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Nanoscale Electromechanics of Ferroelectric and Biological Systems: A New Dimension in Scanning Probe Microscopy Sergei V. Kalinin,1 Brian J. Rodriguez,1 Stephen Jesse,1 Edgar Karapetian,2 Boris Mirman,2 Eugene A. Eliseev,3 and Anna N. Morozovska4 1

Materials Sciences and Technology Division and Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831; email: [email protected]

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Department of Mathematics and Computer Science, Suffolk University, Boston, Massachusetts 02114

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Institute for Problems of Materials Science, National Academy of Science of Ukraine, 03142 Kiev, Ukraine

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V. Lashkaryov Institute of Semiconductor Physics, National Academy of Science of Ukraine, 03028 Kiev, Ukraine

Annu. Rev. Mater. Res. 2007. 37:189–238

Key Words

The Annual Review of Materials Research is online at http://matsci.annualreviews.org

piezoelectricity, flexoelectricity, hysteresis, domain dynamics, polarization switching

This article’s doi: 10.1146/annurev.matsci.37.052506.084323 c 2007 by Annual Reviews. Copyright  All rights reserved 1531-7331/07/0804-0189$20.00

Abstract Functionality of biological and inorganic systems ranging from nonvolatile computer memories and microelectromechanical systems to electromotor proteins and cellular membranes is ultimately based on the intricate coupling between electrical and mechanical phenomena. In the past decade, piezoresponse force microscopy (PFM) has been established as a powerful tool for nanoscale imaging, spectroscopy, and manipulation of ferroelectric and piezoelectric materials. Here, we give an overview of the fundamental image formation mechanism in PFM and summarize recent theoretical and technological advances. In particular, we show that the signal formation in PFM is complementary to that in the scanning tunneling microscopy (STM) and atomic force microscopy (AFM) techniques, and we discuss the implications. We also consider the prospect of extending PFM beyond ferroelectric characterization for quantitative probing of electromechanical behavior in molecular and biological systems and high-resolution probing of static and dynamic polarization switching processes in low-dimensional ferroelectric materials and heterostructures. 189

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INTRODUCTION Nonvolatile ferroelectric random access memory (FeRAM): memory technology based on the existence of stable, switchable polarization states in ferroelectric materials Electromechanical coupling: the mechanical response to an applied electrical stimulus and the electrical response to an applied mechanical stimulus

In the past two decades, the concept of “small is different” has been established for a wide variety of phenomena, including electrical, optical, magnetic, and mechanical properties of materials (1, 2). This has given rise to new areas of research in the physical and biological sciences such as nanomechanics (3), plasmonics (4), nano-optics (5), and molecular electronics (6) as well as to multiple new material and device applications that have led to significant breakthroughs in medicine, biology, physics, and materials science. However, one largely untapped but potentially important area of nanoscience involves the interplay of electricity and mechanics at the nanoscale. Electromechanical interactions underpin a range of systems—from nonvolatile ferroelectric computer random access memories (FeRAM) to micro- and nanoelectromechanical systems (MEMS and NEMS, respectively) to complex electromechanical and electrochemical transformations in functional biomolecules and nanostructures. Coupling between electrical and mechanical phenomena is a universal feature of many inorganic materials and virtually all biological systems. The coupling between electrical and mechanical phenomena in living tissues, discovered by Galvani in 1771 (7), was the harbinger of the modern theory of electricity. Piezoelectricity in inorganic materials has been studied in great detail since the discovery of piezoelectricity in quartz at the end of the nineteenth century. This achievement was made possible by the combination of macroscopic measurements that provided information on properties, diffraction techniques that elucidated atomic structure, and advanced theory (8). However, the spectrum of electromechanical phenomena on the molecular and nanoscale levels is much richer than in the macroscopic systems owing to symmetry breaking at surfaces and interfaces, the lack of constraints imposed by the lattice in low-dimensional and soft condensed-matter systems, and the absence of macroscopic averaging. In this introductory section, we consider the relationship between electromechanical activity, piezoelectricity, and ferroelectricity in inorganic materials; discuss novel electromechanical phenomena emerging in low-dimensional systems; review some aspects of the physics of nanoscale ferroelectric materials; and briefly discuss complex electromechanical couplings in bio- and molecular systems.

Piezoelectricity in Crystals and Polar Molecules The simplest example of linear electromechanical coupling is piezoelectricity, in which the application of stress, X, results in electrical polarization (direct piezoelectric effect, P = d X), whereas the application of electric field, E, results in strain (converse piezoelectric effect, x = dE). From the thermodynamic Maxwell relations, the piezoelectric constant, d, for direct and converse effects is equal; thus, studies of electromechanical response provide insight into the polarizability of material, and vice versa. The atomic origins of piezoelectricity are directly related to the bond and lattice dipoles, as Figure 1 illustrates (9). From simple electrostatic consideration, the piezoelectric constant for a single chemical bond is related to the bond parameters as  d = 2q kl, where q is fractional charge, k is the spring constant, and l is bond length.

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Table 1

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Electromechanical coupling in crystals and molecules Crystal

Single bond

Displacement

xi = s i j X j + d ik Ek

x = (1/k)F/l + (2q /kl)E

Charge

Di = d i j X j + εik Ek

P = (q /k)F + (2q 2 /k)E

For typical bond parameters q = 0.3e, k = 100 N m−1 , l = 1 A, the piezoelectric coupling is d = 9.6 pm V−1 . The piezoelectric coupling is a near-ubiquitous property of polar noncentrosymmetric materials (20 out of 32 crystallographic groups are piezoelectric, and 10 are pyroelectric, i.e., have nonzero polarization) (10). For a linear crystalline piezoelectric and a single chemical bond, the relationship between strain, displacement, and field in tensorial form, using Voigt reduced notation and Einstein summation convention, is summarized in Table 1.

Ferroelectric Materials

Ferroelectric polarization: a spontaneous dipole moment existing owing to the distortion of a crystal lattice that can be switched between two or more equivalent states by the application of electrical field

Pyroelectric materials comprise a subset of piezoelectric materials possessing temperature-dependent spontaneous polarization in the absence of an applied field or strain. The direction of the spontaneous polarization is directly related to the crystal structure. A subset of pyroelectric materials in which the spontaneous polarization and hence the directionality of the electromechanical activity can be switched by external electric (ferroelectric) or mechanical (ferroelastic) stimuli is particularly important for numerous technological applications (8). After the discovery of strong piezoelectricity in ferroelectrics, numerous applications as sensors, actuators, and transducers have emerged (11, 12). In the past decade, the development of deposition techniques for epitaxial ferroelectric thin films and advanced ceramic fabrication has resulted in numerous novel applications such as those in MEMS and NEMS (13–15). The ability of ferroelectric materials to exist in and switch between two or more polarized states and retain polarization for a finite period makes possible their application as the active component of FeRAM (16–18) capacitors. Data storage devices have also been envisioned; in these, epitaxial ferroelectric thin films or ferroelectric nanostructures are accessed by a nanoscale probe or an array of probes to read and write information via ferroelectric polarization switching (19). Polarization-dependent chemical reactivity allows this approach to be extended to lithography (20). Ferroelectric switching may also enable novel devices based on ferroelectric and multiferroic tunneling junctions in which the switchable spontaneous polarization adds a new level of functionality (21). Ferroelectrics are also ideal materials for nonlinear optics applications because the local direction of the polarization is directly related to the electro-optic and electromechanic tensors, and hence the properties of the material can be easily controlled through domain structure engineering. Often, nanoscale disorder results in unusual behavior such as relaxor ferroelectricity in inorganic materials and ferroelectric polymers (22, 23). Strongly enhanced electromechanical properties in relaxors in turn enable new applications for undersea warfare and medical imaging. However, the fundamental nanoscale

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mechanisms behind these properties are still heavily debated (24). Figure 2 summarizes some of the applications of nanoferroelectrics. At the heart of the majority of these applications is ferroelectric polarization switching, and much effort has been put forth toward understanding the fundamental physics involved, including the nucleation mechanism, discrepancies between predicted and observed coercive fields, loss of polarization due to repeated switching cycles (fatigue), domain wall width, and minimum stable domain size. In the case of a capacitor structure with a homogeneous field, the nucleation is initiated at a ferroelectric-electrode interface, and a domain grows as an elongated ellipsoid, as first described by Landauer (25). Remarkably, the nucleation barriers, or the field required to nucleate domains predicted by Landauer theory, are much larger than experimentally observed, suggesting the existence of defect centers at which nucleation is initiated. Several theories have been proposed to address this apparent reduction of the activation barrier, including passive layers (26), tunneling (27), and ferroelectric-electrode coupling (28). Despite more than 50 years of extensive studies, the microscopic origins of this Landauer paradox are still unresolved. Many of the current experimental and theoretical efforts are directed toward the physics of low-dimensional ferroelectrics (29). Traditionally, piezoelectric coupling in nanocrystalline materials was considered to be analogous to macroscopic materials. For ferroelectrics, the studies of the size effect have centered on the paradigm of the ferroelectric bulk and the paraelectric surface layer. The experimental discovery of ferroelectric nanotubes (30, 31) in the past several years has prompted theoretical studies that predict new types of helical polar ordering in nanoferroelectrics (32). In addition, advances in the self-assembly of ferroelectric nanostructures and multiferroic heterostructures have enabled studies on size effects, constrained systems, and multiferroic coupling (30, 33–40).

Landauer paradox: phenomenon in which the electric fields required to induce polarization reversal correspond to unrealistically high values for the activation energy for domain nucleation

Novel Couplings Similar to transport, optical, and magnetic phenomena, the nanoscale can also enable completely new types of electromechanical behavior. As illustrated by the example of a single bond, piezoelectric coupling is a universal feature of polar systems. For example, a dipole layer on the surface, such as an ordered water layer, is expected to be piezoelectric. At the same time, a liquid-like water layer will be nonpiezoelectric. Thus, piezoelectricity can serve as an indication of water state. The significance ←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Figure 1 Electromechanical coupling on the molecular level. (a) The application of an electric field results in bond contraction or expansion. An electric field acting along the molecule axis induces an electrostatic force, Fel = 2qE, and the elongation of the molecule is dl = (F + Fel )/k = F/k + 2q E/k. In terms of strain, x = dl/l. (b) The application of an external force results in a change of the dipole moment. If the bond dipoles are parallel, the material is piezoelectric. (c) If dipoles are antiparallel, the material is nonpiezoelectric. (d ). More complex dipole arrangements give rise to different coupling. Schematics showing the (e) direct and ( f ) converse piezoelectric effect.

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Figure 2 Applications of nanoferroelectrics. (a) Domains can be “written” in ferroelectric materials to represent bits of information for data storage applications, as shown in this nanodomain dot array written by Scanning Nonlinear Dielectric Microscopy in congruent LiTaO3 . Reproduced with permission from Reference 108. Copyright 2006, IOP publishing. (b) Ferroelectrics can be used as the functional basis of nonvolatile FeRAM in consumer electronics (e.g., Sony PlayStation 2). Shown is an isomeric cross section of a ferroelectric memory cell with superimposed reconstruction of the polarization direction. Reproduced with permission from Reference 142. Copyright 2004, American Institute of Physics. (c) Ferroelectric tunneling barriers allow an additional degree of freedom in tunneling device functionality. Reproduced with permission from Reference 21. Copyright 2006, AAAS. (d ) Ferroelectric strontium bismuth tantalate nanotubes (178) can be used in MEMS devices (figure courtesy of M. Alexe). (e) Ferroelectric nanolithography has been used for the fabrication of nanostructures such as silver nanowires on stoichiometric lithium niobate. Reproduced with permission from Reference 167. Copyright 2006, IOP publishing. ( f ) Periodically engineered domain structures (in RbTiOAsO4 ) for applications in nonlinear optics. Reprinted with permission from Reference 179. Copyright 2003, American Institute of Physics. 194

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of this phenomenon is obvious: All biologically relevant surfaces are characterized by the presence of ordered or disordered water layers, which thus will affect the electromechanical activity of the system. In centrosymmetric materials, symmetry breaking at surfaces and interfaces can give rise to surface piezoelectric coupling even in nonpolar materials, and a number of novel electromechanical phenomena, including surface piezoelectricity and flexoelectricity, can exist (41). Even for purely nonpolar materials such as graphene or carbon nanotubes, local curvature will cause a redistribution of electron density and the formation of a curvature-dependent electric dipole and hence result in flexoelectric electromechanical coupling (Figure 3) (S.V. Kalinin & V. Meunier, unpublished data). Similar couplings emerge in soft condensed-matter systems such as liquid crystals and cellular membranes (42). These nanoelectromechanical phenomena will affect other functional properties of a material. For example, the chemical reactivity of piezoelectric semiconductors will be mediated through piezoelectrically induced potential effects of the Fermi level position. Electric fields of the surface water layers during formation and collapse of bubbles are considered one of the origins of the cavitation damage and sonoluminescence and were considered the governing factor determining the feasibility of bubble fusion (43). Another example is triboelectricity, during which mechanical deformations on surface layer give rise to strong electrostatic potentials.

Electromechanical Coupling in Biological and Molecular Systems Even more common and at the same time more difficult to assess are electromechanical coupling phenomena in biological systems. Piezoelectricity is ubiquitous in biopolymers and has been observed in a variety of biological systems, including bones (44–47), teeth (48), wood (49, 50), and seashells (51). Piezoelectric coupling, via mechanical stress that generates the electric potential, may control the mechanisms of local tissue development (52, 53). On the molecular level, this behavior is due to the simultaneous presence of polar bonds and optical activity, which provide sufficient conditions for piezoelectricity. Experimentally, piezoelectric activity of biopolymers is on the order of 1–5 pm V−1 , i.e., comparable to inorganic piezoelectrics such as quartz. Despite the tremendous interest in the role and possible biological significance of piezoelectricity—reflected in multiple publications from 1950 to 1980 (47, 54–61) and in the employment of methodologies of curing fracture, osteoporosis, and other bone diseases—these issues remain largely unresolved. This is due primarily to the complex hierarchical structure of biological systems; for example, that of bone spans seven structural levels, from 25–30-nm-size collagen fibrils and hydroxyapatite platelets through different levels of organization of mineralized fibrils to osteons to the macroarchitecture of cortical or trabecular bone (62). It is this complexity that precludes quantitative measurements and hence understanding of piezoelectricity in biological systems. Notably, piezoelectricity is the simplest form of the electromechanical coupling in biological systems: Linear phenomena such as flexoelectricity of molecular membranes and more complex forms of electromechanical coupling such as

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Figure 3 Electromechanical phenomena on the nanoscale. (a) Nonferroelectric surface layers and helical ordering in ferroelectric nanoclusters, (b) flexoelectricity and electromotor proteins in cellular membranes, ordered water layers in (c) nanocomposites and (d ) on surfaces, and (e) quantum flexoelectricity in graphene.

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voltage-controlled muscular contractions (63), cell electromotility (64), and electromotor proteins (65) are ubiquitous. In many cases, these couplings are the functional bases for processes such as transduction of acoustic signals into electrical pulses in the outer hair cell stereocilia (hearing), transport in ion channels, energy storage in mitochondria, cardiac myocyte activity, etc. Understanding local electromechanics, in addition to electrochemistry, is one of the key challenges in understanding biological systems. To complement biological systems, in the past several years the concept of electrically driven molecular machines, in which electrochemical transformations induce a set of changes in bonding configurations leading to a change in molecular conformations, has emerged (66).

NANOSCALE PROBING OF ELECTROMECHANICAL COUPLING Despite the rich gamut of novel electromechanical phenomena existing on the nanoscale and their relevance for areas as diversified as biological systems, energy harvesting materials, and computer memories, virtually no information is available experimentally. The reasons behind this dearth of information are straightforward— many of these effects manifest on length scales well below micron and often on the order of nanometers, comparable to molecular sizes. Electromechanical measurements at these length scales require either currents on the order of ∼femtoamperes or displacements on the order of ∼picometers probed over areas of ∼10–1000 nm2 , considered impossible until recently. The breakthrough in this direction has come with the advent of piezoresponse force microscopy (PFM) (67–71), which utilizes the converse piezoeffect to image local polarization orientation. Below, we discuss the basic principles of PFM and compare the image formation mechanism in PFM with that of other scanning probe microscopy (SPM) techniques.

Piezoresponse force microscopy (PFM): scanning probe technique based on the detection of the electromechanical response of a material to an applied electrical bias Vector PFM: the real-space reconstruction of electromechanical response vector and local crystallographic orientation from three components of piezoresponse: vertical PFM and two orthogonal lateral PFM components SPM: scanning probe microscopy Piezoelectric surface: a 3-D plot depicting the piezoresponse as a function of the angle between the direction of the applied field and the measurement axis

Principles of PFM PFM is based on the detection of bias-induced piezoelectric surface deformation. The tip is brought into contact with the surface, and the local piezoelectric response is detected as the first harmonic component, A1ω , of the tip deflection, A = A0 + A1ω cos (ωt + ϕ), during the application of the periodic bias Vtip = Vdc + Vac cos (ωt) to the tip. The phase of the electromechanical response of the surface, ϕ, yields information on the polarization direction below the tip. For c− domains (polarization vector oriented normal to the surface and pointing downward), the application of a positive tip bias results in the expansion of the sample, and surface oscillations are in phase with the tip voltage, ϕ = 0. For c+ domains, ϕ = 180◦ . The piezoresponse amplitude, A = A1ω /Vac , given in units of nanometers per volt, defines the local electromechanical activity. PFM images can be conve niently represented as A1ω cos (ϕ) Vac , where A1ω is the amplitude of first harmonic of measured response (in nanometers), provided that the phase signal varies by 180◦ between domains of opposite polarities. Detection of the torsional components of tip

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STM: scanning tunneling microscopy AFM: atomic force microscopy

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vibrations provides information on the in-plane surface displacement, referred to as lateral PFM. A third component of the displacement vector can be determined by imaging the same region of the sample after rotation by 90◦ (72). Provided that the vertical and lateral PFM signals are properly calibrated, the complete electromechanical response vector can be determined, an approach referred to as vector PFM (73). Finally, electromechanical response can be probed as a function of DC bias of the tip, providing information on polarization switching in ferroelectrics as well as more complex electrochemical and electrocapillary processes in the tip-surface junction (74, 75). In the decade since its invention, PFM has been established as a powerful tool for probing local electromechanical activity on the nanometer scale (76–79). Developed originally for imaging domain structures in ferroelectric materials, PFM was later extended to local hysteresis loop spectroscopy (39, 80) and ferroelectric domain patterning for applications such as high-density data storage (81, 82) and ferroelectric lithography (83–85). The broad applicability of PFM to materials such as ferroelectric perovskites, piezoelectric III-V nitrides (86), and, recently, biological systems such as calcified and connective tissues (87–89) has resulted in an increasing number of publications, as illustrated in Figure 4c. Various applications of PFM are shown in Figure 5.

PFM versus Other SPMs The fundamental factors underpinning any SPM method are (a) the tensorial nature of the signal, (b) the signal dependence on contact radius (contact modes) or tip-surface separation (noncontact modes), and (c) the signal dependence on the cantilever spring constant. These factors determine the strategies for instrumentation and technique development and the potential for quantitative measurements. In current-based techniques such as scanning tunneling microscopy (STM) and conductive atomic force microscopy (AFM), the excitation signal (applied bias) and detected signal (current magnitude) are scalar quantities. Hence, the response is scalar, I = V,

1.

where  is the local conductance. The conductance scales with the contact area as  ∼ a 1 in the classical (Maxvell) regime (90) and  ∼ a 2 in the Sharvin regime (91). Hence, image contrast can be induced either by variation in specific resistance or by contact radius, e.g., owing to variations in surface topography (cross talk). When the contact is confined to a single atom in a tunneling regime,  ∼ a 0 . In most applications of conductive AFM, the variations in local conductivities between dissimilar materials (e.g., differently doped regions of a semiconductor) dominate relatively small changes in contact geometry, the condition ensured by operation at large indentation forces on planar substrates. This, in turn, allows for calibration and quantitative measurements using current-based techniques. In force-based techniques, both the excitation signal (force) and the response signal (displacement) are vectors. Hence, the AFM signal is a rank-two tensor. In

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Figure 4 Schematic diagrams of (a) vertical and (b) lateral PFM. (c) Number of published papers based on scanning probe microscopy of ferroelectrics, by year (figure courtesy of A. Gruverman, NCSU). (d ) Schematic relating PFM to other scanning probe techniques.

the coordinate system aligned with the cantilever, where the 1-axis is oriented in the surface plane along the cantilever, the 2-axis is in plane and perpendicular to the cantilever, and the 3-axis is the surface normal, the signal formation mechanism can be represented as ⎛ ⎞ ⎛ ⎞⎛ ⎞ w1 a 11 a 12 a 13 F1 ⎜ ⎟ ⎜ ⎟⎜ ⎟ = 2. w a a a F ⎝ 2 ⎠ ⎝ 21 22 23 ⎠ ⎝ 2 ⎠ . w3 a 31 a 32 a 33 F3

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Figure 5 Applications of PFM for (a) domain imaging (73), (b) domain patterning, (c) studies of domain dynamics (115), (d ) phase transformations, (e) data storage, ( f ) spectroscopy (B.J. Rodriguez, S. Jesse, & S.V. Kalinin, unpublished data), and (g) switching spectroscopy mapping (158). b and g reprinted with permission from References 115 and 158, copyright American Institute of Physics.

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The component of the stiffness tensor a33 is probed by conventional vertical AFM and atomic force acoustic microscopy (AFAM), whereas a22 is probed by lateral force microscopy. The off-diagonal component, a23 and a32 , represents coupling between the vertical and lateral signals. However, a conventional cantilever sensor does not allow the separation of the normal and longitudinal force components, both of which result in a flexural deformation of the cantilever. Hence, SPM flexural, fl, and torsional, tr, responses are related to force components as ⎛ ⎞ F  1 fl αa 11 + βa 31 αa 12 + βa 32 αa 13 + βa 33 ⎜ ⎟ = 3. ⎝ F2 ⎠ , tr χa 21 χa 22 χa 23 F3 where α, β, and χ are proportionality coefficients dependent on cantilever geometry and detector calibration. In the continuum mechanics limit, the contact stiffnesses are proportional to the contact radius, a i j ∼ a 1 . When the contact area is a single molecule, as in protein unfolding spectroscopy, or a single atom, as in atomic resolution imaging, a i j ∼ a 0 . Hence, quantitative force measurements are generally limited to when the contact geometry is well characterized, as in nanoindentation techniques, or is weakly dependent on the probe, as in molecular unfolding spectroscopy. In other cases, the changes in contact stiffness represent the convolution of variation in local elastic properties and surface topography (topographic cross talk). Most force sensors have well-defined resonant behavior: The resonant frequencies are determined by the spring constant of the system and tip-surface junction. This enables a broad range of imaging techniques based on the resonant enhancement of cantilever oscillations, frequency tracking methods, etc. In particular, contact resonant frequency in techniques such as vertical and lateral AFAM is directly related to the contact stiffnesses as determined by local elastic properties (primary signal) and topography (cross talk). Finally, in PFM the excitation signal is scalar, whereas the electromechanical response of the surface is a vector. Hence, the PFM response is a vector, ⎛ ⎞ ⎛ ⎞ w1 d1 ⎜ ⎟ ⎜ ⎟ 4. ⎝w2 ⎠ = ⎝d 2 ⎠V. w3 d3 Here, the response components d i describe the electromechanical coupling in the material in the point contact geometry. As for AFM, the vertical and longitudinal surface displacements contribute to the flexural vibration of the cantilever, and strategies to overcome this limitation are considered below. The distinctive feature of the electromechanical response is dependence on contact area. From simple dimensionality considerations, the electromechanical response does not depend on the contact area, d i ∼ a 0 , as confirmed by rigorous theory for special classes of materials symmetry (92, 93). This weak dependence of the PFM signal on the contact area suggests that (a) electromechanical measurements are intrinsically quantitative (if good tip-surface contact is achieved) and do not require extensive probe calibration and that (b) signal is

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EFM: electrostatic force microscopy KPFM: Kelvin probe force microscopy

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relatively insensitive to topographic cross talk. However, resonance enhancement cannot be directly employed in PFM because the resonance frequency is determined by mechanical, rather then electromechanical, properties of material. Finally, quadratic bias dependence of electrostatic forces, as compared with linear piezoelectric interactions, suggests that the electromechanical signal cannot be unambiguously distinguished from the electrostatic signal. These considerations determine limitations and suggest strategies for improving PFM resolution and sensitivity, as discussed below (see section on Future Directions in PFM).

PFM ON PIEZOELECTRIC MATERIALS The primary factors determining the imaging mechanism in PFM are (a) voltagedependent contact mechanics of the tip-surface junction, including the relationship between local surface displacement and material properties, and displacements in the vicinity of microstructural elements such as domain walls, (b) dynamic behavior of the cantilever, including the relationship between surface displacement vector and torsional and flexural vibrations of the cantilever, and (c) the electroelastic field structure inside the material that determines spatial resolution and switching processes. Below, we consider the contrast formation mechanism, cantilever dynamics and resonance enhancement, and the role of electrostatic interactions in PFM.

Contact Mechanics of Piezoelectric Indentation General formulation. Traditionally, principles and physical underpinnings of SPM techniques can be conveniently understood using force-distance curves (Figure 6a). Depending on the dominant interactions, AFM can be sensitive to elastic interactions (e.g., AFAM), Van der Waals interactions (noncontact AFM), and magnetic and electrostatic forces [e.g., electrostatic force microscopy (EFM)]. A similar approach can be used for voltage modulation techniques such as PFM. However, here the system is described by two independent variables, tip-surface separation and tip bias, giving rise to force-distance-bias surface, as depicted in Figure 6b. In the noncontact regime, the tip-surface forces are purely capacitive, and the shape of the surface is described by Fnc = C z (z)(Vtip − Vsurf )2 , where C z (z) is the tip-surface capacitance gradient. Kelvin probe force microscopy (KPFM) is sensitive to the voltage derivative of  the force, ∂ Fnc ∂ Vtip . The known functional form of this dependence renders KPFM readily interpretable and relatively insensitive to topographic artifacts. At the same time, techniques such as EFM are sensitive to the distance derivative of capacitance, and the presence of the (unknown) C z (z) term makes quantitative interpretation of EFM more challenging, at the same time rendering technique more sensitive to topographic cross talk. In the contact regime, the imaging mechanism of SPM is ultimately controlled by the shape of the force-distance-bias surface, i.e., Fc = Fc (h, Vtip ), where h is the indentation depth. The image formation mechanism in various SPM techniques can be related to the derivatives of this surface. For example, in the small signal approximation the PFM signal is given by (∂h/∂ Vtip ) F = const , the AFAM signal is related

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Figure 6 (a) Force-based SPM can be conveniently described via a force-distance curve showing the regimes in which contact (C), noncontact (NC), intermittent contact (IC), and interleave imaging are performed. Also shown are domains of repulsive and attractive tip-surface interactions. (b) Voltage modulation SPMs can be described using a force-distance-bias surface. In the small signal limit, the signal in techniques such as PFM, AFAM, EFM, and KPFM is directly related to the derivative in bias or distance direction.

to (∂h/∂ F )V = const , and the ultrasonic force microscopy (UFM) signal is determined 2 by (∂ 2 h/∂ F )V = const , as illustrated in Figure 6b. The image formation mechanism in recently developed frequency-mixing techniques [heterodyne electrostatic-UFM (HUEFM)] (94) is controlled by mixed derivatives of the force-distance-bias surface, ∂ 2 h/∂ F ∂ V. Therefore, understanding the functional dependence of Fc (h, Vtip ) is the key element for the quantitative interpretation of SPM on piezo- and ferroelectric materials. Exact solution for contact electromechanics. The rigorous solution of the piezoelectric indentation problem, i.e., the Fc (h, Vtip ) dependence, is currently available only for transversally isotropic material in the continuum mechanics limit (95–97), corresponding to the case of c domains in tetragonal or hexagonal ferroelectrics. In this limit, the electric field generated outside the contact area is neglected owing to the large difference in dielectric constants between the piezoelectric and ambience (Figure 7a). Karapetian and colleagues (92, 93) have derived stiffness relations linking the applied force, P, and the concentrated charge, Q, with indenter displacement, w0 , indenter potential, ψ0 , indenter geometry, and materials properties. The solutions for flat, spherical, and conical indenter geometries have the following phenomenological structure: 2 P = θ (h n+1 C1∗ + (n + 1)h n ψ0 C3∗ ), 5. π 2 6. Q = θ (−h n+1 C3∗ + (n + 1)h n ψ0 C4∗ ). π

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Figure 7 Calculation of PFM response in (a) rigorous and (b) decoupled models. (c) Coordinate system in the decoupled theory.

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where h is the total indenter displacement, θ is a geometric factor [θ = a for flat indenters, θ = (2/3)R1/2 for spherical indenters, and θ = (1/π) tan α for conical indenters] and n = 0 for flat, n = 1/2 for spherical, and n = 1 for conical indenters, respectively. These stiffness relations provide an extension of the corresponding results of Hertzian mechanics and continuum electrostatics to the transversely isotropic piezoelectric medium. From this analysis, the maximum information that can be obtained from SPM or nanoindentation is limited to the indentation elastic stiffness, C1∗ , the indentation piezocoefficient, C3∗ , and the indentation dielectric constant, C4∗ . Indentation stiffnesses are complex functions of electroelastic constants of the material, Ci∗ = Ci∗ (c i j , e i j , εi j ), where c i j are elastic stiffnesses, e i j are piezoelectric constants, and εi j are dielectric constants. Experimentally, AFAM and UFM response is determined by C1∗ , whereas PFM is sensitive to C3∗ /C1∗ . For most materials, C3∗ /C1∗ ∼ d 33 √ and C4∗ ≈ ε11 ε33 (92, 93). The electroelastic fields produced by the indentor rapidly adopt the form of the point charge/point force at distances comparable to the contact radius. Decoupling approximation in contact electromechanics. The necessity for calculating the PFM signal for materials of general symmetry, as well as calculating the response at microstructural elements such as domain walls, cylindrical domains, and topographically inhomogeneous ferroelectrics such as nanoparticles, has stimulated theoretical attempts to derive approximate solutions for PFM contrast. A general approach for the calculation of the electromechanical response is based on the decoupling approximation. In this case, (a) the electric field in the material is calculated using a rigid electrostatic model (no piezoelectric coupling, d i j k = e i j k = 0), (b) the strain or stress field is calculated using constitutive relations for a piezoelectric material, Xi j = Ek e ki j , and (c) the displacement field is evaluated using the appropriate Green’s function for an isotropic or anisotropic solid. Ganpule et al. (98) originally suggested a simplified 1-D version of the decoupled model to account for the effect of 90◦ domain walls on PFM imaging. Agronin et al. (99) adapted a similar 1-D approach to yield closed-form solutions for the PFM signal. Felten et al. (100) developed the 3-D version of the decoupled model, using the analytical form for the corresponding Green’s function. Independently, Scrymgeour & Gopalan (101) used the finite element method to model PFM signals across domain walls. Recently, Kalinin et al. (102), Eliseev et al. (103), and Morozovska et al. (104) applied the decoupled theory to derive analytical expressions for PFM response on materials of low symmetry, derive analytical expressions for PFM resolution function and domain wall profiles, and interpret PFM spectroscopy data, as described below. In the decoupled approximation, the PFM signal, i.e., the surface displacement

 ui x, y at location x induced by the tip at position y = ( y1 , y2 ), is given by 

ui (x, y) =

∞



∞



∞

∂Gi j (x1 − ξ1 , x2 − ξ2 , ξ3 ) ∂ξk −∞ −∞ 0 ×El (ξ)c kj mn dln m (y1 + ξ1 , y2 + ξ2 , ξ3 ). d ξ1

d ξ2

d ξ3

7.

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Here coordinate x = (x1 , x2 , z) is linked to the indentor apex, and coordinates y = ( y1 , y2 ) denote indentor position in the sample coordinate system y (Figure 7b). The integration is performed over the spatial coordinates (ξ1 , ξ2 , ξ3 ). Coefficients dmnk and cjlmn are position-dependent components of the piezoelectric strain constant and elastic stiffness tensors, respectively. Ek (x) is the electric field strength distribution produced by the probe. The Green’s function for a semi-infinite medium G3 j (x − ξ ) links the eigenstrains c jlmn d mnk Ek to the displacement field. This approach is rigorous for materials with small piezoelectric coefficients. A simple estimation of the decoupling approximation applicability is based on the value of the square of the dimensionless electromechanical coupling coefficients ki2j = (d i j )2 /(s j j εii ). For instance, 2 2 2 for BaTiO3 , k15 ≈ 0.32, k31 ≈ 0.10, and k33 ≈ 0.31; for the ceramics lead zirconate 2 2 2 titanate (PZT) 6B, k15 ≈ 0.14, k31 ≈ 0.02, and k33 ≈ 0.13; and for a quartz single 2 crystal, k11 ≈ 0.01. This suggests that the error in the decoupling approximation does not exceed ∼30% for strongly piezoelectric materials and is on the order of ∼1% for weak piezoelectrics.

PZT: lead zirconate titanate

Approximate solution for piezoresponse of homogeneous media. The use of the decoupling approximation reduces an extremely complex coupled contact mechanics problem to the solutions of much simpler electrostatic and mechanical Green’s function problems and numerical integration of the result. The dielectric and particularly elastic properties described by positively defined second- and fourth-rank tensors (invariant with respect to 180◦ rotation) are necessarily more isotropic than the piezoelectric properties described by third-rank tensors (antisymmetric with respect to 180◦ rotation). Hence, elastic and dielectric properties of a material can often be approximated as isotropic. In this case, integrals in Equation 7 can be evaluated, and surface displacement can be written in the form ui (x) = VQ Wi jlk (x) d kjl , where the tensor Wi jlk (x) is symmetrical on the transposition of the indexes j and l. In Voigt notation, the displacements are (102) u 1 (0) = VQ (W111 d 11 + W121 d 12 + W131 d 13 + W153 d 35 + W162 d 26 ) ,

8a.

u 2 (0) = VQ (W121 d 21 + W111 d 22 + W131 d 23 + W153 d 34 + W162 d 16 ) ,

8b.

u 3 (0) = VQ (W313 (d 31 + d 32 ) + W333 d 33 + W351 (d 24 + d 15 )) .

8c.

The nonzero elements of the tensor Wiαk are W111 = −(13 + 4ν)/32, W121 = (1 − 12ν)/32, W131 = −1/8, W153 = −3/8, W162 = −(7 − 4ν)/32, W313 = −(1 + 4ν)/8, W333 = −3/4, W351 = −1/8. Here VQ is the electrostatic potential in the point x = (0, 0, 0) produced by a probe represented by the set of image charges located on a vertical line. Thus, the response is shown to be proportional to the potential induced by the tip on the surface. The latter fundamental result was generalized as response theorems (103). 1. Response theorem 1: For a transversally isotropic piezoelectric solid in an isotropic elastic approximation and an arbitrary point-charge distribution in the tip (not necessarily constrained to a single line), the vertical surface displacement is proportional to the surface potential induced by tip charges in the point of contact.

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2. Response theorem 2: For an anisotropic piezoelectric solid in the limit of dielectric and elastic isotropy, the vertical and lateral PFM signals are proportional to the potential on the surface induced by the tip if the tip charges and the point of contact are located on the same line along the surface normal. For lower material symmetries, the analytical expressions of the displacement fields induced by the point charge allow response calculations for asymmetric tips or probes with special geometry (103). Orientational imaging. In piezoelectric materials, the strong orientation dependence of electromechanical response opens the pathway to orientation imaging. Briefly, the orientation of a molecule or crystallographic lattice with respect to the laboratory coordinate system is given by three Euler angles (θ, ϕ, ψ). The relationship between the d i j k tensor in the laboratory coordinate system and the d i0j k tensor 0 in the crystal coordinate system is d i j k = Ail A j m Akn dlmn , where Ai j (θ, ϕ, ψ) is the rotation matrix (10). Experimentally, PFM measures three components of the response vector. Hence, local crystallographic orientation can be determined from the solutions of Equation 8, which can graphically be represented as response surfaces. As an example, we compare vertical displacement u 3 surfaces with piezoelectric tensors d33 surfaces for the tetragonal PbTiO3 and trigonal LiTaO3 model systems in Figure 8 (103). In this analysis the dielectric properties of the material were assumed to be close to isotropic, and hence the electric field distribution is insensitive to sample orientation. However, a similar analysis can be performed for full dielectric and elastic anisotropy.

Linear Imaging Theory in PFM The remarkable characteristic of the signal formation mechanism in PFM given by Equation 7 is its linearity. In particular, if the sample is uniform in the z-direction on the scale of the penetration depth of electric field, i.e., c jlmn d mnk (x, z) ≈ c jlmn d mnk (x), vertical surface displacement below the tip, i.e., vertical PFM signal, can be rewritten as

 ∞  ∞



 ∂ u 3 0, y = d mnk y − ξ c jlmn Ek (−ξ1 , −ξ2 , z) G3 j (ξ1 , ξ2 , z) d z d ξ1 d ξ2 , ∂ξl −∞ z=0 9. i.e., as a convolution of a function describing the spatial distribution of material properties, d mnk (x), and a resolution function related to probe parameters (integral in parenthesis). Image formation described by Equation 9 belongs to the class of so-called linear imaging mechanisms. Linearity of the PFM image formation mechanism allows one to 1. establish the unambiguous definitions for spatial resolution (minimal feature size required for quantitative response measurements) and information limits (minimal detectable feature size) and its dependence on tip geometry and materials properties, suggesting strategies for high-resolution imaging;

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Figure 8 The dependence of (a) displacement u 3 and (b) piezoelectric tensor component e 33 for PbTiO3 on Euler’s angle θ in the laboratory coordinate system, ν = 0.3. The dependence of (c) piezoelectric tensor component e 33 and (d ) displacement u 3 for LiTaO3 on Euler’s angles ϕ, θ in the laboratory coordinate system, ν = 0.25.

2. develop the pathways for calibration of tip geometry, using appropriate standard (e.g., domain wall) for quantitative data interpretation; 3. interpret the imaging and spectroscopy data in terms of intrinsic domain wall widths and the size of nascent domain below the tip; and 4. deconvolute the ideal image from experimental data and establish applicability limits and errors associated with such a deconvolution process. Kalinin et al. (105) have recently developed the phenomenological resolution theory in PFM. Morozovska et al. (104) have developed a corresponding analytical theory. Although a full summary of these results is well beyond the scope of this review, here we summarize two particularly useful results of linear theory, namely the

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analytical expression for 180◦ domain wall profile in the point-charge tip limit u 3 (s , Q, d ) =

Q {g 313 (s ) d 31 + g 351 (s ) d 15 + g 333 (s ) d 33 } , 2πε0 (εe + κ) d

10.

where s = (a − a 0 )/d is the coordinate along the domain wall normalized by chargesurface separation. The functions g i j k (s ) are simple analytical functions of the dielectric anisotropy factor. Equation 10 allows the parameters of the set of image charges {Q m , d m } N representing the tip (S. Jesse, S.V. Kalinin, B.J. Rodriguez, E.A. Eliseev, & A.N. Morozovska, unpublished data) to be determined, i.e., allows probe calibration. In the context of PFM spectroscopy, the signal in the center of a cylindrical domain can be found as 



 3 1 4 d 15 3πd − 8r π d − 8r eff d 33 ≈− d 33 + + ν d 31 + . 11. 4 3 3 πd + 8r 4 3πd + 8r Here r is the domain radius, and d is charge-surface separation. Equation 11 is derived for weak dielectric anisotropy (γ = 1). Through Equation 11 the geometric parameters of the domain formed below the tip can be determined, allowing the deconvolution of the PFM spectroscopy data. The combination of Equations 10 and 11 allows self-consistent analysis of PFM switching measurements, in which geometric parameters of the probe derived from the domain wall profile are used to quantitatively reconstruct the size of the cylindrical domain formed below the tip in the spectroscopic measurements.

Cantilever Dynamics and Effect of Electrostatic Forces Jesse et al. (106) have performed a complete analysis of the dynamic image formation mechanisms in vector PFM, including (a) the local vertical surface displacement translated to the tip, (b) the longitudinal, in-plane surface displacement along the cantilever axis, (c) the lateral surface displacement, in-plane and perpendicular to the cantilever axis, (d ) the local electrostatic force acting on the tip, and (e) the distributed electrostatic force acting on the cantilever. Jesse et al. have also described an approach for deconvolution of electrostatic and electromechanical contributions based on the 2D response–DC bias–frequency spectroscopy. B. Mirman (unpublished data) derived the solution in the presence of surface, cantilever, and hydrodynamics damping. In particular, in the case of transversally isotropic material (for which displacement vector and tip axis are normal to the surface), the response in the low-frequency regime is a sum of electromechanical and local and nonlocal electrostatic components as PR =

  Csphere + Ccone d 1ω k1 C + (Vdc − Vs ) + cant (Vdc − Vav ). = αa (h)d 1 Vac k1 + k k1 + k 24k

12.

The first term in Equation 12 is the electromechanical response determined by the effective piezoresponse coefficient of material, d 1 , and the ratio of AC potential on the surface and AC bias applied to the tip (i.e., the potential drop in the tip-surface gap of thickness, h), αa (h). The second and third terms are local and nonlocal contributions due to electrostatic tip-surface and cantilever-surface interactions. The signal transduction between the surface displacements and electrostatic forces and flexural

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Shielded probe: a conducting probe tip surrounded by insulating material and grounded counterelectrode that allows a highly localized electric field to be applied Polarization-dependent photodeposition: process during which an ultraviolet light with an energy larger than the bandgap of the material will generate electron hole pairs that respond to local polarization-induced fields and can take part in oxidation or reduction reactions

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cantilever deflection is governed by the spring constant of the tip-surface junction, k1 , and the spring constant of the cantilever, k. Note the different dependence of electromechanical and electrostatic contributions on cantilever and tip-surface spring constants. The factor of 24 in the nonlocal cantilever contribution depends solely on the cantilever spring constant and originates from the cantilever modes (buckling in which tip position is constant versus flexural mode with tip displacement) induced by electrostatic force acting on the cantilever. Predominantly electromechanical imaging is achieved for good tip-surface contact, αa (h) = 1, and in the absence of electrostatic interactions (e.g., stiff cantilever,   k1 k and k → ∞, or shielded probe, Ccone , Ccant → 0). These requirements have long been established as guidelines for quantitative PFM (78, 79). In this case, the PFM response is determined solely by local piezoelectric properties of material, P R = d 1 . Conversely, in KPFM the tip is not in contact with the surface, k1 = 0 and αa (h) → 0, resulting in a purely electrostatic response. The analysis of frequency dynamics has shown that the frequency-dependent PFM signal is a linear combination of contributions that have different frequency dependence and can be maximized by an appropriate choice of driving frequency. At the same time, the resonance frequencies are determined solely by the elastic properties of the material. Therefore, tracking the resonant frequency as a function of tip position provides information on local elastic properties, which is similar to frequency detection in AFAM (107).

LOCAL POLARIZATION SWITCHING IN FERROELECTRIC MATERIALS BY PFM One factor that has resulted in rapidly growing interest in PFM in the ferroelectric community is the capability for local polarization patterning and domain engineering by tip-induced polarization switching. Nanoscale ferroelectric domain patterning was proposed as the basis for ferroelectric data storage devices (78, 79), with recently demonstrated minimal bit size of ∼8 nm, corresponding to storage density of ∼10 Tbit inch−2 (108). Polarization dependence of chemical reactivity in acid dissolution (109) or metal photodeposition (83) processes allows extending domain patterning to nanofabrication. These new applications, as well as the need for understanding domain dynamics in ferroelectric devices, have led to a number of studies of domain growth processes in the presence of defects and disorder in materials. Grain boundaries play an important role in domain wall pinning (110). Paruch et al. used local studies of domain growth kinetics (82) and domain wall morphology (111) to establish the origins of disorder in ferroelectric material. Dawber et al. (112) interpreted the nonuniform wall morphologies as evidence for skyrmion emission during domain wall motion. Most recently, Agronin et al. (113) observed domain pinning on structural defect. A number of experimental and theoretical studies of domain growth kinetics have been reported (114–118). These studies show that in general domain growth follows an approximately logarithmic dependence on the pulse length and a linear dependence in magnitude (114). There have been several attempts to interpret this behavior in

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terms of the activation field for the domain wall motion and dynamics of systems with frozen disorder. However, the use of simplified point-charge-like models that do not include the effects of the spherical and conical parts of the tip and possible contribution due to the surface charge mobility limits the potential for quantitative data analysis and necessitates further theoretical and experimental studies of these phenomena. In addition to tip-induced switching, PFM can be applied to visualize switching phenomena in ferroelectric capacitor structures, providing insight into polarization switching in a uniform field. A recent study by Gruverman et al. (119) shows that domain nucleation in ferroelectric capacitors during repetitive switching cycles is always initiated at the same defect regions. A recent review by Gruverman & Kholkin (120) summarizes many of these results. A comprehensive analysis of the tip-induced switching process is necessarily based on thermodynamic analysis. Thermodynamics of domain switching in the Landauer approximation (25) for domain shape and point-charge approximation for the tip was given by Abplanalp (117) and fully developed by Molotskii and colleagues (121, 122) and Shvebelman (123). Kalinin et al. (124) showed that this model is applicable only for large domain sizes, although the description of switching on the length scales comparable to the tip radius of curvature and higher-order switching phenomena requires exact electroelastic field structure to be taken into account. For realistic tip geometries, domain nucleation requires a certain threshold bias on the order of 0.1–10 V (25, 125), sufficient to nucleate a domain in the finite electric field of the tip. Using simplified Pade approximations for the free energy, Morozovska et al. (126–128) recently performed a detailed analysis of domain nucleation and growth in ferroelectrics, including the effects of material parameters such as Debye length, surface screening, and tip geometry. The analysis of experimental data suggests that the domain growth process on the late stages of domain evolution can be controlled by the domain wall motion kinetics. Molotskii (129) recently addressed this behavior. However, this area is in the preliminary stage owing to the lack of information on the domain wall mobilities at high electric fields and the large domain wall curvatures and possible contribution of surface-charge diffusion to the apparent domain wall kinetics. Below, we summarize some recent results on the switching thermodynamics and the roles of environmental factors and materials properties in switching.

Domain nucleation: the event of polarization reversal when an oppositely polarized domain is formed in a ferroelectric material

Thermodynamics of Domain Switching The thermodynamics of the switching process can be analyzed from the bias dependence of the free energy of the nascent domain. The free energy for a nucleating domain is  (r, l) = U (r, l) +  S (r, l) +  D (r, l),

13.

where the first term is the interaction energy between the tip-induced electric field and the polarization, the second term is the domain wall energy, and the third term  D (r, l) is the depolarization field energy, including the Landauer contribution

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 DL (r, l) and the depolarization energy  DS (r, l) induced by the surface charges. In the Landauer model of switching, the domain shape is approximated as a halfellipsoid, with the small and large axis equal to r and l, correspondingly. The switching process can be understood from the evolution of free energy surfaces representing the free energy of the domain as a function of domain parameters, as illustrated in Figure 9 (130). For small biases U < US , the free energy is a positively defined monotonic function of domain sizes, corresponding to the absence of stable domain. For biases US < U < Ucr , the local minimum min > 0 arises, corresponding to a metastable domain of sizes rms and l ms . The corresponding domain energy is (rms , l ms ) = Ems . Finally, for U ≥ Ucr , the absolute minimum min < 0 is achieved for req and l eq , corresponding to a thermodynamically stable domain. The value Ucr

15

Domain length (nm)

15

40 10

10

5 10

25 5

2

2 0

1

2

3

4

5

a

0

0

3 1

b

2

3

4

5

Domain radius (nm)

<0 16

5

3

4

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20

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16 4

2000 4

8.05

30 1000

20

4

4

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d

0

e

500

4

10

4 3

2

1500

4

2

1

<0

40

8.05

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16

50

10

0

8

3000

60 32

0

7.03

8

c

70

15

0

9.41

5

5

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25

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11.167

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0

Domain length (nm)

15

100

0

2

4

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8

10

Domain radius (nm)

12

0

0

5

10

15

f

Figure 9 Contour plots of the free energy surface under the voltage increase: (a) US > U = 2V, (b) U = US = 2.344V, (c) US < U = 2.4V, (d ) U = Ucr = 2.467V. Figures near the contours are free energy values in eV. Triangles denote the saddle point (nuclei sizes). Material parameters correspond to the PZT6B solid solution and tip-surface characteristics: εe = 81, R0 = 50 nm, and σ S = −PS . The evolution of the free energy map with surface screening charge density: (e) σ S = 0 (U = Ucr = 9.07V), ( f ) σ S = +0.95PS (U = Ucr = 4087V).

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determines the point at which the homogeneous polarization distribution becomes absolutely unstable. Such “threshold” domain nucleation is similar to the well-known first-order phase transition. The minimum point (either metastable {rms , l ms } or stable {req , l eq }) and the coordinate origin are separated by the saddle point {r S , l S } (i.e., critical nucleus sizes). The corresponding energy (r S , l S ) = Ea is an activation barrier for domain nucleation.

Molotskii Model for Switching Using the Landauer model for domain geometry (25) and a capacitance approximation for the electric field of AFM tip, Molotskii (121) obtained elegant closed-form analytical expressions for the domain size dependence on the applied voltage for when the surface charges were completely compensated by the external ones. The interaction with an external electric field was calculated as if these screening charges were absent. In subsequent work, Molotskii modeled the equilibrium size and kinetics of the cylindrical domain, intergrown through the film (118, 131). He showed that the domain was stable only when the applied voltage exceeded some critical value. For a prolate domain the depolarization field energy was assumed to be proportional to the film thickness. These results are summarized in a contribution by Molotskii et al. in this volume (132).

Morozovska-Eliseev-Kalinin (MEK) model The Morozovska-Eliseev-Kalinin (MEK) model analyzes the signal formation mechanism in piezoresponse force spectroscopy (PFS) by deriving the main parameters of domain nucleation in semi-infinite material and establishing the relationships between domain parameters and the PFM signal, using linear Green’s function theory (124, 126–128). The MEK model extends Molotskii’s approach as follows: 1. The effect of surface depolarization energy, surface screening charges, and finite Debye screening length on domain nucleation, growth, and reversal is established. The analytical expressions for domain size and activation energy for domain nucleation are derived. 2. The analysis is performed using a realistic sphere-plane or disc-plane model for the tip. Alternatively, an original effective point-charge model is proposed. The charge parameters are selected to reproduce tip-induced surface potential and tip radius of curvature. 3. The governing role of surface screening on domain nucleation is established. At the same time, Debye screening (finite conductivity) causes a self-limiting effect of domain sizes at late stages of the growth process. These results have obvious implications for ferroelectric data storage because the self-limiting effect increases the information density. 4. The current limitation of the MEK model is an assumption of no space separation between bound and screening charges. Thus, the screening conditions (σ S > −PS ) lead to the decrease of dragging electrostatic force caused by the

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charged probe. In the case of full screening (σ S → PS ), the dragging force is absent. In particular, the MEK model allowed the role of screening charges on switching process to be elucidated. The static and dynamic properties of screening charges have been recently studied using variable temperature KPFM (133–135). A recent study illustrates the role of these charges on back-switching (136). The evolution of free energy surfaces in Figures 9e and 9f illustrates the role of screening on the thermodynamics of domain formation. In the framework of the MEK model, the activation barrier for nucleation at the onset of domain stability (see Figure 9a) is minimal for complete screening at σ S = −PS (10 eV) and increases up to 105 eV for σ S → +PS . The barrier calculated in the inhomogeneous electric field of the tip is three to seven orders of magnitude lower than the one calculated by Landauer for the homogeneous electric field.

SPECTROSCOPIC MEASUREMENTS OF POLARIZATION REVERSAL PROCESSES IN FERROELECTRICS The primary limitation of kinetic studies of domain growth is the large time required to study domain evolution after multiple switching steps. Moreover, the information is obtained on the domain growth initiated at a single point, thus precluding systematic studies of the microstructural influence on the domain growth process. An alternative approach to study domain dynamics in the PFM experiment is based on local spectroscopic measurements, in which the domain switching and electromechanical detection are performed simultaneously, yielding a local electromechanical hysteresis loop. Birk et al. (137), using an STM-based approach, and Hidaka et al. (71), using an AFM-based approach, first reported the in-field hysteresis loop measurements. In this method, the response is measured simultaneously with application of the DC electric field, resulting in an electrostatic contribution to the signal. To avoid this problem, Guo et al. (138) reported a technique to measure remanent loops. In this latter technique, the response is determined after the DC bias is turned off, minimizing the electrostatic contribution to the signal. However, domain relaxation in the off state is possible. In a parallel development, Roelofs et al. (80) demonstrated the acquisition of both vertical and lateral hysteresis loops. Several groups later used this approach to probe crystallographic orientation and microstructure effects on switching behavior (72, 139–143). Buhlmann (144) recently summarized a large volume of phenomenological data on local PFM spectroscopy in polycrystalline PZT films.

Phenomenological Theory of Hysteresis Loop Formation The progress in experimental methods has stimulated parallel development of theoretical models to relate PFM hysteresis loop parameters and materials properties. A number of such models are based on the interpretation of phenomenological characteristics of PFM hysteresis loops similar to macroscopic polarization-electric field

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(P-E) loops, such as slope, imprint bias, vertical shift, as illustrated in Figure 10. Several groups have analyzed the effect of nonuniform materials properties, including the presence of regions with nonswitchable polarization, on hysteresis loop parameters such as imprint and vertical shift. Saya et al. (145) interpreted, in thin films, vertical shift of the PFM hysteresis loops in terms of a nonswitchable layer. Alexe et al. (146) analyzed the hysteresis loop shape in ferroelectric nanocapacitors capped by a top electrode, obtaining the estimate for switchable volume of a nanocapacitor. Ma & Hesse (147) applied a similar analysis to ferroelectric nanoparticles developed by a self-patterning method (36). In all cases, the results were interpreted in terms of ∼10-nm nonswitchable layers, presumably at the ferroelectric-electrode interface. A number of authors have attempted to relate local PFM loops and macroscopic P-E measurements, often demonstrating good agreement between the two (148). This suggests that despite the fundamentally different mechanism in local and macroscopic switching, there may be deep similarities between tip-induced and macroscopic switching processes. On the basis of Landau theory, Ricinschi et al. (149–151) developed a framework for the analysis of PFM and macroscopic loops, demonstrating an approach to extract local switching characteristics from hysteresis loop shape via first-order reversal curve diagrams. In parallel with tip-induced switching studies, several groups combined local detection by PFM with excitation through the top electrode to study polarization switching in ferroelectric capacitor structures. In this case, the switching field is nearly uniform. Gruveman et al. discovered spatial variability in switching behavior, which was attributed to strain (152) and flexoelectric (153) effects. Subsequent work showed that domain nucleation during repetitive switching cycles was initiated at the same defect regions (119, 154). Additionally, in a few cases, abnormal hysteresis loops having shapes much different than that in Figure 10a have been reported. Abplanalp et al. (155) credited the inversion of electromechanical response to the onset of ferroelectroelastic switching. Harnagea and colleagues (148, 156) attributed the abnormal contrast to the in-plane switching in ferroelectric nanoparticles. Finally, B.J. Rodriguez, S. Jesse, & S.V. Kalinin (unpublished data) and Jesse et al. (157) observed a variety of unusual hysteresis loop shapes, including possible Barkhausen jumps and sharp discontinuities in the hysteresis loop shape typically associated with topographic defects. The rapidly growing number of experimental observations and rapid developments in PFM spectroscopy instrumentation and data acquisition require understanding of not only phenomenological but also quantitative parameters of hysteresis loops, such as numerical value of coercive bias and the nucleation threshold. Kalinin et al. (140) have extended the 1-D model of Ganpule et al. (98) to describe the PFM loop shape in the thermodynamic limit. Kholkin and colleagues (159) have postulated an existence of nucleation bias from PFM loop observations, in agreement with theoretical studies by Abplanalp (117), Kalinin et al. (124), Emelyanov (125), and Morozovska & Eliseev (126, 127). Finally, Jesse et al. (157) have analyzed hysteresis loop shape in kinetic and thermodynamic limits for domain formation in the 1-D approximation.

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P-E: polarization-electric field

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Response

Response R0+

Rs+

Vc–

Vertical shift

V–

Rinit V+

Bias

Bias

Vc+ Rs–

R0–

Horizontal shift (imprint)

b

a Tip Top electrode P

P

Bottom electrode

Bottom electrode

c Built-in field

Nonswitchable layer

Bottom electrode

Bottom electrode

d Figure 10 (a) Vertical and lateral displacements of a hysteresis loop are related to the presence of nonswitchable polarization and the built-in electric field. (b) Forward and reverse coercive voltages, V0+ and V0− , nucleation voltages, Vc+0 and Vc−0 , and forward and reverse saturation and remanent responses, R0+ , R0− , Rs+ , and Rs− , are shown. The work of switching As is defined as the area within the loop. The imprint bias and maximum switchable polarization are defined as Im = V0+ − V0− and Rm = Rs+ − Rs− , respectively. b is reprinted with permission from Reference 157. Copyright 2006, American Institute of Physics. (c,d ) Schematics of loop formation mechanism (c) for the macroscopic and microscopic case and (d ) in the presence of a built-in electric field and a nonswitchable polarization.

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Simplified 1-D Models for PFM Spectroscopy The 1-D model for hysteresis loop formation was developed by Kalinin et al. (140) and later independently by Kholkin (159). It was conjectured that the piezoelectric response is given by PR = d eff {V(0) − 2V(l)}, where V(0) is the potential at the surface, V(l) is the potential at the domain boundary below the tip, and d eff is the effective electromechanical response of the material (92). In the strong indentation regime, potential decays inversely with distance as V (l) = βV (0) a/l, in which β is a proportionality coefficient on the order of unity, l is the distance from the center of the contact area, and a is the contact radius. If one uses the Molotskii theory for bias dependence of domain geometry (121), the PFM hysteresis loop shape is

5βσwall π 2 PR = d eff 1 − , 14. 8Ps Vdc where σwall is the direction-independent domain wall energy and Ps is polarization. In this thermodynamic limit, the shape of the hysteresis loop is determined solely by the domain wall energy and spontaneous polarization. In the complete absence of pinning, the thermodynamic model suggests that the hysteresis will proceed along the solid line in Figure 11a; i.e., the process is reversible, and the domain should grow and shrink instantaneously in response to tip bias (Figure 11c). However, if weak pinning is included in the model, the actual domain wall boundary will lag behind that predicted by Equation 14, and nucleation of a new domain will dominate the hysteresis loop, as illustrated in Figure 11c,d. The saturation and remanent responses in this approximation are equal, and the onset of switching corresponds to the domain nucleation below the tip (Vc+ = Vc− = 0 in the point-charge limit when the field below tip is infinite). From Equation 14, the magnitude of the coercive bias can be derived as V + = V − = 5βσwall π 2 /8Ps , and for BaTiO3 (σ = 7 mJ/m2 , Ps = 0.26 C/m2 ) (20) the estimated coercive bias is V + = 0.166 V (for β = 1). The effect of kinetics of domain wall motion can be accounted in a straightforward manner. Experimentally, domain switching in ferroelectric materials follows the phenomenological dependence r (V, t) = ks V ln t, and domain wall velocity is r˙ (V, t) = ks V/t, where ks is the kinetic constant related to the pinning strength in the material. The envelope of the tip bias is ramped linearly with time, V = bt. Hence, r˙ = ks b and r = ks bt = ks V. Therefore, the domain size is independent of the ramp velocity and is determined by the pinning strength of the material alone. Thus, in the kinetic regime the hysteresis loop is expected to follow the functional form of Equation 14, which provides a robust description of hysteresis loop shape. The coercive bias is now V + = V − = 2βa/ks and is controlled by the pinning in the material.

3-D Model for Hysteresis Loop Formation Morozovska et al. (128, 130) recently described the self-consistent 3-D model for hysteresis loop formation. The approach is based on (a) deriving the main parameters of domain nucleation in semi-infinite material, as discussed above [see MorozovskaEliseev-Kalinin (MEK) Model, above], and (b) establishing the relationships between

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Response

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d Figure 11 (a) Electromechanical hysteresis loop in the no pinning (solid line) and weak pinning (dotted line) limits. (b) Hysteresis loop (solid line) in the thermodynamic and kinetic (dotted line) limits. Forward and reverse coercive voltages, V0+ and V0− , nucleation voltages, Vc+0 and Vc−0 , and forward and reverse saturation and remanent responses, R0+ , R0− , Rs+ , and Rs− , are shown. The work of switching As is defined as the area within the loop. The imprint bias and maximum switchable polarization are defined as Im = V0+ − V0− and Rm = Rs+ − Rs− , respectively. Schematic representation of domain growth for the thermodynamic model (c) without pinning and (d ) with pinning. The numbers correspond to points on the hysteresis curve in a. Arrows indicate domain orientation. Reprinted with permission from Reference 158. Copyright 2006, American Institute of Physics.

domain parameters r and l and piezoresponse signal, using decoupled Green’s function theory (124, 126–128). For an interpretation of experimental data on realistic materials, the third step is (c) calibration of the probe geometry, using the appropriate standard (e.g., domain wall width). This analysis has shown that the electromechanical response in the center of cylindrical domain has the form

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Q 1 (d 31 g 1 (r, l) + d 15 g 2 (r, l) + d 33 g 3 (r, l)) , 2πε0 (εe + κ) d

15.

where the functions g i (r, l) = fi −2wi (r, l) are dependent on the domain sizes, Debye screening length, Poisson ratio, and dielectric anisotropy. Functions g i = − fi in the initial (r = 0, l = 0) and wi = fi in the final (r → ∞, l → ∞) states of the switching eff process. Numerically, the effective piezoresponse d 33 = u 3 /U voltage dependence √ eff obtained within Equation 15 is described by the law d 33 = d ∞ (1 − U0 /U ), U > Ucr . eff The difference with the one d 33 = d ∞ (1 − U0 /U) obtained within the framework of the 1-D (140) model may be related to the dimension of the problem.

Switching Spectroscopy PFM of Polarization Dynamics in Low-Dimensional Ferroelectrics Recently, PFM spectroscopy has been extended to imaging mode through an algorithm for fast (30–100-ms) hysteresis loop measurements developed by Jesse et al. (158). In switching spectroscopy PFM (SS-PFM) (Figure 12), hysteresis loops are acquired in each point of the image and analyzed to yield 2-D maps of imprint, coercive bias, and work of switching, providing a comprehensive description of switching behavior of material in each point. As an example, Figure 13 shows surface topography and effective work-ofswitching map (area of hysteresis loop) for polycrystalline PZT ceramics. Analysis of the data showed that the shape of hysteresis loops at certain locations deviated significantly from the ideal loop shape in Figure 11a. These hysteresis loops having a pronounced “shoulder” are shown in Figure 13c. Loops acquired in adjacent locations show clear similarities in the loop shape (compare Figure 13c and Figure 13d). This excludes the possibility of spurious signals that can not be ruled out in conventional PFM spectroscopy. We attribute the “abnormal” loop shape to the interaction of forming domain with the topographic inhomogeneity, resulting in a bistability in the domain size.

ADVANCED TOPICS IN PFM For more than a decade, PFM has been developed primarily in the context of imaging, spectroscopy, and domain patterning of ferroelectric materials. In this section, we briefly review several recent advances in cross-disciplinary areas related to the interplay between electromechanics, surface chemistry, and biology, including surface effects on PFM, electromechanical imaging of biomaterials, and PFM in a liquid environment.

Surface Effects on PFM Surfaces of ferroelectric materials have a high density of bound polarization charge [e.g., for BaTiO3 (100) surface, 0.26 C m−2 = 0.25e− /u.c.], the sign of which can change under the application of an external bias. This charge can lead to significant

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c Figure 12 (a) In switching spectroscopy PFM (SS-PFM), a local hysteresis loop is collected at each point on N × M mesh. (b) Evolution of a PFM hysteresis loop on epitaxial PZT film with bias window. (c) The single-point probing waveform in SS-PFM and data acquisition sequence. Reprinted with permission from Reference 157. Copyright 2006, American Institute of Physics.

surface reactivity and surface chemistry effects, as has been known for nearly half a century (160–162). The role of adsorbates on the PFM signal has been explored as a function of temperature in barium titanate crystals (163, 164) and also in a vacuum environment after heating (165). The chemical reactivity of ferroelectric surfaces has been exploited with the photodeposition of metal (166) and the fabrication of complex nanostructures (83, 84). Whereas previous photodeposition experiments resulted in

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Figure 13 Topography and SS-PFM images of polycrystalline PZT. (a) Topography (vertical scale is 50 nm). (b) A map of the work of switching (area within the hysteresis loop) of approximately the same location as in a. (c,d ) Hysteresis loops measured at the locations indicated in b. Reprinted with permission from Reference 157. Copyright 2006, American Institute of Physics.

metal deposition on positive domains, recently metal nanowires have been formed along the lithium niobate domain walls via photodeposition (167). Although surface reactivity is likely to strongly affect the electromechanical activity of ferroelectric surfaces in ambient conditions, of great interest are intrinsic surface effects, including surface reconstructions and antiferrodistortive phase transitions (168), that can be studied on in-situ-grown ferroelectrics in ultrahigh vacuum (UHV).

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UHV: ultrahigh vacuum

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PFM of Biomaterials As discussed in the introduction, many biological materials are piezoelectric owing to the combination of polar bonding and optical activity inherent in all biopolymers. To understand biopiezoelectricity on the nanoscale and to relate it to biological functionality and other materials properties, a nanoscale probe of local piezoelectricity is needed. PFM has been used for high-resolution imaging of biological structures, including human teeth, canine femoral cartilage, deer antlers, and butterfly wing scales (88, 169). PFM allows organic and mineral components of biological systems to be differentiated and provides information on materials microstructure and local properties (89). The use of vector PFM may also enable protein orientation to be determined in real space. Figure 14 shows the internal structure and orientation of protein microfibrils, with a spatial resolution of several nanometers in human tooth enamel (170). A similar approach has been used to study electromechnical activity in dentin, cartilage, and butterfly Vanessa virginiensis wings, effectively reproducing Galvani’s experiment on the nanoscale (88, 169, 171).

PFM in a Liquid Environment Future work in the area of relating bio-PFM to biofunctionality necessitates electromechanical imaging in a liquid environment. In addition, liquid imaging eliminates capillary forces and may be beneficial to imaging of ferroelectric materials as well. Recently, PFM of a ferroelectric sample in a liquid environment demonstrated that both long-range electrostatic forces and capillary interactions were minimized in liquid, resulting in a localization of the AC field to the tip-surface junction and allowing 3-nm resolution to be achieved in some cases (172). Imaging at frequencies corresponding to high-order cantilever resonances minimized the viscous damping, added mass effects on cantilever dynamics, and allowed sensitivities comparable to those of ambient conditions. A path to further improvements in resolution to the nanometer level and below by controlling hydrophobic interactions is predicted. The ability to image in solution with improved sensitivity and resolution makes PFM a technique of significant value to the wide bioresearch community by allowing a full range of electrical and electromechanical scanning probe techniques to be employed for the investigation of, e.g., piezoelectricity in biopolymers, molecular orientation, and flexoelectricity in cellular membranes. Liquid PFM may provide novel opportunities for high-resolution studies of ferroelectric materials, imaging of soft electroactive polymer materials, and imaging of biological systems in physiological environments on nanometer and ultimately molecular levels.

FUTURE DIRECTIONS IN PFM In the decade and a half since its invention, PFM has evolved from a relatively obscure technique for domain imaging to a broadly accepted tool for nanoscale characterization, domain patterning, and spectroscopy of ferroelectric material. PFM is

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Figure 14 Electromechanical imaging of tooth enamel. (a) Electromechanical response of the surface to the tip bias is a vector measure, components of which are related to the local orientation of protein molecules. In PFM, detection of vertical (VPFM) and torsional (LPFM) components of cantilever response allows two vector components, in plane and out of plane, to be simultaneously measured. (b) Surface topography (vertical scale 20 nm) and (c) elasticity map of 400 × 400 nm2 region on enamel surface (vertical scale is 6% of the average signal). (d ) VPFM and (e) LPFM images of the same region as a and b with a modulation bias of 10 Vpp applied to the tip. The vertical scale for d is −7.5 pm V−1 to 7.5 pm V−1 (the vertical scale for e is not calibrated). ( f ) Vector PFM map of local electromechanical response (maximum is 7.5 pm V−1 ). Color indicates the orientation of the electromechanical response vector, whereas the intensity provides the magnitude (color wheel diagram). (g,h) Semiquantitative maps of local molecular orientation. Reprinted from Reference 170 with permission from Elsevier.

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applicable to imaging of other piezoelectric functional materials such as III-V nitrides, providing information on crystal polarity. The recently demonstrated potential of PFM for sub-10-nm imaging of electromechanically active proteins in calcified and connective tissues (88, 89, 169, 170) suggests much broader applications in the biological community, in which the techniques for studying the ultrastructure of tissues on the submicron level are scarce. Moreover, the pronounced difference in signal formation mechanism between PFM and force- and current-based SPMs suggests that there is a tremendous potential for further development. In this section, we summarize some of the unresolved challenges and opportunities in PFM.

Probe Development Further progress of PFM necessitates the development of probes that (a) are stable and form reliable contact with the surface, (b) have electrostatic shielding (173, 174) to minimize electrostatic force and allow reliable imaging in conductive liquid environments, and (c) have multiple electrodes to create an in-plane field with controllable structure.

Instrumentation The difference in PFM image formation mechanisms compared with those of AFM and STM imposes certain requirements on the SPM platform. In particular, PFM requires the detection of small (∼picometer-scale) probe displacements in response to large (∼10 Vpp ) periodic biases. This requires special attention to the shielding of the microscope cabling to prevent cross talk between the driving and the detected signals. In particular, until recently most commercial microscopes required home-built tip holders to avoid the cross talk; to a large extent the limited research on PFM in a UHV environment is related to the difficulties associated with similar modifications for UHV operation. The second limitation is related to the difficulty of using standard phase-locked loop–based frequency-tracking methods for PFM and the dependence of resonance signal both on local damping and electromechanical activity. This limitation can be overcome through the use of a recently developed band excitation method based on nonsinusoidal excitation signals (175).

Resolution There is no fundamental limitation on achieving subnanometer and potentially atomic resolution in PFM. As illustrated by the single chemical bond example above (see section on Nanoscale Probing of Electromechanical Coupling), the expected level of response is well within the detection limit of modern SPM systems. The primary difficulty in achieving this goal is the minimization of the electrostatic response to the measured signal and the precise control of tip-surface contact area required to achieve molecular and atomic resolution.

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3-D Probes As discussed above, the electromechanical response of the surface is generally a vector having three independent components. The beam-deflection system used in most commercial SPMs allows only normal and lateral components of the response to be detected, necessitating imaging with sample rotation; also, longitudinal displacement of the surface can contribute to the vertical signal. In addition, the sensitivities of vertical and lateral PFM are generally different, necessitating complex and time-consuming calibration. This limitation is a fundamental characteristic of a cantilever-based force sensor, and the use of alternative configurations for a force sensor, e.g., recently introduced 3-D SPM (176, 177) or

Figure 15 Roadmap for development of SPM probes and imaging modes. Shown is the probe evolution from a simple etched STM tip (current probe) to an AFM cantilever (force and current) to 3-D probes with a force-sensing integrated readout and active tip (FIRAT) and shielded probes. In parallel, data acquisition methods have evolved from static detection (STM and contact AFM) to constant frequency (intermittent contact AFM, AFAM, etc.) and frequency tracking (noncontact AFM) to more complex waveforms. Images reproduced with permission from References 180 and 181.

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three-axis nanoindentors (W.C. Oliver, MTS Corporation, private communications), can provide a future alternative. Figure 15 summarizes the roadmap for the development of PFM based on future advances in probe, software, and instrumental platforms.

Figure 16 Electromechanical phenomena on all length scales in inorganic, biological, and macromolecular systems. In inorganic systems, properties measured on the macroscopic scale can be related to the atom structure determined by the diffraction methods on the atomic level. This approach is not applicable for complex biological systems, necessitating local scanning probe microscopy studies of properties from the molecular to macroscopic levels. Similar problems arise in the context of macromolecular polymeric systems. All images except the protein membrane bacteriorhodopsin image are reproduced from References 182–185. The protein membrane bacteriorhodopsin image was made by Dr. Emad Tajkhorshid, using VMD, and is owned by the Theoretical and Computational Biophysics Group, NIH Resource for Macromolecular Modeling and Bioinformatics, at the Beckman Institute, University of Illinois at Urbana-Champaign.

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Theory Although significant theoretical effort has been concentrated on understanding transport and elastic properties of materials on the nanoscale, much less is known about the nanoscale electromechanics. Extensive effort will be required to answer questions such as the signal dependence on the contact area in the small contact limit, predict the role of surface states on the signal, and model PFM in soft condensed matter and biosystems.

PFM: BEYOND THE PIEZO- AND FERROELECTRICS In this review, we discuss in detail electromechanical imaging of piezoelectric and ferroelectric materials, domain switching, and spectroscopy on the basis of bulk models for material behavior. However, with an increase in spatial resolution, SPM can become more and more sensitive to the surface properties of material. Further research (Figure 16) will yield an understanding of electromechanical coupling on the nanometer level, establish the role of surface defects in polarization switching (Landauer paradox), and probe nanoscale polarization dynamics in phase-ordered materials and unusual polarization states. In biosystems, PFM potentially opens pathways for studies of electrophysiology on the cellular and molecular levels, such as, for example, examinations of signal propagation in neurons. Ultimately, on the molecular level, PFM can allow reactions and energy transformation pathways to be understood and become the enabling component of molecular electromechanical machines. SUMMARY POINTS 1. Electromechanical coupling is ubiquitous in biological systems, inorganic materials, and molecular systems. The mechanical response to an applied electrical stimulus and the electrical response to an applied mechanical stimulus are the bases for numerous technological applications and accompany many biological processes, and yet little is known about electromechanics on the nanoscale. 2. Since its invention in the early 1990s, piezoresponse force microscopy (PFM) has become the primary tool to measure local electromechanics on the nanoscale, predominantly in ferroelectric materials. The image formation mechanism in PFM is complementary to that in STM and force-based AFM in terms of signal scaling with contact area and the tensorial nature of the detected signal. In particular, the PFM signal is independent of contact area, resulting in relative insensitivity to topographic cross talk. 3. PFM is based on a nanoscale conducting probe at the end of a cantilever that is brought into contact with a sample surface and used to apply a bias and to measure the resulting surface deformation. Both the magnitude and sign of the displacement can be mapped out to generate images of local piezoresponse and polarization orientation.

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4. PFM has evolved during the past decade and now includes several variations, including lateral PFM to measure the in-plane component of polarization, vector PFM to reconstruct the real-space polarization vector from vertical and lateral PFM images, and PFM spectroscopy to measure local electromechanics as a function of applied bias. 5. PFM spectroscopy has been used to generate local hysteresis loops in ferroelectric materials. From these hysteresis loops, information such as imprint and the onset of nucleation can be obtained. This information is important for any technological application based on ferroelectric switching (e.g., FeRAM) and is required for inquiries into the fundamental mechanisms of polarization reversal. Recently, PFM spectroscopy was extended to an imaging mode. This allowed loops to be measured at every point in an image and parameters describing polarization dynamics (e.g., imprint or nucleation bias) to be extracted and represented in 2-D plots. 6. Interpretation of local hysteresis loops, although originally compared with the formation process of macroscopic P-E loops, required the development of a new 3-D model based on the decoupled Green’s function theory. This model allows the size of the nucleating domain during the acquisition of the hysteresis loop to be determined. 7. The universal presence of piezoelectricity in biopolymers enables applications of PFM for sub-10-nm imaging of the ultrastructure of biological structures such as calcified and connective tissues. Further applications of PFM and liquid PFM for biological systems can uncover fundamental mechanisms of flexoelectricity in cellular membranes, electromotive proteins, ion channels, etc. 8. The achievement of molecular and ultimately atomic resolution in PFM, although theoretically possible, will require advances in AFM instrumentation, data acquisition, and operation modes, probe design, and theory.

ACKNOWLEDGMENTS Research sponsored by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy, under Contract DE-AC0500OR22725 with Oak Ridge National Laboratory, managed and operated by UTBattelle, LLC.

LITERATURE CITED 1. Waser R, ed. 2003. Nanoelectronics and Information Technology: Advanced Electronic Materials and Novel Devices. Darmstadt: Wiley-VCH 2. National Nanotechnology Initiative. 2006. http://www.nano.gov/

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3. Chuang TJ, Anderson PM, Wu MK, Hsieh S, eds. 2006. Nanomechanics of Materials and Structures. Berlin: Springer 4. Masuhara H, Kawata S, eds. 2006. Nanoplasmonics, Vol 2: From Fundamentals to Applications. Amsterdam: Elsevier 5. Kawata S, Ohtsu M, Irie M, eds. 2002. Nano-Optics. Berlin: Springer 6. Lazarev PI, ed. 2004. Molecular Electronics. Berlin: Springer 7. Galvani L. 1791. Commentary on the effect of electricity on muscular motion. Transl. RM Green, 1953, Cambridge, MA (from Latin) 8. Lines ME, Glass AM. 1977. Principles and Applications of Ferroelectric and Related Materials. Oxford, UK: Clarendon Press 9. Cleland AN. 2003. Foundations of Nanomechanics: From Solid State Theory to Device Applications. Berlin: Springer 10. Newnham RE. 2005. Properties of Materials: Anisotropy, Symmetry, Structure. New York: Oxford Univ. Press 11. Hench LL, West JK. 1990. Principles of Electronic Ceramics. New York: Wiley 12. Setter N, Colla EL. 1993. Ferroelectric Ceramics. Basel: Birkhauser Verlag 13. Polla DL, Francis LF. 1998. Processing and characterization of piezoelectric materials and integration into microelectromechanical systems. Annu. Rev. Mater. Sci. 28:563–97 14. Dabbs DM, Aksay IA. 2000. Self-assembled ceramics produced by complexfluid templation. Annu. Rev. Phys. Chem. 51:601–22 15. Schonholzer UP, Gauckler LJ. 1999. Ceramic parts patterned in the micrometer range. Adv. Mater. 11:630–32 16. Palkar VR, Purandare SC, Pinto R. 1999. Ferroelectric thin films of PbTiO3 on silicon. J. Phys. D 32:R1–18 17. Suzuki M. 1995. Review on future ferroelectric nonvolatile memory— FERAM—from the point-of-view of epitaxial oxide thin films. J. Ceram. Soc. Jpn. 103:1099–111 18. Scott J. 2000. Ferroelectric Memories. Berlin: Springer Verlag 19. Hong S, Park N. 2007. Resistive probe storage: read/write mechanism. In Scanning Probe Microscopy: Electrical and Electromechanical Phenomena on the Nanoscale, ed. SV Kalinin, A Gruverman, in press. New York: Springer 20. Jona F, Shirane G. 1993. Ferroelectric Crystals. New York: Dover Publications 21. Tsymbal EY, Kohlstedt H. 2006. Tunneling across a ferroelectric. Science 313:181–83 22. Cross IE. 1987. Relaxor ferroelectrics. Ferroelectrics 76:241–67 23. Samara SA. 2003. The relaxation properties of compositionally disordered ABO3 perovskites. J. Phys. Condens. Matter 15:R367–411 24. Kutnjak Z, Petzelt J, Blinc R. 2006. The giant electromechanical response in ferroelectric relaxors as a critical phenomenon. Nature 441:956–59 25. Landauer R. 1957. Electrostatic considerations in BaTiO3 domain formation during polarization reversal. J. Appl. Phys. 28:227–34 26. Bratkovsky AM, Levanyuk AP. 2000. Easy collective polarization switching in ferroelectrics. Phys. Rev. Lett. 85:4614–17 27. Molotskii M, Kris R, Rosenman G. 2000. Fluctuon effects in ferroelectric polarization switching. J. Appl. Phys. 88:5318–27

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35. Multiferroic self-assembly fabrication holds promise for further device applications utilizing the coupling between polarization, magnetization, and strain order parameters.

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28. Gerra G, Tagantsev AK, Setter N. 2005. Surface-stimulated nucleation of reverse domains in ferroelectrics. Phys. Rev. Lett. 94:107602 29. Scott JF. 2006. Nanoferroelectrics: statics and dynamics. J. Phys. Condens. Matter 18:R361–86 30. Luo Y, Szafraniak I, Zakharov ND, Nagarajan V, Steinhart M, et al. 2003. Nanoshell tubes of ferroelectric lead zirconate titanate and barium titanate. Appl. Phys. Lett. 83:440–42 31. Urban JJ, Spanier JE, Lian OY, Yun WS, Park H. 2003. Single-crystalline barium titanate nanowires. Adv. Mater. 15:423–26 32. Naumov II, Bellaiche L, Fu HX. 2004. Unusual phase transitions in ferroelectric nanodisks and nanorods. Nature 432:737–40 33. Ederer C, Spaldin NA. 2004. Magnetoelectrics: a new route to magnetic ferroelectrics. Nat. Mater. 3:849–51 34. Lottermoser L, Lonkai T, Amann U, Hohlwein D, Ihringer J, Fiebig M. 2004. Magnetic phase control by an electric field. Nature 430:541–44 35. Zheng H, Wang J, Lofland SE, Ma Z, Mohaddes-Ardabili L, et al. 2004. Multiferroic BaTiO3 -CoFe2 O4 nanostructures. Science 303:661–63 36. Szafraniak I, Harnagea C, Scholz R, Bhattacharyya S, Hesse D, Alexe M. 2003. Ferroelectric epitaxial nanocrystals obtained by a self-patterning method. Appl. Phys. Lett. 83:2211–23 ¨ 37. Lee SK, Lee W, Alexe M, Nielsch K, Hesse D, Gosele U. 2005. Well-ordered large-area arrays of epitaxial ferroelectric (Bi,La)4 Ti3 O12 nanostructures fabricated by gold nanotube-membrane lithography. Appl. Phys. Lett. 86:152906 ¨ 38. Alexe M, Harnagea C, Hesse D, Gosele U. 1999. Patterning and switching of nanosize ferroelectric memory cells. Appl. Phys. Lett. 75:1793–95 39. Alexe M, Gruverman A, Harnagea C, Zakharov ND, Pignolet A, et al. 1999. Switching properties of self-assembled ferroelectric memory cells. Appl. Phys. Lett. 75:1158–60 40. Alexe M, Scott JF, Curran C, Zakharov ND, Hesse D, Pignolet A. 1998. Selfpatterning nano-electrodes on ferroelectric thin films for gigabit memory applications. Appl. Phys. Lett. 73:1592–94 41. Tagantsev AK. 1986. Piezoelectricity and flexoelectricity in crystalline dielectrics. Phys. Rev. B 34:5883–89 42. Petrov AG. 2002. Flexoelectricity of model and living membranes. Biochim. Biophys. Acta 1561:1–25 43. Ohta T. 2003. Processes and key properties of pool materials for “bubble fusion”? Int. J. Hydrog. Energy 28:1273–78 44. Fukada E, Yasuda I. 1957. On the piezoelectric effect of bone. J. Phys. Soc. Jpn. 12:1158–62 45. Yasuda L. 1957. On the piezoelectric activity of bone. J. Jpn. Orthop. Surg. Soc. 28:267 46. Lang SB. 1966. Pyroelectric effect in bone and tendon. Nature 212:704–5 47. Anderson JC, Eriksson C. 1970. Piezoelectric properties of dry and wet bone. Nature 227:491–92 48. Marino AA, Gross BD. 1989. Piezoelectricity in cementum, dentine and bone. Archs. Oral Biol. 34:507–9

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49. Fukada E. 1955. Piezoelectricity of wood. J. Phys. Jpn. 10:149 50. Bazhenov VA. 1961. Piezoelectric Properties of Wood. New York: Consultants Bureau 51. Fukada E. 1995. Piezoelectricity of biopolymers. Biorheology 32:593–609 52. Bassett CAL. 1968. Biologic significance of piezoelectricity. Calc. Tiss. Res. 1:252–72 53. Marino AA, Becker RO. 1970. Piezoelectric effect and growth control in bone. Nature 228:473–74 54. Smith JW. 1968. Molecular pattern in native collagen. Nature 219:157– 58 55. Reinish GB, Nowick AS. 1975. Piezoelectric properties of bone as functions of moisture content. Nature 253:626–27 56. Marino AA, Becker RO. 1975. Piezoelectricity in hydrated frozen bone and tendon. Nature 253:627–28 57. Athenstadt H. 1970. Permanent longitudinal electric polarization and pyroelectric behaviour of collagenous structures and nervous tissue in man and other vertebrates. Nature 228:830–34 58. Anderson JC, Eriksson C. 1968. Electrical properties of wet collagen. Nature 218:166–68 59. Lang SB. 1969. Elastic coefficients of animal bone. Science 165:287–88 60. Morris RW, Kittleman LR. 1967. Piezoelectric property of otoliths. Science 158:368–70 61. Andrew C, Bassett L, Pawluk RJ. 1972. Electrical behavior of cartilage during loading. Science 178:982–83 62. Weiner S, Wagner HD. 1998. The material bone: structure mechanical function relations. Annu. Rev. Mater. Sci. 28:271–98 63. Plonsey R, Barr RC. 2000. Bioelectricity: A Quantitative Approach. New York: Kluwer 64. Raphael RM, Popel AS, Brownell WE. 2000. A membrane bending model of outer hair cell electromotility. Biophys. J. 78:2844–62 65. Liberman MC, Gao J, He DZZ, Wu X, Jia S, Zuo J. 2002. Prestin is required for electromotility of the outer hair cell and for the cochlear amplifier. Nature 419:300–4 66. Balzani V, Credi A, Raymo FM, Stoddart JF. 2000. Artificial molecular machines. Angew. Chem. Int. Ed. 39:3348–91 ¨ 67. Gunther P, Dransfeld K. 1992. Local poling of ferroelectric polymers by scanning force microscopy. Appl. Phys. Lett. 61:1137–39 68. Franke K, Besold J, Haessler W, Seegebarth C. 1994. Modification and detection of domains on ferroelectric PZT films by scanning force microscopy. Surf. Sci. Lett. 302:L283–88 69. Gruverman A, Auciello O, Tokumoto H. 1996. Scanning force microscopy for the study of domain structure in ferroelectric thin films. J. Vac. Sci. Technol. B 14:602–5 70. Gruverman A, Auciello O, Tokumoto H. 1996. Nanoscale investigation of fatigue effects in Pb(Zr,Ti)O3 films. Appl. Phys. Lett. 69:3191–93

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67. The pioneering work introducing the concept of electromechanical detection in SPM.

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71. One of the first demonstrations of PFM on inorganic ferroelectric materials and ferroelectric domain writing in inorganic ferroelectric thin films.

72. Introduced the concept of lateral PFM and demonstrated the complete reconstruction of surface crystallography from three components of the electromechanical response vector.

82. Laid the foundation of using PFM as a tool for fundamental studies of polarization reversal processes in ferroelectrics.

83. Established a novel method for metal nanostructure fabrication, referred to as ferroelectric lithography.

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71. Hidaka T, Maruyama T, Saitoh M, Mikoshiba N, Shimizu M, et al. 1996. Formation and observation of 50 nm polarized domains in PbZr1−x Tix O3 thin film using scanning probe microscope. Appl. Phys. Lett. 68:2358–59 ˜ ˜ J. ¨ ¨ 72. Eng LM, Guntherodt HJ, Schneider GA, Kopke U, Munoz Saldana 1999. Nanoscale reconstruction of surface crystallography from threedimensional polarization distribution in ferroelectric barium–titanate ceramics. Appl. Phys. Lett. 74:233–35 73. Kalinin SV, Rodriguez BJ, Jesse S, Shin S, Baddorf AP, et al. 2006. Vector piezoresponse force microscopy. Microsc. Microanal. 12:206–20 74. Verdaguer A, Sacha GM, Bluhm H, Salmeron M. 2006. Molecular structure of water at interfaces: wetting at the nanometer scale. Chem. Rev. 106:1478–510 75. Sacha GM, Verdaguer A, Salmeron M. 2006. Induced water condensation and bridge formation by electric fields in atomic force microscopy. J. Phys. Chem. B 110:14870–73 76. Gruverman A, Auciello O, Tokumoto H. 1998. Imaging and control of domain structures in ferroelectric thin films via scanning force microscopy. Annu. Rev. Mater. Sci. 28:101–23 77. Eng LM, Grafstrom S, Loppacher Ch, Schlaphof F, Trogisch S, et al. 2001. 3-Dimensional electric field distribution above and below dielectric (ferroelectric) surfaces: nanoscale measurements and manipulation with scanning probe microscopy. Adv. Solid State Phys. 41:287–98 78. Alexe M, Gruverman A, eds. 2004. Nanoscale Characterization of Ferroelectric Materials. Berlin: Springer 79. Hong S, ed. 2004. Nanoscale Phenomena in Ferroelectric Thin Films. Norwell, MA: Kluwer Acad. ¨ 80. Roelofs A, Bottger U, Waser R, Schlaphof F, Trogisch S, Eng LM. 2000. Differentiating 180◦ and 90◦ switching of ferroelectric domains with threedimensional piezoresponse force microscopy. Appl. Phys. Lett. 77:3444–46 81. Shin H, Hong S, Moon J, Jeon JU. 2002. Read/write mechanisms and data storage system using atomic force microscopy and MEMS technology. Ultramicroscopy 91:103–10 82. Tybell T, Paruch P, Giamarchi T, Triscone JM. 2002. Domain wall creep in epitaxial ferroelectric Pb(Zr0.2 Ti0.8 )O3 3 thin films. Phys. Rev. Lett. 89:097601 83. Kalinin SV, Bonnell DA, Alvarez T, Lei X, Hu Z, Ferris JH. 2002. Atomic polarization and local reactivity on ferroelectric surfaces: a new route toward complex nanostructures. Nano Lett. 2:589–93 84. Kalinin SV, Bonnell DA, Alvarez T, Lei X, Hu Z, et al. 2004. Ferroelectric lithography of multicomponent nanostructures. Adv. Mater. 16:795–99 85. Terabe K, Nakamura M, Takekawa S, Kitamura K, Higuchi S, et al. 2003. Microscale to nanoscale ferroelectric domain and surface engineering of a nearstoichiometric LiNbO3 crystal. Appl. Phys. Lett. 82:433–35 86. Rodriguez BJ, Gruverman A, Kingon AI, Nemanich RJ, Ambacher O. 2002. Piezoresponse force microscopy for polarity imaging of GaN. Appl. Phys. Lett. 80:4166–68

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87. Halperin C, Mutchnik S, Agronin A, Molotskii M, Urenski P, et al. 2004. Piezoelectric effect in human bones studied in nanometer scale. Nano Lett. 4:1253–56 88. Kalinin SV, Rodriguez BJ, Jesse S, Thundat T, Gruverman A. 2005. Electromechanical imaging of biological systems with sub10 nm resolution. Appl. Phys. Lett. 87:053901 89. Rodriguez BJ, Kalinin SV, Shin J, Jesse S, Grichko V, et al. 2006. Electromechanical probing of biomaterials by scanning probe microscopy. J. Struct. Biol. 153:151–59 90. Lebedev NN, Skal’skaya IP, Uflyand Ya S. 1966. Problems in Mathematical Physics. New York: Pergamon Press 91. Sharvin YV. 1965. On a possible method to study Fermi surfaces. Zh. Eksp. Teor. Fiz. 48:984–85 92. Kalinin SV, Karapetian E, Kachanov M. 2004. Nanoelectromechanics of piezoresponse force microscopy. Phys. Rev. B 70:184101 93. Karapetian E, Kachanov M, Kalinin SV. 2005. Nanoelectromechanics of piezoelectric indentation and applications to scanning probe microscopies of ferroelectric materials. Philos. Mag. 85:1017–51 94. Huey BD. 2004. SPM measurements of ferroelectrics at MHz frequencies. Nanoscale Phenomena in Ferroelectric Thin Films, ed. S Hong, pp. 239–62. New York: Kluwer 95. Giannakopoulos AE, Suresh S. 1999. Theory of indentation of piezoelectric materials. Acta Mater. 47:2153–64 96. Chen WQ, Ding HJ. 1999. Indentation of a transversely isotropic piezoelectric half-space by a rigid sphere. Acta Mech. Solida Sin. 12:114–20 97. Karapetian E, Sevostianov I, Kachanov M. 2000. Point force and point electric charge in infinite and semi-infinite transversely isotropic piezoelectric solids. Philos. Mag. B 80:331–59 98. Ganpule CS, Nagarajan V, Li H, Ogale AS, Steinhauer DE, et al. 2000. Role of 90◦ domains in lead zirconate titanate thin films. Appl. Phys. Lett. 77:292–94 99. Agronin A, Molotskii M, Rosenwaks Y, Strassburg E, Boag A, et al. 2005. Nanoscale piezoelectric coefficient measurements in ionic conducting ferroelectrics. J. Appl. Phys. 97:084312 ˜ Saldana ˜ J, Kalinin SV. 2004. Modeling and 100. Felten F, Schneider GA, Munoz measurement of surface displacements in BaTiO3 bulk material in piezoresponse force microscopy. J. Appl. Phys. 96:563–68 101. Scrymgeour DA, Gopalan V. 2005. Nanoscale piezoelectric response across a single antiparallel ferroelectric domain wall. Phys. Rev. B 72:024103 102. Kalinin SV, Eliseev EA, Morozovska AN. 2006. Materials contrast in piezoresponse force microscopy. Appl. Phys. Lett. 88:232904 103. Eliseev EA, Kalinin SV, Jesse S, Bravina SL, Morozovska AN. 2007. Electromechanical detection in scanning probe microscopy: tip models and materials contrast. http://www.arxiv.org; cond-mat/0607543 104. Morozovska AN, Bravina SL, Eliseev EA, Kalinin SV. 2007. Resolution function theory in piezoresponse force microscopy: domain wall profile, spatial resolution, and tip calibration. http://www.arxiv.org; cond-mat/0608289

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88. Electromechanical activity of biological systems such as calcified and connective tissues was mapped with sub-10-nm resolution, demonstrating the potential of PFM for imaging biological systems.

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108. The 8-nm bit size reported here renders ferroelectrics promising materials for high-density (∼10 Tbit inch−2 ) data storage.

122. Elegantly analyzed ferroelectric domain switching in a PFM experiment, extending early results of Landauer (25) to the point contact geometry.

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105. Kalinin SV, Jesse S, Rodriguez BJ, Shin J, Baddorf AP, et al. 2006. Spatial resolution, information limit, and contrast transfer in piezoresponse force microscopy. Nanotechnology 17:3400–11 106. Jesse S, Baddorf AP, Kalinin SV. 2006. Dynamic behavior in piezoresponse force microscopy. Nanotechnology 17:1615–28 107. Rabe U. 2006. Atomic force acoustic microscopy. In Applied Scanning Probe Methods, Vol II, ed. B Bhushan, H Fuchs, pp. 37–90. New York: Springer 108. Cho Y, Hashimoto S, Odagawa N, Tanaka K, Hiranaga Y. 2006. Nanodomain manipulation for ultrahigh density ferroelectric data storage. Nanotechnology 17:S137–41 109. Nassau K, Levinstein HJ, Loiacono GM. 1965. The domain structure and etching of ferroelectric lithium niobate. Appl. Phys. Lett. 6:228–29 110. Kholkin AL, Bdikin IK, Shvartsman VV, Orlova A, Kiselev D, et al. 2004. Local electromechanical properties of ferroelectric materials for piezoelectric applications. Mater. Res. Soc. Symp. Proc. 839E, O7.6 111. Paruch P, Giamarchi T, Triscone JM. 2005. Domain wall roughness in epitaxial ferroelectric PbZr0.2 Ti0.8 O3 thin films. Phys. Rev. Lett. 94:197601 112. Dawber M, Gruverman A, Scott JF. 2006. Skyrmion model of nano-domain nucleation in ferroelectrics and ferromagnets. J. Phys. Condens. Matter 18:L71– 79 113. Agronin A, Rosenwaks Y, Rosenman G. 2006. Direct observation of pinning centers in ferroelectrics. Appl. Phys. Lett. 88:072911 114. Woo J, Hong S, Setter N, Shin H, Jeon JU, et al. 2001. Quantitative analysis of the bit size dependence on the pulse width and pulse voltage in ferroelectric memory devices using atomic force microscopy. J. Vac. Sci. Technol. B 19:818–24 115. Rodriguez BJ, Nemanich RJ, Kingon A, Gruverman A, Kalinin SV, et al. 2005. Domain growth kinetics in lithium niobate single crystals studied by piezoresponse force microscopy. Appl. Phys. Lett. 86:012906 116. Molotskii MI, Shvebelman MM. 2005. Dynamics of ferroelectric domain formation in an atomic force microscope. Philos. Mag. 85:1637–55 117. Abplanalp M. 2001. Piezoresponse scanning force microscopy of ferroelectric domains. PhD thesis. Swiss Fed. Inst. Technol., Zurich 118. Agronin A, Molotskii M, Rosenwaks Y, Rosenman G, Rodriguez BJ, et al. 2006. Dynamics of ferroelectric domain growth in the field of atomic force microscope. J. Appl. Phys. 99:104102 119. Gruverman A, Rodriguez BJ, Dehoff C, Waldrep JD, Kingon AI, et al. 2005. Direct studies of domain switching dynamics in thin film ferroelectric capacitors. Appl. Phys. Lett. 87:082902 120. Gruverman A, Kholkin A. 2006. Nanoscale ferroelectrics: processing, characterization and future trends. Rep. Prog. Phys. 69:2443–74 121. Molotskii M. 2003. Generation of ferroelectric domains in atomic force microscope. J. Appl. Phys. 93:6234–37 122. Molotskii M, Agronin A, Urenski P, Shvebelman M, Rosenman G, Rosenwaks Y. 2003. Ferroelectric domain breakdown. Phys. Rev. Lett. 90:107601 123. Shvebelman M. 2005. Static and dynamic properties of ferroelectric domains studied by atomic force microscopy. PhD thesis. Tel Aviv University, Tel Aviv

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124. Kalinin SV, Gruverman A, Rodriguez BJ, Shin J, Baddorf AP, et al. 2005. Nanoelectromechanics of polarization switching in piezoresponse force microscopy. J. Appl. Phys. 97:074305 125. Emelyanov AY. 2005. Coherent ferroelectric switching by atomic force microscopy. Phys. Rev. B 71:132102 126. Morozovska AN, Eliseev EA. 2006. Screening and size effects on the nanodomain tailoring in ferroelectrics semiconductors. Phys. Rev. B 73:104440 127. Morozovska AN, Eliseev EA. 2006. The study of screening phenomena under the nano-domain formation in ferroelectric semiconductors. Phys. Status Solidi B 243:1996–2011 128. Morozovska AN, Eliseev EA, Kalinin SV. 2006. Domain nucleation and hysteresis loop shape in piezoresponse force spectroscopy. Appl. Phys. Lett. 89:192901 129. Molotskii M. 2003. Forward motion of ferroelectric domain walls. Philos. Mag. Lett. 83:763–67 130. Morozovska AN, Svechnikov SV, Eliseev EA, Jesse S, Rodriguez BJ, Kalinin SV. 2007. Piezoresponse force spectroscopy of ferroelectric materials. http:// www.arxiv.org; cond-mat/0610764 131. Molotskii M. 2005. Generation of ferroelectric domains in films using atomic force microscope. J. Appl. Phys. 97:014109 132. Molotskii M, Rosenwaks Y, Rosenman G. 2007. Ferroelectric domain breakdown. Annu. Rev. Mater. Res. 37:in press 133. Kalinin SV, Bonnell DA. 2000. Effect of phase transition on the surface potential of the BaTiO3 (100) surface by variable temperature scanning surface potential microscopy. J. Appl. Phys. 87:3950–57 134. Kalinin SV, Johnson CY, Bonnell DA. 2002. Domain polarity and temperature induced potential inversion on the BaTiO3 (100) surface. J. Appl. Phys. 91:3816– 23 135. Kalinin SV, Bonnell DA. 2001. Local potential and polarization screening on ferroelectric surfaces. Phys. Rev. B 63:125411 ¨ 136. Buhlmann S, Colla E, Muralt P. 2005. Polarization reversal due to charge injection in ferroelectric films. Phys. Rev. B 72:214120 137. Birk H, Glatz-Reichenbach J, Li-Jie Schreck E, Dransfeld K. 1991. The local piezoelectric activity of thin polymer films observed by scanning tunneling microscopy. J. Vac. Sci. Technol. B 9:1162–65 138. Guo HY, Xu JB, Wilson IH, Xie Z, Luo EZ, et al. 2002. Study of domain stability on (Pb0.76 Ca0.24 )TiO3 thin films using piezoresponse microscopy. Appl. Phys. Lett. 81:715–17 139. Kim ID, Avrahami Y, Tuller HL, Park YB, Dicken MJ, Atwater HA. 2005. Study of orientation effect on nanoscale polarization in BaTiO3 thin films using piezoresponse force microscopy. Appl. Phys. Lett. 86:192907 140. Kalinin SV, Gruverman A, Bonnell DA. 2004. Quantitative analysis of nanoscale switching in SrBi2 Ta2 O9 thin films by piezoresponse force microscopy. Appl. Phys. Lett. 85:795–97 ¨ 141. Eng LM, Guntherodt HJ, Rosenman G, Skliar A, Oron M, et al. 1998. Nondestructive imaging and characterization of ferroelectric domains in periodically poled crystals. J. Appl. Phys. 83:5973–77

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142. Rodriguez BJ, Gruverman A, Kingon AI, Nemanich RJ, Cross JS. 2004. Threedimensional high-resolution reconstruction of polarization in ferroelectric capacitors by piezoresponse force microscopy. J. Appl. Phys. 95:1958–62 143. Desfeux R, Legrand C, Da Costa A, Chateigner D, Bouregba R, Poullain G. 2006. Correlation between local hysteresis and crystallite orientation in PZT thin films deposited on Si and MgO substrates. Surf. Sci. 600:219–28 ¨ 144. Buhlmann S. 2004. Patterned and self-assembled ferroelectric nano-structures ob´ tained by epitaxial growth and e-beam lithography. PhD thesis. Ecole Polytech. F´ed., Lausanne, Switzerland 145. Saya Y, Watanabe S, Kawai M, Yamada H, Matsushige K. 2000. Investigation of nonswitching regions in ferroelectric thin films using scanning force microscopy. Jpn. J. Appl. Phys. 39:3799–803 146. Alexe M, Harnagea C, Hesse D, Gosele U. 2001. Polarization imprint and size effects in mesoscopic ferroelectric structures. Appl. Phys. Lett. 79:242–44 147. Ma W, Hesse D. 2004. Polarization imprint in ordered arrays of epitaxial ferroelectric nanostructures. Appl. Phys. Lett. 84:2871–73 148. Harnagea C, Pignolet A, Alexe M, Hesse D, Gosele U. 2000. Quantitative ferroelectric characterization of single submicron grains in Bi-layered perovskite thin films. Appl. Phys. A 70:261–67 149. Ricinschi D, Mitoseriu L, Stancu A, Postolache P, Okuyama M. 2004. Analysis of the switching characteristics of PZT films by first order reversal curve diagrams. Integr. Ferroelec. 67:103–15 150. Ricinschi D, Noda M, Okuyama M, Ishibashi Y, Iwata M, Mitoseriu L. 2003. A Landau-theory-based computational study of in-plane and out-of-plane polarization components role in switching of ferroelectric thin films. J. Korean Phys. Soc. 42:S1232–36 151. Ricinschi D, Okuyama M. 2002. Nanoscale spatial correlation of piezoelectric displacement hysteresis loops of PZT films in the fresh and fatigued states. Integr. Ferroelec. 50:149–58 152. Gruverman A, Rodriguez BJ, Kingon AI, Nemanich RJ, Cross JS, Tsukada M. 2003. Spatial inhomogeneity of imprint and switching behavior in ferroelectric capacitors. Appl. Phys. Lett. 82:3071–73 153. Gruverman A, Rodriguez BJ, Kingon AI, Nemanich RJ, Tagantsev AK, et al. 2003. Mechanical stress effect on imprint behavior of integrated ferroelectric capacitors. Appl. Phys. Lett. 83:728–30 154. Dehoff C, Rodriguez BJ, Kingon AI, Nemanich RJ, Gruverman A, Cross JS. 2005. Atomic force microscopy-based experimental setup for studying domain switching dynamics in ferroelectric capacitors. Rev. Sci. Instrum. 76:023708 ¨ 155. Abplanalp M, Fousek J, Gunter P. 2001. Higher order ferroic switching induced by scanning force microscopy. Phys. Rev. Lett. 86:5799–802 156. Harnagea C. 2001. Local piezoelectric response and domain structures in ferroelectric thin films investigated by voltage-modulated force microscopy. PhD thesis. MartinLuther-Univ. Halle-Wittenberg, Halle 157. Jesse S, Lee HN, Kalinin SV. 2006. Quantitative mapping of switching behavior in piezoresponse force microscopy. Rev. Sci. Instrum. 77:073702

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158. Jesse S, Baddorf AP, Kalinin SV. 2006. Switching spectroscopy piezoresponse force microscopy of ferroelectric materials. Appl. Phys. Lett. 88:062908 159. Wu A, Vilarinho PM, Shvartsman VV, Suchaneck G, Kholkin AL. 2005. Domain populations in lead zirconate titanate thin films of different compositions via piezoresponse force microscopy. Nanotechnology 16(11):2587–95 160. Pearson GL, Feldmann WL. 1958. Powder-pattern techniques for delineating ferroelectric domain structures. J. Phys. Chem. Solids 9:28–30 161. Distler GI, Konstantinova VP, Gerasimov VM, Tolmacheva GA. 1968. Interaction of defect and domain structures of triglycine sulphate crystals in ferroelectric and paraelectric states. Nature 218:762–65 162. Distler GI. 1968. The real structure of crystals and selective nucleation at surface local long range active centres. J. Cryst. Growth 3:175–79 163. Kalinin SV, Bonnell DA. 2002. Imaging mechanism of piezoresponse force microscopy of ferroelectric surfaces. Phys. Rev. B 65:125408 164. Kalinin SV, Bonnell DA. 2001. Temperature dependence of polarization and charge dynamics on the BaTiO3 (100) surface by scanning probe microscopy. Appl. Phys. Lett. 78:1116–18 165. Peter F, Szot K, Waser R, Reichenberg B, Tiedke S, Szade J. 2004. Piezoresponse in the light of surface adsorbates: relevance of defined surface conditions for perovskite materials. Appl. Phys. Lett. 85:2896–98 166. Giocondi JL, Rohrer GS. 2001. Spatially selective photochemical reduction of silver on the surface of ferroelectric barium titanate. Chem. Mater. 13:241–42 167. Hanson JN, Rodriguez BJ, Nemanich RJ, Gruverman A. 2006. Fabrication of metallic nanowires on a ferroelectric template via photochemical reaction. Nanotechnology 17:4946–49 168. Munkholm A, Streiffer SK, Ramana Murty MV, Eastman JA, Thompson C, et al. 2002. Antiferrodistortive reconstruction of the PbTiO3 (001) surface. Phys. Rev. Lett. 88:016101 169. Gruverman A, Rodriguez BJ, Kalinin SV. 2006. BioPFM. In Scanning Probe Microscopy: Electrical and Electromechanical Phenomena on the Nanoscale, ed. SV Kalinin, A Gruverman, in press. New York: Springer Verlag 170. Kalinin SV, Rodriguez BJ, Shin J, Jesse S, Grichko V, et al. 2006. Bioelectromechanical imaging by scanning probe microscopy: Galvani’s experiment at the nanoscale. Ultramicroscopy 106:334–40 171. Physics Update. 2006. Phys. Today Jan 2006:9 172. Rodriguez BJ, Jesse S, Baddorf AP, Kalinin SV. 2006. High resolution electromechanical imaging of ferroelectric materials in a liquid environment by piezoresponse force microscopy. Phys. Rev. Lett. 96:237602 173. Stephenson K, Smith T. 2006. Electrochemical SPM: fundamentals and applications. In Scanning Probe Microscopy: Electrical and Electromechanical Phenomena on the Nanoscale, ed. SV Kalinin, A Gruverman, pp. 280–314. New York: Springer Verlag 174. Rosner BT, van der Weide DW. 2002. High-frequency near-field microscopy. Rev. Sci. Instrum. 73:2505–25 175. Kalinin SV, Jesse S. 2006. U.S. Patent Appl. No. 11/513,348

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172. Demonstrated high-resolution PFM imaging in a liquid environment, opening the pathway to electromechanical probing of biological systems in physiological environments.

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176. Onaran AG, Balantekin M, Lee W, Hughes WL, Buchine BA, et al. 2006. A new atomic force microscope probe with force sensing integrated readout and active tip. Rev. Sci. Instrum. 77:023501 177. Dienwiebel M, de Kuyper E, Crama L, Frenken JWM, Heimberg JA, et al. 2005. Design and performance of a high-resolution frictional force microscope with quantitative three-dimensional force sensitivity. Rev. Sci. Instrum. 76:043704 178. Morrison FD, Luo Y, Szafraniak I, Nagarajan V, Wehrspohn RB, et al. 2003. Ferroelectric nanotubes. Rev. Adv. Mater. Sci. 4:114–22 179. Rosenman G, Urenski P, Agronin A, Rosenwaks Y, Molotskii M. 2003. Submicron ferroelectric domain structures tailored by high-voltage scanning probe microscopy. Appl. Phys. Lett. 82:103–5 180. Onaran AG, Balantekin M, Lee W, Hughes WL, Buchine BA, et al. 2006. A new atomic force microscope probe with force sensing integrated readout and active tip. Rev. Sci. Instrum. 77:023501 181. Frederix PLTM, Gullo MR, Akiyama T, Tonin A, de Rooij NF, et al. 2005. Assessment of insulated conductive cantilevers for biology and electrochemistry. Nanotechnology 16:997–1005 182. Precision Acoustics Ltd. www.acoustics.co.uk/5Transducers.jpg 183. Murayama E. www.op.titech.ac.jp/lab/okui/murayam/pvdf.jpg 184. Bai M, Poulsen M, Ducharme S. 2006. Effects of annealing conditions on ferroelectric nanomesa self-assembly. J. Phys. Condens. Matter 18:7383–92 185. Qu H, Yao W, Garcia T, Zhang J, Sorokin AV, et al. 2003. Nanoscale polarization manipulation and conductance switching in ultrathin films of a ferroelectric copolymer. Appl. Phys. Lett. 82:4322–24

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