Determination of tip–sample interaction forces from measured dynamic force spectroscopy curves
Introduction
Atomic force microscopy (AFM) is a powerful tool for the investigation of morphologies and mechanical properties of surfaces. The major achievement of AFM lies in its ability to measure tiny forces acting between a sharp tip and a sample. Even atomic and molecular resolution can now be obtained using dynamic modes of operation 1, 2, 3, 4, 5, 6. Although dynamic scanning force microscopy is very sensitive to tip–sample interaction forces, it is not straightforward to relate the forces quantitatively to the measured data (i.e., frequency shifts, vibration amplitudes, etc.) Often, the detection mechanisms do not even allow a straightforward distinction between regimes where either only attractive forces between tip and sample are probed or where repulsive forces are involved. The question has arisen, which of the two really lead to high resolution imaging [7].
A quantitative analysis of the forces and how they are detected in different modes of operation seems desirable. Apart from characterising the imaging process, this could lead to an understanding of material properties down to nanometer scale. Ideally, the complete force characteristics should be deduced from measured curves of the operational parameters.
The forces acting between tip and sample are often measured indirectly through the frequency Δf shift of a cantilever vibrating with large amplitudes. The presented analysis is based on this technique, often referred to as frequency modulation (FM) technique [8].
Using large amplitudes, an exact analytical description is not possible as the tip moves through a large part of the non-linear potential in each cycle. This problem can be solved by applying small vibration amplitudes in a more complicated experimental procedure [9]. However, with the help of either analytical approximations [10] or numerical methods 11, 12 Δf(d)-curves can be related to force–distance dependencies. The further development of the simulation in Refs. 11, 12 allows the calculation of force–distance dependencies directly from measured Δf(d)-curves. The force–distance dependencies acquired in this way include attractive and repulsive forces and the transition between the two. This will allow a quantitative comparison with theoretical models which are widely used for theoretical approaches to scanning probe techniques [13]. Alternatively, for given models, sample properties such as surface elasticity and adhesion could be determined quantitatively.
Section snippets
Experimental
In the presented experiments, we used a commercial ultrahigh vacuum (UHV) AFM involving the FM-technique (UHV AFM/STM, Omicron Instruments). Here, the cantilever determines the resonance frequency within a self-excited feedback circuit. The movement of the tip results in an AC-signal of the position-sensitive detector. This signal is phase-shifted, amplified and used to excite the shaker-piezo. The frequency of the vibrating cantilever adjusts in response to the tip–sample interaction forces f=f
Calculations
Our analysis is based on two basic assumptions: Firstly, the movement of the tip is one-dimensional, and can be assumed to be harmonic for the free cantilever.1 Secondly, it is assumed that elastic forces influence the oscillation
Results and discussion
The reconstruction method is demonstrated for the case of the experimental Δf(d)-curve given in Fig. 2a for a silicon tip and a silicon sample (both passivated). Many Δf(d)-curves have been recorded in order to check for reproducibility. Note that only relative displacements d′ and not an absolute value for d is known. The force curve F(z′) shown in Fig. 2b was calculated after smoothing the Δf(d)-curve. The d-axis and the z′-axis in (a) and (b) have been plotted in such a way that (b) shows
Conclusion
We have presented a method which could lead to a quantitative understanding of dynamic force spectroscopy curves. The forces can be deduced for a given setpoint (Δf) and the imaging process can be characterised, especially the transition from purely attractive to partially repulsive interaction regimes.
The reconstructed force curves exhibit a significant dependence on different sample materials. The method could be used for a quantitative local analysis of material properties.
The complete force
Acknowledgements
We would like to thank Daniel Krüger for many enjoyable and fruitful discussions.
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