Electromechanical detection in scanning probe microscopy: Tip modelsand materials contrastEugene A. Eliseev, Sergei V. Kalinin, Stephen Jesse, Svetlana L. Bravina, and Anna N. Morozovska Citation: J. Appl. Phys. 102, 014109 (2007); doi: 10.1063/1.2749463 View online: http://dx.doi.org/10.1063/1.2749463 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v102/i1 Published by the American Institute of Physics. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Electromechanical detection in scanning probe microscopy:Tip models and materials contrast

Eugene A. Eliseeva�,b�

Institute for Problems of Materials Science, National Academy of Science of Ukraine, 3, Krjijanovskogo,03142 Kiev, Ukraine

Sergei V. Kalinina�,c� and Stephen JesseMaterials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831

Svetlana L. BravinaInstitute of Physics, National Academy of Science of Ukraine, 4, pr. Nauki, 03028 Kiev, Ukraine

Anna N. MorozovskaV. Lashkaryov Institute of Semiconductor Physics, National Academy of Science of Ukraine, 41,Prospect Nauki, 03028 Kiev, Ukraine

�Received 20 February 2007; accepted 14 May 2007; published online 11 July 2007�

The rapid development of nanoscience and nanotechnology in the last two decades was stimulatedby the emergence of scanning probe microscopy techniques capable of accessing local materialproperties, including transport, mechanical, and electromechanical behaviors, on the nanoscale.Here, we analyze the general principles of electromechanical probing by piezoresponse forcemicroscopy �PFM�, a scanning probe technique applicable to a broad range of piezoelectric andferroelectric materials. The relationship between vertical and lateral PFM signals and materialproperties is derived analytically for two cases: transversally isotropic piezoelectric materials in thelimit of weak elastic anisotropy, and anisotropic piezoelectric materials in the limit of weak elasticand dielectric anisotropies. The integral representations for PFM response for fully anisotropicmaterial are also obtained. The image formation mechanism for conventional �e.g., sphere and cone�and multipole tips corresponding to emerging shielded and strip-line-type probes is analyzed.Possible applications for orientation imaging on the nanoscale and molecular resolution imaging arediscussed. © 2007 American Institute of Physics. �DOI: 10.1063/1.2749463�

I. INTRODUCTION

Rapid progress in nanoscience and nanotechnology inthe last two decades has been stimulated and also necessi-tates further development of tools capable of addressing ma-terial properties on the nanoscale.1 Following the develop-ment of scanning tunneling microscopy2 �STM� and atomicforce microscopy3 �AFM� techniques that allowed visualiz-ing and manipulating matter on the atomic level, a number offorce and current based scanning probe microscopy �SPM�techniques were developed to address properties such as con-ductance, elasticity, adhesion, etc., on the nanoscale.4,5 Cur-rently, the central paradigms of existing SPM techniques arebased on detection of current induced by the bias applied tothe probe, and cantilever displacement induced by a forceacting on the tip. The third possibility, electromechanical de-tection of surface displacements due to piezoelectric andelectrostrictive effects induced by bias applied to the probetip, is realized in piezoresponse force microscopy �PFM�.These three detection mechanisms can be implemented onboth SPM and nanoindentor-based platforms. Finally, detec-tion of current induced by force applied to the probe is lim-ited by the smallness of the corresponding capacitance and

has not been realized in SPM. However, such measurementshave been realized on nanoindentor-based platforms.6,7

PFM was originally developed for imaging, spectros-copy, and modification of ferroelectric materials with strongelectromechanical coupling coefficients �20–2000 pm/V�.8–10 The ability to measure vertical and lateral compo-nents of the electromechanical response vector, perform localpolarization switching, and measure local hysteresis loops�PFM spectroscopy� has attracted broad attention to thistechnique and resulted in a rapidly increasing number ofpublications.11,12 Currently, PFM is one of the most powerfultools for nanoscale characterization of ferroelectric materials.However, until recently, it was believed that PFM was lim-ited to ferroelectric materials, representing a relatively minorclass of inorganic materials and, with a few exceptions �e.g.,polyvinilidendifluoride and its copolymers�, nonexistent inmacromolecular materials and biopolymers.

However, piezoelectric coupling between electrical andmechanical phenomena is extremely common in inorganicmaterials �20 out of 32 symmetry classes are piezoelectric�and ubiquitous in biological polymers due to the presence ofpolar bonds and optical activity. Hence, PFM is a naturaltechnique for high-resolution imaging of these systems. Oneof the limitations in PFM of such materials is low electro-mechanical coupling coefficients, typically one to two ordersof magnitude below that of ferroelectrics. However, the highvertical resolution, inherent to all SPM techniques, combined

a�Authors to whom correspondence should be addressed.b�Electronic mail: eliseev@i.com.uac�Electronic mail: sergei2@ornl.gov

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with large ��1–10 Vpp� modulation amplitudes allows localmeasurement of electromechanical coupling even in materi-als with small piezoelectric coefficients, e.g., III-V nitrides13

and biopolymers.14,15 In fact, the primary limitations in PFMimaging of weakly piezoelectric materials are not the small-ness of the corresponding response magnitude, but the linearcontribution to the PFM contrast due to capacitive tip-surface forces and the inability to use standard phase-lockedloop based circuitry for resonance enhancement of weakelectromechanical signal.16 However, both of these limita-tions can be circumvented by electrically shielded probes,imaging in liquid environment,17 and improved control andsignal acquisition routines.

Symmetry breaking at surfaces and interfaces shouldgive rise to piezoelectric coupling even in nonpolar materi-als, and a number of electromechanical phenomena, includ-ing surface piezoelectricity and flexoelectricity, have beenpredicted.18

To summarize, electromechanical coupling is extremelycommon on the nanoscale. PFM is a natural tool to addressthese phenomena, in turn necessitating the understanding ofthe relationship between signal formation mechanisms, ma-terial properties, and tip parameters. The relationship be-tween the surface and tip displacement amplitudes,19 cantile-ver dynamics in PFM,20 and mechanisms for electrostaticforce contribution21,22 have been analyzed in detail. How-ever, due to its inherent complexity, voltage-dependent tip-surface contact mechanics, the key element that relates ap-plied modulation and measured response, is available onlyfor a limited class of transversally isotropic �i.e., rotationallyinvariant in the surface plane� piezoelectric materials andsimple tip geometries.

In this paper, we discuss in detail the image formationmechanism of piezoresponse force microscopy for piezoelec-trically anisotropic materials. The paper is organized as fol-lows. Decoupling approximation for PFM is discussed inSec. II A. Electrostatic field structure for arbitrary dielectricanisotropy is analyzed in Sec. II B. The relationship betweenthe PFM signal and material properties is analyzed using alinearized decoupled Green’s function approach in Sec. II C.PFM response calculated for different multipole models oftip is considered in Sec. II D. Orientation imaging by PFM isdiscussed in Sec. III.

II. MATERIALS CONTRAST IN PFM

A. Decoupling approximation

In general, a calculation of the electromechanical re-sponse induced by a biased tip requires the solution of acoupled electromechanical indentation problem, currentlyavailable only for uniform transversally isotropic case.23,24

This solution is further limited to the strong indentation case,in which the fields generated outside the contact area areneglected. While this approximation is valid for large contactareas �classical contact�, for small contacts the electrostaticfield produced by the part of the tip not in contact with thesurface can provide a significant contribution to the electro-mechanical response. This behavior is similar to the transi-

tion from Hertzian contact mechanics valid for macroscopiccontacts to Dugdale-Maugis mechanics for nanoscalecontacts.25,26

An alternative approach for the calculation of the elec-tromechanical response is based on the decoupling approxi-mation. In this case, the electric field in the material is cal-culated using a rigid electrostatic model �no piezoelectriccoupling�, the strain or stress field is calculated using consti-tutive relations for piezoelectric material, and the displace-ment field is evaluated using the appropriate Green’s func-tion for an isotropic or anisotropic solid. This approach isrigorous for the materials with small piezoelectric coeffi-cients. A simple estimation of the decoupling approximationapplicability is based on the value of the square of the di-mensionless electromechanical coupling coefficients kij

2

= �dij�2 / �sjj�i� without summation on repeated indexes,where dij are the strain piezoelectric coefficients, sij are thecomponents of elastic compliance tensor, and �i is the dielec-tric susceptibility tensor components in Voigt representation�see also Appendix A in Ref. 27�. For instance, for BaTiO3:k15

2 �0.32, k312 �0.10, and k33

2 �0.31; for the ceramicsPZT6B: k15

2 �0.14, k312 �0.02, and k33

2 �0.13; and for aquartz single crystal: k11

2 �0.01.A simplified one-dimensional �1D� version of the decou-

pled model was suggested by Ganpule et al.28 to account forthe effect of 90° domain walls on PFM imaging. A similar1D approach was adapted by Agronin et al.29 to yield closed-form solutions for the PFM signal. The three-dimensionalversion of this approach was developed by Felten et al.30

using the analytical form for the corresponding Green’s func-tion. Independently, Scrymgeour and Gopalan31 have usedthe finite element method to model PFM signals across do-main walls.

Note that analytical closed-form relationships betweenmaterials parameters, tip geometry, and surface displace-ments, if obtainable, provide the advantage of �a� under-standing the fundamental physics of the image formationmechanism, �b� can be expanded to more complex cases inelementary fashion, and �c� allow for solution of reciprocalproblem. Conversely, numerical approach is preferentialwhen exact, rather than approximate, answer is required.Since the exact geometric parameters of the system in SPMexperiment are generally unknown, here we derive the ana-lytical solutions for PFM image formation mechanism.

The PFM contrast for different tip geometries is ana-lyzed using the Green’s function approach originally sug-gested by Felten et al.30 and extended by recent analysis byKalinin et al..32 Closed-form expressions for the PFM signal,including relative contributions from individual elements ofthe piezoelectric constant tensor, elastic properties, and theeffects of dielectric anisotropy on the PFM signal, are de-rived. Throughout the derivations, we implicitly assume thatthe piezoelectric properties are uniform on the length scale ofelectric field penetration in the material and unaffected bysurface phenomena.

B. Electric field distributions

The initial step in calculating the electromechanical re-sponse in the decoupled approximation is the determination

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of the electric field distributions. While for isotropic33 andtransversally isotropic34 materials, the solution can be ob-

tained using simple image charge method �Fig. 1�, the field issignificantly more complex in materials with lower symme-try. Here we analyze the case of the full dielectric anisotropy.

The potential distribution V�r� inside the half space �z�0� in the anisotropic half space with dielectric permittivity,�ij, induced by a point charge, Q, located at the distance, d, isfound in the Fourier representation in Ref. 27,

V�x1,x2,x3� =Q

2��0�

−�

�

dk1�−�

�

dk2exp�− ik1x1 − ik2x2�

2�

�exp�− d�k1

2 + k22 − x3��k1,k2��

��k12 + k2

2 + �i�31k1 + i�32k2 + �33���. �1�

The Fourier representation Ek�k1 ,k2 ,x3� of the electric fieldacquires the form

E1,2�k1,k2,x3� = ik1,2Q exp�− kd − x3��k��

2��0k + �i�31k1 + i�32k2 + �33��k��,

�2�

E3�k1,k2,x3� = ��k�Q exp�− kd − x3��k��

2��0k + �i�31k1 + i�32k2 + �33��k��,

where

� = ����11�33 − �312 �k1

2 − 2��31�32 − �12�33�k1k2 + ��22�33 − �322 �k2

2 − i��31k1 + �32k2��/�33

and k= �k1 ,k2�, k��k12+k2

2. The square root is real for anyreal wave vector �k1 , k2� since the dielectric constant tensor,�ij, is positively defined.

For a transversally isotropic dielectric material ��ij

=�ii�ij ,�11=�22��33� Eq. �1� is simplified as VQ�x�=Q / �2��0��+1��2+ �x3 /+d�2�, where �x1

2+x22= is ra-

dial coordinate, �=��33�11 is the effective dielectric con-stant, and =��33/�11 is the dielectric anisotropy factor.

C. Electromechanical response to a pointcharge

The decoupling Green’s function approach developed byFelten et al.30 is based on the consecutive calculation of the�1� electric field for a rigid dielectric �i.e., in supposition thattensors of strain piezoelectric coefficients are zero, dijk=0�,�2� stress field Xij =−Ekdklmcmlij in piezoelectric material, and�3� mechanical displacement field for a nonpiezoelectric,elastic material. This approach, while not rigorous, signifi-cantly simplifies the problem and, in particular, allows theeffective symmetry of the elastic, dielectric, and piezoelec-tric properties of a material to be varied independently.32 Inthe framework of this model, the displacement vector ui�x� atposition x is

ui�x� = �0

�

d�3�−�

�

d�2�−�

�

d�1�Gij�x,��

��kEl���ckjmndlnm,

�3a�

where � is the coordinate system related to the material,Gij�x ,�� is the Green’s function relating displacement to theapplied point force, dklm are the strain piezoelectric coeffi-cients, clmij are elastic stiffness coefficients, and the Einsteinsummation convention is used. Ek��� is the electric field pro-duced by the probe.

Since for the many important cases �dielectrically aniso-tropic or semiconducting media, thin films, etc.� the electricfield can be easily found in the form of Fourier integral, it isconvenient to rewrite �3a� as follows:

ui�x1,x2,0� = �−�

�

dk1�−�

�

dk2 exp�− ik1x1 − ik2x2�

· �0

�

d�3Gij,k�k1,k2,�3�El�k1,k2,�3�ckjmndlnm.

�3b�

Here Gij,l�k1 ,k2 ,�3� is Fourier representation of the Green’s

function gradient27 and Ek�k1 ,k2 ,�3� is Fourier representationof the electric field.

FIG. 1. �Color online� Tip representation using image charge distribution inthe PFM experiment.

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In the early version of the theory,27,32 the PFM response was calculated as a function of the piezoelectric stress coefficients,eijk. However, sensitivity analysis of the exact solution for the transversally isotropic homogeneous system23,24 has shown thatpiezoelectric response actually depends primarily on piezoelectric strain components, dijk, and dielectric anisotropy factor, =��33/�11. In this �ckjmn ,dijk ,�ij� representation piezoresponse is virtually independent on elastic properties. Hence, here wederive theory utilizing the piezoelectric strain elements.

For most ferroelectric perovskites, the symmetry of the elastic properties can be approximated as cubic �anisotropy ofelastic properties is much smaller than those of the dielectric and piezoelectric properties�, and therefore, the approximation ofelastic isotropy is used. The Green’s function for an isotropic, semi-infinite half plane is 35–37

Gij�x1,x2,x3 = 0,��

= ��1 + ��/2�Y��ij/R + �xi − �i��xj − � j�/R3 + �1 − 2��/�R + �3���ij − �xi − �i��xj − � j�/R�R + �3��, i, j � 3

�1 + ���xi − �i�/2�Y�− �3/R3 ± �1 − 2��/R�R + �3��,“+” � j = 1,2 and i = 3

− “�” i = 1,2 and j = 3

�1 + ��/2�Y�2�1 − ��/R + �32/R3�, i = j = 3,

�4�

where R=��x1−�1�2+ �x2−�2�2+�32, Y is Young’s modulus,

and � is the Poisson ratio. Corresponding expressions fortransversally isotropic materials are available elsewhere.38

Finally, for lower material symmetries, closed-form repre-sentations for elastic Green’s functions are generally unavail-able, but approximate solutions can be derived.35

Using these equations the integration on � and k in Eq.�3b� can be done analytically. Equation �3b� is then reducedto the onefold integral on the angle which can be expressedin terms of elliptic integrals. Thus, the calculation for themedia with arbitrary dielectric anisotropy is not much morecomplicated than in the case of a transversely isotropic di-electric media and can be done numerically. Moreover,Green’s functions for the anisotropic half space with generalboundary conditions are available in Fourierrepresentation.39

1. Transversally isotropic dielectric material

For the special case of a transversally isotropic material,the PFM response can be calculated in analytical form as-suming weak elastic anisotropy. After lengthy manipulations,Eq. �3� is integrated to yield the normal displacement of thesurface �x3=0� as32

u3�� =Q

2��0�1 + ��d31f1�� + d15f2�� + d33f3��

�2 + d2, �5�

where the functions f i�� depend only on the dielectric an-isotropy, , and the Poisson ratio, �, of the material. TheVoigt representation, d31�d311,d33�d333,d15�2d113, isused whenever possible. The contributions of different piezo-electric constants are additive and the corresponding func-tions f i�� are

f1�� = −1 + 2�1 + ��

�1 + �2 , �6a�

f2�� = −2

�1 + �2 , �6b�

f3�� = −1 + 2

�1 + �2 . �6c�

The vertical PFM signal is determined by the surfacedisplacement at the position of the tip, u3�0�. The in-planecomponents of surface displacement are �u1 ,u2�

u1 = x1u�,d�, u2 = x2u�,d� ,

�7�

u�,d� =Q

2��0�1 + ��d31g1�� + d15g2�� + d33g3��

�2 + d2��2 + d2 + d�.

The functions gi�� that determine the contributions of dif-ferent piezoelectric constants to overall response are

g1�� = − + 2�1 + ��

�1 + �2 , �8a�

g2�� = −

�1 + �2 , �8b�

g3�� = −2 +

�1 + �2 . �8c�

The polar components of the in-plane displacement vectoru= �x1u1+x2u2� / and u = �−x2u1+x1u2� / are u��=u� ,d� and u �0. The in-plane displacements at contactare u1�0�=u2�0�=0, and hence the lateral PFM signals arezero in agreement with the rotational symmetry of the sys-tem.

In most materials, the Poisson ratio is ��0.3. Hence, thefunctions f i�� and gi�� that determine the contributions ofthe piezoelectric constants din to the overall signal dependprimarily on the dielectric anisotropy of the material, . Formost ferroelectric oxides �0.3−1, while �10.5 and 2.3for Rochelle salt and triglycine sulfate, respectively.

Shown in Figs. 2�a� and 2�b� are plots of the functionsf i�� that determine contributions of the piezoelectric con-stants dnm to the vertical displacement, u3. The function f3��varies relatively weakly with in the interval � �0,2�, in-dicating the weak dependence of the longitudinal contribu-

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tion, d33, to signal. For large dielectric anisotropies →�and f3��→0. Conversely, f2��, which determines the con-tribution of constants d15 to the signal, increases with in-crease. Finally, contribution f1�� of d31 is essential only at�1.

Shown in Figs. 2�c� and 2�d� are the functions gi�� thatdetermine relative contributions of piezoelectric constantsdnm to the overall signal u� ,d�. The function g3�� decayswith , indicating the decreasing contribution of d33 to thesignal. Conversely, contribution of d15, g2��, has maximumat =1. Function g1�� varies weakly for 0��1. For �1, all gi�� decrease.

The point-charge response in Eqs. �5� and �7� can beextended to realistic tip geometries using an appropriate im-age charge model, e.g., an image charge series for a sphericaltips or a line charge model for conical tips.40 In particular,from the similarity between Eq. �1� for the potential distri-bution induced on the surface induced by a point charge andEq. �5� for surface displacement induced by a point charge,we derive the following universal principle.

Response Theorem 1: For a transversally isotropic pi-ezoelectric solid in an isotropic elastic approximation and anarbitrary point charge distribution in the tip �not necessarilyconstrained to a single line�, the vertical surface displace-ment is proportional to the surface potential VQ�� inducedby tip charges in the point of contact given by Eq. �5� and

u3�� = VQ���d31f1�� + d15f2�� + d33f3��� . �9�

This is no longer the case for the in-plane components. Inthis case, the surface displacement fields are more complex,

u1�x1,x2� = WQ1�x1,x2��d31g1�� + d15g2�� + d33g3��� ,

�10�

u2�x1,x2� = WQ2�x1,x2��d31g1�� + d15g2�� + d33g3��� ,

�11�

where

WQ1�x1,x2� =1

2��0�1 + �� i

Qi

�x12 + x2

2 + di2

�x1

��x12 + x2

2 + di2 + di�

, �12a�

WQ2�x1,x2� =1

2��0�1 + �� i

Qi

�x12 + x2

2 + di2

�x2

��x12 + x2

2 + di2 + di�

, �12b�

and Qi and di are the effective charges values and z coordi-nates, respectively.

The dependence of the vertical surface response only onthe potential induced on the surface if the tip charges arelocated on the same line along the surface normal also ap-plies for PFM signals for materials with lower symmetries,as will be analyzed below.

2. General piezoelectric anisotropy

One of the key problems in PFM is determining the re-sponse for a fully anisotropic material, in which case bothnormal and in-plane components of the surface displacementcan be nonzero. Note that this case corresponds both to ma-terials with low symmetry �e.g., triclinic and monoclinic� and

FIG. 2. �Color online� ��a� and �c�� Plots of the func-tions f i�� and gi�� for �=0.3 in the �0,1� interval cor-responding to most ferroelectric oxides. ��b� and �d��Plots in the logarithmic scale in the �10−2, 102� intervalillustrating the asymptotic behavior of the dissimilarcontributions with dielectric anisotropy.

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tetragonal perovskites for the case when the orientation ofthe crystallographic c axis and the surface normal do notcoincide.

In the case of general piezoelectric anisotropy, all ele-ments of the stress piezoelectric tensor, djlk, can be nonzero,necessitating the evaluation of all integrals in Eq. �3a�. Thevector of surface displacement ui�x� induced by the electricfield is

ui�x� = VQ�0�Rijlk�x�dklj , �13�

where VQ�0�=Q / �2��0��+1�d� is potential on the surfacebelow the charge. The components of tensor Rijlk�x� are

Rimnl�x� =1

VQ�0��0

�

d�3�−�

�

d�2�−�

�

d�1

��Gij�x,��

��kEl���ckjmn. �14�

Thus Eq. �14� provides a comprehensive description ofPFM image formation mechanism on a fully anisotropic pi-

ezoelectric material in the decoupled approximation. The el-ements of the response tensor Rijlk�x� can be evaluated nu-merically or analytically under simplifying assumptions onmaterials symmetry, as discussed below.

In order to obtain an exact analytical solution forRijlk�x�, all integrations were carried out for x3=0. For sim-plicity and to elucidate the underpinnings of the contrast for-mation mechanism in PFM of anisotropic materials, the ana-lytical calculations are performed for dielectrically isotropicmaterials, i.e., =1. Exact analytical expressions for nonzerodisplacement field components Rijlk�x1 ,x2 ,x3=0� can befound with the help of relationship Rimnl�x�=Uijkl�x�ckjmn /VQ�0� where tensor Uijkl�x� components arelisted in Appendix C of Ref. 27. Note that this approxima-tion, while not rigorous, is a good approximation for a broadclass of weakly anisotropic materials such as LiTaO3 andpoled ferroelectric ceramics in arbitrary orientations with re-spect to the surface normal. For stronger dielectric anisotro-pies, Eq. �14� can be evaluated numerically using exact ex-

FIG. 3. �Color online� Possible structures for multipoletips. Shown are 3D views and schematics of ��a� and�b�� strip-line tips, ��c� and �d�� quadrupole tip, and ��e�and �f�� shielded tip. A strip-line tip can be used tocreate an in-plane dipole field, V1=−V2=Vac cos �t. Aquadrupole tip can be used to create a quadrupolar elec-tric field, V1=V2=Vac cos �t, V3=V4=−Vac cos �t, orrotating in-plane dipole field, V1=−V2=Vac cos �t, V3

=−V4=Vac sin �t. A shielded tip can be used to localizefield and minimize electrostatic contribution to PFMsignal, V1=Vac cos �t, V2=0.

FIG. 4. �Color online� �a� Schematicof the dipole tip. Surface displacementcomponents u1,3�x1 ,x2� and u1�0,0� in�b� x1 and �c� x2 directions for p /d=0.01. �d� Surface displacements vsdipole-surface separation d �arbitraryunits�.

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pressions for the electrostatic potential. The effects of elasticanisotropy can be evaluated numerically using rigorous ex-pressions for elastic Green’s function.

The electromechanical response in PFM is determinedby the components of Eq. �14� evaluated at the origin,Rijlk�0,0 ,0�. Here, we introduced Voigt matrix notations onindices j and l without any factors, while for the piezoelec-tric tensor factors were introduced as dk�=dkjl for �=1¯3and dk�=2dkjl for �=4¯6, so that the relationshipRijlk�x�dklj �Ri�k�x�dk� is satisfied.41 In reduced notation, thesurface displacement below the tip is

u1�0� = VQ�0��R111d11 + R121d12 + R131d13 + R153d35

+ R162d26� , �15�

u2�0� = VQ�0��R121d21 + R111d22 + R131d23 + R153d34

+ R162d16� , �16�

u3�0� = VQ�0��R313�d31 + d32� + R333d33 + R351�d24

+ d15�� . �17�

Here the nontrivial elements of tensor Ri�k depend only onthe Poisson coefficient as follows: R111=−�13+4�� /32,R121= �1−12�� /32, R131=− 1

8 , R153=− 38 , R162=−�7−4�� /32,

R313=−�1+4�� /8, R333=− 34 , and R351=− 1

8 . Note that for asingle point charge the potential on the surface below scalesas VQ�0��1/d. Furthermore, from Eqs. �15� to �17� we de-rive the following.

Response Theorem 2: For an anisotropic piezoelectricsolid in the limit of dielectric and elastic isotropies, the ver-tical and lateral PFM signals are proportional to the potentialon the surface induced by the tip if the tip charges and thepoint of contact are located on the same line along the sur-face normal. This condition corresponds to, e.g., sphere andline charge models for the tip.

It should be noted that for ferroelectrics of 3m symmetry�such as LiTaO3� with piezoelectric tensor containing non-zero components d22, d21=−d22, and d16=−2d22, these com-ponent contribution is exactly zero since −R121+R111

−2R162�0. This is also true for dielectrically anisotropic fer-roelectrics with �1 �such as LiNbO3�.

Due to the complex functional form of the displacementfunctions Ri�k�x�, this is no longer the case when the imagecharge distribution in the tip and point of contact does notfall on the same line, and analyses of several important cases

are given in Sec. II D. Moreover, the formulas in Appendix Cof Ref. 27 allow the effects of tip asymmetry on response tobe evaluated.

D. PFM response for multipole tips

Considered above were simple tip models correspondingto uniform conductive SPM tips, which can be represented asthe combination of point and linear charges �see, e.g., recentpaper of Sacha et al.42�. The conventional sphere-planemodel is considered in detail in Ref. 27. In this and subse-quent sections, we consider more specific models, corre-sponding to SPM tips with shielding or formed by strip linesunder different biases �see Figs. 3�e� and 3�c�, respectively�.

In these cases, the tip can no longer be represented by asurface of constant potential or image charges of the samesign; rather, more complex potential distributions are re-quired. In addition, these models allow estimation of the con-tribution of higher-order multipole moments to the surfaceresponse even for conventional tips, e.g., to allow consider-ation of the effects of tip asymmetry. Here, we analyze thePFM signal formation mechanism for tips with complexelectrostatics modeled using multipole representations forthe tip field.

1. Dipole tip model

The nontrivial behavior can be expected if the electricfield below the tip has a large in-plane component. Suchfields can be created by standard, pyramidal AFM tips withstrip-line-type electrodes where one side is biased positivelyand the other negatively �Figs. 3�a� and 3�b��. Similar tipswith four independent electrodes can be used to create dipo-lar electric field rotating in the surface plane, where the cor-responding torsional or flexural component of cantilever re-sponse is measured.

The electric field in this case can be modeled using anin-plane dipole with moment P0=2Q0p �Fig. 4� formed bycharges ±Q0 located at distances d from the sample surfaceat distance p from each other. The dipole axis is oriented in xdirection.

For transversally isotropic material, the surface displace-ment fields can be obtained from Eqs. �9�–�11�, where func-tions VQ�x1 ,x2� and WQ1,2�x1 ,x2� are substituted by

VQ�x1,x2� �P0

2��0�� + 1�x1

��x12 + x2

2 + d2�3+ O�p3� ,

�18a�

WQ1�x1,x2� � −P0

2��0�� + 1�d�x2

2 + d2� + �− x12 + x2

2 + d2��x12 + x2

2 + d2

��x12 + x2

2 + d2�3�d + �x12 + x2

2 + d2�2+ O�p3� , �18b�

WQ2�x1,x2� � −P0

2��0�� + 1�x1x2�d + 2�x1

2 + x22 + d2�

��x12 + x2

2 + d2�3�d + �x12 + x2

2 + d2�2+ O�p3� . �18c�

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Assuming that the tip-surface contact corresponds to thecenter of the dipole, the potential at the contact, derived fromEq. �18a�, is zero, VQ�0,0�=0, and hence, the normal com-ponent of the electromechanical response u3�0,0� is zero.The surface displacement u3�x1 ,x2� is maximal in the points�x1= ± p ,x2=0� and has different signs. However, for such atip, the lateral displacement PFM signal related to in-planecomponent of surface displacement, u1�x1 ,x2�, is now non-zero and can be derived as

u1�0,0� � −P0

2��0�� + 1�

��d31g1�� + d15g2�� + d33g3���

4d2 . �19�

The second in-plane component is u2�0,0�=0, as expectedfrom symmetry considerations. Equation �19� allows estima-tion of the role of tip asymmetry on the in-plane contrast ontetragonal materials, once the corresponding equivalentcharge model is constructed and corresponding dipole mo-ment is known. Note that that the distance dependence of theresponse is 1 /d2 �see Fig. 4�d��.

Thus, the use of the dipolar tips potentially allows forprobing of different combinations of piezoelectric constants

than traditional tips, thus providing additional informationabout the material. The use of a four-pole tip �Fig. 3�b��allows for the generations of a rotating dipole that can po-tentially provide additional operational modes in PFM.

2. Quadrupole tip model

A promising approach for increasing resolution andminimizing the electrostatic force contribution in PFM isbased on the use of shielded tips, as shown in Fig. 3�f�. Fromsymmetry considerations, the field produced by such a tipwill have monopole and quadrupole components, as consid-ered here. We model the tip with the charge 2Q0 located inthe point �0,0 ,−d� and charges −Q0 are located in crosspoints �±p ,0 ,−d�. Thus, for p�d we can introduce the qua-druple moment K0=2Q0p2, see Fig. 5�a�.

The electromechanical response of the surface for thetransversally isotropic case can be found in a way similar tothe dipolar tip from Eqs. �9�–�11�, where functions VQ�x1 ,x2�and WQ1,2�x1 ,x2� are now given by

VQ�x1,x2� �K0

4��0�� + 1�d2 − 2x1

2 + x22

��x12 + x2

2 + d2�5, �20a�

WQ1�x1,x2� �K0

4��0�� + 1�x1�9d�x2

2 + d2��x12 + x2

2 + d2 + 9d4 + 4d2x12 + 15d2x2

2 − 2x14 + 4x1

2x22 + 6x2

4�

��x12 + x2

2 + d2�5��x12 + x2

2 + d2 + d�3, �20b�

WQ2�x1,x2� �K0

4��0�� + 1�x2�3d�− 2x1

2 + x22 + d2��x1

2 + x22 + d2 + 3d4 − 6d2x1

2 + 5d2x22 − 6x1

4 − 4x12x2

2 + 2x24�

��x12 + x2

2 + d2�5��x12 + x2

2 + d2 + d�3. �20c�

From Eqs. �20a�–�20c�, the lateral displacements belowthe tip are u2�0,0�=u1�0,0�=0, in agreement with the sym-metry of the problem. However, the vertical PFM response isnow �see Figs. 5�b� and 5�c��

u3�0,0� � �2

d−

2�p2 + d2

� �K0

d3 . �21�

The approximate Eq. �21� represents the potential on thesurface as a function of separation and quadruple moments.

III. ORIENTATION DEPENDENCE

Based on the analysis of PFM, the image formationmechanism in Sec. II, we discuss the implications for imag-ing piezoelectric materials. The orientation dependence ofthe PFM signal and the potential for molecular and crystal-lographic orientational imaging are discussed below.

A unique feature of PFM is that in the ideal case, thesignal is independent of the tip-surface contact area and isdetermined solely by material properties. Furthermore, if the

FIG. 5. �Color online� �a� Schematicof the quadrupole tip and normalizeddisplacements in �b� x1 and �c� x2 di-rections for p /d=0.01.

014109-8 Eliseev et al. J. Appl. Phys. 102, 014109 �2007�

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contact nonideality leads to a potential drop between the tipand the surface, all of the response components are reducedproportionately. Finally, in the 3D vector PFM experiment,all three components of the electromechanical response vec-tor can be determined. It has been suggested that these fac-tors allow vector PFM to be applied to mapping crystallo-graphic and molecular orientations on the nanoscale.19,43

Briefly, the orientation of a solid body in 3D space is givenby three Euler angles �Fig. 6�. In PFM, all three componentsof the displacement vector are measured, which providethree independent equations from which the local Eulerangles can be recovered.

The relationship between the dijk tensor in the laboratorycoordinate system and the dijk

0 tensor in the crystal coordinatesystem is41

dijk = AilAjmAkndlmn0 , �22�

where Aij is rotation matrix. The displacement componentsare related to the piezoelectric strain tensor components, asanalyzed above.

In the particular case of a uniform field that correspondsto systems with a continuous top electrode, the displacementcomponents detected from vertical and lateral PFM signalsare �u1 ,u2 ,u3�= �d35,d34,d33�V, where d33 is the longitudinaland d34 and d35 the shear components of the strain piezoelec-tric constant tensor.19 In the PFM geometry, the electric field

produced by the tip is strongly nonuniform and the responsecomponents are given by Eqs. �15�–�17�. Here, we analyzethe applicability of these equations to the orientational de-pendence of the PFM data and compare it with the uniformfield approximation. As model systems, we consider tetrago-nal PbTiO3 and trigonal LiTaO3. In the coordinate systemoriented along the crystal c axis, the piezoelectric strain di�

0

tensor for these materials is

di�0 = � 0 0 0 0 d15

0 − 2d220

− d220 d22

0 0 d150 0 0

d310 d31

0 d330 0 0 0

� . �23a�

For tetragonal crystals and poled ceramics, d220 �0. For cases

of an arbitrary crystal orientation, the response componentsare

d33 = �d150 + d31

0 �sin2 � cos � + d330 cos3 �

+ d220 cos �1 − cos 2 �sin3 � , �23b�

d34 = sin �− �d310 − d33

0 + �d150 + d31

0 − d330 �cos 2��cos �

+ 2d220 �− sin � sin 3 + cos � cos � cos 3 �sin �� ,

�23c�

d35 = sin �− �d310 − d33

0 + �d150 + d31

0 − d330 �cos 2��sin �

+ 2d220 �cos � sin 3 + sin � cos � cos 3 �sin �� .

�23d�

Note that for cases of tetragonal symmetry �i.e., PbTiO3� theresponse is independent of , indicative of the rotationalsymmetry along the 3-axis.

The piezoelectric tensor component d33 and vertical dis-placement u3 below the tip versus the orientation of the crys-tallographic axes with respect to the laboratory coordinatesystem for LiTaO3 crystal is shown in Figs. 7�a� and 7�b�,respectively. From the data, the angular dependence of ver-tical displacement u3 is smoother, more isotropic, and much

FIG. 6. �Color online� Coordinate transformations for the transition from thecrystal to the laboratory coordinate systems. �a� Counterclockwise rotationof � about axis 3. �b� Counterclockwise rotation of � about axis 1�. �c�Counterclockwise rotation of � about axis 3�.

FIG. 7. �Color online� The dependence of �a� piezoelec-tric tensor component, d33, and �b� displacement, u3, forLiTaO3 on Euler’s angles and � in the laboratorycoordinate system. The components of piezotensor arethe following: d22

0 =8.5, d150 =26, d31

0 =−3, and d330 =9.2

in pm/V. Poisson’s ratio is �=0.25. Note that u3 is in-dependent of the Euler angle �.

014109-9 Eliseev et al. J. Appl. Phys. 102, 014109 �2007�

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more convex than the one for d33. The maximum value of u3

corresponds to the polar direction �=0, whereas d33 reachesthe maximum for ��0. From comparisons of Figs. 7�a� and7�b�, it is clear that there are significant differences in thenumerical values of the response components. While the di-rections in which the response is zero and the overall anti-symmetric character of the response are the same, as im-posed by symmetry consideration, there is a significantvariability in the numerical values of the response. Even thedirections in which the response is maximal can differ. Thisdifference in orientational dependences of d33� ,�� and

u3� ,�� is due to the fact that according to Eq. �23a�, theelectromechanical response is a sum of contributions due todissimilar elements of the piezoelectric constant tensor,

u3� ,�� � R313�d31� ,�� + d32� ,��� + R333d33� ,��

+ R351�d24� ,�� + d15� ,��� . �24�

Despite the fact that R333 is several times larger than R351

and R131, for LiTaO3 value of d330 is several times smaller

than d150 so that the contributions of R333d33 and R351�d24

+d15� to the vertical displacement u3 are almost equal near

FIG. 8. �Color online� The depen-dence of �a� displacement u3 and �b�piezoelectric tensor component d33 forPbTiO3 on Euler’s angle � in the labo-ratory coordinate system. The compo-nents of the piezotensor in the crystal-lographic coordinate system are thefollowing: d15

0 =61, d310 =−25, and d33

0

=117 in pm/V. Poisson’s ratio is �=0.3.

FIG. 9. �Color online� �a� Displace-ment u1, and �b� d35 and �c� d11 vs Eu-ler’s angles � and � in the laboratorycoordinate system for PbTiO3. The up-per row side represents the 3D view,the middle row is the cross sections at�=30° ,60° ,90° �curves 1, 2, 3�, andthe bottom row is the conical sectionsat �=90° ,60° ,45° ,30° �curves 1, 2,3, 4�.

014109-10 Eliseev et al. J. Appl. Phys. 102, 014109 �2007�

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the polar axis ���0�. However, angular dependences of d33

and d24+d15 are different, the most pronounced minima ofone surface correspond to maxima of another and vice versa.Therefore the total displacement u3 angular dependence be-comes smoother.

The dependence of the piezoelectric tensor componentd35 and horizontal displacement u1 below the tip versus theorientation of the crystallographic axes with respect to thelaboratory coordinate system for a LiTaO3 crystal is given inRef. 27. A common feature is that the u1 angular distributionis smoother and much more convex than that of d35 sinceboth d35 and d11 can contribute equally to u1�0� since R153

and R111 have close values, but their angular dependences arecomplementary.

Vertical displacement u3 below the tip and the piezoelec-tric tensor component d33 versus the orientation of crystallo-graphic axes with respect to the laboratory coordinate systemfor a PbTiO3 crystal are shown in Figs. 8�a� and 8�b�. Notethat d33 and u3 are independent of the Euler angles and �.It is seen that the maximum values of u3 and d33 correspondto the polar direction �=0.

The horizontal displacement component, u1, below thetip, and piezoelectric tensor components d35 and d11 versusthe orientation of the crystallographic axes with respect tothe laboratory coordinate system for PbTiO3 are shown inFig. 9.

It should be noted that both the displacement vector andpiezoelectric tensor are independent on angle due to thetetragonal symmetry of PbTiO3. It is seen from Fig. 9�a� thatu1 has broad maxima at �= ±90°, �=90°, whereas d35

reaches the maxima at ��135° and ��45°. Again, one canexplain this fact by the influence of component d11 that hasrather narrow maxima at �= ±90°, �=90°. Note that thecomponent u2 can be found as u2� ,� ,��=u1� ,� ,900−��.

In the analysis above, the dielectric properties of thematerial were assumed to be close to isotropic and hence theelectric field distribution is insensitive to sample orientation.The effect of the orientational dependence of the dielectricproperties can be incorporated in a straightforward mannerusing the analysis in Sec. II.

From the analysis above, the widely used uniform fieldapproximation for the PFM signals, �u1 ,u2 ,u3���d35,d34,d33�VQ, provides qualitative information aboutsymmetry and order of magnitude of piezoelectric response.However, quantitative information such as the position ofextreme points and numerical values of the response is lostin the case when components of piezoelectric tensor withdifferent angular dependences have close values.

IV. SUMMARY

The applicability of linear decoupling approximation forthe calculations of PFM signal is analyzed. The analyticalexpressions for the PFM signal for different multipole tipmodels were obtained for transversally isotropic and aniso-tropic piezoelectric materials as linear combinations of pi-ezoelectric constants. The general formulas for PFM signalfor materials with full piezoelectric and dielectric anisotro-

pies are derived. This provides the description of vertical andlateral PFM signal formation mechanism for anisotropic ma-terials.

The dependence of the PFM signal on crystallographicorientation has been analyzed to provide a framework fordeconvolution of local crystallographic or molecular orienta-tion from the three components of electromechanical re-sponse vector measured by PFM.

ACKNOWLEDGMENT

This research is sponsored in part �through two of theauthors, S.V.K. and S.J.� by the Division of Materials Sci-ences and Engineering, Office of Basic Energy Sciences,U.S. Department of Energy, under Contract No. DE-AC05-00OR22725 with Oak Ridge National Laboratory, managedand operated by UT-Battelle, LLC.

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