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.     

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.     

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.     

Nonlocal beam models for buckling of nanobeams using state-space method regarding different boundary conditions

Buckling analysis of nanobeams is investigated using nonlocal continuum beam models of the different classical beam theories namely as Euler-Bernoulli beam theory (EBT), Timoshenko beam theory (TBT), and Levinson beam theory (LBT). To this end, Eringen’s equations of nonlocal elasticity are incorporated into the classical beam theories for buckling of nanobeams with rectangular cross-section. In contrast to the classical theories, the nonlocal elastic beam models developed here have the capability to predict critical buckling loads that allowing for the inclusion of size effects. The values of critical buckling loads corresponding to four commonly used boundary conditions are obtained using state-space method. The results are presented for different geometric parameters, boundary conditions, and values of nonlocal parameter to show the effects of each of them in detail. Then the results are fitted with those of molecular dynamics simulations through a nonlinear least square fitting procedure to find the appropriate values of nonlocal parameter for the buckling analysis of nanobeams relevant to each type of nonlocal beam model and boundary conditions.analysis.     

Vibration of nonuniform carbon nanotube with attached mass via nonlocal Timoshenko beam theory

This paper studies the vibrational behavior of nonuniform single-walled carbon nanotube (SWCNT) carrying a nanoparticle. A nonuniform cantilever beam with a concentrated mass at the free end is analyzed according to the nonlocal Timoshenko beam theory. A governing equation of a nonuniform SWCNT with attached mass is established. The transfer function method incorporating with the perturbation method is utilized to obtain the resonant frequencies of a vibrating nonlocal cantilever-mass system. The effects of the nonlocal parameter, taper ratio and attached mass on the natural frequencies and frequency shifts are discussed. Obtained results indicate that the sensitivity of the frequency shifts on the attached mass increases when the length-to-diameter ratio decreases. Tapered SWCNT possesses higher fundamental frequencies if the taper ratio becomes larger.     

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.     

Atomic force microscope characterization of a resonating nanocantilever

An atomic force microscope (AFM) is used as a nanometer-scale resolution tool for the characterization of the electromechanical behaviour of a resonant cantilever-based mass sensor. The cantilever is actuated electrostatically by applying DC and AC voltages from a driver electrode placed closely parallel to the cantilever. In order to minimize the interaction between AFM probe and the resonating transducer cantilever, the AFM is operated in a dynamic non-contact mode, using oscillation amplitudes corresponding to a low force regime. The dependence of the static cantilever deflection on DC voltage and of the oscillation amplitude on the frequency of the AC voltage is measured by this technique and the results are fitted by a simple non-linear electromechanical model.     

Frequency shift of a nanomechanical sensor carrying a nanoparticle using nonlocal Timoshenko beam theory

The frequency shift of a nanomechanical sensor carrying a nanoparticle is studied. A bridged single-walled carbon nanotube (SWCNT) carrying a nanoparticle is modeled as a clamped micro-beam with a concentrated micro-mass at any position. Based on the nonlocal Timoshenko theory of beams, which incorporates size effects into the classical theory, the natural frequencies of the nanomechanical sensor are derived using the transfer function method. The effects of the mass and position of the nanoparticle on the frequency shift are discussed. In the absence of the nonlocal effect, the frequencies are reduced to the results of the classical model, in agreement with those using the finite element method. The obtained results show that when the mass of the attached nanoparticle increases or its location is close to the beam center, the natural frequency decreases, but the shift in frequency increases. The effect of the nonlocal parameter on the frequency shift is significant. Decreasing the length-to-diameter ratio also increases the frequency shift. The natural frequencies and shifts are strongly affected by rotary inertia, and the nonlocal Timoshenko beam model is more adequate than the nonlocal Euler-Bernoulli beam model for short nanomechanical sensors. The obtained results are helpful in the design of SWCNT-based resonator as nanomechanical mass sensor.     

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.     

Vibration suppression of atomic-force microscopy cantilevers covered by a piezoelectric layer with tensile force

In this paper, vibration suppression of a micro-beam covered by a piezoelectric layer is studied. The micro-beam is modeled with the specific attention to its application in AFM. The AFM micro-beam is a cantilever one which is stimulated close to its natural frequency by applying a harmonic voltage to the piezoelectric layer. The beam is an Euler-Bernoulli beam which abbeys Kelvin-Voigt model. Using such model supplies the comparison between elastic and viscoelastic beams; and one of the most important properties of viscoelastic materials, damping effect can readily be investigated. The pump provides an axial load with the result that it suppresses the vibrations. First, the vibration equations are extracted using Lagrangian and extended Hamiltonian method in vertical, longitudinal, as well as torsional directions and are discretized by exploiting the Galerkin mode summation approach. The discretized time-domain equations are solved by the aid of the Runge-Kutta method. The viscoelastic beam is compared with the elastic one, and the effects of damping ratio on vibration responses are presented. Additionally, the effects of micro-pump load, excitation voltage, and initial twist angle are investigated on the amplitude of vibration and natural frequency of system. It is observed that viscoelasticity of beam and axial load of the pump reduce vibrations and provide uniform time-domain responses without beatings.     

The effect of off-end tip distance on the nanomanipulation based on rectangular and V-shape cantilevered AFMs

The AFM system, which is used as a nanomanipulator, includes a probe consistent of a cantilever and a tapered tip. In cantilevers, the tip can be located in different distances from the cantilever free end. This causes to change in stiffness of the cantilever and therefore changing in pushing force of the nanomanipulation. In this paper, the effect of the tip distance on the cantilever stiffness is studied using the equations of Hazel, and Neumeister and Ducker (ND), and a new equation to correct the torsional stiffness of V-shaped cantilevers (VSC) is proposed, which is based on the ND equation. Then, the effect of distance on pushing force of AFM-based nanomanipulations with rectangular cantilevered (RC) and VSC AFMs is simulated. The obtained results using proposed equation show that increasing of distance causes to non-linear increment of torsional stiffness of VSC. Error of the proposed equation is achieved less than 3% in comparison with result of torsional stiffness equation of ND. Moreover, it is observed that the torsional stiffness of VSC predicted by Hazel’s equation is considerably inaccurate. In nanomanipulation studies, the necessary pushing forces of nanoparticle motion are increased by increment of distance, for both types of cantilevers (RC and VSC). Moreover, critical time for RC AFM increases, but in the case of VSC AFM, the critical time decreases at first, then it is almost constant at a limited range of d, and finally it starts to increase by increasing the distance.     

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.     

Torsional buckling of a DWCNT embedded on winkler and pasternak foundations using nonlocal theory

The small-scale effect on the torsional buckling of a double-walled carbon nanotube (DWCNT) embedded on Winkler and Pasternak foundations is investigated in this study using the theory of nonlocal elasticity. The effects of the surrounding elastic medium, such as the spring constant of the Winkler type and the shear constant of the Pasternak type, as well as the van der Waals (vdW) forces between the inner and the outer nanotubes are taken into account. Finally, based on the theory of nonlocal elasticity and by employing the continuum models, an elastic double-shell model is presented for the nonlocal torsional buckling load of a DWCNT. It is seen from the results that the shear constant of the Pasternak type increases the nonlocal critical torsional buckling load, while the difference between the presence and the absence of the shear constant of the Pasternak type becomes large. It is shown that the nonlocal critical buckling load is lower than the local critical buckling load. Moreover, a simplified analysis is carried out to estimate the nonlocal critical torque for the torsional buckling of a DWCNT.     

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.     

Improving tapping mode atomic force microscopy with piezoelectric cantilevers

This article summarizes improvements to the speed, simplicity and versatility of tapping mode atomic force microscopy (AFM). Improvements are enabled by a piezoelectric microcantilever with a sharp silicon tip and a thin, low-stress zinc oxide (ZnO) film to both actuate and sense deflection. First, we demonstrate self-sensing tapping mode without laser detection. Similar previous work has been limited by unoptimized probe tips, cantilever thicknesses, and stress in the piezoelectric films. Tests indicate self-sensing amplitude resolution is as good or better than optical detection, with double the sensitivity, using the same type of cantilever. Second, we demonstrate self-oscillating tapping mode AFM. The cantilever's integrated piezoelectric film serves as the frequency-determining component of an oscillator circuit. The circuit oscillates the cantilever near its resonant frequency by applying positive feedback to the film. We present images and force-distance curves using both self-sensing and self-oscillating techniques. Finally, high-speed tapping mode imaging in liquid, where electric components of the cantilever require insulation, is demonstrated. Three cantilever coating schemes are tested. The insulated microactuator is used to simultaneously vibrate and actuate the cantilever over topographical features. Preliminary images in water and saline are presented, including one taken at 75.5 μm/s—a threefold improvement in bandwidth versus conventional piezotube actuators.     

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.     

Non-optical tip–sample distance control method for scanning near-field optical microscopy using a piezoresistive micro cantilever

A piezoresistive micro cantilever is applied to monitor the displacement of an optical fibre probe and to control tip–sample distance. The piezoresistive cantilever was originally made for a self-sensitive atomic force microscopy (AFM) probe and has dimensions of 400 µm length, 50 µm width and 5 µm thickness with a resistive strain sensor at the bottom of the cantilever. We attach the piezoresistive cantilever tip to the upper side of a vibrating bent optical fibre probe and monitor the resistance change amplitude of the strain sensor caused by the optical fibre displacement. By using this resistance change to control the tip–sample distance, the two-cantilever system successfully provides topographic and near-field optical images of standard samples in a scanning near-field optical microscopy (SNOM)/AFM system. A resonant characteristic of the two-cantilever system is also simulated using a mechanical model, and the results of simulation correspond to the experimental results of resonance characteristics.     

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.     

Use of AFM for imaging and measurement of the mechanical properties of light-convertible organelles in plants

We obtained topographic images of etioplasts and chloroplasts and measured their elasticity in a physiological buffer using an atomic force microscope (AFM) and found a possible correlation between the morphological and mechanical properties during the light conversion of etioplasts to chloroplasts. Alcian blue 8GX dye was found to be effective for immobilizing the plant organelles stably on a glass surface for AFM experiments. We employed the tapping-mode AFM with a cantilever soft enough to measure the elasticity of the organelles in a liquid solution. The best images of soft, spherical organelles were obtained using the tapping-mode AFM with oscillation at the thermal vibration frequency of the cantilever of around 3 kHz. Whereas etioplasts were found to be smooth-surfaced and stiff against compression by the AFM tip, before light conversion to chloroplasts, they became rough-surfaced and mechanically soft after exposure to light. The elasticity of etioplasts was 20 times higher than that of chloroplasts, probably reflecting changes in their inner structures.     

Calibration of AFM cantilever spring constants

In this paper we present two simple, reliable and readily applicable methods for calibrating cantilevers and measuring the thickness of thin gold films. The spring constant calibration requires knowledge of the Young's modulus, density of the cantilever and resonant frequency. The thickness of thin gold layers was determined by measuring changes in the resonant frequency and Q-factor of beam shaped AFM cantilevers before and after coating.The techniques for measuring the spring constant and thin film thickness provide accuracy on the order of 10-15%.