Exact solution of the frequency shift in dynamic force microscopy

The exact frequency shift of an AFM non-uniform probe with an elastically restrained root, subjected to van der Waals force, is derived. The original distributed system is considered and then its exact fundamental solutions and the general frequency equation are derived. Results are compared with those by the force gradient method and the perturbation method. The effects of several parameters on the sensitivity of measurement are investigated. Results show that the interpretation of frequency shift by using the force gradient method is unsatisfactory. The smaller the amplitude of oscillation and the tip–surface distance are, the larger the frequency shift. The design of a taper beam is recommended for increasing the sensitivity of measurement.     

Study on topographic images of Sn/Si(1 1 1)-(√3 × √3)R30° surface by non-contact AFM

Various contrast of topographic images depending on a state of a tip apex on Sn/Si(1 1 1)-(√3 × √3)R30° surface was investigated using a low temperature non-contact AFM. With the type A tip, the image of the ring-type Sn, composed of six Sn atoms surrounding substitutional Si defect, was observed when the frequency shift (∣Δf∣) was small (the tip-sample distance, Z_tip-sample, was long), while the ring-type Sn was not observed and all the Sn atoms have the same contrast when ∣Δf∣ was large (Z_tip-sample was short). On the other hand, with the type B tip, modified from the type A tip by the tip-sample contact, the image of the ring-type Sn atoms was not observed regardless of variation of Δf. It is the first experimental result on the low temperature NC-AFM observation in the Sn/Si(1 1 1) system, which depends on short-range chemical bonding force or electrostatic force acting between the tip and the sample surface. In addition, the substitutional Si defects on the surface were seen as a dim spot or were not seen, also depending on the tip state.     

General theory of microscopic dynamical response in surface probe microscopy: from imaging to dissipation

We present a general theory of atomistic dynamical response in surface probe microscopy when two solid surfaces move with respect to each other in close proximity, when atomic instabilities are likely to occur. These instabilities result in a bistable potential energy surface, leading to temperature dependent atomic scale topography and damping (dissipation) images. The theory is illustrated on noncontact atomic force microscopy and enables us to calculate, on the same footing, both the frequency shift and the excitation signal amplitude for tip oscillations. We show, using atomistic simulations, how dissipation occurs through reversible jumps of a surface atom between the minima when a tip is close to the surface, resulting in dissipated energies of 1.6 eV. We also demonstrate that atomic instabilities lead to jumps in the frequency shift that are smoothed out with increasing temperature.     

Hard disk magnetic domain nano-spatial resolution imaging by using a near-field scanning microwave microscope with an AFM probe tip

A near-field scanning microwave microscope (NSMM) incorporating an atomic force microscope (AFM) probe tip was used for the direct imaging of magnetic domains of a hard disk under an external magnetic field. We directly imaged the magnetic domain changes by measuring the change of reflection coefficient S₁₁ of the NSMM at an operating frequency near 4.4 GHz. Comparison was made to the magnetic force microscope (MFM) image. Using the AFM probe tip coupled to the tuning fork distance control system enabled nano-spatial resolution. The NSMM incorporating an AFM tip offers a reliable means for quantitative measurement of magnetic domains with nano-scale resolution and high sensitivity.     

Analysis of the number of hydrogen bond groups of a multiwalled carbon nanotube probe tip for chemical force microscopy

In this paper, we describe a statistical method of quantification of the number of functional groups at the contact area of a probe tip for atomic force microscopy from the result of repetitive pull-off force measurements. We have investigated laboratory-made carbon nanotube (CNT) probe tips to apply them for chemical force microscopy because limited number of functional groups at the tip-end is expected. Using a CNT tip, we conducted repetitive pull-off force measurements against a self-assembled monolayer terminated with carboxyl group and analyzed them in terms of the number of hydrogen bond groups at the CNT tip. The elementary hydrogen bond rupture force quantum in n-decane medium was estimated to be 84.2 ± 0.5 pN in the present system. Thus it was revealed that only a couple of hydrogen bond groups of the CNT tip were participating in hydrogen bonding with the sample on an average in this experimental system.     

Theory for the effect of the tip–surface interaction potential on atomic resolution in forced vibration system of noncontact AFM

We present a perturbation theory which enables us to understand the physics of the cantilever-forced vibration in noncontact atomic-force microscopy (nc-AFM). Analytical expressions of the resonance curve and frequency shift are given. This theory is applied to the model system with a van der Waals tip–surface interaction potential. Based on this case study, it is elucidated how the resonance frequency shift is analytically described by an integral of the tip–surface interaction force. Then nc-AFM image of Si(111) 7×7 surface is calculated by the present theory. It is examined that this theory works as an algorithm for nc-AFM image simulator.     

Imaging of chemical reactivity and buckled dimers on Si(100)2×1 reconstructed surface with noncontact AFM

We have investigated the force interactions between the Si tip and the Si(100)2×1 reconstructed surface in the noncontact atomic-force microscopy (AFM) measurement. We observed two types of frequency shift curves without and with discontinuity, similar to the Si(111)7×7 surface. The image contrast changes drastically whether the frequency shift curve shows discontinuity or not. In the case of the frequency shift curves without discontinuity, the noncontact AFM images almost reflect the surface topography including dimers and adsorbates. In the case of the frequency shift curves with discontinuity, they reflect strongly the chemical reactivity of surface. Furthermore, in the case of the frequency shift curves without discontinuity, for the first time, the stabilize-buckling of dimers induced by a defect can be observed. This suggests that the force interactions during the noncontact AFM measurement hardly influence the surface dynamics.     

Rayleigh wave dispersion and attenuation on a statistically rough free surface of a hexagonal crystal

Dispersion and attenuation of Rayleigh surface acoustic waves on a statistically rough free surface of a Z-cut hexagonal crystal were analytically studied using a modified mean-field method within the perturbation theory. Numerical calculations were carried out in the frequency range accessible for the perturbation theory using expressions for the real and imaginary parts of the complex frequency shift of Rayleigh waves caused by a slight surface roughness. The Rayleigh wave dispersion and attenuation in the Z-cut hexagonal crystal were shown to coincide qualitatively with those in an isotropic medium, differing only quantitatively. In the long-wavelength limit λ?a, where a is the lateral roughness correlation length, explicit analytical expressions for the relative change in the phase velocity and the inverse damping depth of Rayleigh waves were derived and used in numerical calculations.     

How to measure energy dissipation in dynamic mode atomic force microscopy

When studying a mechanical system like an atomic force microscope (AFM) in dynamic mode it is intuitive and instructive to analyse the forces involved in tip–sample interaction. A different but complementary approach is based on analysing the energy that is dissipated when the tip periodically interacts with the sample surface. This method does not require solving the differential equation of motion for the oscillating cantilever, but is based entirely on the analysis of the energy flow in and out of the dynamic system. Therefore the problem of finding a realistic model to describe the tip–sample interaction in terms of non-linear force–distance dependencies and damping effects is omitted. Instead, it is possible to determine the energy dissipated by the tip–sample interaction directly by measuring such quantities as oscillation amplitude, frequency, phase shift and drive amplitude. The method proved to be important when interpreting phase data obtained in tapping mode, but is also applicable to a variety of scanning probe microscopes operating in different dynamic modes. Additional electronics were designed to allow a direct mapping of local energy dissipation while scanning a sample surface. By applying this technique to the cross-section of a polymer blend a material specific contrast was observed.     

Frequency shifts of cantilever in AFM

The frequency shift and frequency shift image of cantilever in AFM have been studied by numerical integration of the equation of motion of cantilever for silicon tip with rutile TiO₂(0 0 1) surface in UHV conditions and by the Hamaker summation method for the tip-surface interaction forces. The effects of the excitation frequency at the cantilever base and the equilibrium position of the tip on the frequency shift have been calculated and the results showed the same phenomena as those measured, e.g., the frequency shift increased dramatically or rapidly before the contact point and was then almost level off after the contact point. The effects of scanning speed and the initial closest distance of tip to the contact point have been calculated at different excitation frequencies at the cantilever base and the results showed that proper frequency shift image could be obtained either by noncontact mode at the excitation frequency slightly less than the resonance frequency of free cantilever, or by tapping mode at the excitation frequency a few times smaller than the resonance frequency of free cantilever.     

Non-perturbative approach to high-index-contrast variations in electromagnetic systems

We present a method that formally calculates exact frequency shifts of an electromagnetic field for arbitrary changes in the refractive index. The possible refractive index changes include both anisotropic changes and boundary shifts. Degenerate eigenmode frequencies pose no problems in the presented method. The approach relies on operator algebra to derive an equation for the frequency shifts, which eventually turn out in a simple and physically sound form. Numerically the equations are well-behaved, easy implementable, and can be solved very fast. Like in perturbation theory a reference system is first considered, which then subsequently is used to solve another related, but different system. For our method precision is only limited by the reference system basis functions and the error induced in frequency is of second order for first-order basis set error. As an example we apply our method to the problem of variations in the air-hole diameter in a photonic crystal fiber.     

Micromagnetic analysis of unusual, V-shaped domain transitions in MnAs nanowires

V-shaped domain transitions in α‐MnAs nanowires were investigated by micromagnetic simulations. These rather unusual domain patterns are commonly observed experimentally by surface-sensitive magnetic imaging techniques. It has been speculated that the accompanying inclined domain walls in MnAs are the result of either an exchange biasing effect between ferromagnetic α‐MnAs wires and antiferromagnetic β‐MnAs wires or possibly due to competing exchange mechanisms in MnAs. Here we present evidence that these domain features are in fact transitions between three-dimensional flux-closure domains of opposite chirality and can therefore rule out the involvement of an antiferromagnetic biasing effect or anisotropic exchange. The formation of the energetically unfavorable V-shaped domain transitions is discussed in the context of the magneto-structural phase transition of the sample.     

Atomic resolution in dynamic force microscopy across steps on Si(1 1 1)7×7

In this note we report the first observation of salient features of the Si(1 1 1)7×7 reconstructed surface across monatomic steps by dynamic atomic force microscopy (AFM) in ultrahigh vacuum (UHV). Simultaneous measurements of the resonance frequency shift Δf of the Si-cantilever and of the mean tunneling current ī _i from the cleaned Si tip indicate a restricted range for stable imaging with true atomic resolution. The corresponding characteristics vs. distance reveal why feedback control via Δf is problematic, whereas it is as successful as in conventional STM via ī _i .     

Study of Probe-Surface Interaction in Shear-Force Microscopy: Effects of Humidity and Lateral Spring Constant

The shear force between a glass probe and a mica surface has been investigated as a function of the relative humidity, H, and the lateral spring constant of the probe, K. It was found that the interaction length D_o decreases with increasing H and exhibits a sharp drop around H=40%. With increase in K from 5 to 40 N/m, D_o gradually increases, although this feature was absent when a probe with a softer tip-end was used. The latter result indicates that the shear force in an atmospheric condition is not a remote force but results from some contact between the tip and the surface. Our results that D_o is independent of the oscillating amplitude and that the resonance curve of the probe is almost symmetric except in close vicinity to the surface are not in accord with the force model proposed recently, i.e., the knocking mechanism. It is proposed that the probe can vibrate even if the probe touches the surface, and that the resonance frequency increases steeply as the contact tightens. Theoretical estimation of the contribution of noncontact forces is also described.This paper was originally presented at the seventh Meeting on Near Field Optics, which was held on July 1, 1998 at Nagoya University, Nagoya, organized by Research Group on Near Field Optics, the Optical Society of Japan, an affiliate of the Japan Society of Applied Physics.     

Lipid membrane: inelastic deformation of surface structure by an atomic force microscope

The stability of the 1,2-Dioleoyl-sn-Glycero-3-[phospho-rac-1-Glycerol-Na] liposome in the liquid crystalline state have been investigated using an atomic force microscope (AFM). We have observed the inelastic deformation of the sample surface. The AFM tip causes persistent deformation of the surface of the lipid membrane, in which some of the lipid molecules are eventually pushed or dragged by the AFM tip. The experiment shows how the surface structure of the lipid membrane can be created by the interaction between the AFM tip and lipid membrane. When the operating force exceeds 10^-8 N, it leads to large deformations of the surface. A square region of about 1×1μm² is created by the scanning probe on the surface. When the operating force is between 10^-11N and 10^-8N, it can image the topography of the surface of the lipid membrane. The stability of the sample is related to the concentration of the medium in which the sample is prepared.     

Image dipole approach and polarization effects in scanning near-field optical microscopy

A coupled-dipole approach is proposed in order to study the coupling between the probe tip and the rough sample in SNOM. In the present model both the optical probe tip and the sample protrusions are represented by polarizable dipole spheres. The induced polarization effects on the sample surface can be replaced by the image dipoles in the circumstance of quasi-static electromagnetic field approximation. Applying the radiation theory of the dipole, we have established a set of self-consistent equations to describe the field distribution at the sites of the probe tip and the sample protrusions. The results are completely the same as those obtained by means of the dyadic electromagnetic propagator formalism and also the derivation procedure is relatively simple. This method permits us to analyze the physical mechanisms of the interaction between the probe tip and the rough surface in SNOM intuitively. Based on this approach, we further discuss the influence of polarization of the incident light on the imaging quality. The calculating result shows that the shape and the contrast of the images of the sample are both sensitive to the field polarization, and the z-polarized mode is proved to give better resolution in SNOM.     

Theoretical simulation of Kelvin probe force microscopy for Si surfaces by taking account of chemical forces

A new method of theoretical simulation for Kelvin probe force microscopy (KPFM) imaging on semiconductor or metal samples is proposed. The method is based on a partitioned real space (PR) density functional based tight binding (DFTB) calculation of the electronic states to determine the multi-pole electro-static force, which is augmented with the chemical force obtained by a perturbation treatment of the orbital hybridization. With the PR-DFTB method, the change of the total energy is calculated together with the induced charge distribution in the tip and the sample by their approach under an applied bias voltage, and the KPFM images, namely the patterns of local contact potential difference (LCPD) distribution, are obtained with the minimum condition of the interaction force. However, since the interaction force is due to electro-static multi-poles, the spatial resolution of the KPFM images obtained by PR-DFTB is limited to the nano-scale range and an atom-scale resolution cannot be attained. By introducing an additional chemical force, i.e., the force due to the orbital hybridization, we succeeded in reproducing atom-scale resolution of KPFM images. Case studies are performed for clean and impurity embedded Si surfaces with Si tip models.     

Two-Dimensional Numerical Simulations of Photon Scanning Tunneling Microscopy: Fourier Modal Method and R-Matrix Algorithm

Numerical simulation of a two-dimensional probe–object system for photon scanning tunneling microscope is presented. The R-matrix propagation algorithm incorporated into the Fourier modal method was used to achieve an extended capability for modeling of a realistic system consisting of both a probe and a sample. The type of the mode guided through the dielectric probe and the coupling of the near-field to fundamental guiding mode in the probe are discussed. The influence of the probe parameters on the near-field images is investigated. Three different probe shapes were simulated in the constant height scanning mode. The transmitted flux intensity through the probe was found to be strongly dependent on the tip shape. The analysis shows a good agreement of the obtained results with the available theoretical works and confirming experimental results. The proposed numerical scheme can find applications for near-field probe characterization and provides an understanding of the degree of perturbation introduced by a probe tip in the experiment.     

采用DDS的近场扫描光学显微镜探针-样品的纳米距离检测

近场扫描光学显微术中, 近场距离的检测和控制是需要解决的核心技术之一. 本文研究了基于DDS驱动的压电传感器, 在一个压电陶瓷片上, 电极被分成相同的两部分, 分别用于振动驱动和振幅检测. 近场扫描的光纤探针固定于此压电陶瓷片上. 振动驱动信号采用DDS, 在样品的远场时, 可以通过频率扫描得到误差在0.006 Hz以内的压电陶瓷片谐振频率驱动信号, 而当光纤探针处于样品的近场距离之内时, 压电陶瓷片的谐振频率偏离驱动信号频率, 振幅明显减小, 从而检测出近场距离. 高精度振动驱动源DDS和高灵敏度压电传感器的采用提高了检测灵敏度和工作稳定性.     

Nanotribology of self-assembled monolayer with a probe tip investigated using molecular dynamics simulations

The nanotribology of an alkanethiol self-assembled monolayer (SAM) under tilt contact with a scanning probe tip is studied using molecular dynamics (MD) simulations. The tilt contact is described in terms of the tilt angle and the magnitude of the specimen–tip separation. The effects of tilt angle and magnitude of the specimen–tip separation on the normal force, friction force, friction coefficient, shear strength of the tip–SAM junction, and self-recovery characteristics are evaluated during the scanning probe tip process at a temperature of 300 K. The simulation results clearly show that the magnitudes and periods of the normal force and friction force increase with decreasing magnitude of the specimen–tip separation due to a large change of the tilt angle of the SAM chains during the deformation and recovery stages. For scanning and indentation processes, the effect of the tilt angle of the probe tip on the normal force is more significant than that on the friction force for the SAM. The behaviors of interfacial contact forces, friction coefficient, and shear strength strongly depend on the number of interacting atoms and the contact area, which increases with decreasing magnitude of the specimen–tip separation and increasing tilt angle of the probe tip. The self-recovery of SAM is significantly affected by the magnitude of the specimen–tip separation; the recovery ability of SAM is worse for magnitude of the specimen–tip separation below −0.9 nm with a large tilt angle of the probe tip.