves are marked as QL, QFS, QSS respectively

increased, the phase velocity of antisymmetric mode A0 is considerably increased, for example up 1284.96 go 1645.58 m/s in the [110] direction, but the square of EMCC (Fig. 2) is decreased up 2.3% to 0.8%. Values of the EMCC square were calculated as in the case of SAW propagation, i.e. when the metallization of one free surface of piezoelectric plate has taken into account [11]. For Ai mode propagating along the [110] direction of the (001) plane there is the EMCC square's maximal value equal to 0.5% (h x f = 1500 m/s). Note that the EMCC square's qualitative behavior for the first order modes is similar to the zero order ones, but its numerical values are considerably less.

If E || X1, dependences for av coefficients of A0 mode are similar to ones for Rayleigh surface acoustic wave [12]. When the A0 mode is propagating along the [100] direction, its av coefficients are considerably increased up -4.17 • 10-11 to -2.5 • 10-10 m/V by the variation of (h x f) quantity up 500 to 3000 m/s. For the A1 mode maximal av values are reached in the [110] propagation direction, in particular av = -6.7• 10-11 m/V (h x f = 2500 m/s).

Note that as a result of dc electric field application orthogonal to the sagittal plane (E || X2), the [100] and [010] propagation directions, undistinguished for the undisturbed crystal, become unequal ones. This effect is the consequence of the 23 point symmetry peculiarity of the given crystal, since there is a difference between components of nonlinear properties which are responsible for the dc electric field influence on the phase velocities, for example C155 = C166, em = , H12 = H21. It can point to the fact that these components are equal in all other piezoelectric crystals of cubic symmetry. So av = -6.1 • 10-12 m/V and av = -7.3 • 10-12 m/V for the A0 modes in the [100] and [010] directions (Fig.2, c). In the case when both main surfaces of the plate are coated by metal, the dc electric field application along X3 axis leads to increasing of av coefficients as a result of the thickness increasing. In particular A0 mode ([110] propagation direction) av coefficient varies up -2.58 • 10-10 to -3.71 • 10-10 m/V (Fig. 2).

One of the distinctive peculiarities of wave's propagation in the (001) crystalline plane is the hybridization effect. There are some coupled modes having the energy exchange.

Fig. 2. The square of EMCC and av coefficients of bulk waves and the A0 modes propagating in the (001) plane of Bi12 CeO20 crystal: a) the square of EMCC ; b) av coefficients (E || X1 ); c) av coefficients (E || X2); d) av coefficients (E || X3 ). A number of the curve corresponds to the values h xf (m/s): 1- h x f = 500; 2- hx f = 1000; 3 - h x f = 1500; 4 - h x f = 2000; 5 - h x f = 2500; 6 - h x f = 3000. Points are marked the experimental av coefficients [2]

21

M=

W12 + W W1 + W2

(11)

where W12 + W21 — a complete mutual energy of two coupled modes (time average); W1 + W2 — a complete energy of acoustic wave. Without electric field the hybridization effect between the S0 h SH0 modes exists only in the thick plates if h x f > 2000 m/s (Fig. 3). It should be noted that

Fig. 3. a) Hybridization coefficient M (E || X1) for: 1 — S0-SHi modes (h x f = 2000 m/s); 2 — S0-SH0 (h x f = 2500 m/s). b) Phase velocities of acoustic waves. Designations of curves are in accordance with Fig. 2

Fig. 4. Square of EMCC (a) and av coefficients of the S0 mode propagating in the (001) plane of Bi12GeO20 crystal: b) — E || X1 ; c) — E || X2; d) — E || X3. A number of the curve corresponds to the values h x f (m/s): 1 — h x f = 500; 2 — h x f = 1000; 3 — h x f = 1500; 4 — h x f = 2000; 5 — h x f = 2500; 6 — h x f = 3000

the hybridization takes place in the point of the equality of phase velocities of the S0 and SH0 modes with the phase velocity of the QFS bulk acoustic wave. Hybridization regions are shown

by vertical lines on insets of Fig. 3, b. The E—application along the X1 or X2 axes amplifies the hybridization effect, and av values are increased in accordance with exponential function and reach the maximal quantities (Fig. 4, 5). If the electric field is applied along the X1 or X2 axes the hybridization effect arises between the So and SH modes lacking in the undisturbed case (Fig. 3). The E—application along the X3 axis leads to the hybridization effect decreasing. Maximal av values for the S0 mode are reached in the [110] propagation direction of the (001) plane when E || X1 increasing up —1.9 • 10-12 to 4.34 • 10-10 m/V by the increment of (h x f) value (Fig. 4)

If E || X3 maximal av values take place in the [110] direction: av = 2.99 • 10-10 m/V (h x f = 3000 m/s). Square of EMCC of the So mode is considerably increased in the [110] direction if (h x f ) values are increased: up 0.47 % to 0.84 % when hxf = 500 h 3000 m/s respectively. Hybridization effect leads to exponential dependence of the av coefficient and changes the EMCC coefficients in the hybridization region. When E || X1 anisotropy of av coefficient for the SH0 mode is the similar one to the S0 mode (Fig. 5). When dc electric field is applied along the X3 or X2 axes the av coefficient values don't depend on (h x f ) values excluding the hybridization region between the S0 h SH0 modes. Maximal EMCC values (2.8%) take place in the [100] and [010] directions of the (001) crystalline plane.

The av coefficients of the S1, A1, and SH1 first modes propagating in the (001) plane are shown in [14]. Distinctive peculiarity for higher order modes is the extreme behavior of av coefficient in the region when (h x f ) value is close to critical one, and the appearance of acoustic modes of higher order becomes possible. In this case a small variation in the plate configuration and material properties of the crystal changes considerably the phase velocity. In particular for the A1 mode propagating along the [110] direction there is maximal value av = -12.32 • 10-10 m/V (E || X3 ). Note that there is the influence of the hybridization effect which leads to the exponential dependence of the av coefficient.

Fig. 5. Square of EMCC (a) and av coefficients of the SH0 mode propagating in the (001) plane of Bi12CeO20 crystal: b) - E || X ; c) - E || X2; d) - E || X3. A number of the curve corresponds to the values h x f (m/s): 1 - h x f = 500; 2 - h x f = 1000; 3 - h x f = 1500; 4 - h x f = 2000; 5 - h x f = 2500; 6 - h x f = 3000

2.2. Acoustic Wave Propagation in the (110) Crystalline Plane

For this plane the anisotropy of phase velocities is shown in [14] and the maximal values of the av coefficients are realized if E || X1. The E—application along the X3 axis has a minimal effect on the phase velocities of the A0, S0 and SH0 modes (Fig. 6, 7). Maximal value

av = -5.8 • 10 10 m/V for S0 mode takes place in the direction oriented relative to the [001] axis under the angle ^ = 29° (h x f = 1000 m/s) (Fig. 7). Distinctive features of the av coefficients

[110] [111] 45 (p/ [001] [110] [111] 45 (p?° [001] [110] [111] 45 ^ ° [001]

ab c

Fig. 6. The av coefficients of the A0 mode propagating in the (110) plane of Bi12GeO20 crystal: a) E || X1; b) E || X2; c) E || X3. Designations of curves are in accordance with Fig. 2.

Fig. 7. The av coefficients of the SH0 (a) and S0 (b) modes propagating in the (110) plane of Bi12GeO20 crystal. Designations of curves are in accordance with Fig. 2

in the (110) plane for the bulk acoustic waves are defined by the splitting of the tangent type acoustic axis coinciding with the [001] crystalline direction (twofold axis of symmetry) in the undisturbed state and by the displacement of the conic type acoustic axis coinciding with the [111] crystalline direction (threefold axis of symmetry) in the undisturbed state [2]. Extreme values of the av coefficients are shown in the Table I.

Table I. Maximal and minimal values of the av coefficients of Lamb and SH waves in the bismuth germanium oxide crystal

Crystalline Mode DC electric field direction Angle, h x f, m/s av, 10-11 m/V

plane

(001) Ao E || X3 45 3000 -39.0

E || X2 47 1000 -0.126

(001) SH0 E || Xi 0 3000 19.9

E || X2 43 1500 0.0

(001) So E || X3 45 3000 -29.0

E || X3 90 1500 0.001

(001) Ai E X3 46 1000 -122.4

E X2 89 2500 -0.08

(001) SHi E Xi 90 3000 38.6

E X3 0 2500 -0.6

(001) Si E X3 42 2000 18.4

E X2 43 2500 -0.01

(110) Ao E Xi 90 2500 37.1

E X2 25 500 -0.06

(110) SHo E Xi 90 1000 -50.3

E X2 42 500 0.3

(110) So E Xi 29 1000 -58.1

E X3 90 1000 0.01

Conclusion

Thus, the anisotropy of homogeneous dc electric field influence on the different types of acoustic waves in the bismuth germanium oxide piezoelectric crystal plate has been investigated by means of computer simulation. Detail analysis of the dispersive behavior of zero and first order Lamb and SH modes has been carried out. Crystalline directions with extreme dc electric field influence have found. It was shown that the acoustic modes interaction can arise as a consequence of dc electric field action in the some directions. The obtained data can be useful to design the controlling devices of acoustoelectronics.

This paper was supported by the Russia President's Program on leading scientific schools support (grant № 4645.2010.2).

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S.I.Burkov, O.P.Zolotova, B.P.Sorokin, K.S.Aleksandrov, Anisotropy of dc electric field influence on acoustic wave propagation in piezoelectric plate, http://arxiv.org/abs/1008.2058.

Анизотропия влияния постоянного электрического поля на распространение акустических волн в пьезоэлектрической пластине

Сергей И.Бурков Ольга П.Золотова

_Борис П.Сорокин

Кирилл С.Александров

Рассмотрена анизотропия влияния однородного электрического поля Е на характеристики и условия распространения волн различных типов в пьезоэлектрической пластине германосилле-нита.

Ключевые слова: внешнее электрическое поле, волна Лэмба.