**Shear Mode Piezoelectric Thin Film Resonators**

Takahiko Yanagitani

*Nagoya Institute of Technology Japan* 

### **1. Introduction**

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### **1.1 Shear mode bulk acoustic wave devices and sensors**

Acoustic microsensor technique, well known as QCM (Quartz crystal microbalance) or TSM (Thickness shear mode) sensor, is an effective method to detect small mass loading on the sensor surface. This sensor can be operated even in liquid by using shear mode resonance. Therefore, shear mode piezoelectric film resonators are attractive for liquid microsensor technique such as biosensors and immunosensors.

Shear wave has some unique features compared with the longitudinal wave, for example, it has extremely low velocity in the liquid. Longitudinal wave velocity in the water is 1492.6 m/s, whereas, shear wave velocity in the water is 20-60 m/s at 20-200 MHz (Matsumoto et al., 2000). Therefore, shear mode vibrating solid maintains its vibration even in the liquid, because the difference of acoustic impedance which determines the refection coefficient of solid / liquid interface is very large in the case of shear wave.

The complex refection coefficient Γ of the interface is given as

$$
\Gamma = \frac{\mathbf{Z}\_l - \mathbf{Z}\_s}{\mathbf{Z}\_l + \mathbf{Z}\_s} \tag{1}
$$

where Zs and Z*l* are the complex acoustic impedance of solid and liquid. Complex acoustic impedance can be written as

$$Z = R + jX = \left(\,\,\rho\left(c + jo\eta\right)\right)^{1/2} \tag{2}$$

*R* and *X* represent the real part and imaginary part of the acoustic impedance and ρ, *c* and η represent mass density, stiffness constant and viscosity in the medium, respectively. Acoustic wave equation gives dispersion relation of

$$\left(\frac{\alpha\nu}{v} - j\alpha\right)^2 \left(c + j\alpha\eta\right) = \rho\alpha^2\tag{3}$$

where *v* is velocity and α is attenuation factor (B. A. Auld, 1973). According to (2) and (3), acoustic impedance gives

$$R = \frac{\rho \, v \, o^2}{o\rho^2 + \alpha^2 v^2}, \; X = \frac{\rho \, v^2 \, o\alpha \alpha}{o\rho^2 + \alpha^2 v^2} \tag{4}$$

Shear Mode Piezoelectric Thin Film Resonators 503

The analytical model of a thin film resonator is shown in Fig. 1. The electric field is applied in the *x*3 direction. The c-axis is assumed to lie in the *x*1-*x*3 plane and be inclined at an angle

Wurtzite piezoelectric film

Acoustic wave

β

[ ]

′ transforms as

The 6×6 transformation matrix of coefficients *M* is defined as

222 222 222

piezoelectric constant tensors *c*′ and *e*′ are obtained:

*a*

ε

*zx xx zy xy zz xz xy zz xz zy xz zx xx*

*aa aa aa aa aa aa aa*

The physical constants of the crystal in each direction are determined by the transformed coordinate of each constant tensor. Bond's method (Bond, 1943) for transforming the elastic and piezoelectric constant tensor is introduced below, which can be applied to the constant tensor with abbreviated subscript notation. For example, the transformation matrix [*a*] of a

> cos 0 sin 01 0 sin 0 cos

<sup>−</sup> <sup>=</sup>

β

β

[ ] [ ][ ][ ] .

 ε

*xx xy xz xy xz xz xx xx xy yx yy yz yy yz yz yx yx yy zx zx zz zy zz zz zx zx zy yx zx yy zy yz zz yy zz yz zy yx zz yz zx yy zx yx zy*

*aa aa aa aa aa aa aa aa aa*

*xx yx xy yy xz yz xy yz xz yy xz yx xx yz xx yy xy yx*

+++

*aa aa aa aa aa aa aa aa aa*

Finally, using the above transformation matrix, transformed elastic constant and

<sup>=</sup> +++

*a a a aa aa aa a a a aa aa aa a a a aa aa aa*

with respect to the *x*3 direction.

δ*L*

*x*3

*u* (S)

Fig. 1. Analytical model of a thin film resonator

ε

clockwise rotation through an angle

The dielectric constant

[ ]

*M*

*u* (L)

β

*x*1

*c-*axis

(5)

*aa aa*

+

(7)

′ = *a a* (6)

*x*3

β

*u*3

about the *x*2-axis is described by:

 β

 β

*T*

222 222 222

+ + *zz xx zy xy zx*

Substrate

Electrode

*u*1

Longitudinal and shear wave velocities of water were reported as 1492.6 m/s (Kushibiki et al., 1995) and 35 m/s (Matsumoto et al., 2000), respectively, at 100 MHz. Attenuations of longitudinal and shear wave in the water were also measured to be α/*f* 2 = 2.26×10-14 neper·s2/m (Kushibiki et al, 1995) and α/*f* 2 = 2.12×10-9 neper·s2/m, (Matsumoto et al., 2000) respectively. By substituting these values into Eq. (4), the complex longitudinal wave and shear wave acoustic impedance of the water can be estimated to be 1489000+*j*800 N·s/m3 and 14510+*j*17340 N·s/m3 at 100 MHz, respectively.

From these values and Eq. (1), when quartz resonator is immersed in water, the reflection coefficient of acoustic energy <sup>2</sup> Γ in an X-cut quartz vibrating in thickness extensional mode (Zs= 15.23×10-6 N·s/m3) is estimated to be only 68 % whereas that in an AT-cut quartz vibrating in thickness shear mode (Zs= 8.795×10-6 N·s/m3) is 98 %. This is because an AT-cut quartz has been used as a QCM or TSM sensor operating in liquid. Sensitivity of the QCM mass sensor is determined by the ratio of the mass and the entire mass of the vibrating part in the sensor, at constant sensor active area (Sauerbrey, 1959). Therefore, it is important to decrease thickness of the vibrating part of sensor. Shear mode thin film is promising for high sensitivity mass sensor.

### **1.2 Piezoelectric thin film for shear mode excitation**

Piezoelectric thin film, which excites shear wave, is expected to provide higher sensitivity and IC compatibility, but it is not straightforward. To excite shear wave by standard sandwiched electrode configuration, polarization axis in the film must be tilted or parallel to the film plane. Although perovskite ferroelectric films have large piezoelectricity, their polarization axis is generally normal to the film surface due to the nature of crystal growth, difficultly of in-plane polarization treatment and domain control. Trigonal piezoelectric material such as LiNbO3, LiTaO3 and quartz are difficult to crystallize (tend to form amorphous structure) or to obtain a strong preferred orientation in polycrystalline film.

6mm wurtzite AlN and ZnO film can be easily crystallized, but they tend to develop their polarization axis (c-axis) perpendicular to the substrate plane. This c-axis oriented film cannot excite shear wave in the case of standard sandwiched electrode structure.

Crystalline orientation control for both in-plane and out-of-plane direction is necessary to excite shear wave. One solution is to use an epitaxial growth technique. However, the combinations of the shear mode piezoelectric film and substrate are limited due to the lattice mismatch. a-plane ZnO or AlN/r-plane sapphire (Mitsuyu et al., 1980; Wittstruck et al., 2003), a-plane ZnO/42º Y-X LiTaO3 (Nakamura et al., 2000) where c-axis in the film is parallel to the substrate plane have been reported.

Ion beam orientation control technique (Yanagitani & Kiuchi, 2007c), which enables in-plane and out-of-plane orientation without use of epitaxial growth, is introduced in the third section. This technique is a good candidate for obtaining c-axis parallel films which excites pure shear wave without any excitation of longitudinal wave.

### **2. Electromechanical coupling properties of wurtzite crystal**

Elastic and piezoelectric properties of wurtzite crystals vary with direction due to the crystal anisotropy. Electromechanical coupling changes as a function of the angle between the caxis and the applied electric field direction (Foster et al., 1968; Auld, 1973).

Longitudinal and shear wave velocities of water were reported as 1492.6 m/s (Kushibiki et al., 1995) and 35 m/s (Matsumoto et al., 2000), respectively, at 100 MHz. Attenuations of

respectively. By substituting these values into Eq. (4), the complex longitudinal wave and shear wave acoustic impedance of the water can be estimated to be 1489000+*j*800 N·s/m3

From these values and Eq. (1), when quartz resonator is immersed in water, the reflection

(Zs= 15.23×10-6 N·s/m3) is estimated to be only 68 % whereas that in an AT-cut quartz vibrating in thickness shear mode (Zs= 8.795×10-6 N·s/m3) is 98 %. This is because an AT-cut quartz has been used as a QCM or TSM sensor operating in liquid. Sensitivity of the QCM mass sensor is determined by the ratio of the mass and the entire mass of the vibrating part in the sensor, at constant sensor active area (Sauerbrey, 1959). Therefore, it is important to decrease thickness of the vibrating part of sensor. Shear mode thin film is promising for high

Piezoelectric thin film, which excites shear wave, is expected to provide higher sensitivity and IC compatibility, but it is not straightforward. To excite shear wave by standard sandwiched electrode configuration, polarization axis in the film must be tilted or parallel to the film plane. Although perovskite ferroelectric films have large piezoelectricity, their polarization axis is generally normal to the film surface due to the nature of crystal growth, difficultly of in-plane polarization treatment and domain control. Trigonal piezoelectric material such as LiNbO3, LiTaO3 and quartz are difficult to crystallize (tend to form amorphous structure) or to obtain a strong preferred orientation in polycrystalline

6mm wurtzite AlN and ZnO film can be easily crystallized, but they tend to develop their polarization axis (c-axis) perpendicular to the substrate plane. This c-axis oriented film

Crystalline orientation control for both in-plane and out-of-plane direction is necessary to excite shear wave. One solution is to use an epitaxial growth technique. However, the combinations of the shear mode piezoelectric film and substrate are limited due to the lattice mismatch. a-plane ZnO or AlN/r-plane sapphire (Mitsuyu et al., 1980; Wittstruck et al., 2003), a-plane ZnO/42º Y-X LiTaO3 (Nakamura et al., 2000) where c-axis in the film is

Ion beam orientation control technique (Yanagitani & Kiuchi, 2007c), which enables in-plane and out-of-plane orientation without use of epitaxial growth, is introduced in the third section. This technique is a good candidate for obtaining c-axis parallel films which excites

Elastic and piezoelectric properties of wurtzite crystals vary with direction due to the crystal anisotropy. Electromechanical coupling changes as a function of the angle between the c-

cannot excite shear wave in the case of standard sandwiched electrode structure.

α

/*f* 2 = 2.12×10-9 neper·s2/m, (Matsumoto et al., 2000)

Γ in an X-cut quartz vibrating in thickness extensional mode

/*f* 2 = 2.26×10-14

longitudinal and shear wave in the water were also measured to be

α

neper·s2/m (Kushibiki et al, 1995) and

coefficient of acoustic energy <sup>2</sup>

sensitivity mass sensor.

film.

and 14510+*j*17340 N·s/m3 at 100 MHz, respectively.

**1.2 Piezoelectric thin film for shear mode excitation** 

parallel to the substrate plane have been reported.

pure shear wave without any excitation of longitudinal wave.

**2. Electromechanical coupling properties of wurtzite crystal** 

axis and the applied electric field direction (Foster et al., 1968; Auld, 1973).

The analytical model of a thin film resonator is shown in Fig. 1. The electric field is applied in the *x*3 direction. The c-axis is assumed to lie in the *x*1-*x*3 plane and be inclined at an angle β with respect to the *x*3 direction.

Fig. 1. Analytical model of a thin film resonator

The physical constants of the crystal in each direction are determined by the transformed coordinate of each constant tensor. Bond's method (Bond, 1943) for transforming the elastic and piezoelectric constant tensor is introduced below, which can be applied to the constant tensor with abbreviated subscript notation. For example, the transformation matrix [*a*] of a clockwise rotation through an angle β about the *x*2-axis is described by:

$$\begin{aligned} \begin{bmatrix} a \end{bmatrix} = \begin{bmatrix} \cos \beta & 0 & -\sin \beta \\ 0 & 1 & 0 \\ \sin \beta & 0 & \cos \beta \end{bmatrix} \end{aligned} \tag{5}$$

The dielectric constant ε′ transforms as

$$\begin{bmatrix} \varepsilon' \end{bmatrix} = \begin{bmatrix} a \end{bmatrix} \begin{bmatrix} \varepsilon \end{bmatrix} \begin{bmatrix} a \end{bmatrix}^T. \tag{6}$$

The 6×6 transformation matrix of coefficients *M* is defined as

$$\begin{bmatrix} \begin{bmatrix} a\_{xx}^{2} & a\_{xy}^{2} & a\_{zx}^{2} & 2a\_{xy}a\_{zx} & 2a\_{xx}a\_{xx} & 2a\_{xx}a\_{xy} \\ a\_{yx}^{2} & a\_{yy}^{2} & a\_{yz}^{2} & 2a\_{yy}a\_{yz} & 2a\_{yz}a\_{yx} & 2a\_{yx}a\_{yy} \\ a\_{zx}^{2} & a\_{zx}^{2} & a\_{zz}^{2} & 2a\_{zy}a\_{zz} & 2a\_{zx}a\_{zx} & 2a\_{zx}a\_{zy} \\ a\_{yx}a\_{zx} & a\_{yy}a\_{zy} & a\_{yz}a\_{zz} & a\_{yy}a\_{zz} + a\_{yx}a\_{zy} & a\_{yx}a\_{zz} + a\_{yx}a\_{zx} & a\_{yy}a\_{zx} + a\_{yy}a\_{zy} \\ a\_{zx}a\_{xx} & a\_{zy}a\_{xy} & a\_{zx}a\_{xz} & a\_{xy}a\_{zz} + a\_{zx}a\_{zy} & a\_{xx}a\_{zz} + a\_{xx}a\_{zy} & a\_{xy}a\_{yy} + a\_{yy}a\_{yz} \end{bmatrix} \tag{7}$$

Finally, using the above transformation matrix, transformed elastic constant and piezoelectric constant tensors *c*′ and *e*′ are obtained:

Shear Mode Piezoelectric Thin Film Resonators 505

Equation (10c) describes a pure shear wave with a *u*2 displacement component in the *x*<sup>2</sup>

and (10b) represent a quasi-longitudinal wave and quasi-shear wave. These waves

*<sup>x</sup> u B jt*

 = −

*<sup>v</sup> <sup>C</sup>* ω

2

0

*S*

ε

ε

ε

of a quasi-longitudinal wave and quasi-shear wave:

ρρ

3 3

+ −

 ω

*A B* = − (18)

2 2 2

β

1

ε

( )<sup>2</sup> 33 33 33 33 , *DE S cc e* = + ′′ ′

( )<sup>2</sup> 55 55 35 33 . *DE S cc e* = + ′′ ′

35 35 33 35 33 ( ) , *D E <sup>S</sup> c c ee* = + ′ ′′ ′

A, B and C are all nonzero when the coefficient matrix in Eq. (14) is zero. From this condition,

*DD DD D L S cc cc c*

+ − =± +

Figure 2 shows the calculated results of phase velocity of a quasi-longitudinal wave and

direction. Physical constants in a ZnO single crystal reported by Smith were used in the

3 1 ( ) 2 ( )

= −+ −

1 2 1 2 *B A*

exp exp

*x x u B jt B jt V V C C*

are given by

, 33 55 33 55 35 2 2

ρ

ϕ

1 1 2

*u A A*

ω

1 2

quasi-shear wave for a ZnO crystal as function of the angle

, which are coupled with each other. It is well known that Eqs (10a),

3

0 0,

ρ

(13)

(14)

(15)

(16)

between the c-axis and *x*<sup>3</sup>

(17)

. Eqs. (10a)

direction and propagates along the *x*3 direction with a phase velocity of 44 *c*

<sup>3</sup> exp

Substituting Eq. (13) into Eqs. (11) and (12), the simultaneous equations are obtained

*D*

35 33 33

*cv c A c cv B ee C*

ρ

<sup>−</sup> − = − −

1

ϕ

*u A*

2 55 35

ρ

*D D*

35 33

incorporate *u*1, *u*3, and

where

ϕ

and (10b) have plane-wave solutions:

we obtain the phase velocity *v* (*L, S*)

calculation (Smith, 1969).

and

The general solutions for *u*1, *u*3 and

ϕ

( )

*v*

$$\begin{bmatrix} \mathbf{c'} \end{bmatrix} = \begin{bmatrix} M \end{bmatrix} \begin{bmatrix} \mathbf{c} \end{bmatrix} \begin{bmatrix} M \end{bmatrix}^T \text{ / } \begin{bmatrix} \mathbf{c'} \end{bmatrix} = \begin{bmatrix} M \end{bmatrix} \begin{bmatrix} \mathbf{c} \end{bmatrix} \begin{bmatrix} \mathbf{c} \end{bmatrix} \begin{bmatrix} M \end{bmatrix}^T \tag{8}$$

In the *x*2 axis rotation of a hexagonal (6mm) crystal, the transformed stiffness and piezoelectric constant tensors *c*′ and *e*′ are given by

$$
\begin{bmatrix} c'\_{11} & c'\_{12} & c'\_{13} & 0 & c'\_{15} & 0 \\ c'\_{12} & c'\_{22} & c'\_{23} & 0 & c'\_{25} & 0 \\ c'\_{13} & c'\_{23} & c'\_{33} & 0 & c'\_{35} & 0 \\ 0 & 0 & 0 & c'\_{44} & 0 & c'\_{46} \\ c'\_{15} & c'\_{23} & c'\_{35} & 0 & c'\_{55} & 0 \\ 0 & 0 & 0 & c'\_{46} & 0 & c'\_{66} \end{bmatrix}, \begin{bmatrix} e' \\ e' \end{bmatrix} = \begin{bmatrix} e'\_{11} & e'\_{12} & e'\_{13} & 0 & e'\_{15} & 0 \\ 0 & 0 & 0 & e'\_{24} & 0 & e'\_{26} \\ e'\_{31} & e'\_{32} & e'\_{33} & 0 & e'\_{35} & 0 \\ \end{bmatrix} \tag{9}
$$

In case, wave propagation toward *x*3 direction is only focused, the term of 1 ∂ ∂*x* and <sup>2</sup> ∂ ∂*x* can be ignored. Thus, the wave motion equation for the *x*3 direction is given by mechanical displacement component *u*1, *u*2 and *u*3:

$$\frac{\partial \, T\_{31}}{\partial x\_3} = \rho \frac{\partial^2 u\_1}{\partial t^2} \tag{10a}$$

$$\frac{\partial}{\partial \mathbf{u}\_3} \frac{T\_{33}}{\partial x\_3} = \rho \frac{\partial^2 u\_3}{\partial t^2} \tag{10b}$$

$$\frac{\partial}{\partial x\_3} \frac{T\_{32}}{\partial x\_3} = \rho \frac{\partial^2 u\_2}{\partial t^2} \tag{10c}$$

where

$$T\_{31} = c\_{\rm 95}^{\prime \, E} \frac{\partial \mu\_1}{\partial \mathbf{x}\_3} + c\_{\rm 35}^{\prime \, E} \frac{\partial \mu\_3}{\partial \mathbf{x}\_3} + c\_{\rm 39}^{\prime} \frac{\partial \rho}{\partial \mathbf{x}\_3} \tag{11a}$$

$$T\_{33} = c\_{33}^{\prime \, \, \,} \frac{\partial \, u\_1}{\partial \, \, \, \_3} + c\_{33}^{\prime \, \, \, \, \partial} \frac{\partial \, u\_3}{\partial \, \, \, \_3} + c\_{33}^{\prime \, \, \, \, \, \partial} \frac{\partial \, \phi}{\partial \, \, \, \_3} \tag{11b}$$

$$T\_{32} = c\_{44}^{\prime \hat{E}} \frac{\partial u\_2}{\partial \mathbf{x}\_3} \tag{11c}$$

As div *D* = 0, the electrostatic equation is given by

$$\frac{\partial D\_3}{\partial \mathbf{x}\_3} = \boldsymbol{e}\_{35}^{\prime} \frac{\partial^2 \boldsymbol{u}\_1}{\partial \mathbf{x}\_3^2} + \boldsymbol{e}\_{33}^{\prime} \frac{\partial^2 \boldsymbol{u}\_3}{\partial \mathbf{x}\_3^2} - \boldsymbol{\varepsilon}\_{33}^{\prime S} \frac{\partial^2 \boldsymbol{\mathcal{E}}}{\partial \mathbf{x}\_3^2} = \mathbf{0} \tag{12}$$

In Eqs. (10)-(12), *T*31 and *T*33 are stress components, *D*3 is the electric displacement, *c*33*<sup>E</sup>*, *c*35*<sup>E</sup>* and *c*55E are the stiffness constants with constant electric field, *e*33 and *e*35 are piezoelectric constants, ε33S and ε35S are dielectric constants with constant strain, and ϕ is the electric potential.

Equation (10c) describes a pure shear wave with a *u*2 displacement component in the *x*<sup>2</sup> direction and propagates along the *x*3 direction with a phase velocity of 44 *c* ρ . Eqs. (10a) and (10b) represent a quasi-longitudinal wave and quasi-shear wave. These waves incorporate *u*1, *u*3, and ϕ, which are coupled with each other. It is well known that Eqs (10a), and (10b) have plane-wave solutions:

$$
\begin{pmatrix} u\_1 \\ u\_3 \\ \varphi \end{pmatrix} = \begin{pmatrix} A \\ B \\ C \end{pmatrix} \exp\left\{ j\rho \left( t - \frac{\mathbf{x}\_3}{v} \right) \right\} \tag{13}
$$

Substituting Eq. (13) into Eqs. (11) and (12), the simultaneous equations are obtained

$$
\begin{pmatrix} c\_{33}^{D} - \rho \upsilon^{2} & c\_{33}^{D} & 0 \\ \overline{c}\_{33} & c\_{33}^{D} - \rho \upsilon^{2} & 0 \\ -c\_{33} & -c\_{33} & \mathcal{E}\_{33}^{\mathbb{S}} \end{pmatrix} \begin{pmatrix} A \\ B \\ \mathbb{C} \end{pmatrix} = \mathbf{0},\tag{14}
$$

where

504 Acoustic Waves – From Microdevices to Helioseismology

*c McM* ′ = , [ ] [ ][][ ]*<sup>T</sup>*

In the *x*2 axis rotation of a hexagonal (6mm) crystal, the transformed stiffness and

In case, wave propagation toward *x*3 direction is only focused, the term of 1 ∂ ∂*x* and <sup>2</sup> ∂ ∂*x* can be ignored. Thus, the wave motion equation for the *x*3 direction is given by

> *T u x t* ρ

*T u x t* ρ

*T u x t* ρ

1 3 31 55 35 35

1 3 33 35 33 33

32 44

3 1 3

∂ ∂∂ ∂

*<sup>E</sup> <sup>u</sup> T c*

3

3

3

2 31 1 2

2 33 3 2

2 32 2 2

333 ∂∂ ∂ =++ ′′′ ∂∂∂ *E E u u Tc c e*

333 ∂∂ ∂ =++ ′′′ ∂∂∂ *E E u u Tc c e*

2

3

ε

*x*

22 2

35 222 33 33 33 33 <sup>0</sup> *<sup>D</sup> u u <sup>S</sup> e e xxxx*

35S are dielectric constants with constant strain, and

In Eqs. (10)-(12), *T*31 and *T*33 are stress components, *D*3 is the electric displacement, *c*33*<sup>E</sup>*, *c*35*<sup>E</sup>* and *c*55E are the stiffness constants with constant electric field, *e*33 and *e*35 are piezoelectric

*xxx*

ϕ

ϕ

ε

= + −= ′′′ ∂∂∂∂ (12)

*xxx*

*e MeM* ′ = (8)

24 26

(9)

0 0

0 0

, [ ] 11 12 13 <sup>15</sup>

31 32 33 35

<sup>∂</sup> <sup>∂</sup> <sup>=</sup> ∂ ∂ (10a)

∂ ∂ <sup>=</sup> ∂ ∂ (10b)

<sup>∂</sup> <sup>∂</sup> <sup>=</sup> ∂ ∂ (10c)

<sup>∂</sup> <sup>=</sup> ′ ∂ (11c)

(11a)

(11b)

ϕ

is the electric

000 0

*eee e e ee eee e* ′′′ ′ ′ ′′ <sup>=</sup> ′′′ ′

[ ] [ ][][ ]*<sup>T</sup>*

44 46

*c c*

0 0

0 0 0 0 0 0

46 66

*c c*

piezoelectric constant tensors *c*′ and *e*′ are given by

11 12 13 15 12 22 23 25 13 23 33 35

*ccc c ccc c ccc c*

15 25 35 55

*ccc c*

mechanical displacement component *u*1, *u*2 and *u*3:

As div *D* = 0, the electrostatic equation is given by

ε

000 0

000 0

′′′ ′ ′′′ ′ ′′′ ′ ′ <sup>=</sup> ′ ′ ′′′ ′ ′ ′

[ ]

*c*

where

constants,

potential.

ε33S and

$$\begin{aligned} \mathbf{c}\_{33}^{D} &= \mathbf{c}\_{33}^{\prime E} + \left(\mathbf{e}\_{33}^{\prime}\right)^{2} \Big/ \mathbf{e}\_{33}^{\prime S} \\\\ \mathbf{c}\_{33}^{D} &= \mathbf{c}\_{33}^{\prime E} + \left(\mathbf{e}\_{33}^{\prime} \, \mathbf{e}\_{33}^{\prime}\right) \Big/ \mathbf{e}\_{33}^{\prime S} \\\\ \mathbf{c}\_{33}^{D} &= \mathbf{c}\_{33}^{\prime E} + \left(\mathbf{e}\_{33}^{\prime}\right)^{2} \Big/ \mathbf{e}\_{33}^{\prime S} \end{aligned} \tag{15}$$

A, B and C are all nonzero when the coefficient matrix in Eq. (14) is zero. From this condition, we obtain the phase velocity *v* (*L, S*) of a quasi-longitudinal wave and quasi-shear wave:

$$\boldsymbol{w}^{(L,S)} = \left[\frac{\boldsymbol{c}\_{33}^{D} + \boldsymbol{c}\_{93}^{D}}{2\rho} \pm \sqrt{\left(\frac{\boldsymbol{c}\_{33}^{D} - \boldsymbol{c}\_{93}^{D}}{2\rho}\right)^{2} + \left(\frac{\boldsymbol{c}\_{33}^{D}}{\rho}\right)^{2}}\right]^{\frac{1}{2}}\tag{16}$$

Figure 2 shows the calculated results of phase velocity of a quasi-longitudinal wave and quasi-shear wave for a ZnO crystal as function of the angle β between the c-axis and *x*<sup>3</sup> direction. Physical constants in a ZnO single crystal reported by Smith were used in the calculation (Smith, 1969).

The general solutions for *u*1, *u*3 and ϕ are given by

$$
\begin{pmatrix} u\_1 \\ u\_3 \\ \rho \end{pmatrix} = \begin{pmatrix} A\_1 \\ B\_1 \\ C\_1 \end{pmatrix} \exp\left\{ j\rho \left( t - \frac{\mathbf{x}\_3}{V^{(\ast)}} \right) \right\} + \begin{pmatrix} A\_2 \\ B\_2 \\ C\_2 \end{pmatrix} \exp\left\{ j\rho \left( t - \frac{\mathbf{x}\_3}{V^{(\ast)}} \right) \right\} \tag{17}
$$

and

$$\frac{B\_1}{A\_1} = -\frac{A\_2}{B\_2} \tag{18}$$

Shear Mode Piezoelectric Thin Film Resonators 507

(a) Quasi-thickness extensional mode Quasi-thickness shear mode

> 0 20 40 60 80 β(deg.)

(b) Quasi-thickness extensional mode Quasi-thickness shear mode

> 0 20 40 60 80 β(deg.)

 between the wave displacement *u* and the *x* direction and (b) electromechanical coupling coefficient of the quasi-longitudinal and quasi-shear waves

due to mode conversion in the reflection plane. This induces the decrease of Q value. Both of

however, it is difficult to adjust such as large c-axis tilt angle in a large area deposition. One

field-orientation combination. One is to apply the cross-electric field to c-axis parallel film by sandwiched electrode (Yanagitani et al., 2007d), and the other is to apply the in-plane electric field to c-axis normal film by IDT electrode (Corso et al., 2007; Milyutin et al., 2008, 2010). Of course, the latter is the easiest way to obtain pure shear mode because deposition

δ

between the c-axis and *x*3 direction

S = 0º. Pure shear mode excitation can be achieved by two electric

S values of 0.38º can be obtained at

= 90º) resonator to satisfy both the conditions of no

β = 43º,

β

β


0.4

0.3

0.2

*k*33or

δ

option is to use a pure-shear-mode (

extensional coupling and

for the ZnO crystal as function of the angle

the no extensional mode coupling and small

δ

Fig. 3. (a) Angle

*k*15

0.1

0.0

δ

(deg.)

Fig. 2. Phase velocity of quasi-longitudinal wave and quasi-shear wave for a ZnO crystal as function of the angle βbetween the c-axis and *x*3 direction

is derived from Eqs. (14) and (16). It can be seen that the displacement components of the quasi-longitudinal wave and quasi-shear wave are perpendicular to each other. From Eqs. (14) and (16), the angle δ*<sup>L</sup>* between the quasi-longitudinal wave displacement *u*3 and the *x*<sup>3</sup> direction and the angle δS between the quasi-shear wave displacement *u*1 and the *x*<sup>1</sup> direction are given by

$$\mathcal{S}\_{\rm L} = \tan^{-1} \left( \frac{A\_{\rm 1}}{B\_{\rm 1}} \right), \; \mathcal{S}\_{\rm S} = \tan^{-1} \left( \frac{B\_{\rm 2}}{A\_{\rm 2}} \right) \tag{19}$$

The extensional and shear effective piezoelectric constants *e* (*L*) *eff* and *e* (*S*) *eff* are defined as

$$e\_{\rm eff}^{(\rm L)} = e\_{\rm 35}' \sin \delta\_{\rm L} + e\_{\rm 33}' \cos \delta\_{\rm L} \; \; e\_{\rm eff}^{(\rm s)} = e\_{\rm 35}' \cos \delta\_{\rm s} - e\_{\rm 33}' \sin \delta\_{\rm s} \tag{20}$$

Thus, the quasi-longitudinal and quasi-shear-mode electromechanical coupling coefficients *k*(*<sup>L</sup>*) (transformed *k*33) and *k*(*<sup>S</sup>*) (transformed *k*15) are

$$\left(\left(k^{(\mathcal{S})}\right)^2 = \left(e^{(\mathcal{S})}\right)^2 \Big/ \Big/ e'\_{33}\,\rho\left(V^{(-)}\right)^2\,,\Big/\left(k^{(\mathcal{L})}\right)^2 = \left(e^{(\mathcal{L})}\right)^2\Big/e'\_{33}\,\rho\left(V^{(+)}\right)^2\tag{21}$$

Finally, Figs. 3 (a) and (b) show the calculated angle δ and the electromechanical coupling coefficients (*k* values) of the quasi-longitudinal and quasi-shear waves for the ZnO crystal as function of the angle β(Foster et al., 1968)

From these figures, we can see a relatively large shear-mode electromechanical coupling *k*<sup>15</sup> = 0.39 at c-axis tilt angle of β = 28º. Several author reported FBAR (film bulk acoustic resonator)-type viscosity sensor and biosensor, consisting of c-axis tilted wurtzite films (Weber et al., 2006; Link et al., 2007; Wingqvist et al., 2007, 2009, 2010; Yanagitani, 2010, 2011a). However, the thickness extensional mode (longitudinal wave mode) also has the coupling of *k*33 = 0.155 and the displacement inclination angle of δS = 4.1º at angle of β = 28º. This indicates that the resonator excites both thickness extensional and shear mode (longitudinal and shear wave modes), and the shear displacement direction is not perpendicular to the propagation direction. Larger δS values may result in energy leakage

Quasi-shear wave

Quasi-longitudinal wave

Shear wave velocity (m/s)

(19)

*eff* are defined as

(20)

0 10 20 30 40 50 60 70 80 90 β(deg.)

Fig. 2. Phase velocity of quasi-longitudinal wave and quasi-shear wave for a ZnO crystal as

is derived from Eqs. (14) and (16). It can be seen that the displacement components of the quasi-longitudinal wave and quasi-shear wave are perpendicular to each other. From Eqs.

*<sup>L</sup>* between the quasi-longitudinal wave displacement *u*3 and the *x*<sup>3</sup>

, <sup>1</sup> <sup>2</sup>

<sup>−</sup> <sup>=</sup>

*<sup>S</sup>* tan

δ

S between the quasi-shear wave displacement *u*1 and the *x*<sup>1</sup>

2

*eff* and *e* (*S*)

33

ρ

<sup>+</sup> = ′ (21).

and the electromechanical coupling

S = 4.1º at angle of

S values may result in energy leakage

β= 28º.

 δ

<sup>35</sup> <sup>33</sup> cos sin *<sup>S</sup> eff S S ee e* = − ′ ′ δ

*L L k e eV*

= 28º. Several author reported FBAR (film bulk acoustic

δ

δ

*B A*

between the c-axis and *x*3 direction

1 1 1

*A B*

 δ, ( )

Thus, the quasi-longitudinal and quasi-shear-mode electromechanical coupling coefficients

coefficients (*k* values) of the quasi-longitudinal and quasi-shear waves for the ZnO crystal as

From these figures, we can see a relatively large shear-mode electromechanical coupling *k*<sup>15</sup>

resonator)-type viscosity sensor and biosensor, consisting of c-axis tilted wurtzite films (Weber et al., 2006; Link et al., 2007; Wingqvist et al., 2007, 2009, 2010; Yanagitani, 2010, 2011a). However, the thickness extensional mode (longitudinal wave mode) also has the

This indicates that the resonator excites both thickness extensional and shear mode (longitudinal and shear wave modes), and the shear displacement direction is not

δ

<sup>−</sup> <sup>=</sup> ′ , ( ) ( ) ( ) ( ) ( ) 2 2 <sup>2</sup> ( )

*<sup>L</sup>* tan

<sup>−</sup> <sup>=</sup>

δ

The extensional and shear effective piezoelectric constants *e* (*L*)

()() ( ) 22 2 () () ( ) 33

(Foster et al., 1968)

β

coupling of *k*33 = 0.155 and the displacement inclination angle of

ρ

*S S k e eV*

Finally, Figs. 3 (a) and (b) show the calculated angle

perpendicular to the propagation direction. Larger

β

<sup>35</sup> <sup>33</sup> sin cos *<sup>L</sup> eff L L ee e* = + ′ ′ δ

6400

6300

6200

Longitudinal wave velocity (m/s)

function of the angle

(14) and (16), the angle

direction are given by

function of the angle

= 0.39 at c-axis tilt angle of

direction and the angle

6100

6000

5900

β

δ

( )

*k*(*<sup>L</sup>*) (transformed *k*33) and *k*(*<sup>S</sup>*) (transformed *k*15) are

δ

Fig. 3. (a) Angle δ between the wave displacement *u* and the *x* direction and (b) electromechanical coupling coefficient of the quasi-longitudinal and quasi-shear waves for the ZnO crystal as function of the angle βbetween the c-axis and *x*3 direction

due to mode conversion in the reflection plane. This induces the decrease of Q value. Both of the no extensional mode coupling and small δS values of 0.38º can be obtained at β = 43º, however, it is difficult to adjust such as large c-axis tilt angle in a large area deposition. One option is to use a pure-shear-mode (β = 90º) resonator to satisfy both the conditions of no extensional coupling and δS = 0º. Pure shear mode excitation can be achieved by two electric field-orientation combination. One is to apply the cross-electric field to c-axis parallel film by sandwiched electrode (Yanagitani et al., 2007d), and the other is to apply the in-plane electric field to c-axis normal film by IDT electrode (Corso et al., 2007; Milyutin et al., 2008, 2010). Of course, the latter is the easiest way to obtain pure shear mode because deposition

Shear Mode Piezoelectric Thin Film Resonators 509

Ion energy A: Large ion flux B: Small ion flux

0-5 μA/cm2

30-50 μA/cm2

A/cm2 140

A/cm2 130

A/cm2 120

Cu 111

25 30 35 40 45 50 55 60

(deg.)

scan XRD patterns of the ZnO films deposited without ion irradiation, and with

2θ

ion irradiation of 0-1 keV with "Large ion flux" and "Small ion flux" (Yanagitani & Kiuchi,

Figure 5 shows the XRD patterns of the samples deposited under the conditions that various RF and DC bias are applied to the substrate. Although any dramatic change in usual (0001)

μA/cm2

μA/cm2

μA/cm2

Without ion irradiation

(A) (B)

(A) (B)

(A) (B)

(A) (B)

(A) (B)

(A) (B)

(A) : Large ion flux (B) : Small ion flux

> Ion energy: 0.05 keV

> > 0.25 keV

0.75 keV

0.5 keV

1 keV

μ

μ

μ

0.05 keV 0.25 keV

26

24

22

20

18

16

14

Intensity (kcps)

12

10

8

6

4

2

0

Fig. 4. 2θ–ω

2007c)

0.5 keV 190

0.75 keV 220

1.0 keV 240

1010

Table 1. Ion current densities in "Large ion flux" and "Small ion flux"

0002

1011

technique of c-axis normal film has been well established, but effective electrometrical coupling is weak (*keff*=0.04-0.06) (Corso et al., 2007; Milyutin et al., 2008). The former has large electrometrical coupling (*k*15=0.24) (Yanagitani et al., 2007a), and recently the c-axis parallel oriented film can be easily obtained by using ion beam orientation control technique (presented in next section), even in a large area (Kawamoto et al., 2010).
