A direct method to calculate tip–sample forces from frequency shifts in frequency-modulation atomic force microscopy

Frequency-modulation atomic force microscopy (FMAFM) has proven to be a powerful method for imaging surfaces with true atomic resolution. However, the tip–sample forces are not directly accessible by FMAFM. Here, an algorithm to recover the tip–sample forces from the frequency shift curve is introduced and demonstrated with experimental data. Also, an intuititive connection between frequency shift Δf and tip–sample force gradient kts that simplifies the calculation of FMAFM images is established: Δf is a convolution of kts with a semispherical weight function.