Model-independent quantitative measurement of nanomechanical oscillator vibrations using electron-microscope linescans

Nanoscale mechanical resonators are highly sensitive devices and, therefore, for application as highly sensitive mass balances, they are potentially superior to micromachined cantilevers. The absolute measurement of nanoscale displacements of such resonators remains a challenge, however, since the optical signal reflected from a cantilever whose dimensions are sub-wavelength is at best very weak. We describe a technique for quantitative analysis and fitting of scanning-electron microscope (SEM) linescans across a cantilever resonator, involving deconvolution from the vibrating resonator profile using the stationary resonator profile. This enables determination of the absolute amplitude of nanomechanical cantilever oscillations even when the oscillation amplitude is much smaller than the cantilever width. This technique is independent of any model of secondary-electron emission from the resonator and is, therefore, applicable to resonators with arbitrary geometry and material inhomogeneity. We demonstrate the technique using focussed-ion-beam–deposited tungsten cantilevers of radius ∼60–170 nm inside a field-emission SEM, with excitation of the cantilever by a piezoelectric actuator allowing measurement of the full frequency response. Oscillation amplitudes approaching the size of the primary electron-beam can be resolved. We further show that the optimum electron-beam scan speed is determined by a compromise between deflection of the cantilever at low scan speeds and limited spatial resolution at high scan speeds. Our technique will be an important tool for use in precise characterization of nanomechanical resonator devices