## Resonance enhancement: a tale of two data analysis methods

## The Benefit and Challenge of using Resonance Enhancement

Resonance enhancement technique is the fastlane toward raising signal well above the noise floor to increase signal-to-noise ratio (SNR) and dynamical range. Such an approach is common for Scanning Probe Microscopy (SPM) or Nano/Micro Electro-Mechanical Systems (MEMS) which rely on mechanical amplification from a vibrating resonator coupled to the signal of interest. Similar approaches can be found in Electron Spin Resonance (ESR) [1] or using Radio-Frequency (RF) resonators at much higher frequency than their mechanical counter-part, but still exhibiting similar response behavior. With the advent of quantum and nanotechnology, which need to measure signals at the thermal noise or even quantum limit, it becomes even more relevant to benefit from such amplification methods.

At the same time, for quantitative analysis, the amplification factor needs to remain constant over the course of the measurements, which can easily be achieved with closed-loop techniques, provided all the information of interest can be contained within this frequency span. For open-loop techniques, or when the interaction is too complex or too fast for narrow band demodulation, the transfer function needs to be known at all time to attribute any measured change to the actual signal of interest and not to some change in the resonator itself. Non-quantitative analysis will lead to artifacts or unreliable results.

In this blog, we review 2 data analysis methods that can broadly be distinguished as Smart versus Big Data analysis, or in other words as real-time data crunching versus full data acquisition. The former methods rely on analysis in the frequency domain while the later in the time domain but both methods are intended to provide quantitative analysis.

## Time vs Frequency Domain Analysis

In term of instrumentation, a useful representation for dynamical signal analysis is the Frequency versus Time map, which can be regarded as well as the Demodulation speed versus Digitizer (Analog-to-Digital Convertor) map. Here is a simple sketch for Zurich Instruments main product categories:

*Figure 1: Representation of demodulation bandwidth versus frequency map for various lock-in amplifiers and digitizers (not to scale). For imaging applications, high sustainable data transfer rate is needed for fast and deep analysis.*

The red boxes in Figure 1 sketch the volume of information accessible from a Data Crunching methods versus Big Data approaches. On the one hand, we have a narrow band approach, which can be centered around any input frequency range, with reduced information content, while on the other hand, we have a wide-band approach of Big Data acquisition to stream a much larger volume of information. Some of the key parameters of either method relies in the sustainable amount of data stream. In a smart data approach, most of the information is already crunched by real-time computation on the instrument and therefore necessitates lower transfer rate and file size while for big data collection, higher transfer rate and file size are needed since the computation is then performed in a post-processing step.

## Orthogonal Analysis: in-phase and in-quadrature

Let’s now move on to the so-called smart analysis example, which often makes use of closed-loop operation. At resonance, the frequency response of any resonators exhibit maximum amplitude and phase flip. The resonant phase is typically of –90° (not accounting any delay in the signal path), which corresponds to an inflection point where the slope has the highest gain. While the physical observables are the amplitude and phase as measured by lock-in techniques, the parameter settings are the driving force, controlled by the frequency and drive amplitude of a sinewave. Locking the resonant phase with a Phase Locked Loop (PLL), allows for orthogonal analysis, since at all time the phase is locked at –90°. In term of physical observables, this means that conservative (in-phase) and dissipative (in-quadrature) interactions can be unambiguously separated with no crosstalk, apart from the PID error.*Figure 2: Locking the phase for othorgonal analysis (closed-loop operation). The in-phase and out-of phase component can carry independant information. At the same time, as long as the phase is well locked at resonance, the amplitude can be maximized at constant amplification factor (well-known transfer function).
*

This enhancement is not only beneficial to the actual resonance phase but also to any sideband modulation happening within the natural bandwidth of the resonance (f/2Q) either in an Amplitude Modulated (AM) or Frequency Modulated (FM) fashion, which may carry additional information. A quantitative analysis of dynamical response in SPM in the simple harmonic oscillator (SHO) approximation, is provided in the reference list for steady state [2] as well as more generalized version for transient phenomena [3].

A further application example of such orthogonal analysis, is Dissipation Kelvin Probe Force Microscopy modes (D-KPFM) where the electrostatic contribution from the interaction can be projected onto the dissipation axis, hence de-coupled from mechanical modes and therefore topography. Such methods can be sensitive to either the static force [4] or the force gradient [5].

## Transfer Function: Need for Quantitative Analysis

Resonance enhancement greatly improves the measured signal, often by several orders of magnitude, but how quantitative is this? In other words, can we really attribute the change in measured amplitude or phase to the signal of interest? Ideally, one wants the transfer function to be constant within the bandwidth of interest. Keeping the measured amplitude constant at the resonance guarantees that the measurement is always performed at its peak. The typical closed loop transfer function of a PID is therefore a simple low-pass filter: flat over a large frequency range (i.e., the closed-loop bandwidth) until a certain cut-off frequency above which the response diminishes as the PID cannot respond fast enough anymore. Such closed-loop bandwidth can be measured experimentally as well [6]. Within this closed-loop bandwidth, the gain is constant and therefore any change within this frequency band can be solely attributed to the DUT behavior. If the transfer function is more complicated than a simple Low-Pass Filter (LPF) response, because of more complex interaction of mixing products between different resonances, for instance, then the transfer function needs to be recorded at the same time for post-processing.

Depending on the complexity of the interaction and of the coupling to the resonator, one frequency measurement might carry all the information needed. Some higher harmonics or higher eigenmodes analysis can also be carried similarly to extend it to multifrequency analysis. But regardless of the number of frequencies taken into account, this remains a Frequency Domain (FD) analysis with limited time-resolution.

## Big Data Acquisition: The Big Picture

All of the above analysis is particularly well-suited when a single, well-behaved resonance carries the bulk of the physical information. In other words, the energy transferred from the driving force is perfectly coupled to the resonator and does not leak or couple to other modes. This is particularly true for well-defined systems, for instance in vacuum or with high Q-factor that concentrates most of the energy and information in a narrow band. For more complex systems such as multiferroic materials, with many contributing forces to the interaction, with often competing mechanisms, the coupling of some driving force with the mechanical system can spread over large band of frequencies with non-negligible background force and parasitic behaviors (e.g., from a cavity, other mechanical coupling, …). This is where the frequency domain analysis, and the SHO approximation, reaches some limitation and a deeper analysis is required. In SPM, this means better understanding the complete tip trajectory, from the time domain (TD) analysis. This comes at the expense of more massive data analysis and complex drive excitation that goes beyond simple sine wave and linear approximations.

*Figure 3: Big data acquisition: An Arbitrary Waveform or Chirp will excite all mechanical modes of a mechanical resonator, whose response will be digitized and stored in a matrix-like format for further analysis. Image courtesy of Ehsan Esfahani, see Reference [9]*

For Big Data collection, the excitation source can take any frequency span and shape by use of an Arbitrary Waveform Generator (AWG), which will couple to all mechanical modes of the resonator interacting with the sample. The full time-domain response is recorded with no filters in digitizer mode to capture the continuous excursion of the AFM probe [7]. All further data analysis is then fully reversible in contrast to the noise and information reduction operated in standard lock-in measurements that provides one amplitude and one phase sample at every pixel. Synchronization of arbitrary waveform generation (AWG) and data acquisition (DAQ) makes it possible to use an identical sampling rate and number of data points, allowing for simplified data analysis and leakage-free frequency analysis (by FFT) during post-processing [8]. During scanning, the trigger to output the AWG waveform can also match the start of every line. Once the complete information acquisition is stored in an appropriate data format, matrix diagonalization can then reduce dimensionality of the data to the main eigenvectors, for instance making use of Principle Component Analysis (PCA). Further signal processing and analysis can then lead to data integration either from a physics-based approach (e.g. applying a Simple Harmonic Oscillator model) or an information-based theory analysis [9].

## Conclusion

In this blog, we reviewed two resonance enhancement techniques, one that relies on a frequency-domain analysis and one that relies on a time-domain analysis. Both methods can lead to quantitative physical interpretations, which is particularly useful in the fields of SPM and MEMS but can also be applied to any resonance-based measurement. Zurich Instruments offers great tools to perform fast real-time processing of the key signal of interest from a lock-in and closed-loop techniques but also offers a complete suite of arbitrary signal generation and fast data streaming of a raw digitized input, stored in a matrix-like format, that is easier to handle in post-processing. The user therefore remains in control of the complete data acquisition and analysis process.

### References:

[1] T. Choi *et al,* Nature Nanotechnology **12**, p 420–424 (2017)

[2] H. Söngen *et al.*, J. Phys. Condens. Matter **29** (2017) 274001

[3] T. Wagner, J. Appl. Phys. **125**, 044301 (2019)

[4] Y. Miyahara *et al,* Phys. Rev. Applied **4**, 054011 (2015)

[5] Y. Miyahara *et al,* Appl. Phys. Lett. **110**, 163103 (2017)

[6] https://www.zhinst.com/blogs/romain/how-to-measure-pll-transfer-function/

[7] S. Somnath* et al, Nature Communications*, **7**, p. 13290, (2016)

[8] https://www.zhinst.com/blogs/brunok/frm-with-chirps/

[9] Ehsan Nasr Esfahani, PhD Dissertation, Developing Advanced Atomic Force Microscopy Techniques for Probing Coupled Phenomena in Functional Materials, University of Washington, (2018). Supervisor: Professor Jiangyu Li