Dynamic simulation of atomic force microscope cantilevers oscillating

Dynamic simulation of atomic force microscope cantilevers oscillating in liquid Michael James Martin, Hosam K. Fathy, and Brian H. Houston Citation: J. Appl. Phys. 104, 044316 (2008); doi: 10.1063/1.2970154 View online: http://dx.doi.org/10.1063/1.2970154 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v104/i4 Published by the American Institute of Physics.

Related Articles Prototype cantilevers for quantitative lateral force microscopy Rev. Sci. Instrum. 82, 093706 (2011) Atomic force microscopy at ambient and liquid conditions with stiff sensors and small amplitudes Rev. Sci. Instrum. 82, 093703 (2011) Distortion in the thermal noise spectrum and quality factor of nanomechanical devices due to finite frequency resolution with applications to the atomic force microscope Rev. Sci. Instrum. 82, 095104 (2011) Mapping of conservative and dissipative interactions in bimodal atomic force microscopy using open-loop and phaselocked-loop control of the higher eigenmode Appl. Phys. Lett. 99, 074103 (2011) Comment on “Thermal frequency noise in dynamic scanning force microscopy” [J. Appl. Phys. 109, 024310 (2011)] J. Appl. Phys. 110, 036107 (2011)

Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

Downloaded 08 Oct 2011 to 140.116.210.6. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

JOURNAL OF APPLIED PHYSICS 104, 044316 共2008兲

Dynamic simulation of atomic force microscope cantilevers oscillating in liquid Michael James Martin,1,a兲 Hosam K. Fathy,2 and Brian H. Houston1 1

Naval Research Laboratory, Code 7136, 4555 Overlook Avenue SW, Washington, DC 20375-5320, USA Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA

2

共Received 19 November 2007; accepted 20 June 2008; published online 28 August 2008兲 To simulate the behavior of an atomic force microscope 共AFM兲 operating in liquid, a lumped-parameter model of a 40⫻ 5 ␮m2 thick silicon cantilever with natural frequencies ranging from 3.0 to 6.0⫻ 105 rad/ s was combined with a transient Navier–Stokes solver. The equations of motion were solved simultaneously with the time-dependent flow field. The simulations successfully capture known characteristics of the AFM in liquid, including large viscous losses, reduced peak resonant frequencies, and frequency-dependent damping. From these simulations, the transfer function G共s兲 of the system was obtained. While the transfer function shares many of the characteristics of a second-order system at higher frequencies, the frequency-dependent damping means that a second-order model cannot be applied. The viscous damping of the system is investigated in greater depth. A phase difference between the peak velocity and peak damping force is observed. Both the phase difference and the magnitude of the damping are shown to be functions of the excitation frequency. Finally, the damping is shown to be strongly dependent on the liquid viscosity and weakly dependent on the liquid density. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2970154兴 I. INTRODUCTION

Operation of the atomic force microscope 共AFM兲 in tapping, or vibrating, mode1 constitutes a dynamic system dependent on feedback control. When operated in vacuum or air, damping is relatively low,2 greatly simplifying the control process.3,4 However, the AFM can also be operated in liquids,5,6 allowing the study of living cells7 and biological molecules.8 Operation in liquid greatly complicates operation of such systems. While problems of actuation9 and system integration10 have been addressed, the damping increases, limiting the quality factor of the system.11 A similar set of issues has been encountered in the use of micromachined cantilevers for biosensing.12–15 Among the proposed solutions to this problem is to increase the sophistication of the feedback control systems used, an approach often referred to as Q control.16 This approach has been the subject of extensive experimental and theoretical study.7,8,14,17–19 This, in turn, has led to more rigorous attempts to apply a control theory to the AFM.20–23 Rigorous application of control theory requires understanding of the transfer function G共s兲 of the system. Previous efforts to determine the transfer function through analysis have focused on lightly damped systems.24–26 However, an AFM operating in liquid will encounter heavy damping. Dynamic simulation of an AFM operating in liquid has the potential to determine the transfer function, as well as give insight into the physical mechanisms affecting AFM in liquids. The present work attempts to study the transfer function

for a micromachined cantilever vibrating through a heavy fluid by integrating a lumped-parameter model of the cantilever with a full solution of the Navier–Stokes equations for the surrounding fluid. The simulation method is outlined, and the results for a 40 ␮m ⫻ 5 ␮m thick silicon cantilever vibrating in water are presented. The transfer function for this system is then characterized. Additional simulations provide insight into how the transfer function will change based on fluid properties as the system is operated in fluids other than water. II. SYSTEM MODEL

Lumped-parameter models have proven effective in the determination of the quality factor of micro- and nanoscale cantilevers vibrating in viscous fluids.2,27–30 Recent work has shown that a full solution of the Navier–Stokes equations around a vibrating beam can also capture the phase lag in such a system.31 Similar models have also been used for the dynamic behavior of AFMs and are valid as long as higher order harmonics are not excited.32 In these models, the geometry of the microcantilever of length l, width b, thickness d, and tip displacement ⌬y 共shown in Fig. 1兲 is reduced to the two-dimensional cross section shown in Fig. 2.

a兲

Present address: Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA. Electronic mail: [email protected].

0021-8979/2008/104共4兲/044316/8/$23.00

104, 044316-1

FIG. 1. Cantilever geometry. © 2008 American Institute of Physics

Downloaded 08 Oct 2011 to 140.116.210.6. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

044316-2

J. Appl. Phys. 104, 044316 共2008兲

Martin, Fathy, and Houston

FIG. 2. Cantilever cross section.

The cantilever motion can be described using a maximum displacement and a frequency: v共t兲 = A␻ cos共␻t兲,

共1兲

where A is the amplitude of vibration and ␻ is the vibrational frequency. Solving the Navier–Stokes equations around the cantilever will allow determination of the force FD on the cantilever. Integration of the work done on the fluid by the beam over 1 cycle is then used to determine the quality factor of the beam. This approach must be modified to determine the transfer function for an applied force. The cross section shown in Fig. 2 is incorporated into the lumped-parameter model of the vibrating beam shown as Fig. 3. F共t兲 represents the actuation force. FD is the fluidic damping force. The cross section of the beam is attached to a spring, which creates a force Fs proportional to the displacement from an initial position y o. The equation of motion of the system is given by meff

d2y = F共t兲 − FD − keff共y − y o兲, dt2

共2兲

where m is the mass of the system per unit length and keff is the effective stiffness of the system. The actuation force is given in terms of a peak force Fo and a frequency ␻: F共t兲 = Fo sin共␻t兲.

共3兲

For analysis of the fluid-structure interaction, the drag force is often treated as a linear function of the velocity,33 which allows the system to be simplified to the classical massdamper-spring system described by Eq. 共4兲, which is often used to model the AFM.34–36 In many cases, an effective damping coefficient ceff is used to give the equivalent of linear damping; m

d2y dy − keff共y − y o兲. 2 = F共t兲 − ceff dt dt

共4兲

However, this approach does not capture the phase difference between the peak velocity and the peak fluidic force that has been observed in the numerical simulation of fluidic damping in resonant microsystems. To capture the effect of this phase difference, a simulation approach, similar to that taken for fluid-structure interaction at the macroscale, is proposed. In the revised approach, the equations of motion are coupled with a full solution of the Navier–Stokes equation for the flow around the cantilever. For a two-dimensional simulation of the threedimensional cantilever geometry, an effective mass and stiffness must be chosen that incorporate as much of the threedimensional physics as possible. The cantilever is assumed to be excited in the first mode of vibration, where the natural frequency is given by

␻n =

kn l

2

EI , m

共5兲

where kn is the mode constant, l is the length of the cantilever, E is the elastic modulus of the material, I is the moment of inertia, and m is the mass per unit length.37 For a cantilever in the first mode of vibration, the mode constant is equal to 1.875. The moment of inertia I is given by I=

bd3 , 12

共6兲

where b is the width of the cantilever and d is the thickness.38 The mass per unit length will be given by m = ␳sbd,

共7兲

where ␳s is the density of the cantilever. The mode shape is given by

冋 冉 冊 冉 冊册

A共z兲 = 0.4485Ao sin 1.875

z z − sinh 1.875 L L

,

共8兲

where Ao is the amplitude of vibration at the tip of the cantilever. Simplifying this model from three to two dimensions requires consistent selection of an effective mass meff and an effective stiffness keff. The two should be selected so that the natural frequency of the two-dimensional system is the same as that of the three-dimensional cantilever. This sets the condition

␻n =

FIG. 3. Model of oscillating microbeam.

冉 冊冑

冑 冉 冊冑 keff kn = meff l

2

EI , m

共9兲

The convention from the vibration theory is to select the stiffness based on the response to an actuation force and then use Eq. 共9兲 to estimate the effective mass.38,39 Modeling of the actuation, tip, and fluidic forces for AFM in a viscous fluid using a lumped-parameter model is complicated by the fact that a vibrating cantilever has different responses and stiffnesses relative to actuation, tip, and fluidic forces. Any stiffness keff selected for a lumpedparameter model of AFM will introduce an error in how at least one of these forces is modeled. Figure 4 shows three

Downloaded 08 Oct 2011 to 140.116.210.6. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

044316-3

J. Appl. Phys. 104, 044316 共2008兲

Martin, Fathy, and Houston

⌬y =

fl4 f = , 8EI ka

共11兲

where ka is the stiffness relative to the actuation force.40 The stiffness is given by40 keff = ka =

8EI . l4

共12兲

By setting the displacements equal to each other, we can show that this effective stiffness is proportional to the stiffness based on tip deflection divided by the length: 3 ktip . 8 l

keff =

FIG. 4. 共a兲. Cantilever geometry for tip loading. 共b兲 Cantilever geometry for uniform loading. 共c兲 Cantilever geometry for fluidic loading.

loading models: one in which the force is applied at the tip, one in which the force is applied uniformly across the cantilever, and one in which the force is proportional to the displacement. The tip force can be considered as a concentrated force acting at the end of the cantilever, as shown in Fig. 4共a兲. This will give the reported stiffness for an AFM cantilever as

⌬y =

Ftipl3 Ftip = , 3EI ktip

共10兲

where Ftip is the tip force and ktip is the stiffness relative to the tip force.40 Modeling of the actuation force is more complex. A variety of actuation methods are used for AFM, including mechanical actuation at the base of the cantilever,41 magnetic actuation,9,42 and thermal actuation.43,44 This problem is further complicated by the fact that magnetic actuation can be accomplished either through a magnetic field acting on a magnetic particle on the tip42 or on magnetic material distributed along the length of the cantilever.9 Thermal actuation can be accomplished either through deposition of heat at the cantilever base43 or through uniform heating along the length of the cantilever.44 Both of these approaches will create a distributed force along the length of the cantilever, although computing the exact distribution requires extensive thermal analysis. In this simulation, the actuation force is uniformly distributed along the width of the beam, as shown in Fig. 4共b兲. This simple assumed distribution of actuation force aids in this paper’s analyses, but it has not been validated against experimental AFM results. The problem of modeling actuation force distribution in AFM and using these results in lumped-parameter models remains an open research question. The displacement based on a uniform actuation force per unit length, f, is given by

共13兲

The third force is the fluidic force. The distribution of the fluidic force will be proportional to the local amplitude of vibration or the mode shape of the cantilever, as is shown in Fig. 4共c兲. While as a first approximation the undamped mode shape is shown, previous work suggests that the quality factor and the system performance are independent of the mode shape.27 Because the purpose of this study is to determine the system response relative to an actuation force, the stiffness relative to force was used as the effective stiffness. This choice of conventions for handling the stiffness creates a source of error in handling the response of the system to fluidic and tip forces. This error is not unique to the outlined approach. Some form of this error exists in all lumpedparameter models of damped systems, including the models currently used for atomic force microscopy. Because experience in modeling of damping in micromachined resonators shows that the relative damping and the quality factor are independent of the mode shape and magnitude of excitation, it is assumed that the error in handling the distribution of fluidic forces along the cantilever relative to the stiffness is small. This issue may require additional theoretical investigation using a three-dimensional simulation. Once a stiffness is determined, the effective mass of the system is determined using the Rayleigh method.39 The effective stiffness from Eq. 共12兲 can be substituted into Eq. 共9兲 to determine the effective mass of the system:

␻n =

冑 冑 keff = meff

冉 冊冑

8EI kn 4 = meffl l

2

EI . m

共14兲

This yields an expression for the effective mass of the system: meff =

8m k4n

.

共15兲

III. SIMULATION METHOD

The solution algorithm consists of performing the following computations at each time step: 共1兲 Solution of the velocity and pressure using a timedependent Navier–Stokes algorithm. 共2兲 Computation of the fluid force, spring force, and actuation force on the cantilever.

Downloaded 08 Oct 2011 to 140.116.210.6. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

044316-4

J. Appl. Phys. 104, 044316 共2008兲

Martin, Fathy, and Houston

TABLE I. Cantilever stiffness and resonant frequency.

FIG. 5. Computational geometry with boundary conditions.

共3兲 Computation of the acceleration, velocity, and position of the cantilever. 共4兲 Updating of fluid boundary conditions based on the new cantilever velocity. The Navier–Stokes equations are solved using a transient computational fluid dynamics code using the geometry and boundary conditions shown in Fig. 5. The code uses the transient marker-and-cell method45,46 capturing the oscillatory nature of the flow. A uniform mesh with 100 cells in the x-direction, 120 cells in the y-direction, and a height 20 times the cantilever thickness and five times the cantilever width was used to calculate the flow at each time step. To reduce the computational time, the symmetry plane of the geometry is used. The pressure at the boundary is kept constant. The velocity boundary conditions for the flow are based on the cantilever motion: 共16兲

u = 0,

冏 冏

dy v= dt

共17兲

, cantilever

where u is the velocity in the x-direction and v is the velocity in the y-direction. At each time step n, a fluid force is calculated by integrating the force over the surface of the cantilever: FD =

冕

Pdx −

top

=2

冋冕

冕

Pdx − 2

bottom

b/2

⌬P共x兲dx − ␮

冕

d/2

␻n 共rad/s兲

ktip 共N/m兲

keff 共N / m / m兲

1 2 3 4 5 6 7

3.0⫻ 105 3.5⫻ 105 4.0⫻ 105 4.5⫻ 105 5.0⫻ 105 5.5⫻ 105 6.0⫻ 105

3.81 4.80 5.87 7.00 8.20 9.46 10.78

2.71⫻ 104 3.69⫻ 104 4.83⫻ 104 6.11⫻ 104 7.54⫻ 104 9.12⫻ 104 1.09⫻ 105

vn+1 = vn + an⌬t,

共21兲

y n+1 = y n + vn⌬t,

共22兲

where an, vn, and y n represent the acceleration, velocity, and position at each time step, and ⌬t is the size of the time step. The Euler method was selected because its simplicity allows it to be combined easily with the time marching of the fluid solver. The flow boundary condition 共17兲 is updated using the new velocity, and the computation is repeated until the vibration achieves steady state. The maximum time step for the simulation is set by the stability limits of the Navier–Stokes solver. For a markerand-cell marker at low Reynolds numbers, this is given by ⌬tmax = 0.5

␳f 共⌬ 兲2 , ␮ f min

共23兲

where ␳ f and ␮ f are the fluid density and viscosity, and ⌬min is the minimum size of the fluid mesh.46 This value is typically much smaller than the time scale given by the natural frequency of the structure, meaning that it is the limiting time step in the simulation. IV. TRANSFER FUNCTION RESULTS

The transfer function was analyzed for seven cases, where the natural frequency was altered by changing the effective stiffness of the structure, as described in Table I. An excitation force Fo of 0.001 N/m was applied to the cantilever cross section and water with a density of 999.2 kg/ m3 and a viscosity of 0.001 129 kg/ m s.47 The

␶dy

side

−d/2

0

冕

Case

册

dv dy . dx

共18兲

The actuation force is given by Eq. 共3兲, and the spring force is computed using Eq. 共19兲, Fspring = keff共y n − y o兲.

共19兲

These forces are then combined to obtain the equations of motion, which are integrated numerically using an Euler method: an =

d2y F共tn兲 − FD − keff共y n − y o兲 , = dt2 m

共20兲

FIG. 6. Maximum amplitude as a function of excitation frequency.

Downloaded 08 Oct 2011 to 140.116.210.6. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

044316-5

J. Appl. Phys. 104, 044316 共2008兲

Martin, Fathy, and Houston

FIG. 9. Nyquist diagram. FIG. 7. Gain as a function of excitation frequency.

beam was assumed to be silicon with a density of 2330 kg/ m3 and an elastic modulus of 160 GPa.48 The cantilever was 40⫻ 5 ␮m2. The maximum displacement as a function of excitation frequency is shown in Fig. 6. The results show many of the characteristics observed experimentally,8 including a large shift in the peak frequency away from the resonant frequency in air, low amplitudes of vibration, and higher displacements at low frequencies than observed at high frequencies. This result is nondimensionalized into K using49 K = y maxkeff/Fo .

共24兲

istics of the classic second-order damped system, including a high-frequency roll-off slope of −40 dB per decade and a high-frequency phase shift of −180°. However, due to the frequency-dependent damping, an analytical second order will not accurately model the system response at frequencies below the resonant peak. Figure 9 suggests that because the 共−1, 0兲 vector is not encircled, the Nyquist stability criterion is satisfied for the system.49 These results show the damping to be more complex than a simple dash-pot model. The dynamics of damping at different frequencies can be discussed by viewing results for one cantilever at different excitation frequencies.

The gain Adb and the normalized frequency s are given by

V. DAMPING RESULTS

Adb = 20 log10共K兲,

共25兲

s = ␻/␻n .

共26兲

The Bode plots, showing the gain and phase lag versus the normalized frequency, are shown in Figs. 7 and 8. Figure 9 is the Nyquist plot, showing the real and imaginary portions of the transfer function. The peak frequencies for these cases are listed in Table II. Several conclusions about the physical nature of the system can be reached from these results. Figures 7 and 8 suggest that a transfer function will have many of the character-

Figures 10共a兲–10共c兲 show the driving force, spring force, and fluid drag as a function of ␻t for a cantilever with an undamped natural frequency of 4.0⫻ 105 s−1. The results for three excitation frequencies are shown: 2.4⫻ 105, the peak frequency of 2.94⫻ 105, and 4.00⫻ 105 rad/ s. These sample results illustrate several characteristics of the fluid damping force on the cantilever. First, they show that the damping forces are of the same order of magnitude as the actuation force and spring force in the system, showing that the system is heavily damped. These figures also suggest that the fluid force is not in phase with the fluid velocity. A phase plot of fluid force versus velocity is shown in Fig. 11, confirming this observation. If the position and velocity of the cantilever are given by y共t兲 = y max sin共␻t + ␸兲,

共27兲

TABLE II. Frequency shift.

Case 1 2 3 4 5 6 7

␻n 共rad/s兲

␻d 共rad/s兲 5

3.0⫻ 10 3.5⫻ 105 4.0⫻ 105 4.5⫻ 105 5.0⫻ 105 5.5⫻ 105 6.0⫻ 105

␻d / ␻n 4

5.43⫻ 10 2.01⫻ 105 2.94⫻ 105 3.66⫻ 105 4.30⫻ 105 4.90⫻ 105 5.48⫻ 105

FIG. 8. Phase lag as a function of excitation frequency.

Downloaded 08 Oct 2011 to 140.116.210.6. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

0.181 0.575 0.735 0.813 0.859 0.891 0.913

044316-6

J. Appl. Phys. 104, 044316 共2008兲

Martin, Fathy, and Houston

FIG. 11. Phase plot of force vs velocity.

the damping models collapse onto one curve suggests that the results for one cantilever geometry vibrating in liquid can be generalized across a wide range of natural frequencies. VI. EFFECT OF FLUID PROPERTIES

FIG. 10. 共a兲. Force vs angle for the cantilever at 2.4⫻ 105 rad/ s. 共b兲 Force vs angle for the cantilever at 2.94⫻ 105 rad/ s. 共c兲 Force vs angle for the cantilever at 4.000⫻ 105 rad/ s.

v共t兲 = y max␻ cos共␻t + ␸兲,

To evaluate the effect of fluid properties on the system, two additional sets of simulations were performed using a cantilever with a natural frequency of 4.0⫻ 105 rad/ s. In the first set of simulations, the viscosity was varied between 0.4 and 1.4 times the viscosity of water and the resonant frequency ␻r was extracted from the simulation. These ratios of resonant frequency to the natural frequency as a function of viscosity are shown in Fig. 14. As expected, the frequency shifts and the damping approach zero as viscosity approaches zero. At lower viscosities, the resonant frequency appears to be dependent on the viscosity. However, above a critical viscosity, the resonant frequency decreases dramatically as the system approaches critical damping. At this point, the viscosity does not have a strong effect on the resonant frequency. While holding the viscosity constant, the fluid density was changed to specific gravities ranging from 0.2 to 3.0 and the resonant frequency ␻r was extracted from the simulation. These ratios of the resonant frequency to the natural frequency as a function of viscosity are shown in Fig. 15. These results show that the frequency shift is a weak function of the fluid density. This suggests that there is a

共28兲

then the fluid force Fd can then be written using the velocity, with an additional phase ␪ added: Fd共t兲 = C f b␮y max␻ cos共␻t + ␸ + ␪兲.

共29兲

C f and ␪ collapse onto one curve for all cases studied, as shown in Figs. 12 and 13. Figures 12 and 13 show C f and ␪ and a function of frequency for the cases studied. Figure 12 shows that the magnitude of the damping is dependent on the frequency at low frequencies and may approach an asymptote at higher frequencies. Figure 13 shows that the peak force leads the the peak velocity at low frequencies but lags the peak velocity at higher frequencies. Finally, the fact that

FIG. 12. C f vs frequency.

Downloaded 08 Oct 2011 to 140.116.210.6. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

044316-7

J. Appl. Phys. 104, 044316 共2008兲

Martin, Fathy, and Houston

FIG. 13. Theta vs frequency.

FIG. 15. Frequency shift as a function of density.

slight, but measurable, Reynolds number dependence on the flow around a micromachined cantilever in liquid. An estimated Reynolds number for the flow can be found using Re =

␳bUmax ␳b共y max␻兲 = . ␮ ␮

共30兲

For the 40 ␮m cantilever vibrating at 3.69⫻ 105 rad/ s in water, the Reynolds number is estimated as approximately 0.3. To achieve a truly linear damping, the Reynolds number must be in the Stokes flow regime or be much less than 1. This condition is clearly not satisfied for an AFM operating in liquid, causing a density effect and frequency-dependent damping to appear. VII. CONCLUSIONS

This paper demonstrated that a lumped-parameter model could be integrated with a transient Navier–Stokes solver to simulate the behavior of an AFM oscillating in liquid. The simulation captures the experimentally observed characteristics of cantilevers vibrating in liquid, including a large shift in peak displacement from the natural frequency. While the damping can theoretically be reduced to linear damping with an applied phase shift, the magnitude and phase shift must be determined computationally. However,

FIG. 14. Frequency shift as a function of viscosity.

the results for one cantilever stiffness can be applied across a range of cantilever stiffnesses and natural frequencies. Finally, the effect of fluid properties on the system is computed. The damping appears to be a strong function of the fluid viscosity and a weak function of the fluid density. This can be explained by the fact that the cantilever has a Reynolds number well above the limit for linear Stokes flow values. Finally, these results show that the peak displacement and the peak actuation force are anywhere from 10° to 70° out of phase with each other for an AFM cantilever vibrating in liquid. This is different from the value of 90° expected for a lightly damped resonator, such as an AFM cantilever operating in air or vacuum. Because of this, control algorithms designed for lightly damped AFM may not be ideal for AFMs operating in liquid. ACKNOWLEDGMENTS

The authors acknowledge financial support from the Office of Naval Research. Simulations were performed at the Army Research Laboratory Major Shared Resource Center, an initiative of the DOD High Performance Computing Modernization Program. 1

Q. Zhong, D. Inniss, K. Kjoller, and V. B. Elings, Surf. Sci. Lett. 290, L688 共1993兲. 2 J. E. Sader, J. Appl. Phys. 84, 64 共1998兲. 3 M. Ashhab, M. V. Salapaka, M. Dahleh, and I. Mezić, Automatica 35, 1663 共1999兲. 4 R.-F. Fung and S.-C. Huang, ASME J. Vibr. Acoust. 123, 502 共2001兲. 5 P. K. Hansma, J. P. Cleveland, M. Radmacher, D. A. Walters, P. E. Hillner, M. Bezanilla, M. Fritz, D. Vie, H. G. Hansma, C. B. Prater, J. Massie, L. Fukunaga, J. Gurley, and V. B. Elings, Appl. Phys. Lett. 64, 1738 共1994兲. 6 C. A. J. Putman, K. O. Van der Werf, B. G. De Grooth, N. F. Van Hulst, and J. Greve, Appl. Phys. Lett. 64, 2454 共1994兲. 7 J. Tamayo, A. D. L. Humphris, R. J. Owen, and M. J. Miles, Biophys. J. 81, 526 共2001兲. 8 A. Grant and L. McDonnell, Ultramicroscopy 97, 177 共2003兲. 9 W. Han, S. M. Lindsay, and T. Jing, Appl. Phys. Lett. 69, 4111 共1996兲. 10 G. E. Fantner, G. Schitter, J. H. Kindt, T. Ivanov, K. Ivanova, R. Pater, N. Holten-Andersen, J. Adams, P. J. Thurner, I. W. Rangelow, and P. K. Hansma, Ultramicroscopy 106, 881 共2006兲. 11 D. A. Walters, J. P. Cleveland, N. H. Thomson, P. K. Hansma, M. A. Wendman, G. Gurley, and V. B. Elings, Rev. Sci. Instrum. 67, 3583 共1996兲. 12 N. V. Lavrik, M. J. Sepaniak, and P. G. Datskos, Rev. Sci. Instrum. 75,

Downloaded 08 Oct 2011 to 140.116.210.6. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

044316-8

2229 共2004兲. S. S. Verbridge, L. M. Bellan, J. M. Parpia, and H. G. Craighead, Nano Lett. 6, 2109 共2006兲. 14 J. Tamayo, A. D. L. Humphris, A. M. Malloy, and M. J. Miles, Ultramicroscopy 86, 167 共2001兲. 15 T. Braun, V. Barwich, M. K. Ghatkesar, A. H. Bredekamp, C. Gerber, M. Hegner, and H. P. Lang, Phys. Rev. E 72, 031907 共2005兲. 16 B. Anczykowski, J. P. Cleveland, D. Krüger, V. B. Elings, and H. Fuchs, Appl. Phys. A: Mater. Sci. Process. 66, S885 共1998兲. 17 cR. D. Jäggi, A. Franco-Obregón, P. Studeras, and K. Ensslin, Appl. Phys. Lett. 79, 135 共2001兲. 18 H. Hölscher and U. D. Schrarz, Appl. Phys. Lett. 89, 073117 共2006兲. 19 J. Tamayo, A. D. L. Humphris, and M. J. Miles, Appl. Phys. Lett. 77, 582 共2000兲. 20 U. Dürig, H. R. Steinauer, and N. Blanc, J. Appl. Phys. 82, 3641 共1997兲. 21 R. W. Stark, G. Schitter, M. Stark, R. Guckenberger, and A. Stemmer, Phys. Rev. B 69, 085412 共2004兲. 22 K. Yamasue and T. Hikihara, Rev. Sci. Instrum. 77, 053703 共2006兲. 23 G. Schitter, F. Allwöwer, and A. Stemmer, Nanotechnology 15, 108 共2004兲. 24 F. J. Rubio-Sierra, R. M. Vásquez, and R. W. Stark, IEEE Trans. Nanotechnol. 5, 692 共2006兲. 25 R. M. Vásquez, F. J. Rubio-Sierra, and R. W. Stark, Nanotechnology 18, 1 共2007兲. 26 M. Stark, R. Guckenberger, A. Stemmer, and R. W. Stark, J. Appl. Phys. 98, 114904 共2005兲. 27 F. R. Blom, S. Bouwstra, M. Elwenspoek, and J. H. J. Fluitman, J. Vac. Sci. Technol. B 10, 19 共1992兲. 28 R. B. Bhiladvala and Z. J. Wang, Phys. Rev. E 69, 036307 共2004兲. 29 M. J. Martin and B. H. Houston, Proceedings of the AIAA 46th Aerospace Sciences Meeting and Exhibit, 2008, Paper No. 2008-0690. 30 S. Rast, C. Wattinger, U. Gysin, and E. Meyer, Rev. Sci. Instrum. 71, 2772 共2000兲. 13

J. Appl. Phys. 104, 044316 共2008兲

Martin, Fathy, and Houston

W. Zhang and K. Turner, Sens. Actuators, A 134, 594 共2007兲. T. R. Rodríguez and R. García, Appl. Phys. Lett. 80, 1646 共2002兲. 33 R. D. Blevins, Flow-Induced Vibration, 2nd ed. 共Kreiger, Malabar, FL, 1994兲. 34 R. García and R. Pérez, Surf. Sci. Rep. 47, 197 共2002兲. 35 G. Y. Chen, R. J. Warmack, P. I. Oden, and T. Thundat, J. Vac. Sci. Technol. B 14, 1313 共1996兲. 36 J. Legleiter and T. Kowalewski, Appl. Phys. Lett. 87, 163120 共2005兲. 37 W. Weaver, S. P. Timoshenko, and D. H. Young, Vibration Problems in Engineering 共Wiley, New York, 1990兲. 38 W. T. Thomson, Theory of Vibrations with Applications 共Prentice-Hall, Englewood Cliffs, NJ, 1993兲. 39 J. W. Strutt and L. Rayleigh, The Theory of Sound 共Dover, New York, 1937兲, Vol. 1. 40 W. C. Young and R. Bydynas, Roark’s Formulas for Stress and Strain 共McGraw-Hill, New York, 1989兲. 41 T. Sulchek, R. Hseih, J. D. Adams, S. C. Minne, C. F. Quate, and D. M. Adderton, Rev. Sci. Instrum. 71, 2097 共2000兲. 42 E. L. Florin, M. Radmacher, B. Fleck, and H. E. Gaub, Rev. Sci. Instrum. 65, 639 共1994兲. 43 R. Pedrak, Tzv. Ivanov, K. Ivanova, T. Gotszalk, N. Abedinov, I. W. Rangelow, K. Edinger, T. Schenkel, and P. Hudek, J. Vac. Sci. Technol. B 21, 3102 共2003兲. 44 J. Lee and W. P. King, Rev. Sci. Instrum. 78, 126102 共2007兲. 45 F. H. Harlow and J. E. Welch, Phys. Fluids 8, 2182 共1965兲. 46 C. A. J. Fletcher, Specific Techniques for Different Flow Categories, 2nd ed., Computational Techniques for Fluid Dynamics Vol. II 共SpringerVerlag, New York, 1991兲. 47 R. L. Panton, Incompressible Flow, 3rd ed. 共Wiley, Hoboken, NJ, 2005兲. 48 N. Maluf, An Introduction to Microelectromechanical Systems Engineering 共Arech House, Boston, MA, 2000兲. 49 G. F. Franklin, J. D. Powell, and A. Emani-Naeini, Feedback Control of Dynamic Systems, 4th ed. 共Prentice-Hall, Upper Saddle River, NJ, 2003兲. 31 32

Downloaded 08 Oct 2011 to 140.116.210.6. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions