Torsional Sensitivity and Resonance Tuning in Width-Varying AFM Microcantilevers.

The torsional vibration of atomic force microscope (AFM) cantilevers is key to high-resolution and high-sensitivity measurements. However, standard models often fail to accurately describe the dynamics of width-varying geometries. In this study, we present an exact analytical model for computing torsional resonance frequencies and mode shapes of overhang- and T-shaped microcantilevers. Our predictions match experimental torsional-to-flexural frequency ratios within 5%, resolving long-standing discrepancies. We uncover the emergence of multiple spatial maxima in higher-order modes and demonstrate how overhang geometry allows tunable frequency shifts. Crucially, we derive a sensitivity function that quantifies the dependence of modal response on tip-surface coupling stiffness, revealing nontrivial geometry-dependent trends. These results offer clear design principles for enhancing AFM sensitivity via geometric control, providing a robust theoretical basis for optimizing next-generation microcantilever probes.