ANRV315-MR37-07 ARI 30 May 2007 21:54

Nanoscale Electromechanicsof Ferroelectric and BiologicalSystems: A New Dimensionin 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

1Materials Sciences and Technology Division and Center for Nanophase MaterialsSciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831;email: sergei2@ornl.gov2Department of Mathematics and Computer Science, Suffolk University, Boston,Massachusetts 021143Institute for Problems of Materials Science, National Academy of Science ofUkraine, 03142 Kiev, Ukraine4V. Lashkaryov Institute of Semiconductor Physics, National Academy of Science ofUkraine, 03028 Kiev, Ukraine

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

First published online as a Review in Advance onMarch 27, 2007

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

This article’s doi:10.1146/annurev.matsci.37.052506.084323

Copyright c© 2007 by Annual Reviews.All rights reserved

1531-7331/07/0804-0189$20.00

Key Words

piezoelectricity, flexoelectricity, hysteresis, domain dynamics,polarization switching

AbstractFunctionality of biological and inorganic systems ranging from non-volatile computer memories and microelectromechanical systems toelectromotor proteins and cellular membranes is ultimately basedon the intricate coupling between electrical and mechanical phe-nomena. In the past decade, piezoresponse force microscopy (PFM)has been established as a powerful tool for nanoscale imaging, spec-troscopy, and manipulation of ferroelectric and piezoelectric ma-terials. Here, we give an overview of the fundamental image for-mation mechanism in PFM and summarize recent theoretical andtechnological advances. In particular, we show that the signal for-mation in PFM is complementary to that in the scanning tunnelingmicroscopy (STM) and atomic force microscopy (AFM) techniques,and we discuss the implications. We also consider the prospect of ex-tending PFM beyond ferroelectric characterization for quantitativeprobing of electromechanical behavior in molecular and biologicalsystems and high-resolution probing of static and dynamic polariza-tion switching processes in low-dimensional ferroelectric materialsand heterostructures.

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Nonvolatile ferroelectricrandom access memory(FeRAM): memorytechnology based on theexistence of stable,switchable polarizationstates in ferroelectricmaterials

Electromechanicalcoupling: the mechanicalresponse to an appliedelectrical stimulus and theelectrical response to anapplied mechanical stimulus

INTRODUCTION

In the past two decades, the concept of “small is different” has been established fora wide variety of phenomena, including electrical, optical, magnetic, and mechanicalproperties of materials (1, 2). This has given rise to new areas of research in the physicaland biological sciences such as nanomechanics (3), plasmonics (4), nano-optics (5), andmolecular electronics (6) as well as to multiple new material and device applicationsthat have led to significant breakthroughs in medicine, biology, physics, and materialsscience. However, one largely untapped but potentially important area of nanoscienceinvolves the interplay of electricity and mechanics at the nanoscale. Electromechanicalinteractions underpin a range of systems—from nonvolatile ferroelectric computerrandom access memories (FeRAM) to micro- and nanoelectromechanical systems(MEMS and NEMS, respectively) to complex electromechanical and electrochemicaltransformations in functional biomolecules and nanostructures.

Coupling between electrical and mechanical phenomena is a universal feature ofmany inorganic materials and virtually all biological systems. The coupling betweenelectrical and mechanical phenomena in living tissues, discovered by Galvani in 1771(7), was the harbinger of the modern theory of electricity. Piezoelectricity in inor-ganic materials has been studied in great detail since the discovery of piezoelectricityin quartz at the end of the nineteenth century. This achievement was made possible bythe combination of macroscopic measurements that provided information on prop-erties, diffraction techniques that elucidated atomic structure, and advanced theory(8). However, the spectrum of electromechanical phenomena on the molecular andnanoscale levels is much richer than in the macroscopic systems owing to symmetrybreaking at surfaces and interfaces, the lack of constraints imposed by the lattice inlow-dimensional and soft condensed-matter systems, and the absence of macroscopicaveraging. In this introductory section, we consider the relationship between elec-tromechanical 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 brieflydiscuss complex electromechanical couplings in bio- and molecular systems.

Piezoelectricity in Crystals and Polar Molecules

The simplest example of linear electromechanical coupling is piezoelectricity, inwhich the application of stress, X, results in electrical polarization (direct piezo-electric 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 ofelectromechanical response provide insight into the polarizability of material, andvice versa.

The atomic origins of piezoelectricity are directly related to the bond and latticedipoles, as Figure 1 illustrates (9). From simple electrostatic considerations, thepiezoelectric constant for a single chemical bond is related to the bond parameters asd = 2q

/kl , where q is fractional charge, k is the spring constant, and l is bond length.

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Table 1 Electromechanical coupling in crystals and molecules

Crystal Single bondDisplacement xi = s i j X j + dik Ek x = (1/k)F/ l + (2q/kl)ECharge Di = di j X j + εik Ek P = (q/k)F + (2q2/k)E

Ferroelectric polarization:a spontaneous dipolemoment existing owing tothe distortion of a crystallattice that can be switchedbetween two or moreequivalent states by theapplication of electrical field

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

Ferroelectric Materials

Pyroelectric materials comprise a subset of piezoelectric materials possessingtemperature-dependent spontaneous polarization in the absence of an applied fieldor strain. The direction of the spontaneous polarization is directly related to thecrystal structure. A subset of pyroelectric materials in which the spontaneous polar-ization and hence the directionality of the electromechanical activity can be switchedby external electric (ferroelectric) or mechanical (ferroelastic) stimuli is particularlyimportant for numerous technological applications (8).

After the discovery of strong piezoelectricity in ferroelectrics, numerous applica-tions as sensors, actuators, and transducers have emerged (11, 12). In the past decade,the development of deposition techniques for epitaxial ferroelectric thin films andadvanced ceramic fabrication has resulted in numerous novel applications such asthose in MEMS and NEMS (13–15). The ability of ferroelectric materials to exist inand switch between two or more polarized states and retain polarization for a finiteperiod makes possible their application as the active component of FeRAM (16–18)capacitors. Data storage devices have also been envisioned; in these, epitaxial ferro-electric thin films or ferroelectric nanostructures are accessed by a nanoscale probeor an array of probes to read and write information via ferroelectric polarizationswitching (19). Polarization-dependent chemical reactivity allows this approach to beextended to lithography (20). Ferroelectric switching may also enable novel devicesbased on ferroelectric and multiferroic tunneling junctions in which the switchablespontaneous polarization adds a new level of functionality (21). Ferroelectrics are alsoideal materials for nonlinear optics applications because the local direction of the po-larization is directly related to the electro-optic and electromechanic tensors, andhence the properties of the material can be easily controlled through domain struc-ture engineering. Often, nanoscale disorder results in unusual behavior such as relaxorferroelectricity in inorganic materials and ferroelectric polymers (22, 23). Stronglyenhanced electromechanical properties in relaxors in turn enable new applicationsfor undersea warfare and medical imaging. However, the fundamental nanoscale

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+ + +

– – –

F

E

–q +qk

a

Charge = force x d33 e Strain = bias x d33 f

c Antiparallel d Helical

Disorder Order b

F

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Landauer paradox:phenomenon in which theelectric fields required toinduce polarization reversalcorrespond to unrealisticallyhigh values for theactivation energy fordomain nucleation

mechanisms behind these properties are still heavily debated (24). Figure 2 sum-marizes some of the applications of nanoferroelectrics.

At the heart of the majority of these applications is ferroelectric polarizationswitching, and much effort has been put forth toward understanding the fundamentalphysics involved, including the nucleation mechanism, discrepancies between pre-dicted and observed coercive fields, loss of polarization due to repeated switchingcycles (fatigue), domain wall width, and minimum stable domain size. In the caseof a capacitor structure with a homogeneous field, the nucleation is initiated at aferroelectric-electrode interface, and a domain grows as an elongated ellipsoid, asfirst described by Landauer (25). Remarkably, the nucleation barriers, or the fieldrequired to nucleate domains predicted by Landauer theory, are much larger thanexperimentally observed, suggesting the existence of defect centers at which nu-cleation is initiated. Several theories have been proposed to address this apparentreduction of the activation barrier, including passive layers (26), tunneling (27), andferroelectric-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 thephysics of low-dimensional ferroelectrics (29). Traditionally, piezoelectric couplingin nanocrystalline materials was considered to be analogous to macroscopic materials.For ferroelectrics, the studies of the size effect have centered on the paradigm of theferroelectric bulk and the paraelectric surface layer. The experimental discovery offerroelectric nanotubes (30, 31) in the past several years has prompted theoreticalstudies that predict new types of helical polar ordering in nanoferroelectrics (32). Inaddition, advances in the self-assembly of ferroelectric nanostructures and multifer-roic heterostructures have enabled studies on size effects, constrained systems, andmultiferroic coupling (30, 33–40).

Novel Couplings

Similar to transport, optical, and magnetic phenomena, the nanoscale can also enablecompletely new types of electromechanical behavior. As illustrated by the exampleof a single bond, piezoelectric coupling is a universal feature of polar systems. Forexample, a dipole layer on the surface, such as an ordered water layer, is expected tobe 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 of

←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−Figure 1Electromechanical coupling on the molecular level. (a) The application of an electric fieldresults in bond contraction or expansion. An electric field acting along the molecule axisinduces an electrostatic force, Fel = 2qE, and the elongation of the molecule isdl = (F + Fel )/k = F/k + 2q E/k. In terms of strain, x = dl/ l . (b) The application of anexternal force results in a change of the dipole moment. If the bond dipoles are parallel, thematerial 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 2Applications of nanoferroelectrics. (a) Domains can be “written” in ferroelectric materials torepresent bits of information for data storage applications, as shown in this nanodomain dotarray 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 consumerelectronics (e.g., Sony PlayStation 2). Shown is an isomeric cross section of a ferroelectricmemory cell with superimposed reconstruction of the polarization direction. Reproduced withpermission from Reference 142. Copyright 2004, American Institute of Physics.(c) Ferroelectric tunneling barriers allow an additional degree of freedom in tunneling devicefunctionality. 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 thefabrication 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 nonlinearoptics. Reprinted with permission from Reference 179. Copyright 2003, American Institute ofPhysics.

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this phenomenon is obvious: All biologically relevant surfaces are characterized by thepresence of ordered or disordered water layers, which will affect the electromechanicalactivity of the system.

In centrosymmetric materials, symmetry breaking at surfaces and interfaces cangive rise to surface piezoelectric coupling even in nonpolar materials, and a numberof novel electromechanical phenomena, including surface piezoelectricity and flexo-electricity, were predicted (41). Even for purely nonpolar materials such as grapheneor carbon nanotubes, local curvature will cause a redistribution of the electron den-sity and the formation of a curvature-dependent electric dipole and hence result inflexoelectric electromechanical coupling (Figure 3) (S.V. Kalinin & V. Meunier, un-published data). Similar couplings emerge in soft condensed-matter systems such asliquid crystals and cellular membranes (42).

These nanoelectromechanical phenomena will affect other functional propertiesof a material. For example, the chemical reactivity of piezoelectric semiconductorswill be mediated through piezoelectrically induced potential effects of the Fermi levelposition. Electric fields of the surface water layers during formation and collapse ofbubbles are considered one of the origins of the cavitation damage and sonolumines-cence and were considered the governing factor determining the feasibility of bubblefusion (43). Another example is triboelectricity, during which mechanical deforma-tions 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 electrome-chanical coupling phenomena in biological systems. Piezoelectricity is ubiquitous inbiopolymers 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 me-chanical stress that generates the electric potential, may control the mechanisms oflocal tissue development (52, 53). On the molecular level, this behavior is due to thesimultaneous presence of polar bonds and optical activity, which provide sufficientconditions for piezoelectricity. Experimentally, piezoelectric activity of biopolymersis on the order of 1–5 pm V−1, i.e., comparable to inorganic piezoelectrics such asquartz. Despite the tremendous interest in the role and possible biological signifi-cance of piezoelectricity—reflected in multiple publications from 1950 to 1980 (47,54–61) and in the employment of methodologies for curing fracture, osteoporosis,and other bone diseases—these issues remain largely unresolved. This is due primarilyto the complex hierarchical structure of biological systems; for example, that of bonespans seven structural levels, from 25–30-nm-size collagen fibrils and hydroxyapatiteplatelets through different levels of organization of mineralized fibrils to osteons tothe macroarchitecture of cortical or trabecular bone (62). It is this complexity thatprecludes quantitative measurements and hence understanding of piezoelectricity inbiological systems.

Notably, piezoelectricity is the simplest form of the electromechanical cou-pling in biological systems: Linear phenomena such as flexoelectricity of molec-ular membranes and more complex forms of electromechanical coupling such as

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

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Piezoresponse forcemicroscopy (PFM):scanning probe techniquebased on the detection ofthe electromechanicalresponse of a material to anapplied electrical bias

Vector PFM: thereal-space reconstruction ofelectromechanical responsevector and localcrystallographic orientationfrom three components ofpiezoresponse: vertical PFMand two orthogonal lateralPFM components

SPM: scanning probemicroscopy

Piezoelectric surface: a3-D plot depicting thepiezoresponse as a functionof the angle between thedirection of the applied fieldand the measurement axis

voltage-controlled muscular contractions (63), cell electromotility (64), and electro-motor proteins (65) are ubiquitous. In many cases, these couplings are the functionalbases for processes such as transduction of acoustic signals into electrical pulses in theouter hair cell stereocilia (hearing), transport in ion channels, energy storage in mi-tochondria, cardiac myocyte activity, etc. Understanding local electromechanics, inaddition to electrochemistry, is one of the key challenges in understanding biologicalsystems.

To complement biological systems, in the past several years the concept of elec-trically driven molecular machines, in which electrochemical transformations inducea set of changes in bonding configurations leading to a change in molecular confor-mations, has emerged (66).

NANOSCALE PROBING OF ELECTROMECHANICALCOUPLING

Despite the rich gamut of novel electromechanical phenomena existing on thenanoscale and their relevance for areas as diversified as biological systems, energyharvesting materials, and computer memories, virtually no information is availableexperimentally. The reasons behind this dearth of information are straightforward—many of these effects manifest on length scales well below micron and often on theorder of nanometers, comparable to molecular sizes. Electromechanical measure-ments at these length scales require either currents on the order of ∼femtoamperesor displacements on the order of ∼picometers probed over areas of ∼10–1000 nm2,considered impossible until recently. The breakthrough in this direction has comewith the advent of piezoresponse force microscopy (PFM) (67–71), which utilizes theconverse piezoeffect to image local polarization orientation. Below, we discuss thebasic principles of PFM and compare the image formation mechanism in PFM withthat of other scanning probe microscopy (SPM) techniques.

Principles of PFM

PFM is based on the detection of bias-induced piezoelectric surface deforma-tion. The tip is brought into contact with the surface, and the local piezoelec-tric response is detected as the first harmonic component, A1ω, of the tip de-flection, A = A0 + A1ωcos (ωt + ϕ), during the application of the periodic biasVtip = Vdc + Vaccos (ωt) to the tip. The phase of the electromechanical responseof the surface, ϕ, yields information on the polarization direction below the tip. Forc− domains (polarization vector oriented normal to the surface and pointing down-ward), 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 nanometersper volt, defines the local electromechanical activity. PFM images can be conve-niently represented as A1ωcos (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 tunnelingmicroscopy

AFM: atomic forcemicroscopy

vibrations provides information on the in-plane surface displacement, referred to aslateral PFM. A third component of the displacement vector can be determined byimaging the same region of the sample after rotation by 90◦ (72). Provided that thevertical and lateral PFM signals are properly calibrated, the complete electromechan-ical 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 thetip, providing information on polarization switching in ferroelectrics as well as morecomplex 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 forprobing local electromechanical activity on the nanometer scale (76–79). Developedoriginally for imaging domain structures in ferroelectric materials, PFM was laterextended to local hysteresis loop spectroscopy (39, 80) and ferroelectric domain pat-terning for applications such as high-density data storage (81, 82) and ferroelectriclithography (83–85). The broad applicability of PFM to materials such as ferroelectricperovskites, piezoelectric III-V nitrides (86), and, recently, biological systems suchas calcified and connective tissues (87–89) has resulted in an increasing number ofpublications, as illustrated in Figure 4c. Various applications of PFM are shown inFigure 5.

PFM versus Other SPMs

The fundamental factors underpinning any SPM method are (a) the tensorial nature ofthe signal, (b) the signal dependence on contact radius (contact modes) or tip-surfaceseparation (noncontact modes), and (c) the signal dependence on the cantilever springconstant. These factors determine the strategies for instrumentation and techniquedevelopment and the potential for quantitative measurements.

In current-based techniques such as scanning tunneling microscopy (STM) andconductive atomic force microscopy (AFM), the excitation signal (applied bias) anddetected signal (current magnitude) are scalar quantities. Hence, the response isscalar,

I = �V, 1.

where � is the local conductance. The conductance scales with the contact area as� ∼ a1 in the classical (Maxvell) regime (90) and � ∼ a2 in the Sharvin regime(91). Hence, image contrast can be induced either by variation in specific resistanceor 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, � ∼ a0. Inmost applications of conductive AFM, the variations in local conductivities betweendissimilar materials (e.g., differently doped regions of a semiconductor) dominaterelatively small changes in contact geometry, the condition ensured by operation atlarge indentation forces on planar substrates. This, in turn, allows for calibration andquantitative measurements using current-based techniques.

In force-based techniques, both the excitation signal (force) and the responsesignal (displacement) are vectors. Hence, the AFM signal is a rank-two tensor. In

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Figure 4Schematic diagrams of (a) vertical and (b) lateral PFM. (c) Number of published papers basedon 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 thesurface plane along the cantilever, the 2-axis is in plane and perpendicular to thecantilever, and the 3-axis is the surface normal, the signal formation mechanism canbe represented as ⎛

⎜⎝w1

w2

w3

⎞⎟⎠ =

⎛⎜⎝a11 a12 a13

a21 a22 a23

a31 a32 a33

⎞⎟⎠

⎛⎜⎝F1

F2

F3

⎞⎟⎠ . 2.

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Figure 5Applications of PFM for (a) domain imaging (73), (b) domain patterning, (c) studies of domaindynamics (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).c and g reprinted with permission from References 115 and 158, copyright American Instituteof Physics.

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The component of the stiffness tensor a33 is probed by conventional vertical AFMand atomic force acoustic microscopy (AFAM), whereas a22 is probed by lateral forcemicroscopy. The off-diagonal component, a23 and a32, represents coupling betweenthe vertical and lateral signals. However, a conventional cantilever sensor does notallow the separation of the normal and longitudinal force components, both of whichresult in a flexural deformation of the cantilever. Hence, SPM flexural, fl, and tor-sional, tr , responses are related to force components as

(fltr

)=

(αa11 + βa31 αa12 + βa32 αa13 + βa33

χa21 χa22 χa23

) ⎛⎜⎝F1

F2

F3

⎞⎟⎠ , 3.

where α, β, and χ are proportionality coefficients dependent on cantilever geometryand detector calibration.

In the continuum mechanics limit, the contact stiffnesses are proportional to thecontact radius, ai j ∼ a1. When the contact area is a single molecule, as in proteinunfolding spectroscopy, or a single atom, as in atomic resolution imaging, ai j ∼ a0.Hence, quantitative force measurements are generally limited to when the contactgeometry is well characterized, as in nanoindentation techniques, or is weakly depen-dent on the probe, as in molecular unfolding spectroscopy. In other cases, the changesin contact stiffness represent the convolution of variation in local elastic propertiesand surface topography (topographic cross talk).

Most force sensors have well-defined resonant behavior: Corresponding resonantfrequencies are determined by the spring constant of the system and tip-surfacejunction. This enables a broad range of imaging techniques based on the resonantenhancement of cantilever oscillations, frequency tracking methods, etc. In particular,contact resonant frequency in techniques such as vertical and lateral AFAM is directlyrelated to the contact stiffnesses as determined by local elastic properties (primarysignal) and topography (cross talk).

Finally, in PFM the excitation signal is scalar, whereas the electromechanical re-sponse of the surface is a vector. Hence, the PFM response is a vector,⎛

⎜⎝w1

w2

w3

⎞⎟⎠ =

⎛⎜⎝d1

d2

d3

⎞⎟⎠V. 4.

Here, the response components di describe the electromechanical coupling in the ma-terial in the point contact geometry. As for AFM, the vertical and longitudinal surfacedisplacements contribute to the flexural vibration of the cantilever, and strategies toovercome this limitation are considered below. The distinctive feature of the elec-tromechanical response is dependence on contact area. From simple dimensionalityconsiderations, the electromechanical response does not depend on the contact area,di ∼ a0, 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) elec-tromechanical measurements are intrinsically quantitative (if good tip-surface con-tact is achieved) and do not require extensive probe calibration and that (b) signal is

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EFM: electrostatic forcemicroscopy

KPFM: Kelvin probe forcemicroscopy

relatively insensitive to topographic cross talk. However, resonance enhancement can-not be directly employed in PFM because the resonance frequency is determined bymechanical, rather than electromechanical, properties of material. Finally, quadraticbias dependence of electrostatic forces, as compared with linear piezoelectric inter-actions, suggests that the electromechanical signal cannot be unambiguously distin-guished from the electrostatic signal. These considerations determine limitations andsuggest 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) voltage-dependent contact mechanics of the tip-surface junction, including the relationshipbetween local surface displacement and material properties, and displacements in thevicinity of microstructural elements such as domain walls, (b) dynamic behavior of thecantilever, including the relationship between surface displacement vector and tor-sional and flexural vibrations of the cantilever, and (c) the electroelastic field structureinside the material that determines spatial resolution and switching processes. Below,we consider the contrast formation mechanism, cantilever dynamics and resonanceenhancement, and the role of electrostatic interactions in PFM.

Contact Mechanics of Piezoelectric Indentation

General formulation. Traditionally, principles and physical underpinnings of SPMtechniques can be conveniently understood using force-distance curves (Figure 6a).Depending on the dominant interactions, AFM can be sensitive to elastic interac-tions (e.g., AFAM), Van der Waals interactions (noncontact AFM), and magnetic andelectrostatic forces [e.g., electrostatic force microscopy (EFM)]. A similar approachcan be used for voltage modulation techniques such as PFM. However, here the sys-tem 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 noncontactregime, the tip-surface forces are purely capacitive, and the shape of the surface isdescribed by Fnc = C ′

z(z)(Vtip − Vsurf)2, where C ′z(z) is the tip-surface capacitance gra-

dient. Kelvin probe force microscopy (KPFM) is sensitive to the voltage derivative ofthe force, ∂ Fnc

/∂Vtip. The known functional form of this dependence renders KPFM

readily interpretable and relatively insensitive to topographic artifacts. At the sametime, 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 ofEFM more challenging, at the same time rendering technique more sensitive to to-pographic cross talk.

In the contact regime, the imaging mechanism of SPM is ultimately controlledby the shape of the force-distance-bias surface, i.e., Fc = Fc (h, Vtip), where h isthe indentation depth. The image formation mechanism in various SPM techniquescan be related to the derivatives of this surface. For example, in the small signalapproximation 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 theregimes in which contact (C), noncontact (NC), intermittent contact (IC), and interleaveimaging are performed. Also shown are domains of repulsive and attractive tip-surfaceinteractions. (b) Voltage modulation SPMs can be described using a force-distance-biassurface. In the small signal limit, the signal in techniques such as PFM, AFAM, EFM, andKPFM is directly related to the derivative in bias or distance direction.

to (∂h/∂ F )V = const, and the ultrasonic force microscopy (UFM) signal is determinedby (∂2h/∂ F 2)V = const, as illustrated in Figure 6b. The image formation mechanismin recently developed frequency-mixing techniques [heterodyne electrostatic-UFM(HUEFM)] (94) is controlled by mixed derivatives of the force-distance-bias surface,∂2h/∂ F∂V . Therefore, understanding the functional dependence of Fc (h, Vtip) is thekey element for the quantitative interpretation of SPM on piezo- and ferroelectricmaterials.

Exact solution for contact electromechanics. The rigorous solution of the piezo-electric indentation problem, i.e., the Fc (h, Vtip) dependence, is currently availableonly for transversally isotropic material in the continuum mechanics limit (95–97),corresponding to the case of c domains in tetragonal or hexagonal ferroelectrics. Inthis limit, the electric field generated outside the contact area is neglected owing tothe large difference in dielectric constants between the piezoelectric and ambience(Figure 7a). Karapetian and colleagues (92, 93) have derived stiffness relations linkingthe applied force, P , and the concentrated charge, Q, with indenter displacement, w0,indenter potential, ψ0, indenter geometry, and materials properties. The solutions forflat, spherical, and conical indenter geometries have the following phenomenologicalstructure:

P = 2π

θ (hn+1C∗1 + (n + 1)hnψ0C∗

3 ), 5.

Q = 2π

θ (−hn+1C∗3 + (n + 1)hnψ0C∗

4 ), 6.

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

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where h is the total indenter displacement, θ is a geometric factor [θ = a for flatindenters, θ = (2/3)R1/2 for spherical indenters, and θ = (1/π ) tan α for conicalindenters] 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 ofHertzian mechanics and continuum electrostatics to the transversely isotropic piezo-electric medium. From this analysis, the maximum information that can be obtainedfrom SPM or nanoindentation is limited to the indentation elastic stiffness, C∗

1 , theindentation piezocoefficient, C∗

3 , and the indentation dielectric constant, C∗4 . Inden-

tation stiffnesses are complex functions of electroelastic constants of the material,C∗

i = C∗i (c i j , ei j , εi j ), where c i j are elastic stiffnesses, ei j are piezoelectric constants,

and εi j are dielectric constants. Experimentally, AFAM and UFM response is deter-mined by C∗

1 , whereas PFM is sensitive to C∗3/C∗

1 . For most materials, C∗3/C∗

1 ∼ d33

and C∗4 ≈ √

ε11ε33 (92, 93). The electroelastic fields produced by the indentor rapidlyadopt the form of the point charge/point force at distances comparable to the contactradius.

Decoupling approximation in contact electromechanics. The necessity for cal-culating the PFM signal for materials of general symmetry, as well as calculating theresponse at microstructural elements such as domain walls, cylindrical domains, andtopographically inhomogeneous ferroelectrics such as nanoparticles, has stimulatedtheoretical attempts to derive approximate solutions for PFM contrast.

A general approach for the calculation of the electromechanical response is basedon the decoupling approximation. In this case, (a) the electric field in the material iscalculated using a rigid electrostatic model (no piezoelectric coupling, di jk = ei jk = 0),(b) the strain or stress field is calculated using constitutive relations for a piezoelec-tric material, Xi j = Ekeki j , and (c) the displacement field is evaluated using the ap-propriate 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 accountfor the effect of 90◦ domain walls on PFM imaging. Agronin et al. (99) adapteda similar 1-D approach to yield closed-form solutions for the PFM signal. Feltenet al. (100) developed the 3-D version of the decoupled model, using the analyt-ical form for the corresponding Green’s function. Independently, Scrymgeour &Gopalan (101) used the finite element method to model PFM signals across domainwalls. 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 onmaterials of low symmetry, derive analytical expressions for PFM resolution func-tion and domain wall profiles, and interpret PFM spectroscopy data, as describedbelow.

In the decoupled approximation, the PFM signal, i.e., the surface displacementui

(x, y

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

ui (x, y) =∫ ∞

−∞dξ1

∫ ∞

−∞dξ2

∫ ∞

0dξ3

∂Gi j (x1 − ξ1, x2 − ξ2, ξ3)∂ξk

×El (ξ)c kjmndln m(y1 + ξ1, y2 + ξ2, ξ3). 7.

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PZT: lead zirconatetitanate

Here coordinate x = (x1, x2, z) is linked to the indentor apex, and coordinatesy = ( 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 andelastic stiffness tensors, respectively. Ek (x) is the electric field distribution producedby the probe. The Green’s function for a semi-infinite medium G3 j (x − ξ ) links theeigenstrains c j lmndmnk Ek to the displacement field. This approach is rigorous for ma-terials with small piezoelectric coefficients. A simple estimation of the decouplingapproximation applicability is based on the value of the square of the dimensionlesselectromechanical coupling coefficients k2

i j = (di j )2/(s j j εi i ). For instance, for BaTiO3,k2

15 ≈ 0.32, k231 ≈ 0.10, and k2

33 ≈ 0.31; for the ceramics lead zirconate titanate(PZT) 6B, k2

15 ≈ 0.14, k231 ≈ 0.02, and k2

33 ≈ 0.13; and for a quartz single crystal,k2

11 ≈ 0.01. This suggests that the error in the decoupling approximation does notexceed ∼30% for strongly piezoelectric materials and is on the order of ∼1% for weakpiezoelectrics.

Approximate solution for piezoresponse of homogeneous media. The use of the de-coupling approximation reduces an extremely complex coupled contact mechanicsproblem to the solutions of much simpler electrostatic and mechanical Green’s func-tion problems and numerical integration of the result. The dielectric and particularlyelastic properties described by positively defined second- and fourth-rank tensors (in-variant with respect to 180◦ rotation) are necessarily more isotropic than the piezo-electric properties described by third-rank tensors (antisymmetric with respect to180◦ rotation). Hence, elastic and dielectric properties of a material can often beapproximated as isotropic. In this case, integrals in Equation 7 can be evaluated, andsurface displacement can be written in the form ui (x) = VQWi jlk (x) dkjl , where thetensor Wi jlk (x) is symmetrical on the transposition of the indexes j and l . In Voigtnotation, the displacements are (102)

u1 (0) = VQ (W111d11 + W121d12 + W131d13 + W153d35 + W162d26) , 8a.

u2 (0) = VQ (W121d21 + W111d22 + W131d23 + W153d34 + W162d16) , 8b.

u3 (0) = VQ (W313 (d31 + d32) + W333d33 + W351 (d24 + d15)) . 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 pointx = (0, 0, 0) produced by a probe represented by the set of image charges locatedon a vertical line. Thus, the response is shown to be proportional to the potentialinduced by the tip on the surface. The latter fundamental result was generalized asresponse theorems (103).

1. Response theorem 1: For a transversally isotropic piezoelectric solid in anisotropic elastic approximation and an arbitrary point-charge distribution inthe tip (not necessarily constrained to a single line), the vertical surface dis-placement is proportional to the surface potential induced by tip charges in thepoint of contact.

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2. Response theorem 2: For an anisotropic piezoelectric solid in the limit of dielec-tric and elastic isotropy, the vertical and lateral PFM signals are proportional tothe potential on the surface induced by the tip if the tip charges and the pointof contact are located on the same line along the surface normal.

For lower material symmetries, the analytical expressions of the displacementfields induced by the point charge allow response calculations for asymmetric tips orprobes with special geometry (103).

Orientational imaging. In piezoelectric materials, the strong orientation depen-dence of the electromechanical response opens the pathway to orientation imaging.Briefly, the orientation of a molecule or crystallographic lattice with respect to thelaboratory coordinate system is given by three Euler angles (θ, ϕ, ψ). The relation-ship between the di jk tensor in the laboratory coordinate system and the d 0

i j k tensorin the crystal coordinate system is di jk = Ail Ajm Aknd 0

lmn, where Ai j (θ, ϕ, ψ) is the ro-tation matrix (10). Experimentally, PFM measures three components of the responsevector. Hence, local crystallographic orientation can be determined from the solu-tions of Equation 8, which can graphically be represented as response surfaces. As anexample, we compare vertical displacement u3 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 beclose to isotropic, and hence the electric field distribution is insensitive to sample ori-entation. However, a similar analysis can be performed for full dielectric and elasticanisotropy.

Linear Imaging Theory in PFM

The remarkable characteristic of the signal formation mechanism in PFM given byEquation 7 is its linearity. In particular, if the sample is uniform in the z-direction onthe scale of the penetration depth of electric field, i.e., c jlmndmnk(x, z) ≈ c jlmndmnk(x),vertical surface displacement below the tip, i.e., vertical PFM signal, can be rewrittenas

u3(0, y

) =∫ ∞

−∞dmnk

(y − ξ

) (∫ ∞

z=0c j lmn Ek (−ξ1, −ξ2, z)

∂

∂ξlG3 j (ξ1, ξ2, z) d z

)dξ1dξ2,

9.i.e., as a convolution of a function describing the spatial distribution of materialproperties, dmnk (x), and a resolution function related to probe parameters (integralin parenthesis).

Image formation described by Equation 9 belongs to the class of so-called linearimaging mechanisms. Linearity of the PFM image formation mechanism allows oneto

1. establish the unambiguous definitions for spatial resolution (minimal featuresize required for quantitative response measurements) and information lim-its (minimal detectable feature size) and its dependence on tip geometry andmaterials properties, suggesting strategies for high-resolution imaging;

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Figure 8The dependence of (a) displacement u3 and (b) piezoelectric tensor component e33 forPbTiO3 on Euler’s angle θ in the laboratory coordinate system, ν = 0.3. The dependence of(c) piezoelectric tensor component e33 and (d ) displacement u3 for LiTaO3 on Euler’s anglesϕ, 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 wallwidths and the size of nascent domain below the tip; and

4. deconvolute the ideal image from experimental data and establish applicabilitylimits and errors associated with such a deconvolution process.

Kalinin et al. (105) have recently developed the phenomenological resolutiontheory in PFM. Morozovska et al. (104) have developed a corresponding analyticaltheory. Although a full summary of these results is well beyond the scope of thisreview, 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

u3 (s , Q, d ) = Q2πε0 (εe + κ) d

{g313 (s ) d31 + g351 (s ) d15 + g333 (s ) d33} , 10.

where s = (a − a0)/d is the coordinate along the domain wall normalized by charge-surface separation. The functions gi jk (s ) are simple analytical functions of the dielec-tric anisotropy factor. Equation 10 allows the parameters of the set of image charges{Qm, dm}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 domaincan be found as

d eff33 ≈ −

[34

(d33 +

(13

+ 43ν

)d31

)πd − 8rπd + 8r

+ d15

43πd − 8r3πd + 8r

]. 11.

Here r is the domain radius, and d is charge-surface separation. Equation 11 is de-rived for weak dielectric anisotropy (γ = 1). Through Equation 11 the geometricparameters of the domain formed below the tip can be determined, allowing the de-convolution of the PFM spectroscopy data. The combination of Equations 10 and 11allows self-consistent analysis of PFM switching measurements, in which geometricparameters of the probe derived from the domain wall profile are used to quanti-tatively reconstruct the size of the cylindrical domain formed below the tip in thespectroscopic measurements.

Cantilever Dynamics and Effect of Electrostatic Forces

Jesse et al. (106) have performed a complete analysis of the dynamic image formationmechanisms in vector PFM, including (a) the local vertical surface displacementtranslated to the tip, (b) the longitudinal, in-plane surface displacement along thecantilever axis, (c) the lateral surface displacement, in-plane and perpendicular to thecantilever axis, (d ) the local electrostatic force acting on the tip, and (e) the distributedelectrostatic force acting on the cantilever. Jesse et al. have also described an approachfor deconvolution of electrostatic and electromechanical contributions based on the 2-D response–DC bias–frequency spectroscopy. B. Mirman (unpublished data) derivedthe solution in the presence of surface, cantilever, and hydrodynamics damping.

In particular, in the case of transversally isotropic material (for which displacementvector and tip axis are normal to the surface), the response in the low-frequency regimeis a sum of electromechanical and local and nonlocal electrostatic components as

PR = d1ω

Vac= αa (h)d1

k1

k1 + k+

C ′sphere + C ′

cone

k1 + k(Vdc − Vs ) + C ′

cant

24k(Vdc − Vav). 12.

The first term in Equation 12 is the electromechanical response determined bythe effective piezoresponse coefficient of material, d1, and the ratio of AC potentialon the surface and AC bias applied to the tip (i.e., the potential drop in the tip-surfacegap of thickness, h), αa (h). The second and third terms are local and nonlocal contri-butions due to electrostatic tip-surface and cantilever-surface interactions. The signaltransduction between the surface displacements and electrostatic forces and flexural

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Shielded probe: aconducting probe tipsurrounded by insulatingmaterial and groundedcounterelectrode that allowsa highly localized electricfield to be applied

Polarization-dependentphotodeposition: processduring which an ultravioletlight with an energy largerthan the bandgap of thematerial will generateelectron hole pairs thatrespond to localpolarization-induced fieldsand can take part inoxidation or reductionreactions

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 elec-tromechanical and electrostatic contributions on cantilever and tip-surface springconstants. The factor of 24 in the nonlocal cantilever contribution depends solely onthe cantilever spring constant and originates from the cantilever modes (buckling inwhich tip position is constant versus flexural mode with tip displacement) induced byelectrostatic force acting on the cantilever.

Predominantly electromechanical imaging is achieved for good tip-surface con-tact, αa (h) = 1, and in the absence of electrostatic interactions (e.g., stiff cantilever,k1 k and k → ∞, or shielded probe, C ′

cone, C ′cant → 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 = d1. Conversely, in KPFM the tip is not in contact with the surface, k1 = 0 andαa (h) → 0, resulting in a purely electrostatic response.

The analysis of frequency dynamics has shown that the frequency-dependent PFMsignal is a linear combination of contributions that have different frequency depen-dence and can be maximized by an appropriate choice of driving frequency. At thesame time, the resonance frequencies are determined solely by the elastic propertiesof the material. Therefore, tracking the resonant frequency as a function of tip po-sition provides information on local elastic properties, which is similar to frequencydetection in AFAM (107).

LOCAL POLARIZATION SWITCHING IN FERROELECTRICMATERIALS BY PFM

One factor that has resulted in rapidly growing interest in PFM in the ferroelectriccommunity is the capability for local polarization patterning and domain engineer-ing by tip-induced polarization switching. Nanoscale ferroelectric domain patterningwas proposed as the basis for ferroelectric data storage devices (78, 79), with recentlydemonstrated minimal bit size of ∼8 nm, corresponding to storage density of ∼10Tbit inch−2 (108). Polarization dependence of chemical reactivity in acid dissolution(109) or metal photodeposition (83) processes allows extending domain patterningto nanofabrication. These new applications, as well as the need for understandingdomain dynamics in ferroelectric devices, have led to a number of studies of do-main growth processes in the presence of defects and disorder in materials. Grainboundaries play an important role in domain wall pinning (110). Paruch et al. usedlocal studies of domain growth kinetics (82) and domain wall morphology (111) toestablish the origins of disorder in ferroelectric material. Dawber et al. (112) inter-preted the nonuniform wall morphologies as evidence for skyrmion emission duringdomain wall motion. Most recently, Agronin et al. (113) observed domain pinning ona structural defect.

A number of experimental and theoretical studies of domain growth kinetics havebeen reported (114–118). These studies show that in general domain growth followsan approximately logarithmic dependence on the pulse length and a linear dependencein magnitude (114). There have been several attempts to interpret this behavior in

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Domain nucleation: theevent of polarizationreversal when an oppositelypolarized domain is formedin a ferroelectric material

terms of the activation field for the domain wall motion and dynamics of systemswith frozen disorder. However, the use of simplified point-charge-like models thatdo not include the effects of the spherical and conical parts of the tip and possiblecontribution due to the surface charge mobility limits the potential for quantitativedata analysis and necessitates further theoretical and experimental studies of thesephenomena.

In addition to tip-induced switching, PFM can be applied to visualize switchingphenomena in ferroelectric capacitor structures, providing insight into polarizationswitching in a uniform field. A recent study by Gruverman et al. (119) shows thatdomain nucleation in ferroelectric capacitors during repetitive switching cycles isalways 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 basedon thermodynamic analysis. Thermodynamics of domain switching in the Landauerapproximation (25) for domain shape and point-charge approximation for the tipwas 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 applicablefor large domain sizes, although the description of switching on the length scalescomparable to the tip radius of curvature and higher-order switching phenomenarequires exact electroelastic field structure to be taken into account. For realistictip geometries, domain nucleation requires a certain threshold bias on the order of0.1–10 V (25, 125), sufficient to nucleate a domain in the finite electric field of thetip. Using simplified Pade approximations for the free energy, Morozovska et al.(126–128) recently performed a detailed analysis of domain nucleation and growthin 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 onthe late stages of domain evolution can be controlled by the domain wall motionkinetics. Molotskii (129) recently addressed this behavior. However, this area is in thepreliminary stage owing to the lack of information on the domain wall mobilities athigh electric fields and the large domain wall curvatures and possible contribution ofsurface-charge diffusion to the apparent domain wall kinetics. Below, we summarizesome recent results on the switching thermodynamics and the roles of environmentalfactors and materials properties in switching.

Thermodynamics of Domain Switching

The thermodynamics of the switching process can be analyzed from the bias depen-dence of the free energy of the nascent domain. The free energy for a nucleatingdomain 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 fieldand the polarization, the second term is the domain wall energy, and the thirdterm �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 half-ellipsoid, with the small and large axes equal to r and l, correspondingly.

The switching process can be understood from the evolution of free energy sur-faces 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 positivelydefined monotonic function of domain sizes, corresponding to the absence of stabledomain. For biases US < U < Ucr, the local minimum �min > 0 arises, correspond-ing to a metastable domain of sizes rms and lms. The corresponding domain energy is�(rms, lms) = Ems. Finally, for U ≥ Ucr, the absolute minimum �min < 0 is achievedfor req and leq, corresponding to a thermodynamically stable domain. The value Ucr

a b c

d e f

Do

main

len

gth

(n

m)

Do

main

len

gth

(n

m)

Domain radius (nm)

Domain radius (nm)

00 1 2 3 4 5

5

10

2

510

25

50

100

15

00 1 2 3 4 5

5

10

4

16

8.05

32

4

<0

15

00 2 4 6 8 10 12

20

10

30

50

40

60

4

16

4

<0

70

00 5 10 15 20 25

1000

500

2000

1500

2500

4

8.05

16

4

3000

00 1 2 3 4 5

5

10

2

5

11.167

20

40

15

00 1 2 3 4 5

5

10

37.03

8

9.41

25

25

8

7.03

15

Figure 9Contour 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 contoursare free energy values in eV. Triangles denote the saddle point (nuclei sizes). Materialparameters correspond to the PZT6B solid solution and tip-surface characteristics: εe = 81,R0 = 50 nm, and σS = −PS. The evolution of the free energy map with surface screeningcharge density: (e) σS = 0 (U = Ucr = 9.07V), ( f ) σS = +0.95PS (U = Ucr = 4087V).

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determines the point at which the homogeneous polarization distribution becomesabsolutely unstable. Such “threshold” domain nucleation is similar to the well-knownfirst-order phase transition. The minimum point (either metastable {rms, lms} or stable{req, leq}) and the coordinate origin are separated by the saddle point {rS, lS} (i.e., crit-ical nucleus sizes). The corresponding energy �(rS, lS) = Ea is an activation barrierfor domain nucleation.

Molotskii Model for Switching

Using the Landauer model for domain geometry (25) and a capacitance approxima-tion for the electric field of AFM tip, Molotskii (121) obtained elegant closed-formanalytical expressions for the domain size dependence on the applied voltage for whenthe surface charges were completely compensated by the external ones. The interac-tion with an external electric field was calculated as if these screening charges wereabsent. In subsequent work, Molotskii modeled the equilibrium size and kinetics ofthe cylindrical domain, intergrown through the film (118, 131). He showed that thedomain was stable only when the applied voltage exceeded some critical value. Fora prolate domain the depolarization field energy was assumed to be proportional tothe 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 mech-anism in piezoresponse force spectroscopy (PFS) by deriving the main parameters ofdomain nucleation in semi-infinite material and establishing the relationships be-tween 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 fi-nite Debye screening length on domain nucleation, growth, and reversal isestablished. The analytical expressions for domain size and activation energyfor domain nucleation are derived.

2. The analysis is performed using a realistic sphere-plane or disc-plane modelfor the tip. Alternatively, an original effective point-charge model is proposed.The charge parameters are selected to reproduce tip-induced surface potentialand 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-limitingeffect of domain sizes at late stages of the growth process. These results have ob-vious implications for ferroelectric data storage because the self-limiting effectincreases the information density.

4. The current limitation of the MEK model is an assumption of no space sepa-ration between bound and screening charges. Thus, the screening conditions(σS > −PS) lead to the decrease of dragging electrostatic force caused by the

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charged probe. In the case of full screening (σS → PS), the dragging force isabsent.

In particular, the MEK model allowed the role of screening charges on switchingprocess to be elucidated. The static and dynamic properties of screening chargeshave been recently studied using variable temperature KPFM (133–135). A recentstudy illustrates the role of these charges on back-switching (136). The evolutionof free energy surfaces in Figures 9e and 9f illustrates the role of screening on thethermodynamics of domain formation. In the framework of the MEK model, theactivation barrier for nucleation at the onset of domain stability (see Figure 9a) isminimal for complete screening at σS = −PS (10 eV) and increases up to 105 eV forσS → +PS. The barrier calculated in the inhomogeneous electric field of the tip isthree to seven orders of magnitude lower than the one calculated by Landauer forthe homogeneous electric field.

SPECTROSCOPIC MEASUREMENTS OF POLARIZATIONREVERSAL PROCESSES IN FERROELECTRICS

The primary limitation of kinetic studies of domain growth is the large time requiredto study domain evolution after multiple switching steps. Moreover, the informationis obtained on the domain growth initiated at a single point, thus precluding sys-tematic studies of the microstructural influence on the domain growth process. Analternative approach to study domain dynamics in the PFM experiment is based onlocal spectroscopic measurements, in which the domain switching and electrome-chanical detection are performed simultaneously, yielding a local electromechanicalhysteresis loop.

Birk et al. (137), using an STM-based approach, and Hidaka et al. (71), using anAFM-based approach, first reported the in-field hysteresis loop measurements. Inthis method, the response is measured simultaneously with application of the DCelectric field, resulting in an electrostatic contribution to the signal. To avoid thisproblem, Guo et al. (138) reported a technique to measure remanent loops. In thislatter technique, the response is determined after the DC bias is turned off, minimiz-ing the electrostatic contribution to the signal. However, domain relaxation in theoff state is possible. In a parallel development, Roelofs et al. (80) demonstrated theacquisition of both vertical and lateral hysteresis loops. Several groups later used thisapproach to probe crystallographic orientation and microstructure effects on switch-ing behavior (72, 139–143). Buhlmann (144) recently summarized a large volume ofphenomenological 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 theo-retical models to relate PFM hysteresis loop parameters and materials properties. Anumber of such models are based on the interpretation of phenomenological char-acteristics of PFM hysteresis loops similar to macroscopic polarization-electric field

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P-E: polarization-electricfield

(P-E) loops, such as slope, imprint bias, vertical shift, as illustrated in Figure 10. Sev-eral groups have analyzed the effect of nonuniform materials properties, includingthe presence of regions with nonswitchable polarization, on hysteresis loop param-eters 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. Alexeet al. (146) analyzed the hysteresis loop shape in ferroelectric nanocapacitors cappedby a top electrode, obtaining the estimate for switchable volume of a nanocapacitor.Ma & Hesse (147) applied a similar analysis to ferroelectric nanoparticles developedby 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 macroscopicP-E measurements, often demonstrating good agreement between the two (148).This suggests that despite the fundamentally different mechanism in local and macro-scopic switching, there may be deep similarities between tip-induced and macroscopicswitching processes. On the basis of Landau theory, Ricinschi et al. (149–151) de-veloped a framework for the analysis of PFM and macroscopic loops, demonstratingan approach to extract local switching characteristics from hysteresis loop shape viafirst-order reversal curve diagrams.

In parallel with tip-induced switching studies, several groups combined local de-tection by PFM with excitation through the top electrode to study polarization switch-ing in ferroelectric capacitor structures. In this case, the switching field is nearly uni-form. Gruveman et al. discovered spatial variability in switching behavior, which wasattributed to strain (152) and flexoelectric (153) effects. Subsequent work showed thatdomain nucleation during repetitive switching cycles was initiated at the same defectregions (119, 154).

Additionally, in a few cases, abnormal hysteresis loops having shapes much differ-ent than that in Figure 10a have been reported. Abplanalp et al. (155) credited theinversion of electromechanical response to the onset of ferroelectroelastic switch-ing. Harnagea and colleagues (148, 156) attributed the abnormal contrast to thein-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 unusualhysteresis loop shapes, including possible Barkhausen jumps and sharp discontinuitiesin the hysteresis loop shape typically associated with topographic defects.

The rapidly growing number of experimental observations and rapid develop-ments in PFM spectroscopy instrumentation and data acquisition require under-standing of not only phenomenological but also quantitative parameters of hysteresisloops, such as numerical value of coercive bias and the nucleation threshold. Kalininet al. (140) have extended the 1-D model of Ganpule et al. (98) to describe the PFMloop shape in the thermodynamic limit. Kholkin and colleagues (159) have postu-lated an existence of nucleation bias from PFM loop observations, in agreement withtheoretical studies by Abplanalp (117), Kalinin et al. (124), Emelyanov (125), andMorozovska & Eliseev (126, 127). Finally, Jesse et al. (157) have analyzed hysteresisloop shape in kinetic and thermodynamic limits for domain formation in the 1-Dapproximation.

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Figure 10(a) Vertical and lateral displacements of a hysteresis loop are related to the presence ofnonswitchable polarization and the built-in electric field. (b) Forward and reverse coercivevoltages, V+

0 and V−0 , nucleation voltages, V+

c 0 and V−c 0, and forward and reverse saturation and

remanent responses, R+0 , R−

0 , R+s , and R−

s , are shown. The work of switching As is defined asthe area within the loop. The imprint bias and maximum switchable polarization are defined asIm = V+

0 − V−0 and Rm = R+

s − R−s , respectively. b is reprinted with permission from

Reference 157. Copyright 2006, American Institute of Physics. (c,d ) Schematics of loopformation mechanism (c) for the macroscopic and microscopic case and (d ) in the presence of abuilt-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 piezoelectricresponse is given by PR = deff{V(0) − 2V(l)}, where V(0) is the potential at thesurface, V(l) is the potential at the domain boundary below the tip, and deff is theeffective electromechanical response of the material (92). In the strong indentationregime, potential decays inversely with distance as V (l) = βV (0) a/ l , in which β is aproportionality coefficient on the order of unity, l is the distance from the center ofthe contact area, and a is the contact radius. If one uses the Molotskii theory for biasdependence of domain geometry (121), the PFM hysteresis loop shape is

PR = deff

(1 − 5βσwallπ

2

8Ps Vdc

), 14.

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 solelyby the domain wall energy and spontaneous polarization. In the complete absence ofpinning, the thermodynamic model suggests that the hysteresis will proceed along thesolid line in Figure 11a; i.e., the process is reversible, and the domain should growand shrink instantaneously in response to tip bias (Figure 11c). However, if weakpinning is included in the model, the actual domain wall boundary will lag behindthat predicted by Equation 14, and nucleation of a new domain will dominate thehysteresis loop, as illustrated in Figure 11c,d. The saturation and remanent responsesin this approximation are equal, and the onset of switching corresponds to the domainnucleation below the tip (V+

c = V−c = 0 in the point-charge limit when the field below

tip is infinite). From Equation 14, the magnitude of the coercive bias can be derivedas V+ = V− = 5βσwallπ

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 straightfor-ward manner. Experimentally, domain switching in ferroelectric materials followsthe phenomenological dependence r (V, t) = ks V ln t, and domain wall velocity isr (V, t) = ks V/t, where ks is the kinetic constant related to the pinning strength inthe 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 ofthe 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 functionalform 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 thematerial.

3-D Model for Hysteresis Loop Formation

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

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Figure 11(a) Electromechanical hysteresis loop in the no pinning (solid line) and weak pinning (dottedline) limits. (b) Hysteresis loop (solid line) in the thermodynamic and kinetic (dotted line) limits.Forward and reverse coercive voltages, V+

0 and V−0 , nucleation voltages, V+

c 0 and V−c 0, and

forward and reverse saturation and remanent responses, R+0 , R−

0 , R+s , and R−

s , are shown. Thework of switching As is defined as the area within the loop. The imprint bias and maximumswitchable polarization are defined as Im = V+

0 − V−0 and Rm = R+

s − R−s , respectively.

Schematic representation of domain growth for the thermodynamic model (c) without pinningand (d ) with pinning. The numbers correspond to points on the hysteresis curve in a. Arrowsindicate 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 func-tion theory (124, 126–128). For an interpretation of experimental data on realisticmaterials, the third step is (c) calibration of the probe geometry, using the appropriatestandard (e.g., domain wall width).

This analysis has shown that the electromechanical response in the center ofcylindrical domain has the form

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ui3 (r, l) = Q

2πε0 (εe + κ)1d

(d31g1 (r, l) + d15g2 (r, l) + d33g3 (r, l)) , 15.

where the functions gi (r, l) = fi −2wi (r, l) are dependent on the domain sizes, Debyescreening length, Poisson ratio, and dielectric anisotropy. Functions gi = − fi in theinitial (r = 0, l = 0) and wi = fi in the final (r → ∞, l → ∞) states of the switchingprocess. Numerically, the effective piezoresponse d eff

33 = u3/U voltage dependenceobtained within Equation 15 is described by the law d eff

33 = d∞(1 − √U0/U ), U > Ucr.

The difference with the one d eff33 = 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 inLow-Dimensional Ferroelectrics

Recently, PFM spectroscopy has been extended to an imaging mode through an al-gorithm for fast (30–100-ms) hysteresis loop measurements developed by Jesse et al.(158). In switching spectroscopy PFM (SS-PFM) (Figure 12), hysteresis loops areacquired in each point of the image and analyzed to yield 2-D maps of imprint, coer-cive bias, and work of switching, providing a comprehensive description of switchingbehavior of material in each point.

As an example, Figure 13 shows surface topography and effective work-of-switching map (area of hysteresis loop) for polycrystalline PZT ceramics. Analysisof the data showed that the shape of hysteresis loops at certain locations deviatedsignificantly from the ideal loop shape in Figure 11a. These hysteresis loops havinga pronounced “shoulder” are shown in Figure 13c. Loops acquired in adjacent loca-tions show clear similarities in the loop shape (compare Figure 13c and Figure 13d).This excludes the possibility of spurious signals that cannot be ruled out in conven-tional PFM spectroscopy. We attribute the “abnormal” loop shape to the interactionof forming domain with the topographic inhomogeneity, resulting in a bistability inthe domain size.

ADVANCED TOPICS IN PFM

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

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 canchange under the application of an external bias. This charge can lead to significant

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–2.0–40 40200–20

–1.5

–1.0

–0.5

0.0

0.5

1.0

1.5

Resp

on

se (

a.u

.)Bias (Vdc)a

c

b

Vtip = Vdc + Vac sinωt

Highτ1 τ2

Low

Time

Am

plitu

de

τ3δτ

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

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

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Figure 13Topography and SS-PFM images of polycrystalline PZT. (a) Topography (vertical scale is50 nm). (b) A map of the work of switching (area within the hysteresis loop) of approximatelythe 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 ofPhysics.

UHV: ultrahigh vacuum

metal deposition on positive domains, recently metal nanowires have been formedalong the lithium niobate domain walls via photodeposition (167).

Although surface reactivity is likely to strongly affect the electromechanical activityof ferroelectric surfaces in ambient conditions, of great interest are intrinsic surfaceeffects, 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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PFM of Biomaterials

As discussed in the introduction, many biological materials are piezoelectric owingto the combination of polar bonding and optical activity inherent in all biopolymers.To understand biopiezoelectricity on the nanoscale and to relate it to biological func-tionality and other materials properties, a nanoscale probe of local piezoelectricityis needed. PFM has been used for high-resolution imaging of biological structures,including human teeth, canine femoral cartilage, deer antlers, and butterfly wingscales (88, 169). PFM allows organic and mineral components of biological systemsto be differentiated and provides information on materials microstructure and localproperties (89). The use of vector PFM may also enable protein orientation to bedetermined in real space. Figure 14 shows the internal structure and orientation ofprotein microfibrils, with a spatial resolution of several nanometers in human toothenamel (170). A similar approach has been used to study electromechnical activityin dentin, cartilage, and butterfly Vanessa virginiensis wings, effectively reproducingGalvani’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 elec-tromechanical imaging in a liquid environment. In addition, liquid imaging elim-inates capillary forces and may be beneficial to imaging of ferroelectric materialsas well. Recently, PFM of a ferroelectric sample in a liquid environment demon-strated that both long-range electrostatic forces and capillary interactions were min-imized in liquid, resulting in a localization of the AC field to the tip-surface junc-tion and allowing 3-nm resolution to be achieved in some cases (172). Imaging atfrequencies corresponding to high-order cantilever resonances minimized the vis-cous damping, added mass effects on cantilever dynamics, and allowed sensitivitiescomparable to those of ambient conditions. A path to further improvements in res-olution to the nanometer level and below by controlling hydrophobic interactions ispredicted.

The ability to image in solution with improved sensitivity and resolution makesPFM a technique of significant value to the wide bioresearch community by allow-ing a full range of electrical and electromechanical scanning probe techniques to beemployed for the investigation of, e.g., piezoelectricity in biopolymers, molecular ori-entation, and flexoelectricity in cellular membranes. Liquid PFM may provide novelopportunities for high-resolution studies of ferroelectric materials, imaging of softelectroactive polymer materials, and imaging of biological systems in physiologicalenvironments 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 ob-scure technique for domain imaging to a broadly accepted tool for nanoscale char-acterization, domain patterning, and spectroscopy of ferroelectric materials. PFM is

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Figure 14Electromechanical imaging of tooth enamel. (a) Electromechanical response of the surface tothe tip bias is a vector measure, components of which are related to the local orientation ofprotein molecules. In PFM, detection of vertical (VPFM) and torsional (LPFM) componentsof cantilever response allows two vector components, in plane and out of plane, to besimultaneously measured. (b) Surface topography (vertical scale 20 nm) and (c) elasticity mapof 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 10Vpp applied to the tip. The vertical scale for d is −7.5 pm V−1 to 7.5 pm V−1 (the vertical scalefor e is not calibrated). ( f ) Vector PFM map of local electromechanical response (maximum is7.5 pm V−1). Color indicates the orientation of the electromechanical response vector, whereasthe intensity provides the magnitude (color wheel diagram). (g,h) Semiquantitative maps oflocal 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 ofPFM for sub-10-nm imaging of electromechanically active proteins in calcified andconnective tissues (88, 89, 169, 170) suggests much broader applications in the bio-logical community, in which the techniques for studying the ultrastructure of tissueson the submicron level are scarce. Moreover, the pronounced difference in signalformation mechanism between PFM and force- and current-based SPMs suggeststhat there is a tremendous potential for further development. In this section, wesummarize some of the unresolved challenges and opportunities in PFM.

Probe Development

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

Instrumentation

The difference in PFM image formation mechanisms compared with those of AFMand STM imposes certain requirements on the SPM platform. In particular, PFMrequires the detection of small (∼picometer-scale) probe displacements in response tolarge (∼10 Vpp) periodic biases. This requires special attention to the shielding of themicroscope cabling to prevent cross talk between the driving and the detected signals.In particular, until recently most commercial microscopes required home-built tipholders to avoid the cross talk; to a large extent the limited research on PFM in aUHV environment is related to the difficulties associated with similar modificationsfor UHV operation.

The second limitation is related to the difficulty of using standard phase-lockedloop–based frequency-tracking methods for PFM and the dependence of resonancesignal both on local damping and electromechanical activity. This limitation can beovercome through the use of a recently developed band excitation method based onnonsinusoidal excitation signals (175).

Resolution

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

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3-D Probes

As discussed above, the electromechanical response of the surface is generally avector having three independent components. The beam-deflection system usedin most commercial SPMs allows only normal and lateral components of the re-sponse to be detected, necessitating imaging with sample rotation; also, longitudinaldisplacement of the surface can contribute to the vertical signal. In addition,the sensitivities of vertical and lateral PFM are generally different, necessitat-ing complex and time-consuming calibration. This limitation is a fundamentalcharacteristic of a cantilever-based force sensor, and the use of alternative con-figurations for a force sensor, e.g., recently introduced 3-D SPM (176, 177) or

Figure 15Roadmap for development of SPM probes and imaging modes. Shown is the probe evolutionfrom a simple etched STM tip (current probe) to an AFM cantilever (force and current) to3-D probes with a force-sensing integrated readout and active tip (FIRAT) and shieldedprobes. In parallel, data acquisition methods have evolved from static detection (STM andcontact AFM) to constant frequency (intermittent contact AFM, AFAM, etc.) and frequencytracking (noncontact AFM) to more complex waveforms. Images reproduced with permissionfrom References 180 and 181.

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

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

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Theory

Although significant theoretical effort has been concentrated on understanding trans-port and elastic properties of materials on the nanoscale, much less is known aboutthe nanoscale electromechanics. Extensive effort will be required to answer questionssuch as the signal dependence on the contact area in the small contact limit, predictthe role of surface states on the signal, and model PFM in soft condensed matter andbiosystems.

PFM: BEYOND THE PIEZO- AND FERROELECTRICS

In this review, we discuss in detail electromechanical imaging of piezoelectric andferroelectric materials, domain switching, and spectroscopy on the basis of bulk mod-els for material behavior. However, with an increase in spatial resolution, SPM canbecome more and more sensitive to the surface properties of material. Further re-search (Figure 16) will yield an understanding of electromechanical coupling onthe nanometer level, establish the role of surface defects in polarization switching(Landauer paradox), and probe nanoscale polarization dynamics in phase-orderedmaterials and unusual polarization states.

In biosystems, PFM opens pathways for studies of electrophysiology on the cellularand molecular levels, such as, for example, examinations of signal propagation inneurons. Ultimately, on the molecular level, PFM can allow reactions and energytransformation pathways to be understood and become the enabling component ofmolecular electromechanical machines.

SUMMARY POINTS

1. Electromechanical coupling is ubiquitous in biological systems, inorganicmaterials, and molecular systems. The mechanical response to an appliedelectrical stimulus and the electrical response to an applied mechanical stim-ulus are the bases for numerous technological applications and accompanymany biological processes, and yet little is known about electromechanicson the nanoscale.

2. Since its invention in the early 1990s, piezoresponse force microscopy (PFM)has become the primary tool to measure local electromechanics on thenanoscale, predominantly in ferroelectric materials. The image formationmechanism in PFM is complementary to that in STM and force-based AFMin terms of signal scaling with contact area and the tensorial nature of thedetected 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 cantileverthat is brought into contact with a sample surface and used to apply a biasand to measure the resulting surface deformation. Both the magnitude andsign of the displacement can be mapped out to generate images of localpiezoresponse 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 verticaland lateral PFM images, and PFM spectroscopy to measure local electrome-chanics as a function of applied bias.

5. PFM spectroscopy has been used to generate local hysteresis loops in ferro-electric materials. From these hysteresis loops, information such as imprintand the onset of nucleation can be obtained. This information is impor-tant for any technological application based on ferroelectric switching (e.g.,FeRAM) and is required for inquiries into the fundamental mechanisms ofpolarization reversal. Recently, PFM spectroscopy was extended to an imag-ing mode. This allowed loops to be measured at every point in an image andparameters describing polarization dynamics (e.g., imprint or nucleationbias) to be extracted and represented in 2-D plots.

6. Interpretation of local hysteresis loops, although originally compared withthe formation process of macroscopic P-E loops, required the developmentof a new 3-D model based on the decoupled Green’s function theory. Thismodel allows the size of the nucleating domain during the acquisition of thehysteresis loop to be determined.

7. The universal presence of piezoelectricity in biopolymers enables applica-tions of PFM for sub-10-nm imaging of the ultrastructure of biologicalstructures such as calcified and connective tissues. Further applications ofPFM and liquid PFM for biological systems can uncover fundamental mech-anisms of flexoelectricity in cellular membranes, electromotive proteins, ionchannels, etc.

8. The achievement of molecular and ultimately atomic resolution in PFM,although theoretically possible, will require advances in AFM instrumenta-tion, data acquisition and operation modes, probe design, and theory.

ACKNOWLEDGMENTS

Research sponsored by the Division of Materials Sciences and Engineering, Officeof Basic Energy Sciences, U.S. Department of Energy, under Contract DE-AC05-00OR22725 with Oak Ridge National Laboratory, managed and operated by UT-Battelle, LLC.

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Annual Review ofMaterials Research

Volume 37, 2007Contents

MATERIALS CHARACTERIZATION

Low-Temperature Degradation of Zirconia and Implications forBiomedical ImplantsJerome Chevalier, Laurent Gremillard, and Sylvain Deville � � � � � � � � � � � � � � � � � � � � � � � � � � � � �1

Single-Molecule Micromanipulation TechniquesK. C. Neuman, T. Lionnet, and J.-F. Allemand � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 33

Spin-Polarized Scanning Tunneling Microscopy of MagneticStructures and Antiferromagnetic Thin FilmsWulf Wulfhekel and Jürgen Kirschner � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 69

Microscale Characterization of Mechanical PropertiesK. J. Hemker and W. N. Sharpe, Jr. � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 93

Three-Dimensional Atom-Probe Tomography: Advances andApplicationsDavid N. Seidman � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �127

The Study of Nanovolumes of Amorphous Materials Using ElectronScatteringDavid J. H. Cockayne � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �159

Nanoscale Electromechanics of Ferroelectric and Biological Systems:A New Dimension in Scanning Probe MicroscopySergei V. Kalinin, Brian J. Rodriguez, Stephen Jesse, Edgar Karapetian,

Boris Mirman, Eugene A. Eliseev, and Anna N. Morozovska � � � � � � � � � � � � � � � � � � � � � � �189

AFM and Acoustics: Fast, Quantitative Nanomechanical MappingBryan D. Huey � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �351

Electron Holography: Applications to Materials QuestionsHannes Lichte, Petr Formanek, Andreas Lenk, Martin Linck,

Christopher Matzeck, Michael Lehmann, and Paul Simon � � � � � � � � � � � � � � � � � � � � � � � � � �539

Three-Dimensional Characterization of Microstructure by ElectronBack-Scatter DiffractionAnthony D. Rollett, S.-B. Lee, R. Campman, and G.S. Rohrer � � � � � � � � � � � � � � � � � � � � � � �627

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Atom Probe Tomography of Electronic MaterialsThomas F. Kelly, David J. Larson, Keith Thompson, Roger L. Alvis,

Joseph H. Bunton, Jesse D. Olson, and Brian P. Gorman � � � � � � � � � � � � � � � � � � � � � � � � � � � �681

Electron Holography: Phase Imaging with Nanometer ResolutionMartha R. McCartney and David J. Smith � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �729

FERROELECTRICS AND RELATED MATERIALS, David R. Clarkeand Venkatraman Gopalan, Guest Editors

Atomic-Level Simulation of Ferroelectricity in Oxides: Current Statusand OpportunitiesSimon R. Phillpot, Susan B. Sinnott, and Aravind Asthagiri � � � � � � � � � � � � � � � � � � � � � � � � �239

Ferroelectric Domain BreakdownMichel Molotskii, Yossi Rosenwaks, and Gil Rosenman � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �271

Local Structure of Ferroelectric MaterialsT. Egami � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �297

Terahertz PolaritonicsT. Feurer, Nikolay S. Stoyanov, David W. Ward, Joshua C. Vaughan,

Eric R. Statz, and Keith A. Nelson � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �317

Spiral Magnets as MagnetoelectricsT. Kimura � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �387

Universal Domain Wall Dynamics in Disordered Ferroic MaterialsW. Kleemann � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �415

Defect–Domain Wall Interactions in Trigonal FerroelectricsVenkatraman Gopalan, Volkmar Dierolf, and David A. Scrymgeour � � � � � � � � � � � � � � � � �449

Influence of Electric Field and Mechanical Stresses on the Fracture ofFerroelectricsGerold A. Schneider � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �491

Strain Tuning of Ferroelectric Thin FilmsDarrell G. Schlom, Long-Qing Chen, Chang-Beom Eom, Karin M. Rabe,

Stephen K. Streiffer, and Jean-Marc Triscone � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �589

Ferroelectric Epitaxial Thin Films for Integrated OpticsBruce W. Wessels � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �659

Index

Cumulative Index of Contributing Authors, Volumes 33–37 � � � � � � � � � � � � � � � � � � � � � � � �769

Errata

An online log of corrections to Annual Review of Materials Research chapters (if any,1997 to the present) may be found at http://matsci.annualreviews.org/errata.shtml

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