The dynamical properties of an oscillating tip-cantilever system are now widely used in the field of scanning force microscopy. The aim of the present work is to get analytical expressions describing the nonlinear dynamical properties of the oscillator in noncontact and intermittent contact situations in the tapping mode. Three situations are investigated: the pure attractive interaction, the pure repulsive interaction, and a mixing of the two. The analytical solutions obtained allow general trends to be extracted: the noncontact and the intermittent contact show a very discriminate variation of the phase. Therefore the measurement of the phase becomes a simple way to identify whether or not the tip touches the surface during the oscillating period. It is also found that the key parameter governing the structure of the dynamical properties is the product of the quality factor by a reduced stiffness. In the attractive regime, the reduced stiffness is the ratio of an attractive effective stiffness and the cantilever one. In the repulsive regime, the reduced stiffness is the ratio between the contact stiffness and the cantilever one. The quality factor plays an important role. For large values of the quality factor; it is predicted that a pure topography can be obtained whatever the value of the contact stiffness. For a smaller quality factor, the oscillator becomes more sensitive to change of the local mechanical properties. As a direct consequence, varying the quality factor, for example with a vacuum chamber, would be a very interesting way to investigate soft materials either to access topographic information or nanomechanical properties. ͓͔͑͒

An analytical model of the electrostatic force between the tip of a noncontact atomic force microscope ͑nc-AFM͒ and the ͑001͒ surface of an ionic crystal is reported. The model is able to account for the atomic contrast of the local contact potential difference ͑CPD͒ observed, while nc-AFM-based Kelvin probe force microscopy ͑KPFM͒ experiments. With the goal in mind to put in evidence this short-range electrostatic force, the Madelung potential arising at the surface of the ionic crystal is primarily derived. The expression of the force, which is deduced, can be split into two major contributions: the first stands for the coupling between the microscopic structure of the tip apex and the capacitor formed between the tip, the ionic crystal, and the counterelectrode and the second term depicts the influence of the Madelung surface potential on the mesoscopic part of the tip, independent of its microscopic structure. The former has the lateral periodicity of the Madelung surface potential, whereas the latter only acts as a static component, which shifts the total force. These short-range electrostatic forces are in the range of 10 pN. Beyond the dielectric properties of the crystal, a major effect, which is responsible for the atomic contrast of the KPFM signal, is the ionic polarization of the sample due to the influence of the tip/counterelectrode capacitor. When explicitly considering the crystal polarization, an analytical expression of the bias voltage to be applied on the tip to compensate for the local CPD, i.e., to cancel the short-range electrostatic force, is derived. The compensated CPD has the lateral periodicity of the Madelung surface potential. However, the strong dependence on the tip geometry, the applied modulation voltage, and the tip-sample distance, which can even lead to an overestimation of the real surface potential, makes quantitative KPFM measurements of the local CPD extremely difficult.

For the first time, ordered polar molecules confined in monolayer-deep rectangular pits produced on an alkali halide surface by electron irradiation have been resolved at room temperature by non-contact atomic force microscopy. Molecules self-assemble in a specific fashion inside pits of width smaller than 15 nm. By contrast no ordered aggregates of molecules are observed on flat terraces. Conclusions regarding nucleation and ordering mechanisms are drawn. Trapping in pits as small as 2 nm opens a route to address single molecules.

In this paper the dynamic behavior of an oscillating tip-microlever system at the proximity of a surface is discussed. We show that the nonlinear behavior of the oscillator is able to explain the high sensitivity of the oscillating tip microlever and 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. PUBLISHED IN Phys. Rev. Lett. 82(17), 3388-3391 (1999).PACS numbers: 07.79. Lh, 61.16.Ch, 68.35.Bs A few years ago, the resonant noncontact mode (hereafter noted NC-AFM) was developed to map tip surface interaction as shifts of the resonance frequency [1,2]. Since then, numerous experimental efforts have been performed showing that measurements of shifts were able to produce images at the atomic scale [3][4][5][6][7]. The "routine" achievement of the atomic resolution with the NC-AFM [7] is thought of as a real breakthrough in the field of the scanning probe microscopy. Moreover, the use of a high vibrating amplitude of the cantilever (CL) with the NC-AFM and the development of the intermittent contact (the so-called tapping mode) [8][9][10] have boosted theoretical works dedicated to a nonlinear analysis [11][12][13][14].Before going a step further, it is worth discussing general ideas at a qualitative level. Leaving aside the technical point that the use of a high amplitude reduces the influence of the fluctuations in frequency [2], a first, counterintuitive, result is the use of a large vibrating amplitude allowing the atomic resolution to be achieved. Typically, the amplitudes used are 100 times greater than the vertical motion of the surface required, maintaining a given value of the frequency shift. The most common and shared idea is that to probe an attractive field with an oscillator requires the use of a small amplitude in order to keep the force field nearly constant throughout the vibrating amplitude. The idea is that shifts in resonance frequency are uniquely due to the gradient force variations.An image with the atomic resolution based on a repulsive interaction appears quite easily understandable, but getting the same resolution when the attractive van der Waals forces are involved is very puzzling. The van der Waals atom-atom interaction and the finite size of the interacting objects lead to a tip-sample interaction smoothly varying as a function of the CL-surface distance. For instance, the sphere-plan interaction leads to an attractive force with a d 22 power law. The force gradient dependence, with a power law d 23 , gives a too smooth variation of the resonance frequency and is unable to predict the observed shape of the frequency changes as a function of the tip-sample distance.

The adsorption of two kinds of porphyrin (Cu-TBPP) and perylene (PTCDA) derived organic molecules deposited on KBr and Al 2 O 3 surfaces has been studied by non-contact force microscopy in ultra-high vacuum, our goal being the assembly of ordered molecular arrangements on insulating surfaces at room temperature. On a Cu(100) surface, well ordered islands of Cu-TBPP molecules were successfully imaged. On KBr and Al 2 O 3 surfaces, it was found that the same molecules aggregate in small clusters at step edges, rather than forming ordered monolayers. First measurements with PTCDA on KBr show that nanometre-scale rectangular pits in the surface can act as traps to confine small molecular assemblies.