Higher-Order Electromechanical Response of Thin Films by

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 12, december 2006 2309

Higher-Order Electromechanical Response ofThin Films by Contact ResonancePiezoresponse Force Microscopy

Catalin Harnagea, Member, IEEE, Alain Pignolet, Member, IEEE, Marin Alexe, and Dietrich Hesse

Abstract—Piezoresponse scanning force microscopy(PFM) has turned into an established technique for imag-ing ferroelectric domains in ferroelectric thin films. At leastfor soft cantilevers, the piezoresponse signal is not only de-pendent on the elastic properties of the material under in-vestigation but also on the elastic properties of the can-tilever. Due to this dependency, the cantilever responseand, therefore, the measured properties depend on the fre-quency of the small alternating current (AC) testing volt-age. At the contact resonance, the cantilever response ismaximum, and this increased sensitivity can be used to de-tect very small signals or to decrease the voltage appliedto the sample. We have shown that by using the hystereticferroelectric switching, it is possible to separate the sig-nal into its components (viz. electromechanical and elec-trostatic contributions). Additionally, the measurement fre-quency can be tuned such that the second and third har-monics of the electromechanical response can be detectedat the cantilever resonance, providing information aboutthe higher-order electromechanical coefficients. We assumethat this nonlinear behavior seen in local and macroscopicmeasurements is rooted in the nonlinearity of the dielectricpermittivity.

Our results are of crucial importance for the study offerroelectric and electromechanical properties of nanostruc-tures.

I. Introduction

Scanning probe microscopy (SPM) is now estab-lished as the technique of choice to study the phenom-

ena taking place at the surface of materials. Voltage mod-ulation initially has been introduced in SPM in order todetect surface charges by detecting electric field gradientsor to nullify the electrostatic interaction in Kelvin probemicroscopy [1]. Both techniques operate in noncontact inorder to benefit from the free cantilever resonance to en-hance the measurement sensitivity.

Piezoresponse scanning force microscopy (PFM) hasemerged as a technique for studying ferroelectric phenom-

Manuscript received July 31, 2005; accepted July 26, 2006. Wegratefully acknowledge financial support from the German Min-istry of Education and Research (BMBF; contract No. 13N7986),NanoQuebec (project 101), the Canadian Foundation for Innova-tion (CFI), the National Science and Engineering Council of Canada(NSERC) and the Fond Quebecois pour la Recherche sur la Natureet les Technologies (FQRNT).

C. Harnagea and A. Pignolet are with INRS - Energie, Materiauxet Telecommunications, Universite du Quebec, Varennes, Quebec,Canada (e-mail: [email protected]).

M. Alexe and D. Hesse are with the Max-Planck-Institut furMikrostrukturphysik, Halle(Saale), Germany.

Digital Object Identifier 10.1109/TUFFC.2006.179

ena at the nanoscale [2], because of its ability to bothvisualize the ferroelectric domains and to modify themby locally switching the spontaneous polarization. In thistechnique, a small oscillating, testing voltage is applied be-tween the conductive tip of a scanning probe microscopeand the bottom electrode of a ferroelectric sample. Thisalternating current (AC) voltage induces mechanical oscil-lations of the cantilever, which then are retrieved from theglobal deflection of the cantilever using a lock-in technique.Initially it was proposed that the surface of the ferroelec-tric sample is oscillating due to the converse piezoelectriceffect under the applied voltage, and that these oscilla-tions are transmitted to the cantilever [2]–[5] via the tip.Later, it was shown that the electrostatic interaction be-tween the tip/cantilever and the bottom electrode of thesample plays a significant role in the cantilever vibrationand, therefore, in the formation of the “PFM” contrast [6]–[13]. Because both the ferroelectric-electromechanical andthe electrostatic interactions have been found to contributeto the induced cantilever vibration, the measurement tech-nique should be named voltage modulated scanning forcemicroscopy in contact mode. However, in order to remainconsistent with the wealth of published literature, we willfurther refer to this technique as piezoresponse force mi-croscopy, PFM, and to the signal detected by the lock-inamplifier as piezoresponse.

Generally, it was considered that cantilever resonancesare to be avoided in piezoresponse measurements, in or-der not to disturb the cantilever deflection [2]. Here wewill show that, on the contrary, the cantilever resonancescan be used to enhance the sensitivity of the detection.The frequency dependence of the piezoresponse signal ob-tained from measurement on TGS (triglycine sulphate)single crystals was reported by Labardi et al. [13]. Theyattributed the change in contrast with frequency to thespectral response of the tip-sample junction. However, theydid not discriminate between the piezoelectric and electro-static contributions. Rabe et al. [14] performed compar-ative studies between acoustic force microscopy and thepiezo-mode of an SPM, extracting material elastic proper-ties at the cantilever contact resonance. The elastic prop-erties of the cantilever control the (static) force applied tothe sample, as well as the cantilever dynamical properties(mainly its frequency response). The latter influence themagnitude and the phase of the detected deflection signal.Here we apply the contact resonance for the study of higherorder electromechanical coefficients in ferroelectrics.

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2310 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 12, december 2006

II. Experimental

A. Sample Preparation

Lead zirconate titanate (Pb(ZrxTi1-x)O3 with x =0.3) films were prepared using metalorganic decompo-sition (MOD). The precursor solution was prepared bydissolving stoichiometric amounts of Pb-ethylhexanoate,Ti-isopropoxide, and Zr-n-propoxide (Alfa Aesar GmbH,Karlsruhe, Germany) in ethyl-hexanoic acid and xylene.Gel layers were deposited by spin coating the precursorsolution on standard Pt-coated silicon substrates. Afterdeposition, the precursor films were dried for 5 minutesin air at 70◦C, then pyrolyzed in air at 300◦C for 5 min-utes and then crystallized by conventional thermal anneal-ing in air at 650◦C for 30 minutes. The resulting thick-ness of the films was 100 nm. Subsequent analysis of thesefilms revealed that they did not crystallize completely intothe ferroelectric phase, resulting in rosette-like ferroelec-tric regions surrounded by a pyrochlore phase. However,it turned out that exactly this incomplete crystallizationof the films providing ferroelectric and nonferroelectric re-gions in close proximity are ideal for the analysis of thepiezoresponse measurements performed at the contact res-onance of the cantilever. This configuration permitted thecomparison of the cantilever vibration when the SPM tipis in contact with either ferroelectric grains or with nonfer-roelectric regions. If different samples (i.e., one completelyferroelectric and one completely nonferroelectric) were tobe used, the comparison would have been much more dif-ficult for local measurements and impossible for imaging.

For macroscopic measurements, we prepared thickerfilms (500 nm) by repeating five times the spin coating,the drying and the pyrolysis steps mentioned above. Toreduce the formation of the pyrochlore phase (and, there-fore, to improve the ferroelectric properties) we have added10 at.% Pb (using the same Pb-ethylhexanoate precursoras source) in order to compensate for the Pb evaporationduring the annealing process.

B. Scanning Probe Microscopy Setup

To study the frequency dependence of the vibrationsinduced in the cantilever, in particular at the contact res-onance, as well as to detect the local electromechanicalproperties of the sample, we used the well-known setupfor piezoresponse force microscopy [2]–[15]. Basically, thisis a commercial SPM working in contact mode combinedwith a lock-in amplifier (LIA). The challenge is to detectan oscillation of known frequency but very small amplitudeburied in the deflection signal caused by the topography,whose amplitude is huge in comparison (but at low fre-quency). The global deflection signal from the SPM (usedby the SPM electronics for feedback regulation) is fed tothe input of the LIA. In a LIA, the signal is multiplied bya reference signal Vref = V ref

0 sin (ωreft) and the result isintegrated over several periods of Vref. Theoretically, theresult is nonzero only for the spectral component of the in-

put signal that has the same frequency ω = ωref, and thesame phase as the reference signal. Modern LIAs usuallyhave two channels, detecting signals in quadrature (i.e.,with 90◦ phase difference); therefore, they are in fact verysensitive vector voltmeters with a very narrow band-passfilter at the input, centered at the frequency investigatedωref. For all experiments, the cantilever vibrations wereinduced by applying a small AC voltage between the tipand the bottom electrode of the sample using the internaloscillator of the LIA.

The SPM manufacturers generally provide the connec-tions to contact the tip and the sample (in our case itsbottom electrode) via a signal access module. However,using this connection, a huge coherent pickup signal wasdetected by the LIA, increasing with the measurement fre-quency and becoming stronger than the actual cantileversignal above 100 kHz (this was checked by turning the laserbeam on/off1). For this reason, we connected the sampleand the tip directly without using the signal access mod-ule, considerably improving the detection accuracy.

We performed the measurements using two commercialSPMs: an Autoprobe CP Research, Thermomicroscopes,Sunnyvale, CA, (at the Max Planck Institute in Halle, Ger-many) and a DI-Enviroscope, Veeco Instruments, SantaBarbara, CA, (at INRS-EMT in Varennes, Canada). Al-though the working principle is the same for both systems,they actually differ in the geometry of the laser beam.In Autoprobe, the cantilever axis is perpendicular to theplane of the laser beam; in Enviroscope, the cantilever axisis contained in this plane. This causes some differences inthe calibration parameters of the signals coming from thedetectors. However, careful testing has proven that thesedifferences do not affect the PFM signal.

The AC amplitudes applied ranged from 50 mV up to1.5 V. In order to prevent artifacts coming from nonlin-earities at high-amplitude signals, the excitation voltage isadjusted so that the cantilever vibration was kept within3% of its static deflection (i.e., the deflection in the linearregime). The LIAs used were EG&G Instruments (now Sig-nal Recovery, AMETEK, Advanced Measurement Tech-nology, Wokingham, UK), Model 7260 and its successorModel 7265. They are dual channel LIAs with a digitalsignal processing unit, a working frequency up to 250 kHz,and a dynamic reserve (DR) of 100. The DR is an impor-tant parameter of a LIA, especially when detecting a smallsignal (∼100 µV) out of a large noise. The noise in PFM isthe deflection signal of the SPM with variations in the 1 Vrange. Additionally, this LIA can be set to lock on virtu-ally any harmonic (n < 65536 but such that the workingfrequency does not exceed the maximum working value of250 kHz), a feature we used for detecting the second andthird harmonic of the electromechanical response.

The spring constants of the cantilevers used in theseexperiments ranged from k = 0.02 N/m (soft cantilever)

1Previous works also have met this problem, most probably dueto electromagnetic induction between unshielded transmission linesand/or electronic components; Labardi et al. [14] have subtractedthis pickup as a constant background.

harnagea et al.: electromechanical response of thin films and contact resonance microscopy 2311

Fig. 1. Schematics of the modified Sawyer-Tower circuit.

up to k = 40 N/m (hard cantilever), corresponding to firstfree resonance frequencies of 9 kHz up to 260 kHz. Hystere-sis loop measurements were performed using a computer-controlled Keithley 2400 Digital Source Meter (Cleveland,OH) in series with the AC source.

We used cantilevers from Micromash, (Tallinn, Estonia)models CSC12-, NSC15-, and CSC11-series coated withthe hard semiconductor W2C, because they proved to de-liver the most stable signal over extended periods of time,each tip lasting up to a few days of operation.

C. Macroscopic Measurements

In order to detect the higher-order dielectric coefficientsin our samples, we used a slightly modified Sawyer-Towercircuit as shown in Fig. 1. In this small-signal measure-ment, a testing AC voltage is inducing a small, alternat-ing charge on the electrodes of the two capacitors, and thischarge is detected by the LIA. Simultaneously, a large di-rect current (DC) voltage is used to bias the sample (thevoltage drops mainly on Cx, the typical ratio C0/Cx in ourexperiments being 50–100). As we will see later, the higherorder dielectric coefficients are related to higher harmonicsin the small-signal, dielectric response.

The role of the LIA in this arrangement is to extractonly the signal at the desired frequency, and we used itto detect higher harmonics of the sample’s response. Al-though the setup looks very simple, special attention hasto be paid when choosing and assembling the components;all of them have to be highly linear at the desired work-ing amplitudes and frequencies. The input impedance ofthe LIA has to be much higher than that of C0, the DC-source should have a low AC-impedance, and the AC-source should have a high DC-conductance in order notto introduce additional nonlinearities. We have validatedour setup by replacing the sample with a linear capaci-tor and recording its response, which was indeed linear. Inthese experiments, we used C0 = 53 nF (Cx was ∼0.5 nF),and a Vac with 0.1 Vrms at 9.97 kHz.

All electronic equipment (LIA, AC- and DC-sources)were controlled by a computer using an in-house-writtencontrol application under the TestpointTM platform (Cap-ital Equipment, Bedford, NH). Hysteresis measurementswere performed by cycling the bias in small steps between

Fig. 2. Schematics showing the cantilever bending under its elec-trostatic interaction with the sample in PFM. (a) Noncontact (thecantilever end is free) first mode of oscillation. (b) Noncontact, sec-ond mode of oscillation. (c) Ideal contact, the cantilever is completelyclamped at both ends (case of infinitely hard surfaces), first mode ofoscillation. (d) Contact, second mode. “N” shows the location of anode of the vibration mode.

the desired minimum and maximum limits. The hystere-sis loop of the first harmonic response is in fact a normalC(V) curve, provided that the higher harmonics are muchsmaller than the first. This is because, in contrast to con-ventional bridge measurements of the capacitance, the LIAfilters out all spectral components except the reference fre-quency.

III. Cantilever Resonances

In SPM measurements, a sharp tip is brought in theproximity of a sample surface, and its z-position is moni-tored and controlled while scanning across the sample sur-face (in the plane perpendicular to z). Here we refer onlyto the optical beam detection method, widely used in mostcommercial SPMs to detect a sample topography, and inthis case the tip is mounted at one end of a cantilever andthe other end is fixed.

The cantilever, therefore, is the key element of such anatomic force microscope (AFM) because, first, it providesinformation to the system about the tip vertical displace-ment (via the cantilever deflection). Second, it allows oneto control the interaction between tip and surface. Thismethod works quite well in the quasistatic regime, i.e.,when the tip moves relatively slowly, typically with a re-sponse time of 1 ms (corresponding to vibrations in the1 kHz range) such as in the usual topography imagingcontact mode.

As a force detector, the cantilever is an elastic beamand has its own resonance frequencies corresponding todifferent vibration modes. The first two are illustrated inFig. 2. The cantilever vibrations are governed by elasticand geometrical factors (mass, shape) and by the bound-ary conditions (forces acting on the cantilever). The latterhave to be taken into account in the detection process. This


is especially important when the cantilever is driven intooscillation at or near one of its resonance frequencies, whenthe vibration is most sensitive to changes in boundary con-ditions. Usually, the most important one is the interactionforce between tip and sample. However, for soft cantilevers(elastic constant below 1 N/m), the electrostatic force be-tween the conductive cantilever and the bottom electrodeof the sample produces an additional bending that con-tributes significantly to the PFM signal and, therefore,can drive the cantilever into oscillation. Fig. 2 sketchesthe first two vibrational modes [16] of a cantilever underan electrostatic force. In Fig. 2(a) and (b) we can see themajor difference between the first and second modes ofoscillation for the noncontact case. The sign of the slopeof the cantilever at its free end is opposite for the twomodes. Because the optical system detects the angle (andnot the real position), this means that the AC-signal fromthe cantilever also will be 180◦-phase-shifted in the secondmode of oscillation compared to the first mode. The sameis valid for the contact case, and this affects the interpre-tation of the PFM signal as the phase of this signal givesinformation about the spontaneous polarization.

A. Forces Driving the Cantilever Vibration

The case when the tip is in contact with the surfaceis even more complicated because the boundary conditionat the tip end (i.e., the interaction force between the tipand the sample), as well as the contact properties, haveto be taken into account. Depending on tip/cantilever andsample, we may have or have not the situation shown inFig. 2(c) for an ideal contact, represented by hard samplesand tips mounted on stiff cantilevers. If the sample or tipare soft, the contact generally is described with a springof elastic constant k∗, the contact stiffness [16].

For the case of piezoelectric (or ferroelectric) materials,the geometry of the system is illustrated in Fig. 3. Thetip is coupled to the surface via a spring of effective stiff-ness k∗, which is strained between the tip and the samplesurface. The latter is not fixed, but it is moving with thepiezoelectrically induced displacement h.

Assuming that the tip does not detach from the surface,we can write the interaction forces as (positive z-axis ispointing out of the surface):

Ftip = −k∆z − Fadh − F tipES ,

Fk∗ = k∗ (−∆z∗ + h) ,(1)

where k and ∆z are the equivalent spring constant andvertical deflection of the cantilever, Fadh is the attrac-tive adhesion force acting at the contact, k∗ and ∆z∗ arethe contact stiffness and deformation, and h is the piezo-electrically induced displacement of the surface below thetip. Electrostatic forces (caused by voltages between theconductive cantilever/tip assembly and the bottom elec-trode) act on both tip and cantilever [10], [12]. The gen-eral expression for the electrostatic force (also known asthe Maxwell stress) is:

Fig. 3. Sketch of the forces driving the cantilever into vibration involtage-modulated force microscopy of piezoelectric materials. TheSPM tip is coupled to the surface via the effective contact stiffnessk∗. Below the contact, the sample is deformed due to the piezoelectriceffect by the applied voltage.

F iES = αi

(V i

)2,

V i =(Vdc − CPDi

)+ Vac,

2αi =∂Ci

∂z,

(2)

where i = (tip, cantilever), Vdc is the DC-voltage ap-plied between cantilever/tip and bottom electrode, Cis arethe tip-sample or the cantilever-sample capacitances, andCPDis stand for the contact potential differences and de-pends on the sample and tip/cantilever materials. Theseforces are distributed over the whole area of electrostaticinteraction; and, ideally, the exact field distribution shouldbe taken into account.

The equilibrium condition is obtained by nullifying thetotal momentum (the cantilever does not rotate aroundthe fixed end) and can be used to further calculate k∗ fromcontact mechanics [13], [16]. Considering the tip positionexactly at the cantilever end, F cant

ES uniformly distributedover the cantilever surface and neglecting the shear at thefixed end:

Ftip + Fk∗ = (1/2)F cantES . (3)

In the following, for simplicity, we will consider an in-finitely stiff contact k∗ → ∞ because first, both tip coating(W2C) and sample (ferroelectric ceramic) are hard mate-rials. Second, the static force is relatively small (∼2 nN)because the cantilevers used are soft (k ∼ 0.02–0.2 N/m).This simplification implies that the tip is pinned on thesample surface and rigidly follows the surface movements.Additionally, it means that F tip

ES does not influence thecantilever vibration [9].

In the case of PFM of ferroelectric materials, thesample-induced vibration, h in (1) is to be detected. Thisvibration intervenes in the boundary conditions for thecantilever oscillation, and it contributes more significantlyas the frequency approaches the (contact) resonance ofthe cantilever. The electric field-induced strain in a ferro-electric characterized by the spontaneous polarization PS,permittivity ε, and electrostriction coefficient Q usually isexpressed for the one dimensional case as [17], [18]:

x = QP 2total = QP 2

S + 2QPS(εE) + Q(εE)2. (4)


Considering a uniform electric field below the tip acrossthe sample thickness τ , E = (Vdc − CPDcant + Vac)/τ :

h = xτ = (Qτ)P 2S + 2QPSε

(Vdc − CPDcant + Vac

)+ (Q/τ)ε2 (

Vdc − CPDcant + Vac)2

. (5)

B. Spectral Components of the Driving Forces

As mentioned in the previous section, the bending ofthe SPM cantilever is partly governed by the electrostaticforce acting on the cantilever (uniformly distributed overits surface), and the tip is pinned to the surface follow-ing its vertical displacement h. The optical system detectsthe angular deflection of the cantilever, usually at its end(in the region in which the tip is mounted) because thisalignment ensures the best sensitivity at low frequency.The cantilever is driven into oscillation by applying the acvoltage Vac = V0 sin(ωt) between the tip and the bottomelectrode of the sample. Replacing Vac in (2) and (5), thefollowing relations can be found for the first and secondharmonics [6], [7]:(

F cantES

)ω

= 2αcant (Vdc − CPDcant)V0 sin(ωt),

(6)

hω = 2ε Q[PS + ε

(Vdc − CPDtip) /τ

]V0 sin(ωt),

(7)(F cant

ES)2ω

= −(1/2)αcantV 20 cos(2ωt), (8)

h2ω = −(1/2)(ε2Q/τ

)V 2

0 cos(2ωt). (9)

From these relations, we deduce the following:• The first harmonic depends linearly on Vdc and V0. A

change in sign of the DC-component in (6) is equiva-lent to a 180◦ phase shift. Vdc usually is used to en-hance the signal in classical (noncontact) electrostaticforce microscopy (EFM) (or to nullify the signal inKelvin probe microscopy).

• Ferroelectric, hysteretic switching (explicit) informa-tion is contained only in (7) via PS (Vdc).

• The second harmonics do not explicitly depend on Vdc.• The maximum exponent of the voltage in (5) is two.

Therefore, the highest harmonic generated is the sec-ond, provided that no switching of PS occurs dueto Vac.

It is important to note that, according to the sketch inFig. 3, the electromechanical displacement h (for a positivepiezoelectric coefficient) and the electrostatic force havedifferent signs. Therefore their effects on the cantilevervibration for the first mode (below and around the firstcontact resonance) are opposite. For the harmonic compo-nents, this means that hω,2ω and (F cant

ES )ω,2ω in (6)–(9) arein antiphase. Franke [7] used this relationship to establishthe dominant contribution in piezoresponse characteriza-tion.

C. Frequency Dependence and Vibration Modes

We shall now analyze the cantilever vibration inducedsolely by F cant

ES , both in noncontact and in contact. For this

purpose, we kept the tip fixed 10 µm above or in contactto a certain location on the nonferroelectric (pyrochlore)region of the sample.

Fig. 4 compares the frequency dependence of the elec-trostatic signals in noncontact mode with positive and neg-ative Vdc. This experiment was performed in order to testthe validity of (6) for the electrostatic force. A given ampli-tude of the electrostatic force can be achieved by applyingtwo different bias voltages, one negative and one positive.If the amplitude in (6) does not explicitly depend on fre-quency, the response amplitudes should be equal at anyfrequency. Similarly, the phase shift between the signalsalways should be 180◦. The cantilever used (CSC12f) hada first free resonance frequency of 17 kHz and an esti-mated spring constant of 0.05 N/m. The two bias voltages(V +

dc = 2.5 V and V −dc = −1.5 V) were chosen such that, at

the first resonance, the amplitudes of the signal are equal(this means that CPDcant = 0.5 V in our noncontact ex-periment).

The spectra in Fig. 4 shows that (6) is valid only withina finite range around the first contact resonance. We be-lieve this is due to a cantilever asymmetry that makes thecantilever vibrate differently under opposite Vdc. Below thesecond resonance, the signal under negative bias is smallerthan the signal at positive bias. Above the second reso-nance, the situation is reversed. This behavior could beinduced by the static deflection due to the DC-componentof the electrostatic force, always attractive. At the first res-onance (at 17 kHz), the cantilever vibration mode being asshown in Fig. 2(a), the phase jumps by 180◦, in agreementwith the behavior of a forced oscillator. Below resonanceit is in phase with the excitation, above it is in antiphase.Similarly, at the second resonance (133 kHz), the vibrationmode being as shown in Fig. 2(b), the phase of the signalagain jumps by 180◦.

Between the resonances, we observe regions with verysmall amplitudes (so-called antiresonances) and, therefore,with a high phase noise corresponding to very low angularamplitudes at the cantilever end. We believe that, in thesefrequency regions (e.g., 75–100 kHz), the vibration modeof the cantilever gradually changes from that sketched inFig. 2(a) to that in Fig. 2(b). The amplitude minima do notoccur at the same frequency for positive and negative Vdcbecause the static deflection induces an offset of the can-tilever deflection angle. Above the second resonance, thespectra at positive bias again passes through a minimumthen switches to the third vibration mode. The situationis similar for the case of the contact vibration (not shown).

This supposition, in principle, can be verified experi-mentally. One way is to record the frequency spectra atdifferent Vdc biases, therefore, at different strengths of theasymmetry. A measurement performed with Vdc = +10 Vindeed showed a shift of the minimum toward higherfrequencies, at 175 kHz, compared to the minimum at120 kHz obtained with Vdc = 2.5 V as seen in Fig. 4(a).Another way would be to align the laser beam on the can-tilever such that it is reflected at different positions alongthe cantilever length. In this way, the signal will be pro-


(a) (b)

Fig. 4. Frequency dependence of the oscillations of a soft cantilever (CSC12f, k = 0.05 N/m) when excited electrostatically with positive (◦)and negative (−) Vdc in noncontact. (a) Amplitude, (b) phase.

portional to the angle (local slope) of the cantilever atthat specific location, allowing the analysis of the can-tilever bending. If our supposition is true, the position ofthe minima will depend on the position of the reflectionalong the cantilever. For the low frequency regime, Hueyet al. [19] found that, indeed, the amplitude of the electro-static contribution has a minimum at a specific positionof the laser beam reflection on the cantilever (at ∼60% ofcantilever length from the fixed base).

The DC-bias dependence of the antiresonance frequencyexplained above for the noncontact cantilever vibrationwas very recently observed experimentally and modeledtheoretically for the general case by Jesse et al. [20]. Forthe case of contact, they concluded that piezoresponse con-trast is optimal at frequencies in which the antiresonanceis obtained for highest bias because it implies that theelectrostatic contribution to the overall signal is minimal.Although they do not show theoretical predictions for thenoncontact case, their noncontact measurements above aSiO2 (nonferroelectric) surface seem to confirm our resultpresented above. Even at the second resonance frequency,the nulling bias is not equal to that at the first resonance.This automatically implies that (6), which is the basis forEFM and Kelvin force microscopy imaging modes, is notvalid for any frequency; and these measurements have tobe performed at the first cantilever resonance because thecantilever vibration mode resembles best a vibrating ca-pacitor.

Fig. 5 compares the cantilever vibration spectra whenexcited with the tip above surface (noncontact) with thatwhen the tip is in contact with a nonferroelectric regionfor negative Vdc. For the contact case, we cannot excludea contribution from the field-induced piezoelectricity cor-responding to the linear term in Vdc in (6). However, thiscontribution is much smaller than the electrostatic contri-bution because the permittivity of the pyrochlore is small.

The first contact resonance is about one order of magni-tude higher than the first noncontact resonance, in agree-ment with the calculations performed by Rabe et al. [16].The frequency dependence of the phase clearly shows that

the vibration modes around the first resonance are as il-lustrated in Fig. 2(a) and (c). Below the first noncontactresonance, they are in antiphase. The phase of the vibra-tion in contact is constant until the contact resonance, atwhich it jumps in-phase with its excitation. As we willdiscuss later, this is not the case for the piezoelectric ex-citation.

D. Using the Contact Resonance for PiezoresponseDetection

The first application of the cantilever contact resonanceis to enhance the sensitivity for the detection of the samplepiezoelectric displacements. The response amplitude beingenhanced typically by more than one order of magnitude(see the contact spectrum in Fig. 5), samples with lowerpiezoelectric response and for which the signal-to-noise ra-tio is very low far from resonance, could induce a goodsignal if the excitation frequency is close to the contactresonance. Alternatively, for samples with good piezoelec-tric response, the excitation amplitude can be decreasedby the same factor. This is useful when the sample polarstate is influenced by the AC-voltage applied, as reportedby Okino et al. [21], who successfully detected ferroelec-tric domains in a relaxor single crystal at a temperatureof 175◦C, exactly because they could decrease the excita-tion amplitude at the contact resonance while keeping thesignal-to-noise ratio within acceptable limits.

As discussed in our previous work [22], we observedthat both the ferroelectric and the electrostatic contribu-tions exhibit the same frequency dependence as the contactspectrum curve in Fig. 5 obtained from a nearby, nonfer-roelectric location. This finding has shown that the can-tilever can be used to amplify the piezoresponse as well.Fig. 6 exemplifies this enhancement of the piezoresponsesignal near a contact resonance. From the topographyimage—Fig. 6(a)—we distinguish a flat (nonferroelectric)background (“P”) and clusters (“rosettes”) of ferroelectricgrains about 200 nm in lateral size (“F”). Figs. 6(b) and(c) show the PFM signal from the same area of the sam-


(a) (b)

Fig. 5. Comparison between the frequency dependence of the oscillations of the cantilever from Fig. 4 when excited electrostatically innoncontact (◦) and in contact (−) with a nonferroelectric surface. The Vdc used to enhance the signal had the same polarity (−1.5 V innoncontact, and −2 V in contact). (a) Amplitude, (b) phase.

Fig. 6. Topography (a) and piezoresponse images obtained (b) off-resonance and (c) near contact resonance. (d) Ferroelectric hysteresisobtained from one of the grains, with testing frequencies below (dots)and above (solid line) the contact resonance. Note the change in signof the linear slope at hysteresis saturation, but no change in the signof the hysteresis loop. Scan size is 5 mm, z-scale of (a) is 20 nm, thez-scale of (b) and (c) is the same, 10 mV full scale (representing theLIA input).

ple, obtained with an AC signal at 9.9 kHz (off-resonance)and 135 kHz (near the first contact resonance of this can-tilever), respectively. Both images reveal the same ferro-electric domain structure. Note that the z-scale of Fig. 6(c)is the same as that of Fig. 6(b). In this experiment, work-ing at the contact resonance increased the sensitivity ofthe measurement by a factor 40.

The same enhancement of the ferroelectric response wasconfirmed for local measurements. Piezoelectric hysteresisloops obtained by sweeping Vdc while recording the first

harmonic signal (as described in [15]) showed that the re-manent value (therefore, at zero Vdc) depends on the fre-quency of Vac.

An intriguing question related to the phase of the fer-roelectric contribution remains. To separate the two elec-trostatic and electromechanical contributions, we assumedthe ferroelectric part to be a rectangular hysteresis loopwhile the electrostatic part was taken as the slope of theloop at saturation. The reason for doing this separation isshown in Fig. 6(d). The slope of the hysteresis loop changesits sign when passing through resonance, in agreementwith the frequency dependence under electrostatic exci-tation discussed above. The interesting point, however, iswhy this is not the case for the piezoelectric contribution.If the phase of the cantilever vibration changes by 180◦, wewould expect a change in the sign of the hysteresis loop,too. A negative (“Z-shaped”) hysteresis loop hω (Vdc) isone that has both properties:

• The loop can be converted to a normal one (“S-shaped”) by the transformation hω (Vcd) → −hω (Vdc).

• The loop direction is clockwise (this is to preservecausality, i.e., the relation “cause-effect”. Once the sat-uration is reached with a certain electric field, it canbe switched at the coercive field of opposite polarity).

As we also reported in [22] that the 180◦ phase shift (orsign change) was not observed for the ferroelectric contri-bution when the excitation frequency passes through reso-nance. Later we became aware of the related work by Rabeet al. [14] that mentions a similar observation. The con-trast in acoustic force microscopy changes the sign whenthe excitation frequency is passing through contact reso-nance, although this change does not occur for the contrastin piezomode images of the cantilever. Our first supposi-tion was that this is happening because the ferroelectricpart is acting directly on the tip, and the laser spot isreflected at the same location on the cantilever. But thisdoes not explain the results from [14] because, in their case,both the acoustic response and the piezoelectric responseare excited at the same location, the SPM tip. This isan intriguing finding and deserves further consideration to


elucidate the effects of the cantilever bending and, there-fore, the transfer of the piezoelectric deformation of thesample to the optical detector.

IV. Detection of the Local, Higher Order

Electromechanical Response

Nonlinear ferroelectric phenomena, associated with ahigher-order electromechanical response, were previouslyaddressed by studying the first harmonic piezoelectric re-sponse. Domain walls intersecting the sample surface werefound to cause an increase in the effective piezoelectriccoefficient with the driving amplitude [23]. Shvartsmanet al. [24] explained the high nonlinearity and imprint(domain instability under AC-voltage) in (111)-orientedPZT (45/55) films assuming that charged domain wallsare present below the film surface. Intrinsic nonlinearityor tunability (field dependence of the piezoelectric coeffi-cient) in epitaxial PZT has been explained by Nagarajanet al. [25] using a thermodynamic formalism taking intoaccount the in-plane misfit strain.

A review of the literature shows that SPM-based de-tection of higher-than-first harmonic cantilever responsewas first achieved by Franke et al. [26], who associate thesecond harmonic image with the permittivity squared (rul-ing out the electrostatic contribution). Shortly after, theypropose to extract information about spontaneous polar-ization from the electromechanical response images by di-viding the piezoresponse image (first harmonic response)by the square root of the second harmonic response im-age [27], because they found that electrostriction is domi-nant in the second harmonic signal of a PZT(27/73). Onthe contrary, Zavala et al. [28] assumed that the elec-trostatic force was the main contribution to the secondharmonic signal, estimated the tip-to-sample capacitance,then computed the permittivity. Shao and Bonnell [29] im-aged and reported for the first time the bias dependenceof the second harmonic electromechanical signal from apolymer P(VDF-TrFE) film. We will compare their inter-esting results with ours later in this section. Regardingthe third harmonic in SPM-based measurements, Cho andOhara [30], using a technique which they named scanningnonlinear dielectric microscopy2 [31], detected the fifth-order permittivity (here the order “five” refers to the rankof the tensor, i.e., the fifth derivative of the free energyin the thermodynamic formalism) by extracting the thirdharmonic of the microwave modulation signal. An inge-nious and simple method to detect capacitance variationswas presented by Kobayashi et al. [32] who noticed that,if the capacitance depends on the voltage, ∂C/∂V �= 0,the electrostatic force acting on the cantilever will have a

2This technique turns out to be very similar to the commerciallyavailable scanning capacitance microscopy (SCM) technique. Theonly difference we noticed is the geometry of the electrodes for mi-crowave excitation. In contrast to SCM, the microwave field is appliedbetween the SPM tip and a small ring around the tip, parallel to thesample surface. The AC modulation, however, is applied as usual,between the tip and the sample bottom electrode.

Fig. 7. Higher order electromechanical characterization. The ferro-electric rosette shown in the topography image (a) was previouslypolarized half negatively (left) and positively (right) as can be seenin the first harmonic piezoresponse image (b). (c) and (d) are thesecond and third harmonic response, excited such that the responseof interest is near the contact resonance (in this case 247 kHz). Scansize is 4 µm, z-scale of image (a) is 10 nm, arbitrary units for (b),(c), and (d).

third harmonic component. Measuring the third harmonic,they succeeded in detecting regions with different dopingconcentrations in a silicon test sample.

By using the signal amplification at the contact res-onance, we succeeded in detecting the second and thirdharmonics of the electromechanical response. To achievethis, we used the cantilever as a selective amplifier. The fre-quency of the excitation voltage was chosen to be half or athird of the cantilever’s contact resonance for the detectionof the second harmonic and third harmonic, respectively.The LIA was always detecting the cantilever response atthe contact resonance. Fig. 7 demonstrates the successfuldetection of the second and third order electromechanicalresponse in ferroelectric thin films. In order to facilitate theinterpretation and comparison of the different harmonics,the rosette shown in the topographic image Fig. 7(a) wasfirst polarized as shown in the PFM image (first harmonicresponse) from Fig. 7(b): the left half negatively (polar-ization pointing into the film) and the other half posi-tively, by scanning the area from left to right with a biasVdc = −10 V (left half of the image) and Vdc = +10 V(right half of the image) on the tip. Fig. 7(b) was acquiredafter the poling with a small excitation Vac near the can-tilever resonance (0.1 V, 247 kHz) and proves that po-larization has been switched. The frequency then was de-


creased to 123.5 kHz, and the amplitude was increased to1 V in order to record the second harmonic image shown inFig. 7(c). The third harmonic electromechanical responsewas detected similarly, by setting the LIA to lock on thethird harmonic of the excitation Vac whose frequency wastuned to 82.3 kHz and whose amplitude was increased to1.5 V.

The result is shown in Fig. 7(d). The second harmonicof the electromechanical response is in principle relatedto electrostriction by (9). In fact, the product ε2Q = Mrepresents the field-related electrostriction coefficient [18],[33] used to describe the quadratic dependence of strainon the applied electric field. For highly nonlinear materi-als, such as ferroelectrics, Q is the preferred coefficient [see(4)] because it was proven for many materials of interest(among which the perovskites), to be basically indepen-dent of hydrostatic pressure, temperature [34], and onlyweakly dependent on the electric field [35].

The image in Fig. 7(c), therefore, represents a map ofthe electrostriction coefficient M . As easily noticed bycomparing with the piezoresponse image—Fig. 7(a)—itdoes not depend on the direction of the spontaneous polar-ization because there is no difference in contrast betweenthe left and right halves of the rosette, in agreement withthe fact that M is proportional to the square of permit-tivity.

This relationship between permittivity and M is bet-ter expressed in Fig. 8(a), which shows the bias depen-dence of the second harmonic. The hysteresis loops wereobtained by stopping the x-y scanning and recording thesignal while cycling Vdc with the tip in contact with oneferroelectric single grain. The numerical values were cal-culated using the detector sensitivity measured from the(low frequency) force curve (we obtained a good value of2.3 V/nm for the sensitivity) [15]. We assume that thisvalue does not change, as long as the vibration mode ofthe cantilever is the same. In order to compare the orderof magnitude of the first and second harmonic signals, wedivided the response by the AC-voltage applied for eachloop (0.1 V for the first and 1 V for the second harmonic),although this is not entirely correct because the secondharmonic coefficient should be proportional to the squareof the field. Nevertheless, it turns out that the second har-monic response is about two orders of magnitude weakerthan the piezoelectric response.

Another important point to be discussed is the shapeof the curve. In principle, the permittivity squared hasthe same allure as the permittivity because it is a posi-tive quantity and, indeed, the second harmonic responsein Fig. 8(a) has the appearance of a usual ε(E) curve ofa ferroelectric material. An important property of a fer-roelectric hysteresis loop is the direction of its evolution.The P(E) loop has to be counterclockwise, i.e., the systemreceives energy in order to switch polarization through thecycle (the path integral of PdE has to be negative throughthe complete loop). In terms of ε(E), this implies that thereturning branch from one saturation reaches a maximumafter the field goes through zero, i.e., the direction is as

shown by the arrows in Fig. 8(b). This is not the case,however, in [29], which shows “inverted hysteresis loops,”when switching occurs before the voltage passes throughzero. In other words, the remanent value has an oppo-site sign than the saturation value from which it origi-nated. Inverted magnetic hysteresis loops were previouslyreported and explained by an antiferromagnetic-type ofinteraction between ferromagnetic blocked and superpara-magnetic particles [36]. For the ferroelectric case, however,we believe that the reason for the strange result of Shaoand Bonnell [29] is not the negative electrostriction co-efficient but rather the long relaxation time of polymers,which also means a long switching time (possibly longerthan the cycling period).

To explain the presence of the third harmonic signal,we have to consider nonlinear effects. The simplest way todo this is to expand the (intrinsic) permittivity in (4) witha first order dependence on the electric field [37]:

ε(E) = ε(2) + ε(3)E,

ε(3)(E) =∂ε

∂E,

(10)

with the third order permittivity ε(3) being equivalent tothe well-known tunability of the permittivity κ:

κ =ε(0) − ε(E)

ε(0)= 1 −

ε(3)

ε(2)E. (11)

Replacing ε in (4) and performing similar operations asin Section III-B we find the third harmonic component ofthe piezoelectric displacement as being:

h3ω = −(1/2)(Qτ−2) ε(3)

[ε(2) +

+ 2ε(3)(Vdc − CPDtip) /τ

]V 3

0 sin 3ωt. (12)

This relation clearly shows that the third harmonic sig-nal originates in the nonlinearity of the dielectric permit-tivity. Being an odd-rank tensor, it is expected that ε(3)(E)depends on polarization [31] and, therefore, changes itssign upon switching of polarization. This is indeed ob-served in our experimental results, in both imaging andlocal hysteresis measurements. The third harmonic imagefrom Fig. 7(d) shows an incontestable change in sign ofh3ω between the regions of the rosette poled with oppositevoltages, and it is strongly correlated with the hω (piezore-sponse) image. Local measurements confirm this result.The bias dependence shown in Fig. 8(b) clearly showsswitching of h3ω upon voltage bias cycling and, moreover,it shows a dependence on the bias at saturation, as (12)predicts through the term ε(3) in the parenthesis. However,the dependence of h3ω on the spontaneous polarizationand, therefore, the hysteretic nature of this signal, doesnot appear explicitly in (12), although it is formally con-tained in the derivative of the permittivity with respectto the electric field, as we will demonstrate in the nextsection presenting macroscopic measurements. In an alter-nate approach, we could consider the electrostriction (and


(a) (b)

Fig. 8. Local higher order electromechanical hysteresis loops (a) second and (b) third harmonic cantilever response (solid lines) as a functionof the bias Vdc (each compared with the first harmonic, represented with open circles) obtained from one of the ferroelectric grains. Thefrequency of Vac excitation was such that the response of interest was near the contact resonance (in this case 247 kHz).

not the permittivity) to be nonlinear: We start by addinga higher order electrostriction term in (4):

x = QP 2total + RP 4

total. (13)

The nonlinear term is actually justified if we recall thatthe Q and M coefficients are related via the permittivity,which is nonlinear. Through this approach, however, thethird harmonic component will depend explicitly on spon-taneous polarization via Ptotal = PS + εE. By replacingE = (Vdc + Vac) /τ , and, after performing the calculations,we find the third harmonic term as:

h3ω = −(Rτ−2) ε3[PS +

+ ε(Vdc − CPDtip)

/τ]V 3

0 sin(3ωt). (14)

The hysteretic nature of the third harmonic signal ap-pears explicitly through the term in the square paren-thesis that can be identified with the total polarizationand, therefore, shows hysteresis upon cycling the Vdc bias.Comparing the hysteresis loops in Fig. 8(b), we again inferthat the magnitude of the third harmonic response is morethan two orders of magnitude smaller than the piezoelec-tric response for the Vac used.

V. Detection of Higher Order Dielectric

Response

To confirm the dielectric nonlinearity invoked in theprevious section, we performed macroscopic measurementsusing the modified Sawyer-Tower circuit described inFig. 1. As is well-known, in such a circuit the voltage de-tected is proportional to the charge on the linear capacitorC0. This charge in turn is equal to that induced in the sam-ple Cx by the voltage applied. In the case when Cx � C0,we have:

Cx = (C0/Utotal)UC0 , (15)

where UC0 is the voltage measured by the voltmeter. Ifnow we apply the voltage Utotal = Vdc + Vac and measure

different AC spectral components using a LIA, we can, infact, extract the bias dependence of the nonlinear dielectricresponse of Cx. For the first harmonic we can calculate thelinear capacitance and, therefore, the permittivity from(15), as shown in Fig. 9(a).

If the LIA is now set to lock on the second harmonicof the electric excitation, it will detect the charges gener-ated inside Cx at frequency 2ω. If we assume a first orderdependence of the permittivity on the electric field as in(10), we can write the charge generated at 2ω:

q2ω = C0U2ω (16)

={[

ε(2) + ε(3) (Vdc + Vac) /τ] S

τ(Vdc + Vac)

}2ω

,

with S = 0.24 mm2 being the area of the top electrode. Af-ter performing the computations, we find that the voltagegenerated at 2ω by the nonlinear capacitor is:

U2ω = − 12C0

S

τ2 ε(3)V20 cos(2ωt). (17)

The bias dependence of ε(3) [calculated using (17) fromthe second harmonic dielectric response, U2ω] is shown inFig. 9(b). It unquestionably demonstrates the hystereticnature of the second harmonic dielectric response, there-fore, demonstrating that in (12) the hysteretic behaviorof h3ω is generated by the hysteresis of ε(3), i.e., that thethird harmonic can be detected and is a consequence ofthe nonlinearity of the dielectric permittivity.

The derivative of the permittivity (∂ε/∂E) calculatedfrom Fig. 9(a) compares relatively well at fields close tosaturation, with ε(3) extracted from the second harmonicdielectric response. The difference between the two curves,which appear close to the “coercive fields,” could be phys-ically explained by the fact that in (16) we neglected thevoltage drop (at 2ω) on the external sources. Extrinsiccontributions to the dielectric response, represented by thedomain wall movements [28], may also account for the dif-ferences of the two curves when switching of polarizationoccurs.


Fig. 9. Macroscopic dielectric characterization of a ferroelectric capacitor with a Sawyer-Tower-like circuit. (a) The first harmonic (capaci-tance). (b) ε(3) as function of bias as extracted from the second harmonic charge response, solid line, compared with the derivative of (a),dotted line.

VI. Discussion

A. Other (Nonferroelectric) Possible Origins of NonlinearBehavior in Local Measurements

In Section IV we assumed that the second and thirdorder harmonics detected originate solely in the nonlinearcharacter of the ferroelectric sample under investigation.We should point out, however, that, in the tip-sample-cantilever system (excluding the nonlinearities of the de-tector/AFM electronics), there are at least two other pos-sible origins of nonlinearities.

• Contact nonlinearities. Contact mechanics is nonlin-ear; therefore, higher harmonics could be easily gen-erated. Numerical estimations (see next subsection)show that indeed the piezoelectric vibration of 32 pmis comparable to the estimated static indentation of0.35 nm. Moreover, under the DC-bias applied, thecontact stiffness could change due to the increasedelectrostatic force. This would further increase thenonlinearity. This effect, however, should not cause ahysteretic dependence upon cycling the bias. In ad-dition, as we mentioned in [22], we did not observea change in the resonance frequency with the appliedbias, as should happen if the contact stiffness is mod-ified by the applied voltage that increases the staticload. We also verified that the resonance frequencyis not depending on the amplitude of the AC-voltageand, therefore, we excluded this effect.

• A secondary effect related to the change in the dielec-tric permittivity with bias is the change in the pene-tration depth of the electric field. This change in thefield distribution would induce nonlinear-like behav-ior. However, as in the previous case, this effect shouldnot show a hysteretic dependence on the applied bias.Therefore, we can exclude it.

B. Considerations About the Numerical Values

In Section IV we have shown the successful detectionof the second and third harmonics of the piezoelectric re-

sponse. We proposed that the dielectric nonlinearity is theorigin of the observed signals. In Section V macroscopicmeasurements proved that, indeed, the permittivity of thematerial under investigation is nonlinear. The numericalvalues for the piezoresponse signal reported in Fig. 8 (thehysteresis loops of all three harmonics) do not allow oneto characterize quantitatively the material properties be-cause they were obtained at the cantilever resonance. Atmost, the ratios between the magnitudes of the three har-monics could be compared, provided that the tip/samplecontact junction as well as the cantilever did not changefrom one measurement to another. We emphasize that, atthe contact resonance, it is not the sample response thatincreases by 1–2 orders of magnitude, but the cantileverresponsivity to the forces driving it into oscillation.

In this section we shall estimate the contributions ofthe second and third harmonic components to the (firstharmonic) piezoresponse. Also, we will discuss alternativeorigins of the nonlinearity observed in the local measure-ments.

The electric field induced by the AFM tip inside theferroelectric is highly nonuniform being concentrated inthe proximity of the contact. This fact restricts the appli-cability of the one-dimensional model and requires exactcalculations of the electroelastic fields in the case of inden-tation of piezoelectric materials [38]. A simple parameterthat gives an idea of the electric field uniformity in thesample region below the AFM tip is the factor ξ equal toratio between the electric field at the surface (at the con-tact between tip and sample) and the electric field at thebottom electrode (on the same normal to the film plane).Using a simple electrostatic model [39] we calculated ξas a function of the tip radius, for the following parame-ters: permittivity of the film ε = 100 · ε0, film thicknessτ = 100 nm, potential at the surface Vsurf = 1 V. The tipradius was calculated by fitting the equipotential surfacetangent to the film surface with a sphere. To vary the tipradius, we modified the distance between the point chargeand the surface, and adjusted the charge value so that the


Fig. 10. Ratio between the electric field at the surface and the electricfield at the bottom electrode, below the AFM tip in piezoresponseforce microscopy, using a simple point charge model. A ratio equalto unity means uniform electric field throughout the film thicknessdirectly below the tip.

potential at the surface remained constant. It turned out,however, that the uniformity factor depends on the ratiotip radius/film thickness and not on the tip radius alone,or on the absolute value of the imposed Vsurf. Therefore,the result shown in Fig. 10 is valid for any set of quantities(Vsurf, thickness, tip radius). As expected, decreasing thepermittivity increases the uniformity (ξ closer to 1).

For a reasonable radius of the tip apex of ∼100 nm(equal to the film thickness) the parameter ξ ∼ 100 and,therefore, the electric field is quite far from being uniform.However, there are at least two more significant effects inPFM measurements that influence the permittivity and,therefore, the field uniformity. First, the mechanical stressdue to the contact force in the contact region is of theorder of Fcontact/(contact area) and easily reaches severalgigapascal under a relatively weak force of 100 nN. Thisstress decreases the permittivity of the material in the con-tact region, therefore, decreasing the parameter ξ accord-ing to the phenomenological relation, ∆(1/ε) = 2QX withX being the stress. Estimations of this effect using typ-ical material constants for thin films (electrostriction co-efficient Q = 0.04 m4/C2, ε/ε0 = 100) reveal a decreaseof permittivity by only ∼10% under typical measurementconditions with a medium-soft cantilever. Therefore, thisdecrease does not significantly affect the electric field dis-tribution. Another effect is the dependence of the permit-tivity on the electric field itself, as shown in Fig. 9(a). Thiseffect may become quite important at high fields, caus-ing a decrease by up to 50% of the permittivity, which inturn modifies the field distribution. For the case shown inFig. 10, we find that the parameter ξ decreases by a fac-tor of two (approximately), thus increasing the degree ofuniformity of the electric field below the AFM tip.

From the above discussion, we conclude that the max-imum possible decrease in permittivity due to the experi-mental conditions is not sufficient to achieve an acceptableξ < 10, such that the classic relations could be used to es-timate quantities involved in PFM. The one-dimensionalequations derived were only used as support to explain

TABLE IValues of the Physical Parameters Used for the Numerical

Estimations.

Physical quantity Value Unit

Load force (including adhesion), F0 100 nNTip apex radius, R 100 nmCantilever: Surface, length × width 200 × 40 µm × µm

Tip height 10 µmSample relative permittivity, ε 100 1Sample thickness, τ 100 nmSeparation between the spherical tip 1 nmand the sample surfaceYoung’s modulus: PZT 70 GPa

W2C 500 GPaPoisson’s ratio: PZT 0.3 1

W2C 0.24 1Spontaneous polarization PS , PZT 0.5 C/m2

Electrostriction coefficient Q, PZT 0.04 m4/C2

Third order permittivity [Fig. 9(b)], ε(3) 10−5 pF/VDC voltage, Vdc 1 VAC voltage, Vac 1 V

qualitatively the experimental observations. In the follow-ing we estimate the signals corresponding to the secondand third harmonics by taking into account the nonuni-formity of the electric field below the tip. Because we esti-mate the signal in the vicinity of zero bias field, we use afixed permittivity in the calculations of the field distribu-tion, and we use the linear field dependence given in (10)to calculate the harmonic components.

Because the measurements were done with soft can-tilevers, we assume that the piezoresponse contrast haselectrostatic origin (the so-called “electrostatic regime” de-fined in [12]). Indentation of the surface occurs under theload force F0 (including the adhesion force), and the elec-trostatic forces are acting on both cantilever and tip. Wehave used the simplified model for the tip-sample capaci-tance given in [12], using the parameters shown in Table I.Using the simple hertz model, we find that the static in-dentation into the sample surface is 0.35 nm, the contri-bution of the local force (i.e., electrostatic force acting onthe tip) being 0.24 pm (three orders of magnitude lower)the rest being attributed to nonlocal forces (static forceF0 + electrostatic force due to cantilever capacitance). Forthe first harmonic signal, we find a low response of 1.64 pm,with a local contribution (piezoresponse) of 1.3 pm. Thetotal second harmonic response is estimated at 0.41 pm,out of which 0.33 pm represents the local contribution.Surprisingly, for the third harmonic signal, we obtain notonly a local contribution of 0.3 fm, but also a nonlocalcomponent of 0.1 fm [the “surprise” coming from the factthat the nonlocal electrostatic contribution involves onlythe cantilever-sample capacitance, therefore, only ε0 inter-venes in (2)]. This is actually the effect of the—nonlinear—indentation itself and this induces higher harmonics in theresponse. The ratio between the first and second harmon-ics, therefore, is ∼4, and that between the first and thirdharmonics is ∼103, in disagreement with our results inFig. 8.


However, better estimations of the sample piezoelectricdisplacements were obtained using the one-dimensionalmodel and taking into account the nonuniformity of theelectric field along the z-direction using the formula:

hkω = ω/π

π/ω∫−π/ω

sin(kωt)dt

τ∫0

2QPSε(E)Edz,

with E = E(z) · sin(ωt), E(z) being calculated with thecharge model from [39]. The first harmonic signal is foundto be 32 pm, and the second and third harmonic piezo-electric displacements are found as 6.7 pm (ratio to firstharmonic ∼5) and 0.44 pm [i.e., two orders of magnitudelower than the first harmonic, close to the ratio measuredin Fig. 8(b)]. The simple electromechanical model seemsto describe the measurements somewhat better, althoughit predicts an unusually high second harmonic response.

VII. Conclusions

We have shown that the contact resonance of cantileverin scanning probe microscopy can be used to amplify smallpiezoelectric response signals and increase the sensitivityof the detection by several order of magnitudes; or alter-natively, to allow one to reduce the excitation voltage tomake sure to stay in a linear regime. The cantilever wasused as a selective amplifier to improve the signal-to-noiseratio in voltage modulated SPM. The cantilever resonancecan be used further to locally characterize nonlinear phe-nomena in ferroelectric materials. By tuning the frequencyof the excitation voltage such that the harmonic of inter-est is at the contact resonance of the cantilever, we wereable to both map and establish the local bias dependenceof the second and third harmonic electromechanical re-sponse. The second harmonic, associated with electrostric-tion, shows a bias dependence similar to that of the di-electric permittivity squared. The third harmonic exhibitshysteresis-like behavior similar to that of the spontaneouspolarization. Using simple one-dimensional calculations,we show that the origin of this hysteresis is the first or-der term in the expansion of permittivity as a function offield (ε(3)) or the nonlinear electrostriction. We also provefrom macroscopic measurements that, indeed, ε(3) exhibitsthe predicted hysteresis also at macroscopic scale, and notonly in piezoresponse experiments.

Acknowledgment

We thank the referees for their helpful suggestions thatcontributed to the quality of this work.

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Catalin Harnagea obtained his degree inphysics (1996) from the Alexandru IoanCuza University of Iasi, Romania, and hisPh.D. degree (2001) from Martin LutherUniversity of Halle, Germany. Between 1997and 2003 he worked at Max Planck Insti-tute of Microstructure Physics, Halle, study-ing the properties of ferroelectric thin filmsand nanostructures by means of voltage-modulated force microscopy.

He continued his research at Institut Na-tional de la Recherche Scientifique-Energie,

Materiaux et Telecommunications (INRS-EMT) near Montreal, Que-bec, Canada. His research interests include functional nanostruc-tures, molecular electronics, and scanning probe microscopy. Dr. Har-nagea has coauthored over 50 peer-reviewed publications, contribut-ing significantly to the understanding of the local electromechanicalmeasurements on ferroelectric films and the ferroelectric behavior ofnanostructures.

Alain Pignolet earned his engineeringphysics degree and his Ph.D. degree in physicsfrom Ecole Polytechnique Federale de Lau-sanne (EPFL), Switzerland, in February 1986and April 1992, respectively. Joining the IBMResearch Division (Thomas Watson ResearchCenter, Yorktown Heights, NY) as a Post-doc, he began working on high-permittivityrelaxor materials, and moved to the MaterialsResearch Laboratory, Penn State University,University Park, PA, to continue the research,working mainly with pulsed laser deposition

(PLD), reactive multitarget magnetron, and ion-beam sputtering.Returning to Europe in October 1994, Dr. Pignolet with a

research staff position at Max-Planck-Institut fur Mikrostruktur-physik, Halle/Saale, Germany, set up a laboratory for depositionof ferroelectric thin films using large area PLD. He also worked onthe optimization and better understanding of piezo-response forcemicroscopy.

Now again in North America, Dr. Pignolet has been since April2002 an associate professor at the University of Quebec’s ResearchInstitute INRS-Energie, Materiaux et Telecommunication, Montreal,Quebec, Canada, where he is the head of the Ferroıc Lab and is work-ing on the deposition and growth of thin films and nanostructuresof functional materials, together with their application in microelec-tronics, nanoelectronics, and integrated photonics.

Dr. Pignolet has over 70 publications in the best internationaljournals. He is also co-author of two book chapters on some aspectsof piezoresponse force microscopy, and he holds two patents.

Marin Alexe is with Max Planck Institute ofMicrostructure Physics, Halle, Germany. Heobtained a diploma in physics from Univer-sity of Bucharest, Bucharest, Romania, and aPh.D. degree in solid state physics from theInstitute of Atomic Physics, Bucharest.

His research interests cover nanoscienceand nanotechnology, ferroelectric nanostruc-tures and ultra-thin films, and physics andengineering of complex oxide thin films.

Dietrich Hesse is with Max Planck Insti-tute of Microstructure Physics, Halle, Ger-many. He obtained a diploma in physicsfrom Moscow State University, Russia, and aPh.D. degree in solid state physics from Mar-tin Luther University Halle-Wittenberg, Ger-many.

From 1992 to 1998 he was a member of theCoordinated Science Project “Solids far fromequilibrium,” at the University of Gottingen,Gottingen, Germany. Since 1998 he is an asso-ciated lecturer of solid state physics at Martin

Luther University Halle-Wittenberg, and since 2001 he is a memberof the Scientific Council of the Max Planck Society, Berlin, Germany.

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