Higher-order eigenmodes of qPlus sensors for high resolution dynamic atomic force microscopy

Abstract

The time response of tuning-fork based sensors can be improved by operating them at higher eigenmodes because a measurement takes at least one oscillation cycle in dynamic force microscopy and the oscillation period of the second eigenmode is only about one sixth of the fundamental mode. Here we study the higher-order eigenmodes of quartz qPlus sensors [Bettac et al., Nanotechnology 20, 264009 (2009); Giessibl and Reichling, Nanotechnology 16, S118 (2005); Giessibl, Appl. Phys. Lett. 76, 1470 (2000); and Giessibl, Appl. Phys. Lett. 73, 3956 (1998)], their equivalent stiffness, and piezoelectric sensitivity, while paying special attention to the influence of the mass and rotary inertia of the sensing tip which is attached to the end of the qPlus quartz cantilever. A combination of theoretical modeling and scanning laser Doppler vibrometry is used to study the eigenmodes of qPlus sensors with tungsten tips. We find that the geometry of tungsten tips can greatly influence the shape, equivalent stiffness, and piezoelectric sensitivity of the second eigenmode of the quartz cantilever. At a critical tip length it is possible to theoretically achieve infinite equivalent stiffness and infinite piezoelectric sensitivity when the tip becomes a perfect node of vibration and beyond this critical tip length the second eigenmode loses its vibration node and the trajectory of the tip reverses with respect to the beam curvature. The findings have major implications for optimizing tip geometry for high-resolution imaging with qPlus sensors using higher eigenmodes.