Analysis of the effect of mechanical properties of liquid and geometrical parameters of cantilever on the frequency response function of AFM

The dynamic behavior of atomic force microscopy (AFM) cantilevers in liquid is completely different from its behavior in air due to the applied hydrodynamic force. Exciting cantilever with frequencies close to resonance frequency and primary position alignment are two critical issues that should be considered in deriving frequency response function (FRF). In this paper, the hydrodynamic force has been modeled with string of spheres and the effect of the damping and the added mass on the model has been analyzed. Afterward, this force is applied to the dynamic equation so that the dynamic behavior of AFM cantilevers is studied in liquids by analyzing the effect of some important parameters such as added mass, internal, and fluid damping. By simulations of the dynamic equations for a silicon cantilever, FRF is determined in both air and liquid. In addition, the effects of two significant parameters of liquid mechanical properties (liquid viscosity and density) and geometrical parameters of cantilever on FRF are studied. The results for string of spheres model are compared with the other hydrodynamic model and the experimental data. When length/width ratio decreases, it is found that string of spheres model has a better agreement than the other hydrodynamic model with experimental data.     

Modeling and simulation of AFM cantilever with two piezoelectric layers submerged in liquid over rough surfaces

In this paper, the vibration of an atomic force microscope (AFM) cantilever in tapping mode with two whole piezoelectric layers submerged in liquid medium is investigated. In the performed modeling, the sample surface has been considered as rough, and to show these surface asperities, two models of Rumpf and Rabinovich have been employed for analyzing the attractive Van der Waals force. This paper has been organized in two sections. The first section deals with the functioning of cantilever over rough surfaces, which accompanies the changes of the attractive Van der Waals force, and the second section involves the changes in the Van der Waals force which lead to a change in the liquid medium. The cantilever is totally submerged in the liquid. To show the effect of liquid on cantilever, first, only the cantilever tip is immersed in the liquid and it is dynamically analyzed. Then, the cantilever is totally submerged and then taken out of the liquid, so that the additional mass and damping of the cantilever could be calculated. In these two manners of cantilever immersion in liquid, the effects of the added mass and damping on the cantilever can be measured. When a cantilever vibrates totally in liquid, since the mass and damping of the liquid that is present on the cantilever cannot be determined, first, the cantilever's natural frequency in liquid is estimated in the laboratory and then by using this frequency and the cantilever stiffness (which is not medium-dependent and is always considered as constant), the additional mass and damping of the cantilever are determined.     

Resonance-mode effect on microcantilever mass-sensing performance in air

This research investigates the air drag damping effect of the micromachined cantilevers in different resonance modes on the quality factor, which are operated in ambient air. Based on a simplified dish-string model for air drag force acting on the resonant cantilever, the air drag damping properties of the cantilevers vibrating in different modes are analyzed with theoretic vibration mechanics, which is complemented and further confirmed with finite-element simulation. Four kinds of integrated cantilevers, which resonate in the first flexural mode, the second flexural mode, the first torsional mode, and the second torsional mode, respectively, are designed and fabricated by using micromachining techniques. Finally, biomolecular sensing experiments are carried out to verify the theoretical results obtained before. From both the modeling and experimental results, it can be seen that damping characteristics of the torsional cantilever resonators are generally better than that of the flexural ones, and quality factor of the cantilever resonator in a higher-frequency mode is always superior to that in a lower-frequency one. Among the four kinds of microcantilever resonators operated in our experiments, the one operated in the second flexural modes exhibits the highest Q factor and the best biomass sensing performance.     

Effects of geometrical dimensions and liquid properties on frequency response of resonating microcantilevers in the vicinity of a surface

Frequency response of an atomic force microscopy cantilever immersed in liquid near a surface strongly depends on the hydrodynamic forces specially the squeezed film damping, mechanical properties of the liquid including the dynamic viscosity and the density and the geometrical dimensions of the cantilever. For a slightly inclined magnetically oscillated cantilever with the approximate hydrodynamic forces acting on it, the analytical solution of the equation of motion has already been acquired. In this paper, the effects of geometrical dimensions of the cantilever on the resonance frequency, the motion amplitude and the quality factor are observed and then any increase in the kinematic viscosity of the liquid is studied through the simulation of the oscillatory motion of the cantilever. The acquired amplitude–frequency curves indicate that with an appropriate proportion between the cantilever dimensions, it is possible to optimize the quality factor for extremely small tip-sample separations. Also, if the thickness is increased and the width is reduced with the cross section area being held constant, the resonance will occur at higher frequency and the quality factor will be enhanced. Adding glycerol to water will result in the reduction of resonance frequency of the cantilever near the surface due to the viscous friction and squeezed film damping. Consequently the quality factor is decreased as a result of viscosity increase in the simulations.     

Study of the tip mass and interaction force effects on the frequency response and mode shapes of the AFM cantilever

The vibrational characteristics of an atomic force microscope (AFM) cantilever beam play a key role in dynamic mode of the atomic force microscope. As the oscillating AFM cantilever tip approaches the sample, the tip–sample interaction force influences the cantilever dynamics. In this paper, we present a detailed theoretical analysis of the frequency response and mode shape behavior of a cantilever beam in the dynamic mode subject to changes in the tip mass and the interaction regime between the AFM cantilever system and the sample. We consider a distributed parameter model for AFM and use Euler–Bernoulli method to derive an expression for AFM characteristics equation contains tip mass and interaction force terms. We study the frequency response of AFM cantilever under variations of interaction force between AFM tip and sample. Also, we investigate the effect of tip mass on the frequency response and also the quality factor and spring constant of each eigenmodes of AFM micro-cantilever. In addition, the mode shape analysis of AFM cantilever under variations of tip mass and interaction force is investigated. This will incorporate the presentation of explicit analytical expressions and numerical analysis. The results show that by considering the tip mass, the resonance frequencies of the cantilever are decreased. Also, the tip mass has a significant effect on the mode shape of the higher eigenmodes of the AFM cantilever. Moreover, tip mass affects the quality factor and spring constant of each modes.     

Influence of Poisson's ratio variation on lateral spring constant of atomic force microscopy cantilevers

Atomic force microscopy (AFM) can be used to measure the surface morphologies and the mechanical properties of nanostructures. The force acting on the AFM cantilever can be obtained by multiplying the spring constant of AFM cantilever and the corresponding deformation. To improve the accuracy of force experiments, the spring constant of AFM cantilever must be calibrated carefully. Many methods, such as theoretical equations, the finite element method, and the use of reference cantilever, were reported to obtain the spring constant of AFM cantilevers. For the cantilever made of single crystal, the Poisson's ratio varies with different cantilever-crystal angles. In this paper, the influences of Poisson's ratio variation on the lateral spring constant and axial spring constant of rectangular and V-shaped AFM cantilevers, with different tilt angles and normal forces, were investigated by the finite element analysis. When the cantilever's tilt angle is 20 degrees and the Poisson's ratio varies from 0.02 to 0.4, the finite element results show that the lateral spring constants decrease 11.75% for the rectangular cantilever with 1muN landing force and decrease 18.60% for the V-shaped cantilever with 50nN landing force, respectively. The influence of Poisson's ratio variation on axial spring constant is less than 3% for both rectangular and V-shaped cantilevers. As the tilt angle increases, the axial spring constants for rectangular and V-shaped cantilevers decrease substantially. The results obtained can be used to improve the accuracy of the lateral force measurement when using atomic force microscopy.     

Hydrodynamic damping of a magnetically oscillated cantilever close to a surface

We studied the frequency response of a magnetically driven atomic force microscope (AFM) cantilever close to a sample surface in liquids. Amplitude–frequency (tuning) curves showed pronounced differences in dependence on the tip–sample separation (from 1 to 50 μm), with significant shifts of the resonance peak. A model was developed in which the cantilever was described in a full shape manner and the hydrodynamic forces acting on the cantilever were approximately calculated. The slight inclination of the cantilever to the surface (15°) leads to a force profile along the cantilever. Therefore, the mathematical problem can be strictly solved only numerically. For an approximate analytical solution, the hydrodynamic force profile was approximated by a constant force along the cantilever for large separations and by a point force acting on the tip of the cantilever for small separations. The theoretical results calculated within this model agreed well with the experimental data and allowed to determine the cantilever mass in liquid M^*, the joint mass at the tip end m_t^*, and the coefficient of viscous interaction of the cantilever with free liquid, γ_∞.     

Prototype cantilevers for quantitative lateral force microscopy

Prototype cantilevers are presented that enable quantitative surface force measurements using contact-mode atomic force microscopy (AFM). The "hammerhead" cantilevers facilitate precise optical lever system calibrations for cantilever flexure and torsion, enabling quantifiable adhesion measurements and friction measurements by lateral force microscopy (LFM). Critically, a single hammerhead cantilever of known flexural stiffness and probe length dimension can be used to perform both a system calibration as well as surface force measurements in situ, which greatly increases force measurement precision and accuracy. During LFM calibration mode, a hammerhead cantilever allows an optical lever "torque sensitivity" to be generated for the quantification of LFM friction forces. Precise calibrations were performed on two different AFM instruments, in which torque sensitivity values were specified with sub-percent relative uncertainty. To examine the potential for accurate lateral force measurements using the prototype cantilevers, finite element analysis predicted measurement errors of a few percent or less, which could be reduced via refinement of calibration methodology or cantilever design. The cantilevers are compatible with commercial AFM instrumentation and can be used for other AFM techniques such as contact imaging and dynamic mode measurements.     

Low temperature ultrahigh vacuum noncontact atomic force microscope in the pendulum geometry

A noncontact atomic force microscope (nc-AFM) operating in magnetic fields up to ±7 T and liquid helium temperatures is presented in this article. In many common AFM experiments the cantilever is mounted parallel to the sample surface, while in our system the cantilever is assembled perpendicular to it; the so called pendulum mode of AFM operation. In this mode measurements employing very soft and, therefore, ultrasensitive cantilevers can be performed. The ultrahigh vacuum conditions allow to prepare and transfer cantilevers and samples in a requested manner avoiding surface contamination. We demonstrate the possibility of nc-AFM and Kelvin force probe microscopy imaging in the pendulum mode. Ultrasensitive experiments on small spin ensembles are presented as well.     

The effect of fluid properties and geometrical parameters of cantilever on the frequency response of atomic force microscopy

Nowadays, the atomic force microscopy plays an indispensable role in imaging and manipulation of biological samples. To observe some specific behaviors and biological processes, fast and accurate imaging techniques are required, and one way to speed up the imaging process is to use short cantilevers. For short beams, the Timoshenko model seems to be more accurate compared to other models such as the Euler–Bernoulli. By using the Timoshenko beam model, the effects of rotational inertia and shear deformation are taken into consideration. In this paper, the frequency response of a rectangular atomic force microscope (AFM) in liquid environment has been analyzed by using the Timoshenko beam model. Afterward, since the dynamic response of AFM is influenced by the applied medium, the effects of physical and mechanical properties (e.g., fluid density and viscosity) on the frequency response of the system have been investigated. The frequency responses of the AFM cantilever immersed in various liquids have been compared with one another. And eventually, to study the influence of geometry on the dynamic behavior of AFM, the effect of the cantilever's geometrical parameters (e.g., cantilever length, width and thickness) on the frequency response of the system has been studied.     

Constant tip-surface distance with atomic force microscopy via quality factor feedback

The atomic force microscope (AFM) is a powerful and widely used instrument to image topography and measure forces at the micrometer and nanometer length scale. Because of the high degree of operating accuracy required of the instrument, small thermal and mechanical drifts of the cantilever and piezoactuator systems hamper measurements as the AFM tip drifts spatially relative to the sample surface. To compensate for the drift, we control the tip-surface distance by monitoring the cantilever quality factor (Q) in a closed loop. Brownian thermal fluctuations provide sufficient actuation to accurately determine cantilever Q by fitting the thermal noise spectrum to a Lorentzian function. We show that the cantilever damping is sufficiently affected by the tip-surface distance so that the tip position of soft cantilevers can be maintained within 40 nm of a setpoint in air and within 3 nm in water with 95% reliability. Utilizing this method to hover the tip above a sample surface, we have the capability to study sensitive interactions at the nanometer length scale over long periods of time.     

Improved parallel scan method for nanofriction force measurement with atomic force microscopy

Based on Ruan and Bhushan's study [J. Ruan and B. Bhushan, J. Tribol. 116, 378 (1994)], an improved method for quantitative nano/microfriction force measurements with the atomic force microscope (AFM) is presented. The related theoretical derivation is given in detail. The coefficient of friction can be calculated by scanning in the direction parallel to the long axis of the AFM cantilever. Then conversion factor, which can convert the lateral deflection response of the photodetector into corresponding friction force, is identified with the Meyer and Amer method [G. Meyer and N. M. Ame, Appl. Phys. Lett. 57, 2089 (1990)]. Like Ruan and Bhushan method, the advantage of this approach is that the coefficient of friction can be obtained with the plan-view geometry of AFM cantilevers and some common uncertainties, such as thickness, coating, and material properties, are not necessary. The result of the experiments performed utilizing rectangular cantilevers of different lengths shows that this improved method produces an accurate agreement for cantilevers of different lengths, thus the method can be used to measure nano/microfriction force.     

Exploiting cantilever curvature for noise reduction in atomic force microscopy

Optical beam deflection is a widely used method for detecting the deflection of atomic force microscope (AFM) cantilevers. This paper presents a first order derivation for the angular detection noise density which determines the lower limit for deflection sensing. Surprisingly, the cantilever radius of curvature, commonly not considered, plays a crucial role and can be exploited to decrease angular detection noise. We demonstrate a reduction in angular detection shot noise of more than an order of magnitude on a home-built AFM with a commercial 450 μm long cantilever by exploiting the optical properties of the cantilever curvature caused by the reflective gold coating. Lastly, we demonstrate how cantilever curvature can be responsible for up to 45% of the variability in the measured sensitivity of cantilevers on commercially available AFMs.     

Energy dissipation and dynamic response of an amplitude-modulation atomic-force microscopy subjected to a tip-sample viscous force

In a common environment of atomic force microscopy (AFM), a damping force occurs between a tip and a sample. The influence of damping on the dynamic response of a cantilever must be significant. Moreover, accurate theory is very helpful for the interpretation of a sample's topography and properties. In this study, the effects of damping and nonlinear interatomic tip-sample forces on the dynamic response of an amplitude-formulation AFM are investigated. The damping force is simulated by using the conventional Kelvin-Voigt damping model. The interatomic tip-sample force is the attractive van der Waals force. For consistance with real measurement of a cantilever, the mathematical equations of the beam theory of an AM-AFM are built and its analytical solution is derived. Moreover, an AFM system is also simplified into a mass-spring-damper model. Its exact solution is simple and intuitive. Several relations among the damping ratio, the response ratio, the frequency shift, the energy dissipation and the Q-factor are revealed. It is found that the resonant frequencies and the phase angles determined by the two models are almost same. Significant differences in the resonant quality factors and the response ratios determined by using the two models are also found. Finally, the influences of the variations of several parameters on the error of measuring a sample's topography are investigated.     

Electroplated CoPt magnets for actuation of stiff cantilevers

A cantilever has been microfabricated for use in non-contact Atomic Force Microscopy (AFM) using a very thick magnetic film to actuate the cantilever motion. The thick magnetic block is deposited electrochemically over a defined area of the cantilever. This cantilever is particularly suitable for driving stiff AFM cantilevers in a liquid environment. Clean mechanical resonances are easily observed. Examples are given of a hard (CoPt) magnet of dimension 29 × 21 × 6 μm(3) electroplated on Silicon cantilevers of stiffness ~22 N/m, giving a static displacement of ~0.2 nm in an applied field of 10(-3) T.     

Improved lateral force calibration based on the angle conversion factor in atomic force microscopy

A novel calibration method is proposed for determining lateral forces in atomic force microscopy (AFM), by introducing an angle conversion factor, which is defined as the ratio of the twist angle of a cantilever to the corresponding lateral signal. This factor greatly simplifies the calibration procedures. Once the angle conversion factor is determined in AFM, the lateral force calibration factors of any rectangular cantilever can be obtained by simple computation without further experiments. To determine the angle conversion factor, this study focuses on the determination of the twist angle of a cantilever during lateral force calibration in AFM. Since the twist angle of a cantilever cannot be directly measured in AFM, the angles are obtained by means of the moment balance equations between a rectangular AFM cantilever and a simple commercially available step grating. To eliminate the effect of the adhesive force, the gradients of the lateral signals and the twist angles as a function of normal force are used in calculating the angle conversion factor. To verify reliability and reproducibility of the method, two step gratings with different heights and two different rectangular cantilevers were used in lateral force calibration in AFM. The results showed good agreement, to within 10%. This method was validated by comparing the coefficient of friction of mica so determined with values in the literature.     

Precise atomic force microscope cantilever spring constant calibration using a reference cantilever array

A method for calibrating the stiffness of atomic force microscope (AFM) cantilevers is demonstrated using an array of uniform microfabricated reference cantilevers. A series of force-displacement curves was obtained using a commercial AFM test cantilever on the reference cantilever array, and the data were analyzed using an implied Euler-Bernoulli model to extract the test cantilever spring constant from linear regression fitting. The method offers a factor of 5 improvement over the precision of the usual reference cantilever calibration method and, when combined with the Systeme International traceability potential of the cantilever array, can provide very accurate spring constant calibrations.     

Dynamic analysis of tapping mode atomic force microscope (AFM) for critical dimension measurement

Transient dynamics of tapping mode atomic force microscope (AFM) for critical dimension measurement are analyzed. A simplified nonlinear model of AFM is presented to describe the forced vibration of the micro cantilever-tip system with consideration of both contact and non-contact transient behavior for critical dimension measurement. The governing motion equations of the AFM cantilever system are derived from the developed model. Based on the established dynamic model, motion state of the AFM cantilever system is calculated utilizing the method of averaging with the form of slow flow equations. Further analytical solutions are obtained to reveal the effects of critical parameters on the system dynamic performance. In addition, features of dynamic response of tapping mode AFM in critical dimension measurement are studied, where the effects of equivalent contact stiffness, quality factor and resonance frequency of cantilever on the system dynamic behavior are investigated. Contact behavior between the tip and sample is also analyzed and the frequency drift in contact phase is further explored. Influence of the interaction between the tip and sample on the subsequent non-contact phase is studied with regard to different parameters. The dependence of the minimum amplitude of tip displacement and maximum phase difference on the equivalent contact stiffness, quality factor and resonance frequency are investigated. This study brings further insights into the dynamic characteristics of tapping mode AFM for critical dimension measurement, and thus provides guidelines for the high fidelity tapping mode AFM scanning.     

Kinetostatic model of spring constant ratios for an AFM cantilever with end extended mass

In previous work we showed that the kinetostatic method is very effective in computing the increase in value of the spring constants of an AFM free (with or without added mass) and supported rectangular cantilever for higher mode oscillations relative to their values for natural vibration. We have considered in all previous cases that added mass is a concentrated one. However, the additional mass may be an extended one particularly in the case of a V-shaped cantilever. In this article we consider the influence of the constituent beam’s (leg’s) mutual skew and the altered position of the nodal points in the case when the attached extended triangular (trapezoid) mass of the V-shaped cantilever has a significant moment of rotational inertia and a center of this mass gravity located beyond the constituent beam end. We show that considering these effects in using the kinetostatic model yields results for the ratios of the spring constants at higher modes of oscillation and their values at the first frequency natural vibration for a V-shaped cantilever which are in good agreement with the thermomechanical noise amplitudes obtained by other researchers. This should prove helpful for the proper calibration of V-shaped cantilevers whose application with higher modes oscillation provides increased measurement sensitivity.     

Micro-fabricated mechanical sensors for lateral molecular-force microscopy

Atomic force microscopy (AFM) has been very successful in measuring forces perpendicular to the sample plane. Here, we present the advantages of turning the AFM cantilever 90° in order for it to be perpendicular to the sample. This rotation leads naturally to the detection of in-plane forces with some extra advantages with respect to the AFM orientation. In particular, the use of extremely small (1 μm wide) and soft (k≅10^–5 N/m) micro-fabricated cantilevers is demonstrated by recording their thermal power spectral density in ambient conditions and in liquid. These measurements lead to the complete characterisation of the sensors in terms of their stiffness and resonant frequency. Future applications, which will benefit from the use of this force microscopy technique, are also described.