Waves in fluids and solids Part 4 docx

Surface acoustic waves in multilayer structures In the linear theory of piezoelectricity and in the quasistatic electric approximation the system of differential equations, describing t

Proof a) flows out from considering the right-hand-side of (6.1), it ensures that all the terms

are positive at the assumption of positive definiteness of the elasticity tensor Proof b) also

follows from the right-hand-side of (6.1) by passing to a limit at h   n

Remarks 6.1 a) Expression (6.1)1 for the limiting speed c was apparently obtained for the s1

first time; expression for the limiting speed c was obtained by Kuznetsov (2006) and s2

Kuznetsov and Djeran-Maigre (2008) with a different asymptotic scheme

b) It follows from the right-hand side of (6.3) that the corresponding limiting speed is

independent of physical and geometrical properties of other layers It can be said that the

limiting wave is insensitive to the layers of finite thickness in a contact with a halfspace

c) Assuming in Eq (6.1)1 that the plate is single-layered with n  and taking 1

where  is Poisson’s ratio The plot on Fig.1 shows variation of the longitudinal bulk wave

speed and the limiting speed c versus Poisson’s ratio The plot reveals that in the whole s1

admissible range of   1;1 , the speed c remains substantially lower than the s1

longitudinal bulk wave speed The speed c approaches speed of the shear bulk wave only s1

Soliton-Like Lamb Waves in Layered Media 65

d) For a triple-layered plate with the outer layers of the same physical and geometrical

properties (such a case often occurs in practice) the limiting speed c s1 is

where index 1 is referred to the outer layers, and 2 corresponds to the inner layer Assuming

in Eq (6.7) that h1h2, while other physical properties of the layers have comparable

values, yields coincidence of c s2 with the shear bulk wave speed of the inner layer

Remarks 6.2 a) Expression (6.1)1 for the limiting speed c s1 was apparently obtained for the

first time; expression for the limiting speed c s2 was obtained by Kuznetsov (2006) and

Kuznetsov and Djeran-Maigre (2008) with a different asymptotic scheme

7 Acknowledgements

Authors thank INSA de Lyon (France) and the Russian Foundation for Basic Research

(Grants 08-08-00855 and 09-01-12063) for partial financial support

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3

Surface and Bulk Acoustic Waves in Multilayer Structures

V I Cherednick and M Y Dvoesherstov

Nizhny Novgorod State University

Russia

1 Introduction

The application of various layers on a piezoelectric substrate is a way of improving the

parameters of propagating electroacoustic waves For example, a metal film of certain

thickness may provide the thermal stability of the wave for substrate cuts, corresponding to

a high electromechanical coupling coefficient The overlayer can vary the wave propagation

velocity and, hence, the operating frequency of a device The effect of the environment (gas

or liquid) on the properties of the wave in the layered structure is used in sensors The layer

may protect the piezoelectric substrate against undesired external impacts Multilayer

compositions allow to reduce a velocity dispersion, which is observed in single-layer

structures In multilayer film bulk acoustic wave resonators (FBAR) many layers are

necessary for proper work of such devices Therefore, analysis and optimization of the wave

propagation characteristics in multilayer structures seems to be topical General methods of

numerical calculations of the surface and bulk acoustic wave parameters in arbitrary

multilayer structures are described in this chapter

2 Surface acoustic waves in multilayer structures

In the linear theory of piezoelectricity and in the quasistatic electric approximation the

system of differential equations, describing the mechanical displacements ui along the three

spatial coordinates xi (i = 1, 2, 3) and the electric potential  in the solid piezoelectric

medium, may be written in such view (Campbell and Jones, 1968):

2

2

j k

ijkl kij

i l k i

u u

i l i k

u e

In these equations cijkl is the forth rank tensor of the elastic stiffness constants, eijk is the third

rank tensor of the piezoelectric constants, ij is the second rank tensor of the dielectric

constants,  - the mass density, t – time, and the summation convention for repeated indices

is used The expression (1) contains three equations and (2) gives one more equation, totally

four equations These equations must be solved for each medium of all the multilayer

system, which is shown in Fig 1

Fig 1 Multilayer structure - substrate and M layers

The coordinate axis x1 direction coincides with the wave phase velocity v, the coordinate

axis x3 is normal to the substrate surface and the axis origin is set on this surface, as shown

in Fig 1 A solution of equations (1) and (2) we will seek in the following form:

Here j – amplitudes of the mechanical displacements, 4 – the amplitude of the electric

potential, b i – directional cosines of the wave velocity vector along the corresponding axises,

k = /v = 2/ – the wave number,  – a circular frequency,  – a wavelength Substitution

of (3) into (1) and (2) gives the system of four linear algebraic equations for wave

amplitudes:

2 4

Surface and Bulk Acoustic Waves in Multilayer Structures 71

This equation allows to determine the unknown directional cosine b 3 , if the values v, b 1, and

b 2 are set For flat pseudo-surface acoustic wave the values of the directional cosines are

following:

where  is the wave attenuation coefficient along the propagation direction For surface

acoustic wave the attenuation is absent and = 0 The equation (8) with taking into account

(9) gives the following eighth power polynomial equation with respect to the b value:

Coefficients ai of this equation are represented by very complicated expressions, depending

on material constants of the medium, a phase velocity v, and the attenuation coefficient 

For pseudo-surface acoustic waves ≠ 0 and therefore coefficients ai are complex values For

surface acoustic waves = 0 and coefficients ai are pure real values In this case roots of the

equation (10) are either real or complex conjugated pairs If ≠ 0, roots of the equation (10)

are complex but not conjugated So, solving (numerically certainly) the equation (10), we get

eight roots b (n) (n = 1, 2, …, 8), which are complex values in general case These values are the

eigenvalues of the problem Substituting each of these values into (7) and then into equation

system (6), we can define all four complex amplitudes ( )j n for each root b (n) Values

( )n

j

 represent the eigenvectors of the problem This procedure must be performed for the

substrate and for each layer Found solutions are the partial solutions of the problem or

partial modes

The general solution for each medium is formed as a linear combination of partial solutions

(partial modes) Quantity partial modes in the general solution for each medium must be

equal to quantity of boundary conditions on its surfaces Four boundary conditions on each

surface are used, namely three mechanical and one electrical one The substrate is

semi-infinite, i.e it has only one surface Hence only four partial solutions are required for

forming the general solution for the substrate It means that some procedure of roots

selection is required for substrate For surface acoustic wave four roots with negative

imaginary parts are selected from four complex conjugated pairs This condition of roots

selection corresponds to decreasing of the wave amplitude along the –x3 direction (into the

depth of the substrate), i.e to condition of the localization of the wave near the surface

Practically the procedure of roots sorting with increasing imaginary parts order is

performed and then four first roots are used for forming of the general solution

For pseudo-surface wave roots are not complex conjugated, but they also contain four roots

with negative imaginary part and also these four roots are first in the sorted roots sequence

In this case the roots selection rule is some different Three first roots in the sorted sequence

are selected, but the fourth root of this sequence is replaced with the fifth one (with the

positive imaginary part of minimal value) This condition corresponds to increasing of the

wave amplitude into the depth of the substrate and provide the energy conservation law

satisfaction (wave attenuates along the propagation direction x1 due to nonzero value of  in

the direction cosine b1, see (9)) For high velocity pseudo-surface wave (the second order

pseudo-surface wave or quasi-longitudinal pseudo-surface wave) only two first roots of the

sorted sequence are selected, the third and the fourth roots are replaced with the fifth and

the sixth ones

All these rules of roots selection are applied for substrate only For each layer of the

structure shown in Fig 1 there is no problem of roots selection, because each layer has two

surfaces and all eight roots (all eight partial modes) are used for forming of the general

solution for each layer

One must to note, that in some special cases the quantity of partial modes may be less, than

four for substrate and less, than eight for layers This must be taken into account at forming

of the general solution for corresponding case

So, the general solution for each medium is formed as a linear combination of corresponding

Here m is the medium number, N m = n0 + n1 + … + nm, nm – the quantity of partial modes in

the medium number m (m = 0 corresponds to a substrate, m = 1 corresponds to the 1st layer

etc., N0-1 = n0-1 = 0), Cn – unknown coefficients and a continuous numeration is used for them

(strange upper indices support this continuous numeration here and further)

The substrate is assumed the piezoelectric medium in all the cases and n0 = 4 in general case

(or less in some special cases) There are eight partial modes for each layer in the general

case if it is piezoelectric or six modes in the general case, if the layer is anisotropic

nonpiezoelectric or isotropic medium (dielectric or metal) For isotropic medium the second

component of the mechanical displacement u2 is decoupled with u1 and u3 and may be

arbitrary, for example one can set u2 = 1

Unknown coefficient Cn in (11) and (12) can be determined using the boundary conditions

on all the internal boundaries and on the external surface of the upper layer Unfortunately

it is impossible to formulate boundary conditions in the universal form, applicable to all the

combinations of the substrate and layers materials Therefore we must investigate different

variants of material combinations separately

For piezoelectric layers conditions of continuity of the mechanical displacements, electric

potential, normal components of the stress tensor and the electric displacement must be

satisfied for all the internal boundaries On the external surface of the top layer normal

components of the stress tensor must be equal to zero If this surface is open (free), the

continuity of the normal component of the electric displacement must be satisfied, if this

surface is short circuited, then electric potential must be equal to zero The stress tensor and

electric displacement in piezoelectric medium can be calculated by means of following

expressions:

Surface and Bulk Acoustic Waves in Multilayer Structures 73

In these equations j, k, l = 1, 2, 3, m = 0, 1, 2, … M-1 (not up to M!), where M is the quantity of

layers, x 3(m) = h1 + h2 + … + hm, x3(0) = 0 Equations (15a) represent the continuity of

mechanical displacements, (15b) – the continuity of the stress normal components, (15c) –

the continuity of the electrical potential, (15d) – the continuity of the electric displacement

normal component If surface x3 = x3(m) is short circuited by metal layer of zero thickness,

equations (15c) and (15d) must be changed The right part of the (15c) must be replaced

with zero, the left part of (15d) also must be replaced with zero and the right part of (15d)

must be replaced with the right part of (15c)

The boundary conditions equations for stress on the external surface of the top layer (m = M)

can be obtained from equations (15b) by replacing the right part of this equation with zero

Analogously by replacing the right part with zero the equation (15c) gives electric boundary

condition for the short circuited external surface In order to formulate the boundary

condition on the free external surface, the potential in the free space must be written in the

Here φ (M) is the potential of the external surface (x3 = x3(M)) The potential (16) satisfies

Laplace equation (that can be checked by direct substitution of (16) into this equation) and

vanishes at x3  ∞

The normal component of the electric displacement in the free space:

Here 0 is the dielectric permittivity of the free space Using the expression (17) we can get

the condition of the continuity of the normal component of the electric displacement on the

free (open) external surface:

The system of the boundary conditions equations contains n0 + n1 + n2 + … + nM equations

with the same number of unknown coefficients Cn In general case n0 = 4, n1 = n2 = … = nM = 8

For metal layers mechanical boundary conditions are the same as for the previous case (only

one must take into account, that piezoelectric constants of layers are zero) and the electric

boundary condition is formulated only for the substrate surface:

0

( ) 0 4 1

n n n n





This variant of boundary conditions is also valid, if the first layer is metal and all other

layers are non-piezoelectric dielectrics and metals in an arbitrary combination For this

variant in the general case n0 = 4, n1 = n2 = … = nM = 6

For isotropic dielectric layers the mechanical boundary conditions are the same as for the

previous case Electric boundary conditions became complicated and multi-variant because

any boundary may be either free or short circuited Only the single variant is simple – the

first boundary is short circuited For this variant the electric boundary condition is

presented by the single equation (19), such as for previous case

In general case the dependence of the potential in the free space is defined by equation (16)

and inside the m-th dielectric isotropic layer it must be written as:

Coefficients Am and Bm can be expressed by potentials on the layer boundaries, which

depend on the electric conditions on this boundaries (free or short) Using conditions of the

continuity of the potential and the normal component of the electric displacement one can

exclude all the boundary potentials and express the potential φ(1) in the first layer as

function of x3 This function will content only φ(0)(x3 = 0) – potential on the substrate

surface From the potential φ(1) one can express the normal component of the electric

displacement on the substrate surface and use the condition of the continuity of this value

for formulation of the electric boundary condition equation This is the single equation, but

its view significantly depends on the electric conditions on other boundaries

If all the boundaries are electrically free and there is only the single layer, the equation,

which describes the electric boundary conditions, can be written so:

Surface and Bulk Acoustic Waves in Multilayer Structures 75

Here and hereinafter εm (m = 1, 2, … M) is the relative permittivity of the m-th layer R2 in

(21b) is the recurrent coefficient, which allows to obtain the equation for two layers from

equations (21) for one layer For the single layer R2 = 1, and for two layers:

The recurrent coefficient R3 gives possibility to obtain the equation for three layers from

equation for two layers:

And so on, i.e the equation of electric boundary conditions for m + 1 layers may be obtained

from the equation for m layers by using the recurrent coefficient Rm+1 (RM+1 = 1, if M is the

total number of layers) To obtain the equation for M layers one must write equation for one

layer, then for two layers and so on until the equation for M layers will be obtained

If one of the boundary surfaces x3 = x3(m) is short circuited (metalized), then electric

conditions of all the further boundaries are unimportant, because the electric field outside