Study of the sensitivity and resonant frequency of the torsional modes of an AFM cantilever with a sidewall probe based on a nonlocal elasticity theory

A relationship based on a nonlocal elasticity theory is developed to investigate the torsional sensitivity and resonant frequency of an atomic force microscope (AFM) with assembled cantilever probe (ACP). This ACP comprises a horizontal cantilever and a vertical extension, and a tip located at the free end of the extension, which makes the AFM capable of topography at sidewalls of microstructures. First, the governing differential equations of motion and boundary conditions for dynamic analysis are obtained by a combination of the basic equations of nonlocal elasticity theory and Hamilton's principle. Afterward, a closed‐form expression for the sensitivity of vibration modes has been obtained using the relationship between the resonant frequency and contact stiffness of cantilever and sample. These analysis accounts for a better representation of the torsional behavior of an AFM with sidewall probe where the small‐scale effect are significant. The results of the proposed model are compared with those of classical beam theory. The results show that the sensitivities and resonant frequencies of ACP predicted by the nonlocal elasticity theory are smaller than those obtained by the classical beam theory. Microsc. Res. Tech. 78:408–415, 2015. © 2015 Wiley Periodicals, Inc.     

Study of the sensitivity of the first four flexural modes of an AFM cantilever with a sidewall probe

The resonant frequency and sensitivity of flexural vibration for an atomic force microscope (AFM) cantilever with a sidewall probe have been analyzed. A closed-form expression for the sensitivity of vibration modes has been obtained using the relationship between the resonant frequency and contact stiffness of cantilever and sample. The results show that a sidewall scanning AFM is more sensitive when the contact stiffness is lower and that the first mode is the most sensitive. However, the high-order modes become more sensitive than the low-order modes as the contact stiffness increases. The resonance frequency of an AFM cantilever is low when contact stiffness is small. However, the frequency rapidly increases as contact stiffness increases. In addition, it can be found that the effects of the vertical extension on the sensitivity and the resonant frequency of an AFM cantilever are significant. Decreasing the length of vertical extension can increase the resonance frequency and sensitivity of mode 1 when the contact stiffness is small. However, the situation is reverse when the contact stiffness becomes large.     

Coupled lateral bending-torsional vibration sensitivity of atomic force microscope cantilever

We study the influence of the contact stiffness and the ration between cantilever and tip lengths on the resonance frequencies and sensitivities of lateral cantilever modes. We derive expressions to determine both the effective resonance frequency and the mode sensitivity of an atomic force microscope (AFM) rectangular cantilever. Once the contact stiffness is given, the resonance frequency and the sensitivity of the vibration modes can be obtained from the expression. The results show that each mode has a different resonant frequency to variations in contact stiffness and each frequency increased until it eventually reached a constant value at very high contact stiffness. The low-order vibration modes are more sensitive to vibration than the high-order mode when the contact stiffness is low. However, the situation is reversed when the lateral contact stiffness became higher. Furthermore, increasing the ratio of tip length to cantilever length increases the vibration frequency and the sensitivity of AFM cantilever.     

Dual resonance excitation system for the contact mode of atomic force microscopy

We propose an improved system that enables simultaneous excitation and measurements of at least two resonance frequency spectra of a vibrating atomic force microscopy (AFM) cantilever. With the dual resonance excitation system it is not only possible to excite the cantilever vibrations in different frequency ranges but also to control the excitation amplitude for the individual modes. This system can be used to excite the resonance frequencies of a cantilever that is either free of the tip-sample interactions or engaged in contact with the sample surface. The atomic force acoustic microscopy and principally similar methods utilize resonance frequencies of the AFM cantilever vibrating while in contact with the sample surface to determine its local elastic modulus. As such calculation demands values of at least two resonance frequencies, two or three subsequent measurements of the contact resonance spectra are necessary. Our approach shortens the measurement time by a factor of two and limits the influence of the AFM tip wear on the values of the tip-sample contact stiffness. In addition, it allows for in situ observation of processes transpiring within the AFM tip or the sample during non-elastic interaction, such as tip fracture.     

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.     

MEMS微悬臂梁构件变形规律的AFM试验研究

采用原子力显微镜测试了微悬臂梁构件的弯曲变形规律。发现构件的弹性支撑是试验数据和理论计算出现误差的主要原因。引入支撑扭转等效刚度对探针-样品等效刚度计算公式进行修正。修正结果对大有效长度的构件与实测结果吻合良好。对短有效长度微悬臂梁,其误差原因主要为支撑的影响以及测试探针的大变形,论文进一步提出研究的建议。     

A new model for investigating the flexural vibration of an atomic force microscope cantilever

A new model for the flexural vibration of an atomic force microscope cantilever is proposed, and a closed-form expression is derived. The effects of angle, damping and tip moment of inertia on the resonant frequency were analysed. Because the tip is not exactly located at one end of the cantilever, the cantilever is modelled as two beams. The results show that the frequency first increases with increase in angle and then decreases to a constant value for high values of the angle. Moreover, the damping is increased at lower contact positions. The tip moment of inertia is also sensitive to the resonant frequency at small values for the odd modes and large values for the even modes.     

Evaluating local elasticity of the metal nano-films quantitatively based on referencing approach of atomic force acoustic microscopy

Traditional technique such nanoindenter(NI) can’t measure the local elastic modulus at nano-scale(lateral). Atomic force acoustic microscopy (AFAM) is a dynamic method, which can quantitatively determine indentation modulus by measuring the contact resonance spectra for high order modes of the cantilever. But there are few reports on the effect of experimental factors, such length of cantilever, contact stiffness on measured value. For three different samples, including copper(Cu) film with 110 nm thickness, zinc(Zn) film of 90 nm thickness and glass slides, are prepared and tested, using referencing approach in which measurements are performed on the test and reference samples (it’s elastic modulus is known), and their contact resonance spectra are measured used the AFAM system experimentally. According to the vibration theory, from the lowest two contact resonance frequencies, the tip-sample contact stiffness is calculated, and then the values for the elastic properties of test sample, such as the indentation modulus, are determined. Using AFAM system, the measured indentation modulus of copper nano-film, zinc nano-film and glass slides are 113.53 GPa, 87.92 GPa and 57.04 GPa, which are agreement with literature values MCu=105-130 GPa, MZn=88.44 GPa and MGlass=50-90 GPa. Furthermore, the sensitivity of contact resonance frequency to contact stiffness is analyzed theoretically. The results show that for the cantilevers with the length 160μm, 225μm and 520μm respectively, when contact stiffness increases from 400 N/m to 600 N/m, the increments of first contact resonance frequency are 126 kHz, 93 kHz and 0.6 kHz, which show that the sensitivity of the contact resonance frequency to the contact stiffness reduces with the length of cantilever increasing. The novel method presented can characterize elastic modulus of near surface for nano-film and bulk material, and local elasticity of near surface can be evaluated by optimizing the experimental parameters using the AFAM system.     

Sensitivity optimization of the scanning microdeformation microscope and application to mechanical characterization of soft materials

In this article we present the study of the sensitivity optimization of our system of micromechanical characterization called the scanning microdeformation microscope. The flexural contact modes of vibration of the cantilever have been modeled. We discuss the matching between the cantilever stiffness and the contact stiffness which depends on the sample material. In order to obtain the best sensitivity, the stiffnesses must be the closest one to each other. Because the length of the cantilever directly affects its stiffness, the cantilever geometry can be optimized for different materials. We have validated this study with measurements on a soft material the polydimethylsiloxane with a cantilever optimized for materials of Young's moduli of some megapascals. Experimental results obtained with two different samples have shown the high sensitivity of the method for the measurement of low Young's moduli and have been compared with nanoindentation and dynamic mechanical analysis results.     

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.     

Influence of the tip mass and position on the AFM cantilever dynamics: Coupling between bending,torsion and flexural modes

The effects of the geometrical asymmetric related to tip position as a concentrated mass, on the sensitivity of all three vibration modes, lateral excitation (LE), torsional resonance (TR) and vertical excitation (VE), of an atomic force microscopy (AFM) microcantilever have been analyzed. The effects of the tip mass and its position are studied to report the novel results to estimating the vibration behavior of AFM such as resonance frequency and amplitude of the microcantilever. In this way, to achieve more accurate results, the coupled motion in all three modes is considered. In particular, it is investigated that performing the coupled motion in analysis of AFM microcantilever is almost necessary. It is shown that the tip mass and its position have significant effects on vibrational responses. The results show that considering the tip mass decreases the resonance frequencies particularly on high-order modes. However, dislocating of tip position has an inverse effect that causes an increase in the resonance frequencies. In addition, it has been shown that the amplitude of the AFM microcantilever is affected by the influences of tip and its position. These effects are caused by the interaction between flexural and torsional motion due to the moment of inertia of the tip mass.     

Characteristics of a dynamic atomic force microscopy based on a higher-order resonant silicon cantilever and experiments

In order to improve the sensitivity and scanning speed of the dynamic AFM, a surface scanning method using higher-order resonant cantilever is adopted and investigated based on the higher-order resonance characteristics of the silicon cantilever, and the theoretical analysis and experimental verification on the higher-order resonance characteristics of the corresponding dynamic AFM cantilever are given. In this method, the cantilever is excited to oscillate near to its higher-order resonant frequency which is several times higher than that of the fundamental mode. Then the characteristic changes a lot compared with the first-order resonant cantilever. Because of the changes of the quality factor, amplitude and the mode shape of the cantilever, the higher-order resonant AFM gets higher sensitivity and scanning speed. Based on the home-built tapping-mode AFM experiment system, the resolution and the response time of the first and second order resonance measured by experiment are respectively: 0.83 nm, 0.42 nm; 1265 μs, 573 μs. The higher-order resonance cantilever has higher sensitivity and the dynamic measurement performance of the cantilever is significantly improved from the experimental results. This can be a useful method to develop AFM with high speed and high sensitivity. Besides above, the surface profile of a grating sample and its three-dimensional topography are obtained by the higher-order resonant mode AFM.     

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.     

Topography imaging with a heated atomic force microscope cantilever in tapping mode

This article describes tapping mode atomic force microscopy (AFM) using a heated AFM cantilever. The electrical and thermal responses of the cantilever were investigated while the cantilever oscillated in free space or was in intermittent contact with a surface. The cantilever oscillates at its mechanical resonant frequency, 70.36 kHz, which is much faster than its thermal time constant of 300 micros, and so the cantilever operates in thermal steady state. The thermal impedance between the cantilever heater and the sample was measured through the cantilever temperature signal. Topographical imaging was performed on silicon calibration gratings of height 20 and 100 nm. The obtained topography sensitivity is as high as 200 microVnm and the resolution is as good as 0.5 nmHz(1/2), depending on the cantilever power. The cantilever heating power ranges 0-7 mW, which corresponds to a temperature range of 25-700 degrees C. The imaging was performed entirely using the cantilever thermal signal and no laser or other optics was required. As in conventional AFM, the tapping mode operation demonstrated here can suppress imaging artifacts and enable imaging of soft samples.     

Force modulation with a scanning force microscope: an analysis

A magnetic force modulation microscope (FMM) has been employed to measure the dynamic behavior of a contact between a scanning force microscope (SFM) tip and a surface. Our experimental results show the inefficiency of the classical models (two Kelvin-Voigt elements in parallel). A new model which takes into account the normal and tangential stiffness of the contact, and also the geometrical and mechanical properties of the cantilever which hold the tip, is proposed. This model shows that the natural frequency is sensitive to the normal stiffness, only if the ratio of the normal contact stiffness to the cantilever stiffness is between 0.2 and 200. Above this domain, the natural frequency is sensitive to sliding (Mindlin theory).     

Frequency shifts and analytical solutions of an AFM curved beam

The squeeze damping coefficient between the cantilever of a straight AFM probe and the surface of a biological sample in liquids is inversely proportional to their distance to the third power. Due to the small cantilever-sample distance, the quality factor of AFM in liquid is too small and results in a low signal–noise ratio. In this study, an AFM curved beam is proposed to solve this problem. Results show that the squeeze damping is significantly decreased and thus the quality factor of an AFM curved beam is greatly increased. An effective mass-spring-damper model is presented and its analytical solution is derived. Moreover, the formulas of the resonant quality factor and frequency shift are discovered. In addition to the requirement of the low squeeze damping, high frequency shifts or sensitivities is necessary for accurate measurement. Results indicate that the effects of the arc angle and several parameters on the quality factor and the frequency shifts are significant. The optimum parameters for high quality and frequency shift are also investigated.     

A more comprehensive modeling of atomic force microscope cantilever

This paper focuses on the development of a complete model of an atomic force microscope (AFM) micro-cantilever beam, based on considering the effects of four major factors in modeling the cantilever. They are: rotary inertia and shear deformation of the beam and mass and rotary inertia of the tip. A method based on distributed-parameter modeling approach is proposed to solve the governing equations. The comparisons generally show a very good agreement between the present results and the results of other investigators. As expected, rotary inertia and shear deformation of the beam decrease resonance frequency especially at high ratio of cantilever thickness to its length, and it is relatively more pronounced for higher-order frequencies, than lower ones. Mass and rotary inertia of the tip have similar effects when the mass-ratio of the tip to the cantilever is high. Moreover, the influence of each of these four factors, thickness of the cantilever, density of the tip and inclination of the cantilever on the resonance frequencies has been investigated, separately. It is felt that this work might help the engineers in reducing AFM micro-cantilever design time, by providing insight into the effects of various parameters with the micro-cantilever.     

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.     

Dual frequency atomic force microscopy on charged surfaces

The cantilever is mechanically driven at two resonant frequencies in a bimodal atomic force microscope (AFM). To generate the feedback signal for topography measurement the deflection signal is demodulated at one frequency and for compositional surface mapping at the other. In particular, the second mode amplitude and phase signals are used to map surface forces such as the van der Waals interaction. On electrically charged surfaces both, van der Waals forces and electrostatic forces contribute to the second eigenmode signal. The higher eigenmode signal in bimodal AFM reflects the local distribution of electrical charges. Mechanically driven bimodal AFM thus also provides a valuable tool for compositional mapping based on surface charges.     

Quantitative characterization of crosstalk effects for friction force microscopy with scan-by-probe SPMs

If the photodetector and cantilever of an atomic force microscope (AFM) are not properly adjusted, crosstalk effects will appear. These effects disturb measurements of the absolute vertical and horizontal cantilever deflections, which are involved in friction force microscopy (FFM). A straightforward procedure is proposed to study quantitatively crosstalk effects observed in scan-by-probe SPMs. The advantage of this simple, fast, and accurate procedure is that no hardware change or upgrade is needed. The results indicate that crosstalk effects depend not only on the alignment of the detector but also on the cantilever properties, position, and detection conditions. The measurements may provide information on the origin of the crosstalk effect. After determination of its magnitude, simple correction formulas can be applied to correct the crosstalk effects and then the single-load wedge method, using a commercially available grating, can be employed for accurate calibration of the lateral force.