Calibration of Piezoelectric Positioning ActuatorsUsing a Reference Voltage-to-Displacement TransducerBased on Quartz Tuning Forks

Andres Castellanos-Gomez,1,† Carlos R. Arroyo,1,† Nicolás Agraït,1,2,3 andGabino Rubio-Bollinger1,2*

1Departamento de Física de la Materia Condensada (C–III). Universidad Autónoma de Madrid, Campus de Cantoblanco,28049 Madrid, Spain

2Instituto Universitario de Ciencia de Materiales “Nicolás Cabrera”, Universidad Autónoma de Madrid, Campus deCantoblanco, 28049 Madrid, Spain

3Instituto Madrileño de Estudios Avanzados en Nanociencia, IMDEA-Nanociencia, 28049 Madrid, Spain

Abstract: We use a piezoelectric quartz tuning fork to calibrate the displacement of ceramic piezoelectricscanners that are widely employed in scanning probe microscopy. We measure the static piezoelectric responseof a quartz tuning fork and find it to be highly linear, nonhysteretic and with negligible creep. Theseperformance characteristics, close to those of an ideal transducer, make quartz transducers superior to ceramicpiezoelectric actuators. Furthermore, quartz actuators in the form of a tuning fork have the advantage ofyielding static displacements comparable to those of local probe microscope scanners. We use the staticdisplacement of a quartz tuning fork as a reference to calibrate the three axis displacement of a ceramicpiezoelectric scanner. Although this calibration technique is a nontraceable method, it can be more versatilethan using calibration grids because it enables characterization of the linear and nonlinear response of apiezoelectric scanner in a broad range of displacements, spanning from a fraction of a nanometer to hundredsof nanometers. In addition, the creep and the speed dependent piezoelectric response of ceramic scanners canbe studied in detail.

Key words: scanning probe microscopy, atomic force microscopy, scanning tunneling microscopy, force sensors,piezoelectricity, quartz tuning fork, microscopes, calibration standards, piezoelectric materials

INTRODUCTION

Since the invention of the scanning tunneling microscope~STM! ~Binnig & Rohrer, 1982! and the atomic force micro-scope ~AFM! ~Binnig et al., 1986!, most of the scanningprobe techniques make use of piezoelectric transducers toscan a tip over a surface with subatomic resolution. Anaccurate calibration of the displacements of these transduc-ers as a function of the applied voltage is necessary toobtain quantitative information about the topography ofthe surface under study, which has motivated the develop-ment of several procedures to calibrate the xyz displacementof the piezoelectric scanners such as optical interferometry~Riis et al., 1989!, imaging calibration specimens ~Alliataet al., 1996; Pal & Banerjee, 2000!, and others ~Vieira, 1986;Van de Leemput et al., 1991; Brodowsky et al., 1996!.

In this work we present a method to calibrate thedisplacement of piezoelectric actuators using the static piezo-electric deformation of quartz tuning forks. Although thiscalibration could be done using calibrated patterned sub-strates, the strong nonlinear behavior of the voltage-to-displacement relation over the full displacement range ofpiezoscanners would require a numerous set of such pat-

terned substrates of different length scales or periodicities.As the presented method relies on the comparison betweenthe displacement of the uncalibrated actuator and that of acalibrated quartz tuning fork, which is used as a reference,the calibration for a broad range of displacements can beeasily obtained. Moreover, unlike imaging calibration grids,this method enables calibration of the displacements in thez-direction for microscopes with limited or even withoutxy-scanning capabilities, such as the ones used in somescanning tunneling microscopy break junction experiments~Venkataraman et al., 2006; Hihath & Tao, 2008! or somenanoindenter setups.

We find quartz tuning forks very adequate when used asreference actuators because they meet most of the desirableproperties of an ideal voltage-to-displacement transducer:low aging, hysteresis and creep, and high linearity and ther-mal stability. Moreover, the tuning fork geometry guaranteesdisplacements of the reference voltage-to-distance trans-ducer in an appropriate range ~0.1–100 nm! to calibrate theactuators used in scanning probe microscopy. We have ap-plied the technique presented here to characterize the xyzdisplacements of the piezotube scanner of an AFM.

MATERIALS AND METHODS

While the primary application of quartz tuning forks istheir use as piezoelectric resonators, we characterize their

Received September 10, 2010; accepted December 2, 2011*Corresponding author. E-mail: gabino.rubio@uam.es†Present address: Kavli Institute of Nanoscience, Delft University of Technology,Lorentzweg 1, 2628 CJ Delft, The Netherlands

Microsc. Microanal. 18, 353–358, 2012doi:10.1017/S1431927611012839 Microscopy AND

Microanalysis© MICROSCOPY SOCIETY OF AMERICA 2012

static deformation and obtain their voltage-to-displacementcalibration to determine their adequacy as displacementstandards to calibrate the ceramic piezoelectric scanner oflocal probe microscopes.

Characterization of the Static Displacementof Quartz Tuning ForksCeramic piezoelectric actuators suffer from severe accuracylimitations when used as voltage-to-displacement transduc-ers such as the temperature dependence of their piezoelec-tric properties ~Vandervoort et al., 1993!, aging, nonlinearvoltage-to-displacement response, hysteresis, and creep~Van de Leemput et al., 1991!. These undesirable propertiesrelated to their ceramic structure are not expected forsingle-crystal piezoelectric actuators ~Rubio-Bollinger et al.,2004; Hayashi et al., 2008!. Mass production of quartzsingle-crystal actuators by the electronic and watch indus-tries makes them very reliable, affordable, and easy toobtain. Single crystal quartz, however, has a low piezoelec-tric constant, and thus the shape of the actuator has to becarefully engineered to ensure large enough displacementsto make the actuator useful for calibration of scanningprobe microscopes ~SPMs!. Quartz tuning forks, present inmost wristwatches, yield displacements in ranges from thesubnanometer to hundreds of nanometers and thereforecan be used as reference voltage-to-displacement transduc-ers to calibrate SPMs.

While quartz tuning forks are widely used in local probemicroscopy as high quality factor resonators to measureatomic scale forces ~Karrai & Grober, 1995; Giessibl, 1998;Smit et al., 2007; Castellanos-Gomez et al., 2010, 2012; Zechet al., 2010!, they can also be employed as static piezoelectricdisplacement transducers when a DC voltage is applied be-tween the tuning fork electrodes. To characterize the capabil-ities of quartz tuning forks as piezoelectric transducers, weuse a position sensor based on an AFM setup where thetuning fork is mounted in the sample holder. Once the tip ofa cantilever is in contact with the end of the tuning forkprong, the cantilever deflection follows the motion of thetuning fork. The cantilever deflection is measured fast andaccurately by means of a laser beam deflection setup. Fig-ure 1 shows the deflection of an AFM cantilever in contactwith the tuning fork prong as it is displaced in the z direc-tion ~Fig. 1a! or alternatively using the piezoelectric tubeactuator of the microscope ~Fig. 1b!. The cantilever deflec-tion versus displacement trace measured using the tuningfork actuator shows high linearity and negligible hysteresis~Fig. 1a!. In contrast, the motion of the microscope scannerdisplays a noticeable nonlinearity and creep ~Fig. 1b!. An-other benchmark to characterize the motion of piezoelectricactuators is to follow the time evolution of the displacementafter a sudden change of the applied voltage ~Fig. 1c!. Wefind that the displacement of quartz tuning fork actuatorsclosely follows the ideal steplike displacement, whereas themicroscope scanner transducer suffers from a strong creep,showing a continued displacement over time despite the ap-plied voltage being fixed. This adverse behavior, typical of

piezoelectric ceramics at room temperature, is undesirablefor experiments in which it is necessary to perform accuratedisplacements followed by measurement intervals withoutany movement such as spectroscopic measurements carriedout at different tip-sample distances.

Calibration of the Displacement of a QuartzTuning ForkTo use the quartz tuning fork as reference voltage-to-displacement transducer, one has to determine with accu-racy the deflection of the tuning fork prongs ~Dz! producedby a DC voltage ~VDC! applied to its electrodes. In previousworks this calibration of the tuning fork was done with anoptical interferometer ~Rubio-Bollinger et al., 2004!. Thistechnique, however, requires a dedicated setup capable ofmeasuring deflections in the nanometer range. An alterna-

Figure 1. a: Cantilever deflection as a function of the z axisdisplacement of the quartz tuning fork transducer prong. Thearrows indicate the direction of motion. b: Cantilever deflectionversus the z axis displacement of the microscope piezoelectric tubescanner. c: Actuator displacement versus time traces after theexcitation voltage is suddenly changed. While the quartz tuningfork displacement closely follows the ideal one ~dashed line!, thepiezotube actuator continues its displacement due to creep.

354 Andres Castellanos-Gomez et al.

tive approach can be used taking advantage of the highquality factor Q of the mechanical resonance of the tuningfork whose amplitude ~A! can be as large as several micronsfor an excitation voltage ~Vexc!. As the tuning fork behaveseffectively as a harmonic oscillator ~Castellanos-Gomez et al.,2009!, one can relate the motion of the prongs at resonanceto their static deflection as

Dz

VDC

�1

Q

A

Vexc

. ~1!

Note that although the Q factor can vary with theambient conditions such as the humidity, the product~A{Vexc

�1 !Q�1 and thus the DC calibration of the tuningfork do not change. This is because the DC calibration ofthe tuning fork only depends on the piezoelectric propertiesof the quartz and the geometry of the tuning fork and noton the ambient conditions. To ensure a right tuning forkcalibration, nevertheless, both the Q factor and the ratioA{Vexc

�1 should be measured under the same ambient condi-tions. This approach is convenient because of the largevariety of methods developed to measure the oscillationamplitude of the tuning fork prongs ~Qin & Reifenberger,2007; Simon et al., 2007; Liu et al., 2008!. Here we opt forthe optical inspection of the oscillation using a Canon EOS550D camera attached to a Nikon Eclipse LV-100 opticalmicroscope. We have found that the accuracy of the mea-surement of the oscillation amplitude improves using strobo-scopic illumination ~Castellanos-Gomez et al., 2009, 2011!.To do so, the tuning fork is illuminated with a light emittingdiode modulated at twice the excitation frequency. We ad-just the phase shift between the illumination and the excita-tion voltage to obtain an image that is the superposition ofthe two extremes of an oscillation cycle. Figures 2b–2d showan example of three optical micrographs of the end of atuning fork prong under stroboscopic illumination ~Fig. 2a!acquired for increasing excitation voltage amplitude. Therelationship between the oscillation amplitude of the prongsand the excitation voltage ~Fig. 2f! is linear with a slope of5,496 6 60 nm/V. The inset in Figure 2f shows the mea-sured resonance spectrum from which one can obtain aquality factor Q � 4,667 6 3. Using expression ~1! thecalibration of the tuning fork used as reference transducer isDz/VDC � 1.177 6 0.025 nm/V. Although this calibrationmethod is not traceable, its validity can be checked. This hasbeen done by comparing the DC calibration determinedwith the method proposed here with the one that we haveexperimentally measured with an optical interferometer~Rubio-Bollinger et al., 2004!. It is also important to empha-size that once a tuning fork is calibrated it can be used forseveral years because of the low aging of the quartz piezo-electric properties.

RESULTS

We use a calibrated tuning fork as a reference displacementtransducer to characterize the three axis ~xyz! displacementof the piezotube scanner of a local probe microscope. The

tuning fork is mounted as a sample, and its calibrateddisplacements are monitored by means of the feedbackcontrol loop of the microscope. In particular, we performthe calibration of the piezotube scanner of a Nanotec Elec-tronica ~Madrid, Spain! Cervantes AFM system operated inamplitude modulation AFM mode ~AM-AFM mode!. Thecalibration in the z axis is obtained by measuring theresponse of the feedback control loop for a well-defineddisplacement of the tuning fork in the z direction. Thecalibration in the x or y direction is determined from theanalysis of the shift of a topography profile when the tuningfork is deflected in the x or y direction.

Calibration of the z Displacementof a Piezotube ScannerTo calibrate the z displacement, the reference tuning fork ismounted on the AFM sample holder so that the displace-ment of the prongs is in the z direction ~Fig. 3a!. The AFMtip can be readily positioned at the end of the tuning forkprong with the help of a long working distance opticalmicroscope attached to the AFM and two micrometer screwsthat change the xy position of the cantilever with respect tothe sample. Note that most commercial AFMs have xypositioning capability as well as an optical microscope.Then the amplitude modulation AFM mode ~AM-AFM

Figure 2. a: Schematic diagram of an image of an oscillatingquartz tuning fork under stroboscopic illumination. The image isthe superposition of two oscillation instants phase shifted by 1808.b–c: Optical micrographs of the end of a tuning fork prong understroboscopic illumination with excitation voltage of Vexc � 0.7, 1.8,and 3.2 V e: Detail of the region marked with a rectangle in panel c.d: Measured oscillation amplitude of the prong as a function ofthe excitation voltage and a linear fit ~dashed line!. Inset in panel e:Resonance spectrum and a Lorentzian fit ~dotted line!.

Piezoelectric Tuning Fork as a Calibration Tool 355

mode! is used to keep the tip-prong distance constant. Todetermine the z axis calibration, we record the voltage,applied to the z electrode of the piezotube, necessary tokeep constant the tip-prong distance while we establish awell-defined deflection of the tuning fork prong. The rela-tionship between the prong deflection and the piezotubevoltage is rather linear ~correlation coefficient R2 � 0.997!for a range of 65 nm. From the slope of this relationship,we determine the z axis calibration of the piezotube 10.6 60.6 nm/V. This value is in good agreement with the calibra-tion 11 6 1 nm/V obtained by measuring the height ofmonoatomic steps in graphite ~HOPG!.

A more detailed analysis of the residues of the linear fitshown in Figure 3b reveals that even for these moderatedisplacements of the piezo there is a nonnegligible nonlinearbehavior. A quadratic fit to the data gives a calibration forthe piezo Dz~V ! � �0.31V 2 �10.6V ~with Dz in nm and Vin volts!, with the uncertainty of the coefficients 60.05 and60.6, respectively, and the correlation coefficient R2 �0.9999.

In the configuration shown in Figure 3a, the tuningfork can also be used to adjust the feedback parameters of

the microscope. This can be done by monitoring the outputof the feedback loop while a low frequency ~1–10 Hz!triangular-shape signal is applied to the tuning fork elec-trodes. Then, both proportional and integral gains can beadjusted until the feedback output reliably follows thetriangular shape of the tuning fork displacement. By apply-ing a square signal instead of a triangular signal, the timeresponse and the creep of the piezo scanner can becharacterized.

Calibration of the xy Displacement of aPiezotube ScannerThe calibration of the xy displacement of the piezotubescanner is carried out mounting the reference tuning forkon the AFM sample holder with the axis of the deflection ofthe prongs parallel to the x displacement of the piezotube~Fig. 4a!. The AFM tip is positioned at the end of one prongof the tuning fork, and the AM-AFM mode is used to keep

Figure 3. a: Schematic diagram of the experimental setup used tocalibrate the z axis displacement of the piezotube scanner of anAFM. b: Relationship between the deflection of the tuning forkprong and the voltage applied to the z sector of the piezotube tokeep the tip-prong distance constant.

Figure 4. a: Optical micrograph of the end of a tuning fork prongwith the AFM cantilever positioned on top. The small misalign-ment between the axis of the deflection of the prongs and the xaxis of the piezotube has been taken into account to calibrate thepiezotube. Inset in panel a: Schematic of the tuning fork arrange-ment used to calibrate the x displacement of the AFM piezotube.b: Two consecutive topographic profiles of the prong surfacemeasured before ~solid blue trace! and after ~dashed red trace!establishing a static deflection Dx � 23.46 nm of the tuning forkprong.

356 Andres Castellanos-Gomez et al.

constant the tip-prong distance. The x axis calibration ofthe piezotube can be obtained by acquiring a topographicprofile of the prong surface before and after establishing awell-defined displacement of the tuning fork prong. Fig-ure 4b shows two consecutive topographic profiles mea-sured before and after a displacement of the prong of Dd �23.55 6 0.06 nm. The misalignment between the movementof the piezotube scanner and the prong of the tuning forkhas also been taken into account, resulting in an x displace-ment of the prong Dx � Dd{sin~858! � 23.46 6 0.06 nm.The two profiles are shifted by 0.85 6 0.02 V in the voltageapplied to the x electrode of the piezotube, which gives acalibration of the piezotube of 27.6 6 0.7 nm/V. We findthat this calibration is in agreement with the one obtainedby imaging a microfabricated AFM calibration gratinga

~28 6 1 nm/V!.To study the nonlinear response in xy of the piezoelec-

tric scanner, one can repeat the process described abovewith increasing displacements of the tuning fork prongs.The nonlinear response of the piezo scanner can be ob-tained from the relationship between the displacements ofthe tuning fork prong and the apparent shifts of the topo-graphic profiles. Another possibility would be recording anAFM image without movement in the slow scan axis while alow-frequency triangular wave is applied to the tuning forkelectrodes. In this way one would have a collection oftopographic line profiles shifted from each other followinga triangular shape form. From these profiles one can extractthe voltage-to-displacement relationship in the x or y direc-tions for a broad range of displacements.

CONCLUSIONS

We have characterized the static displacement of piezoelec-tric quartz tuning forks to determine their adequacy asreference voltage-to-displacement transducers. We havefound that, due to their single-crystalline structure, quartztuning forks perform like ideal piezoelectric transducers.Their high voltage-to-displacement linearity and lowhysteresis and creep make these quartz actuators superiorto conventionally used piezoelectric ceramic actuators.Additionally, quartz actuators in the form of tuning forkcan yield displacements in the range required in SPMs ~0.1to 100 nm!. We have presented a method to calibratepiezoelectric scanners, used in scanning probe microscopy,by comparing the displacement of an uncalibrated actua-tor with one of a calibrated tuning fork. We have appliedthis versatile method to characterize the three axis displace-ment of the piezotube scanner of a commercial AFM.There are advantages when employing a tuning fork as acalibrated voltage-to-displacement reference instead ofusing calibration grids. On the one hand, the displace-ment range can be selected, according to the size of the

structures to be characterized, in a broad range spanningfrom a fraction of a nanometer to hundreds of nanometers.On the other hand, a detailed study of the ceramic piezo-electric scanner creep and the speed dependent calibrationcan be performed for experiments that require a precisepositioning of the tip of a microscope. In addition, thetuning fork itself can be used as a supplement in a SPM toperform well-controlled displacements without noticeablecreep.

ACKNOWLEDGMENTS

A.C-G. acknowledges fellowship support from the Comu-nidad de Madrid ~Spain!. This work was supported byMICINN ~Spain!, MAT2008-01735 and MAT2011-25046and CONSOLIDER-INGENIO-2010 “Nanociencia Molecu-lar” CSD-2007-00010 and Comunidad de Madrid “Nano-biomagnet” S2009/MAT-1726.

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