Quantitatively Analyzing Nanoscale The Piezoelectric Coefficients Of Piezoelectric Materials Under PFM Measurement

Piezoelectric materials have been widely used in various microelectronic devices and microelectromechanical systems due to their excellent electromechanical properties.However,with the increasing demand miniaturization and multifunction of piezoelectric materials and devices,it is urgent to characterize their electromechanical properties at the nanoscale and provide important guidance for the design and optimization of related micro-nano materials and devices.Piezoresponse force microscopy(PFM),developed as an important tool in family of atomic force microscopy techniques,has been widely utilized to nondestructively characterize various non-centrosymmetric polar materials including piezoelectric and ferroelectric materials at the nanoscale.Piezoelectric effect is one of the important electromechanical properties of piezoelectric materials,which has direct correlation with piezoelectric coefficients.However,it is still very difficult to quantitatively analyze piezoelectric coefficient via PFM.On the one hand,under contact-mode PFM measurement,the electroelastic interactions between the scanning probe microscopy(SPM)probe and the piezoelectric materials are very complicated due to the simultaneous application of mechanical and electrical loadings,and this makes it difficult to decouple the intrinsic piezoelectric displacement responses from the total displacement responses measured by PFM.On the other hand,the electromechanical properties of the material are unknown during the PFM experiment,it is necessary to further analyze the intrinsic properties of the specimen from the measured displacement response.However,the displacement response measured by PFM couples with elastic,dielectric,and piezoelectric properties.Therefore,it remains a big challenge on quantitative determination of intrinsic piezoelectric coefficients of piezoelectric materials by reversing from the displacement response in PFM experiment.To overcome these difficulties,the framework of electroelastic theory is developed to quantitatively analyze the piezoelectric coefficients in piezoelectric mediums,where the main works and core conclusions are summarized as follows.(1)Considering the contact-mode PFM experiment,the fully coupled electromechanical theory is well established.Here the piezoelectric materials assuming to be transversely isotropic mediums,contact with the three typical conductive SPM probes,including cylindrical probe,conical probe,and spherical probe.Hence,the distribution of the electroelastic fields are derived in piezoelectric materials.The results show that the electric fields under the cylindrical and spherical probe demonstrate singularities at the contact edge,which leads to the stress concentration and causes the comparable larger strain.Furthermore,the finite element method(FEM)is adopted to compare with our constructed fully coupled method,and the results proved that their electromechanical fields agree well at the most part of the area except the contact edge.(2)Based on the fully coupled theoretical framework under contact-mode PFM experiment,we provide a novel approach to determine the intrinsic piezoelectric response from displacement response measured by PFM.It is proved that the effective piezoelectric coefficient obtained by the contact-mode PFM experiment is related to the external loadings,and geometries of the SPM probe,which infers that the distribution of the electroelastic fields affects the intrinsic piezoelectric response.Meanwhile that the effective piezoelectric coefficients have the strong correlations with the polarization orientation in ferroelectric materials.(3)Under non-contact mode PFM experiment,the displacements and potential have nonlinear relation with intrinsic electromechanical properties and the measurement conditions,which enables us to reverse out the piezoelectric coefficients by using the optimization method.After objective function of the displacement and potential are constructed,the optimization is divided into two steps by adopting the differential evolution theory.In the process of optimization,we studied the effects of population size and iteration evolution generations on the piezoelectric coefficients and their discreteness.The results show that the piezoelectric coefficients e33,e15,e31 derived after optimization,agree well with their intrinsic ones.