oscillation amplitude obtained from the frequency noise measurement shown in figure 4. 

## VI. SUMMARY 

In the present work we have revised DSFM frequency detection and have analyzed how the thermal fluctuation of the cantilever is processed by a DSFM detection electronics. We find a general relation for the frequency noise as a function of a bandwidth and oscillation amplitude. This relation is correct for all possible values of parameters, while the relation known so far from the literature is only correct for a particular range. We find that for sufficiently large bandwidth -that is, small time constants of the DSFM detection electronics- essentially all the thermal noise of the cantilever is measured. In this case the width of the resonance peak is the characteristic noise of any DSFM frequency measurement for small oscillation amplitude, while for larger oscillation amplitude the noise decreases linearly with the oscillation amplitude. In the large amplitude and (very) low bandwidth limit our general relation converges within a constant factor, to the relation known from the literature. 

We are convinced that the results presented in this work are relevant for the precise determination of the ultimate limits of DSFM. In particular, our general relation shows that for small oscillation amplitudes the frequency noise does not diverge, but rather converges towards a finite value. Therefore, small oscillation DSFM might be much more competitive than considered up to now. DSFM without external oscillation, that is, driven by thermal noise, might be possible not only for the measurement of tip-sample interaction, but also for imaging applications. Since many high precision measurements SFM measurements -in particular in the field of Electrostatic and Magnetic Force Microscopy- are ultimately based on frequency measurements we believe that the present work will also improve understanding and optimization of these related SFM techniques and shed light on the ultimate limit of Scanning Force Microscopy in different important applications. 

## VII. ACKNOWLEDGMENTS 

The authors acknowledge stimulating discussions with A. Urbina, J. G´ omez, L. Colchero and A. Gil. The authors also thank Atomic Force F&E GmbH, and in particular Mr. Ludger Weisser, for supplying the cantilevers used. This work was supported by the Span- 

