Piezoresponse force microscopy (PFM) has emerged as a powerful tool for research in ferroelectric and piezoelectric materials. While the vertical PFM (VPFM) mode is well understood and applied at a quantitative level, the lateral PFM (LPFM) mode is rarely quantified, mainly due to the lack of a practical calibration methodology. Here by PFM imaging on a LiNbO3 180° domain wall, we demonstrate a convenient way to achieve simultaneous VPFM and LPFM calibrations. Using these calibrations, we perform a full quantitative VPFM and LPFM measurement on a (001)-cut PbTiO3 single crystal. The measured effective piezoelectric coefficients d33eff and d35eff together naturally provide more information on a material's local tensorial electromechanical properties. The proposed approach can be applied to a wide variety of ferroelectric and piezoelectric systems.

Piezoresponse force microscopy (PFM) has been applied as a powerful tool for the characterization of ferroelectric and piezoelectric materials.1 Because a highly localized electric field distribution can be achieved under a biased conductive tip, PFM is especially important in nanoscale characterization and manipulation of domains and domain walls.2–8 To date, quite a few works have reported quantitative vertical PFM (VPFM) measurements, such as studies on the substrate clamping effect in ferroelectric thin film system,9,10 piezoelectric characterization in biomaterials,11,12 and improper ferroelectricity in multiferroic hexagonal manganite.13,14 Nevertheless, the as-measured VPFM piezoelectric response should not be considered as a direct reflection of a material's intrinsic vertical piezoelectric coefficient, d33, because the electric field induced by the conductive atomic force microscope (AFM) tip during the PFM scanning is not uniform, but rather localized and highly inhomogeneous within the material volume underneath the tip. Therefore, we name the as-measured VPFM piezoresponse as the effective piezoelectric coefficientd33eff for future reference. A quantitative VPFM characterization is essentially the measurement of d33eff.

For a full PFM quantification, a proper calibration is necessary. Currently, the following two approaches are commonly used in VPFM calibrations. One is to take advantage of a separate piezoelectric sample whose piezoelectric coefficient d33 is well known, to characterize it with a PFM tip, and to use the same cantilever for further quantification of the unknown sample using the same calibration factors. With electrodes on both sides, the vertical piezoelectric response can be readily calculated under the uniform field approximation. This method can be found in Refs. 9, 10, and 15. An alternative method relies on the force-distance curve to extract the optical lever sensitivity. Examples of this method can be seen in Refs. 12, 16, and 17. Particularly, Nagarajan et al.18 previously tested both of these calibration methods, and they concluded that they are in good agreement. Furthermore, in Ref. 30, the quantitative VPFM measurement based on the calibration from the force-distance curve was in good agreement with the theoretical prediction. Compared with the routine applications of VPFM quantifications, quantitative lateral PFM (LPFM) measurements are not commonly used, mainly due to a lack of well-accepted calibration methodology. To address this issue, Choi et al.19 suggested a method to calibrate the cantilever torsional mode by tracking the lateral signals from both trace and retrace scans of various distances, thus leading to quantitative LPFM measurements. This method was recently significantly improved by Wang et al.20 with a better sensitivity, and it was generalized to calibrations for both buckling and torsional modes, making it possible to determine the exact local polarization orientation for a fully in-plane polarized ferroelectric sample. Compared with these works, we aim to demonstrate a practical method to achieve simultaneous VPFM and LPFM calibrations.

Kalinin et al.21 initially conceived a convenient approach to perform simultaneous VPFM and LPFM calibrations on a (001)-cut BaTiO3 single crystal. This approach takes advantage of the fact that PFM is able to provide an image, not only a single-point measurement. Since the piezoelectric properties of BaTiO3 are well known, both calculations based on the analytical theory22–27 as well as finite element method (FEM)15,28,29 are suitable for the prediction of the effective piezoelectric coefficients measured by PFM. The effective piezoelectric response (deformation) depends not only on material's intrinsic piezoelectric coefficients but also on its full elastic and dielectric constants, which are fully included in the calculations based on FEM simulations. By correlating the calculated values of d33eff on the c-domain and d35eff on the a-domain with the corresponding experimental PFM responses, both VPFM and LPFM calibrations can be achieved. In practice, one single PFM imaging on an area involving both a- and c-domains is sufficient. Unfortunately, no follow-up experimental implementations have been reported so far. Here, we perform the VPFM and LPFM calibrations using this idea.

Previously with analytical theory and FEM simulations,29 we studied the PFM response on a 180° domain twinning with a ↑|↓ polarization configuration in a z-cut LiNbO3 single crystal. Since the theory predicted VPFM and LPFM line-profiles show a reasonably good agreement with experiments,30 it offers us an opportunity to calibrate the VPFM and LPFM response simultaneously on a single LiNbO3 sample. Based on this calibration, we demonstrate a quantitative VPFM and LPFM measurement on a (001)-cut PbTiO3 single crystal. With the FEM simulations, PFM measured effective piezoelectric coefficients d33eff and d35eff are compared with the intrinsic piezoelectric coefficients d15, d31, and d33. The VPFM and LPFM line-profiles across the tilted 90°-domain wall predicted by FEM in this tetragonal ferroelectrics is found to be reasonably consistent with experiments.

This paper is organized as follows. Section II describes the detailed procedures on quantitative PFM measurements on a PbTiO3 single crystal. Section III discusses how quantitative PFM measured effective piezoelectric coefficients d33eff and d35eff provide insight on a material's intrinsic piezoelectric properties. Section IV presents our brief conclusions.

The goal of VPFM and LPFM calibrations is to determine two optical amplification factors: δV and δL. δV is defined in the following expression:31 

AVPFM=δVd33effVAC,
(1)

where AVPFM is the amplitude of the VPFM response, VAC is the amplitude of AC driving voltage, and d33eff is the effective piezoelectric coefficient that can be determined by the calibrated VPFM measurement. Essentially, δV is the conversion factor between the VPFM raw signals (in mV) and vertical displacement (in pm) of the sample surface under the tip. Likewise, the optical amplification factor δL is defined as

ALPFM=δLd35effVAC,
(2)

where ALPFM is the amplitude of the LPFM response and d35eff is the effective piezoelectric coefficient determined by LPFM after calibration. Essentially, δL is the conversion factor between the LPFM raw signals (in mV) and lateral displacement (in pm) of the sample surface under the tip.

Here, the δV and δL of the same cantilever used for later PbTiO3 measurements are initially determined by a PFM imaging on a z-cut LiNbO3 single crystal. The PFM experiments were performed under ambient conditions on a Bruker Icon atomic force microscope (AFM) system using a platinum coated HQ:NSC36/Pt conductive tip from MikroMasch. The incident laser spot is positioned at the free end of the cantilever and approximately right above the protruded tip. The spring constant of the tip is determined to be 1.9 N/m by an energy equipartition method.32 Although the indentation force was set to be 40 nN, we actually did not observe any noticeable changes in the PFM measurements when the indentation force changed from 10 nN to 40 nN. This is in agreement with the PFM measurement in the strong indentation regime.33 Under this condition, the piezoelectric response dominates the measured PFM response. For the convenience of PFM quantifications, the frequency of AC driving voltage is set to be 20 kHz, far below the contact resonance frequency of ∼700 kHz. Under this condition, the tip is in intimate contact with the sample surface and the measured PFM response shows negligible frequency dependence. In order to maximally reduce the level of background, the PFM imagings were performed by applying the modulation voltage on the back electrode of the sample while keeping the conductive AFM tip grounded. This is the same as the way performed in Refs. 13, 20, and 30.

Fig. 1(a) shows the scanning configuration, with the PFM images shown in Figs. 1(b)–1(e). The cantilever axis is aligned to be parallel to the domain wall so that the LPFM response comes purely from the x-component piezoresponse. The fast scan direction is perpendicular to the wall so that any unexpected drift from the piezo scanning stage can be conveniently monitored.

FIG. 1.

(a) Schematics of PFM calibrations on a z-cut periodically poled LiNbO3 single crystal. The domain wall is located at x = 0. The polarization is along the −z direction (−Ps) for the x < 0 domain, and along the +z direction (+Ps) for x > 0 domain. xtip denotes the tip position. The fast scan direction is perpendicular to the domain wall. Panels (b)–(e): PFM images across a 180° domain wall, with the amplitude of AC driving voltage setting to 5 V.

FIG. 1.

(a) Schematics of PFM calibrations on a z-cut periodically poled LiNbO3 single crystal. The domain wall is located at x = 0. The polarization is along the −z direction (−Ps) for the x < 0 domain, and along the +z direction (+Ps) for x > 0 domain. xtip denotes the tip position. The fast scan direction is perpendicular to the domain wall. Panels (b)–(e): PFM images across a 180° domain wall, with the amplitude of AC driving voltage setting to 5 V.

Close modal

After background subtraction, the averaged VPFM and LPFM line-profiles are shown in Fig. 2(a). For a detailed description of background subtraction procedures, see Ref. 30.

FIG. 2.

(a) VPFM and LPFM line-profiles after background subtractions. (b) and (c) The FEM simulations predicted d33eff and d35eff line-profiles across a 180° domain wall in a z-cut LiNbO3 single crystal. Experimental PFM line-profiles in (a) are normalized to the FEM calculation results. Note: FEM results with two different values of tip radius, r = 11 nm and 30 nm, are presented. Excellent fits to the experimental line-profiles are obtained for r = 11 nm tip radius.

FIG. 2.

(a) VPFM and LPFM line-profiles after background subtractions. (b) and (c) The FEM simulations predicted d33eff and d35eff line-profiles across a 180° domain wall in a z-cut LiNbO3 single crystal. Experimental PFM line-profiles in (a) are normalized to the FEM calculation results. Note: FEM results with two different values of tip radius, r = 11 nm and 30 nm, are presented. Excellent fits to the experimental line-profiles are obtained for r = 11 nm tip radius.

Close modal

Since the piezoelectric properties of LiNbO3 are well known,34 the expected d33eff and d35eff from the PFM measurements can be calculated by either analytical theory or FEM simulations.29,30 Here, we use the latter approach. Figures 2(b) and 2(c) show the calculated d33eff and d35eff across the 180° domain wall. By correlating the experimental values with the calculated ones, the calibration factors of δV and δL are determined. Note that the experimental values are obtained by ramping the applied voltage with a linear fit, which was shown to generate more reliable results in our previous work.30 In Figs. 2(b) and 2(c), we show the calculated PFM line-profiles with two different values of tip radius r. While a slight change of r from 11 nm to 30 nm leads to no observable changes of the maximum d33eff and d35eff in the piezoresponse line-profiles (the maximum d33eff and d35eff remain as 12.8 pm/V and 12.0 pm/V, respectively, within the error of ±0.1 pm/V in our simulations), it results in a significant increase in the diffuseness of the PFM line-profiles near the wall. A tip contact radius of r = 11 nm well reproduces the VPFM and LPFM line-profiles simultaneously. The VPFM and LPFM optical amplification factors δV and δL are thus determined to be 0.0172 mV/pm and 0.0615 mV/pm, respectively. Next, we demonstrate the quantitative VPFM and LPFM measurements on a tetragonal ferroelectric single crystal, PbTiO3.

PbTiO3 single crystals were grown using a flux method with PbO flux.35 The sample has a (001) orientation, with the size of 3 mm × 4 mm × 0.4 mm. 90° charge-neutral domain walls are typically found in this tetragonal system. We focus on the so-called a/c/a domain structure, consisting of alternating a domains with polarization parallel to (001) surface and c domains with polarization perpendicular to the (001) surface. Due to the lattice tetragonality and continuity condition at the domain wall, the angle between the spontaneous polarizations in the neighboring two domains is in reality, 2tan−1(c/a) = 93.6°, rather than nominal 90°.36,37 Consequently, surface undulations are expected for a/c/a domain structures, with a bending angle of 3.6°. This is one of the signatures for the a/c/a domain configurations.

Figure 3(a) shows the schematics of PFM imaging on a c/a domain twinning. In the current coordinate system, a mirror symmetry exists in the x-z plane; thus, no piezoresponse is expected in the y-direction. For this reason, the cantilever axis is set to be parallel to the domain wall so that the VPFM response comes purely from the z-component piezoresponse (the possible coupling from the bulking contribution is avoided due to zero y-component piezoresponse), and the LPFM channel measures the x-component piezoresponse. The PFM images in Fig. 3(b) were performed with VAC = 1 V. Typically for a PFM measurement, a higher excitation voltage is preferred for higher signal-to-noise ratio, as is implemented in the PFM calibration step. However, it is important to note that, as the excitation voltage increases, the domain wall dynamics starts to dominate the PFM response near the wall. This behavior is called “correlated switching” and is in detail studied in the work by Aravind et al.38 and will be briefly discussed in Sec. III in this work. For clean PFM line-profiles with minimal domain nucleation dynamics involved, a lower voltage is thus preferred, which is the reason that a small VAC of 1 V is applied here.

FIG. 3.

(a) Schematic of the PFM setup on the c/a domain twin, with a cantilever parallel to the domain wall in a single crystal PbTiO3. The domain wall at the top surface cross-section is set as x = 0. Arrows indicate the polarization directions. (b) Topography and piezoresponse images simultaneously obtained from one PFM imaging. (c) Averaged topography line-profile with γ = 3.68°. (d) Averaged piezoresponse line-profiles from (b). (e) VPFM and LPFM line-profiles after background noise subtractions. Note that the piezoresponse profiles plotted in (d) and (e) have considered the calibrations factors of δV and δL.

FIG. 3.

(a) Schematic of the PFM setup on the c/a domain twin, with a cantilever parallel to the domain wall in a single crystal PbTiO3. The domain wall at the top surface cross-section is set as x = 0. Arrows indicate the polarization directions. (b) Topography and piezoresponse images simultaneously obtained from one PFM imaging. (c) Averaged topography line-profile with γ = 3.68°. (d) Averaged piezoresponse line-profiles from (b). (e) VPFM and LPFM line-profiles after background noise subtractions. Note that the piezoresponse profiles plotted in (d) and (e) have considered the calibrations factors of δV and δL.

Close modal

By averaging each line-scan in the images shown in Fig. 3(b), the topography and piezoresponse line-profiles across the wall were obtained, as shown in Figs. 3(c) and 3(d), respectively. The surface bending angle across this domain wall is γ = 3.68°, which is in good agreement with the aforementioned theoretical value of 3.60°. The measured PFM line-profiles after background subtraction30 are presented in Fig. 3(e). The background-free VPFM and LPFM line-profiles will be used for comparison with the FEM calculated line-profile in Section III.

From Fig. 3(e), the vertical piezoresponse on the c-domain away from the wall is estimated to be ∼34 pm in terms of displacement amplitude, while the lateral response on the a-domain is ∼22 pm, under the condition of VAC = 1 V. By simply assuming that PFM measurement linearly scales with the applied driving voltage VAC, the effective piezoelectric coefficients d33eff and d35eff are readily calculated as 34 pm/V and 22 pm/V, respectively. However, this assumption is not necessarily valid practically and can indeed be spurious for small applied voltages. By ramping VAC from 0 to 4 V, the measured VPFM response on the c-domain and LPFM response on the a-domain are shown in Fig. 4. When VAC < 1 V, the measured PFM response shows a noticeable nonlinear scaling behavior with VAC. Such nonlinear behavior was observed in several prior studies.30,39–42 Although the origin of this nonlinearity is not yet well understood, a general way to avoid this effect is by fitting the linear part. This approach has been proven to give a good description of the effective piezoelectric coefficients d33eff,39–43 

d33eff=1δVdAVPFMdVAC.
(3)
FIG. 4.

The VPFM response on the c-domain and LPFM response on the a-domain in a single crystal PbTiO3 as a function of VAC. The fitting is performed on the linear part (VAC > 2 V).

FIG. 4.

The VPFM response on the c-domain and LPFM response on the a-domain in a single crystal PbTiO3 as a function of VAC. The fitting is performed on the linear part (VAC > 2 V).

Close modal

From the fitting in Fig. 4, d33eff on the c-domain and d35eff on the a-domain are determined to be 97 pm/V and 59 pm/V, respectively. These values are closely related to the intrinsic piezoelectric coefficients (d15, d31, and d33) of PbTiO3 as described in Section III.

In order to check the reproducibility, we also tested our quantitative PFM measurements by using a different type of tip, HQ:NSC19/Cr-Au from MikroMasch. The spring constant of this tip is determined to be 1.2 N/m by an energy equipartition method. In the supplementary material, we also provide the PFM ramping curve similar to Figure 4. The d33eff on the c-domain and d35eff on the a-domain are determined to be 106 pm/V and 56 pm/V, respectively, compared with 97 pm/V and 59 pm/V by using the tip, HQ:NSC36/Pt. This is in reasonably good agreement (within 10% variations) by the PFM quantifications. The minor variation may occur from the system-inherent background.20 

Currently, there is a moderate discrepancy on the reported piezoelectric coefficients of PbTiO3 in the literature. This is mainly due to the fact that a sizable single crystal with a single domain state is not readily obtained because of the large c/a ratio of 1.065.35,36 For the sake of convenience of discussion, we list five independent sets of material properties in Table I, of which three sets are from Brillouin scattering measurements,44,45 one set from resonance measurements46 and the remaining one from the Landau–Ginsburg–Devonshire thermodynamic model.47 The discrepancy is intensively reflected on the piezoelectric coefficients of d31 and d33. The measurements from the resonance method [Set 4 in Table I] may be problematic because the authors later significantly revised their values on the dielectric constants.48 Subsequent Brillouin scattering measurements44,45 show good consistency in the acoustic velocity. However, due to a low sensitivity of acoustic measurements to the piezoelectric coefficients of d31 and d33, the postoptimization procedures lead to large variations in d31 and d33. In comparison, the property set 5 in Table I from the Landau–Ginsburg–Devonshire thermodynamic model is very close to Set I. For these reasons discussed above, we focus on the analysis on the first three sets of material properties in Table I.

TABLE I.

Intrinsic piezoelectric coefficients d15, d31, and d33 (pm/V) of PbTiO3 single crystals from various literature sources.

Set 1aSet 2bSet 3cSet 4dSet 5e
d15 (pm/V) 60.2 63.8 69.6 65 56 
d31 (pm/V) −27.5 −22.8 −49.9 −25 −23 
d33 (pm/V) 83.7 136.6 156.4 117 80 
Set 1aSet 2bSet 3cSet 4dSet 5e
d15 (pm/V) 60.2 63.8 69.6 65 56 
d31 (pm/V) −27.5 −22.8 −49.9 −25 −23 
d33 (pm/V) 83.7 136.6 156.4 117 80 
a

Brillouin scattering measurements from Ref. 44.

b

Material properties set A from Brillouin scattering measurements in Ref. 45.

c

Material properties set B from Brillouin scattering measurements in Ref. 45.

d

Resonance measurements from Ref. 46.

e

Calculated values from the Landau–Ginsburg–Devonshire thermodynamic model from Ref. 47.

FEM calculated effective piezoelectric coefficients d33eff on the c-domain and d35eff on the a-domain as well as our quantitative PFM measured experimental results are listed in Table II. Note that the tip-sample contact radius of 11 nm determined in Section II is applied in the FEM simulations here. Details on the FEM simulations can be found in Ref. 29.

TABLE II.

Effective piezoelectric coefficients d33eff and d35eff (pm/V) from the FEM simulations and a comparison with our quantitative PFM measurements. Note: the experimental values in the parenthesis are from the measurement with a separate tip of HQ:NSC19/Cr-Au.

Set 1Set 2Set 3PFM experiments
d33eff on c-domain (pm/V) 43 83 62 97 (106) 
d35eff on a-domain(pm/V) 57 76 74 59 (56) 
Set 1Set 2Set 3PFM experiments
d33eff on c-domain (pm/V) 43 83 62 97 (106) 
d35eff on a-domain(pm/V) 57 76 74 59 (56) 

By comparing the FEM calculated effective piezoelectric coefficient d33eff in Table II with the corresponding intrinsic piezoelectric coefficient d33 listed in Table I, some general trends are seen: the calculated d33eff is appreciably smaller than d33 in this PbTiO3 system, which is consistent with the prior conclusion by Wang and Chen49 on 9 groups of transversally isotropic piezoelectric materials using various analytical models. This is different from LiNbO3, where d33eff is determined to be significantly larger than d33 by analytical theory23 and FEM modeling,29 and later confirmed by quantitative VPFM measurements.30 The difference between PbTiO3 and LiNbO3 is due to their distinct crystal symmetries and piezoelectric tensors. By comparing the FEM calculations with PFM measurements in Table II, the LPFM determined d35eff is close to that calculated from material property set 1, while VPFM measured d33eff is closer to that calculated from material property set 2. The d33eff and d35eff calculated from material property set 3 have relatively larger disagreement with our PFM measurements, which probably indicates that this set of material property itself is problematic.

From Table II, PFM measurements show that d33eff is evidently larger than d35eff. Based on this criterion, material property set 2 should be the one that best reflects the real piezoelectric properties of PbTiO3. As mentioned above, Brillouin scattering experiments directly measured that the acoustic velocity has a low sensitivity to the piezoelectric coefficients of d31 and d33, thus leaving a relatively large uncertainty. Now with the aid of quantitative VPFM and LPFM measurements, additional information on piezoelectric coefficients is gained. In this case, the quantitative PFM can act as a complementary tool for ascertaining an unknown material's piezoelectric coefficients measured by other techniques.

For PFM techniques based on the contact-resonance enhancement, such as Dual AC Resonance Tracking (DART)50 or Band Excitation (BE),51 quantification is more challenging. The tip-sample contact is not maintained near the resonance and the calibration factor is closely related to the spring constant of the cantilever because it directly affects the cantilever shape near resonance. Very recently, an article by Balke et al.52 is dedicated to the study of quantification of surface displacements and electromechanical phenomena via dynamic atomic force microscopy. Details on the problems and strategies on this topic can be found there.

Since material property set 2 in Table I best describes PbTiO3's intrinsic piezoelectric properties, the effective piezoelectric responses d33eff and d35eff across the c/a domain wall are calculated by the FEM modeling based on this set of material parameters. Figure 5 shows the normalized line-profiles compared with experiments. Overall, it shows a reasonably good agreement between FEM prediction and experiment. The slight disagreement is mainly due to the domain wall motion/bending near the titled c/a domain wall, while in our FEM simulations, a static domain wall is assumed. This domain wall bending effect is more pronounced when the driving voltage is larger. The static domain wall assumption is sufficient to reproduce the experimental PFM line-profiles across the 180° domain wall in a z-cut LiNbO3 crystal when the driving voltage is at VAC = 5 V, as is manifested in Figs. 2(b) and 2(c). However, with higher driving voltage, domain wall bending effect gradually shows up and could even play a dominant role in the PFM response near the wall. This phenomenon has previously been systematically studied in the proximity of a 180° domain wall in LiNbO3 using switching spectroscopy PFM (SS-PFM).38,53 Its long-range effect on the PFM line-profiles is up to 2–3 μm away from the wall at high voltage of 28 V. In contrast, our current PFM line-profile across a tilted c/a wall is obtained with a low driving voltage of VAC = 1 V. The fact that the domain wall bending effect shows up near the wall at such low voltage in PbTiO3 compared with LiNbO3 case demonstrates the high mobility of the tilted c/a domain wall in a tetragonal ferroelectrics. A systematic experimental SS-PFM study and phase-field simulations on the domain nucleation and switching dynamics near the asymmetric c/a and a/c walls in PbTiO3 will be presented elsewhere.

FIG. 5.

Normalized piezoresponse line-profiles from the FEM modeling and experiment. The slight disagreement between FEM and experiment is due to the domain wall motions. The experimental PFM line-profile in the black dashed-circle shows a slight kink compared to the FEM simulation results. This kink actually becomes more pronounced with larger driving voltages. The orange dashed-circled anomalous response is due to the cross-talk with a topography feature, see Fig. 3(b) Topography channel.

FIG. 5.

Normalized piezoresponse line-profiles from the FEM modeling and experiment. The slight disagreement between FEM and experiment is due to the domain wall motions. The experimental PFM line-profile in the black dashed-circle shows a slight kink compared to the FEM simulation results. This kink actually becomes more pronounced with larger driving voltages. The orange dashed-circled anomalous response is due to the cross-talk with a topography feature, see Fig. 3(b) Topography channel.

Close modal

We demonstrated a practical and convenient approach to perform full quantitative VPFM and LPFM measurements. A simultaneous VPFM and LPFM calibration was realized based on an excellent quantitative understanding of PFM imaging on a LiNbO3 180° domain twinning. After calibrations, we performed a full quantitative PFM study on a classic tetragonal ferroelectrics, PbTiO3. With the FEM simulations, quantitative PFM measured effective piezoelectric coefficients d33eff and d35eff are understood in terms of the intrinsic piezoelectric tensor coefficients of PbTiO3. As compared with other techniques such as Brillouin scattering measurements, quantitative PFM measurements provide a unique opportunity to characterize the electromechanical properties on ∼100 nm scale due to the strong localization of electric field under the tip. The method is potentially useful in the characterization of the piezoelectricity in a material system with more complex microstructures involving domains, grains, or composite phases.

See supplementary material for quantitative VPFM and LPFM ramping curves with respect to the excitation voltage VAC on PbTiO3 by using a separate tip, HQ:NSC19/Cr-Au from MikroMasch.

We thank Lei Zhang and Y. Zhao for useful discussions. S.L. and V.G. acknowledge the financial support from the U.S. National Science Foundation (NSF) Grant Nos. DMR 1420620 (Penn State MRSEC Center) and DMR-1210588. S.W.C. was funded by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant GBMF4413 to the Rutgers Center for Emergent Materials. The FEM calculations were carried out on CyberSTAR clusters at the Pennsylvania State University, supported in part through instrumentation funded by NSF through Grant No. OCI-0821527.

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