Tracking Resonance Frequency without Phase: the DFRT Method

For high Q-factor resonator (> 500), it is a common method for SPM application to use a Phase Locked Loop (PLL) to track the change in cantilever frequency: the high phase sensitivity at 90° guarantees accurate measurements and stable PID feedback on the driving oscillator. This is of particular interest for experiment in Vacuum/UHV or with stiff sensors (quartz or MEMS-based resonator) that exhibits low intrinsic dissipation (hence high Q).

On the other hand, the PLL method is not relevant or impossible to implement when:

  • Phase is unstable due to reversed polarization (e.g. Piezoresponse Force Microscopy or PFM)
  • Phase sensitivity is very low due to degraded Q factor because of the environment (e.g. in  liquid) or the sample under investigation (e.g. contact resonance on highly corrugated material).

Here I would like to continue with the same step by step approach on how to set-up the so-called Dual Frequency Resonance Tracking (DFRT) method as was previously described in an earlier blog regarding FM-KPFM method. As a side note, DFRT is easier to set-up than KPFM because one does not need to pay attention to the phase behavior (we deliberately make no use of it for feedback!). Phase will only be used for recording.

In order to set-up such experiment, the following options to the Zurich Instruments HF2LI Lock-in Amplifier are necessary:

  • HF2LI-MF Multifrequency option to excite and demodulate at arbitrary phases and frequencies
  • HF2LI-MOD Modulation option to generate & demodulate AM/FM signals
  • HF2LI-PID QuadPID option to feedback on the difference of demodulated amplitudes

General principle of the DFRT method


If we cannot rely on the phase to measure the resonance frequency, we need to measure the deviation of amplitude slightly above and below the resonance. It sounds a bit like Tapping mode (or Intermittent Contact Mode), which is also an Amplitude Modulation technique, but with the major difference that both sides of the resonance amplitude are measured simultaneously. Schematically, this will look like this:

Blog DFRT-schematics

In the frequency range f2-f1 (to be chosen appropriately), the linear dependance of the difference of amplitude (red curve) will allow for a monotonic behavior of the PID feedback loop and therefore best operating conditions.

Generating Bimodal Excitation


Let’s assume we have already identified the resonance (free or in contact with the surface), which in the example below was sitting at 391kHz. The first step consists in generating a sideband modulation, or bimodal excitation, around the cantilever resonance (i.e. the carrier). With the HF2LI-MOD option this is straightforward: enable the ‘AM Gen + Demod‘ in the Modulation tab and define a modulation range (frequency span f2-f1, around the resonance to stay in the linear regime) that would hit let say 80% of the amplitude on each side of the resonance, in practice it will look like this:

Blog DFRT-bimodal oscillations

In no time, a nice and clean AM modulation will excite the cantilever near its resonance as seen on the internal oscilloscope trace above. All signals are internally generated and added, which will also serve as reference for the first 3 demodulators in the Lock-in MF tab (carrier and 2 sidebands).

Nullifying the Difference of the Sideband Amplitudes


When there are no interaction, the Lock-in Amplitude (R) measured on each side of the resonance (sideband demodulators 2& 3) will give almost the same value. As soon as the cantilever probe an interaction, the amplitude of sidebands will increase/decrease (in opposite phase) because the resonance now shift to a new frequency. If we therefore apply a PID feedback on the difference of the 2 measured amplitudes, with a set-point of zero (no difference), the PID output will drive the generated oscillator to the new frequency at the cantilever resonance. In the PID tab, the value of the absolute difference in amplitude can be selected as input for the excitation feedback:

PID feedback on measured amplitudes

Figure (click to enlarge): PID feedback on difference of amplitudes

The last parameters to optimize will be as usual the P and I gains. In the example above, the cantilever is driven mechanically by a shaker piezo, which require several Volts of excitation to reach a reasonable amplitude response (in mV), that’s why P-gain is so high. One can also use the DUT system model for the PID adjustment, at least for the right order of magnitude. Make sure to enter the correct Gain value for the model as the ratio of measured amplitude to driving excitation (for the sideband amplitude in this case).

Example for Piezoresponse Force Microscopy in Contact Resonance


While the standard PFM method simply records Lock-in Amplitude and Phase (in- and out-of-plane components) at a fixed bias modulation frequency, it is not always clear which frequency generates the best ‘piezo-response’. Taking advantage of the contact resonance enhancement allows to always maximize PFM signal. The DFRT method therefore enables the tracking of the contact resonance even when the actual piezo-response phase flips by 180° (adjacent ferroelectric domains have reversed polarization, which would not be possible to track with a PLL). Below is an example obtained with the method described in this blog.

Blog PFM image

In this particular example, the sidebands (amplitude modulation) serve to track the resonance while the actual carrier signal (phase and amplitude) measures the PFM signal in the best operating conditions. For the SPM Controller, the Z-feedback in contact mode still relies on static cantilever deflection from the photo-detector, while all phase and amplitude information are recorded for imaging via the Auxiliary outputs.


Special thanks to Igor Stolichtnov and Enrico Colla (EPFL) for the excellent experimental conditions in their labs and to Iaroslav Gaponenko (Université de Genève) for brightful ideas on PFM methods and beyond!