Nonlinear flexural vibration of AFM multi-layered piezoelectric microcantilever under tip-sample interaction

M. H. Korayem^†‡ and R. Ghaderi^†

^† Department of Mechanical and Aerospace Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
^‡ Robotic Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran hkorayem@iust.ac.ir

Abstract— In this paper, the vibration of an AFM microcantilever with one and two layers of piecewise piezoelectric is analyzed. With respect to discontinuities in the MC, due to the piezoelectric layer, the non-uniform beam model is used in the modeling of vibrating motion. The vibration analysis of MC is carried out near the sample surface in the non-contact mode. To solve the nonlinear differential equation of motion, both numerical solution and multiple time scales (MTS) method are used and the results are compared. Comparison of results, at the non-contact mode, shows good agreement between the two solving methods at normal equilibrium distances (d≥2nm). MC is modeled with one and two piezoelectric layers; the nonlinear behavior of each MC is studied by investigating the effect of geometrical dimensions of each layer on the nonlinearity of the system. It is then become clear that they can affect the nonlinearity of the system.

Keywords— Multi-Layered Piezoelectric Microcantilever; AFM; Multiple Time Scale Method; Nonlinearity.

NOMENCLATURE
V Microcantilever bending vibration
ρ_i Density
A Cross section area
h_i Layer thickness
W_i Layer width
L Microcantilever length
L₁ Lower piezoelectric length
L₂ Upper piezoelectric length
E_i Linear elastic stiffness coefficient of each layer
Elastic compliance constant at constant electric field
d₂₁ Piezoelectric constant
Piezoelectric stress constant
H_Li Heaviside function
K(x) MC spring constant
P(t) Total input voltage
P_s Applied voltage for controlling the static equilibrium orientation
P_d(t) Applied voltage for controlling the amplitude of vibration
F_ts Nondimensional tip - sample force
H Hamaker constant
σ Typical atomic distance
R Tip radius
Y Tip-sample separation
d Equilibrium distance
Static deflection of MC
u Dynamic deflection of microcantilever
C_e Electrical coupling coefficient
q_n Generalized time-dependent coordinate
U_n n^th mode shapes
μ_n Modal damping terms
ω_n Natural frequency
g_n Coefficients of differential equation of motion
Ω Excitation frequency
ε Bookkeeping parameter
σ Detuning parameter
γ_f Nonlinear coefficient
Neutral axis for the multi layer section

I. INTRODUCTION

Piezoelectric-material-based micro-electromechanical systems (MEMS) are equipments that have been used in a vast range of electromechanical applications during recent years. Using direct and reversed properties of the piezoelectric material, such equipments can be applied as actuators or sensors (Wang et al., 2003; Zhang et al., 2003). Micro-electromechanical systems with one or two piezoelectric layers are classic samples, which are used extensively in acoustic sensors, speakers, amplifiers, micro-pumps, micro-positioning equipments, and many other applications. Such micro-actuators and micro-systems can be used as nano-robots, nanolithography, biological cell and nanoparticles transporters, sensors, and surface imaging actuators in the AFM (Sitti and Hashimoto, 2000; Dong et al., 2007; Rogers et al., 2004; Adams et al., 2005).

Most of the earlier works performed on the piezoelectric MCs emphasized on their actuating and sensing applications. Itoh and Suga (1996) and Lee et al. (1999) made MCs with one and two piezoelectric layers were used for the dynamic scanning force microscope (SFM) in which the piezoelectric layer is used as the MC actuator and sensor. Using this MC, they could provide images with favorable quality, which required no external actuators and sensors. Rogers et al. (2004) used the piezoelectric MC with ZnO layer as the sensor and the actuator for AFM tapping mode.

So far, many analytical models have been presented for beams two piezoelectric layers for applications with macro dimensions in which a piezoelectric layer is used as the actuator and another piezoelectric layer as the sensor; among them are the studies of Costa-Branco and Dente (2004), Liqun et al. (2003) and Kursu et al. (2009). The effects of electrode layers and insulation layer have been neglected in all of these models. Brissaud et al. (2003) presented an analytical model for asymmetric multi-layered beam, which was made of a piezoelectric layer, two electrode layers, and one elastic base layer. Here, the beam was vibrated by the harmonic actuation and it was used as an actuator or a sensor.

The analysis of piezoelectric beams with micro-size dimensions has been considered in earlier works. Liu et al. (2006) presented a general model for static deforma-tion of multi-layer piezoelectric MCs in which each lay-er was made of several sections with different materials. Using this model, MC tip deformation and the charge of piezoelectric layer were analyzed and excitability and measurability of the MC with two piezoelectric layers were compared. Mahmoodi and Jalili (2007) and Mahmoodi and et al. (2009, 2010) modeled the free vibration motion (in the absence of the tip-sample force) of the piezoelectric MC using the non-linear beam model and compared the results of their an-alytical model with the experimental results. Wolf and Gottlieb (2002) studied the vibration motion of a unifo-rm piezoelectric MC in non-contact mode using multi-ple time scale method. Fung and Huang (2000) also modeled the piezoelectric MC in non-contact mode using finite element method.

In this paper, the vibration of an AFM microcantil-ever with one or two piecewise piezoelectric layers is analyzed. With respect to the discontinuities of the MC, due to the present of the piezoelectric layer, modeling of vibrating motion is performed. MC vibration analysis is carried out near the surface of the sample in non-con-tact mode. To solve the nonlinear differential equation of motion, both numerical solution and MTS method are used and the obtained results are compared. Three diffe-rent configurations were selected for MC piezoelectric and MC's nonlinear behavior is studied within these th-ree configurations.

II. DYNAMIC MODELING OF AFM PIEZOELECTRIC MC

In order to perform the dynamic modeling of AFM piezoelectric MC near the sample surface, a discontinuous MC, near the sample surface is considered. MC is studied with three different configurations in terms of the arrangement of the piezoelectric layers -MC with a piecewise piezoelectric layer on it, MC with two piezoelectric layers on one side of base layer, and MC with two piezoelectric layers on both sides of base layer (Fig. 1). Each piezoelectric layer is enclosed with two electrode layers on both sides.

Fig. 1. Different configurations of piezoelectric layers on MC, (a) MC with one piezoelectric layer, (b) MC with two piezoelectric layers on one side, (c) MC with two piezoelectric layers on both sides

Using Hamilton method and taking Euler-Bernoulli theory, one can obtain the AFM piezoelectric MC equation of motion with the presence of the tip-sample force. In Appendix A, the details of solution to drive this equation is given. Regarding Eq. A.7, the equation of motion is explained in non-dimensional form as follows:

(1)

In Eq. (1), C_e is the electromechanical coupling and enters the applied voltage to the piezoelectric layer as an actuator in the equation of motion. Galerkin approximation is used to discretize the differential equa-tion of motion into spatial and temporal functions as follows:

(2)

where q_n is the generalized time-dependent coordin-ates and U_n is the n^th form of the modes. Galerkin ap-proximation for discretizing the MC equation of mot-ion at the presence of the tip-sample force is a meth-od already used by other authors (Delnavaz et al., 2009; 2010; Horng, 2009).

Since the length of the piezoelectric layers on the MC is not necessarily equal to the MC length and the tip may be made narrower to improve deformation measurement (Mahmoodi and Jalili, 2007) the piezo-electric MC is discontinuous. Discontinuities in the beam affect the MC vibration motion through the mode shapes; therefore, to increase the accuracy of the dynamic model, it is necessary to use the disco-ntinuous beam model in the vibration analysis of the beam. With respect to the discontinuities of MC, it can be divided into the uniform beams. By dividing the piezoelectric MCs of Fig. 1 into three uniform beams, the mode shapes can be explained as follows:

	(3)

where and and are unknown values, which can be calculated through the boundary conditions, deformation, slope, bending moment and shearing force continuity as well as normalization condition with respect to mass.

By substituting Eq. (2) into Eq. (1) and taking the inner product of the resulting Equation in U_n(x), and integrating along MC, the ordinary differential equation of motion is obtained as follows:

	(4)

III. PRIMARY RESONANCE RESPONCE

A. Multiple Time Scale Method

MTS method can be used to solve the ordinary differ-ential equation of motion (Eq. 4). The other authors (Delnavaz et al., 2009; 2010) also used this method to solve the MC differential equation of motion (MC without piezoelectric layer), which is under the tip-sample force. In order to use the MTS method, the solution of steady state can be extended as follows:

(5)

where T_n= ε ⁿ t and ε is introduced as a small book keeping parameter to show infinitesimal quantity in the equation. Time derivatives can be expressed as follows:

(6)

where D_n=∂/∂T_n. In order to analyze, the effect of the exciting voltage and modal damping has been scaled to be in the same order with the perturbation problem. In the other words:

(7)

Substituting Eqs. (5-7) into Eq. (4) and separating simi-lar powers of ε, yields:

	(8)
	(9)
	(10)

Solution of Eq. 8 is assumed as follows:

(11)

where A_n is a complex amplitude and cc represents the complex conjugate of the preceding terms. Substituting Eq. (11) into Eq. (9) yields:

(12)

where is the complex conjugate of A_n. To get stea-dy-state solutions, the secular terms of Eq. (12) must be eliminated. In other words, the terms with expressions are eliminated. As a consequence D₁A₁=0. This indicates that A_n= A_n (T₂). Consequently:

(13)

The cantilever is excited with a voltage that is equal to where the exciting frequency (Ω) is close to the natural frequency and it is considered as follows:

(14)

where σ is the detuning parameter which indicates the deviation of excitation frequency to natural frequency. By substituting the assumed excitation into Eq. 10 and eliminating the secular terms, following equation will be obtained:

	(15)
	(16)

γ_f is a criterion for the effect of nonlinearity in the sys-tem which can be introduced as a nonlinear coefficient. Negative values of γ_f indicate the hardening phenomen-on and its positive values show the softening phenomen-on. Since this value is always positive here, it can be concluded that the nonlinear tip-sample force causes softening phenomenon in the frequency response of the system.

To solve Eq. (15), it is better to assume A_n as a polar form:

(17)

where α_n and β_n are the real parameters of the amplitude and phase. By substituting Eq. (17) into Eq. (15) and separating the real and imaginary parts, the following modulation equation of amplitude and frequency can be expressed as:

(18)

where . Equation (18) expresses the modulation of amplitude response of a_n and response phase of τ_n. Since it is of interest to investigate the steady-state response, and values are assumed to be zero. By eliminateing τ_n, the nonlinear frequency response equation is obtained:

(19)

The obtained frequency response equation clearly shows that for some of the values of the detuning para-meter, the number of obtained amplitudes exceeds one, which indicates the softening phenomenon in the frequ-ency response.

B. Numerical Solution

In the numerical Solution, instead of Taylor series expa-nsion of tip-sample force equation, its main equation (Eq. A.2) is used. Therefore, Eq. (4) can be rewritten as follows:

	(20)

To solve Eq. 20, Runga-Kutta method is used by applying ode45 command of Matlab Software.

IV SIMULATION

A. Comparison of MTS Method and Numerical Solution Results

In order to study the vibrating behavior of piezoelectric MC near the surface of the sample, three configurations of MC are considered: MC with one piezoelectric layer, MC with two piezoelectric layers on one side, MC with two piezoelectric layers on both sides. Two Ti/Au elec-trode layers enclose each piezoelectric layer. The geo-metrical specifications of them are presented in Table 1, regarding all the three selected configurations. The req-uired coefficients for the Lennard-Jones model are selected as σ=0.34(nm) and H=10^-18(J).

Table 1 Parameters of simulation for piezoelectric MC

Taylor series expansion of the Lennard-Jones equation around the selected equilibrium distances was used in the analysis of vibrating motion of MC using MTS method. Taylor series expansion of a nonlinear function around a certain point has an appropriate accuracy within certain distances from the selected point; Therefore, MTS method, within the range in which Taylor series expansion does not agreement to the nonlinear function, does not have adequate accuracy. Using Fig. 2, the interval in which Taylor series expansion conforms to the non-linear force function can be determined. In these Figures, Lennard-Jones curve (solid curve) is estimated using its Taylor series expansion (dashed curves) at the equilibrium distances of 10, 5, 2, and 1 nm. Such curves are used in the following analyses in selecting magnitude of exciting voltage, and amplitude of vibrating motion.

Fig. 2. Third order Taylor series approximation of tip-sample interaction force curve at distances (a) d = 10 nm, (b) d =5 nm, (c) d = 2 nm, (d) d = 1 nm

Figure 3 shows the vibrating motion of MC with one piezoelectric layer at equilibrium distances (d) 10, 5, 2, and 1 nm. Exciting voltage was selected for 0.45, 0.055, 0.04 and 0.012 (mV), respectively and exciting frequency was selected equal to the MC's natural frequency. The amplitude of the oscillatory motion obtained through both MTS and numerical solution to solve the non-linear equation of motion. As seen, for 10, 5, 2 nm, the obtained results through the two methods have a good agreement and for the equilibrium distance of 1 nm, little differences arise between the two solutions. As the vibrating motions studied in Fig. 3 are in the non-contact mode, the attractive Van der Waals force has greater effect on the vibrating motion of MC in this region. This force always attracts the MC tip toward the sample. Consequently, the force increases at the very short distances of MC to the sample surface and MC time response curve becomes loses its symmetry around the equilibrium position. With respect to Eqs. (11-19), the time response of the vibrating motion, which is achieved using the MTS method, is always symmetric. It means that the time response asymmetry, which is caused by the van der Waals attractive force, is not achieved through the MTS solution method. However, as Fig. 3-d shows, if the vibrating motion equation (Eq. 20) is solved through the numerical method, the time response will be asymmetrical at the very close distance to the sample surface (d=1 nm); this is due to the intensification of the van der Waals attractive motion. This difference between the two solution methods causes little difference between the time responses obtained from the two methods; this issue can be seen in Fig. 3-d.

Fig. 3. Piezoelectric MC time response through MTS and num-erical solutions at the equilibrium distances of (a) d = 10 nm, (b) d = 5 nm, (c) d = 2 nm, (d) d = 1 nm

B. Studying the Effects of MC's Geometrical Dime-nsions on the Nonlinearity

Tip-sample nonlinear force will lead to the nonlinear vibrating response of MC; with respect to the MTS analytical method, such a nonlinearity by nonlinear coefficient (γ) (Eq. 16) leads to the bending of MC's frequency response curve. Therefore, γ can be used as an index to determine the nonlinearity of the systems. With respect to Eq. 16, this coefficient is not only dependent upon the tip-sample force but also the MC's geometrical dim-ensions. Here, the effect of piezoelectric MC geometri-cal dimensions on γ coefficient is studied in three different configurations, i.e. MC with one piezoelectric layer, with two piezoelectric layers on one side and with two piezoelectric layers on both sides of MC.

Figure 4 shows the effect of different layers thickness of piezoelectric MC on γ. It shows that by increasing the thickness of each layer, the nonlinearity of system and consequently, the effect of nonlinearity of tipsample force on the vibrating motion decreases. Figure 4 shows that the thickness of the base layer has the most effect on γ and thickness of the upper piezoelectric layer, when two piezoelectric layers are on one side, has the least effect on γ.

Fig. 4. Effect of different layers thickness of piezoelectric MC on γ coefficient, (a) MC with two piezoelectric layers on one side, (b) MC with two piezoelectric layers on both sides, (c) MC with one piezoelectric layer (d=1.2 nm)

Figures 5-7 show the effect of the width of base and piezoelectric layers on γ. As these figures show, in all cases, γ decreases with the increase of width of the layer. The results show that if the width of one of the layers is small while the width of another layer changes then changes of γ will be greater. In other words, as MC layers are widened, the effect of the width of other layers on γ decreases.

Fig. 5. Effect of the width of layers on γ - MC with two piezo-electric layers on one side (d=1.2nm)

Fig. 6. Effect of the width of layers on γ -MC with two piez-oelectric layers on both sides (d=1.2nm)

Fig. 7 Effect of the width of base and piezoelectric layers on γ -MC with one piezoelectric layer (d=1.2nm)

Figure 8 shows the effect of the length of MC and piezoelectric layers on γ. The obtained results show that the increase of the length of MC in MC with two piezoelectric layer on one side and MC with one piezoelectric layer lead to the increase of nonlinearity of the system; while the increase of the length of piezoelectric layer in these MCs leads to the decrease of the system's nonlinearity or γ. In the MC with two piezoelectric layers on both sides, the increase of the length of MC and piezoelectric layer has a different effect on the nonlinearity of the system.

Fig. 8 Nondimensional amplitude of MC versus the exciting voltage near the surface of the sample (d=2 nm)

With respect to the results obtained from Figs. 4-7, one can conclude that the increase of geometrical dimensions of MC (except length) and the geometrical dim-ensions of the piezoelectric layers lead to the decrease of nonlinearity of the system or γ. In other words, the increase of MC's stiffness will lead to the decrease of the MC's vibrating motion in terms of the nonlinearity of the system.

C. Effect of the Exciting Voltage on the Vibrating Motion near the Surface of the Sample

Figure 8 shows the vibrating motion amplitude of piezoelectric MC tip (maximum deviation of MC tip from the equilibrium state) as it approaches the surface of the sample. In this figure, the effect of the input voltage to the piezoelectric layer on the amplitude is investigated in three different configurations, i.e. MC with one piezoelectric layer, with two piezoelectric layers on one side (lower piezoelectric layer as the actuator) and with two piezoelectric layers on both sides of MC. The results were obtained using the numerical solution. It shows that tip-sample nonlinear force has caused the amplitude to increase in a nonlinear manner with the increase of the exciting voltage. Such behavior is observed in all the three selected MCs.

V. CONCLUSIONS

Vibrating motion of MC with one and two piezoelectric layers near the surface of the sample and in the non-contact status was studied. Regarding the piecewise piezoelectric layers on the MC, non-uniform beam model was chosen to perform the analysis. Since the tipsurface force is nonlinear, the vibrating behavior of MC is also nonlinear. In the present paper, both the numerical solution based on Runge-Kutta and MTS analytical method are used to solve the nonlinear differential equation of the motion. Comparison of the obtained results shows that there is an appropriate agreement between the two selected solutions. Within the equilibrium distances, which are very close to the surface of the sample, and with the increase of attractive force, time response will be asymmetric. The asymmetry time response can be observed in the numerical solution; however, with respect to the type of equations, it is not achieved through MTS method. This leads to some little differences between the two selected solutions at very short equilibrium distances. With respect to the MTS method, nonlinearity of the system or the effect of the nonlinearity of the force on the frequency response of the vibrating motion can be determined by γ. This coefficient is not only dependent upon the tip-sample force but also it is affected by the geometry of the cantilever. The simulation results show that for all the three selected modes, nonlinearity of the system decreases with the increase of thickness and width of each layer and consequently with the increase of the stiffness of MC.

APPENDIX

Three different configurations are considered to study the vibrating motion of AFM piezoelectric MC - MC with one piezoelectric layer (case I), MC with two piezoelectric layers on one side of the base layer (case II), MC with two piezoelectric layers on both sides of the base layer (case III). As in the formulation of the MC with two piezoelectric layers on one side of the base layer, we can reach the formulation of MC with one piezoelectric layer by setting the thickness of the upper piezoelectric layer and also its upper electrode to zero, here, no independent formulation is presented for the MC with one piezoelectric layer. In the calculations, which will be presented subsequently, L₁ is the length of the lower piezoelectric layer, L₂ is the length of the upper piezoelectric layer and L specifies total length of the MC. In addition, numbering of MC layers is performed from bottom to top.

The piezoelectric MC equation of motion is obtained using Hamilton method and considering the Euler-Bern-oulli theory as follows:

(A.1)

where:

- For case II:

Itoh and Suga (1996) have experimentally studied on the vibrating motion of MC with two piezoelectric layers on one side. They indicated that the upper piezoelectric layer plays a better role in sensing. Therefore, here, it is assumed that the lower piezoelectric layer plays the role of an actuator.

- For case III:

If the upper piezoelectric layer is an actuator:

If the lower piezoelectric layer is an actuator:

It should be noted that the width of piezoelectric layers is equal to the width of their surrounding electr-odes. The force imposed by the surface of the sample to the tip can be expressed according to Lennard-Jones model as follows:

(A.2)

where H, R and σ are Hamaker constant, tip radius, and the atomic size of the sample, respectively. Y indicates the distance between the tip of probe and the surface of the sample.

On the other hand, static deflection of the MC, under the effect of the tip-surface force, can be obtained by the following equation:

(A.3)

where P_s is the voltage that controls the static balance of the tip (Wolf and Gottlieb,2002) The distance between the tip of the probe and the surface of the sample at the static equilibrium is introduced as . Total deformation of MC in terms of time is:

(A.4)

where u(x,t) expresses the dynamic deformation of MC. This way, in Eq. A.2, interaction force can be express-ed as follows, using Taylor series extension around the static equilibrium, regarding the first four terms:

(A.5)

In order to ease calculations and provide a better under-standing, dimensionless variables are introduced and used:

(A.6)

where . In this case, the equation of motion can be expressed by substituting Eqs. A.3-A.5 into Eq. A.1 as follows:

	(A.7)

For simplicity, the symbol (*) was neglected in this equation.

REFERENCES
1. Adams, J.D., L. Manning, B. Rogers, M. Jones and S.C. Minne, "Self-sensing tapping mode atomic force mic-roscopy," Sens. and Act. A., 121, 262-266 (2005).         [ Links ]
2. Brissaud, M., S. Ledren and P. Gonnard, "Modelling of a cantilever non-symmetric piezoelectric bimorph," J. Micromech. Microeng., 13, 832-844 (2003).         [ Links ]
3. Costa Branco, P.J. and J.A. Dente, "On the electromecha-nics of a piezoelectric transducer using a bimorph cantilever undergoing asymmetric sensing and act-uation," Smart Mat. Struct., 13, 631-642 (2004).         [ Links ]
4. Delnavaz, A., S.N. Mahmoodi, N. Jalili, M.M. Ahadian and H. Zohoor, "Nonlinear vibrations of microcant-ilevers subjected to tip-sample interactions: Theory and experiment," J. Appl. Phys. 106, 113510 (2009).         [ Links ]
5. Delnavaz, A., S.N. Mahmoodi, N. Jalili and H. Zohoor, "Linear and non-linear vibration and freq-uency response analyses of microcantilevers subjected to tip-sample interaction," Int. J. of Non-Lin. Mech. 45, 176-185 (2010).         [ Links ]
6. Dong, W., X. Lu, Y. Cui, J. Wang and M. Liu "Fab-rication and characterization of microcanti-lever integrated with PZT thin film sensor and actuator," Th. Sol. Films, 515, 8544-8548 (2007).         [ Links ]
7. Fung, R.F. and S.C. Huang, "Dynamic modeling and vibration analysis of the atomic force micros-cope," ASME J. Vib. Acoust., 123, 502-509 (2001).         [ Links ]
8. Horng, T.-L., "Analyses of vibration responses on nanoscale processing in a liquid using tapping-mode atomic force microscopy," Appl. Sur. Sci., 256, 311-317 (2009).         [ Links ]
9. Itoh, T. and T. Suga, "Self-excited force sensing mic-rocantilevers with piezoelectric thin films for dynamic scanning force," Sens. and Act. A: Phys., 54, 477-481 (1996).         [ Links ]
10. Kursu, O., A. Kruusing, M. Pudas and T. Rahkonen, "Piezoelectric bimorph charge mode force sensor," Sens. Act. A., 153, 42-49 (2009).         [ Links ]
11. Lee, C., T. Itoh and T. Suga, "Self-excited piezoelectric PZT microcantilevers for dynamic SFM-with inherent sensing and actuating capabilities," Sens. and Act. A., 72, 179-188 (1999).         [ Links ]
12. Liqun, D., K. Guiryong, A. Fumihito, F. Toshio, I. Kouichi and T. Yasunori, "Structure design of micro touch sensor array," Sens. and Act. A., 107, 7-13 (2003).         [ Links ]
13. Liu, M.W., J.H. Tong, J. Wang, W.J. Dong, T.H. Cui and L.D. Wang, "Theoretical analysis of the sens-ing and actuating effects of piezoelectric multimor-ph cantilevers," Microsyst. Technol., 12, 335-342 (2006).         [ Links ]
14. Mahmoodi, S.N., M.F. Dagag and N. Jalili, "On the nonlinear-flexural response of piezoelectrically driven microcantilever sensors," Sens. and Act. A, 153, 171-179 (2009).         [ Links ]
15. Mahmoodi, S.N., N. Jalili and M. Ahmadian, "Subharmonics analysis of nonlinear flexural vibrations of piezoelectrically actuated microcantilevers," Non. Dyn., 59, 397-409 (2010).         [ Links ]
16. Mahmoodi, S.N. and N. Jalili, "Non-linear vibrations and frequency response analysis of piezoelectrically driven microcantilevers," Int. J. of Non-Lin. Mech., 42, 577-587 (2007).         [ Links ]
17. Rogers, B., L. Manning, T. Sulchek and J.D. Adams, "Improving tapping mode atomic force microscopy with piezoelectric cantilevers," Ultramicroscopy, 100, 267-276 (2004).         [ Links ]
18. Sitti, M. and H. Hashimoto, "Controlled Pushing of Nano-particles Modeling and Experiments," IEEE/ ASME Trans. on Mech., 5, 2, 199-211 (2000).         [ Links ]
19. Wang, L.P., R.J. Wolf, Y. Wang, K.K. Deng, L. Zou, R.J. Davis and S. Trolier-Mc Kinstry, "Design, fabrication, and measurement of high sensitivity piezoelectric microelectromechanical systems accelemeters," J. MEMs, 12, 433-439 (2003).         [ Links ]
20. Wolf, K. and O. Gottlieb, "Nonlinear dynamics of a noncontacting atomic force microscope cantilever actuated by a piezoelectric layer," J. Appl. Phys., 91, 4701-4709 (2002).         [ Links ]
21. Zhang, Q.Q., S.J. Gross, S. Tadigadapa, T.N. Jackson, F.T. Djuth and S. Trolier-Mc Kinstry, "Lead zirconate titanate films for d₃₃ mode cantilever actuators," Sens. and Act. A., 105, 91-97 (2003).         [ Links ]

Received: March 28, 2012.
Accepted: June 21, 2012.
Recommended by Subject Editor Eduardo Dvorkin