This project, performed in collaboration with Alexei Gruverman (NCSU), Mark Kachanov and Edgar Karapetian (Tufts), addresses the quantitative aspects of ferroelectric behavior on the submicron and nanoscale, including local electromechanical property measurements, ferroelectric size effects, surface effect on polarization, and nanoscale polarization switching. This  is directly related to applications as diversified as nonvolatile ferroelectric memories (FeRAMs),  ferroelectric field effect transistors, ferroelectric heterostructures, and micron- and nanoscale electromechanical sensors and actuators. Understanding the local ferroelectric behavior on such lengthscales is possible only using Scanning Probe Microscopy techniques.
    Rapid development of SPM techniques in the last two decades has given rise to several microscopic techniques capable of accessing ferroelectric properties on the nanoscale, including Piezoresponse Force Microscopy (PFM), Atomic Force Acoustic Microscopy (AFAM), Scanning Near Field Acoustic Microscopy, etc. These techniques allow imaging of ferroelectric domain structures on the 3-10 nm level, providing direct information on localized electromechanical activity. Another broad set of electromechanical SPM applications is based on tip-induced changes in material properties. Application of high voltage or stress to the PFM tip can induce local 180 or 90 polarization switching, providing an approach to engineer and control domain structures at the nanoscale. This approach can potentially be used for high-density ferroelectric storage.

Minimal switched domain size was experimentally demonstrated to be as small as ~ 40 nm, corresponding to recording densities of order of 400 Gb/in². Theoretical predictions suggest that the minimal stable domain size is on the order of several unit cells; combined with demonstrated ferroelectric properties in the films of ~2 unit cell thickness, this paves the way to the nearly-atomic level data storage.

Among the eelctromechanical SPM techniques, the most popular  is Piezoresponse Force Microscopy (PFM), due to ease of implementation, high resolution and its relative insensitivity to topography. Application of periodic bias to the conductive tip in contact with ferroelectric surface results in periodic surface displacement due to inverse piezoelectric effect. Mapping of the amplitude and phase of the displacement allows imaging of ferroelectric domain structures with ~3-10 nm resolution. Therefore, PFM imaging provides direct insight into the nanoelectromechanics of ferro- and piezoelectric materials on the length scales defined by radius of the tip-surface contact and the radius of curvature of the tip.

Piezoresponse Force Microscopy can be implemented in the spectroscopic mode, thus allowing polarization switching phenomena in the form of vertical and lateral electromechanical hysteresis loops to be measured within individual ~100 nm grains. Polarization dependent reactivity of the surface in the acid etching  or metal photodeposition processes ,  can be used to engineer nanoscale structures (ferroelectric lithography). [Bonnell group]. The practical viability of these SPM applications is critically dependent on the minimal stable domain size that can be formed during polarization switching.

Similar issues arise in other electromechanical SPMs, as schematically illustrated below. In PFM, detected is the first harmonic of the mechanical response of the AFM tip induced by the periodic bias applied to the tip. In SNAM, detected is acoustic signal on the sensor induced by the periodic bias applied to the tip. In AFAM, measured is the first harmonic of the mechanical response of the AFM tip induced by the mechanical oscillations applied to the sample. In HUEFM, detected is mechanical response of the AFM tip at the difference frequency between the periodic bias applied to the tip and mechanical excitation applied to the sample.

Image formation mechanism in electromechanical SPMs of ferroelectric and piezoelectric materials is extremely complex an involves coupling between electrical and mechanical phenomena. In Piezoresponse Force Microscopy, the tip acts simultaneously as an electrical excitation source and mechanical detector. In Scanning Near Field Acoustic Microscopy, mechanical sensor detects acoustic waves generated locally by the bias applied to the SPM tip. In Atomic Force Acoustic Microscopy, SPM tip acts as a local mechanical sensor detecting the surface displacements produced by the actuator. In Heterodyne Ultrasonic Electrostatic Force Microscopy, SPM tip acts as a local mechanical sensor detecting the surface displacements produced by the frequency mixing between local electrostatic excitation and global mechanical excitation.

Understanding of image formation mechanisms in electromechanical SPMs requires contact mechanics of the piezoelectric materials for various tip shapes to be understood. Quantitative knowledge of tip-induced electrostatic and elastic fields inside the material is required to predict and control polarization switching processes and determined the minimal switchable domain sizes, etc.
    A number of approaches to model the PFM imaging mechanism and field distribution inside the ferroelectric using the point charge models were undertaken. However, the point charge model is clearly inapplicable for the description of a realistic tip shape when the tip size is comparable with the domain size. More importantly, the point charge model completely ignores strain effects. To address contact mechanics of piezoelectric indentation and quantify electroelastic fields in the ferroelectric materials, we solve problems of the piezoelectric indentation involving indenters of several different shapes.
    Using Correspondence Principle developed by Karapetian and Kachanov, we can solve problems of the piezoelectric indentation for indenters of several different shapes. Exact solutions for the contact mechanics of piezoelectric materials and electroelastic fields are obtained in elementary functions. These solutions then readily lend itself to studies of polarization switching dynamics, strain and charge phenomena in ferroelectrics.

    The first applications for these involve the description of contact mechanics for PFM. The stiffness relations fully describe the indentation process and relate indentation depth, indentation force and bias to the relevant material properties and indenter parameters. This constitutes an extension of Hertzian mechanics to piezoelectric materials.

Matrials properties in the stiffness relations are described by numerical constants C₁, C₃ and C₄, which can be identified as effective Young's modulus, effective piezoelectric constant, and effective dielectric constant for piezoelectric indentation. Notable, these constants do not depend on the indentor shape (while the functional form of the stiffness relations does). Also, similarly to the uniform field case, the same piezoelectric constants relates displacement and electric potential and induced charge and the force, suggesting the deep similarity between the physics of these systems.
    The coupling coefficients C₁, C₃ and C₄ are complex functions of 10 electroelastic constants of material. Relative contribution of individual electroelastic constants can be estimated as shown on the slide above.  C₁ is clearly dominated by the elastic constants,  C₃ is determined predominantely by piezoelectric and dielectric constants and C₄ depends primarily on dielectric constants, which agrees with their interpretation as Young's modulus, piezolectric constant and dielectric constant. The piezoresponse amplitude measure din PFM is determined by the ratio C₁/C₃. We can correlate the PFM amplitude and effective dielectric constant with d₃₃ and averaged dielectri cconstant of material, and for a broad variety of materials we have a linear correlation.

    These stiffness relations provide the relationship between tip displacement, tip potential and indentation force, thus describing the imaging mechanisms of electromechanical SPMs, as illustrated below on an example of Piezoresponse Force Microscopy.

    The electroelastic field distribution below PFM tip for PZT6b material. The use of correspondence principle allows separating electrical and mechanical contributions to the field below the indenter by solving problem with purely electrical and purely mechanical boundary conditions.

Electroelastic fields for BaTiO₃. Notice the difference between the shape of electric potential distribution due to the large anistropy in dielectric tensor of the material.

    The fields at large separations from indenter are expected to follow point charge/ point force type behavior. Shown below is the ratio between exact coupled electroelastic solution and fields in purely elastic and purely electrostatic cases.

    The electroelastic field distributions can be used to numerically analyze polarization switching processes. We have shown that ferroelectric switching can be described using point charge type models. However, more complex electroelastci switching requires exact field distribution to be used.