## Abstract

The paper discusses the properties of the acoustic waves guided by an interface inside piezoelectric media. The interfaces of two types have been considered: (i) an infinitesimally thin metallic layer inserted into homogeneous piezoelectric crystal of arbitrary symmetry; (ii) rigidly bonded crystals whose piezoelectric coefficients differ by sign while the other material constants are identical. Several general theorems have been proved regarding the existence of interface acoustic waves (IAWs) propagating more slowly than bulk waves. In particular, a sufficient condition for the existence of such ‘slow’ IAWs has been derived. The propagation of leaky IAWs has been studied. Special attention has been paid to the analysis of the situation when the imaginary component of the leaky IAW velocity vanishes, resulting in the appearance of non-attenuating IAWs travelling faster than the slow transverse bulk wave. The computations performed for LiNbO_{3} and LiTaO_{3} illustrate general conclusions.

## 1. Introduction

The impedance method has enabled the study of the existence of surface acoustic waves (SAWs) on anisotropic media (Ingebrigsten & Tonning 1968; Lothe & Barnett 1976*a*–*c*; Barnett & Lothe 1985). According to the theorems proved by Lothe & Barnett (1976*a*–*c*) and Barnett & Lothe (1985), at most, one SAW and at most, two SAWs exist on the mechanically free surface of non-piezoelectric and piezoelectric substrates, respectively. SAWs must exist on non-piezoelectric media provided that the slowest limiting bulk wave (LBW) is not exceptional and does not need to exist otherwise. It has also been established that at least one SAW can propagate on the metallized surface of piezoelectric media if the slowest LBW does not meet the conditions of mechanically free metallized surfaces. Alshits *et al*. (1992, 1994*b*) have studied the SAW propagation on half-infinite substrates possessing piezoelectric, piezomagnetic and magnetoelectric properties. Not more than two SAWs appear on such substrates. On the other hand, unlike non-piezoelectric and piezoelectric media, piezoelectric/piezomagnetic substrates support SAW propagation on the mechanically clamped surface.

Acoustic waves on the interface between two solids (interface acoustic waves (IAWs)) are far more difficult to study, as compared with SAWs, by solving the boundary-value problem explicitly. For instance, the equation of the velocity of Stoneley waves in isotropic solids is derivable explicitly. However this equation appears to be of fourth degree with respect to *v*^{2} with rather involved coefficients so that its solvability can be analysed only when the material constants fulfil special relations (Viktorov 1981). The impedance method again appears to be a useful tool, making it possible to draw a number of conclusions regarding IAWs in anisotropic media having various types of contact. It has been found that the interface between two rigidly bonded non-piezoelectric media of arbitrary anisotropy can guide, at most, one IAW (Barnett *et al*. 1985). At the same time, two IAWs can travel on the sliding contact (Barnett *et al*. 1988). In piezoelectric crystals, one, two or three IAWs can appear, depending on the mechanical and electrical boundary conditions on the interface (Abbudi & Barnett 1990; Alshits *et al*. 1994*a*).

This paper discusses the existence of subsonic IAWs guided by: (i) an infinitesimally thin metallic layer inserted into a piezoelectric medium of arbitrary anisotropy and (ii) a ‘180° domain wall’. By ‘180° domain wall’ (180DW) we mean the interface between rigidly bonded crystals whose piezoelectric constants are of opposite sign, with the other material constants being identical. These special cases have not been considered by Abbudi & Barnett (1990) and Alshits *et al*. (1994*a*).

It should be said that numerical computations of IAWs in crystals of various symmetry groups guided by metallic insertion and 180DW have been carried out by Laprus & Danicki (1997), Mozhaev & Weihnacht (1998), Dvoesherstov *et al*. (2002) and Camou *et al*. (2003). As for analytic results, the boundary-value problem can be solved explicitly in full only for purely shear horizontal (SH) IAWs (Maerfeld & Tournois 1971; Kessenikh *et al*. 1972; Peusin & Lissalde 1974; Danicki 1994). Note that Darinskii & Lyubimov (1999) have proved a number of general theorems concerning SH IAW propagation. Note also that a series of studies have recently been performed of the interaction between IAWs and 2D electron gas at hetero junctions of semiconductors (see, for example, Kosevich 1997; Ponomarev & Efros 2001).

## 2. Basic equations

The boundary conditions put relations on the following characteristics of plane partial modes exp i*k*[(*x*+*p*_{α}*z*)−*vt*]: the polarization vector *A*_{α}, the potential *Φ*_{α}, the traction *f*_{α} and the projection *D*_{nα} of the electric displacement *D*_{α} on the normal *n* to the surface. These quantities, together with the decay factors *p*_{α}, satisfy an eigenvalue problem which can be formulated in two forms (Lothe & Barnett 1976*b*,*c*):(2.1)

The vectors are constructed as *ξ*_{Xα}=(*U*_{Xα}, *V*_{Xα})^{T} from four-component vector-columns(2.2)where *L*_{α}=i*k*^{−1}*f*_{α} and *D*_{α}=i*k*^{−1}*D*_{nα}. The matrix is represented in a concise form as(2.3)

In equation (2.3) the symbols of the type (*ab*) stand for 4 × 4 matrices with components (*ab*)_{IJ}=*a*_{k}*E*_{kIJl}*b*_{l}, *I*, *J*=1,…,4,where ** a** and

**are a pair of three-component vectors and**

*b**E*

_{kIJl}=

*c*

_{kIJl}−

*ρv*

^{2}

*m*_{k}

*m*_{l}

*δ*

_{IJ},

*I*,

*J*=1, 2, 3,

*E*

_{k4Jl}=

*e*

_{kJl},

*J*=1, 2, 3,

*E*

_{kI4l}=

*e*

_{lIk},

*I*=1, 2, 3,

*E*

_{k44l}=−

*ϵ*

_{kl}. The unit vector

**lies in the surface indicating the direction of wave propagation, (**

*m**nn*)

^{−1}is the inverse of the matrix (

*nn*). By interchanging the fourth and eighth rows, as well as the fourth and eighth columns in equation (2.3), one obtains .

Taking into account the relation , where has components *T*_{ij}=*T*_{i+4,j+4}=0, *T*_{i+4,j}=*T*_{i,j+4}=*δ*_{ij}, *i*, *j*=1,…,4, one introduces the *ξ*_{Xα} such that(2.4)and, hence, the completeness of *ξ*_{Xα} results in(2.5)where the symbol ‘⊗’ stands for dyadic multiplication.

In what follows, we number non-uniform modes such that , and Im(*p*_{α})>0, *α*≤4, and assign to a pair of uniform (reflected and incident) modes either the subscripts *α* and ‘*α*+4’ or ‘r’ and ‘i’, respectively.

Let us build up the ‘integral’ matrices ,(2.6)(the , can also be introduced as integrals of (2.3) over the orientation angle of the mutually orthogonal vectors ** m** and

**(see Lothe & Barnett 1976**

*n**b*,

*c*)).

Due to (2.4), , *α*=1,…,4, or in full(2.7)

Equation (2.7) allows the introduction of the impedances relatingwhere the *b*_{α} are the amplitudes of the partial modes *α*=1,…,4 (see Ingebrigsten & Tonning 1968; Lothe & Barnett 1976*a*,*b*,*c*):(2.8)

The matrices(2.9)are symmetric, while are antisymmetric; the latter follows from the identity , which can be derived using equations (2.4) and (2.5). In view of these properties the vectorsconstructed from *V*_{Xα} and *U*_{Xα} associated with the modes *α*=5,…,8 are expressed in terms of one another as(2.10)

Note that the are real in the subsonic interval so that are Hermitian matrices. However, at (figure 1), are complex and are not Hermitian.

## 3. Subsonic IAWs

### (a) Metallic insertion

Alshits *et al*. (1994*a*) have shown that not more than two IAWs exist on the metallized rigid contact between two different piezoelectric media. In the present paper, the case of identical media is studied. We will also use the impedance approach but in a different way to that employed by Alshits *et al*. (1994*a*), as the conclusions to be made below cannot be derived directly from the relations used in Alshits *et al*. (1994*a*).

In accordance with our notation, IAWs can involve the modes *α*=1,…,4 in the medium occupying the half-space *z*>0 and the modes *α*=5,…,8 in the medium occupying *z*<0. The boundary conditions require that(3.1)

In terms of the impedance , accounting for equations (2.2), (2.8) and (2.10), the equation of the IAW's velocity can be brought into the form(3.2)where and are the 3×3 upper blocks of the matrices and , respectively; the symbol ‖…‖ stands for the determinant.

At the same time(3.3)

It is known that does not vanish and does not tend to infinity at (Lothe & Barnett 1976*b*,*c*) (the determinant can tend to infinity as *v* approaches limiting velocities). Hence, unless (and in figure 1), equation (3.1) is equivalent to(3.4)Here we have taken advantage of equation (2.9).

There is another way of deriving equation (3.4). Putting *b*_{α}=*Φ*_{α}, *α*=1,…,4, and *b*_{α}=−*Φ*_{α}, *α*=5,…,8, in equation (3.1) we obtain, referring to equation (2.5), that all boundary conditions will be identically fulfilled excluding the vanishing of potential. Referring again to equation (2.5), we conclude that the potential becomes zero from both the sides of the interface when .

Let us analyse the existence of solutions to equation (3.4). The properties of as a function of the velocity *v* within the range have been thoroughly studied by Lothe & Barnett (1976*b*,*c*) in relation to the existence theorems for SAWs: at *v*=0, steadily decreases with increasing *v*, and as if the potential1 *Φ*_{d3} of the LBW at does not equal zero. Hence, when *Φ*_{d3}≠0, the value of goes through zero once in the subsonic interval.

However, if *Φ*_{d3}=0, then at Thus we arrive at the following conclusions:

an infinitesimally thin metallic layer inserted into a piezoelectric medium can guide at most one subsonic IAW;(3.5)

subsonic IAWs must exist if the slowest LBW is piezoactive;(3.6)

subsonic IAWs do not exist if the slowest LBW is not piezoactive.(3.7)

In the latter case, instead of IAWs, there appear to be LBWs propagating along the metallic insertion with zero potential.

The directions supporting bulk waves with zero potential form continuous one parametric sets (lines) in the two dimensional (2D) space of angles specifying the wave normal (Shuvalov & Radowicz 2000), i.e. given one angle, the second angle is found from an equation. A bulk wave appears to be limiting relative to the surfaces (interfaces) parallel to its group velocity. Hence,

the space of the non-existence of IAWs is two-parametric and can be represented as a set of surfaces in the three dimensional (3.8)(3D) space of angles specifying the interface and the direction of the IAWs' propagation in it.

### (b) ‘180° domain wall’

Bi-crystals whose parts differ only by the sign of piezoelectric coefficients when their material constants are referred to a common coordinate system, can be fabricated from two crystals (of the same substance) possessing the plane of symmetry. One of the crystals must be rotated through 180° about the perpendicular to the plane of symmetry before both the crystals are cut along the equivalent planes and the upper (lower) part of crystal 1 is joined to the lower (upper) part of crystal 2.

If the symmetry group does not contain planes of symmetry, then one can use left- and right-hand crystal types of a substance.

The boundary conditions imply(3.9)where the superscripts ‘(u)’ and ‘(l)’ label the vectors **ξ**_{Xα} pertaining to the upper and lower ‘domain’, respectively. Changing the sign of the piezoelectric coefficients changes only the sign of the potential and electric displacements.2 Therefore, the sign of *Φ*_{α} and *D*_{α} is different in and while the other components of and are alike. Taking this fact into account, we seek the coefficients entering (3.5) in the form *b*_{α}=*Φ*_{α} +*γD*_{α} and *b*_{α+4}=−(*Φ*_{α+4}+*γD*_{α+4}), *α*=1,…,4, where *γ* is an unknown to be determined and all the *Φ*_{α}, as well as all the *D*_{α}, pertain to the same part of the structure. In view of (2.5), the mechanical boundary conditions are satisfied identically. The electrical boundary conditions yield two equations:(3.10)These equalities are met simultaneously if the velocity satisfies(3.11)Thus we arrive at the dispersion equation for IAWs.

Note that equation (3.11) describes all possible IAWs, although it has been derived somewhat artificially. The quickest way to verify this fact is to solve the reflection problem using equation (2.5). It will then be seen that the denominator of the coefficients of mode conversion is exactly the function on the left-hand side of equation (3.11). However, equation (3.11) can also be derived using equations (2.7)–(2.10) instead of guessing *b*_{α}, but this is a more involved way.

According to Lothe & Barnett (1976*b*,*c*) is negative when Recollecting the properties of we deduce from (3.11)

that no IAWs exist on 180DW if the LBW is non-piezoactive.(3.12)

The results from §4 (see also Darinskii & Weihnacht 2004) reveal that, given the geometry of propagation, the appearance of IAWs generally is dependent on the material constants and that only one IAW can exist at small piezoelectric coefficients.

Let us choose on the slowest branch a piezoactive bulk wave having group velocity * s* and producing the electric displacement

*. This bulk wave is LBW with respect to the boundary perpendicular to the vector*

**D***‖*

**n***×*

**s***and*

**D***D*

_{d3}=0 while

*Φ*

_{d3}≠0. In this case and are finite at , . On the other hand, as . Hence, at least one subsonic IAW results for such an orientation and within its neighbourhood. Thus,

any piezoelectric media can be used to fabricate ‘domain structures’ supporting subsonic IAWs.(3.13)

It is required that the ‘domains’ be specially oriented.

Consider now a number of particular geometries of propagation. We assume that one of the following four conditions are met:(3.14)

Here, the *x*-axis and the (*x*,*y*)-plane of the current coordinate system are meant (figure 1). For instance, the (0,*θ*,0) and (90,90,*ψ*) cuts3 of 3*m* crystals, such as LiNbO_{3} and LiTaO_{3}, fall into cases 1 and 3, respectively.

The decay parameters come about in pairs (*p*_{α}, *p*_{α+4}=−*p*_{α}) if the media are oriented according to (3.14). As a result, the identity *Φ*_{α}*D*_{α}≡*Φ*_{α+4}*D*_{α+4} holds (Darinskii 2000*a*, 2002). Hence,(3.15)so that equation (3.7) splits into equation (3.4) and(3.16)

Thus, two types of IAW can travel along the ‘symmetric’ 180DW oriented in accordance with equation (3.14). One IAW with the velocity satisfying equation (3.4) does not produce the electric potential on the interface. In turn, equation (3.16) determines the velocity of the IAW which has the vanishing normal component of the electric displacement on the interface. To check the latter we can either employ the relations that the components of the vectors **ξ**_{Xα} obey at orientations (3.14) (Darinskii 2000*a*, 2002), or, the more general way, derive that the dispersion equation for IAWs on the electrically open interface between two rigidly bonded solids (see Alshits *et al*. 1994*a*) simplifies to equation (3.16) when these solids are identical.

As has been pointed out, does not vanish below . Hence, for the geometries of propagation listed in equation (3.14)(3.17)

In cases (3.14) LBWs with *p*_{d3}=0 have either *D*_{d3}=0 or *Φ*_{d3}=0. Therefore, IAWs either exist for any values of the angle *θ* (or *ψ*) specifying orientations (3.14) or exist within particular intervals of this angle or do not appear at all. At the points confining the existence intervals, the shape of the outmost slowness curve changes from convex to concave and the LBW travelling strictly along the *x*-axis is no longer the slowest one. The two slowest LBWs at are piezoactive and IAWs will appear if IAWs do not exist when the curve is convex.

For instance, in (90,90, *ψ*) LiNbO_{3} structures, the slowest bulk wave travelling along the interface (plane of symmetry) is polarized perpendicular to the interface when the *ψ* are such that the associated transonic state is the slowest one, thereby being the upper limit of the subsonic interval. Accordingly, the IAW branch (figure 2) exists only when the outermost slowness surface becomes concave. One sees that the difference between the IAW's velocity and the limiting velocity is not large, although LiNbO_{3} is a fairly strongly piezoelectric material.

## 4. Leaky IAWs

The solubility of equations (3.4) and (3.10) within the intersonic intervals can be analysed by assuming that the piezoeffect is weak. When *e*_{ijk}=0, an LBW freely travels along the ‘interface’. With *e*_{ijk}≠0, the LBW generally stops to meet the boundary conditions. However, there can appear a leaky IAW, the predominant components of which are the non-uniform decaying modes arising from the LBW. The decay factors of these modes are determined by the quantity or near positive or negative curvature transonic states (TS^{(+)} and TS^{(−)}; see figure 1), respectively, where **A**_{d3} is the polarization vector of LBW and (Darinskii 1998, 2000*b*). To the lowest approximation, Δ*p* can be found in terms of the contractionsthat determine the linear term in *e*_{ijk} of potential *Φ*_{d3} and electric displacement *D*_{d3} of the LBW after the piezoeffect is taken into consideration:(4.1)here, * F*=(

**0**,0,

**0**,1)

^{T}and

*=(*

**G****0**,1,

**0**,0)

^{T}are eight-component column vectors (they differ from

*and*

**F***introduced in Darinskii & Weihnacht (2003, 2004) by the factors and , respectively, where*

**G***q*=[

*ϵ*

_{nn}

*ϵ*

_{mm}−(

*ϵ*

_{mn})

^{2}]

^{1/2}and

*ϵ*

_{nn}=

*ϵ*

_{ij}

*n*

_{i}

*n*

_{j},

*ϵ*

_{mm}=

*ϵ*

_{ij}

*m*

_{i}

*m*

_{j}and

*ϵ*

_{nm}=

*ϵ*

_{ij}

*n*

_{i}

*m*

_{j}). The matrix is real. The contractions and are expressed in terms of the contractions and , where is the correction to linear in

*e*

_{ijk}(Darinskii & Weihnacht 2003).

The analysis performed in our previous paper concerns IAWs arising when only one LBW exists at a given limiting velocity (TS of type 1; figure 1*a*). In this section we consider the situation when two TSs of type 1 appear simultaneously (figure 1*b*). Let two pairs of eigensolutions to equation (2.1) (pair 1: (**ξ**_{X3},*p*_{3}) and (**ξ**_{X7},*p*_{7}); pair 2: (**ξ**_{X2},*p*_{2})and (**ξ**_{X6},_{p6})) merge into two degenerate eigensolutions (**ξ**_{Xd3},*p*_{d3}) and (**ξ**_{Xd2},*p*_{d2}), respectively, at . These two degenerate solutions describe two LBWs.

In the vicinity of TSs, using the perturbation theory developed by Darinskii & Weihnacht (2003, 2004) (see also Darinskii 1998, 2000*b*), we can takewhere or (by analogy with Δ*p*). It is also sufficient to neglect the piezoelectric contribution to potential *Φ*_{4} of the ‘electrostatic’ mode putting (see Darinskii & Weihnacht 2003, 2004). We can insert these *Φ*_{2,3,4} into equation (3.4) and discard the potential induced by the ‘acoustic’ mode *α*=1 to obtain(4.2)

If the curvature of the slowness curve at TS is positive, then **ξ**_{d2},_{d3} are real. However, if the curvature is negative, then **ξ**_{d2,d3} are purely imaginary. Hence, and in the first and the second cases, respectively. The quantities Δ*p*_{α} have been introduced such that only the solutions corresponding to Im(Δ*p*_{α})>0 have a physical meaning. Therefore, IAWs must arise on the metallic insertion in the neighbourhood of TS^{(+)} once *e*_{ijk}≠0. IAWs cannot appear near TS^{(−)}.

Similar to the case of a single LBW (Darinskii & Weihnacht 2004), the lowest correction to the IAWs velocity is real and has the order of , where *κ* is the electromechanical coupling coefficient. The imaginary component of the velocity, which characterizes the attenuation of IAWs because of the leakage, has the order . To derive an expression for , higher-order terms with respect to *e*_{ijk} must be included in (4.2).

Applying an analogous approximation to equation (3.11) yields the following equation of the IAW's velocity in 180DW bi-crystals:(4.3)

The solution can exist in the neighbourhood of both TS^{(+)} and TS^{(−)} depending on the material constants.

In the case of ‘symmetric’ 180DW structures (3.8), and and Δ*p*_{2}=Δ*p*_{3}=Δ*p*. Equations (3.4) and (3.10) reduce to(4.4)These equations differ from the equations for the case of a single LBW only by a factor of 2 before . One sees that both TS^{(+)} and TS^{(−)} can give rise to leaky IAWs, the type of IAW depending on the curvature of the slowness curve at TS.

In figure 3, the real and imaginary components of the leaky wave velocity are shown as functions of the angle θ on (0,*θ*,0) interfaces in LiNbO_{3}. The analogous dependences for (0,*θ*,0) interfaces in LiTaO_{3} are depicted in figure 4. It is worth noting that attenuation is weak enough throughout the angle range. Moreover, the imaginary component vanishes twice in LiNbO_{3} at particular values of *θ* so that these orientations support the propagation of a ‘supersonic’ IAW (SIAW1), which does not attenuate because of the energy leakage, although its velocity exceeds the velocity of bulk waves.

At the same time, in LiTaO_{3} reduces to the minimum at *θ*≈−94° rather than vanishes.

Using a method similar to that exploited by Darinskii & Weihnacht (2002) to analyse the number of the conditions that must be satisfied for ‘supersonic’ SAWs to exist, one can show that the existence of SIAW1s on the interface (metallized or not) between two arbitrary solids puts five conditions on the free parameters. However, as applied to ‘metallic insertion’ the number of conditions generally reduces to three. Indeed, since becomes complex valued above , three angles and the velocity are found from the equations and By referring to (2.5)(4.5)in the first intersonic interval; the subscripts ‘r’ and ‘i’ label the reflected and incident waves (upper half-space), respectively, *α*=2, 3, 4 are still non-uniform modes. We have also taken into account that *ξ*_{i} is real and *ξ*_{r} is purely imaginary if the normalization (2.4) is adopted. Thus, three equalities must be fulfilled simultaneously for SIAW1s to appear:(4.6)

The first equation generally allows one to exclude *v*. This fact can be established as follows. Let us take a hexagonal medium with . The metallic layer inserted parallel to the sixfold axis supports SH SIAW1s in the transverse isotropic direction. We can explicitly calculate to obtain that this derivative does not vanish at *v*_{IAW}. Accordingly, after arbitrary changes of orientation and material constants there still remains so that will be solvable with respect to *v*.

The second and third equations put one condition each on the angles, because the directions supporting bulk waves with zero potential form continuous one-parametric subsets (lines) in the 2D space of angles specifying the wave normal. To summarize, the existence space of SIAW1s guided by the metallic insertion can be represented as lines in the 3D space of orientation angles specifying the geometry of propagation of IAWs.

Note also that(4.7)

If the metallic insertion assumes one of the orientations (3.8), then |*Φ*r|≡|*Φ*i| and we have two equations on *v* and one orientation angle. Therefore, it looks natural that SIAW1s realizes in (0,*θ*,0) LiNbO_{3} structures at isolated values of *θ* (figure 3).

Figure 5 depicts the SIAW1s lines we have found in LiNbO_{3}. In this case, we specify the geometry of propagation of SIAW1s by three angles, *α*, *β* and *γ*, measuring the successive rotations about the *z*-, *y*-and *x*-axes, respectively. The *z*-axis coincides with the threefold symmetry axis, the *x*-axis is along the propagation direction and the *y*-axis is perpendicular to the sagittal plane. The dependence of *β* and *γ* on *α* is depicted within the sector −30°≤*α*≤30°. In the sector 30°≤*α*≤60°, one has *β*(*α*)=*β*′(60−*α*) and *γ*(*α*)=−*γ*′(60−*α*), where *β*′(*α*) and *γ*′(*α*) denote the values of functions within 0°≤*α*≤30°. In the sector −60°≤*α*≤0°, one has *β*(*α*)=−*β*″(−*α*) and *γ*(*α*)=*γ*″(−*α*), where *β*″(*α*) and *γ*″(*α*) are the values of functions within the sector 0°≤*α*≤60°. In the other 120° sectors, these curves are of exactly the same shape.

Note, first of all, that the two points of vanishing attenuation shown in figure 3 belong to the same line. This line forms a loop (curve 1–1′).

There are also loops lying completely within 30° sectors (curves 2 and 2′). A third group includes ‘equivalent’ lines in figure 5 (curves 3 and 3′) which are closed within a 120° sector but not −60°≤*α*≤60°.

It has already been mentioned that does not vanish in the case of (0,*θ*,0) LiTaO_{3}. It could be thought that the line of SIAW1s on which the solutions for (0,*θ*,0) cuts lie (loop 1–1′) splits into two loops not crossing the line *α*=0 when the material constants change from LiNbO_{3} to LiTaO_{3} (instead of loop 1–1′ there appear two loops resembling loops 2 and 2′ in figure 5). However, in fact, the transition from LiNbO_{3} to LiTaO_{3} develops in a different manner (figure 6). The material constants of ‘intermediate’ media are determined in terms of the sets of material constants of LiNbO_{3}, and LiTaO_{3} as , where the parameter *x* runs from 0 to 1. The loop at *x*=0 in figure 6 is loop 1–1′ in figure 5. With increasing *x* this loop shrinks, becoming a point at an *x*-value in the interval 0.296<*x*<0.297. SIAW1s do not exist at greater *x*.

In the second intersonic interval, where *rj*, *ij* (*j*=1,2) label two pairs of the bulk reflected/incident waves. As a result, five equations must be obeyed for non-attenuating IAWs to exist in this interval. The four free parameters cannot fulfil five equations unless special conditions are met so that non-attenuating IAWs generally do not appear in the second intersonic interval.

As to ‘supersonic’ IAWs on 180DW, our analysis shows that, generally, at least four conditions must be met for such a wave to appear in the first intersonic interval. Since 180DW ‘leaves’ only four changeable parameters, including velocity, SIAW1s exist only at isolated geometries of propagation. However, the branches of SIAW1s can exist in specific situations. In particular, consider the 180DW structure constructed from 6 mm crystals with anti-parallel sixfold axes. When the interface is parallel to the sixfold axes, the SH IAW propagates along the transverse isotropic direction (Maerfeld & Tournois 1971; Kessenikh *et al*. 1972). This IAW has vanishing potential on the interface. If , the SH IAW is ‘supersonic’. We can change the direction of propagation, keeping the interface parallel to the sixfold axes, or rotate the interface around the perpendicular (*x*-axis) to the sixfold axes, keeping the *x*-axis as the direction of propagation. In both cases we will have a ‘symmetric’ 180DW (cases 4 and 1 in equation (3.8), respectively). The IAW will have zero potential on the interface and remain non-attenuating. The point is that the non-SH polarized transverse bulk wave in 6 mm media is non-piezoactive for all directions of propagation. This wave is the slowest bulk wave once at least around the transverse isotropic geometry of propagation. Thus, the existence space of SIAW1s in the structure being discussed can be represented as two 2D surfaces rather than isolated points. These surfaces intersect. The line of intersection is the ‘axis’ along which the angle specifying the rotation of the interface about the sixfold axes is counted.

## 5. Conclusion

We have shown that an infinitesimally thin metallic layer, being inserted into a piezoelectric medium of arbitrary symmetry, gives rise to not more than one subsonic IAW. The existence criterion for such IAWs has been derived. It states that the subsonic IAW must appear unless the slowest LBW is non-piezoactive. If the slowest LBW is non-piezoactive, the IAW does not come about. The orientations of the interface not supporting subsonic IAWs exist in media of arbitrary symmetry. The non-existence space of IAWs generally is a set of 2D surfaces in the 3D space of orientation angles. However, in 6*mm* (and ∞*mm*) media it can be that IAWs do not exist within 3D regions, because the slowest bulk wave can be non-piezoactive for all directions of propagation (see the end of §4). Instead of subsonic IAWs, there exist ‘supersonic’ IAWs in this 3D region. This is also an exclusion stipulated by the aforementioned specific feature of the transverse bulk wave; as has been proved, the existence space of ‘supersonic’ IAWs on the metallized insertion generally is a set of 1D lines (figure 5). On the other hand, an example has been found of the case when the ‘supersonic’ IAW exists at an isolated orientation.

At most, one subsonic IAW exists in 180DW bi-crystals oriented in accordance with equation (3.8). This IAW has vanishing potential on the interface. IAWs with vanishing normal projection of the electric displacement do not appear in the subsonic region. However, both types of wave can exist in the intersonic intervals, where generally they come about as leaky solutions.

On a 180DWof general type, subsonic IAWs produce non-vanishing potentials and electric displacements on the interface. Therefore, they are sensitive to metallization and can be generated by interdigital transducers.

It is known that the existence of leaky waves strongly affects the reflection of bulk waves from the interface. The analysis of the specific behaviour of the coefficients of plane mode conversion arising because of the resonance excitation of leaky waves will be presented elsewhere.

## Acknowledgments

A.N.D. thanks Deutsche Forschungsgemeinschaft (grant 436RUS 113/645/3-2) and the Leibniz Institute for Solid State and Materials Research, Dresden, for financial support. He also thanks the Russian Foundation for Basic Research for partial financial support. M.W. is grateful to Dr R. Wobst for his programming work.

## Footnotes

↵For the sake of uniformity we assign to an LBW the subscript

*α*=*d*3, as in our earlier papers.↵This rule can be applied to all eight modes, including the ‘electrostatic’ ones with non-vanishing potential as

*e*_{ijk}→0.↵The orientation is specified by Euler angles.

- Received April 7, 2004.
- Accepted June 24, 2004.

- © 2005 The Royal Society