Amplitude Dependence of Resonance Frequency and its Consequences for Scanning Probe Microscopy

Abstract

With recent advances in scanning probe microscopy (SPM), it is now routine to determine the atomic structure of surfaces and molecules while quantifying the local tip-sample interaction potentials. Such quantitative experiments using noncontact frequency modulation atomic force microscopy is based on the accurate measurement of the resonance frequency shift due to the tip-sample interaction. Here, we experimentally show that the resonance frequency of oscillating probes used for SPM experiments change systematically as a function of oscillation amplitude under typical operating conditions. This change in resonance frequency is not due to tip-sample interactions, but rather due to the cantilever strain or geometric effects and thus the resonance frequency is a function of the oscillation amplitude. Our numerical calculations demonstrate that the amplitude dependence of the resonance frequency is an additional yet overlooked systematic error source that can result in nonnegligible errors in measured interaction potentials and forces. Our experimental results and complementary numerical calculations reveal that the frequency shift due to this amplitude dependence needs to be corrected even for experiments with active oscillation amplitude control to be able to quantify the tip-sample interaction potentials and forces with milli-electron volt and pico-Newton resolutions.

Dynamic Responses of Electrically Driven Quartz Tuning Fork and qPlus Sensor: A Comprehensive Electromechanical Model for Quartz Tuning Fork

“…This discrepancy could be attributed either to imperfect bonding between one prong of the qPlus and the supporting wall (Figure 1b), or to the base of the qPlus partially covered with glue. While our model is linear, the nonlinearity shown by Dagdeviren et al [20] could be partially responsible for the discrepancy in Figure 5; the resonance frequency of oscillating probes can change by a few hertz when changing the oscillation amplitude by nearly two orders of magnitude. In addition, one could derive more accurate formulas for U1 and U2 (Equations (11) and (12)) to reduce the difference.…”

“…The electrical responses of the ED-TF (Figure 1a) and ED-qPlus (Figure 1b) are described well by the equivalent circuit model shown in Figure 1c. Note that the equivalent circuit describes the linear motion of a TF or qPlus, although a recent study showed the amplitude dependence of the resonance frequency in such quartz resonators [20]. Therefore, one should consider the nonlinearity when using a quartz sensor to quantify the tip–sample interaction potentials and forces with milli-electron volt and pico-Newton resolutions [20].…”

“…Note that the equivalent circuit describes the linear motion of a TF or qPlus, although a recent study showed the amplitude dependence of the resonance frequency in such quartz resonators [20]. Therefore, one should consider the nonlinearity when using a quartz sensor to quantify the tip–sample interaction potentials and forces with milli-electron volt and pico-Newton resolutions [20]. In the equivalent circuit, the LRC circuit is connected in parallel with a capacitance C0, and thus the total signal Ie is given by the sum of Im and Ic.…”

Enhancement of the CO2 Sensing/Capture through High Cationic Charge in M-ZrO2 (Li+, Mg2+, or Co3+): Experimental and Theoretical Studies

“…The uniform distribution of the homogeneous particles was registered for CoZrO 2 (Figure a), and it is consistent with the histogram (Figure b) showing the distribution of size (around 80–160 nm) of zirconia particles. It means that a maximum size (135 nm) was registered and observing no amplitude error in the mapping (Figure c). , The same observations were noted for LiZrO 2 and MgZrO 2 (see Figure S1).…”

“…It means that a maximum size (135 nm) was registered and observing no amplitude error in the mapping (Figure 5c). 46,47 The same observations were noted for LiZrO 2 and MgZrO 2 (see Figure S1).…”