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Frequenzmodulations-Rasterkraftmikroskopie (FM-AFM)

FM AFM

A cantilever with a sharp tip is excited in resonance by positive feedback, controling a constant amplitude A. For the free cantilever, the eigenfrequency is given by f0 = 2π ( k / m )0.5, k is the spring constant and m is the effective mass of the cantilever. Forces between tip and sample change the frequency f = 2π ( k + kts /m)0.5, where the tip-sample force gradient is given by kts. When kts is small compared to k and essentially constant within the tips trajectory to and from the sample, a frequency change
\Delta f = f_{0} k_{ts} / (2 k)
results. Forces cause a frequency shift, and with a feedback circuit adjusting the sample height such that Δf remains constant, an image is created.

"Classic" frequency-modulation-AFM with cantilevers with small spring stiffness k ≈ 20 N/m and comparatively large amplitudes A ≈ 10 nm enables routine-imaging at atomic resolution. However, theory shows that resolution enhancement is possible by using sub-nm amplitudes with stiff cantilevers. Minimal image noise δz is then proportional to
\delta z = (1+(A / \lambda )^{3/2}) / A
where λ is the range of the tip-sample forces. For chemical forces, λ ≈ 0.1 nm. Optimal resolution is expected for A ≈ λ.

Why didn't people use small amplitudes from the beginning?

  1. 1. The tip-sample forces disturb the cantilever´s oscillation. Jump-to-contact, e.g., is avoided if the restoring force that the cantilever exerts on the tip when it is deflected at amplitude A is larger than the sample force trying to pull the tip towards the sample:
  2. Tip-sample forces are not conservative, i.e. the cantilever loses an amount of energy Ets on its way to the sample and back. This energy has to be supplied by the amplitude controller. The amplitude controller's task - keeping A constant - is easier if Ets is small to the intrinsic energy loss of the cantilever (given by its Q-factor). Thus, we have an additional (conjectural) stability criterion:

Operation with sub-nm amplitudes is thus only possible using very stiff cantilevers (k ≈ 1 kN/m). Traditional silicon cantilevers with such a large stiffness are usually not available, moreover they suffer from two additional disadvantages:

  1. The tips of microfabricated Si cantilevers point in a [001]-crystal direction - an unfavorable orientation.
  2. The eigenfrequency of Si-cantilevers varies strongly with temperature (-58 ppm/K).

A solutions to these problems offers the qPlus Sensor design.

 
Last modified: 25th Sep, 2008 by Webmaster