Nanoscale Ferroelectric Characterization with Heterodyne Megasonic Piezoresponse Force Microscopy

Abstract Piezoresponse force microscopy (PFM), as a powerful nanoscale characterization technique, has been extensively utilized to elucidate diverse underlying physics of ferroelectricity. However, intensive studies of conventional PFM have revealed a growing number of concerns and limitations which are largely challenging its validity and applications. In this study, an advanced PFM technique is reported, namely heterodyne megasonic piezoresponse force microscopy (HM‐PFM), which uses 106 to 108 Hz high‐frequency excitation and heterodyne method to measure the piezoelectric strain at nanoscale. It is found that HM‐PFM can unambiguously provide standard ferroelectric domain and hysteresis loop measurements, and an effective domain characterization with excitation frequency up to ≈110 MHz is demonstrated. Most importantly, owing to the high‐frequency and heterodyne scheme, the contributions from both electrostatic force and electrochemical strain can be significantly minimized in HM‐PFM. Furthermore, a special measurement of difference‐frequency piezoresponse frequency spectrum (DFPFS) is developed on HM‐PFM and a distinct DFPFS characteristic is observed on the materials with piezoelectricity. By performing DFPFS measurement, a truly existed but very weak electromechanical coupling in CH3NH3PbI3 perovskite is revealed. It is believed that HM‐PFM can be an excellent candidate for the ferroelectric or piezoelectric studies where conventional PFM results are highly controversial.


S1. Frequency Response Analysis for Rectangular Cantilever
shows the normalized force constant k n and quality factor Q n as a function of mode number (both are normalized with respect to the first eigenmode). Compared with Q n , k n shows a much faster increase with the mode number, indicating that the resonance is more and more difficult to be stimulated with increasing frequency. The amplitude response has been calculated here to show the cantilever dynamic property.
According to the Euler-Bernoulli beam theory, for a rectangular cantilever, the equation of motion along its longitudinal axis is [1]  The solution of Equation (S3) can be expressed as [2] [ ] [ ] where C 1 , C 2 , C 3 and C 4 are undetermined constants. The amplitude response at the cantilever end is defined as Then using the parameters from AFM tip (240AC-PP, OPUS) and a hydrodynamic damping of 5.06 × 10 -4 kg/(m⸱s) [3] (Table S1) to solve the Equation (S3) to (S6) analytically using MATLAB, the amplitude response A R (s) can be obtained and its magnitude in frequency domain is plotted in Figure S2.

S2. Heterodyne Detection Principle of HM-PFM
Considering both tip and sample surface have a vibration, A t sin(ω t t+ϕ t ) and A s sin(ω s t+ϕ s ) respectively (here use the angular frequency, ω t = 2πf t , and ω s = 2πf s , and similarly hereinafter), the time-dependent tip-sample interaction force is given by Expanding the tip-sample interaction with a Taylor series at z=z 0 up to second order gives To further process the Equation (S8), the tip-sample interaction force can be expressed by the sum of the components with different frequencies in which ω diff = ω s − ω t is the difference frequency. For ferroelectric domains with upward and downward polarization, there is a 180° phase difference between the tip voltage-induced piezoelectric strain, thus the vibration of sample surface has a 180° phase difference between upand downward domains, [4] which are schematically shown in Figure S3. From Equation (S11), when the drive signals applied to the tip and sample remain constant, the difference-frequency tip-sample interaction force on upward and downward ferroelectric domains are calculated by (S12) (S13) Obviously, there is a theoretical 180° phase difference between F ts (z) diff-up and F ts (z) diff-down , implying that

S3. Transfer Functions of Cantilever
To compare the difference of cantilever dynamics under electrostatic force and sample vibration excitations, the cantilever transfer functions have been calculated here. Since the AFM tip is always in contact with sample surface during PFM measurements, the vertical tip-sample coupling is modelled as a spring in parallel with a dashpot (Kelvin-Voigt model) and no lateral contact coupling is considered for simplicity. [5,6] Figure S4 shows the mechanical models used in these calculations. Considering there is only a sample vibration u(t) ( Figure S4b), the cantilever is driven by the local tip-sample interaction, the equation of motion and the corresponding boundary conditions are: [6] ( ) ( ) in which k ts and γ are tip-sample contact stiffness and contact damping constant, respectively. Performing Laplace transform to Equation (S14), and defining the transfer function for sample vibration (the local tip-sample interaction) excitation as  Table 1 and assuming k ts = 100k c = 100×3EI/L 3 , γ = 0.1×(EIm) 0.5 /L (dimensionless damping constant of 0.1) [7] , the calculated transfer function H ts (ω) is plotted in Figure S5.
Performing Laplace transform to Equation (S17), and a particular solution of Equation (S18) is then the transfer function for the electrostatic force excitation can be defined as Using the same parameters and calculation method mentioned above to solve Equation (S18) to (S21), the transfer function in frequency domain H EF (ω) can be obtained. As the relative magnitude between f EF-cant and f EF-tip depends on specific experiment, [4,9] three values of α, 0.001, 0.1 and 10, are used to calculate H EF (ω) and the results are all plotted in Figure S5.
Comparing the transfer function H EF (ω) (dot line in Figure S5) and H ts (ω) (red solid line in Figure   S5)   The key factor that enables the minimization of electrostatic force contribution in HM-PFM is the difference between frequency dependences of cantilever transfer function H ts (ω) and piezoelectric strain.
Here, still using the same parameters and calculation method described in the S3 of this supporting information, and assuming the total electrostatic force is 10 nN, [4] the amplitude of the electrostatic force excited tip vibration A EF can be calculated. Figure S7a shows the calculated A EF as a function of frequency under α = 0.1 and 10. For a typical PFM measurement on PPLN sample, the amplitude of the sample vibration A s is ~10 pm, as the piezoelectric strain does not change apparently within MHz frequency region, the amplitude A s is schematically plotted by a horizontal line in Figure S7a. By comparing A EF and A s , it is obvious that there exists a huge difference between their frequency dependences, A EF gradually decays with increasing excitation frequency, and in particular, A EF attenuates dramatically at off-resonance or anti-resonance states (such as the shadow area). For low frequency, A EF is larger than A s , implying that the electrostatic force contribution is significant or even dominant. But at high frequency, by properly choosing the frequency (e.g. in the shadow area), A EF (with α = 0.1 and 10 both) can be largely attenuated to be much smaller than A s , thus realizing significant minimization of electrostatic force contribution no matter the electrostatic force is dominated by distributed or local part.
Note that the calculation of A EF here is based on multiple assumptions, such as ignoring the internal damping and the frequency dependence of contact damping, the practical attenuation of A EF is much more rapid. Figure S7b shows the experimentally measured tip vibration amplitude (the amplitude of cantilever deflection signal) as a function of excitation frequency. This curve is measured on clean SiO 2 surface by conventional PFM set-up with the same AFM tip used for the calculation (i.e., 240AC-PP, OPUS).
As the SiO 2 is the pure dielectric layer of Si wafer, the measured tip vibration is actually stimulated by the electrostatic force. It is obvious that the practical tip vibration excited by electrostatic force decays very fast, just from the 1 st to the 3 rd eigenmode, the resonant amplitude has already attenuated ~100 times, which highly indicates that at much higher frequency, the practical A EF will further decrease to be far smaller than A s . Therefore, HM-PFM can achieve almost an ideal minimization of electrostatic force.   Figure S8 shows that, in the conventional PFM, the tip voltage stimulated piezoelectric vibration will cause a local varying tip-sample interaction force F ts ′(z 0 )sin(ω s t+ϕ s ), this force will drive the cantilever to vibrate and generate the piezoelectric signal A s-p sin(ω s t+ϕ p-s ) via the cantilever's transfer function H ts (ω). At the same time, the first harmonic electrostatic force C′(V dc −V cpd )V ac sin(ω s t) directly drive the cantilever to vibrate, generating the electrostatic force signal A EF sin(ω s t+ϕ p-EF ) via transfer function H EF (ω). Finally, the piezoelectric signal A s-p sin(ω s t+ϕ p-s ) and electrostatic force signal A EF sin(ω s t+ϕ p-EF ) will vectorially synthesis to the final piezoresponse signal A p sin(ω s t+ϕ p ). Although multiple methods, such as using probes with large force constant and operating at high frequency or higher eigenmodes, [5,10] are proposed to minimize the contribution of electrostatic force, these methods in principle are based on the difference between H ts (ω) and H EF (ω). However, as shown in Figure S5,  Figure S9 shows the contact resonance curves of the 7 samples discussed in the Figure 6 of the main text, in which are all measured by single-frequency PFM with sample drive amplitude of 2 ~ 10 V pp .

S5. Contact Resonance Curves Measured by Single-Frequency PFM
The resonance frequencies are all fitted from the resonance curves by using simple harmonic oscillator model. [12] Displacement Voltage Force

S6. Influence of the Radio-Frequency Radiation
With increasing the excitation frequency of the holder transducer, the radio-frequency radiation around the transducer increases, causing an additional electric field E RF between the tip and substrate. This radiated electric field E RF has the same frequency with the transducer drive. Assuming that E RF is caused by an equivalent AC voltage V RF sin(ω t t) applied between tip and substrate, the total voltage between tip and substate is (ignoring the DC bias and phase here): Then the total electrostatic force under the radiation is given by: From Equation (S24), it is evident that there is a difference-frequency component in * EF F , C′V s V RF cos(ω diff t)/4, which has the same frequency with DFP signal thus it can drive the cantilever and affect the HM-PFM results. However, if the radiation is effectively shielded, V RF will be closed to zero

S8. Influence of Tip-Sample Contact Radius
To check if the HM-PFM result is apparently affected by the tip-sample contact radius, a continuously repeated scanning experiment has been performed. As during continuous tip-sample contact scanning, the tip-sample contact area is expected to be increased due to the tip wearing. Figure S11 shows the HM-PFM amplitude and phase images recorded in the repeated scanning on the same area with fixed measurement conditions. It can be seen that from the 1 st scanning to the 12 th scanning, the spatial resolution is decreased due to the increasing tip-sample contact radius, but the HM-PFM amplitude and phase signals almost keep constant, indicating that the tip-sample contact radius has negligible influence to the HM-PFM signal.

S9. Supplementary Results of MAPbI 3 Perovskite
The fabricated MAPbI 3 film has been verified by performing optical absorption and X-Ray Diffraction (XRD) experiments. Figure S12a and S12b show the obtained UV-vis absorption spectra and XRD pattern, respectively, which confirms that the sample is MAPbI 3 perovskite.    Figure S13d, and the result indicates that V 2 O 5 almost has no electromechanical response thus its influence on the DFPFS measurement of the MAPbI 3 can be neglected. Note that the supplementary DFPFS data in Figure S13c and S13d uses the same amplitude scale with the Figure 7 of the main text.

S10. Measurement of Electrostriction
The HM-PFM developed here can be easily modified to measure the high-order electromechanical coupling, such as the important 2 nd -order coupling of electrostriction. Conventional PFM-based method has been extensively used to study the electrostriction, [14] while this method is typically influenced by the 2 nd -harmonic electrostatic force and Joule heating. Obviously, the 2 nd -harmonic electrostatic force contribution can be largely minimized in HM-PFM-based method by similar means of the 1 st -harmonic electrostatic force. Then due to the periodical temperature variation decays with increasing frequency, [15] the Joule heating-induced thermal strain can also be diminished in HM-PFM-based measurement. Figure S14 shows the modified HM-PFM set-up for the measurement of electrostriction, [16] in which the main modification is the reference signal generation circuit. However, if the reference signal is provided internally by synchronizing the clocks of signal source and lock-in amplifier, no hardware change is needed. As the electrostriction is a quadratic effect, the target electrostrictive vibration locates at the 2 nd -harmonic of the sample drive. Therefore, after heterodyne process, the electrostrictive vibration information i s included in the cantilever deflection signal with frequency of 2f s − f t . In a similar fashion, the 3 rd -order electromechanical coupling information should reside in the signal with frequency of 3f s − f t and the n th -order is in nf s − f t . If the clocks of signal source and lock-in amplifier can be synchronized, then detecting these high-order electromechanical couplings by HM-PFM-based method will be quite straightforward.