Precise measurement of the transverse piezoelectric coefficient for thin films onanisotropic substrateDoo-Man Chun, Masashi Sato, and Isaku Kanno Citation: Journal of Applied Physics 113, 044111 (2013); doi: 10.1063/1.4789347 View online: http://dx.doi.org/10.1063/1.4789347 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/113/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Influence of pulse poling on the piezoelectric property of Pb(Zr0.52,Ti0.48)O3 thin films AIP Advances 4, 117116 (2014); 10.1063/1.4901912 Temperature dependence of the transverse piezoelectric coefficient of thin films and aging effects J. Appl. Phys. 115, 034105 (2014); 10.1063/1.4862045 Surface potential measurement of organic thin film on metal electrodes by dynamic force microscopy using apiezoelectric cantilever J. Appl. Phys. 109, 114306 (2011); 10.1063/1.3585865 Piezoelectric nanoelectromechanical resonators based on aluminum nitride thin films Appl. Phys. Lett. 95, 103111 (2009); 10.1063/1.3216586 In-plane excitation of thin silicon cantilevers using piezoelectric thin films Appl. Phys. Lett. 91, 183510 (2007); 10.1063/1.2805070

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Precise measurement of the transverse piezoelectric coefficientfor thin films on anisotropic substrate

Doo-Man Chun,1,a) Masashi Sato,2 and Isaku Kanno3

1School of Mechanical Engineering, University of Ulsan, Ulsan, 680-749, South Korea2Department of Micro-Engineering, Kyoto University, Kyoto, 606-8501, Japan3Department of Mechanical Engineering, Kobe University, Kobe, 657-8501, Japan

(Received 31 July 2012; accepted 27 December 2012; published online 28 January 2013)

In this study, we propose a reliable measurement method for the effective transverse piezoelectric

coefficient for thin films especially on anisotropic substrate. This coefficient for piezoelectric

Pb(Zr, Ti)O3 (PZT) thin films was calculated by measuring the electric field-induced tip

displacement of unimorph cantilevers composed of PZT thin films and Si substrates. We evaluated

the reliability of the proposed measurement method by comparing it with numerical analysis and

confirmed that the relative error of the piezoelectric coefficient (e31, f) was less than 1%. We

prepared 16 different unimorph cantilevers composed of identical PZT films on different Si beam

geometries that had various substrate thicknesses and cantilever widths. Although the effective

transverse piezoelectric coefficient e31, f of PZT thin films ranged from �6.5 to �14 C/m2 as a

function of the applied voltage, the difference among the 16 samples with an applied voltage of

25 V was within 10%. These results demonstrate that the proposed measurement method has

sufficient reliability and can be used to evaluate the effective transverse piezoelectric coefficient

e31, f of thin films. VC 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4789347]

I. INTRODUCTION

Thin film piezoelectric materials such as Pb(Zr, Ti)O3

(PZT), ZnO, and AlN have been widely investigated in

micro/nano-electromechanical systems (M/NEMS).1–5 The

transverse piezoelectric effect is used for micro-sensors and

actuators using piezoelectric thin films because most of the

piezoelectric micro-components have a membrane, cantile-

ver, or bridge structure.2–5 Several methods for measuring

the transverse piezoelectric coefficient have been reported.

For example, the effective piezoelectric transverse coeffi-

cient e31, f of a PZT thin-film diaphragm was measured using

the energy method, which uses a thin diaphragm for mea-

surement.6 In another study, the piezoelectric transverse

coefficient d31 was measured directly from the change of the

length of PZT micro-cantilevers under an applied DC volt-

age.7 On the other hand, Shepard et al. utilized the wafer

bending method, which measures the electric charge induced

by applying pressure, to evaluate the transverse piezoelectric

coefficient d31. This method uses a uniform pressure rig,

pressure transducer, and electric charge integrator, and it is

suitable for wafer-level testing.8 In 1999, Dubois et al. devel-

oped a measurement method for the e31, f of a PZT thin-film

cantilever by bending the cantilever and measuring the

induced charges.9 Southin et al. suggested another method to

measure the e31, f of PZT films using a conventional d33 me-

ter by indenting PZT on Si.10 In 2003, a measurement

method for determining the transverse piezoelectric coeffi-

cient e31* was proposed by measuring the displacement of a

PZT thin-film cantilever as a function of the applied volt-

age.11–13 Unfortunately, the definition of the transverse pie-

zoelectric coefficient e31* is different from widely used e31, f,

so it is difficult to compare the measured transverse piezo-

electric coefficient values measured by different methods. In

addition, the validity and reliability of this measurement

method has not been sufficiently discussed. Thus, this study

proposes a method for measuring the transverse piezoelectric

characteristics with small piece of PZT thin-film cantilever

using the remnant of wafer dicing, which can be easily

obtained during device fabrication process and discusses the

validity and reliability of the derived piezoelectric coeffi-

cient. To obtain our results, we calculated the effective pie-

zoelectric transverse coefficient from the tip displacement of

unimorph cantilevers. We prepared a variety of specimens,

which were composed of identical PZT films on different Si

cantilevers, and evaluated the validity of the calculated trans-

verse piezoelectric coefficient from the PZT films. We uti-

lized numerical analysis to verify the equation for the

transverse piezoelectric coefficient. Finally, we determined

the optimal sample structure to obtain reliable measurements

and calculated values of the effective transverse piezoelectric

coefficient for piezoelectric thin films.

II. EVALUATION OF MEASUREMENT METHODOF THE TRANSVERSE PIEZOELECTRICCOEFFICIENT FOR THIN FILMS

A. Sample preparation

In this research, a PZT thin film was fabricated on a sili-

con substrate, which is a commonly used material for

MEMS sensors and actuators. The PZT thin film was fabri-

cated via radio frequency (RF) magnetic sputtering. Detailed

material information and the dimensions of the specimen are

summarized in Table I. A cantilever type of specimen was

fabricated to test the measurement method, and the length,

a)Author to whom correspondence should be addressed. Tel: þ82 52 259

2706. E-mail: dmchun@ulsan.ac.kr.

0021-8979/2013/113(4)/044111/9/$30.00 VC 2013 American Institute of Physics113, 044111-1

JOURNAL OF APPLIED PHYSICS 113, 044111 (2013)

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width, and thickness were evaluated to determine the mea-

surement accuracy.

B. Measurement method

In order to obtain the effective piezoelectric transverse

coefficient, we must measure the voltage generated by the

deformation or displacement from an applied voltage. In this

research, the tip displacement was measured by laser Dopp-

ler vibrometry as a function of the electric voltage between

the top and bottom electrodes. The bottom electrode and

PZT thin film were deposited on the whole area by sputter-

ing, and the top electrode was deposited on the specific area

using a shadow mask to avoid any electric contacts with the

clamping fixture. Figure 1 presents a schematic image of the

equipment used to measure the transverse piezoelectric coef-

ficient and the real images of clamping fixture and speci-

mens. For precise positioning, the laser beam was aligned

using in-situ microscope. For the proper clamping, the same

cantilever with the same thickness was positioned in another

side of clamp fixture, and the distance from the clamping

position to the laser beam was controlled by the precision 2-

axis stage. The bottom electrode was directly contacted to

the clamp fixture by removing PZT layer and using silver

paste, and the top electrode was contacted using very thin

gold wire and silver paste. Note that all of the components

shown in Figure 1, such as the power supply and oscillo-

scope, are commonly used in sensor or actuator research.

The structure of the cantilever beam includes the substrate,

bottom electrode, piezoelectric material, and top electrode.

The voltage difference between the top and bottom electro-

des displaces the cantilever tip. Any type of piezoelectric

film on various substrates can be evaluated using this system.

The configuration of the measurement system was described

in a previous study.11–13

In order to calculate the effective piezoelectric trans-

verse coefficient, it is necessary to know the material proper-

ties of the substrate, the tip displacement, and the electric

field applied to the piezoelectric material. Hence, the equa-

tion for the effective piezoelectric transverse coefficient can

be derived using the constituent equations for the piezoelec-

tric effect and the pure bending model of a plate. The constit-

uent equations of the piezoelectric effect can be written as

Sa ¼ sEabTb þ diaEi; Di ¼ diaTa þ eT

ijEj

a; b ¼ 1; 2;…; 6 and i; j ¼ 1; 2; 3;(1)

where Sa, Di, Ei, and Ta are vector forms of the strain, the

electric displacement, the electric field, and the stress,

respectively. The compliance matrix subjected to a constant

electric field is denoted by sEab, the piezoelectric coefficients

by dia, and the permittivity under constant stress by eTij .

Because the electric field is applied in the 3 direction,

the constituent equations can be expressed as

D3 ¼ e3E3 þ d31ðT1 þ T2Þ þ d33T3; (2)

S1 ¼ sE11T1 þ sE

12T2 þ sE13T3 þ d31E3; (3)

S2 ¼ sE12T1 þ sE

11T2 þ sE13T3 þ d31E3; (4)

S3 ¼ sE13T1 þ sE

13T2 þ sE33T3 þ d33E3: (5)

The stress component T3 becomes zero because the film is

free to move in the 3 direction (T3 ¼ 0). When the small pi-

ece of the cantilever beam is modeled as in Figure 2(a), the

stress components T1 and T2 have the same amplitudes due

to the beam’s geometric symmetry and isotropic piezoelec-

tric material property in the 1 and 2 directions (T1 ¼ T2).14

And, the electric field can be expressed as Eq. (6), where V

is the applied voltage between top and bottom electrodes of

piezoelectric thin film, and hp is the thickness of piezoelec-

tric thin film

E3 ¼ V=hp: (6)

In addition, the piezoelectric film is much thinner than the

substrate (hs � hp), so the piezoelectric film will create large

bending and small extension or small contraction. Conse-

quently, the deformation pattern cannot be considered a pure

bending model of beam, but it can be considered a pure

bending model of plate. Figure 2(b) shows a rectangular

plate with a uniformly distributed bending moment. The

mid-surface displacement of the rectangular plate with pure

bending can written as

TABLE I. Sample information.

Length of cantilever 20 mm

Width of cantilever 1, 2, 3, 4 mm

Electrode material Pt (Top), Pt/Ti/SiO2

(Bottom)

Material of piezoelectric thin film Lead zirconate titanate

(Pb[Zr0.52Ti0.48]O3)

Substrate material Silicon wafer (6 in.)

Substrate crystalline orientation along cantilever h100i h110iThickness of piezoelectric thin film 2 lm

Thickness of substrate without PZT 356, 512 lm

FIG. 1. Equipment used to measure the transverse piezoelectric coefficient:

(a) schematic image and (b) real images of clamping fixture and cantilevers

with different width.

044111-2 Chun, Sato, and Kanno J. Appl. Phys. 113, 044111 (2013)

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wðx; yÞ ¼ � 6ðMb � �MaÞYh3

x2 � 6ðMa � �MbÞYh3

y2; (7)

where w is the displacement of the mid-surface of the rectan-

gular plate, Ma and Mb are bending moments per unit length,

v is Poisson’s ratio of the plate, Y is Young’s modulus of the

plate, h is the thickness of the plate, and x and y are the posi-

tions of measurement.15

The origin of the rectangular plate can be set as the

clamping area of the cantilever beam. Then, the displace-

ment of the cantilever tip can be measured from the end of

the cantilever beam at the center of the beam width so that xand y become L (the length of the cantilever beam) and zero,

respectively. Moreover, the bending moment per unit length

can be expressed by Eq. (8), and the displacement of the can-

tilever tip d can be summarized by Eq. (9)

M ¼ �hpT1

hs

2; (8)

d ¼ wðL; 0Þ ¼ 3hpT1ð1� �sÞYsh2

s

L2: (9)

The length of upper side of cantilever contracts due to the

contraction of piezoelectric thin film (DL ¼ LS1), and the

length of lower side of cantilever becomes almost constant.

And, since the curvature radius is large and the displacement

is very small, the displacement of the cantilever tip can be

expressed using the length and bending angle of the cantile-

ver via the following geometric constraint (Figure 3):

sinh � h ¼ d=L; LS1 ¼ hsh; (10)

S1 ¼ hsd=L2: (11)

Using Eqs. (3), (6), (9), and (11), the piezoelectric coefficient

d31 can be derived as

d31 ¼ �h2

s YsðsE11 þ sE

12Þ3ð1� �sÞL2V

d� 1� hp

hs

ð1� �sÞYsðsE

11 þ sE11Þ

� �: (12)

Then, sE11 ¼ 1=Yp, sE

12 ¼ ��p=Yp, and hs � hp simplify such

that

d31 ¼ �h2

s YsðsE11 þ sE

12Þ3ð1� �sÞL2V

d� 1� hp

hs

Ypð1� �sÞYsð1� �pÞ

� �

ffi � h2s YsðsE

11 þ sE12Þ

3ð1� �sÞL2Vd: (13)

However, the mechanical properties of a piezoelectric thin

film are not easy to measure, so Eqs. (14) and (15) demon-

strate how to obtain the effective transverse piezoelectric

coefficient e31, f of a piezoelectric thin film without these me-

chanical properties

e31; f ¼d31

sE11 þ sE

12

; (14)

e31; f ¼ �h2

s Ys

3ð1� �sÞL2Vd: (15)

Note that Eq. (15) expresses the effective transverse piezo-

electric coefficient in terms of the material properties of the

substrate, the thickness of the substrate, the length of the can-

tilever, the applied voltage on the piezoelectric thin film, and

the measured tip displacement.

C. Numerical analysis

We used the commercial finite element method (FEM)-

based numerical analysis software package, ANSYS, to assess

the reliability of the measurement method for the effective

transverse piezoelectric coefficient.16 The well-known typi-

cal material properties of PZT and silicon were input for

the numerical analysis, and the input piezoelectric property

e31, f was compared with the calculated piezoelectric prop-

erty using Eqs. (14) and (15). The ratio of the difference

between the input piezoelectric property and the calculated

piezoelectric property from the tip displacement over input

piezoelectric property indicated the relative error of the

derived equation for the effective transverse piezoelectric

coefficient. The relative error was expressed in Eq. (16). In

order to confirm the reliability of this equation, the ANSYS

software analyzed the behavior of the piezoelectric mate-

rial. Note that piezoelectric materials, especially the PZT

material, have been widely researched and analyzed inFIG. 3. Geometric constraint.

FIG. 2. (a) Schematic view of the simplified

cantilever beam and (b) a rectangular plate

model with a uniformly distributed bending

moment.

044111-3 Chun, Sato, and Kanno J. Appl. Phys. 113, 044111 (2013)

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various manners.17,18 Table II shows the material properties

of the substrate and PZT material. The tip displacement

was obtained using the numerical analysis results, as shown

in Figure 4.

Relative Error ¼����Calculated e31; f � input e31; f

input e31; f

����: (16)

The equation for the effective transverse piezoelectric coeffi-

cient was derived using a few assumptions for simplification.

First, the equation ignored the mechanical properties of the

piezoelectric thin film, making the thickness ratio between

the substrate and the piezoelectric thin film very large

(hs � hp). Moreover, the cantilever was modeled as a pure

bending beam without clamping, but in reality, one edge of

the cantilever was fixed, suggesting that the cantilever width

may affect the bending of the plate. Consequently, the effects

of the thickness ratio and the width of the cantilever need to

be assessed. In addition, since the substrate material, silicon,

is anisotropic, the effect of its crystalline orientation should

be evaluated because the Young’s modulus and Poisson’s ra-

tio of silicon depend on the crystalline orientation. More-

over, the initial bending of the cantilever caused by residual

stress may affect the actual tip displacement. But in this

study, the initial bending was not considered because the

effect of residual stress can be small with thick substrate,

and no significant transverse bending was observed. The

effect of residual stress will be studied as a future research.

First, the effect of the cantilever width was evaluated.

For the numerical analysis, the length of the cantilever beam

was 17 mm, and its width ranged from 0.5 to 17 mm. These

values were identical to the experimental specimens and are

comparable to the chip size in various micro-devices. The

crystalline orientation of the silicon wafer along the cantile-

ver was selected as [100] or [110] because these orientations

have been widely utilized. Figure 5 shows the effect of the

cantilever width. The solid line is the input piezoelectric

property of PZT and the points are calculated values: A

larger cantilever width corresponds to a more substantial rel-

ative error. The dashed line, which is parallel to the solid

line, represents a 1% relative error. The large observed error

may be due to the assumption of a rectangular plate model

with a uniformly distributed bending moment. The model

was fixed at the origin point, but the numerical analysis

accounted for the clamping of the cantilever to one face of

the plate. In other words, the model allowed the clamped

TABLE II. Material properties.19–21

Material Material property name Material property

PZT Density (kg/m3) 7590

Relative permeability 180

Relative permittivity x: 1180, y: 730, z: 1180

Piezoelectric strain coefficient (pC/N) d33: 223 d31: �93.5 (e31, f: �9.61 C/m2) d51: 494

Compliance coefficient (pm2/N) 13:8 �5:8 �4:7 0 0 0

�5:8 17:1 �5:8 0 0 0

�4:07 �5:8 13:8 0 0 0

0 0 0 48:2 0 0

0 0 0 0 48:2 0

0 0 0 0 0 38:4

26666664

37777775

Si Density (kg/m3) 2331

Stiffness matrix (GPa) Crystal orientation along the cantilever: Si (100)

166 63:9 63:9 0 0 0

63:9 166 63:9 0 0 0

63:9 63:9 166 0 0 0

0 0 0 79:6 0 0

0 0 0 0 79:6 0

0 0 0 0 0 79:6

26666664

37777775

Crystal orientation along the cantilever: Si (110)

195 35:0 64:0 0 0 0

35:0 195 64:0 0 0 0

64:0 64:0 166 0 0 0

0 0 0 80:0 0 0

0 0 0 0 80:0 0

0 0 0 0 0 51:0

26666664

37777775

FIG. 4. Modeling and meshing of the cantilever beam.

044111-4 Chun, Sato, and Kanno J. Appl. Phys. 113, 044111 (2013)

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edge of the cantilever to bend freely. Thus, a large cantilever

width will inhibit a reliable measurement of the effective

transverse piezoelectric coefficient because the rectangular

plate model assumes a uniformly distributed bending

moment. According to the numerical analysis, a cantilever

with a width of 1 mm showed less than 1% error, regardless

of the crystalline orientation.

Second, the effect of the thickness ratio between the

substrate and piezoelectric film was evaluated. The numeri-

cal analysis was carried out using various substrate thick-

nesses, including 10, 20, 100, 356, and 512 lm. The

fabricated PZT thickness was 2 lm, and the cantilever length

was 17 mm. The cantilever width was set to 1 mm to mini-

mize its effect, which is shown in Figure 5. Figure 6 shows

the results of the numerical analysis. The smaller substrate

thickness produced a larger relative error. For example,

when the thickness ratio of the piezoelectric thin film to the

substrate was 2:100, the relative error was 1.35%. Note that

the derived equation for the effective transverse piezoelectric

coefficient ignored the structural effect of the piezoelectric

thin film because the thickness ratio between the substrate

and the film was very large (hs � hp), so a small substrate

thickness produced a large relative error. In this case, the

stiffness of piezoelectric thin film must be considered.

Third, the effect of the crystal orientation of the substrate

along the cantilever direction was evaluated. One of the typi-

cal substrates for MEMS devices using PZT thin films is a sili-

con wafer. Silicon has an anisotropic material property, so the

derived equation must be satisfied regardless of the substrate’s

crystal orientation. Figure 7 shows the calculated e31, f values

and the input e31, fvalue (�9.61 C/m2). In particular, it shows

that the difference between the calculated e31, f and the input

FIG. 6. Effect of the thickness ratio between

the PZT thin film and the substrate.

FIG. 5. Effect of cantilever width.

FIG. 7. Effect of the substrate crystal orientation along the cantilever.

FIG. 8. XRD patterns for PZT layers on substrates with thicknesses of 356

and 512 lm.

044111-5 Chun, Sato, and Kanno J. Appl. Phys. 113, 044111 (2013)

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e31, f changed according to the substrate crystal orientation,

but the relative error remained less than 1%.

Next, numerical analysis was used to validate the equa-

tion for the effective transverse piezoelectric coefficient. The

input e31, f value was compared with the calculated e31, f

value based on the numerically calculated tip displacement

under the given electric field and Eq. (13). The relative error

between the input e31, f and the calculated e31, f was found to

be less than 1% regardless of the substrate crystal orientation

for samples that were properly prepared—i.e., with an appro-

priate substrate width and thickness ratio between the sub-

strate and piezoelectric thin film. As noted above, a large

cantilever width and a thin substrate thickness can induce

large rates of error.

D. Experiment

Experimental samples with different cantilever widths,

substrate crystalline orientations, and substrate thicknesses

were prepared to confirm the proposed measurement method.

PZT thin films were formed on silicon substrates with differ-

ent substrate thicknesses (356 and 512 lm) under the identi-

cal process conditions; then dicing was performed for two

different crystalline orientations of the silicon substrate—

(100) and (110)—along the length direction of the cantilever

for four different widths (1, 2, 3, and 4 mm).

1. Evaluation of PZT deposition

The fabricated PZT thin films with thicknesses of about

2 lm were assessed in order to confirm the uniformity of the

PZT thin films on two different substrates. The dielectric

properties of the PZT thin films were measured using an in-

ductance-capacitance-resistance (LCR) meter. Five platinum

top electrodes with diameters of 1.0 mm were deposited on

the PZT thin films. The average relative constants, er, of the

samples with widths of 356 and 512 lm were 924 and 912,

respectively. The average dielectric losses, tan d, of the sam-

ples with widths of 356 and 512 lm were 2.51% and 2.42%,

respectively.

The crystalline structure of the PZT thin films was meas-

ured using X-ray diffraction (XRD). Figure 8 shows the

FIG. 10. Comparison of piezoelectric properties

for different substrate crystal orientations along

the cantilever as well as for different substrate

widths.

FIG. 9. P-E hysteresis of PZT layers on sub-

strates with thicknesses of 356 and 512 lm.

044111-6 Chun, Sato, and Kanno J. Appl. Phys. 113, 044111 (2013)

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XRD patterns of the PZT films with widths of 356 and

512 lm. There are several peaks for the perovskite phases of

PZT (001), (101), (111), and (002) and for the pyrochlore

phase of PZT. Even though a small pyrochlore phase was

observed, the PZT thin film showed good piezoelectric prop-

erties. In addition, the polarization-electric field (P-E) hyster-

esis loop was measured using a Sawyer-Tower circuit.

Figure 9 shows the P-E hysteresis results for two different

substrates. The XRD patterns and the P-E hysteresis results

were very similar regardless of the substrate thickness. Con-

sequently, these results indicate that the PZT thin films can

be reproduced consistently, and the two PZT thin films had

very similar material properties.

2. Tip displacement measurement and e31, f

calculation

The tip displacements according to the applied voltage,

substrate thickness, substrate width, and substrate crystal ori-

entation were measured for the e31, f calculation. The applied

voltages ranged from 5 to 25 V, and the frequency of the

power supply was 500 Hz. The selected frequency was much

smaller than the natural frequency of the prepared cantilever.

Voltages higher than 25 V occasionally created electric leak-

age. Before the tip displacement measurement, 25 V (12.5

MV/m) was applied for poling, and the polarization direction

was the same as the applied voltage direction. All data are

summarized in Figure 10 according to the substrate thick-

ness, crystalline orientation along the cantilever, and cantile-

ver width. Different conditions produced results with similar

values, but deviations were apparent, especially for low

applied voltages. Figure 11 shows the average values for all

of the data with minimum and maximum values according to

the applied voltages. The low voltage result showed a rela-

tively large deviation. Low voltages produce only small tip

displacements, so a small measurement error can greatly

influence the relative error. Therefore, the results with the

highest applied voltage (25 V) were used to analyze the

effects of the substrate thickness, width, and crystal orienta-

tion. The average values and their corresponding standard

deviations as a function of the substrate thickness, width,

and crystal orientation are summarized in Tables III–V. In

addition, all data are summarized in Figure 12. The maxi-

mum difference between the average values from different

experimental conditions is less than one sigma of the stand-

ard deviation, so it is difficult to determine the effects of the

sample conditions. In addition, the maximum difference

between the measured data is less than 10% of the difference

from the average value. Thus, these numerical analysis

results suggest that the proposed measurement method can

reliably measure an effective transverse piezoelectric coeffi-

cient regardless of the substrate crystal orientation for the

substrate thicknesses and widths given here.

III. DISCUSSION

Both the experimental results and the numerical analysis

of the proposed measurement method for the effective piezo-

electric transverse coefficient showed good reliability. How-

ever, certain guidelines should be followed to obtain

reproducible results—e.g., thin substrates, wide cantilevers, or

low applied voltages can produce unreliable measurements.

Equations (17) and (18) present previous versions of

e31* that are very similar to the measurement method and

derived equation discussed in this paper11

e31� ¼ � d31

sE11

; (17)

e31� ¼ h2

s Ys

3L2Vd: (18)

The difference between the two measurement methods is

purely in their theoretical representations. For example,

FIG. 11. Average values of measured piezoelectric properties with minimum

and maximum values.

TABLE III. Average values and standard deviations according to substrate

thickness.

Thickness 356 mm Thickness 512 mm All data

Average (C/m2) �13.5 �13.8 �13.7

Standard deviation (C/m2) 0.44 0.44 0.46

TABLE IV. Average values and standard deviations according to substrate

crystal orientation.

Si (100) Si (110) All data

Average (C/m2) �13.6 �13.7 �13.7

Standard deviation (C/m2) 0.48 0.44 0.46

TABLE V. Average values and standard deviations according to substrate

width.

Width

1 mm

Width

2 mm

Width

3 mm

Width

4 mm

All

data

Average (C/m2) �13.5 �13.5 �13.7 �13.9 �13.7

Standard

deviation (C/m2)

0.44 0.52 0.31 0.43 0.46

044111-7 Chun, Sato, and Kanno J. Appl. Phys. 113, 044111 (2013)

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Eq. (17) for e31* does not contain the Poisson ratio for the

substrate. In addition, e31* does not require the material

property for the Poisson ratio, s12. If the Poisson ratio of the

substrate equals the Poisson ratio of the piezoelectric thin

film, then e31* can represent the piezoelectric material prop-

erty because the derived d31 from e31* is the same as the real

value. However, if the Poisson ratio of the substrate is not

the same as the Poisson ratio of the piezoelectric thin film,

then the d31 derived from e31* may contain errors according

to Eqs. (13) and (14). In fact, Figure 13 shows the numerical

analysis results for e31* and e31, f for different Si crystal ori-

entations along the cantilever. As the Si crystal orientation

changed, the modulus and Poisson ratio also changed. Con-

sequently, the calculated piezoelectric property of the PZT

thin film, e31*, was altered for the different Si crystal orienta-

tions, as shown in Figure 13. The Poisson ratio for silicon

(001) ranged from 0.064 to 0.279 according to the Si crystal

orientation along the cantilever.20 The difference in Pois-

son’s ratio directly affected the e31*. The proposed measure-

ment method based on e31, f produced consistent results

regardless of the Si crystal orientation. The experimental

results also revealed the same problem with e31*. Regardless

of the substrate crystal orientation, the piezoelectric property

of the same PZT thin film should not change. However, the

measurement method for e31* generated different results for

the same PZT thin films with different substrate crystal ori-

entations, as shown in Figure 14.

IV. CONCLUSIONS

In this research, a method for measuring the effective

piezoelectric transverse coefficient e31, f was proposed to

evaluate the piezoelectric property of a thin film considering

the anisotropic material properties of substrate. The effective

transverse piezoelectric coefficient of piezoelectric PZT thin

films was calculated by measuring the tip displacement of

unimorph cantilevers of PZT thin films and Si substrates. By

comparing the calculated results with those from the numeri-

cal analysis, we evaluated the reliability of the proposed

measurement method, and we confirmed that the relative

error of the piezoelectric coefficient (e31, f) was less than 1%

regardless of the substrate crystalline orientation. For experi-

mental verification, 16 different unimorph cantilevers com-

posed of identical PZT films on various Si beam geometries

that had different substrate thicknesses and widths were

measured. Although the effective transverse piezoelectric

coefficient e31, f ranged from �6.5 to �14 C/m2 as a function

of the applied voltage, the difference between the 16 samples

measured using a 25 V applied voltage was within 10%.

These results demonstrate that the proposed measurement

method has sufficient reliability and can be utilized to evalu-

ate the effective transverse piezoelectric coefficient e31, f of

thin films. In addition, the measurement method showed the

precise result regardless of the different substrate crystal

orientations.

FIG. 12. Comparison of piezoelectric properties

with an applied voltage of þ25 V: (a) substrate

thickness of 356 lm and (b) substrate thickness

of 512 lm.

FIG. 13. Comparison of numerical results for e31* and e31, f for different Si

crystal orientations along the cantilever.

FIG. 14. Comparison of experimental results between e31* and e31, f for dif-

ferent applied voltages and different Si crystal orientations along the

cantilever.

044111-8 Chun, Sato, and Kanno J. Appl. Phys. 113, 044111 (2013)

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ACKNOWLEDGMENTS

This work was supported by the 2012 Research Fund of

University of Ulsan.

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