Eigenvalue Veering in Quartz Tuning Fork Sensors and its Effect on Dynamic Atomic Force Microscopy

Quartz tuning fork (QTF) sensors offer an attractive alternative to traditional silicon microcantilevers for sensing applications in dynamic atomic force microscopy (DAFM). The QTF sensor consists of two identical, weakly-coupled tines with a sharp tip affixed to the distal end of one tine. The fundamental anti-phase mode of the QTF achieves a stable resonant frequency with a high Quality factor making it ideal for DAFM applications in which a small shift in the resonant frequency is linked to a tip-sample force. The addition of the tip-sample force also breaks the symmetry of the QTF leading to a classic eigenvalue veering scenario. The eigenvalue veering and accompanying mode localization phenomena violate the standard DAFM modeling assumptions which treat the addition of the tip-sample force as a small perturbation to a single-degree-of-freedom oscillator. We find that the eigenvalue veering can contribute a systematic error in force measurements on the order of 20%. Methodology for correcting the systematic error caused by eigenvalue veering is proposed.