Performance Analysis of Resonantly Driven Piezoelectric Sensors Operating in Amplitude Mode and Phase Mode

Piezoelectric layers coupled to micromechanical resonators serve as the basis for sensors to detect a variety of different physical quantities. In contrast to passive sensors, actively operated sensors exploit a detuning of the resonance frequency caused by the signal to be measured. To detect the time-varying resonance frequency, the piezoelectric resonator is resonantly excited by a voltage, with this signal being modulated in both amplitude and phase by the signal to be measured. At the same time, the sensor signal is impaired by amplitude noise and phase noise caused by sensor-intrinsic noise sources that limit the reachable detectivities. This leads to the question of the optimum excitation frequency and the optimum readout type for such sensors. In this article, based on the fundamental properties of micromechanical resonators, a detailed analysis of the performance of piezoelectric resonators in amplitude mode and phase mode is presented. In particular, the sensitivities, the noise behavior, and the resulting limits of detection (LOD) are considered and analytical expressions are derived. For the first time, not only the influence of a static measurand is analyzed, but also the dynamic operation, i.e., physical quantities to be detected that quickly change over time. Accordingly, frequency-dependent limits of detection can be derived in the form of amplitude spectral densities. It is shown that the low-frequency LOD in phase mode is always about 6 dB better than the LOD in amplitude mode. In addition, the bandwidth, in terms of detectivity, is generally significantly larger in phase mode and never worse compared with the amplitude mode.

Noise in NC-AFM measurements with significant tip-sample interaction

The frequency shift noise in non-contact atomic force microscopy (NC-AFM) imaging and spectroscopy consists of thermal noise and detection system noise with an additional contribution from amplitude noise if there are significant tip-sample interactions. The total noise power spectral density DΔf (fm) is, however, not just the sum of these noise contributions. Instead its magnitude and spectral characteristics are determined by the strongly non-linear tip-sample interaction, by the coupling between the amplitude and tip-sample distance control loops of the NC-AFM system as well as by the characteristics of the phase locked loop (PLL) detector used for frequency demodulation. Here, we measure DΔf (fm) for various NC-AFM parameter settings representing realistic measurement conditions and compare experimental data to simulations based on a model of the NC-AFM system that includes the tip-sample interaction. The good agreement between predicted and measured noise spectra confirms that the model covers the relevant noise contributions and interactions. Results yield a general understanding of noise generation and propagation in the NC-AFM and provide a quantitative prediction of noise for given experimental parameters. We derive strategies for noise-optimised imaging and spectroscopy and outline a full optimisation procedure for the instrumentation and control loops.