Influence of bias electric field on elastic waves propagation in piezoelectric layered structures

Accepted Manuscript Short communication Influence of bias electric field on elastic waves propagation in piezoelectric layered structures S.I. Burkov, O.P. Zolotova, B.P. Sorokin PII: DOI: Reference:

S0041-624X(13)00085-1 http://dx.doi.org/10.1016/j.ultras.2013.03.009 ULTRAS 4568

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Ultrasonics

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11 February 2013 20 March 2013 20 March 2013

Please cite this article as: S.I. Burkov, O.P. Zolotova, B.P. Sorokin, Influence of bias electric field on elastic waves propagation in piezoelectric layered structures, Ultrasonics (2013), doi: http://dx.doi.org/10.1016/j.ultras. 2013.03.009

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Influence of bias electric field on elastic waves propagation in piezoelectric layered structures S.I. Burkov1, O.P. Zolotova2, B.P. Sorokin3 1) Condensed Matter Physics Department, Siberian Federal University, 79 Svobodny Ave., 660041 Krasnoyarsk, Russian Federation 2) Siberian State Aerospace University, 31 Krasnoyarsky Rabochy Ave., 660014 Krasnoyarsk, Russian Federation 3) Technological Institute for Superhard and Novel Carbon Materials, 7a Central’naya Str., 142190, Moscow, Troitsk, Russian Federation e-mail: [email protected], [email protected] Abstract – Theoretical and computer investigations of acoustic wave propagation in piezoelectric layered structures, subjected to the dc electric field influence have been fulfilled. Analysis of the dispersive parameters of elastic waves propagation in the BGO/fused silica and fused silica/LiNbO3 piezoelectric layered structures for a number of variants of dc electric field application has been executed. Transformation of bulk acoustic wave into SAW type mode under the dc electric field influence has been found. Possibility to control the permission or prohibition of the wave propagation by the dc electric field application and the appropriate choice of the layer and substrate materials has been discussed. Keywords: piezoelectric layered structure, Love wave, dc electric field influence. PACS: 43.25.Fe; 43.35.Cg; 77.65.-j INTRODUCTION Propagation of elastic waves in isotropic structures such as "layer/substrate" was examined in detail in [1, 2]. Development of acoustic devices based on the Love dispersion waves (SH-waves) has reinvigorated the research efforts on the way of SH-wave properties investigation in layered structures with different types of layers such as metal, dielectric, and piezoelectric layer of finite thickness, deposited on the semi-infinite substrate [3-8]. But in these works mainly tetragonal or hexagonal crystals as a piezoelectric medium have taken into consideration. Strict solution of the existence of elastic waves in piezoelectric bi-plates has been obtained in [9]. In the paper [10] there was developed a framework based on the concept of the impedance matrix, which is convenient for studying the subsonic velocity spectrum of a coated substrate in the case of general anisotropy of substrate and layer materials. The external influence of uniform electric field E on the characteristics and propagation conditions of bulk (BAW) and surface acoustic waves in piezoelectric crystals has been considered in [11, 12]. Additionally the peculiarities of Lamb wave propagation under the E influence have been investigated by the authors [13-15]. In this paper, the influence of dc electric field on the dispersion characteristics of Rayleigh and Love waves in a layered piezoelectric structure has been investigated in details by means of theoretical approach as well as computer simulation.

ELASTIC WAVE PROPAGATION IN LAYERED PIEZOELECTRIC STRUCTURES UNDER THE INFLUENCE OF HOMOGENEOUS DC ELECTRIC FIELD Based on the theory developed in [12], let’s write the basic equations describing the effect of dc electric field on the propagation conditions of acoustic waves in a piezoelectric medium. Referring to the initial coordinate system, the small-amplitude wave equations in the uniformly deformed acentric crystals and electrostatic equations have the form [12]:

ρ0UI = τIK , K , D M , M = 0, * τIK = CIKPQ ηPQ - e*NIK E N ,

(1)

* D N = e*NIKηIK + ε MN E M .

There are introduced the values to be used as: ρ0 is the crystalline density in the undeformed (initial) state, U A is the unit vector of the dynamical elastic displacement, τAB is the thermodynamical stress tensor, ηAB is the strain tensor, and DM is the vector of electric displacement. Here and after the time-dependent variables are marked by "tilde" symbol. The comma after the subscript denotes spatial derivative, and coordinate Latin indices vary from 1 to 3. Here and further, the rule of summation over repeated indices is used. Effective elastic, piezoelectric and dielectric constants which are the linear functions of dc electric field are defined by: * E E C ABKL = C ABKL + ( C ABKLQR d JQR − eJABKL ) M J E ,

e*NAB = eNAB + ( eNABKL d JKL + H NJAB ) M J E ,

(2)

* ε NM = ε ηNM + ( H NMAB d PAB + ε ηNMP ) M P E.

E Here E is the dc electric field strength, MJ is the unit vector of E direction; C ABKL , eNAB

( d JKL )

and ε ηNM are the

E second-order elastic, piezoelectric and clamped dielectric material constants, respectively; C ABKLQR , eNABKL , and

ε ηNMP are the third-order elastic, nonlinear piezoelectric and nonlinear dielectric constants, respectively, and H NMAB is the tensor of electrostriction. Let the X3 axis of working coordinate system is directed along the outer normal to the surface of the layer, occupying the 0 ≤ X3 ≤ h space, and the X1 axis coincides with the wave propagation direction. Here h is the layer thickness. Dynamical elastic displacements and electrostatic potentials are taken in the conventional plane wave form.

Under the action of dc electric field on piezoelectric crystal the components of Green-Christoffel tensor can be written as [11]: * E M J E ⎤⎦ k A k D , Γ BC = ⎡⎣C ABCD + 2d JFC C ABFD

ΓC 4 = e*ADC k A k D , (3)

Γ 4C = ΓC 4 + 2eAPD d JPC M J Ek A k D , Γ 44 = −ε IJ* k I k J . Here kA is the wave vector.

In general, the total magnitudes of the elastic displacement and electrical potential both in thin layer and substrate have the form of linear combinations of partial waves:

U I = ∑ Cn( L , S )α I( n ) exp ⎡⎣i ( k1 x1 + k3( n ) x3 - ωt ) ⎤⎦ , n

Φ = ∑C

α

( L,S ) 4

(n) 4

n

exp ⎡⎣i ( k1 x1 + k3 x3 - ωt ) ⎤⎦ .

(4)

(n)

Here and below superscripts (S) and (L) denote the substrate and layer terms, respectively; Cn and αI values are the weight coefficients and partial wave amplitudes. Wave propagation in a piezoelectric layered structure under the E influence must satisfy the boundary condition as the zero normal components of the stress tensor at the crystal-vacuum interface. The continuity of the tangential components of the electric field is ensured by the condition of continuity of the electric potential ϕ, and the condition of continuity of the normal component of the electrical displacement vector. All such conditions can be written as:

τ 3( LA) = 0 x = h , 3

( L) 3

D

= D ( vac )

U A( S ) = U A( L )

τ 3( SA) = τ 3( LA)

x3 = h

x3 = 0

x3 = 0

D3( S ) = D3( L )

,

,

ϕ ( L ) = ϕ ( vac ) ϕ (S ) = ϕ ( L)

x3 = h

x3 = 0

,

, (5)

,

x3 = 0

.

In a manner the similar [10], substitution of the solutions in the form of homogeneous plane waves into equations (1) led to the system of the homogeneous equations to be used for the search of the eigenvalues and eigenvectors. Thereafter the general solution compiled as the sum of the partial waves was substituted into the boundary conditions (5) in order to obtain the system of equations as [16]:

8

∑ a( ) ⎡⎣( C n

*( L ) 3 KPI

n =1

8

∑ a ⎡⎣( e (n)

( L) ( L) + 2d JPF C3EKFI M J E ) k I( n )α P( n ) + eP*(3LK) k P( n )α 4( n ) ⎤⎦ exp ( ik3( n ) h ) = 0,

( L) ( L) + 2d JKP e3 PL M J E ) k L( n )α K( n ) − ( ε 3*(KL ) k K( n ) − iε 0 ) α 4( n ) ⎤⎦ exp ( ik3( n ) h ) = 0,

*( L ) 3 KL

n =1

8

∑a

α 4( n ) exp(ik3( n ) h) − a0 exp(ik1 h) = 0,

(n)

n =1

4

8

m =1

n =1

∑ b( m ) β I( m ) − ∑ a ( n )α I( n ) = 0, 4

∑ b( ) ⎡⎣( C m

m =1

*( S ) I 3 KL

4

8

m =1

n =1

∑ b( m ) β 4( m ) − ∑ a ( n )α 4( n ) = 0,

(6)

(S ) (S ) + 2d AKF C3EIFL M A E ) qL( m ) β K( m ) + e3*(PIS ) qP( m ) β 4( m ) ⎤⎦ −

8

n ( L) ( L) −∑ a ( ) ⎡⎣( CI*(3LKL) + 2d AKF C3EIFL M A E ) k L( n )α K( n ) + e3*(PIL ) k P( n )α 4( n ) ⎤⎦ = 0, n =1 4

∑b m =1

(m)

(S ) (S ) ⎡( e3*(KLS ) + 2d AKP e3 PL M A E ) qL( m ) β K( m ) − ε 3*(KS ) qK( m ) β 4( m ) ⎤⎦ − ⎣

8

( L) ( L) e3 PL M A E ) k L( n )α K( n ) − ε 3*(KL ) k K( n )α 4( n ) ⎤⎦ = 0. −∑ a ( n ) ⎡⎣( e3*(KLL ) + 2d AKP n =1

Here α K(n ) , a ( n ) , and k L(n ) are the amplitudes, its weight coefficients, and wave vectors of the n-th partial wave (n = 1, ..., 8) in the piezoelectric layer; β K(m ) , b ( m ) , and q P(m ) are the same ones for the m-th partial wave (m = 1, ..., 4) in the substrate, respectively. Now we have the system of the twelve equations relative to the twelve variables a ( n ) and b ( m ) to be used for computer simulations. Calculations of the elastic wave parameters have been performed by the standard method of partial waves [11]. Briefly the algorithm can be written as: 1) substitution of the homogeneous plane waves into equations (1) has given us a characteristic parametric equation; 2) by characteristic equation at given k1( n ) ≡ q1( m ) ≡ k the k3( n ) and q3( m ) parameters as well as the α K( n ) and β K( m ) partial wave polarization vectors have been defined; 3) finally, it is necessary to find the only such k vector value which gives us the zero value of the (12×12) determinant of the system (4). The last allows to calculate the phase velocity of elastic wave. Note that the relations (6) have been derived at the assumptions about the influence of homogeneous dc electric field on the crystal layer without edge effects (one-dimension approximation). It has been taken into account all changes in the configuration of an anisotropic continuum, in particular, changes of the sample form (geometric nonlinearity) due to direct piezoelectric effect, and changes of the material constants (2) (physical

nonlinearity) under the influence of strong dc electric field. The complete solution of the system (6) can be found by numerical methods only.

NUMERICAL EXAMPLES AND DISCUSSION As a measure of dc electric field influence on the phase velocities of Rayleigh and Love waves there is convenient to introduce the controlling coefficients [17, 18]:

αv =

1 ⎛ Δv ⎞ . ⎜ ⎟ v(0) ⎝ ΔE ⎠ΔE →0

(7)

Electromechanical coupling coefficient (EMCC) of Rayleigh and Love waves has been calculated by conventional relation [2]:

K2 = 2

v - vs . v

(8)

where v is the phase velocity on the free surface, and vs is associated with electrically shorted surface. When piezoelectric crystal was used as a thin layer on dielectric substrate, it was supposed that metal film was placed on x3 = h piezoelectric layer surface. But if piezoelectric crystal and dielectric material were used as a substrate and a layer, respectively, the infinitely thin conducting film was placed on the x3 = 0 surface between dielectric layer and piezoelectric substrate. The configurations of layered structures to be used for calculation are shown on the fig. 1a and fig. 3a. On the basis of equations (1) - (6) in a layered piezoelectric structures for a few modes of Rayleigh and Love waves the calculation of such parameters as the phase velocities, EMCC and αv coefficients has been carried out. As an example let’s take piezoelectric crystal Bi12GeO20 (BGO) belonging to the (23) cubic symmetry group, and its material coefficients of linear and nonlinear electromechanical properties are taken from [4, 7]. Fig. 1 shows the dispersion curves of phase velocities, EMCCs and αv coefficients of Rayleigh and Love waves depending on the h×f value when waves are propagated along the [100] direction of (001) BGO/fused silica layered structure. In such structure there is the condition of the existence of Love elastic wave in the layer [10], i.e. phase velocity of the shear bulk wave in the layer has the value less than that within a substrate: vBGO < vsub. Pure Love wave has a piezoelectric activity and Rayleigh wave is non-piezoactive one. Interval of permissible phase velocities of Love waves lies between the values of the shear wave velocities of fused silica and BGO. Rayleigh wave phase velocity at the h×f increasing tends to the Rayleigh wave phase velocity of BGO (1625.92 m/s). The maximum value of K2 = 2.9 % at h×f = 1000 m/s takes place for zero Love mode.

If dc electric field E||[010] directed orthogonally to the sagittal plane is applied to the considered layered structure, the new elastic, piezoelectric, and dielectric constants appear as a consequence of the changing of the effective crystalline symmetry down to monoclinic one: * C15* = (C155 d14 − e134 ) E; C35 = (C166 d14 − e124 ) E; * C 46 = (C 456 d14 − e156 ) E; e16* = (e156 d14 + H 44 ) E;

ε 13* = ( H 44 d14 + ε 123 ) E.

(9)

When dc electric field is applied along the E||[010], i.e. orthogonally to the sagittal plane, the modes of elastic waves remain the pure ones as it follows analyzing (3). The αv coefficient of Love wave at the h×f increasing tends to αv = -3·10-11 m/V for the slow shear bulk acoustic wave of BGO, and αv coefficient of all the Rayleigh type waves at the h×f increasing tends to αv = -6.9·10-12 m/V value as the same one for the fast shear wave of BGO. There is the remarkable peculiarity of BAW propagation in BGO single crystal as the removing of degeneration for shear waves velocities. Initially the [100] crystalline direction is the acoustic axes of tangential type with Poincare index n = ±1. As a result of E influence the initial acoustic axes splits into two acoustic axis of conical type with n = ±1/2, and two independent quasi-shear slow (QSS) and fast (QFS) waves of linear polarizations should be propagated [19]. Need to clarify that in the linear case at E = 0 within the BGO layer there are propagated bulk longitudinal and shear degenerate waves. The E⎪⎢[010] BGO application orthogonally to sagittal plane has a result that the shear degenerate wave transforms into the quasi-shear horizontal (QSH) and quasi-shear vertical (QSV) surface type waves. In BGO single crystal when shear wave propagates along the [100] direction under the influence E⎪⎢[010], the phase velocity variations are too small, because the appropriate coefficients have the small magnitudes as αv = 7·10-13 m/V and αv = -5.7·10-13 m/V (Table 1) for the QFS and QSS waves, respectively [11, 12]. But in the BGO/fused silica layered piezoelectric structure under the influence E⎪⎢[010] the QSS wave which in this case has the SH-wave properties, transforms into SAW type mode with αv = 2,4·10-11 m/V. Such QSH-wave is a non-dispersive one, and its polarization U2 remains to be constant within the layer and rapidly decays into the substrate depth (Fig. 2a). Thus, this mode can be classified as a version of the “Anisimkin Jr. mode” (AN). Note that AN mode as defined in [20 - 22] has the basic properties such as а dominant longitudinal displacement (U1 >> U2, U3, and U1 ≈ constant within the plate or layer), or a dominant shear displacement for shear-horizontal QSH (U2 >> U1, U3) and shear-vertical QSV (U3 >> U1, U2) waves [23]. Moreover, an approximate equality of the BAW and AN phase velocities should be observed. In our case the two attributes are satisfied for the QSH-mode

only. On the contrary, QFS wave transforms into the QSV mode with a dominant displacement U3 > U1 and U3 changes within the piezoelectric layer.



The influence of dc electric field applied along the [100]⎪⎢ k direction changes more significantly the phase velocities of shear waves (Table 1). As a result shear wave is transformed into the QSH and QSV modes [23, 24]. Though αv coefficient for QSH-mode has the a much more value than the relevant coefficient for the shear BAW mode in a single crystal, the U2 dominant displacement has not constant magnitude within the crystalline layer (Fig. 2b). Let’s consider the acoustic wave propagation in the fused silica/Y-cut LiNbO3 layered structure. Figure 3 shows the dispersion curves of phase velocities, EMCCs and αv coefficients of Rayleigh and Love waves depending on the h×f production when acoustic waves are propagated along the [100] direction of LiNbO3. The material coefficients of linear and nonlinear electromechanical properties for LiNbO3 single crystal were published by authors [25]. Values of EMCCs were calculated under the presence of infinitesimal thin metal layer deposited between the fused silica layer and LiNbO3 substrate, i.e. without changing of mechanical boundary conditions (5). Phase velocity value of the fundamental (zero) Rayleigh modes R0 with the increasing of h×f production tends to the SAW phase velocity in fused silica equal to 3405 m/s. Phase velocity values of the Love and Sezawa modes are in the limits of velocities between the slow shear bulk acoustic wave of LiNbO3 and the shear wave of the fused silica. All the modes have the piezoelectric activity, and K2 = 1.3 % at h×f = 100 m/s for the R0 Rayleigh mode and K2 = 6.9 % at h×f = 1600 m/s for the L0 SH-mode maximum values are observed. Note that the influence of electric field on the phase velocities variation is significantly more for the Rayleigh modes than for the Love modes. So there are the controlling coefficients αv = -8.6·10-11 m/V and αv = 1.5·10-11 m/V at h×f = 1400 m/s and E||[010] for the fundamental modes of the Rayleigh and SH waves, respectively. However, it should be noted that it is possible to choose the parameters of an isotropic medium in a way that the electric field application to above-mentioned layered structure leads to the permission or prohibition of the wave propagation. In particular, along the Euler (90°, -90°, 25.1°) LiNbO3 direction the phase velocity of the QSS wave is equal vQSS = 3758.78 m/s, and the phase velocity of shear wave in fused silica is equal vS = 3759.84 m/s. Thus, there is fulfilled the inequality that the shear wave phase velocity in the layer has a greater value than it in the substrate: vS > vQSS. It means that the existence of elastic waves in such layered structure is not possible [10], except for the zero Rayleigh mode (Fig. 4a). When the dc electric field is applied along the direction of the QSS wave propagation with αv = 7.12·10-11 m/V, the phase velocity of this wave will be equal to vQSS = 3803.2 m/s at

E =5·106 V/m, i.e., there is satisfied already the condition vS < vQSS. Consequently, it becomes possible the existence both the Love and higher-order Rayleigh modes, including Sezawa wave (Fig. 4b). CONCLUSION Results presented allow on the base of derived dispersion equations and boundary conditions the definition of the elastic wave propagation parameters such as phase velocities, electromechanical coupling coefficients, and controlling coefficients in the layered piezoelectric structures subjected to the dc electric field influence. Naturally, the computer simulation to be executed it is necessary to know the linear and non-linear electromechanical properties of materials. The analysis of the dispersive parameters of elastic waves propagation in the BGO/fused silica and fused silica/LiNbO3 piezoelectric layered structures for a number of variants of dc electric field application has been executed. The dc electric field influence resulting of changes in the effective crystalline symmetry can lead to the transformation of bulk acoustic wave into the SAW type wave. For example, shear wave of BAW type in BGO/fused silica layered structure can be transformed into the wave like the Anisimkin Jr. mode. Note that there is the possibility to control the permission or prohibition of the wave propagation by the dc electric field application and the appropriate choice of the layer and substrate materials. ACKNOWLEDGMENTS This work has supported by the Russian Ministry of Education under the Federal Program "Research and Development on Priority Directions of Scientific and Technological Complex of Russia for 2007-2013" (State Contract № 16.513.12.3025). REFERENCES [1] I.A. Viktorov, Rayleigh and Lamb waves, Plenum Press, New York, 1967. [2] G.W. Farnell, E.L. Adler, Elastic wave propagation in thin layers, in: W.P. Mason, R.N. Thurston (Eds.), Physical Acoustics, vol. 9, Academic Press, New York, 1972, pp. 35–127. [3] R.G. Curtis, M. Redwood, Transverse surface waves on a piezoelectric material carrying a metal layer of finite thickness, J. Appl. Phys. 44 (5) (1973) 2002–2007. [4] Q. Wang, Wave propagation in a piezoelectric coupled solid medium, J. Appl. Mech. 69 (6) (2002) 819–824. [5] P. Vivekanandaa, V.R. Srinivasamoorthy, Surface-waves in thin-film coatings on piezoelectric half-space and on piezoelectric plate belonging to the (622) crystal class, J. Appl. Phys. 59 (4) (1986) 1301–1304. [6] P.J. Kielczynski, W. Pajewski, M. Szalewski, Shear-horizontal surface-waves on piezoelectric ceramics with depolarized surface layer, IEEE TUFFC 36 (2) (1989) 287–293.

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Fig. 1. Dispersive dependences of the Rayleigh and Love waves parameters as the functions of the h×f production for the waves propagating along the [100] direction in (001) BGO/fused silica layered structure: a) layered piezoelectric structure; b) phase velocities, c) EMCCs; d) αv at E||[010]. Phase velocities of slow shear bulk acoustic waves in the layer and the substrate are denoted as SSBGO and Ssub, respectively; slow quasi-shear bulk wave in the layer is denoted as QSSBGO. The Ri and Li curves indicate the modes of Rayleigh and Love (SH) type, respectively

Fig. 2. The depth profiles of displacement for the waves propagating along the [100] direction in (001) BGO/fused silica layered structure: a) QSH (solid line) and QSV (dashed line) modes at h×f = 2000 m/s, and E||[010]; b) QSH (solid line) and QSV (dashed line) modes at h×f = 1500 m/s and E||[100]

Fig. 3. Dispersive dependences of the Rayleigh and Love waves parameters as the functions of the h×f production for the waves propagating along the [100] direction in fused silica/Y-cut LiNbO3 layered structure: a) layered piezoelectric structure; b) phase velocities, c) EMCCs; d) αv at E||[100]; e) αv at E||[001]; f) αv at E||[010]. Shear bulk acoustic waves in the layer and the substrate are denoted as Slayer and SSNL, respectively. The Ri and Li curves indicate the modes of Rayleigh and Love (SH) type, respectively

Fig. 4. Dispersive dependences of the Rayleigh and Love waves parameters as the functions of the h×f production for the waves propagating along the (90°, -90°, 25.1°) direction of Y-cut LiNbO3 at the fused silica/LiNbO3 layered structure: a) phase velocities at E = 0; b) phase velocities at E = 5·106 V/m

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Table 1. Controlling coefficients αv for the BAW modes propagating along the [100] direction in (001) BGO/fused silica layered structure DC electric field direction BGO/fused silica BGO single crystal

[010]

BGO/fused silica BGO single crystal

[100]

-11

QL 0.014 0.014 1.58 1.4

αv, 10

m/V QFS 0.06 0.08

QSS 2.38 -0.06

90 43.8

-47 -45.9

The BGO/fused silica and fused silica/LiNbO3 piezoelectric layered structures are investigated The dispersion equations for Love waves propagating in the studied structure have been written DC electric field influence tends to the transformation of BAW into SAW type mode The electric field application to layered structure leads to the permission or prohibition of the wave propagation