  We measure the mechanical thermal noise of soft silicon atomic force
microscopy cantilevers. Using an interferometric setup, we have a resolution
down to 1E-14 m/rtHz on a wide spectral range (3 Hz to 1E5 Hz). The low
frequency behavior depends dramatically on the presence of a reflective
coating: almost flat spectrums for uncoated cantilevers versus 1/f like trend
for coated ones. The addition of a viscoelastic term in models of the
mechanical system can account for this observation. Use of Kramers-Kronig
relations validate this approach with a complete determination of the response
of the cantilever: a power law with a small coefficient is found for the
frequency dependence of viscoelasticity due to the coating, whereas the viscous
damping due to the surrounding atmosphere is accurately described by the Sader
model.
