Relationship between the non linear dynamic behaviour of an oscillating tip–microlever system and the contrast at the atomic scale

In this paper, the dynamic behaviour of an oscillating tip–microlever system at the proximity of a surface is discussed. The attractive tip–surface interaction is simply described with a Van der Waals dispersive term and a sphere–plane geometry. We show that the non linear behaviour of the oscillator is able to explain the observed shifts of the resonance frequency as a function of the tip–surface distance without the need of introducing a particular short range force.     

Contrast mechanism in non-contact SFM imaging of ionic surfaces

We analyse the mechanisms of contrast formation in NC-SFM imaging of ionic surfaces and calculate constant frequency shift scanlines of the perfect surfaces of NaCl and MgO. Non-contact SFM operation is modelled in a perturbed oscillator model using an atomistic simulation technique for force–field calculations. We demonstrate that the contrast in NC-SFM imaging of ionic surfaces is based on the interplay between the van der Waals interaction and the electrostatic interaction of the tip with the surface potential and the local surface polarisation induced by the tip. The results emphasise the importance of the tip-induced relaxation of the surface ions in the tip–surface interaction and image contrast.     

Simultaneous imaging of tunneling current and damping energy by noncontact-AFM in ultra-high vacuum

We have simultaneously observed the images of tunneling current and damping energy with the nc-AFM (noncontact atomic force microscopy) image of Si(111)7ǻ. When inverted contrast is observed in the constant frequency shift nc-AFM image, the current image is not inverted, and vice versa. On the other hand, the damping images show a contrast similar to that of the nc-AFM images; the damping decreases at a narrow separation between the tip and the sample. This possibly indicates that the damping decreases as the attractive interaction increases under a constant oscillation amplitude of the cantilever. To keep the oscillation amplitude constant under an attractive interaction between a tip and a sample, the total energy of the cantilever oscillation is reduced and the oscillation velocity of the cantilever decreases. An explanation is proposed that the change in energy dissipation occurs in the cantilever oscillation, depending on the oscillation velocity of the cantilever, and the value is estimated from a simple model.     

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.     

Calculation of the frequency shift in dynamic force microscopy

A theoretical study of the quality and the range of validity of different numerical and analytical methods to calculate the frequency shift in dynamic force microscopy is presented. By comparison with exact results obtained by the numerical solution of the equation of motion, it is demonstrated that the commonly used interpretation of the frequency shift as a measure for the force gradient of the tip–sample interaction force is only valid for very small oscillation amplitudes and leads to misinterpretations in most practical cases. Perturbation theory, however, allows the derivation of useful analytic approximations.     

A simplified but intuitive analytical model for intermittent-contact-mode force microscopy based on Hertzian mechanics

The forces acting on the substrate in intermittent-contact-mode (IC mode, tapping mode) atomic force microscopy are not accessible to a direct measurement. For an estimation of these forces, a simple analytical model is developed by considering only the shift of the cantilever resonance frequency caused by Hertzian (contact) forces. Based on the relationship between frequency shift and tip–sample force for large-amplitude frequency-modulation atomic force microscopy, amplitude and phase versus distance curves are calculated for the intermittent contact mode, and the forces on the substrate are calculated. The results show a qualitative agreement with numerical calculations, yielding typical maximal forces of 50–150 nN. When working above the unperturbed resonance, forces are found to be significantly larger than below the resonance.     

Imitation of non-contact mode while scanning in the presence of an electric double layer?

Non-contact atomic force microscopy (NCAFM) minimizes the physical interaction between the AFM tip and the surface of interest. Several recent studies have reported observation of single atom defects using this technique. The repulsive force is presumably the primary interatomic force (cf. our paper on pseudo-non-contact mode in this issue) responsible for the reported atomic resolution in these studies. The combination of these factors, minimal tip–sample deformation and repulsive force interaction, are responsible for the observation of the single atom defects. In the present study, we show that similar resolution can be achieved utilizing the same two factors but which employs scanning in a surfactant. The method decreases the tip–sample interaction by eliminating the attractive forces between the tip and sample. The surfactant solution induces an electrical double-layer (EDL) on the surface of the tip and sample. This EDL creates additional repulsion that is distributed over a large area, and hence does not contribute noticeably to the image contrast during scanning. However, it does compensate for the high pressures normally experienced by the tip in the absence of surfactant. In addition, the presence of the EDL enhances tip stability during the image scan. This method has been tested on surfaces of such minerals as mica, chlorite, and anhydrite.     

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.     

High-resolution imaging of C60 molecules using tuning-fork-based non-contact atomic force microscopy

Recent advances in non-contact atomic force microscopy (nc-AFM) have led to the possibility of achieving unprecedented resolution within molecular structures, accomplished by probing short-range repulsive interaction forces. Here we investigate C(60) molecules adsorbed on KBr(111) and Cu(111) by tuning-fork-based nc-AFM. First, measurements of C(60) deposited on KBr(001) were conducted in cryogenic conditions revealing highly resolved nc-AFM images of the self-assembly. Using constant-frequency shift mode as well as three-dimensional spectroscopic measurements, we observe that the relatively weak molecule-substrate interaction generally leads to the disruption of molecular assembled structures when the tip is probing the short-range force regime. This particular issue hindered us in resolving the chemical structure of this molecule on the KBr surface. To obtain a better anchoring of C(60) molecules, nc-AFM measurements were performed on Cu(111). Sub-molecular resolutions within the molecules was achieved which allowed a direct and unambiguous visualization of their orientations on the supporting substrate. Furthermore, three-dimensional spectroscopic measurements of simultaneous force and current have been performed above the single molecules giving information of the C(60) molecular orientation as well as its local conductivity. We further discuss the different imaging modes in nc-AFM such as constant-frequency shift nc-AFM, constant-height nc-AFM and constant-current nc-AFM as well as three-dimensional spectroscopic measurement (3D-DFS) employed to achieve such resolution at the sub-molecular scale.     

The influence of tip–sample interaction on step fluctuations on Ag(111)

The fluctuations of the position of monatomic steps on Ag(111) are investigated by scanning tunneling microscopy (STM). We analyze the influence of tip–sample interaction by varying the gap impedance over more than two orders of magnitude. For tunneling tips providing a weak tip–sample interaction, we show that the step position autocorrelation function remains essentially unaltered. In this unperturbed case, the kinetics of step fluctuations are found to be dominated by one-dimensional mass transport. For larger variations of the tip–sample distance or for less favorable tip configurations, we observe a tip-induced increase of the step fluctuations. Our measurements suggest that this effect is caused rather by short-range forces than by the electric field in the tunneling gap.     

Theoretical evaluation of the frequency shift and dissipated power in noncontact atomic force microscopy

An exact analytical formula of the frequency shift and dissipated power in noncontact atomic force microscopy (nc-AFM) is obtained, in the case of a tip-surface interaction force with an inverse power law. The frequency shift for various tip shapes is evaluated. It is confirmed that the tip-height dependence of the analytical frequency shift shows very good agreement with that of the experimental one. The tip-height dependence of the dissipated power derived from a Langevin equation approach is also evaluated.     

Optimizing phase imaging via dynamic force curves

Tapping mode (TM, also called intermittent contact mode) atomic force microscopy (AFM) has been routinely used in many laboratories. However, consistent or deliberate control of measuring conditions and interpretation of results are often difficult. In this article, we demonstrate how measurement parameters (drive frequency, cantilever stiffness and oscillation amplitude) affect the tapping tip's state. This has been done by systematic dynamic force measurements performed on mica and polystyrene surfaces together with computer simulations. Our study shows the following results. (1) Weaker cantilevers, smaller amplitude and higher drive frequency (around the resonance) lead to an extension of the attractive region (greater phase lag) in amplitude–phase–distance curves and thus can help to achieve stable high-setpoint TM imaging with minimal tip–sample pressure. (2) Bistability of tapping tips often exists and may cause height artefacts if the setpoint falls in the bistable region. (3) Tapping tips with high vibrating energy (stiff cantilevers and large amplitude) driven at resonance are only slightly perturbed by tip–sample interactions and usually remain monostable during the sweep of the scanner position. This can help to achieve good phase contrast without significant artefacts when the setpoint falls in a continuous negative–positive phase shift transition region. (4) Low energy cantilevers (compliant cantilevers and small amplitude) usually result in large phase shift and can be used to acquire large phase contrast images. However, height artefacts will occur when the setpoint falls in the bistable region usually existing for such cantilevers. (5) Computer simulations are useful in understanding the bistability in dynamic force curves and determining either material properties or the optimal imaging parameters.     

Non-contact AFM images of a C60 molecule adsorbed on the Si(0 0 1) surface: An ab initio method

A theoretical non-contact atomic force microscope (nc-AFM) image is simulated for an individual C₆₀ molecule adsorbed on the clean Si(0 0 1) surface. The ability to identify the intramolecular features of the molecule though nc-AFM imaging would allow many of the different configurations the molecule may take when adsorbed, to be identified. This would be an important preliminary step in the manipulation of the molecule along the surface, as the initial configuration that the molecule is in will affect the periodicity of the tip trajectory during the manipulation event (see D.L. Keeling et al., Phys. Rev. Lett. 94 (2005) #146104). Presented in this paper are simulated images and force curves above important sites for key stable configurations of the C₆₀ molecule. It is shown that in principle it is possible to distinguish between the different configurations of the molecule using nc-AFM. An ab initio DFT method was used to accurately describe the chemical interaction between the adsorbate and the Si tip, which has a single dangling bond at its apex.     

Simultaneous imaging of the In and As sublattice on InAs(110)-(1×1) with dynamic scanning force microscopy

Distance-dependent dynamic scanning force microscopy (SFM) measurements of InAs(110)-(1×1) acquired in ultrahigh vacuum at low temperatures are presented. On this surface, the atoms of the As sublattice are lifted by 80 pm with respect to the In sublattice and terminate the surface. Thus, since in most dynamic SFM images only protrusions with the periodicity of one sublattice are observed, these protrusions are correlated with the positions of the As atoms. However, under certain conditions, an additional contrast is visible which can be attributed to an interaction between the foremost tip atoms and the In atoms. Possible contrast mechanisms are discussed in terms of tip–sample distance and tip structure.     

STM/nc-AFM investigation of (n×6) reconstructed GaAs(0 0 1) surface

We have investigated the n×6 reconstructed GaAs(0 0 1) surface with scanning tunneling and non-contact atomic force microscopy techniques (STM/nc-AFM). For the first time atomically resolved nc-AFM images of that surface are shown. The images confirm the presence of rows of arsenic dimers in the topmost layer as predicted by the current model of n×6 reconstructed surface. However, in contrast to previous reports we found that postulated As dimer sites are not fully occupied. Moreover, the images suggest that ×6 symmetry is present on the surface even in absence of the dimers. We show that due to probing of different surface properties nc-AFM and STM are complementary tools for complex surfaces investigation.     

Calculation of the optimal imaging parameters for frequency modulation atomic force microscopy

True atomic resolution of conductors and insulators is now routinely obtained in vacuum by frequency modulation atomic force microscopy. So far, the imaging parameters (i.e., eigenfrequency, stiffness and oscillation amplitude of the cantilever, frequency shift) which result in optimal spatial resolution for a given cantilever and sample have been found empirically. Here, we calculate the optimal set of parameters from first principles as a function of the tip–sample system. The result shows that the either the acquisition rate or the signal-to-noise ratio could be increased by up to two orders of magnitude by using stiffer cantilevers and smaller amplitudes than are in use today.     

Theory of electronic light scattering in lateral superlattices with Rashba spin–orbit coupling under quantizing electric and magnetic fields

The influence of an in-plane electric and out-of-plane magnetic field on the electronic light scattering is calculated for a lateral semiconductor superlattice within Rashba spin–orbit interaction. Sharp resonances are predicted to appear when the Raman shift matches one frequency of the Wannier–Stark ladder. The spin–orbit interaction gives rise to a dispersion of the exact one-particle eigenstates and an associated finite width of the Raman line, which can be tuned by the electric and magnetic field. When the Bloch frequency is located in this Raman line, a Fano resonance is 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.     

Determination of tip–sample interaction forces from measured dynamic force spectroscopy curves

The forces between a sharp tip and a sample are characteristic for different sample materials. A new method for quantifying the elastic tip–sample interaction forces from measured frequency vs. distance curves is presented. The dynamic force–spectroscopy curves investigated were obtained by dynamic force microscopy under ultrahigh vacuum (UHV) conditions for large vibration amplitudes with commercial levers/tips. The full non-linear force–distance relationship is deduced via a numerical algorithm, where the equation of motion describing the oscillation of the tip is solved explicitly. The elastic force distance dependence can be determined by fitting the results of a computer simulation to experimental frequency vs. distance data. The obtained force–distance curves can be compared quantitatively with theoretical models.     

Influence of the sample-probe coupling on the resolution of transmitted near-field optical image

Numerical simulation of photon scanning tunneling microscopy is presented to study the near-field distribution in the vicinity a dielectric surface with one-dimensional sub-wavelength structures. Multiple scattering between the probe tip and the sample has been taken into account implicitly by matching electromagnetic boundary conditions at interfaces. The near-field intensity in transmission mode through two ridges on surface has been modeled in order to analyze the resolution of the system. The effects on the signal by the sample-tip coupling, the polarization of the incident light, and the angle of incidence are investigated. We find that the capability to recognize the feature will be improved when the tip–object interaction is strong.