An Accurate Method for the Determination of Complex

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 51, no. 2, february 2004

227

An Accurate Method for the Determination of Complex Coefficients of Single Crystal Piezoelectric Resonators I: Theory Xiao-Hong Du, Member, IEEE, Qing-Ming Wang, Member, IEEE, and Kenji Uchino, Member, IEEE Abstract—This paper presents the theoretical derivation for a method of accurately determining the complex coefficients of single crystal piezoelectric materials. The vibration equations are analytically solved for a piezoelectric plate and a piezoelectric thin rod in an arbitrary crystallographic orientation. The solutions provide the distributions of the stresses and the vibration velocities inside the sample and the electrical impedance between the electrodes. Based on the analysis of the solutions, the complex dielectric, elastic, and piezoelectric coefficients are determined from the impedance or admittance measurements near the resonance frequencies. The measurement for LiNbO3 single crystals is used as an example to demonstrate the experiment and calculation procedures and to indicate the comparisons with previous methods.

I. Introduction ecently, some relaxor-based ferroelectric single crystals such as Pb(Zn1/3 Nb2/3 )O3 –PbTiO3 and Pb(Mg1/3 Nb2/3 )O3 –PbTiO3 have received great attention due to their extremely high electromechanical coupling factors (higher than 90%) and large piezoelectric coefficients (larger than 1500 pC/N) [1]–[3]. They are potentially used for transducers and other ultrasonic devices [3], [4]. On the other hand, these materials usually exhibit low mechanical quality factors that degrade the performance of the devices made from them, especially high-frequency transducers. Thus, accurate determination of the coefficients and the dissipation of these materials is essential to the successful utilization of them. Several methods have been developed to determine the coefficients of piezoelectric materials. The measurement methods for low loss materials have been summarized in IEEE standards [5]. Those methods have been successfully used to determine the coefficients of materials with high mechanical quality factors, such as Alpha-Quartz [6], lithium tantalate and lithium niobate single crystals [7], [8], barium titanate single crystals [9]–[12], lead titanate single crystals [13], and lead titanate zirconate ceramics [14]–[16], etc. However, significant errors may be intro-

R

Manuscript received January 21, 2003; accepted August 19, 2003. X.-H. Du is with Ramtron International Corporation, Colorado Springs, CO 80921. Q.-M. Wang is with the Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh, PA 15261 (e-mail: [email protected]). K. Uchino is with the Materials Research Institute, The Pennsylvania State University, University Park, PA 16802.

duced when the IEEE standards are directly applied to lossy materials, as discussed in [17] and [18]. The dissipation of the materials can be phenomenologically expressed by complex dielectric, piezoelectric, and elastic coefficients [19]. This treatment is valid when the material is driven under a small sinusoidal electric field and useful for constructing the transfer functions for various devices. Some methods have been reported for measuring the complex coefficients for piezoelectric ceramics. Holland et al. [17] proposed the gain-bandwidth method that is based on the approximation of the impedance or the admittance by the first term of its partial fraction expansion (or the Mittag Leffler expansion). The error of this approximation is dependent on the frequency and the elastic loss tangent φ = s11 /s11 (symbols used in this paper are described in Table I). We found that the error is about 2.5% for φ = 0.1 and 5% for φ = 0.2 at f = fs and rapidly increases as the frequency deviates from fs [18]. Thus, the approximation is not valid at the half-power frequencies when the mechanical coupling factor is high and the mechanical quality factor is low. In addition, neglecting the dielectric loss and the phase of k31 in [17] also introduces some model error. Thus, the method can be applied for materials with moderate loss tangents and low mechanical coupling factors. Smits [20] developed an iterative method that obtains the complex coefficients by forcing the admittance equation to satisfy the measured values of the admittance Y (ω) at three different frequencies. However, the values of the admittance Y (ω) (or the impedance Z(ω)) are vulnerable to random errors and deviations of the vibration from the ideal state. Thus, the accuracy of the results from this method is dependent on the errors relative to the magnitude of the admittance, and is thus sensitive to the choice of the points [21], [22]. Sherrit et al. [21] developed a non-iterative method by introducing complex frequencies fs and fp into impedance expressions or into a polynomial approximation of k31 in terms of frequency ratios. This treatment implicitly predetermines that the phase of the electromechanical coupling factor is dependent only on the mechanical loss. Thus, this method can be applied for materials with low dielectric and piezoelectric losses. Kwok et al. [22] compared the above methods and introduced a method with curve fitting. It should be mentioned that random and model errors might degrade methods based on curve fitting. Although these methods can be used to determine some coefficients of single crystals that are associated with sim-

c 2004 IEEE 0885–3010/$20.00

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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 51, no. 2, february 2004 TABLE I List of Symbols. T S E D ρ qe f and ω c s e d εS εT l = [l1 l2 l3 ]t la and lb Tl = T · l n = [n1 n2 n3 ]t x1 , x2 , and x3 V Λ v Vs ∂ ∂l l a

: : : : : : : : : : : : : : : : : : : : : :

stress (N/m2 ) strain electric field intensity (V/m) electric displacement (C/m2 ) density of the sample (kg/m3 ) electric charge density (C/m3 ) frequency (Hz) and angular frequency (rad/sec) elastic stiffness under constant electric field (N/m2 ) elastic compliance under constant electric field (m2 /N) piezoelectric constant (C/m2 ) piezoelectric constant (C/N) permittivity under constant strain (F/m) permittivity under constant stress (F/m) a unit vector along the wave propagation direction unit vectors which form orthogonal system with l traction force on a plane normal to l (N/m2 ) a unit vector along the electric field direction space coordinates (m) phase velocity in direction l (m/s) inverse of the phase velocity, i.e. 1/V, in direction l (s/m) vibration velocity or particle velocity (m/s) voltage source (V)

: the directional derivative in direction l : the distance from the origin in direction l. : half of the thickness for a plate, or half of the length for a bar.

A bold symbol stands for a matrix or a vector. Its components are denoted by the same symbol with subscripts; for example, Tij is the ij component of T. It is understood that all the coefficients are complex. We apply “prime” and “double prime” to a quantity to indicate its real and imaginary parts, respectively.

ple vibration modes, new methods are needed to obtain the complete set of complex coefficients of single crystals. The previous methods for single crystals have not considered the dissipation of the materials. In addition, some cutting orientations, which cause no problems for Alpha-Quartz, lithium tantalate, and lithium niobate single crystals, may cause some problems for relaxor ferroelectrics. For example, the rotated Y-cut used in [7] and [8] cause polarization depoling in the relaxor materials. The objective of the present work is to develop a method that overcomes the shortcomings of the previous methods mentioned above, is valid for the whole range of quality and electromechanical coupling factors, and is applied for piezoelectric single crystals and ceramics. We will present the analytical derivations for the measurements with plates and rods. The plates are usually used for determining the materials constants in a relatively high frequency range whereas the rods are used for the constants in a relatively low frequency range.

II. Field Equations and Piezoelectric Constitutive Equations In order to develop a general method for the determination of complex dielectric, elastic, and piezoelectric coefficients of piezoelectric resonators, we start our discussion from the general acoustic field equations and piezoelectric constitutive equations.

A. Acoustic Field Equations in Time Domain We start from the acoustic field equations and piezoelectric constitution equations. The derivations for these equations can be found in [23], [24]. Under zero body force, the acoustic field equations for piezoelectric resonators can be expressed as: ∂v ∂t ∂S ∇s v = ∂t

∇·T= ρ

(1) (2)

The electric field equations under zero magnetic field are ∂D = −J ∂t ∇ · D = qe ∇ϕ = −E

(3) (4) (5)

where the bold symbols represent tensors or matrices. The tensors can be either in dyadic forms or in the compressed matrix notations. If dyadic forms are used, the differential operators in the equations are standard gradients or divergences defined in tensor calculus. If the matrix notations are used, then ∇· in (1) is defined as ⎡ ⎤ ∂/∂x1 0 0 0 ∂/∂x3 ∂/∂x2 ∇· = ⎣ 0 ∂/∂x2 0 ∂/∂x3 0 ∂/∂x1 ⎦ 0 0 ∂/∂x3 ∂/∂x2 ∂/∂x1 0 (6)

du et al.: method for determination of complex coefficients i: theory

∇s is the transposition of ∇· in (6), i.e., ∇s = (∇·)t

(7)

⎤ ∂ ⎢ ∂x1 ⎥ ⎢ ∂ ⎥ ⎥ ⎢ ∇=⎢ ⎥. ⎢ ∂x2 ⎥ ⎣ ∂ ⎦ ∂x3

(8)

and ∇ in (4) and (5) is

⎡

The electric charge density qe in (4) may not be zero for ferroelectric with multi-domains where there is charge accumulation on the domain boundaries. In the present work, we assume that the electric field is low and has no influence on the domain pattern. Thus, qe is a constant independent from time and is virtually 0 in the frequency domain. It should be noted that this is not true when an electric field higher than the coercive field is applied. B. Acoustic Field Equations in Frequency Domain For sinusoidal steady states, (1)–(3) have equivalent expressions in the frequency domain, ∇ · T = jωρv ∇s v = jωS J = −jωD

(9) (10) (11)

C. Piezoelectric Constitutive Equations For linear piezoelectrics, the general constitutive equations, given by ANSI/IEEE 176-1987 [5], describe relations between mechanical and electric variables, i.e., mechanical stress T, mechanical strain S, electrical field E, and electrical displacement D. Because the permittivity of the free space is very small, the difference between P (electric polarization) and D (electric displacement) can be safely neglected when the electric field is in a normal range. Depending on the variable choice, four sets of constitutive relations are defined, which can be expressed in either matrix or tensor form. We can express each electromechanical pair of the quantities in terms of the others. The following are the four sets of relations expressed in matrix form: T = cE S − et E S

(12)

D = eS + ε E

(13)

t

T =c S−h D

(14)

S

(15)

t

(16)

D

E = −hS + β D E

S=s T+d E T

D = dT + ε E

(17)

S = sD T + g t D

(18)

E = −gT + β T D

(19)

where the superscript t indicates the transposition operation of matrices, and superscripts E, D, T , and S represent

229

constant electric field, constant electric displacement, constant stress, and constant strain conditions, respectively. There are two ways to understand these equations. From the viewpoint of thermodynamics, each pair of the equations describes the internal connections among the coefficients of the terms in the free energy expressions under specific boundary conditions. From the engineering point of view, any linear system can be mathematically described by a set of linear equations either in the time domain or in the frequency domain. The latter interpretation leads us to extend the elastic, piezoelectric, and dielectric coefficients to complex values for lossy materials in the sinusoidal steady states or, more generally, to transfer functions in the transient states.

III. Vibration of a Thin Piezoelectric Rod The measurement of the extensional modes of a thin piezoelectric rod has been described in [5]. It was believed that all the elastic and piezoelectric constants could be obtained with a sufficient number of rods in various cutting orientations [5], [16]. However, this may not be practically possible for some materials. The reasons are: 1) It is not true that there is always a pure extensional mode in an arbitrary cutting orientation. As we will see later, longitudinal and shear modes may be coupled in a thin rod. In this situation, the measurement method based on the assumption of a simple extensional mode will cause significant model errors. 2) For some materials, the polarization de-poling may be severe during sample preparation for some cutting orientations, and it is difficult to pole the sample again after the sample is cut in a direction canted an angle from the poling direction. 3) The sample cutting in some orientations requires special equipment and introduces errors that cause inconsistency among the measurement results from different samples. In this section, we will generally study the vibrations of a thin piezoelectric rod in an arbitrary crystallographic orientation and derive the impedance or admittance expressions. The results can be used to design the simplest measurements for various materials. A. Stress and Traction Force in a Thin Bar A thin piezoelectric bar cut in direction l is schematically shown in Fig. 1. Let l = [l1 l2 l3 ]t be the unit vector along the length direction of the bar and la and lb be unit vectors which form orthogonal system with l. We assume that the length of the bar is much larger than the width and the thickness such as the stress on any plane normal to la or lb is zero when length vibration is excited, i.e., Tla la = Tlb lb = Tla lb = 0.

(20)

Thus, the stress tensor can be expressed as T = Tla l (la ⊗ l + l ⊗ la ) + Tlb l (lb ⊗ l + l ⊗ lb ) + Tll l ⊗ l. (21)

230


B. Vibration Equations of a Thin Piezoelectric Bar If we neglect the variations of the stress along the width and the thickness and combine (9) and (25), we obtain ∂Tl = jωρv ∂l

(30)

With the aid of (29), (10) can be rewritten as ∂v = jω(L − M)S ∂l

We adopt the following piezoelectric constitutive equations:

Fig. 1. Illustration of a piezoelectric thin rod.

We include the bases for the second order tensors in (21) in order to distinguish them from the conventional crystallography coordinate systems. The traction force on a plane normal to l is Tl = T · l = Tla l la + Tlb l lb + Tll l.

S = sE T + dt E T

D = dT + ε E

(22)

(23)

∂ 2 Tl = −ω 2 ρ(L − M)s(L − M)t Tl − ω 2 ρBE ∂l2

Tij = (Tl )i lj + li (Tl )j − (l · Tl ) li lj

(24)

where (Tl )i is i-th component of the traction force Tl . Eq. (24) is applicable for an arbitrary orthogonal coordinate system as long as the components of the stress tensor T and the components of the direction vector l are referred to the same bases. It can be proven that the stress T and the traction force Tl on a plane normal to l satisfy the following relationships by translating (22) and (23) into matrix forms Tl = LT

(25)

and T = (L − M) Ll t

where L and M are defined, ⎡ l1 0 L = ⎣ 0 l2 0 0 and

respectively, by ⎤ 0 0 l3 l2 0 l3 0 l1 ⎦ l3 l2 l1 0

(26)

(27)

⎡

⎤⎡ 2 ⎤ l1 0 0 l1 − 1 l22 l32 l2 l3 l1 l3 l1 l2 M = ⎣ 0 l2 0 ⎦ ⎣ l12 l22 − 1 l32 l2 l3 l1 l3 l1 l2 ⎦ , 0 0 l3 l12 l22 l32 − 1 l2 l3 l1 l3 l1 l2 (28)

L and M satisfy L(L − M)t = I where I is the identity matrix.

(29)

(33)

(34)

and Dn = Bt Tl + εn E

In terms of the tensor components, (23) becomes

(32)

Combining (26) and (30)–(33), we have

Thus T can be expressed in terms of traction force Tl by T = Tl ⊗ l + l ⊗ Tl − (l · Tl ) l ⊗ l.

(31)

(35)

where B = (L − M) dt n, εn = nt εT n, Dn = nt D, and n is the unit vector along the electric field E. We found that it is convenient to classify the above boundary problems according to the direction n of the electric field relative to the wave propagation direction l. C. Vibration of a Piezoelectric Bar with Electric Field Parallel to l When the electrodes are normal to l, we assume that Dn is independent from l. Substituting (35) into (34), we obtain ∂ 2 Tl ω2ρ = −ω 2 ρATl − BDn 2 ∂l εn

(36)

where A = (L − M)s(L − M)t − BBt /εn and B = (L − M) dt n, A is generally a symmetric complex matrix. It may not be diagonalizable. Thus, the existence of three principle directions is not guaranteed when the losses in the system are significant. If this is the case, the validation of the expressions of losses by complex quantities should be reexamined. Usually, expressing losses by complex quantities is a concept limited in the frequency domain and is hardly mapped to the time domain. However, if we consider the dissipation as a small perturbation to the ideal lossless system, the three principle directions, i.e., the three eigenvectors of A, should exit. From now on, we assume that our systems satisfy this condition. If we assume that A is diagonalizable and consider that the last term in (36) is independent from l, the solution of


⎡

1 ⎢ λ1 ⎢ 1 ⎢ Tl (l) = P⎢ εn ⎢ ⎢ ⎣

cosh(jωΛ1 l) −1 cosh(jωΛ1 a) 0

⎢ ⎢ ⎢ 1 v(l) = P⎢ ρεn ⎢ ⎢ ⎣

Vs =

⎤

1 λ2

0 cosh(jωΛ2 l) −1 cosh(jωΛ2 a)

0

⎡

231

0

0

1 λ3


⎥ ⎥ ⎥ −1 ⎥ P BDn ⎥ ⎥ ⎦

⎤ ρ sinh(jωΛ1 l) 0 0 ⎥ λ1 cosh(jωΛ1 a) ⎥ ⎥ −1 ρ sinh(jωΛ2 l) ⎥ P BDn 0 0 ⎥ λ2 cosh(jωΛ2 a) ⎥ ρ sinh(jωΛ3 l) ⎦ 0 0 λ3 cosh(jωΛ3 a)

(37)

(38)

a

E(l)dl ⎧ ⎫ ⎤ ⎡ 1 tan(ωΛ1 a) ⎪ ⎪ ⎪ ⎪ −1 0 0 ⎪ ⎪ ⎪ ⎥ ⎢ λ1 ⎪ ⎪ ⎪ ωΛ1 a ⎪ ⎥ ⎢ ⎪ ⎨ ⎬ ⎥ a) 2aDn 1 t ⎢ 1 tan(ωΛ 2 −1 ⎥ ⎢ = 1 − B P⎢ 0 −1 0 P B . ⎥ ⎪ εn ⎪ εn λ2 ωΛ2 a ⎪ ⎢ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪ ⎦ ⎣ 1 tan(ωΛ3 a) ⎪ ⎪ ⎪ ⎪ 0 0 −1 ⎩ ⎭ λ3 ωΛ3 a −a

(36) subject √ to Tl |l=±a = 0 is shown in (37) (see above) where Λi = ρλi , P = [P1 P2 P3 ] and λi and Pi are the i-th eigenvalue and the i-th eigenvector of A, respectively. Substituting (37) into (30) and ⎡ (35), ⎤ we obtain (38) and ξ1 (39) (see above). Let P−1 B = ⎣ξ2 ⎦; then the impedance ξ3 between the electrodes can be expressed as: 3 1 tan(ωΛi a) 2 Z(ω) = 1− Ki −1 jωCn ωΛi a i=1 3 1 Ki2 tan(ωΛi a) 2 = 1 + KΣ 1 − (40) jωCn 1 + KΣ2 ωΛia i=1 ξ2

n where Ki2 = λi εi n , KΣ2 = K12 + K22 + K32 , Cn = εndA , An is n the area of the electrodes, and dn is the distance between the two electrodes.

D. Vibration of a Piezoelectric Bar with Electric Field Perpendicular to l

(39)

where A = (L − M)sE (L − M)t and B = (L − M) dt n. The solution of (41) subject to Tl |l=±a = 0 is (42) (see √ next page) where Λi = ρλi , P = [P1 P2 P3 ] and λi and Pi are the i-th eigenvalue and the i-th eigenvector of A, respectively. Substituting (42) into (30) and (35), we obtain (43) and (44) (see next page). ⎡ ⎤ ξ1 Let P−1 B = ⎣ξ2 ⎦; then the admittance between the ξ3 electrodes can be expressed as 3 tan(ωΛi a) 2 Y (ω) = jωCn 1 + Ki −1 ωΛi a i=1 3 2 K tan(ωΛ a) i i 1+ (45) = jωCn 1 − KΣ2 2 1 − K ωΛi a Σ i=1 ξ2

n where Ki2 = λi εi n , KΣ2 = K12 + K22 + K32 , Cn = εndA , An is n the area of the electrodes, and dn is the distance between the two electrodes.

When the electrodes are parallel to l, we assume that E is independent from l. Eq. (34) can be rewritten as

IV. Vibrations of a Piezoelectric Plate

∂ 2 Tl = −ω 2 ρATl − ω 2 ρBE ∂l2

The propagation of plane elastic waves in piezoelectric materials has been described in [5], [23], and [24]. The

(41)

232


⎡

1 ⎢ λ1 ⎢ ⎢ Tl (l) = P ⎢ ⎢ ⎢ ⎣

cosh(jωΛ1 l) −1 cosh(jωΛ1 a) 0

⎢ ⎢ 1 ⎢ v(l) = P ⎢ ρ ⎢ ⎢ ⎣

1 λ2


0

⎡

In = w

⎤

0

0

1 λ3


⎥ ⎥ ⎥ −1 ⎥ P BE ⎥ ⎥ ⎦

⎤ ρ sinh(jωΛ1 l) 0 0 ⎥ λ1 cosh(jωΛ1 a) ⎥ ⎥ −1 ρ sinh(jωΛ2 l) ⎥ P BE 0 0 ⎥ λ2 cosh(jωΛ2 a) ⎥ ρ sinh(jωΛ3 l) ⎦ 0 0 λ3 cosh(jωΛ3 a)

(42)

(43)

a

Dn (t) dl ⎧ ⎫ ⎡ ⎤ 1 tan(ωΛ1 a) ⎪ ⎪ ⎪ ⎪ − 1 0 0 ⎪ ⎪ ⎥ ⎪ ⎢ λ1 ⎪ ⎪ ⎪ ωΛ a 1 ⎥ ⎪ ⎢ ⎪ ⎨ ⎬ ⎥ a) 1 tan(ωΛ 1 t ⎢ 2 −1 ⎥ P B . (44) = 2awεn E 1 + B P ⎢ 0 − 1 0 ⎥ ⎢ ⎪ ⎪ εn λ2 ωΛ2 a ⎪ ⎢ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪ ⎦ ⎣ 1 tan(ωΛ a) ⎪ ⎪ 3 ⎪ ⎪ 0 0 −1 ⎩ ⎭ λ3 ωΛ3 a −a

dispersion relations for the elastic waves have been investigated in these previous works. However, the specific solutions of the vibration equations subject to different boundary conditions and different electrical excitations provide more information about the stress distribution and vibration velocity within the samples, which are important in electromechanical property measurement as well as in device design. Here we classify the vibration problems according to the electrical excitation direction relative to the wave propagation direction and determine the corresponding solutions for the boundary problems. The advantage of this classification is that we can obtain uniform mathematical descriptions for the vibrations of piezoelectric plates and bars. Our solutions are based on the assumption that the thickness vibrations of a plate can be closely approximated by uniform plane waves when the lateral dimensions of the plate are much larger than its thickness. The difference between this assumption and the real situation can be easily seen on the edges. For a finite plate, the stresses on the edges are zeros, but a uniform plane wave implies that the stresses on the edges are equal to the stresses at the center of the plate and thus non-zeros. Nevertheless, it has been widely accepted that “the edges have small effects on the behavior of the fundamental thickness resonance”

when the lateral dimensions satisfy the conditions mentioned above [24]. A. Vibration Equations of a Thin Piezoelectric Plate Again, we start from the field equations and piezoelectric constitutive equations in the notation of Auld [24]. (9), (10) and (4) are re-listed here for convenience, ∇ · T = jωρv ∇s v = jωS ∇ · D = qe

(46) (47) (48)

We adopt the following piezoelectric constitutive equations: T = cE S − et E S

D = eS + ε E

(49) (50)

When the lateral dimension of the plate is much larger than its thickness, the thickness vibration is assumed to be a uniform plane wave. Let l = [l1 l2 l3 ]t be the unit vector normal to the main surface of the plate and la and lb be unit vectors which form orthogonal system with l, as shown in Fig. 2; we then have ∂Ti ∂Ti = lk ∂xk ∂l

(51)


233

l, the solution of (57) subject to Tl |l=±a = 0 is (60) (see ρ next page) where Λi = λi , P = [P1 P2 P3 ], and λi and Pi are the i-th eigenvalue and the i-th eigenvector of A, respectively. Substituting (60) into (57) and (58), ⎡ ⎤we obtain (61) and ξ1 (62) (see next page). Let P−1 B = ⎣ξ2 ⎦ and consider that ξ3 P is close to an orthogonal matrix, i.e., the Pt ≈ P−1 ; the impedance between the electrodes can be expressed as 1 Z(ω) = jωCn

1−

Fig. 2. Illustration of a piezoelectric plate.

3

tan(ωΛi a) Ki2 ωΛi a i=1

(63)

ξ2

Recall that L in (27) and consider that the traction force Tl = T · l = LT; we obtain ∇·T= L

∂(T · l) ∂Tl ∂T = = ∂l ∂l ∂l

(52)

C. Vibration of a Piezoelectric Bar with Electric Field Perpendicular to l

(53)

In this case, the electrodes are on the planes perpendicular to la or lb . We assume that E is independent from l. Combining (53)–(55) and (50), we have

Eqs. (46), (47), and (49) can then be rewritten as ∂Tl = jωρv ∂l ∂v Lt = jωS ∂l Tl = LcS − Let E

(54) A (55)

Let n be a unit vector along the electric field E. In the following two sections, we will separately consider two simple ways to electrically excite the vibration of the plate: the electric field is parallel to l (i.e., n // l) and perpendicular to l (i.e., n ⊥ l). B. Vibration of a Piezoelectric Plate with Electric Field Parallel to l When the electrodes are on the main plane (i.e., n = l), we assume that the electric field E is independent from the lateral coordinates, i.e., E = E(l)n. Eq. (50) implies that D is also independent from the lateral coordinates since S has a uniform plane wave form. Thus, (48) becomes t

∂(n D)/∂l = 0.

(56)

Combining (53)–(55) and (50), we obtain ∂ 2 Tl ω2ρ A 2 = −ω 2 ρTl − BDn ∂l εn 1 ∂Tl v= jωρ ∂l 1 1 ∂v E(l) = Dn − Bt εn jωεn ∂l

n where Ki2 = λi εi n , Cn = εndA , An is the area of the elecn trodes, and dn is the distance between the two electrodes.

(57) (58) (59)

where A = LcE Lt + BBt /εn , B = Let n, εn = ntεS n, and Dn = nt D. If we assume that A is diagonalizable and consider that (56) implies that the last term in (57) is independent from

∂ 2 Tl = −ω 2 ρTl − ω 2 ρBE ∂l2 1 ∂Tl v= jωρ ∂l ∂v Jn = Bt + jωεn E ∂l

(64) (65) (66)

where A = LcE Lt , B = Let n, E = |E|, Jn = jωnt D, and εn = nt εS n. The solution of (64), subject to Tl |l=±a = 0 is (67) (see next page) where Λi = λρi , P = [P1 P2 P3 ], and λi and Pi are the i-th eigenvalue and the i-th eigenvector of A, respectively. Substituting (67) into (65) and (66), we obtain (68) (see next page) and

In = w

a

−a

Jn dl = jωCn Vs + wBt (v(a) − v(−a)) (69)

n where Cn = εndA , An is the area of the electrodes, and dn n is the distance between the two electrodes. Thus, the electric admittance ⎡ can ⎤ be expressed as (70) (see next page). ξ1 Let P−1 B = ⎣ξ2 ⎦; then ξ3

Y (ω) = jωCn

where Ki2 =

ξi2 λi εn .

1+

3

tan(ωΛi a) Ki2 ωΛi a i=1

(71)

234


⎤ ⎡ cosh(jωΛ l) 1 −1 0 0 ⎥ ⎢ cosh(jωΛ1 a) ⎥ 1 ⎢ cosh(jωΛ2 l) ⎥ −1 ⎢ −1 0 Tl (l) = P⎢ 0 ⎥ P BDn ⎥ εn ⎢ cosh(jωΛ2 a) ⎦ ⎣ cosh(jωΛ3 l) 0 0 −1 cosh(jωΛ3 a)

(60)

⎡ sinh(jωΛ l) ⎤ 1 Λ1 0 0 ⎢ cosh(jωΛ1 a) ⎥ ⎢ ⎥ sinh(jωΛ2 l) 1 ⎢ ⎥ −1 0 Λ2 0 v(l) = P⎢ ⎥ P BDn ⎥ cosh(jωΛ2 a) ρεn ⎢ ⎣ sinh(jωΛ3 l) ⎦ 0 0 Λ3 cosh(jωΛ3 a)

(61)

⎧ ⎫ ⎡ ⎤ 1 tan(ωΛ1 a) ⎪ ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎢ λ1 ωΛ1 a ⎥ ⎪ ⎪ ⎪

a ⎨ ⎢ ⎥ ⎬ 2aDn 1 t ⎢ 1 tan(ωΛ2 a) ⎥ −1 1 − B P⎢ Vs = E(l)dl = 0 0 ⎥P B . ⎢ ⎥ ⎪ εn ⎪ εn λ2 ωΛ2 a −a ⎪ ⎪ ⎪ ⎪ ⎣ ⎪ ⎪ 1 tan(ωΛ3 a) ⎦ ⎪ ⎪ ⎩ ⎭ 0 0 λ3 ωΛ3 a

(62)

⎤ ⎡ cosh(jωΛ l) 1 −1 0 0 ⎥ ⎢ cosh(jωΛ1 a) ⎥ ⎢ cosh(jωΛ2 l) ⎥ −1 ⎢ 0 −1 0 Tl (l) = P ⎢ ⎥ P BE ⎥ ⎢ cosh(jωΛ2 a) ⎦ ⎣ cosh(jωΛ3 l) 0 0 −1 cosh(jωΛ3 a)

(67)

⎡ ⎢ 1 ⎢ ⎢ v(l) = P ⎢ ρ ⎢ ⎣

Λ1

sinh(jωΛ1 l) cosh(jωΛ1 a)

⎤ 0

0

sinh(jωΛ2 l) Λ2 cosh(jωΛ2 a)

0

0

0

⎥ ⎥ ⎥ −1 0 ⎥ P BE ⎥ sinh(jωΛ3 l) ⎦ Λ3 cosh(jωΛ3 a)

⎧ ⎫ ⎡ ⎤ 1 tan(ωΛ1 a) ⎪ ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎢ λ1 ωΛ1 a ⎥ ⎪ ⎪ ⎪ ⎨ ⎢ ⎥ ⎬ In 1 t ⎢ 1 tan(ωΛ2 a) ⎥ −1 Y = = jωCn 1 + B P ⎢ 0 0 ⎥P B . ⎪ ⎢ ⎥ ⎪ Vs εn λ2 ωΛ2 a ⎪ ⎪ ⎪ ⎪ ⎣ ⎪ ⎪ 1 tan(ωΛ3 a) ⎦ ⎪ ⎪ ⎩ ⎭ 0 0 λ3 ωΛ3 a

(68)

(70)


V. Determination of the Coefficients from Impedance Measurements In this section, we develop a general method to determine the complex elastic, piezoelectric, and dielectric coefficients from the electric impedance and admittance measurements. This approach has been applied for the accurate determination of the complex coefficients d31 , s11 , and ε33 of piezoelectric ceramic resonators in an earlier paper [18]. Over the years, it has been realized that it is much more difficult to accurately determine the imaginary parts of the coefficients than their real parts. The difficulties come from at least three aspects. First, the imaginary parts are usually much smaller than the real parts and there is little information that is sensitive only to the imaginary parts. Second, there may be compensations among the coefficients; for example, overestimation on the dielectric tangent loss may be compensated by overestimation on the piezoelectric loss. As a result, there may be no obvious differences between the measured and the calculated values of impedance. Thus, it is critical to choose the points of the impedance and to select a curve-fitting method if a measurement method is based on the values of impedance. Third, random errors may be in the same order of magnitude as the imaginary parts. As we mentioned at the beginning, the values of the impedance or admittance are vulnerable to random errors. Because of these uncertainty factors, we will not use any curve-fitting method, but try to extract the coefficients from the information that is reliable and sensitive to the coefficients. The impedance or admittance for a thin bar or a thin plate is expressed by (40), (45), (63), or (71). These equations are very similar to each other. Actually, according to the electric field parallel to or perpendicular to the specifically defined direction vector l, we can express the impedance or the admittance, respectively, as 3 1 2 tan(ωΛi a) Z(ω) = 1− ki (72) jωC ωΛi a i=1 and

Y (ω) = jωC

3

tan(ωΛi a) 1+ ki2 ωΛi a i=1

(73)

where C, ki , and Λi are some complex constants that are known functions of the materials coefficients for a specified vibration mode, as derived above. We will present the method to determine the constants C, ki , and Λi from the impedance measurement. The method for the admittance measurement is similar. For simplicity, we assume that these constants are not dependent on the frequency in the vicinity of a resonance point. To eliminate the frequency dependence of the first term in (72), we normalize the impedance by 1/ω. Thus, (72) becomes z(ω) = ωZ(jω) =

3 1 tan(ωΛi a) +j ηi jC ωΛi a i=1

(74)

235

where ηi =

ki2 . C

(75)

Considering tan(x + jy) =

sin(2x) + j sinh(2y) , cos(2x) + cosh(2y)

(76)

we rewrite (74) to A sin(2ωΛ a) + A sinh(2ωΛ a) 1 i i i i − jC i=1 cos(2ωΛi a) + cosh(2ωΛi a) 3

z(ω) =

+j

3 A sin(2ωA a) − A sinh(2ωΛ a) i

i=1

i

i

(77)

i

cos(2ωΛia) + cosh(2ωΛi a)

where Ai =

ηi = Ai + jAi . ωΛi a

(78)

It should be mentioned that, in the vicinity of a resonance point, Ai could be considered as frequency-independent. From (77), we can see that the derivative of z(ω) with respect to ω is independent of C. Near the i-th mode, the derivative of the real part of z(ω) is given by (79) (see next page). The second term in the numerator is dominant because the magnitudes of the imaginary parts of the constants are much less than those of their real parts. Thus, the zero point of this derivative is close to the frequency corresponding to ωΛi a = π. Therefore, we have Λi =

1 4fiR max a

(80)

where fiR max is the frequency of the maximum real part of z(ω) at the i-th mode. Near fiR max , the first term in the numerator is at least the second order of the imaginary parts. Thus, the errors from (80) can be neglected when the imaginary parts are much less than their real parts. If we define qmi = |Λi /Λi | and let the derivative of the imaginary part X of z(ω) be zero, we find that qm can be accurately determined from [18]: cosh(π/qmi ) − 1 π2 − 2 (1 + 1/qmi ) cosh(π/qmi ) 8

fiX max − fiX min fiR max

2 =0 (81)

and the imaginary part of Λi is given by Λi = sign(fiX max − fiX min )

Λi qm

(82)

where fiX max and fiX min are the frequencies of the maximum and the minimum of the imaginary part of z(ω) at the i-th mode. From the definition, qmi is related to the effective mechanical quality factor Qmi for the i-th mode. Approximately, qmi = 2Qmi.

236


∂R (Λ A + Λi Ai )[1 + cos(2ωΛi a) cosh(2ωΛi a)] + (Λi Ai − Λi Ai ) sin(2ωΛi a) sinh(2ωΛi a) = −2a i i . 2 ∂ω [cos(2ωΛi a) + cosh(2ωΛi a)]

The real part of Ai is calculated from the slope of the imaginary part X of z(ω) at fiR max . 1 − cosh(π/qmi ) ∂X Ai = 4πΛi a ∂f f =fiR max (83) π ∂X ≈− 8aΛ q 2 ∂f i mi

f =fiR max

The determination of the imaginary part of Ai is much more difficult. This can be seen from the expression of z(ω). In (77), the imaginary part Ai is multiplied by small quantities at the frequencies near fiR max . Thus, the terms containing Ai are easily overridden by the errors from the other terms containing Ai . This implies that relatively large variations on the imaginary parts of the piezoelectric coefficients do not cause obvious changes in the calculated impedance in the vicinity of resonance. In this sense, the imaginary parts of the piezoelectric coefficients are not “observable” from the impedance measurements and have been neglected for a long time. We may determine the imaginary part Ai from the asymmetry of the impedance over the sidebands. The asymmetry of the sidebands is used to detect nonlinear effects in [19]. However, we can see from (77) that the dissipation also introduces the asymmetry of the impedance around the frequency fiR max . It is interesting to take a close look at (79). The first term in the numerator is an even function of (f − fiR max ), whereas the second term is an odd function if the hyperbolic functions are assumed to be constants over the sidebands. Thus, if we add two ∂R values ∂R ∂f (fiR max − ∆f ) + ∂f (fiR max + ∆f ), the second term is canceled and the first term, which contains Ai , is doubled. From this idea, we obtain Ai from N ∂R k=1

∂f

(−k) · α(−k) +

−

4πa(Λi Ai

+

∂R (k) · α(k) = ∂f

Λi Ai )

N (β(−k) + β(k))

(84)

where

+ cosh (4π(fiR max + k · ∆f )Λi a) β(k) = 1 + cos (4π(fiR max + k · ∆f )Λi a) · cosh (4π(fiR max + k · ∆f )Λi a)

VI. Summary We have derived the general expressions of impedance or admittance for a piezoelectric thin bar resonator and a plate resonator. Two unit vectors l and n have been defined in our derivation: for a piezoelectric thin bar resonator, l is defined along the length direction; for a piezoelectric plate resonator, l is perpendicular to the major surfaces of the plate; and n is along the direction of the electric field. For both types of piezoelectric resonators, the resonator samples can be further classified into two types of configurations: the electric field parallel to l, and electric field perpendicular to l. When the electric field is parallel to l, the impedance of the resonators can be expressed by (72); when the electric field is perpendicular to l, the admittance of resonators can be expressed by (73). The complex constants C, ki , and Λi in the impedance (or admittance) equations are functions of the materials coefficients for a specified vibration mode. Therefore, the elastic, dielectric, and piezoelectric coefficients of piezoelectric resonators can be determined from the impedance (or admittance) measurements. The real and imaginary parts of the inverse phase velocity Λi can be determined from three critical frequencies corresponding to the maximum of the normalized resistance, and the maximum and the minimum of the normalized reactance [(80)–(82)]. The electromechanical coupling factor ki can be calculated from the intermediate quantities ηi and Ai , where the real part and imaginary part of Ai is calculated from (83) and (84), and ηi from (78). C in (72) can be determined by capacitance measurements. Finally, the elastic and piezoelectric coefficients are derived from the definitions of ki and Λi . The design of the measurements and the applications of this method to the determination of materials coefficients for single crystals LiNbO3 are presented in a later paper [25]. References

k=1

∂R ∂R (k) = (fiR max + k · ∆f ) ∂f ∂f α(k) = cos (4π(fiR max + k · ∆f )Λi a)

(79)

(85)

2 (86) (87)

where ∆f is the measurement increment step of frequency or its multiples.

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du et al.: method for determination of complex coefficients i: theory [7] A. W. Warner, M. Onoe, and G. A. Coquin, “Determination of elastic and piezoelectric constants for crystals in class (3m),” J. Acoust. Soc. Amer., vol. 42, no. 6, pp. 1223–1231, 1967. [8] R. T. Smith and F. S. Welsh, “Temperature dependence of elastic, piezoelectric, and dielectric constants of lithium tantalate and lithium niobate,” J. Appl. Phys., vol. 42, pp. 2219–2230, 1971. [9] D. Berlincourt and H. Jaffe, “Elastic and piezoelectric coefficients of single-crystal barium titanate,” Phys. Rev., vol. 111, no. 1, pp. 143–148, 1958. [10] M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Gunter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B, vol. 50, no. 9, pp. 5941–5949, 1994. [11] Z. Li, S.-K. Chan, M. H. Grimsditch, and E. S. Zouboulis, “Elastic and electromechanical properties of tetragonal BaTiO3 single crystals,” J. Appl. Phys., vol. 70, pp. 7327–7332, 1991. [12] A. Schaefer, H. Schmitt, and A. Dorr, “Elastic and piezoelectric coefficients of TSSG barium titanate single crystals,” Ferroelectrics, vol. 69, pp. 253–266, 1986. [13] Z. Li, M. Grimsditch, X. Xu, and S.-K. Chan, “The elastic, piezoelectric and dielectric constants of tetragonal PbTiO3 single crystals,” Ferroelectrics, vol. 141, pp. 313–325, 1993. [14] D. A. Berlincourt, C. Cmolik, and H. Jaffe, “Piezoelectric properties of polycrystalline lead titanate zirconate compositions,” in Proc. IRE, vol. 48, pp. 220–229, 1959. [15] M. Onoe, H. F. Tiersten, and A. H. Meitzler, “Shift in the location of resonance frequencies caused by large electromechanical coupling in thickness-mode resonators,” J. Acoust. Soc. Amer., vol. 35, no. 1, pp. 36–42, 1963. [16] D. Berlincourt, “Piezoelectric crystals and ceramics,” in Ultrasonic Transducer Materials. New York: Plenum, 1971, pp. 63– 124. [17] R. Holland and E. EerNisse, “Accurate measurement of coefficients in a ferroelectric ceramic,” IEEE Trans. Sonics Ultrason., vol. SU-16, no. 4, pp. 173–181, 1969. [18] X. H. Du, Q.-M. Wang, and K. Uchino, “Accurate determination of complex materials coefficients of piezoelectric resonators,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 50, no. 3, pp. 312–320, 2003. [19] R. Holland, “Representation of dielectric, elastic, and piezoelectric losses by complex coefficients,” IEEE Trans. Sonics Ultrason., vol. SU-14, no. 1, pp. 18–20, 1967. [20] J. G. Smits, “Iterative method for accurate determination of the real and imaginary parts of the materials coefficients of piezoelectric ceramics,” IEEE Trans. Sonics Ultrason., vol. SU-23, no. 6, pp. 393–402, 1976. [21] S. Sherrit, H. D. Wiederick, and B. K. Mukherjee, “Non-iterative evaluation of the real and imaginary material constants of piezoelectric resonators,” Ferroelectrics, vol. 134, pp. 111–119, 1992. [22] K. W. Kwok, H. L. W. Chan, and C. L. Choy, “Evaluation of the material parameters of piezoelectric materials by various methods,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 44, no. 4, pp. 733–742, 1997. [23] H. F. Tiersten, Linear Piezoelectric Plates Vibrations. New York: Plenum, 1969. [24] B. A. Auld, Acoustic Fields and Waves in Solids. vol. 1, New York: Wiley, 1973. [25] X. H. Du, Q.-M. Wang, and K. Uchino, “An accurate method for the determination of complex coefficients of single crystal piezoelectric resonators II: Design of measurement and experiments,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 51, no. 2, pp. 238–248, Feb. 2004.

Xiao-Hong Du (M’99) received the Ph.D. degree in electrical engineering and the M.S. degree in computer science and engineering from the Pennsylvania State University at University Park, PA, in 2000 and 1999, respectively. He also received the M.S. degree in mathematics from New Mexico Institute of Mining and Technology at Socorro, NM, and the B.S. degree in electrical engineering from Huazhong University of Science and Technology, Wuhan, China.

237

His research interests include electromagnetic and acoustic fields and waves; characterization of piezoelectric materials; microelectromechanical systems (MEMS); microelectronic devices; VLSI circuit design and simulation; piezoelectric thin films, single crystals, and ceramics for resonator, actuator, and transducer applications. He is now a design engineer in VLSI circuits at Ramtron International Corporation, Colorado Springs.

Qing-Ming Wang (M’00) is an assistant professor in the Department of Mechanical Engineering, the University of Pittsburgh, Pennsylvania. He received the B.S. and M.S. degrees in Materials Science and Engineering from Tsinghua University, Beijing, China, in 1987 and 1989, respectively, and the Ph.D. degree in Materials from the Pennsylvania State University in 1998. Prior to joining the University of Pittsburgh, Dr. Wang was an R&D engineer and materials scientist in Lexmark International, Inc., Lexington, Kentucky, where he worked on piezoelectric and electrostatic microactuators for inkjet printhead development. From 1990 to 1992, he worked as a development engineer in a technology company in Beijing where he participated in the research and development of electronic materials and piezoelectric devices. From 1992 to 1994, he was a research assistant in the New Mexico Institute of Mining and Technology working on nickel-zinc ferrite and ferrite/polymer composites for EMI filter application. From 1994 to 1998, he was a graduate assistant in the Materials Research Laboratory of the Pennsylvania State University working toward his Ph.D. degree in the areas of piezoelectric ceramic actuators for low frequency active noise cancellation and vibration damping, and thin film materials for microactuator and microsensor applications. Dr. Wang’s primary research interests are in microelectromechanical systems (MEMS) and microfabrication; smart materials; and piezoelectric/electrostrictive ceramics, thin films, and composites for electromechanical transducer, actuator, and sensor applications. He is a member of IEEE, IEEE-UFFC, the Materials Research Society (MRS), ASME, and the American Ceramic Society.

Kenji Uchino (M’89) received his Ph.D. in Physical Electronics at the Tokyo Institute of Technology in Japan. He also received his M.S. in Physical Electronics and B.S. in Physics at the institute. Dr. Uchino was presented with the outstanding research award by the Pennsylvania State Engineering Society in 1996, along with other numerous awards throughout his technical career. He is currently a professor of Electrical Engineering as well as the director of the International Center for Actuators and Transducers at Pennsylvania State University. He has also been recognized for his positions including: A professional committee member of the Space Shuttle Utilizing Committee in Japan, a research associate for the Tokyo Institute of Technology, a research associate of Materials Research at Pennsylvania State University, and other distinguishing positions adding to his strong background and professional experience. Dr. Uchino’s research interests lie in the field of solid-state physics, especially in the areas of ferroelectrics and piezoelectrics. His work also includes basic research on materials and device development of solid-state actuators and sensors to precision positioners such as ultrasonic motors and smart structures. Dr. Uchino is a member of several technical societies including the Japanese Technology Transfer Association, the American Ceramic Society, IEEE (the Institute of Electrical and Electronics Engineers), and the New York Academy of Sciences. He is Executive Associate Editor of the Journal of Intelligent Materials Systems and Structures (VPI).