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## III. THERMAL FLUCTUATION OF THE CANTILEVER POSITION 

In addition to the “coherent” signal a$_{ex}$ ( t ) induced by the external excitation of the cantilever, in the present context also the “incoherent” contribution due to thermal noise, a$_{th}$ ( t ), is important[18]. The total motion of the cantilever is thus a ( t ) = a$_{ex}$ ( t ) + a$_{th}$ ( t ). When this motion is transduced into an electrical signal, the position detector will add some instrumental noise n ( t ), the total signal entering the DSFM detection unit is then u$_{DSFM}$ ( t ) = e ( a$_{ex}$ ( t ) + a$_{th}$ ( t )) + n ( t ), where e is a constant that describes the sensitivity (unit:V/nm) of the photodetector electronics. The key question is now: how does this signal pass the DSFM detection unit and what is the final noise of the frequency output? 

The Equipartition Theorem discussed above (relation 1) relates the total thermal dis- placement to the force constant of the cantilever and the temperature of the system. How- ever, since the DSFM - detection system performs non-trivial processing of the input signal, the precise spectrum of the thermal noise has to be taken into account in order to cal- culate the noise of the frequency output and thus the precise limit of DSFM. A detailed discussion of thermal noise in SFM set-ups is out of the scope of the present work (see, for example[9, 10, 18, 24]). In a simple picture, the noise density of the cantilever motion can be obtained by observing that the Equipartition Theorem (equation 1) relates the mean energy of the cantilever with the thermal energy kT . Assuming that the effective thermal noise “force” driving the cantilever has a constant spectral noise density α$_{th}$ (unit: nm/ √ Hz ), this coupling strength and the mechanical gain G ( ν ) of the cantilever (relation 17) define the spectral response of the thermal spectrum a$_{th}$ ( ν ) of the cantilever: a$_{th}$ ( ν ) = α$_{th}$ G ( ν ). The Equipartition Theorem (relation 1) then implies: 

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The second integral gives ν$_{0}$Q π/ 2, therefore the thermal noise density α$_{th}$ is 

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The thermal noise density defines the coupling of the thermal bath into the SFM system. It depends on the quality factor and thus on the dissipation properties of the system. As the quality factor increases, thermal fluctuations induce less “thermal force” on the system, that is, coupling of the thermal bath and the tip-sample system becomes weaker. 

