Oak Ridge, TN 37831 Piezoresponse Force M icroscopy (PFM ) has emerged as a primary tool for imaging, domain engineering, and switching spectroscopy on ferroelectric materials. Quantitative interpretation of PFM data including measurements of the intrinsic width of the domain walls, geometric parameters of the domain below the tip in local hysteresis loop measurements, as * Permanent address: V.Lashkaryov Institute of Semiconductor Physics, National Academy of Science of Ukraine, 41, pr. Nauki, 03028 Kiev, Ukraine † Corresponding author, sergei2@ornl.gov 2 well as interpretation of switching and coercive biases in terms of materials properties and switching mechanisms, requires reliable knowledge on electrostatic field structure produced by the tip. Using linear imaging theory, we develop a theoretical approach for interpretation of these measurements and determination of tip parameters from a calibration standard. The resolution and object transfer functions in PFM are derived and effect of materials parameters on resolution is determined. Closed form solutions for domain wall profiles in vertical and lateral PFM and signal from cylindrical domain in transversally isotropic piezoelectric are derived for point-charge and sphere-plane geometry of the tip. 3

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The rapid development of nanoscience and nanotechnology in the last two decades was stimulated by the emergence of scanning probe microscopy (SPM) techniques capable of accessing local material properties, including transport, mechanical, and electromechanical behavior on the nanoscale. Here, we analyze the general principles of electromechanical probing by piezoresponse force microscopy (PFM), a scanning probe technique applicable to a broad range of piezoelectric and ferroelectric materials. The physics of image formation in PFM is compared to Scanning Tunneling Microscopy and Atomic Force Microscopy in terms of the tensorial nature of excitation and the detection signals and signal dependence on the tipsurface contact area. It is shown that its insensitivity to contact area, capability for vector detection, and strong orientational dependence render this technique a distinct class of SPM.The relationship between vertical and lateral PFM signals and material properties are derived analytically for two cases: transversally-isotropic piezoelectric materials in the limit of weak elastic anisotropy, and anisotropic piezoelectric materials in the limit of weak elastic and dielectric anisotropies. The integral representations for PFM response for fully anisotropic material are also obtained. The image formation mechanism for conventional (e.g., sphere and cone) and multipole tips corresponding to emerging shielded and strip-line type probes are analyzed. Resolution limits in PFM and possible applications for orientation imaging on the nanoscale and molecular resolution imaging are discussed.

Frequency dependent dynamic electromechanical response of the mixed ionic-electronic conductor film to a periodic electric bias is analyzed for different electronic and ionic boundary conditions. Dynamic effects of mobile ions concentration (stoichiometry contribution), charge state of acceptors (donors), electron concentration (electron-phonon coupling via the deformation potential) and flexoelectric effect contribution are discussed. A variety of possible nonlinear dynamic electromechanical response of MIEC films including quasi-elliptic curves, asymmetric hysteresis-like loops with pronounced memory window and butterfly-like curves are calculated. The electromechanical response of ionic semiconductor is predicted to be a powerful descriptor of local valence states, band structure and electronphonon correlations that can be readily measured in the nanoscale volumes and in the presence of strong electronic conductivity. Keywords: thin films of ionic semiconductors, dynamic electromechanical response, deformation potential, flexoelectric effect. 0 = = h J J d c d c 0 = = h J J d c d c .Other parameters are listed in the capture to Fig. 4.

Electromechanical hysteresis loop formation in piezoresponse force microscopy of thin ferroelectric films is studied with special emphasis on the effects of tip size and film thickness, as well as dependence on the tip voltage frequency. Here, we use a combination of Landau-Ginzburg-Devonshire (LGD) theory for the description of the local polarization reversal with decoupling approximation for the calculation of the local piezoresponse loops shape, coercive voltages and amplitude. LGD approach enables addressing both thermodynamics and kinetics of hysteresis loop formation. In contrast to the "rigid" ferroelectric approximation, this approach allows for the piezoelectric tensor components dependence on the ferroelectric polarization and dielectric permittivity. This model rationalizes the non-classical shape of the dynamic piezoelectric force microscopy (PFM) loops.