For small oscillation amplitudes, ie, amplitudes much smaller than characteristic length scale of the interaction force between the tip and sample, it is well established that the frequency shift is proportional to the force gradient:

Albrecht et al, J. Appl. Phys. 69, 668 (1991)

For arbitary oscillation amplitudes, the following relationship holds (Giessbl, F.J., Phys. Rev. B 56, 16010 (1997):

where a is the amplitude of oscillation.

However, this relationship does not provide an intuitive connection between frequency shift and force, except for the small amplitude limit.

We have mathematically analyzed the behavior of the frequency shift as a function of oscillation amplitude and established that, in the large amplitude limit, the frequency shift is proportional to the half fractional integral of the force, or equivalently, the half fractional derivative of the energy. This finding concurs with the observation that the frequency shift is intermediate to the force and energy for specific force laws. Our analysis also shows that FM AFM should be interpreted in general as a fractional differential operator, where the order of the derivative/integral is determined only by the oscillation amplitude relative to the length scale of the interaction.

The analysis has yielded simple yet accurate analytical formulas that enable the direct determination of the interaction force and energy from the measured frequency shift. These formulas are valid for any oscillation amplitude and can be used with any force law.

Actual (solid line) and recovered (dashed line) Lennard-Jones force law using the new arbitrary amplitude formula. A spectrum of oscillation amplitudes ranging from a/l = 0.1 to a/l = 10 were used. z is the tip-sample separation and l is the separation where the attractive force is maximum.

Download the Mathematica code here.

Our linked publications:

J. E. Sader and S. P. Jarvis, "Accurate Formulas for Interaction Force and Energy in Frequency Modulation Force Spectroscopy", Appl. Phys. Lett., 84 , 1801-1803 (2004).

J. E. Sader and S. P. Jarvis, "Interpretation of frequency modulation atomic force microscopy using fractional calculus", Phys. Rev. B, vol 70, 012303 (2004).