Influence of the tip mass on the tip-sample interactions in TM-AFM

This paper focuses on the influences of the tip mass ratio (the ratio of the tip mass to the cantilever mass), on the excitation of higher oscillation eigenmodes and also on the tip-sample interaction forces in tapping mode atomic force microscopy (TM-AFM). A precise model for the cantilever dynamics capable of accurate simulations is essential for the investigation of the tip mass effects on the interaction forces. In the present work, the finite element method (FEM) is used for modeling the AFM cantilever to consider the oscillations of higher eigenmodes oscillations. In addition, molecular dynamics (MD) is used to calculate precise data for the tip-sample force as a function of tip vertical position with respect to the sample. The results demonstrate that in the presence of nonlinear tip-sample interaction forces, the tip mass ratio plays a significant role in the excitations of higher eigenmodes and also in the normal force applied on the surface. Furthermore, it has been shown that the difference between responses of the FEM and point-mass models in different system operational conditions is highly affected by the tip mass ratio.     

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_∞.     

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.     

Dynamic spring constants for higher flexural modes of cantilever plates with applications to atomic force microscopy

In atomic force microscopy (AFM) a sharp tip fixed close to the free end of a cantilever beam interacts with a surface. The interaction can be described by a point-mass model of an equivalent oscillator with a single spring located at the position of the tip. However, other spring constants have to be used to describe the oscillation behavior correctly if forces are acting on the cantilever over an extended lateral range. A point-mass model is then no longer valid. In the present study we derive expressions for the spring constants of cantilevers that can interact with any part of their plan view area along the beam and for all flexural modes. The equations describe the oscillation behavior in the corresponding mass model and are based on the eigenfrequencies and modal shapes of the free cantilever. The results are of high practical relevance, for example if an AFM is operated in a higher flexural mode, if the tip is not located at the free end of the cantilever beam, or if the external conservative forces affecting cantilever movement are not restricted to a single point. The limitations of the approach are discussed.     

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.     

Experimental and theoretical studies of frequency response function of dagger-shaped atomic force microscope cantilever in different immersion environments

In this study, the amplitude of frequency response functions of vertical and rotational displacements and resonant frequency of a dagger-shaped atomic force microscope cantilever have been investigated. To increase the accuracy of theoretical model, all necessary details for cantilever and sample surface have been taken into account. In this paper, carbon tetrachloride (CCL₄), methanol, acetone, water and air have been considered as the environments. In the most cases, presence and absence of tip–sample interaction force have studied. For a sample cantilever immersed in air, both of the Euler–Bernoulli and Timoshenko beam theories have been compared. The results indicate that the tip–sample interaction force raises the resonant frequency. Increasing the liquid viscosity leads to a decrease in the resonant frequency and the amplitude of frequency response functions of vertical and rotational displacements. Increasing the rectangular and tapered parts lengths, decreases the resonant frequency and amplitude of frequency response functions of vertical and rotational displacements. By increasing the cantilever thickness the resonant frequency and amplitude of frequency response functions of vertical and rotational displacements increases. Theoretical model for air and water has been compared with experimental work. Results show good agreement.     

Mechanically engraved mica surface using the atomic force microscope tip facilitates return to a specific sample location

By controlling the interaction between the atomic force microscope tip and mica, patterns of different sizes and shape have been produced on the surface of mica. Using these operator-constructed patterns as a reliable marker, the original scanned sample location can be re-located and imaged again on the same mica surface by atomic force microscopy (AFM). This location technique can be used to find the same object again even if the sample was removed from the AFM instrument or the sample was imaged in a different mode.     

Development of eddy current microscopy for high resolution electrical conductivity imaging using atomic force microscopy

We present a high resolution electrical conductivity imaging technique based on the principles of eddy current and atomic force microscopy (AFM). An electromagnetic coil is used to generate eddy currents in an electrically conducting material. The eddy currents generated in the conducting sample are detected and measured with a magnetic tip attached to a flexible cantilever of an AFM. The eddy current generation and its interaction with the magnetic tip cantilever are theoretically modeled using monopole approximation. The model is used to estimate the eddy current force between the magnetic tip and the electrically conducting sample. The theoretical model is also used to choose a magnetic tip-cantilever system with appropriate magnetic field and spring constant to facilitate the design of a high resolution electrical conductivity imaging system. The force between the tip and the sample due to eddy currents is measured as a function of the separation distance and compared to the model in a single crystal copper. Images of electrical conductivity variations in a polycrystalline dual phase titanium alloy (Ti-6Al-4V) sample are obtained by scanning the magnetic tip-cantilever held at a standoff distance from the sample surface. The contrast in the image is explained based on the electrical conductivity and eddy current force between the magnetic tip and the sample. The spatial resolution of the eddy current imaging system is determined by imaging carbon nanofibers in a polymer matrix. The advantages, limitations, and applications of the technique are discussed.     

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.     

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

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

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.     

Computational models of a nano probe tip for static behaviors

It is difficult to predict the measurement bias arising from the compliance of the atomic force microscope (AFM) probe. The issue becomes particularly important in this situation where nanometer uncertainties are sought for measurements with dimensional probes composed of flexible carbon nanotubes mounted on AFM cantilevers. We have developed a finite element model for simulating the mechanical behavior of AFM cantilevers with carbon nanotubes attached. Spring constants of both the nanotube and cantilever in two directions are calculated using the finite element method with known Young's moduli of both silicon and multiwall nanotube as input data. Compliance of the nanotube-attached AFM probe tip may be calculated from the set of spring constants. This paper presents static models that together provide a basis to estimate uncertainties in linewidth measurement using nanotubes. In particular, the interaction between a multiwall nanotube tip and a silicon sample is modeled using the Lennard-Jones theory. Snap-in and snap-out of the probe tip in a scanning mode are calculated by integrating the compliance of the probe and the sample-tip interacting force model. Cantilever and probe tip deflections and points of contact are derived for both horizontal scanning of a plateau and vertically scanning of a wall. The finite element method and the Lennard-Jones model provide a means to analyze the interaction of the probe and sample and measurement uncertainty, including actual deflection and the gap between the probe tip and the measured sample surface.     

Experimental observation of contact mode cantilever dynamics with nanosecond resolution

We report the use of a laser Doppler vibrometer to measure the motion of an atomic force microscope contact mode cantilever during continuous line scans of a mica surface. With a sufficiently high density of measurement points the dynamics of the entire cantilever beam, from the apex to the base, can be reconstructed. We demonstrate nanosecond resolution of both rectangular and triangular cantilevers. This technique permits visualization and quantitative measurements of both the normal and lateral tip sample interactions for the first and higher order eigenmodes. The ability to derive quantitative lateral force measurements is of interest to the field of microtribology/nanotribology while the comprehensive understanding of the cantilever's dynamics also aids new cantilever designs and simulations.     

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.     

Molecular dynamics simulation of the contact process in AFM surface observations

In the present work, several molecular dynamics simulations have been performed to clarify dynamically the contact mechanism between the specimen surface and probe tip in surface observations by an atomic force microscope (SFM) or friction force microscope (FFM). In the simulation, a three‐dimensional model is proposed where the specimen and the probe are assumed to consist of monocrystalline copper and rigid diamond or a carbon atom, respectively. The effect of the cantilever stiffness of the AFM/FFM is also taken into consideration. The surface observation process is simulated on a well‐defined Cu{100} surface. From the simulation results it has been verified that the surface images and the two‐dimensional atomic‐scale stick‐slip phenomenon, just as is the case for real AFM/FFM surface observations, can be detected from the spring force acting on the cantilever. From the evaluation of the behaviour of specimen surface atoms, the importance of the specimen stiffness in deciding the cantilever properties can also be understood. The influence of the probe tip shape on the force images is also evaluated. From the results it can be verified that the behaviour of the specimen surface atoms as well as the solid surface images in AFM/FFM surface observations can be understood using the molecular dynamics simulation of the model presented.     

Contact mechanics and tip shape in AFM-based nanomechanical measurements

Stiffness-load curves obtained in quantitative atomic force acoustic microscopy (AFAM) measurements depend on both the elastic properties of the sample and the geometry of the atomic force microscope (AFM) tip. The geometry of silicon AFM tips changes when used in contact mode, affecting measurement accuracy. To study the influence of tip geometry, we subjected ten AFM tips to the same series of AFAM measurements. Changes in tip shape were observed in the scanning electron microscope (SEM) between individual AFAM tests. Because all of the AFAM measurements were performed on the same sample, variations in AFAM stiffness-load curves were attributed to differences in tip geometry. Contact-mechanics models that assumed simple tip geometries were used to analyze the AFAM data, but the calculated values for tip dimensions did not agree with those provided by SEM images. Therefore, we used a power-law approach that allows for a nonspherical tip geometry. We found that after several AFAM measurements, the geometry of the tips at the very end is intermediate between those of a flat punch and a hemisphere. These results indicate that the nanoscale tip-sample contact cannot easily be described in terms of simple, ideal geometries.     

Nanotribology: tip–sample wear under adhesive contact

In this report, the irreversible variation of mass of the probe tip of an atomic force microscope (AFM) is considered from theoretical and numerical points of view through statistical methods. The tip–sample interaction due to the intermittent-contact operating mode of an AFM is modelled as a double-well potential where the wear mechanism, which reveals itself as mass sticking to the probe tip, is described as a transition between the two potential wells. We evaluate the interaction of a silicon nitride AFM/FFM tip with gold in order to compare the results with those obtained from previous experimental and numerical studies.     

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.     

Direct measurement of cantilever spring constants and correction for cantilever irregularities using an instrumented indenter

A method is presented that allows direct measurement of a wide range of spring constants of cantilevers using an indentation instrument with an integrated optical microscope. An uncertainty of less than 10% can be achieved for spring constants from 0.1 to 10(2) Nm. The technique makes it possible to measure the spring constant at any desired location on a cantilever of any shape, particularly at the tip location of an atomic force microscopy cantilever. The article also demonstrates a technique to detect and correct apparent length anomalies of cantilevers by analyzing spring constants at multiple positions.     

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.