Distortion in the thermal noise spectrum and quality factor of nanomechanical devices due to finite frequency resolution with applications to the atomic force microscope

Abstract

The thermal noise spectrum of nanomechanical devices is commonly used to characterize their mechanical properties and energy dissipation. This spectrum is measured from finite time series of Brownian motion of the device, which is windowed and Fourier transformed. Here, we present a theoretical and experimental investigation of the effect of such finite sampling on the measured device quality factor. We prove that if no spectral window is used, the thermal noise spectrum retains its original Lorentzian distribution but with a reduced quality factor, indicating an apparent enhancement in energy dissipation. A simple analytical formula is derived connecting the true and measured quality factors - this enables extraction of the true device quality factor from measured data. Common windows used to reduce spectral leakage are found to distort the (true) Lorentzian shape, potentially making fitting problematic. These findings are expected to be of particular importance for devices with high quality factors, where spectral resolution can be limited in practice. Comparison and validation using measurements on atomic force microscope cantilevers are presented.