Top electrode size effects in the piezoresponse force microscopy of piezoelectric thin films attached to a rigid substrate

Abstract

In order to avoid the highly concentrated electric field induced beneath the sharp tip, the technique using a top coating electrode in the piezoresponse force microscopy (PFM) has been developed to detect the piezoelectric coefficients. Reliable theory should be erected to explain the broadly reported top electrode size effects and relate the responses with material constants. In this paper, the surface displacement, electric potential inside the film, electric charge and effective piezoelectric coefficient are expressed as a set of integral equations. Analytical solutions are obtained for two limiting cases, i.e., half space (HS) and infinitely thin film (IT). The effective piezoelectric coefficient of the HS case is proved to be the same as that from the PFM of a piezoelectric half plane without a top coating. For the IT case, it is identical to the well-known piezoelectric coefficient result of piezoelectric thin film clamped between flat rigid electrodes subject to homogeneous external electric field. For PZT4 thin layer, numerical results reveal that the surface displacement obviously decreases and the electric potential distributions inside the film become more and more homogeneous as the electrode radius to film thickness ratio (a/t) enlarges. The electric charge dramatically increases while the effective piezoelectric coefficient evidently decreases and they both transfer from the HS solutions to the IT results when a/t varies from 0.001 to 20. The transition occurs at about a/t = 1 in agreement with the experimental observations. A critical top electrode size, i.e., a/t > 10, is obtained and applicable to other piezoelectric materials. Under such circumstances, one can readily gain the piezoelectric coefficients e33, d33 and the dielectric coefficient if other mechanical coefficients and one piezoelectric constant are known a prior.

Extraction of Electromechanical Coefficients from Capacitance–Voltage Measurements of Unannealed Solution‐Processed KNN Thin Films: Effects of Frequency and Electrostatic and Mechanical Deformation

“…In general, this problem of undesirable poling of the thin film during the measurement changes as the dimensions of the top electrode change. [ 13,14 ] We note that localized poling exists as an issue even for PFM, which is often deemed to provide an overestimate of the actual piezoelectric coefficient ( d 33 ) in the case of thin‐film characterization. [ 15,42 ] In principle, the application of a local electric field exerted by the PFM cantilever can locally pole the domains by rotating the polarization direction of the domains in the vicinity of the cantilever/thin‐film system (due to inhomogeneous electric field).…”

“…[15,42] In principle, the application of a local electric field exerted by the PFM cantilever can locally pole the domains by rotating the polarization direction of the domains in the vicinity of the cantilever/thin-film system (due to inhomogeneous electric field). [13,14] A range of limiting values for true d 33 : an overestimate from PFM and a possible underestimate from CV measurements can thus be inferred to constrain possible d 33 measurements.…”

“…[5] Methods such as Berlincourt, [6] laser Doppler vibrometry (LDV), [7] double-beam interferometry, [8] and piezoresponse force microscopy (PFM) [9] have been standard methods of piezoelectric coefficient characterization. Issues such as substrate bending, [10,11] elastic coupling, [5] electrostatic artifacts between cantilever and substrate, [11,12] top electrode size effects, [13,14] large leakage currents, [15] concentrated inhomogeneous electric fields, [16] and local poling [17] complicate precise measurements. Further, thin-film based measurements (except PFM) usually are known to underestimate parameters like d 33 due to substrate clamping effects; when compared with the bulk material, [5,18] PFM, on the other hand, has been found to overestimate electromechanical coefficients d 33 .…”

Piezoresponse in Ferroelectric Materials under Uniform Electric Field of Electrodes

“…PFM allows measuring the piezoresponse in the micro and nano-objects where macroscopic measurement techniques fail and PFM becomes the only way to quantify the material’s piezoelectric coefficients [ 12 , 13 , 14 , 15 , 16 , 17 ]. Despite the rapid development of the PFM technique within the last 30 years, many issues still remain, such as the contribution from the electrostatic force [ 18 , 19 , 20 ] and other parasitic effects [ 11 , 21 , 22 , 23 ], as well as the difficulties of the PFM response quantification, i.e., understanding the relationship between the measured surface displacement and components of the piezoelectric tensor [ 17 , 24 , 25 , 26 , 27 , 28 ].…”

“…In both cases, the solution of the problem of electromechanical response quantification in the applied electric field is close to the conventional laser interferometry problem of the response under the action of the uniform or close-to-the uniform electric field. Contrary to the highly non-uniform electric field from the SPM probe, where the solution is usually limited to the case of the uniform elastic properties or needs numerical calculations [ 24 , 26 , 27 , 36 , 37 ], the piezoelectric response in the uniform electric field allows an accurate solution. Lefki and Dormans derived the solution [ 38 ] for the case of the tetragonal (001) oriented ferroelectric materials bonded to the rigid substrate, as it is imperative to the interpretation of the Doppler vibrometer and laser interferometry data [ 39 , 40 , 41 ].…”