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A resonant method for determining mechanical properties of Si3N4 and SiO2 thin films
Materials Letters 61 (2007) 3089 – 3092 www.elsevier.com/locate/matlet
A resonant method for determining mechanical properties of Si3N4 and SiO2 thin films Ching-Liang Dai ⁎, Yuan-Ming Chang Department of Mechanical Engineering, National Chung Hsing University, Taichung, 402 Taiwan, ROC Received 25 September 2006; accepted 2 November 2006 Available online 27 November 2006
Abstract This study investigates the measurement of Poisson's ratio and Young's modulus of silicon dioxide (SiO2) and silicon nitride (Si3N4) thin films using a resonant method. Two thin films, which are SiO2 and Si3N4, are fabricated as the specimens of microcantilever beams and plates using the bulk micromachining. The resonant frequency of the cantilever beams and plates is measured using a laser interferometer. The Young's modulus of thin films can be calculated from the resonant frequency of the cantilever beams, and the Poisson's ratio of thin films is determined by the frequency of the cantilever plates. Experimental results show that the Poisson's ratios of SiO2 and Si3N4 are 0.16 and 0.26, respectively, and the Young's moduli of SiO2 and Si3N4 are, respectively, 55.6 GPa and 131.6 GPa. © 2006 Elsevier B.V. All rights reserved. Keywords: Young's modulus; Poisson's ratio; Thin films
1. Introduction The Young's modulus and Poisson's ratio are two important mechanical properties of thin films. The static or dynamic behavior of several micromechanical sensors and microactuators, such as accelerometers [1] and RF micromechanical switches [2], depends on the mechanical properties of thin films. The Young's modulus and Poisson's ratio of thin films depend on the microstructure (grain size, orientation, and density) of the films, which is determined by the specific deposition conditions. The microstructure of thin films changes with the heat cycle, resulting in a change in the mechanical characteristics. It is difficult to evaluate the Young's modulus and Poisson's ratio of thin films using the simulation method according to the deposition conditions of the films. Therefore, the mechanical properties of thin films are always determined by experimental measurement methods [3–6]. In this work, we propose a resonant method, which is based on a PZT vibrator and a laser interferometer, to determine the Poisson's ratio and Young's modulus of thin films. The method ⁎ Corresponding author. E-mail address: [email protected] (C.-L. Dai). 0167-577X/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2006.11.031
has high accuracy owing to the nano-scale resolution of the PZT vibrator and laser interferometer. The PZT vibrator excites the cantilever beams and plates to generate vibration, and the laser interferometer measures the resonant frequency of the cantilever beams and plates. The Young's modulus and Poisson's ratio of thin films can be evaluated according to the resonant frequency of the cantilever beams and plates, respectively. 2. Principle Two micromachined structures, microcantilever beam and plate, are applied to measure the resonant frequencies and to evaluate the Young's modulus and Poisson's ratio of thin films. 2.1. Evaluating the Young's modulus of thin films The equation of motion for the free vibration of a uniform beam can be expressed as [7] EI
A4 w A2 w ðx; tÞ þ qA ðx; tÞ ¼ 0 Ax4 At 2
ð1Þ
where E is the Young's modulus; I is the moment of inertia of the beam cross section; w(x,t) is the transverse displacement of
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the beam; ρ is the mass density of the beam and A is the crosssectional area of the beam. The first natural frequency of cantilever beam can be expressed as [7] sffiffiffiffiffiffiffiffiffiffiffi EI x ¼ ð1:875Þ qAL4 2
ð2Þ
where L is the length of the cantilever beam. According to Eq. (2), we know that the Young's modulus, E, depends on the first resonant frequency, ω. The beams are made from thin films. The parameters of the beams, I, A, L and ρ, are also given. Therefore, the Young's modulus of thin films can be evaluated by Eq. (2) if the first resonant frequency of the beams is measured. 2.2. Evaluating the Poisson's ratio of thin films The classical differential equation of motion for the free vibration of a rectangular plate is given by [8] Dj4 w þ q
A2 w ¼0 At 2
ð3Þ
where w is the transverse deflection of the plate; ρ is the mass density per unit area of the plate; ∇4 is the biharmonic differential operator; t is the time and D is the flexural rigidity of the plate. Rossi and Laura [9] obtained the natural
Fig. 2. SEM photograph of SiO2 cantilever beams.
frequencies of the cantilever plate using the finite element method. The first natural frequency of the cantilever plate, ω1, for a / b = 2 can be written as [9] X1 x1 ¼ 2 a
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eh2 12qð1−r2 Þ
ð4Þ
where Ω1 represents the first natural frequency coefficients of the cantilever plate; E is the Young's modulus; h is the thickness of the plate; ν is the Poisson's ratio of the plate; ρ is the mass density per unit area of the plate; a and b are the length and width of the plate, respectively. According to Eq. (4), we know that the Poisson's ratio, ν, depends on the first resonant frequency, ω1. The plates are made from thin films. The parameters of the plates, E, h, a, Ω1 and ρ, are given. Therefore, the Poisson ratio of thin films can be evaluated by Eq. (4) if the first resonant frequency of the plates is measured. 3. Fabrication of specimens
Fig. 1. Process flow of microcantilever beams and plates; (a) depositing LPCVD SiO2 or Si3N4 layer, (b) spinning the photoresist, (c) patterning the LPCVD SiO2 or Si3N4 layer and (d) etching the silicon substrate by KOH.
Four kinds of specimens are fabricated — SiO2 cantilever beam, Si3N4 cantilever beam, SiO2 cantilever plates and Si3N4 cantilever plates. Fig. 1 depicts the process flow of the chips. First, low-pressure chemical vapor deposition (LPCVD) is used to deposit SiO2 or Si3N4 films on an orientation (100) silicon substrate, as shown in Fig. 1(a). Second, the photoresist (AZ5214) is spun on the SiO2 or Si3N4 films, and patterned by photolithography. The shapes of the cantilever beams and plates are formed in the photoresist, as illustrated in Fig. 1(b). Third, CF4/O2 reactive ion etching (RIE) is use to etch the SiO2 or Si3N4 films. The cantilever beams and plates are patterned in the SiO2 or Si3N4 films, and then the photoresist is striped, as shown in Fig. 1(c). Finally, the anisotopic etchant KOH heated to temperature of 80 °C is utilized to etch the silicon substrate, and released the suspended microstructures of the cantilever plates and beams, as illustrated in Fig. 1(d). Fig. 2 displays the SEM (Scanning electron micrograph) photograph of the SiO2 cantilever beams after the process.
C.-L. Dai, Y.-M. Chang / Materials Letters 61 (2007) 3089–3092
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Fig. 3. Experimental setup for measuring the resonant frequency of micromachined structures.
4. Results and discussion The resonant frequency of the cantilever beams and plates requires to be measured to evaluate the Young's modulus and Poisson's ratio of thin films. Fig. 3 illustrates the experimental setup that contains a function generator, a power amplifier, an oscilloscope, a PZT vibrator, a laser interferometer, an optical vibrometer and a computer for measuring the resonant frequency of the cantilever beams and plates. The PZT vibrator and the laser interferometer have nano-scale resolution. The tested sample mounts on the PZT vibrator. The function generator sends a swept sinusoidal signal in the frequency range from zero to 500 kHz. The swept sinusoidal signal is amplified through the power amplifier, and then sent to the PZT vibrator exciting the vibration of the tested specimens. The response of the tested specimens is detected using the laser interferometer. Finally, the tested results are recorded in the computer. The frequency response of the SiO2 and Si3N4 cantilever beams with different lengths were measured. The first frequency response measured of all SiO2 and Si3N4 cantilever beams is shown in Table 1. The width of all SiO2 and Si3N4 cantilever beams is 10 μm. The thickness of all SiO2 and Si3N4 cantilever beams is about 0.9 μm and
1.4 μm, respectively. The mass density of the SiO2 and Si3N4 are approximately 2160 kg/m3 and 3000 kg/m3, respectively [10,11]. Substituting the above first resonant frequency, dimensions and mass density of the SiO2 and Si3N4 cantilever beams into Eq. (2), the Young's modulus of the SiO2 and Si3N4 films can be yielded and the result is shown in Table 1. The average Young's modulus of the SiO2 and Si3N4 films was 55.6 GPa and 131.6 GPa, respectively. The Young's modulus of the SiO2 and Si3N4 films reported by Petersen [12] and Stark et al. [13] was 57–92 GPa and 129 GPa, respectively. Obviously, the present experiment results in good agreement with literature values. The length and thickness of the SiO2 cantilever plate are 100 μm and 0.9 μm, respectively. The mass density of the SiO2 is about 2160 kg/m3. Substituting the above values and E = 55.6 GPa into Eq. (4), the relation between the Poisson's ratio and the first resonant frequency of the SiO2 cantilever plate can be yielded, as shown in Fig. 4. The measured result showed that the first resonant frequency of the SiO2 cantilever plate was 74.3 kHz. By checking Fig. 4, the Poisson's ratio of SiO2 film with a resonant frequency of 74.3 kHz was 0.16. The Poisson's ratio of Si3N4 films can also be obtained using a similar procedure. The length and thickness of the Si3N4 cantilever plate are 100 μm and 1.4 μm, respectively. The mass density of the Si3N4 is about 3000 kg/m3. Substituting the above values and E = 131.6 GPa into Eq. (4), the relation between the Poisson's ratio and the first resonant frequency of the Si3N4 cantilever plate can be yielded, as shown in Fig. 5. The measured result depicted that the first resonant frequency of the Si3N4 cantilever plate was 152.8 kHz. By checking Fig. 5, the Poisson's ratio of Si3N4 film with a resonant frequency of 152.8 kHz was 0.26. Literatures [11,13] showed that the Poisson's ratio of the SiO2 and Si3N4 was about 0.17 and 0.28,
Table 1 Resonant frequency (RF) and Young's modulus (E) of cantilever beams with different lengths Beam length (μm)
Fig. 4. Relation between Poisson's ratio and resonant frequency of SiO2 cantilever plate.
100 150 200 Average
SiO2
Si3N4
SiO2
Si3N4
RF (kHz)
RF (kHz)
E (GPa)
E (GPa)
73.78 32.30 18.70
150.28 65.11 38.10
55.6 54.0 57.2 55.6
132.5 125.9 136.3 131.6
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Experimental results showed that the Poisson's ratios of SiO2 and Si3N4 were 0.16 and 0.26, respectively, and the Young's moduli of SiO2 and Si3N4 were, respectively, 55.6 GPa and 131.6 GPa. Acknowledgements The authors would like to thank the National Center for High-Performance Computing (NCHC) for chip simulation, the National Chip Implementation Center (CIC) for chip fabrication and the National Science Council of the Republic of China for financially supporting this research under Contract no. NSC 952221-E-005-043-MY2. References Fig. 5. Relation between Poisson's ratio and resonant frequency of the Si3N4 cantilever plate.
respectively. The experimental results are matched with literature values.
5. Conclusion The Poisson's ratio and Young's modulus of SiO2 and Si3N4 thin films using the resonant method have been measured. The micromachined SiO2 and Si3N4 cantilever beams and plates were fabricated on a chip using the bulk micromachining. The resonant frequency of the cantilever beams and plates were measured using a laser interferometer. The Young's modulus of the thin film was calculated from the resonant frequency of the cantilever beams, and the Poisson's ratio of thin film was determined from the resonant frequency of the cantilever plates.
[1] D.O. King, M.C.L. Ward, K.M. Brunson, D.J. Hamilton, Sens. Actuators, A 68 (1998) 238–243. [2] C.L. Dai, J.H. Chen, Microsystem Technol. 12 (2006) 1143–1151. [3] G. Alcala, P. Skeldon, G.E. Thompson, A.B. Mann, H. Habazaki, K. Shimizu, Nanotechnology 13 (2002) 451–455. [4] E. Harry, A. Rouzaud, M. Ignat, P. Juliet, Thin Solid films 332 (1998) 195–201. [5] B. Cros, E. Gat, J.M. Saurel, J. Non-cryst. Solid 209 (1997) 273–282. [6] C. Luo, T.W. Schneider, R.C. White, J. Currie, M. Paranjape, J. Micromechanics Microengineering 13 (2002) 129–133. [7] S.S. Rao, Mechanical Vibrations, Addison-Wesley, Massachusetts, 1990. [8] A.W. Leissa, J. Sound Vib. 31 (1973) 257–293. [9] R.E. Rossi, P.A.A. Laura, J. Sound Vib. 195 (1996) 142–148. [10] Y.S. Ju, K.E. Goodson, J. Appl. Phys. 85 (1999) 7130–7134. [11] S.D. Senturia, Microsystem Design, Kluwer Academic Publishers, Boston, 2001. [12] K.E. Petersen, IEEE Trans. Electron Devices 25 (1978) 1241–1250. [13] R.W. Stark, G. Schitter, A. Stemmer, Phys. Rev. 68 (2003) 085401.
A resonant method for determining mechanical properties of Si3N4 and SiO2 thin films Ching-Liang Dai ⁎, Yuan-Ming Chang Department of Mechanical Engineering, National Chung Hsing University, Taichung, 402 Taiwan, ROC Received 25 September 2006; accepted 2 November 2006 Available online 27 November 2006
Abstract This study investigates the measurement of Poisson's ratio and Young's modulus of silicon dioxide (SiO2) and silicon nitride (Si3N4) thin films using a resonant method. Two thin films, which are SiO2 and Si3N4, are fabricated as the specimens of microcantilever beams and plates using the bulk micromachining. The resonant frequency of the cantilever beams and plates is measured using a laser interferometer. The Young's modulus of thin films can be calculated from the resonant frequency of the cantilever beams, and the Poisson's ratio of thin films is determined by the frequency of the cantilever plates. Experimental results show that the Poisson's ratios of SiO2 and Si3N4 are 0.16 and 0.26, respectively, and the Young's moduli of SiO2 and Si3N4 are, respectively, 55.6 GPa and 131.6 GPa. © 2006 Elsevier B.V. All rights reserved. Keywords: Young's modulus; Poisson's ratio; Thin films
1. Introduction The Young's modulus and Poisson's ratio are two important mechanical properties of thin films. The static or dynamic behavior of several micromechanical sensors and microactuators, such as accelerometers [1] and RF micromechanical switches [2], depends on the mechanical properties of thin films. The Young's modulus and Poisson's ratio of thin films depend on the microstructure (grain size, orientation, and density) of the films, which is determined by the specific deposition conditions. The microstructure of thin films changes with the heat cycle, resulting in a change in the mechanical characteristics. It is difficult to evaluate the Young's modulus and Poisson's ratio of thin films using the simulation method according to the deposition conditions of the films. Therefore, the mechanical properties of thin films are always determined by experimental measurement methods [3–6]. In this work, we propose a resonant method, which is based on a PZT vibrator and a laser interferometer, to determine the Poisson's ratio and Young's modulus of thin films. The method ⁎ Corresponding author. E-mail address: [email protected] (C.-L. Dai). 0167-577X/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2006.11.031
has high accuracy owing to the nano-scale resolution of the PZT vibrator and laser interferometer. The PZT vibrator excites the cantilever beams and plates to generate vibration, and the laser interferometer measures the resonant frequency of the cantilever beams and plates. The Young's modulus and Poisson's ratio of thin films can be evaluated according to the resonant frequency of the cantilever beams and plates, respectively. 2. Principle Two micromachined structures, microcantilever beam and plate, are applied to measure the resonant frequencies and to evaluate the Young's modulus and Poisson's ratio of thin films. 2.1. Evaluating the Young's modulus of thin films The equation of motion for the free vibration of a uniform beam can be expressed as [7] EI
A4 w A2 w ðx; tÞ þ qA ðx; tÞ ¼ 0 Ax4 At 2
ð1Þ
where E is the Young's modulus; I is the moment of inertia of the beam cross section; w(x,t) is the transverse displacement of
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the beam; ρ is the mass density of the beam and A is the crosssectional area of the beam. The first natural frequency of cantilever beam can be expressed as [7] sffiffiffiffiffiffiffiffiffiffiffi EI x ¼ ð1:875Þ qAL4 2
ð2Þ
where L is the length of the cantilever beam. According to Eq. (2), we know that the Young's modulus, E, depends on the first resonant frequency, ω. The beams are made from thin films. The parameters of the beams, I, A, L and ρ, are also given. Therefore, the Young's modulus of thin films can be evaluated by Eq. (2) if the first resonant frequency of the beams is measured. 2.2. Evaluating the Poisson's ratio of thin films The classical differential equation of motion for the free vibration of a rectangular plate is given by [8] Dj4 w þ q
A2 w ¼0 At 2
ð3Þ
where w is the transverse deflection of the plate; ρ is the mass density per unit area of the plate; ∇4 is the biharmonic differential operator; t is the time and D is the flexural rigidity of the plate. Rossi and Laura [9] obtained the natural
Fig. 2. SEM photograph of SiO2 cantilever beams.
frequencies of the cantilever plate using the finite element method. The first natural frequency of the cantilever plate, ω1, for a / b = 2 can be written as [9] X1 x1 ¼ 2 a
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eh2 12qð1−r2 Þ
ð4Þ
where Ω1 represents the first natural frequency coefficients of the cantilever plate; E is the Young's modulus; h is the thickness of the plate; ν is the Poisson's ratio of the plate; ρ is the mass density per unit area of the plate; a and b are the length and width of the plate, respectively. According to Eq. (4), we know that the Poisson's ratio, ν, depends on the first resonant frequency, ω1. The plates are made from thin films. The parameters of the plates, E, h, a, Ω1 and ρ, are given. Therefore, the Poisson ratio of thin films can be evaluated by Eq. (4) if the first resonant frequency of the plates is measured. 3. Fabrication of specimens
Fig. 1. Process flow of microcantilever beams and plates; (a) depositing LPCVD SiO2 or Si3N4 layer, (b) spinning the photoresist, (c) patterning the LPCVD SiO2 or Si3N4 layer and (d) etching the silicon substrate by KOH.
Four kinds of specimens are fabricated — SiO2 cantilever beam, Si3N4 cantilever beam, SiO2 cantilever plates and Si3N4 cantilever plates. Fig. 1 depicts the process flow of the chips. First, low-pressure chemical vapor deposition (LPCVD) is used to deposit SiO2 or Si3N4 films on an orientation (100) silicon substrate, as shown in Fig. 1(a). Second, the photoresist (AZ5214) is spun on the SiO2 or Si3N4 films, and patterned by photolithography. The shapes of the cantilever beams and plates are formed in the photoresist, as illustrated in Fig. 1(b). Third, CF4/O2 reactive ion etching (RIE) is use to etch the SiO2 or Si3N4 films. The cantilever beams and plates are patterned in the SiO2 or Si3N4 films, and then the photoresist is striped, as shown in Fig. 1(c). Finally, the anisotopic etchant KOH heated to temperature of 80 °C is utilized to etch the silicon substrate, and released the suspended microstructures of the cantilever plates and beams, as illustrated in Fig. 1(d). Fig. 2 displays the SEM (Scanning electron micrograph) photograph of the SiO2 cantilever beams after the process.
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Fig. 3. Experimental setup for measuring the resonant frequency of micromachined structures.
4. Results and discussion The resonant frequency of the cantilever beams and plates requires to be measured to evaluate the Young's modulus and Poisson's ratio of thin films. Fig. 3 illustrates the experimental setup that contains a function generator, a power amplifier, an oscilloscope, a PZT vibrator, a laser interferometer, an optical vibrometer and a computer for measuring the resonant frequency of the cantilever beams and plates. The PZT vibrator and the laser interferometer have nano-scale resolution. The tested sample mounts on the PZT vibrator. The function generator sends a swept sinusoidal signal in the frequency range from zero to 500 kHz. The swept sinusoidal signal is amplified through the power amplifier, and then sent to the PZT vibrator exciting the vibration of the tested specimens. The response of the tested specimens is detected using the laser interferometer. Finally, the tested results are recorded in the computer. The frequency response of the SiO2 and Si3N4 cantilever beams with different lengths were measured. The first frequency response measured of all SiO2 and Si3N4 cantilever beams is shown in Table 1. The width of all SiO2 and Si3N4 cantilever beams is 10 μm. The thickness of all SiO2 and Si3N4 cantilever beams is about 0.9 μm and
1.4 μm, respectively. The mass density of the SiO2 and Si3N4 are approximately 2160 kg/m3 and 3000 kg/m3, respectively [10,11]. Substituting the above first resonant frequency, dimensions and mass density of the SiO2 and Si3N4 cantilever beams into Eq. (2), the Young's modulus of the SiO2 and Si3N4 films can be yielded and the result is shown in Table 1. The average Young's modulus of the SiO2 and Si3N4 films was 55.6 GPa and 131.6 GPa, respectively. The Young's modulus of the SiO2 and Si3N4 films reported by Petersen [12] and Stark et al. [13] was 57–92 GPa and 129 GPa, respectively. Obviously, the present experiment results in good agreement with literature values. The length and thickness of the SiO2 cantilever plate are 100 μm and 0.9 μm, respectively. The mass density of the SiO2 is about 2160 kg/m3. Substituting the above values and E = 55.6 GPa into Eq. (4), the relation between the Poisson's ratio and the first resonant frequency of the SiO2 cantilever plate can be yielded, as shown in Fig. 4. The measured result showed that the first resonant frequency of the SiO2 cantilever plate was 74.3 kHz. By checking Fig. 4, the Poisson's ratio of SiO2 film with a resonant frequency of 74.3 kHz was 0.16. The Poisson's ratio of Si3N4 films can also be obtained using a similar procedure. The length and thickness of the Si3N4 cantilever plate are 100 μm and 1.4 μm, respectively. The mass density of the Si3N4 is about 3000 kg/m3. Substituting the above values and E = 131.6 GPa into Eq. (4), the relation between the Poisson's ratio and the first resonant frequency of the Si3N4 cantilever plate can be yielded, as shown in Fig. 5. The measured result depicted that the first resonant frequency of the Si3N4 cantilever plate was 152.8 kHz. By checking Fig. 5, the Poisson's ratio of Si3N4 film with a resonant frequency of 152.8 kHz was 0.26. Literatures [11,13] showed that the Poisson's ratio of the SiO2 and Si3N4 was about 0.17 and 0.28,
Table 1 Resonant frequency (RF) and Young's modulus (E) of cantilever beams with different lengths Beam length (μm)
Fig. 4. Relation between Poisson's ratio and resonant frequency of SiO2 cantilever plate.
100 150 200 Average
SiO2
Si3N4
SiO2
Si3N4
RF (kHz)
RF (kHz)
E (GPa)
E (GPa)
73.78 32.30 18.70
150.28 65.11 38.10
55.6 54.0 57.2 55.6
132.5 125.9 136.3 131.6
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Experimental results showed that the Poisson's ratios of SiO2 and Si3N4 were 0.16 and 0.26, respectively, and the Young's moduli of SiO2 and Si3N4 were, respectively, 55.6 GPa and 131.6 GPa. Acknowledgements The authors would like to thank the National Center for High-Performance Computing (NCHC) for chip simulation, the National Chip Implementation Center (CIC) for chip fabrication and the National Science Council of the Republic of China for financially supporting this research under Contract no. NSC 952221-E-005-043-MY2. References Fig. 5. Relation between Poisson's ratio and resonant frequency of the Si3N4 cantilever plate.
respectively. The experimental results are matched with literature values.
5. Conclusion The Poisson's ratio and Young's modulus of SiO2 and Si3N4 thin films using the resonant method have been measured. The micromachined SiO2 and Si3N4 cantilever beams and plates were fabricated on a chip using the bulk micromachining. The resonant frequency of the cantilever beams and plates were measured using a laser interferometer. The Young's modulus of the thin film was calculated from the resonant frequency of the cantilever beams, and the Poisson's ratio of thin film was determined from the resonant frequency of the cantilever plates.
[1] D.O. King, M.C.L. Ward, K.M. Brunson, D.J. Hamilton, Sens. Actuators, A 68 (1998) 238–243. [2] C.L. Dai, J.H. Chen, Microsystem Technol. 12 (2006) 1143–1151. [3] G. Alcala, P. Skeldon, G.E. Thompson, A.B. Mann, H. Habazaki, K. Shimizu, Nanotechnology 13 (2002) 451–455. [4] E. Harry, A. Rouzaud, M. Ignat, P. Juliet, Thin Solid films 332 (1998) 195–201. [5] B. Cros, E. Gat, J.M. Saurel, J. Non-cryst. Solid 209 (1997) 273–282. [6] C. Luo, T.W. Schneider, R.C. White, J. Currie, M. Paranjape, J. Micromechanics Microengineering 13 (2002) 129–133. [7] S.S. Rao, Mechanical Vibrations, Addison-Wesley, Massachusetts, 1990. [8] A.W. Leissa, J. Sound Vib. 31 (1973) 257–293. [9] R.E. Rossi, P.A.A. Laura, J. Sound Vib. 195 (1996) 142–148. [10] Y.S. Ju, K.E. Goodson, J. Appl. Phys. 85 (1999) 7130–7134. [11] S.D. Senturia, Microsystem Design, Kluwer Academic Publishers, Boston, 2001. [12] K.E. Petersen, IEEE Trans. Electron Devices 25 (1978) 1241–1250. [13] R.W. Stark, G. Schitter, A. Stemmer, Phys. Rev. 68 (2003) 085401.