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the other way, the frequency shift is the control parameter for tip-sample distance and the oscillation amplitude -dissipation energy- is measured as secondary channel. At the moment DSFM is the most sensitive SFM technique to measure tip-sample interaction, which is detected as a shift of the resonance frequency of the system. Therefore, with regard to the ultimate limits of SFM a key issue -to be discussed in the present work- is to understand in detail the effect of thermal fluctuations on DSFM detection schemes and in particular on frequency detection. At present, the thermal noise density of a frequency measurement is assumed to be[17, 18, 20] 

<img src='content_image/349076.jpg'>

with Q the quality factor, ν$_{0}$ the resonance frequency, a the (root mean square!) oscillation amplitude and bw the bandwidth of the measurement. Note that this relation diverges for small oscillation amplitude and is proportional to the square root of the measurement bandwidth. 

In the present work we will revise in detail how thermal fluctuation limits the measure- ment of frequency shift. We find that neither for large bandwidth measurements nor for low oscillation amplitude relation 2 is correct. We present a general relation valid for all ranges of amplitude and bandwidth that agrees with relation 2 in the low bandwidth and large amplitude range. Finally, we present experimental data that unambiguously proves the validity of the general relation obtained in this work. 

## II. FREQUENCY DETECTION IN DYNAMIC SCANNING FORCE MI- CROSCOPY 

A thorough discussion of frequency detection and DSFM is out of the scope of the present work (see the excellent original works[17, 18] or recent reviews by Garcia and Perez[19] and Giessibl[20]), nevertheless we think it is important to revise some of its basic principles. The focus of this revision is rather on a comprehensive physical explanation of the technique than on a profound analysis of the underlying physiscs and statistical mechanics (see[18]) or the electronic details of its implementation(see [18, 21]). 

An externally driven SFM cantilever -usually driven by inertial forces- is a textbook example of a harmonic oscillator. In many aspects SFM and modern gravitational wave 

