Tip-Sample Forces in Atomic Force Microscopy: Interplay between Theory and Experiment
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Tip-Sample Forces in Atomic Force Microscopy: Interplay between Theory and Experiment Sergey Belikov, John Alexander, Craig Wall, and Sergei Magonov NT-MDT Development Inc., 430 W. Warner Road, Tempe, AZ 85284, U.S.A. ABSTRACT Several examples of Atomic Force Microscopy imaging in the oscillatory resonant and non-resonant modes are analyzed with a theoretical description of tip-sample force interactions. The problems of high-resolution imaging and compositional mapping of heterogeneous polymers are considered. The interplay with theory helps the experiment optimization and rational understanding of the image contrast. INTRODUCTION Over the past 20 years Atomic Force Microscopy (AFM) has evolved from a highresolution and low-force imaging tool to a family of comprehensive techniques also covering local measurements of different sample properties. The theoretical understanding of tip-sample interactions, however, lags far behind practical experience and instrumental developments in AFM. This situation needs to be changed and several aspects of interplay between the theory and experiment are outlined below. There are several DC and AC AFM modes, in which the probe deflection or changes of its oscillation parameters are utilized for instrument control and data harvesting. A rational understanding of tip-sample forces in these modes can be elucidated by solving the Euler-Bernoulli equations that describe the oscillating probe interacting with a sample. The Krylov-Bogoliubov-Mitropolsky asymptotic approach based on separation of the fast and slow changing variables led to the solution expressed by two equations containing four unknowns (central probe position - Zc, imaging amplitude, frequency, phase - θ ). By fixing two of the unknowns, one can realize the amplitude modulation (AM), frequency modulation (FM) and other modes [1]. In AM these equations are as follows: 1 sin θ = N
cosθ = −
π
∫ [Fa − Fr ](Z c + Asp cos y )sin ydy + 0
1 N
π
∫ [F
a
+ Fr ](Z c + Asp cos y )cos ydy
Asp A0
(1) (2)
0
Where A0 – the free probe amplitude, Asp – the set-point amplitude; Zc – a central position of the cantilever over a surface during oscillations, and N = πA0 k Q ( k and Q – spring constant and quality factor of the probe at 1st flexural resonance), Fa and Fr – the tip-sample forces experienced by a probe in the approach and retract to the surface parts of the oscillatory cycles. The integrals in (1) - Isin and in (2) - Icos can be connoted as the dissipative and conservation integrals, respectively. In the case of conservative forces the dissipative integral is nullified and the first equation is simplified. Knowledge of a relationship between the force and specific surface properties, is essential for their quantitative extraction from AFM data. Besides the classification of the AFM oscillatory modes, this theoretical method was also
implemented in an AFM simulator, which has been beneficial in generating atomic-scale images with single defects, for tips of different sizes, and various force levels [2]. The general nat
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