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.     

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.     

High-resolution constant-height imaging with apertured silicon cantilever probes

We present high-resolution aperture probes based on non-contact silicon atomic force microscopy (AFM) cantilevers for simultaneous AFM and near-infrared scanning near-field optical microscopy (SNOM). For use in near-field optical microscopy, conventional AFM cantilevers are modified by covering their tip side with an opaque aluminium layer. To fabricate an aperture, this metal layer is opened at the end of the polyhedral probe using focused ion beams (FIB). Here we show that apertures of less than 50 nm can be obtained using this technique, which actually yield a resolution of about 50 nm, corresponding to λ/20 at the wavelength used. To exclude artefacts induced by distance control, we work in constant-height mode. Our attention is particularly focused on the distance dependence of resolution and to the influence of slight cantilever bending on the optical images when scanning at such low scan heights, where first small attractive forces exerted on the cantilever become detectable.     

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.     

Fabrication of ionic liquid thin film by nano-inkjet printing method using atomic force microscope cantilever tip

We demonstrate the fabrication of thin films of ionic liquid (IL), 1-butyl-3-methyl-imidazolium tetrafluoborate, by nano-inkjet printing method using an atomic force microscope (AFM) cantilever. The IL filled in a pyramidal hollow of the AFM cantilever tip was extracted from an aperture at the bottom of the hollow and deposited onto a Pt substrate when the bias voltage was applied between the cantilever and the substrate. We succeeded in fabricating IL thin films with a thickness of 4 nm. The areas and thicknesses of IL thin films were controlled by the fabrication conditions in this method, which is also useful for the investigations of nanometer-scale properties of ionic liquid.     

An interferometric platform for studying AFM probe deflection

This paper describes an interferometric platform for measuring the full-field deflection of atomic force microscope (AFM) probes and generic cantilevers during quasi-static loading. The platform consists of a scanning white light interferometer (SWLI), holders for the cantilevers, a translation stage, a rotation (tip-tilt) stage, and an adapter plate to connect these items to the SWLI table. Visualization of cantilever bending behavior is demonstrated for snap-in against a rigid surface, cantilever-on-cantilever tests, and a damaged AFM probe. A new approach to normal force calculation using a polynomial fit to the cantilever deflection profile is also presented and verified experimentally. The method requires only the coefficient for the third order (cubic) term from the fit to the deflection profile, the elastic modulus, and the area moment of inertia for the cantilever under test.     

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.     

Analysis of dynamic cantilever behavior in tapping mode atomic force microscopy

Tapping mode atomic force microscopy (AFM) provides phase images in addition to height and amplitude images. Although the behavior of tapping mode AFM has been investigated using mathematical modeling, comprehensive understanding of the behavior of tapping mode AFM still poses a significant challenge to the AFM community, involving issues such as the correct interpretation of the phase images. In this paper, the cantilever's dynamic behavior in tapping mode AFM is studied through a three dimensional finite element method. The cantilever's dynamic displacement responses are firstly obtained via simulation under different tip‐sample separations, and for different tip‐sample interaction forces, such as elastic force, adhesion force, viscosity force, and the van der Waals force, which correspond to the cantilever's action upon various different representative computer‐generated test samples. Simulated results show that the dynamic cantilever displacement response can be divided into three zones: a free vibration zone, a transition zone, and a contact vibration zone. Phase trajectory, phase shift, transition time, pseudo stable amplitude, and frequency changes are then analyzed from the dynamic displacement responses that are obtained. Finally, experiments are carried out on a real AFM system to support the findings of the simulations. Microsc. Res. Tech. 78:935–946, 2015. © 2015 Wiley Periodicals, Inc.     

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.     

Step response measurement of AFM cantilever for analysis of frequency-resolved viscoelasticity

Extension of AFM-based viscoelasticity measurement into a frequency-resolved analysis is attempted. A cantilever immersed into and interacting with distilled water was employed for the trial system. Using a home-built wideband magnetic excitation AFM, a step force with a transient time less than 1 μs is applied to the AFM cantilever and its deflection is measured. The 1st and 2nd mode resonance ringing of the cantilever was suppressed using quality-factor-control technique, so that the measurement system becomes equivalent to driving a resonance-free virtual cantilever within the bandwidth limited by the surviving 3rd mode resonance. From the obtained response of the cantilever deflection, a frequency-dependent complex compliance of the cantilever-water system was derived in a frequency range of 1–100 kHz. Effect of water confining between the tip and a mica substrate is discussed.     

Defining the parameters of a cantilever tip AFM by reference structure

A method of measurement and control of atomic force microscope (AFM) probe parameters is offered. The AFM real cantilever parameters are defined.     

Mapping of lateral vibration of the tip in atomic force microscopy at the torsional resonance of the cantilever

Lateral vibration of the tip in atomic force microscopy was mapped at the torsional resonance of the cantilever by attaching a shear piezo element at the base of the cantilever or under the sample. Fixed frequency excitation and self-excitation of torsional motion were implemented. The lateral vibration utilized as measured by an optical lever was in the order of 10 pm to 3 nm, and its frequency approximately 450 kHz for a contact-mode silicon nitride cantilever. The amplitude and phase of the torsional motion of the cantilever was measured by a lock-in-amplifier or a rectifier and plotted in x and y as the sample was raster scanned. The imaging technique gave contrast between graphite terraces, self-assembled monolayer domains, silicon and silicon dioxide, graphite and mica. Changing contrast was observed as silicon islands oxidized in atmosphere, showing that the imaging technique can detect change in lateral tip mobility due to changes occurring near the surface. Torsional self-excitation showed nanometric features of self-assembled monolayer islands due to different lateral dissipation. Dependence of torsional resonance frequency on excitation amplitude, and contrast change due to driving frequency around resonance were observed.     

Investigation of influence of gas atmosphere and pressure upon non-contact atomic force microscopy

Resonance measurements and atomic force microscopy (AFM) observations were carried out by the non-contact AFM operating in various gas atmospheres (hydrogen, helium, nitrogen and argon) over the range of pressures from 0.1 to 1.1 MPa. In each atmosphere, the resonance frequency of the AFM cantilever depended on the pressure of gases studied. The plots of the relative resonance frequency at a constant pressure vs. the gas density gave a straight line. It was found that the characteristic of the resonance frequency for the AFM cantilever were dependent on the density of the gas species. The resolution of the AFM was hardly influenced by the gas atmosphere under the present experimental conditions.     

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.     

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.     

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.     

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_∞.     

Atomic force microscopy imaging using a tip-on-chip: opening the door to integrated near field nanotools

We describe in detail how atomic force microscopy (AFM) images can be routinely achieved with macroscopic silicon-based chips integrating mesoscopic tips, paving the way for the development of new near field devices combining AFM imaging with any kind of functionality integrated on a chip. The chips have been glued at the end of the free prong of 100 kHz quartz tuning forks mounted in Qplus configuration. Numerical simulations by modal analysis have been carried out to clarify the nature of the vibration modes observed in the experimental spectra. It is shown that two low frequency modes can be used to drive the system and scan the surface with a great stability in amplitude modulation as well as in frequency modulation AFM under ultrahigh vacuum. The AFM capabilities are demonstrated through a series of examples including phase and dissipation contrast imaging, force spectroscopy measurements, and investigations of soft samples in weak interaction with the substrate. The lateral resolution with the tips grown by focused ion beam deposition already matches the one achieved in standard amplitude modulation mode AFM experiments.     

A New AFM Nanotribology Method Using a T-Shape Cantilever with an Off-Axis Tip for Friction Coefficient Measurement with Minimized Abbé Error

A new AFM (atomic force microscopy) nanotribology method using a T-shape cantilever with an off-axis tip (Nat Nanotechnol 2:507–514, 2007) has been developed for measuring friction coefficient at nanometer scale. In this method, signals due to both bending and twisting of the T-shape AFM cantilever are detected simultaneously. For a T-shape AFM cantilever, the bending is caused by the normal load and the twisting is caused by both the normal and the lateral loads. The twisting generated by the normal load is calibrated in advance. Consequently, the twisting only due to the lateral load can be decoupled from the total lateral voltage signal. And the friction coefficient can be finally determined based on a conversion relationship between the normal and lateral voltage signals of the AFM photodetector. A practical procedure for minimizing Abbé error in friction coefficient measurement has also been introduced. The proposed new method is simple and accurate, and requires the least operation for friction coefficient measurement at nanometer scale.