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.     

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.     

Sensitivity and resonant frequency of an AFM with sidewall and top-surface probes for both flexural and torsional modes

The resonant frequencies and flexural sensitivities of an atomic force microscope (AFM) with assembled cantilever probe (ACP) are studied. This ACP comprises a horizontal cantilever, a vertical extension and two tips located at the free ends of the cantilever and the extension, which makes the AFM capable of simultaneous topography at top surface and sidewalls of microstructures especially microgears, which consequently leads to a time-saving swift scanning process. In this work, the effects of the sample surface contact stiffness and the geometrical parameters such as the ratio of the vertical extension length to the horizontal cantilever length and the distance of the vertical extension from clamped end of the horizontal cantilever on both flexural and torsional resonant frequencies and sensitivities are assessed. These geometrical effects are illustrated in some figures. The results show that the low-order vibration modes are more sensitive for low values of the contact stiffness, but the situation is reversed for high values.     

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.     

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.     

多模态动态原子力显微镜系统

动态原子力显微镜(atomic force microscope,AFM)是通过检测悬臂谐振状态的变化来对物体表面形貌进行测量的。通过对谐振状态的三种因素即振幅、相位、频率的检测,动态AFM可以分为三种工作模式,即振幅反馈、相位反馈与频率反馈模式,这三种反馈模式有着不同的扫描特点。基于硅悬臂具有高阶谐振的特性,动态原子力显微镜可以在悬臂工作于高阶谐振状态时对物体进行扫描。综合上述工作模式研制了一套多模态动态AFM,可以在三种反馈模式、不同阶谐振状态下对物体进行扫描测量。利用该系统在不同反馈模式、不同阶谐振状态下进行了扫描测试,结果显示,系统在各模式下具有亚纳米分辨力,其中在相位反馈模式,悬臂二阶谐振时可达到最优灵敏度与分辨力,分别为17.5V/μm和0.29nm,在最优灵敏度与分辨力状态下对光栅试样进行了三维扫描,得到光栅的三维形貌图。     

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.     

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.     

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.     

Suppression of spurious vibration of cantilever in atomic force microscopy by enhancement of bending rigidity of cantilever chip substrate

To improve the precision of dynamic atomic force microscopy (AFM) using cantilever vibration spectra, a simple but effective method for suppressing spurious response (SR) was developed. The dominant origin of SR was identified to be the bending vibration of the cantilever substrate, by the analysis of the frequency of SR. Although a rigid cover pressing the whole surface of the substrate suppressed SR, the utility was insufficient. Then, a method of enhancing the bending rigidity of the substrate by gluing a rigid plate (clamping plate, CP) to the substrate was developed. This chip can be used with an ordinary cantilever holder, so that the reproducibility of SR suppression when attaching and detaching the cantilever chip to the holder was improved. To verify its utility, the evaluation of a microdevice electrode was performed by ultrasonic atomic force microscopy. The delamination at a submicron depth was visualized and the detailed variation of the delamination was evaluated for the first time using clear resonance spectra. The CP method will particularly contribute to improving dynamic-mode AFM, in which resonance spectra with a low quality factor are used, such as noncontact mode AFM in liquid or contact resonance mode AFM. The effect of the CP can be achieved by fabricating a substrate with a thick plate beforehand.     

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 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.     

Vibration sensitivity of the scanning near-field optical microscope with a tapered optical fiber probe

In this paper the Rayleigh-Ritz method was used to study the scanning near-field optical microscope (SNOM) with a tapered optical fiber probe's flexural and axial sensitivity to vibration. Not only the contact stiffness but also the geometric parameters of the probe can influence the flexural and axial sensitivity to vibration. According to the analysis, the lateral and axial contact stiffness had a significant effect on the sensitivity of vibration of the SNOM's probe, each mode had a different level of sensitivity and in the first mode the tapered optical fiber probe was the most acceptive to higher levels of flexural and axial vibration. Generally, when the contact stiffness was lower, the tapered probe was more sensitive to higher levels of both axial and flexural vibration than the uniform probe. However, the situation was reversed when the contact stiffness was larger. Furthermore, the effect that the probe's length and its tapered angle had on the SNOM's probe axial and flexural vibration were significant and these two conditions should be incorporated into the design of new SNOM probes.     

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.     

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.     

Vibration reduction control of a voice coil motor (VCM) nano scanner

This paper presents vibration reduction control of a voice coil motor (VCM) nano scanner. We had developed a VCM scanner. The scanner has flexure hinges structure. However, the VCM nano scanner has some problems of thermal drift and small damping compared to the PZT driven nano scanner. Especially, the small damping coefficient of the VCM scanner causes mechanical vibration when the control input signal is near to the resonance frequencies. Additionally, disturbance to the VCM scanner and electronic noise in the sensor also cause the mechanical vibration when they are near to the resonant frequencies. The mechanical vibration reduces the servo bandwidth as well as the accuracy, which deteriorates the AFM image of the samples. We design a pre-filter to reduce the signal applied to the VCM nano scanner and electronic noise in the sensor whose frequency is closed the resonant frequency of the VCM nano scanner. We measure the time and frequency response of the VCM scanner without using the pre-filter and with using the pre-filter. Finally, the topology images of a bare wafer are measured and compared using the AFM.     

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.     

Investigating ultra-thin lubricant layers using resonant friction force microscopy

The ultrasonic friction mode of an atomic force microscope is a scanning probe technique allowing one to analyze the load and velocity dependence of friction. The technique is based on evaluation of the resonance behavior of an AFM cantilever when in contact with a vibrating sample surface. The effect of load and lateral displacement of the sample surface on the shape of the torsional resonance spectra of the AFM cantilever is evaluated under dry and lubricated sliding conditions. A characteristic flattening of the torsional resonance curve has been observed at large surface displacements, resulting from the onset of sliding friction in the AFM cantilever–sample surface contact. An analytical model describing torsional cantilever vibrations in Hertzian contact with a sample surface is presented, and numerical simulations have been carried out in order to confirm that the flattening of the resonance curve occurs with the onset of the sliding friction in the contact.     

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.     

Advances in hybrid laser joining

This paper presents vibration reduction control of a voice coil motor (VCM)-driven actuator for SPM applications. We had developed a VCM nanoscanner. The scanner has flexure hinges structure. However, the VCM nanoscanner has some problems of thermal drift and small damping compared to the PZT driven nanoscanner. Especially, the small damping coefficient of the VCM nanoscanner causes mechanical vibration when the control input signal is near to the resonance frequencies of the scanner. Additionally, disturbance to the VCM scanner and electronic noise in the sensor also causes the mechanical vibration when they are near to the resonant frequencies. The mechanical vibration reduces the servo bandwidth as well as the accuracy, which deteriorates the AFM image of the samples. We design input shaping prefilter to reduce the signal applied to the VCM nanoscanner and electronic noise in the sensor whose frequency is close to the resonant frequency of the VCM nanoscanner. We measure the time and frequency response of the VCM scanner without using the prefilter and with using the prefilter. Finally, the topology images of a bare wafer are measured and compared using the AFM.