Surface and Pseudo-Surface Acoustic Waves

in Multi-Layered Structures

V. I. Cherednick and M. Yu. Dvoesherstov

Nizhny Novgorod State University

Russia, 603950, N. Novgorod, Gagarin av. 23

Abstract - The general problem of calculation of propagation characteristics of surface and pseudo-surface acoustic waves on the piezoelectric substrate with some layers of different materials on the substrate surface is considered. Some concrete results are presented, which illustrate possibilities of improvement of the basic wave characteristics by means of various layers.

  1. INTRODUCTION

Surface and pseudo-surface acoustic wave (SAW and PSAW) devices with layered and multi-layered structures are widely used nowdays, because such structures allow to improve the device properties and solve some problems which can not be solved without layers (for example, gas sensors).

In this work the influence of various materials on the piezoelectric substrate on the propagation characteristics of the surface and pseudo-surface acoustic waves is considered. The general equations, including the wave equations and the boundary conditions equations for multi-layered systems are formulated. Various materials as layers are considered, in particular crystal layers, metallic layers and non-piezoelectric dielectric layers.

Some new results are obtained for some concrete combinations of the substrate and layers (quartz, lithium niobate, metals, dielectrics). All the basic results are represented as two dimensional contour maps, those show dependencies of the propagation characteristics on two parameters (e.g. layer thickness and Euler angle).

It is shown, that some combinations of the substrate and the layer (layers) allow to improve some propagation characteristics. In particular, some cuts and orientations, which are not temperature stable, can became such by means of the metallic layer evaporation (for example, Al, Au and others on YX-quartz). The piezoelectric layer with the large coupling coefficient on the piezoelectric substrate with the high temperature stability allows to get the SAW device with high coupling coefficient and excellent temperature properties. For example, the layer of lithium niobate on the quartz substrate allows to obtain the coupling coefficient almost 4 % with the zero temperature coefficient of delay (or frequency).

The dielectric layer with the metallic one allows to get protection of the crystal surface against external mechanical, chemical and electrical influences without of the significant deterioration of the wave propagation properties.

The metallic layer allows also significantly reduce the pro-

pagation attenuation of the first and second order pseudo-surface acoustic waves.

II. WAVE EQUATIONS

The wave equation system in piezoelectric medium in quasi-static approaching is following:

(1)

i, j, k, l = 1, 2, 3

Here Xi – co-ordinates (X1 – along the propagation direction, X2 – along transversal direction along the surface, X3 – along the normal to the surface – see Fig. 1), cijkl, eijk, εij – tensors of the elastic, piezoelectric and dielectric constants respectively in the working co-ordinate system, ρ – the mass density, ui – mechanical displacements, φ – the electric potential. Twice repeated indexes in (1) and hereinafter mean summation.

Equations (1) must be solved for each medium of the system, shown in Fig. 1.

According to technique, described in [1], the solution of the system (1) can be written as:

uj = αjexp(ikb3X3)exp[ik(b1X1 – vt)] j = 1, 2, 3

φ = α4exp(ikb3X3)exp[(ik(b1X1 – vt)]. (2)

X3

LayerMhM

Layer 2h2

Layer 1 h1

Substrate 0 X1

Fig. 1. Multi-layered structure - substrate and M layers.

Here αj – magnitudes of mechanical displacements along the X1, X2 and X3 axes, α4 – magnitude of the electric potential, - the wave number, λ – the wavelength, ω – frequency, v – the phase velocity, t – time, b1 and b3 – coefficients, which define dependencies of the wave magnitudes on the X1 and X3 axes respectively.

Magnitudes and coefficients b3 must be determined for each medium. The phase velocity and the coefficient b1 must be the same for all media (substrate and layers). The coefficient b1 can be presented as:

b1 = 1 + iδ, (3)

where δ is the real positive value – attenuation along the propagation direction. For SAWs δ = 0, for PSAWs δ > 0.

Substitution of (2) into (1) gives in general case the 8th order polynomial equation relatively b3, which gives 8 complex roots b3(n) for fixed value of the velocity (and attenuation δ for PSAWs). Each root gives the partial solution (mode) of type (2). The general solution in each medium is formed as a linear combination of such partial modes:

i, j = 1, 2, 3

(4)

Here b1(n) = 1 + iδ, b2(n) = 0 for all media, b3(n) – roots, which are individual for each medium, Nm = n0 + n1 + … + nm, nm – the quantity of partial modes in the medium number m (m = 0 – substrate, m = 1 – the 1st layer etc., N0-1 = n0-1 = 0), Cn – unknown coefficients. Twice repeated index i in (4) means a summation.

The substrate is the piezoelectric medium in all the cases and n0 = 4 in general case (or less in some special cases). For SAWs only roots with negative imaginary parts from 4 pairs of complex conjugate roots are selected for (4). For 1st order PSAWs the 4th root with positive imaginary part is selected. For 2nd order PSAWs the 3rd and 4th roots with positive imaginary part are selected.

For each layer all the roots are included in combination (4). There are 8 partial modes for each layer in the general case if it is piezoelectric or 6 modes in the general case, if the layer is isotropic medium (dielectric or metal).

III. BOUNDARY CONDITIONS

Unknown coefficient Cn in (4) 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.

A. Piezoelectric layers.

In this case 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 [2]. On the external surface of the upper layer normal components of the stress tensor must be 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 zero.

The stress tensor and electric displacement in piezoelectric medium can be calculated by means of following expressions:

(5)

(6)

Expressions (4) – (6) allow to get such boundary conditions equations:

(7a)

(7b)

(7c)

(7d)

In these equations m = 0, 1, 2, … M-1, where M is the quantity of layers, X3(m) = h1 + h2 + … + hm, X3(0) = 0. Equations (7a) represents the continuity of mechanical displacements, (7b) – of the stress, (7c) – of the potential, (7d) – of electric displacement. If some surface X3 = X3(m) is short circuited, equations (7c) and (7d) must be changed. The right part of the (7c) must be replaced with zero, the left part of (7d) also must be replaced with zero and the right part of (7d) must be replaced with the right part of (7c).

The boundary condition equation for stress on the external surface of the upper layer (m = M) can be obtained from equation (7b) by replacing the right part of this equation by zero. Analogously the equation (7c) 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 following form:

(8)

Here φ(M) is the potential of the external surface (X3 = X3(M)).

The condition of the continuity of the normal component of the electric displacement on the external surface gives in this case the following equation:

(9)

Here ε0 is the permittivity of the free space.

The system of the boundary condition equations contains n0 + n1 + n2 + … + nM equations with the same number of unknown coefficients Cn. In general case n0 = 4, n1 = n2 = … = nM = 8.

  1. Metallic layers

In this case 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 are formulated only for the substrate surface:

(10)

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.

  1. 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 boundary condition is presented by the equation (10) independently of all the rest boundary conditions.

In general case the dependence of the potential in the free space is defined by equation (8) and inside the m-th layer it must be written as:

(11)

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. It is the single equation, but its view significantly depends on the electric conditions on other boundaries.

All the boundaries are electrically free (not short circuited)

If there is the single layer, the equation, which describes the electric boundary conditions, can be written so:

(12a)

where

(12b)

Here and hereinafter εm (m = 1, 2, … M) is the relative permittivity of the m-th layer. R2 in (12b) is the recurrent coefficient, which allows to obtain the equation for two layers from equations (12) for one layer:

(13)

I.e. for two layers the electric boundary condition has the follow view:

(14a)

where:

(14b)

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

(15)

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 usage of the recurrent coefficient Rm+1 (RM = 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.

Short circuited surface

If one of the boundary surfaces X3 = X3(m) is short circuited, then electric conditions of all the further boundaries are unimportant, because the electric field outside the short circuited surface (X3 > X3(m)) is equal to zero. The same result will be, if the layer m + 1 is metal and all the further layers are metals and dielectrics in arbitrary combination.

To obtain the electric boundary condition equation in this case one has to get the equation for m layers with electrically free boundaries as described above. Then one must remain in the expression for Sm (for the last layer before the short circuited surface) only the first term ch(kb1hm) and the second term, which contains Rm+1, replace with zero. The equation, obtained so, corresponds to the zero potential on the surface X3 = X3(m). For example, for case then the second boundary is short circuited, i.e. φ(2) = 0, the boundary condition equation coincides with (14a), but S2 = ch(kb1h2) must be set in this equation instead of (14b).

IV. SOLVING OF BOUNDARY CONDITION EQUATIONS

After formulating of all the boundary condition equations the system of linear equations for determination of coefficients Cn can be obtained:

a11C1 + a12C2 + … + a1NCN = 0

a21C1 + a22C2 + … + a2NCN = 0 (16)

……………………………….

aN1C1 + aN2C2 + …+ aNNCN = 0

The determinant of this system must be equal to zero:

|d|2 = 0(17)

The equation (17) allows to determine the phase velocity (and attenuation for PSAWs) and all the basic propagation characteristics in particular coupling coefficient, temperature coefficient of delay, power flow angle, diffraction parameter and some others.

Another method for solving of the boundary condition equations is based on the Adler’s technique [2]. For example, for external surface of the upper layer one can define some value, named effective permittivity (all the layers are piezoelectric):

(18)

Substituting expressions for the electric displacement and potential into (18), one can obtain the electric boundary condition equation for the external surface:

(19)

Here top value in the right part corresponds to the free surface and bottom one – to the short circuited surface. The top part of the (19) is completely equivalent to the equation (9) and the bottom part is equivalent to the equation (7c) with zero right part.

After some modification the effective permittivity method may be applied to any boundary surface (all the layers below this surface must be piezoelectric).

In this work the technique, described in [3], was used for solving equations (17) and (19).

  1. SOME RESULTS

Figs. 2 and 3 show dependencies of the temperature coefficient of delay (tcd) on quartz with single Al and Au layer respectively on the second Euler angle and on the relative layer thickness. Material constants for quartz are taken from [4], for Al and Au – from [5]. One can see in Figs. 2 and 3, that negative values of tcd can be compensated by metallic layer. For example, orientation YX-quartz (0o,90o,0o) becomes thermostable if h/ = 0.061 for Al layer and YX-quartzkeeps the temperature stability in range 0.027 ≤ h/ ≤ 0.032 for Au layer.

Fig. 4 shows influence of the single layer of the isotropic SiO2 (smelted quartz) on the temperature properties of quartz. Material constants of SiO2 are taken from [5]. Influence of the SiO2 layer on the temperature properties of quartz is rather weak, as it is seen from Fig. 4, hence the SiO2 layer can be used for protection of quartz surface against external mechanical and chemical influences without deterioration of the thermal stability. Evaporation of the thin metal layer on the SiO2 layer can guarantee an additional protection against external electric fields. Influence of two layers (SiO2 layer on the substrate and Al layer above SiO2) on the temperature properties of ST-X quartz(0o,132.75o,0o) is illustrated by

Fig. 5. Dependence on the Al layer thickness is rather strong therefore this thickness must be small to prevent its influence on the thermostability. Fig. 6 shows influence of SiO2 and Al layers on the coupling coefficient for ST- X quartz. IDTs are placed on the quartz surface under the SiO2 layer. As one

Fig. 2. Al layer on quartz. Dependence of tcd (ppm/oC) on the 2nd Euler angle  (deg) and the relative layer thickness h/. The first and third Euler angles are equal to zero.

Fig. 3. Au layer on quartz.

can see in Fig. 6, coupling coefficient reaches its maximal value for h1/ = 0.11 – 0.13 and decreases slowly when the second layer thickness increases.

The layer of one piezoelectric with large coupling coefficient on the surface of other piezoelectric with high thermostability can guarantee combination of both these properties. Fig. 7 shows dependence of the phase velocity Vo on the free surface, tcd and coupling coefficient K2 (IDTs on the layer

Fig. 4. SiO2 layer on quartz.

Fig. 5. SiO2 and Al layers on ST-X quartz. Dependence of tcd (ppm/oC) on thickness. Relative thickness of SiO2 is h1/, Al – h2/.

surface) on the layer thickness of LiNbO3 (0o,38o,0o) on quartz substrate (0o,100o,0o). Material constants for LiNbO3 are taken from [6]. It is seen from Fig. 7 that K2 = 3.85 % and tcd ≈ 0 near h/ = 0.26. So the niobate lithium layer on quartz can guarantee combination of large K2 and high thermostability.

Fig. 6. SiO2 and Al layers on ST-X quartz. Dependence of coupling coefficient (%) on thickness. Relative thickness of SiO2 is h1/, Al – h2/.

Fig. 7. Dependencies of velocity Vo, tcd and coupling coefficient K2 on the relative layer thickness for LiNbO3 layer on quartz substrate.

At last Fig. 8 shows influence of the metallic layer on the PSAW properties. Dependencies of the phase velocity and propagation attenuation for short circuited surface on the Al layer thickness for the 2nd order PSAW on LiNbO3 (0o,-49o,0o) are shown in Fig. 8. One can see, that during increasing of h/ from 0 to about 0.047 attenuation decreases from 2.865 dB/λ to the value about 10-3 dB/λ. It can be explained by change of propagation conditions because of metallic layer presence. The declination angle of the wave propagation decreases and propagation attenuation decreases too.

Fig. 8. Dependencies of phase velocity and propagation attenuation for the 2nd order PSAW on LiNbO3 (0o,-49o,0o).

VI. CONCLUSION

The solution of the problem, considered in this work, allows to calculate various variants of multi-layered structures, including layers of piezoelectric, metals and dielectric on the semi-infinite piezoelectric substrate. Some new results are obtained, which illustrate influence of layers of some materials on propagation characteristics of surface and pseudo-surface acoustic waves. Usage of some layer materials for protection of the substrate surface is also possible without deterioration of wave characteristics. The possibility of improvement of some wave characteristics is shown, for example, thermostability, electromechanical coupling coefficient and propagation attenuation.

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