Spatial horizons in amplitude and frequency modulation atomic force microscopy

In dynamic atomic force microscopy (AFM) the cantilever is vibrated and its dynamics are monitored to probe the sample with nanoscale and atomic resolution. Amplitude and frequency modulation atomic force microscopy (AM-AFM and FM-AFM) have established themselves as the most powerful methods in the field. Nevertheless, it is still debatable whether one or the other technique is preferred in a given medium or experiment. Here, we quantitatively establish and compare the limitations in resolution of both techniques by introducing the concept of spatial horizon (SH) and quantifying it. The SH is the limiting spatial boundary beyond which collective atomic interactions do not affect the detection parameters of a given feedback system. We show that while an FM-AFM feedback can resolve a single atom or atomic defect where an AM feedback might fail, relative contrast is in fact equivalent for both feedback systems. That is, if the AM feedback could detect sufficiently small amplitude shifts and there was no noise, the detection of single atoms or atomic defects would be equivalent in AM-AFM and FM-AFM.