Characterization of strongly coupled quartz tuning fork sensors for precision force measurement in atomic force microscopy

Miniaturized quartz tuning forks (QTFs) have been adopted as force sensors for non-contact atomic force microscopy (AFM). However, the coupled oscillation behaviors of the QTF prongs are not well understood, preventing quantitative measurement of the nanoscale tip-sample interaction forces. This article presents a lumped model that accurately delineates the coupled mechanical oscillations of QTF prongs, establishing rigorous relationships between experimental observables and tip-sample interaction forces. The first-order resonance spectra of a commercial QTF were fully characterized by correlating its piezoelectric response with the actual mechanical oscillation measured with a Fabry-Pérot interferometer. In order to uniquely determine the modeling parameters (i.e., the effective masses, spring constants, and damping constants), the experimental results were compared with the lumped model predictions while masses were added to one prong. The results reveal that the QTF’s center of mass is highly damped, preventing the observation of a symmetric resonance mode. In addition, the mass loading experiment demonstrates that the mechanical oscillations of the QTF prongs are strongly coupled, accounting for 59% (84%) of the effective stiffness at the in-plane (out-of-plane), antisymmetric resonance mode. We believe that the obtained QTF characterization results will pave the way for quantitative measurements of non-contact interaction forces in QTF-based AFM platforms, significantly improving the precision and reliability of nanoscale force measurements.