2, University of Colorado Boulder, Boulder, Colorado, United States
In bias induced strain measurements such as piezoresponse force microscopy and electrochemical strain microscopy, the contact resonance (CR) of the surface-coupled cantilever-sample system can serve as an amplifier of cantilever motion, providing orders of magnitude improvement in strain sensitivity compared to off-resonance measurements. However, use of the CR can significantly complicate relative and absolute quantification of AC strain because the optical lever sensitivity of the CR differs dramatically from the quasistatic bending shape typically used for optical calibration. Furthermore, the CR optical lever sensitivity varies with the CR frequency, which itself is a function of sample modulus, sample curvature, and contact area, all of which can vary spatially over a scan area. The net result is that a response that appears as contrast in AC strain amplitude can be predominantly associated with a change in the mechanical boundary conditions of the probe, rather than actual electromechanical function. Recently, the concept of a shape factor to relate the CR amplitude to the actual tip motion has been developed by Balke et al. through use of an Euler-Bernoulli beam model. However, epistemic uncertainties between the beam model and the experiment may still result in unacceptable errors. Here, we introduce an experimental method with a calibrated reference artifact to calculate the shape factor across the range of relevant boundary conditions that the unknown sample may exhibit. The reference artifact is a high-frequency (fundamental resonance >2 MHz) ultrafast cantilever affixed to an ultrasound transducer. The base motion of the cantilever is quantitatively calibrated using a Michelson interferometer while the transducer is driven at frequencies encompassing the CR frequencies of the unknown cantilever-sample system. When an unknown cantilever with precisely placed detection-laser is brought into contact with the reference cantilever, the resultant contact resonance is monotonically related to the position of the unknown cantilever’s tip on the reference cantilever beam (i.e. positions closer to the free-end result in lower stiffness and lower CR frequency, closer to the base result in higher stiffness and higher frequency). Here, we show that the relation between a known sample-motion drive-force and the CR-amplitude (i.e. the CR shape factor) can be continuously measured. We then apply the calibrated cantilever to measure quantitative surface strain on a piezoelectric material.