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

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


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][2][3][4][5][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×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 (Bi 12 GeO 20 ) piezoelectric crystal under the influence of dc electric field has been investigated by means of computer simulation.

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]: Here ρ 0 is the density of crystal taken in non-deformed (initial) state, i Ũ is the vector of dynamic elastic displacements, τ ik is the tensor of thermo-dynamical stresses and D m 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: where η AB is the deformation tensor and effective elastic, piezoelectric, dielectric constants are defined by: In (3)  Let's use coordinate system X 3 axis directs along the external normal to the surface of a media occupying the space h ≥X 3 ≥ 0, and the wave propagation direction coincides with X 1 axis. Plane waves propagating in the piezoelectric plate are taken in the form: where α i and α 4 are amplitudes of elastic wave and electric potential ϕ concerned closely with the wave, and k j are components of wave propagation vector. Taking into account (2) and ( where Green-Christoffel tensor has the form: 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 "crystalvacuum". 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: Here the upper index «I» is concerned to the half-space X 3 > h and index «II» -to the halfspace X 3 < 0. Substituting the solutions (4) into equations (7) and neglecting of the terms which are proportional E 2 (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: Here the index n = 1,…4 corresponds to the number of one of the partial waves (4) and С n are the amplitude coefficients of partial waves.
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].

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 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 X 1 , X 2 and X 3 axes.
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 X 1 axis, i.e. along the acoustic wave propagation direction, or along X 2 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 X 3   when the metallization of one free surface of piezoelectric plate has taken into account [11]. For A 1 mode propagating along the [110] direction of the (001) plane there is the EMCC square's maximal value equal to 0.5% (h×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||X 1 , dependences for α v coefficients of A 0 mode are similar to ones for Rayleigh surface acoustic wave [12]. When the A 0 mode is propagating along the [100] direction, its α v coefficients are considerably increased up -4.17·10 -11 to -2.5·10 -10 m/V by the variation of (h×f) quantity up 500 to 3000 m/s. ties (Fig. 4, 5). If the electric field is applied along the X 1 or X 2 axes the hybridization effect arises between the S 0 and SH 1 modes lacking in the undisturbed case (Fig. 3). The E-application along the X 3 axis leads to the hybridization effect decreasing. Maximal α v values for the S 0 mode are reached in the [110] propagation direction of the (001) plane when E||X 1 increasing up -1.9·10 -12 to 4.34·10 -10 m/V by the increment of (h×f) value (Fig. 4).
If E||X 3