Abstract
In dynamic atomic force microscopy the cantilever is excited at a driving frequency which is close to the resonance frequency of the free cantilever. Due to the interaction between tip and the surface, the resonance frequency of the cantilever changes. As shown in this chapter, an attractive force between tip and sample leads to a lower resonance frequency of the cantilever, while for repulsive tip-sample forces the resonance frequency increases.
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Notes
- 1.
Actually, this is not strictly true: As shown later it is not the sign of the force, but rather the sign of the force gradient that determines the direction of the resonance frequency shift.
- 2.
The tip length is set to zero in order to avoid an additional offset length.
- 3.
Since the two springs attach to the tip from above and below one might think that this should lead to a subtraction of the spring constants. Here we show that the spring constants indeed add up. As indicated in Fig. 13.2 the cantilever spring under the influence of a tip-sample force can be approximated by a cantilever effective mass held by two springs (i.e. the cantilever spring k and the spring \(k'\) representing the tip-sample interaction). In static equilibrium, \(z=0\), the forces of both springs compensate as \(F_{k} + F_{k'} = 0\). If the cantilever is moved by \(\Delta z\) during the oscillation, Fig. 13.2b shows that the force components relative to the forces in static equilibrium point in the same direction for both springs and \(\Delta F = \Delta F_k + \Delta F_{k'} = - (k + k') \Delta z\) results. Thus, the spring constants k and \(k'\) combine to \(k_\mathrm {eff} = k + k'\).
- 4.
In the spring model the force \(F_\mathrm {ts} \left( d\right) \) can be considered arising form an offset stretch of the tip-sample spring.
- 5.
Technically the driving signal can be considered as a carrier signal which is modulated by a low-frequency (quasi-DC) amplitude signal (deviations from the desired amplitude setpoint due to the sample topography). Then the task of the lock-in amplifier is the demodulation of the low frequency amplitude signal. The term AM demodulation is traditionally used in connection with the audio signal detection/demodulation in AM radio receivers. This is the reason why the term AM detection is used for this detection scheme.
- 6.
This value for the frequency shift was chosen as it leads to half of the original amplitude in the steady-state.
- 7.
References
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Y. Martin, C.C. Williams, H.K. Wickramasinghe, Atomic force microscope force mapping and profiling on a sub 100 Å scale. J. Appl. Phys. 61, 4723 (1987). https://doi.org/10.1063/1.338807
R. Garcia, Amplitude Modulation Atomic Force Microscopy, 1st edn. (WileyVCH, Weinheim, 2010). https://doi.org/10.1002/9783527632183. ISBN:9783527408344
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Voigtländer, B. (2019). Amplitude Modulation (AM) Mode in Dynamic Atomic Force Microscopy. In: Atomic Force Microscopy. NanoScience and Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-13654-3_13
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DOI: https://doi.org/10.1007/978-3-030-13654-3_13
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