Federbalkendynamik in der elektrochemischen Verformungsmikroskopie

This thesis mainly analyzes the cantilever dynamics in electrochemical strain microscopy (ESM) measurements. First, a comprehensive model was developed and described mathematically. Next, its implications for practical measurements and their analysis was discussed theoretically and demonstrated practically. It was shown how to quantify the phase and amplitude of the signal and how different excitation types can be distinguished. Additionally, it was demonstrated that insufficient frequency tracking leads to systematic errors in dual AC resonance tracking measurements. In the first part, a model for the cantilever dynamics in ESM was developed. This model was then applied to quantify the phase between the applied AC bias and the resulting strain or electrostatic forces. In ESM measurements, an AC bias is applied to the cantilever, leading to ionic and electronic transport within the sample. This causes a periodical sample strain as well as periodical electrostatic forces. Both excite the cantilever oscillation which is measured with an optical beam deflection setup. In order to obtain the real strain or electrostatic force, the cantilever dynamics have to be analyzed. In this paper, a detailed model of the cantilever dynamics was developed and described mathematically. The model combines properties of previous models and contains additional aspects, making it much more detailed than previous models. The most important novelties are the inclusion of the tip mass as well as a non-uniform electrostatic excitation of the lever. The model was then used to quantify the phase between the applied AC bias and the resulting strain or electrostatic forces. The measured phase between the AC bias and the measured signal also includes a phase delay between the excitation and the cantilever motion as well as an instrumental phase offset. The model shows that the delay between excitation and cantilever motion is inversely proportional to the damping strength and can be neglected for typical measurements. In the next step, a sample with a known phase between AC bias and strain was used to determine the instrumental phase offset. Using different cantilevers, the instrumental phase offset was found to depend linearly on the frequency. In further measurements it was demonstrated how the instrumental phase offset and, if necessary, also the phase delay caused by damping can be subtracted from the measured phase. The remaining signal is the desired phase between AC bias and strain or electrostatic forces. In the second part, the resonance peak analysis techniques dual AC resonance tracking (DART) and band excitation (BE) were compared. During the analysis, correlations between the measured contact resonance frequency and drive amplitude were found. Using the model from the first part, it was shown that both the contact resonance frequency and the sensitivity of the measured signal towards the excitation strength are determined by the contact stiffness. Thus, the non-uniform measured amplitude is not caused by a non-uniform excitation strength, but by a non-uniform sensitivity. The resulting correlations between contact resonance frequency and measured amplitude are different for the different excitation types. When comparing the experimental correlation with theoretical predictions, it was found that both on a mixed ionic-electronic conductor with low electronic conductivity and on a purely ion conducting glass ceramic, the electrostatic excitation dominates the signal. For the ion-conducting glass ceramic, the time constants for double layer formation (leading to electrostatic excitation) and chemical diffusion (leading to sample displacement) were estimated and it was found that only electrostatic excitation is expected. In the third part, the sensitivity of the measured signal towards the excitation strength was analyzed in more detail. First, a procedure for obtaining simulation parameters from experimental quantities was developed. The lever mass and tip mass are calculated from the first and second free resonance. The contact parameters can then be adjusted to match the observed contact resonance properties. Next, the shape of the cantilever oscillation was analyzed as a function of contact stiffness. We then compared the excitation power to the dissipated power. This allowed us to derive formulas for the sensitivity as a function of the cantilever oscillation shape. Thus, we could explain mathematically, why the sensitivities towards the different excitation types depends on the contact stiffness in different ways. Consequently, the four excitation types show different correlations between contact resonance frequency and apparent amplitude. A comparison between experimental and simulated correlations then allowed us to correctly determine the excitation type on both ferroelectric LiNbO3 and ion conducting soda-lime glass. This analysis was extended to higher bending modes, where the expected correlations were confirmed. Another way for identifying the excitation type is by comparing the relative sensitivities between the different bending modes. Again, both samples showed the expected behavior, with the soda-lime glass requiring a proper distinction between local and non-local electrostatic excitation. Additionally, the sensitivity of the signal towards non-local electrostatic excitation is strongly reduced in higher bending modes. This is caused by the nodes in the cantilever oscillation shape. Because of these nodes, only part of the cantilever can oscillate in phase with the excitation and the other part counteracts this motion. The non-local electrostatic excitation is usually an undesired signal contribution, as it does not contain information about the sample beneath the tip. Measuring in higher bending modes suppresses this contribution. In the fourth part, errors caused by insufficient frequency tracking in DART measurements are analyzed. In DART measurements, the amplitude and phase are each measured at two frequencies close to the resonance frequency. From the four measured quantities, the four parameters of the damped harmonic oscillator (DHO) model are calculated: drive amplitude, drive phase, resonance frequency and amplification factor. Additionally, the excitation frequencies are adjusted according to the resonance frequency. For a system ideally described by the DHO model, the obtained DHO parameters should be independent of the exact excitation frequencies. However, experimental resonance peaks deviate from the DHO model, whereas the model developed in the first part offers a better description. This model predicts errors in the calculated drive amplitude and calculated amplification factor, unless the excitation frequencies are symmetric around the resonance frequency (tracking error). The errors for drive amplitude and amplification factor are equal in magnitude but with opposite sign and are both proportional to the tracking error. This prediction was confirmed on ferroelectric LiNbO3. The deviations from the DHO model and the magnitude of the resulting error are both determined by changes in the cantilever oscillation shape. Next, the influence of tracking errors was demonstrated on the cathode material LiCoO2. The contact resonance frequency is inhomogeneous across the sample. During the scan, the excitation frequencies could not be adjusted instantaneously, leading to tracking errors. These tracking errors lead to artifacts in the calculated drive amplitude, which can be identified by comparing it to the tracking error and amplification factor. This comparison is crucial for DART measurements in order to distinguish real effects from tracking error artifacts.