The real-time and accurate measurement of tip-sample interaction forces in Tapping Mode Atomic Force Microscopy (TM-AFM) is a remaining challenge. This obstruction fundamentally stems from the causality of the physical systems. Since i) the input of the dynamic systems propagates to the output with some delay, and ii) , multiple different inputs can generate the same output, there exist no measurement or estimation technique that can estimate the force input of the systems in real-time without phase and amplitude distortion. However, an approximate and delayed estimation can still be possible. This article presents a general-purpose algorithm which aims to estimate an approximation of the force input of TM-AFM with minimum delay and error. For this reason, first, the input estimation problem is converted to an ill-posed state observation problem. Then, a Tikhonov-like regularization technique is applied to eliminate the ill-conditioning and estimate the force input using a linear Kalman filter. The proposed input observer is remarkably robust, real-time in the order of the sampling frequency, and applicable for any Linear Time Invariant (LTI) system with a (semi-) periodic process. Simulation and experimental results show that using the proposed algorithm with a wide-band AFM probe; one can determine the tip-sample forces with only a few percent error and a delay in the order of sampling time. Unlike the existing force estimation techniques for AFM, this algorithm does not require any prior knowledge of the force-distance relationship which can be very beneficial for the closed-loop control of AFM.@en

The technique of Atomic Force Microscopy (AFM) is one of the major inventions of the twentieth century which substantially contributed to our understanding of the nanoscale world. In contrast to other microscopy techniques, the AFM does not operate based on the electromagnetic waves, but nano-mechanical interactions between the sample surface and a sharp probe. Therefore, its resolution is not fundamentally limited to the diffraction limit of light, but the sharpness of the probe tip which can be as small as a few atoms. The images and data obtained by AFM have had crucial importance for the scientists in the fields of biology, material science, and experimental physics. However, AFM experiments have always involved some challenges. Particularly, the limited imaging speed, and the probability of damaging the samples hinder scientists from extracting the necessary information on the samples. Besides its applications as a research tool, the AFM could potentially solve some of the challenges in semiconductor industry as a metrology and inspection tool, however, the aforementioned limitations are even more restrictive for any industrial use. Therefore, it is imperative to develop apparatus and methods which can increase the speed and reliability of AFM. In this thesis, we try to understand the physics of AFM and contribute to its development towards a potential industrial and clinical tool, from the perspective of dynamics and control of its cantilever. @en

Many investigations have focused on steady-state nonlinear dynamics of cantilevers in tapping mode atomic force microscopy (TM-AFM). However, a transient dynamic model—which is essential for a model-based control design—is still missing. In this paper, we derive a mathematical model which covers both the transient and steady-state behavior. The steady-state response of the proposed model has been validated with existing theories. Its transient response, however, which is not covered with existing theories, has been successfully verified with experiments. Besides enabling model-based control design for TM-AFM, this model can explain the high-end aspects of AFM such as speed limitation, image quality, and eventual chaotic behavior.

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The origin of amplitude reduction in Tapping Mode Atomic Force Microscopy (TM-AFM) is typically attributed to the shift in resonance frequency of the cantilever due to the nonlinear tip-sample interactions. In this paper, we present a different insight into the same problem which, besides explaining the amplitude reduction mechanism, provides a simple reasoning for the relationship between tip-sample interactions and operation parameters (amplitude and frequency). The proposed formulation, which attributes the amplitude reduction to an interference between the tip-sample and dither force, only deals with the linear part of the system; however, it fully agrees with experimental results and numerical solutions of the full nonlinear model of TM-AFM.

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Due to the harmonic motion of the cantilever in Tapping Mode Atomic Force Microscopy, it is seemingly impossible to estimate the tip-sample interactions from the motion of the cantilever. Not directly observing the interaction force, it is possible to damage the surface or the tip by applying an excessive mechanical load. The tip-sample interactions scale with the effective stiffness of the
probe. Thus, the reduction of the mechanical load is usually limited by the manufacturability of low stiffness probes. However, the one-to-one relationship between spring constant and applied force only holds when higher modes of the cantilever are not excited. In this paper, it is shown that, by passively tuning higher modes of the cantilever, it is possible to reduce the peak repulsive force.
These tuned probes can be dynamically more compliant than conventional probes with the same static spring constant. Both theoretical and experimental results show that a proper tuning of dynamic modes of cantilevers reduces the contact load and increases the sensitivity considerably.
Moreover, due to the contribution of higher modes, the tuned cantilevers provide more information on the tip-sample interaction. This extra information from the higher harmonics can be used for mapping and possibly identification of material properties of samples.@en

This paper investigates the closed-loop dynamics of the Tapping Mode Atomic Force Microscopy using a new mathematical model based on the averaging method in Cartesian coordinates. Experimental and numerical observations show that the emergence of chaos in conventional tapping mode AFM strictly limits the imaging speed. We show that, if the controller of AFM is tuned to be faster than a certain threshold, the closed-loop system exhibits a chaotic behavior. The presence of chaos in the closed-loop dynamics is confirmed via bifurcation diagrams, Poincaré sections, and Lyapunov exponents. Unlike the previously detected chaos due to attractive forces in the AFM, which can be circumvented via simple changes in operation parameters, this newly identified chaos is seemingly inevitable and imposes an upper limit for the closed-loop bandwidth of the AFM.

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The maximum amount of repulsive force applied to the surface plays a very important role in damage of tip or sample in Atomic Force Microscopy(AFM). So far, many investigations have focused on peak repulsive forces in tapping mode AFM in steady state conditions. However, it is known that AFM could be more damaging in transient conditions. In high-speed scanning, and in presence of 3D nano structures (such as FinFET), the changes in topography appear in time intervals shorter than the response time of the cantilever. In this case, the tip may crush into the sample by exerting much higher forces than for the same cantilever-sample distance in steady state situations. In this study the effects of steep upward steps in topography on the tip-sample interactions have been investigated, and it has been found that the order(s) of magnitude higher forces can be applied. The information on the worst case scenario obtained by this method can be used for selection of operation parameters and probe design to minimize damage in high-speed imaging. The numerically obtained results have been verified with the previous works in steady state regime. Based on this investigation the maximum safe scanning speed has been obtained for a case study@en