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 ͑2009͒; Giessibl and Reichling, Nanotechnology 16, S118 ͑2005͒; Giessibl, Appl. Phys. Lett. 76, 1470 ͑2000͒; and Giessibl, Appl. Phys. Lett. 73, 3956 ͑1998͔͒, 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.

Calculation of the effect of tip geometry on noncontact atomic force microscopy using a qPlus sensor

“…This becomes particularly important if calibration is done by thermal tuning, or if force measurements are performed by using higher modes [11]. The elastic potential energy of the tine is given by…”

“…The qPlus sensor is unusual for an AFM sensor in that it is constructed from a quartz tuning fork and has a tip comparable in length to the tine [10]. Hence, the macroscopic geometry of the tip also cannot be ignored [11]. …”

“…Measurements of the spring constants of qPlus sensors have produced conflicting results [4,13], which highlights the need for more detailed analysis. Tung et al [11] have shown that the dimensions of conical tips also have large effects on the dynamics of higher eigenmodes, suggesting that careful consideration of tip geometry is necessary for sensors operated in the second eigenmode or above.…”

Amplitude dependence of image quality in atomically-resolved bimodal atomic force microscopy

“…7 Theoretically, the effective stiffness of the second flexural mode is about 40 times higher than in the first flexural mode, and the resonance frequency is about 6.2 times higher. 8,9 This can be implemented with bimodal AFM, 10,11 in which the first flexural mode is excited at a large amplitude and the second flexural mode at a small amplitude to detect short range interactions.…”

Note: In situ cleavage of crystallographic oriented tips for scanning probe microscopy

“…Therefore, the tips are relatively short after cleavage-a desirable result because long tips lead to complications in particular for operation at higher flexural modes. 16 The deflection during cleavage is measured by recording the cleavage process with a video camera. The maximal possible bending of the free prong corresponds to a force of approximately 240 mN.…”