Low temperature and high magnetic field performance of a commercial piezo-actuator probed via laser interferometry
LLow temperature and high magnetic field performance of a commercial piezo-actuator probed via laserinterferometry
R. Adhikari, a) K. Doesinger, P. Lindner, B. Faina, and A. Bonanni b) Institut f¨ur Halbleiter-und-Festk¨orperphysik, Johannes Kepler University, Altenbergerstr. 69, A-4040 Linz,Austria (Dated: October 23, 2020)
The advances in the fields of scanning probe microscopy, scanning tunneling spectroscopy, point contactspectroscopy and point contact Andreev reflection spectroscopy to study the properties of conventional andquantum materials at cryogenic conditions have prompted the development of nanopositioners and nanoscan-ners with enhanced spatial resolution. Piezoelectric-actuator stacks as nanopositioners with working strokes > µ m and positioning resolution ∼ (1-10) nm are desirable for both basic research and industrial appli-cations. However, information on the performance of most commercial piezoelectric-actuators in cryogenicenvironment and in the presence of magnetic fields in excess of 5 T is generally not available. In particular,the magnitude, rate and the associated hysteresis of the piezo-displacement at cryogenic temperatures arethe most relevant parameters that determine whether a particular piezoelectric-actuator can be used as ananopositioner. Here, the design and realization of an experimental set-up based on interferometric tech-niques to characterize a commercial piezoelectric-actuator over a temperature range of 2 K ≤ T ≤
260 K andmagnetic fields up to 6 T is presented. The studied piezoelectric-actuator has a maximum displacement of30 µ m at room temperature for a maximum driving voltage of 75 V, which reduces to 1 . µ m with an absolutehysteresis of (9 . ± .
3) nm at T = 2 K. The magnetic field is shown to have no substantial effect on thepiezo properties of the studied piezoelectric-actuator stack. I. INTRODUCTION
The development of precise nanopositioning systemswith spatial resolution ∼ (1-10) nm and time constant ∼ (10-100) µ s is relevant for both basic and applied re-search, as well as for industrial applications . The po-sitioning resolution of conventional actuator systems in-cluding hydraulic and ac/dc motors is too coarse for mostmodern technologies, even though these actuators areable to provide large output force and working strokes.The working stroke of an actuator is defined as its lineardisplacement under dynamic conditions. With the recentdevelopment of actuators based on piezoelectric materi-als i.e. piezoelectric-actuator (PEA), it is now possibleto achieve spatial resolution of a few nm . The appli-cation of PEA is wide spread in basic research fields andin industrial sectors, from high resolution scanning probemicroscopy (SPM) , to optical systems for astronomy and aerospace industry . The working stroke of a sin-gle piezoelectric element is generally limited to few µ meven at room temperature (RT). Alternative approacheshave improved the working strokes of hybrid PEA tofew centimeters , but for most commercial PEA, the lowworking strokes limit their applications.A major application area for the PEA is repre-sented by scanning probe measurement systems suchas atomic force microscopy (AFM) , scanning tunnel-ing microscopy (STM) , and scanning tunnel-ing spectroscopy (STS) . The recent developmentsin the fields of STM , STS and in particular of a) Electronic mail: [email protected] b) Electronic mail: [email protected] point contact spectroscopy (PCS) including point con-tact Andreev reflection spectroscopy (PCAR) ,have underlined the relevance of PEA for nanoposi-tioning applications. With the emergence of new fam-ilies of quantum materials encompassing topologicalinsulators , topological crystalline insulators , topo-logical superconductors , Weyl and Dirac semimetals and unconventional superconductors including heavyfermionic systems , PCS and PCAR have proven tobe efficient spectroscopic tools to study these materialsystems . Recent investigations of topological crys-talline insulators like Pb − x Sn x Se and Pb − x Sn x Te havepointed at the presence of Majorana fermion-like exci-tations at the atomic steps of the epitaxial layers .Most PCS and PCAR set-ups reported in literature are static, with the sample kept fixed while the tipis the only dynamic component . Therefore, by in-troducing a dynamic mode to the static PCAR set-up, a lateral degree of freedom in the sample plane isadded and makes it possible to map the sample sur-face. This improvement is expected to open new per-spectives for the characterization of quantum materi-als. In particular, the use of PEA-based nanoposition-ing systems along the sample plane promises to facilitatethe mapping of surfaces of bulk specimens, thin filmsand of 2D layers such as (beyond-) graphene systems .The scanning mode PCAR or scanning PCAR (SP-CAR) is an exclusive tool for mapping exotic quantumphases and phenomena like Majorana fermions , weaklink Josephson effect , Meissner and mixed phases ofconventional and unconventional superconductors, oddfrequency superconductivity and Yu-Shiba-Rusinovstates in magnetically doped superconductors .The scanning degree of freedom of any physical probe1 a r X i v : . [ c ond - m a t . m t r l - s c i ] O c t round V pp piezo m k l l kmm+k l U (a) (b)(c) (d)(e) (f) S E E C S E E max Figure 1. (a) Schematic of a PEA stack where multiple PZT layers are mechanically switched in series and electrically inparallel configuration. (b) Commercial PEA stack Model:PK2JUP2 from Thorlabs – considered in this work. (c) PEA stack:unloaded, mass-loaded and spring-loaded . (d) Hysteresis loops for an unloaded piezo, for a spring-loaded piezo, for a mass-loaded piezo and for a simultaneously spring- and mass-loaded piezo. (e) Strain S in a piezo-element as a function of appliedelectric field E . (f) Operation of a PEA stack when operated under the conditions indicated by the rectangle in Fig. 1(e). can be accessed by employing either an electromechan-ical motor or a PEA stack. For applications at cryo-genic temperatures, the PEA stacks are preferred overelectromechanical motors, since their compactness andphysical dimensions favor the integration with state-of- the-art cryostats. In addition, PEA stacks can be con-trolled by using high precision electronics. However, onemajor challenge in employing PEA stacks as scannersis represented by their limited displacements at cryo-genic temperatures. For most commercial PEA stacks,2he absolute displacement reduces with temperature T and can shrink to (5-10)% of the maximum displacementat RT. Most available PEAs have a maximum displace-ment (20-30) µ m for an applied voltage of (75V-150V)at RT. Moreover, creep and ferroelectric hysteresis as-sociated with a PEA can be detrimental to the applica-tion of commercial PEA stacks in cryogenic regime. Thepiezo-hysteresis as a function of T determines whethera particular PEA stack can be employed as a positionscanner with nm resolution, in particular in techniquesfor mapping surfaces, like STM and SPCAR. Minimal orzero hysteresis of the piezo-scanner is desirable for bothspectroscopic and microscopic mapping of surfaces. How-ever, for most commercial PEA stacks, specifications ofthe PEA stacks at cryogenic conditions are generally notavailable.The absolute displacements of PEAs can be measuredby mechanical, electrical or optical means. Due to en-hanced signal-to-noise ratio, optical techniques are ad-vantageous over mechanical or electrical sensors such asflexural hinges and strain gauges. In addition, light-based measurements are compatible with cryogenic ap-plications, which can pose challenges to strain gauge orflexural hinge based sensors . Optical interferometrictechniques have been successfully applied for measure-ment of nanometer level displacements such as those re-lated to the detection of gravitational waves . Here,an indigenously designed and fabricated interferometricset-up for the estimation of displacement, hysteresis andcreep of a commercial PEA stack are reported. The mea-surements have been carried out over the temperaturerange 2 K < T <
260 K, both in the absence and as afunction of an applied magnetic field. The absolute dis-placement of the PEA stack is evaluated to be 25 . µ mat RT, while for T = 2 K an absolute displacement of1 . µ m has been estimated for the maximum allowedvoltage of 75 V. Both the displacement and hysteresisof the PEA stack are found to decrease with the T andan absolute residual hysteresis of (9 . ± .
3) nm is esti-mated at T = 2 K. II. PIEZO-ACTUATORS: FUNDAMENTALSA. Piezoelectricity: sensors and actuators
If mechanical stress is applied to a piezoelectric ma-terial, an accumulation of opposite electric charges oc-curs at the opposite faces of the specimen. The effectis related to the formation of electric dipole moments insolids and is known to take place in crystalline systemswith inversion asymmetry and in polar materials. Thepolarization density (cid:126)P for such systems is the sum of thedipole moments over the volume of the unit cell. In thepiezoelectric effect, the applied mechanical stress leadsto a change in (cid:126)P and the phenomenon of piezoelectric-ity depends on the orientation of (cid:126)P within the crystal, on the spatial symmetry of the lattice and on the mag-nitude of the applied mechanical stress. Out of the 21crystal space groups that are noncentrosymmetric, bar-ring the cubic class, all other 20 crystal classes exhibitdirect piezoelectricity. Half of these 20 space group crys-tals are known to be polar crystals showing spontaneouspolarization, i.e. for the polar crystals (cid:126)P (cid:54) = 0 in theabsence of an external mechanical stress. However, fornon-polar crystals, the piezoelectric behavior is triggeredonly upon application of external stress. The applicationof an external mechanical stress to any piezoelectric ma-terial leads to the accumulation of charges, which in turngenerates an electric voltage .Following the Onsager’s reciprocity theorem , theinverse piezoelectric effect in which an applied electri-cal voltage leads to a mechanical strain also emerges.The existence of an inverse piezoelectric effect was postu-lated by Gabriel Lippmann in 1881 and was subsequentlydemonstrated by the Curie brothers . The first applica-tions of inverse piezoelectric effect were in ultrasonic sys-tems for underwater imaging and communication. Theimplementation of sensors and actuators based on thepiezoelectric effect is widely found in force and acceler-ation sensors, microphones, 3D printing technology, andmedical technology B. Piezo-actuators: materials
Most of the current piezo-actuators are produced fromdonor doped lead zirconium titanate (PZT), which hasa ferroelectric transition at T = 623 K. and it is cate-gorized as a soft piezoelectric material . The PZTs aregenerally optimized for: (i) a large coupling factor, whichis a parameter defined as the ratio between the mechan-ical energy generated by the piezo and the electrical en-ergy (due to the applied voltage and the piezo-capacity);(ii) the charge coefficient, i.e. the ratio between the straindeveloped in the actuator and the applied voltage . Byachieving significantly high coupling factors and chargecoefficients, the maximum displacement of PZT basedactuators can be as high as 100 µ m at RT.The schematic of a commercial PEA stack, like the onestudied in this work , is shown in Fig. 1(a). An advan-tage of employing a stack of PZT elements compared to asingle PZT chip is that the maximum displacement is notlimited by the material. In the case of the stacked chiparchitecture of PEAs, every alternate PZT chip, is sin-tered with silver and connected to every alternate layerforming two electrodes for the application of a voltage.For an ac voltage, the polarization of the PZT is alsoalternated. A photo of the particular PEA stack used inthis work is provided in Fig. 1(b).3 igure 2. Sketch of the experimental set-up: A piezo-stage inside a cryostat shifts vertically a mirror. The resultingdisplacement is measured using an interferometer. The whole set-up is vibration damped. C. Piezo-actuators: mechanical and thermalproperties
Most piezoelectric materials are prone to mechan-ical damage due to the brittle nature of ceramiccompounds . In the case of PZT, the ceramic has acompressive strength of ∼
250 MPa. But at ∼
10 MPaa mechanical depolarization of the material occurs, af-fecting the lifetime of the PZT-based actuator. For thisreason, the PEA is generally preloaded as little as pos-sible to ensure that the tensile strength of the piezo isnot exceeded during dynamic operation. The tensilestrength of a material is defined as the stress that thematerial can withstand while being stretched or pulledbefore failure. For PZT-based piezo-actuators the tensilestrength is estimated to be ∼ (5-10) MPa. The three cat-egories of preloads generally employed for the dynamicoperation of a PEA, namely: (i) mass-loaded, (ii) spring-loaded and (iii) (mass+spring)-loaded are reported inFig. 1(c) together with the unloaded PEA stack as refer-ence. A schematic diagram of the corresponding piezo-displacements under the above mentioned preloads is given in Fig. 1(d). When the PEA is loaded with a mass,it shifts by a constant length, the amplitude of which de-pends on the load mass and on the piezo-stiffness. Thiscategory of PEA displacement is represented by the mass-loaded loop in Fig. 1(d). For the spring-loaded loop, at U = 0 V the spring does not apply any force to the piezoand thus the spring-loaded piezo has the same length l as the unloaded one. With increasing piezo length,the spring force also increases, resulting in a relativelysmaller steepness of the spring-loaded loop in compari-son to the unloaded loop. D. Piezo-actuators: Hysteresis
The hysteresis of the piezo-displacement as a functionof the applied electric field is relevant for piezo-actuatorsand must be considered before choosing it as a nanopo-sitioner or nanoscanner in scanning probe set-ups .The typical hysteresis behavior of a PEA as a function ofan applied external electric field E is shown in Fig. 1(e).The polarization of the PEA is antiparallel w.r.t E , re-4 c)(d)(a) (b)1 10 100 10000.20.40.60.81.0 Frequency (Hz) A m p li fic a t i on ( a r b . un i t s ) (e) X1-1X1-2X1-3X1-4Power SupplyV+V-PEGND C1100 nFV++V+ - X3-1X3-2IN GND C2U132 +IN-IN + LTC6090CS8E-5 V+V-74 L E D OUT X2-1X2-2X2-3GNDPER1R2R3R4R5R6R7R9R8
Figure 3. (a) Assembled piezo-amplifier driven by a myDAQ. (b) Elements of the piezo-amplifier circuit: (1) mains filter, (2)power supply board, (3) power supply stabilization, (4) amplifier board, and (5) electrical connectors. (c) Circuit diagram ofthe piezo-amplifier. (d) PCB layout of the piezo-amplifier. (e) Amplification of the piezo-amplifier as a function of the drivingfrequency. sulting in a reduction of the strain with increasing E .The electric field is increased from 0 to the critical field E C along the curve 1. At E = E C , the polarization flipsgiving rise to a rapidly increasing strain, as indicated bycurve 2, until a saturation is reached for E (cid:29) E C . Fordecreasing E , the strain decreases until E = 0 wherethe polarization is again antiparallel to E . This behav-ior of the PEA is represented by the curves 3 and 4. However, at E = − E C , the polarization flips again andthe strain increases and saturates, according to curve 5.A decrease in E results in reduction of the strain until E = 0, where the segment 6 closes the loop for the en-tire cycle of E . When a piezoelectric material is used asPEA, the flipping of the polarization is generally inhib-ited by not applying the maximum voltage range allowedfor the material. This operation is shown in Fig. 1(f),5hich represents a closed loop operation of the segments3 and 4 in Fig. 1(e). When the electric field is applied in acyclic operation from 0 to a maximum E max , a hysteresisis observed in the resulting strain of the piezo-actuator.The hysteresis is defined as: H = [∆ S ] E max2 [ S ] E max (1)where ∆ S is the piezo-displacement. For an unipolar op-eration of the piezo-actuator, the applied electric fieldis always positive, as shown by the loop represented bythe dotted lines in Fig. 1(f), while the outermost loop,represented by the solid line, is an example of a bipolaroperation. A bipolar operation results in an enhanceddisplacement range as compared to an unipolar opera-tion, but also in a broader hysteresis, which represents adrawback for applications requiring nanopositioning. III. DEVELOPMENT OF THEEXPERIMENTAL SET-UP
An in-depth account of the design and fabrication ofthe interferometry set-up for measuring the PEA dis-placement as a function of T and µ H is presented in thissection. The experimental assembly comprises: (i) theelectronics and (ii) the mechanical block. The electron-ics segment includes the piezo-amplifier, the photodiode-amplifier and the connector board. The components ofthe mechanical block are: the interferometer, the piezo-stage and the vibration damping assembly. A schematicrepresentation of the experimental set-up is shown inFig. 2. The complete experimental assembly is designedon a 1.0 m long hollow cylindrical sample holder rod(SHR) made out of G10 fiber material. The SHR is de-signed to fit into a Janis Super Variable Temperature7TM-SVM cryostat equipped with a 7 T superconduct-ing magnet. On one end of the SHR, a mechanical stageis attached which accommodates the PEA on a stage(piezo-stage). The piezo-stage is positioned at the centerof the 7 T superconducting solenoid magnet, as shown inFig. 2. On the other end of the SHR, the optical assemblyfor the Michelson interferometer and for the photodiode-detector is mounted. The SHR connects the interferome-ter platform to the piezo-stage in a stiff manner. All wiresfor the electrical connections to the piezo-stage are placedwithin the hollow SHR. The entire cryostat assembly isplaced on an indigenously designed vibration dampingmechanism in order to reduce ground vibrations. A. The electronics block
The electronics block of the experimental set-up con-sists of a piezo-amplifier used as the power source forthe PEA stack, the photodiode-amplifier for detecting (a)(b)(c)
100 1000 100000.60.81.0 Frequency (Hz) P ho t od i ode a m p li t ude ( a r b . un i t s ) Figure 4. (a) Photodiode-amplifier board in the groundedhousing. (b) Circuit diagram of the photodiode-amplifierboard. (c) Frequency response of the photodiode-amplifier. the signal from the interference fringes and an electron-ics connector board. All the electrical and electronic de-vices designed and used here are controlled via a myDAQfrom National Instruments. The myDAQ is connected toa computer through an Universal Serial Bus (USB). Byusing the myDAQ, the displacement of the PEA stack canbe estimated from the output voltage of the photodiode-amplifier. The piezo-displacement is controlled by apply-ing an analog ac voltage to the piezo-amplifier input.6 a) (b) Figure 5. (a) Connector board and the myDAQ. (b) Circuit diagram of the fabricated connector board.
1. The PEA stack
The PEA stack used in this work is a commercial dis-crete low voltage PEA stack with a 75 V of maximumload voltage . The zero load response time of the PEAstack lies in the sub milliseconds range and a free strokedisplacement of 30 µ m at RT. The stack consists of multi-ple 75 V maximum voltage based piezoelectric chips fab-ricated from sintered PZT ceramic. The PEA stack issuitable for open-loop circuit set-ups and for originalequipment manufacturers. The stack is equipped withpre-soldered wires and with flat ceramic plates for easyintegration into an experimental set-up.
2. Piezo-amplifier
As mentioned in the previous subsection, the PEAstack chosen for this work has an operating voltage rangeof (0-75) V. However, the maximum output voltage fromthe analog output of the myDAQ is limited to ±
10 V.Hence, an amplifier is necessary to exploit the maximummovement range of the PEA stack. In order to achive aspatial resolution as low as ∼ ≤
25 mV, which is the minimum voltage requiredfor a piezo-disaplacement ∼ ac voltage from the mains filter into anunstabilized dc voltage. Using the stabilizer 3, the dc voltage is stabilized to provide a low noise supply volt-age for the amplifier itself. The amplifier board indi-cated as 4, is in turn controlled by the input and out-put connectors labelled as 5. The circuit diagram andthe printed circuit board (PCB) layout of the ampli-fier are shown in Fig. 3(c) and Fig. 3(d), respectively.The amplifier realized here is an electrometer amplifierbased on the low-noise-high-voltage operational amplifier(OPAMP) LTC6090. By designing the electronic circuitof the amplifier so that the amplifications is stable with T , the thermal drifts and fluctuations of the amplifiercan be kept at a minimum. The thermal stability of theamplifier is achieved by using eight nearly equal resis-tors of ∼
17 kΩ as feedback voltage dividers, as shownin Fig. 3(c). This circuit arrangement ensures that thesame power is dissipated by each resistor and that thereis no thermal gradient in the feedback voltage divider.Therefore, the ratio between the output and input volt-ages of the feedback voltage divider are independent of T . The purpose of this amplifier is to control the PEAstack, which is a capacitive load. A capacitive load onthe output of an OPAMP introduces a negative phaseshift, thereby reducing the phase margin and turning theamplifier into an oscillating amplifier. In order to makethe circuit stable, a lead compensation (R1,C2) adds apositive phase shift to the curcuit. In the experiments,the PEA stack is driven at frequencies f <
25 Hz, andtherefore no cooling of the OPAMP is required. Thefrequency response of the piezo-amplifier is reported inFig. 3(e), where the amplification of the circuit is mea-sured by sweeping f from 1 Hz to 5000 Hz. The amplifierbandwidth is estimated to be 600 Hz. Since the PEAstack is operated at f = 25 Hz, the measured bandwidthof 600 Hz is sufficient for the application of the PEAstack as nanopositioner and nanoscanner. An increase inbandwidth greater than 600 Hz would augment the noiseof the amplifier. A noise level ∼ µ V RMS has been es-7 igure 6. (left) Schematic of the interferometer inside the cryostat showing the mechanical and optical components. (right)The photographs of the optical parts are also shown. timated for the piezo-amplifier. Both the stabilizer andthe amplifier are placed inside a grounded tin-plated steelbox for shielding against external electromagnetic radia-tion.
3. Photodiode-amplifier
Photodiodes are used as the detectors in light based in-terefence experiments where the interference of twocoherent light beams produces concentric and alternatebright and dark fringes. In general, the intensity of lightat a particular position of the interference pattern is mea-sured by a photodiode designed to operate at the wave-length of the used light source. In the set-up describedhere, the photodiode is connected to the amplifier circuitboard. The entire photodiode assembly is placed insidean electrically grounded housing and is connected to theconnector board through two shielded cables, as shownin Fig. 4(a). The photodiode-amplifier converts the op-tical intensity pattern into an electrical signal. The cir-cuit of the photodiode-amplifier is reported in Fig. 4(b).The photodiode-amplifier is designed based on a singleLMC6081 OPAMP which has a low input bias current. The circuit converts the photocurrent of the photodiodeinto a voltage. The amplification is determined by theresistor R3, while the capacitor C1 ensures stability ofthe circuit by limiting the bandwidth. Since the max-imum supply voltage of LMC6081 is 16 V, the ±
15 Voutput voltage from the myDAQ, must be lowered bythe two Zener diodes D1 and D2 and the resistors R1and R2, as shown in Fig. 4(b). The frequency responseof the photodiode-amplifier is measured by illuminatingthe photodiode with a modulated light beam. The ampli-tude of the output signal as a function of the modulatingfrequency is reported in Fig. 4(c). The estimated criticalfrequency of 10 kHz fits well with the chosen values ofthe circuit components R3 and C1, which are 1 MΩ and12 pF, respectively.
4. Connector board
A connector board serves as the hub between thepiezo-amplifier, the photodiode-amplifier and the my-DAQ and is shown in Fig. 5(a). The connector boardis directly plugged into the connectors of the myDAQ.The circuit diagram of the fabricated circuit board used8 a) (b) (c) Figure 7. CAD sketch of the piezo-stage: (a)-(1) base, (2) slide, (3) springs pressing the slide to the base, (4) holder forM1, (5) heater. (b)-(6) piezoelectric-actuator, (7) ceramic case, (8) hemispherical end, (9) tightening block, (10) slide mount.(c)-(11) ball-bearing, (12) upper and lower tracks. in this work is reported in Fig. 5(b). The ±
15 V out-put of the myDAQ is directly connected to the termi-nal ( ±
15 V OUT) for powering the photodiode-amplifier.The output of the photodiode-amplifier is connected tothe PHOTO AMP IN terminal on the connector board.The output signal amplified by a factor of 11 is achievedby using a OPA2277 dual OPAMP in order to make use ofthe full range of the myDAQ’s analog-to-digital-converter(ADC) while maintaining a bandwidth of 100 kHz. Theinput and output of the piezo-amplifier are connectedto the PIEZO AMP OUT and to the PIEZO AMP INterminals, respectively. The output signal of the piezo-amplifier is attenuated by a factor of 11, buffered by theOPAMP and read by the second channel of the ADC.The PEA stack is connected to the PIEZO OUT termi-nal, which is switched parallel to the PIEZO AMP IN.The output noise of the myDAQ is further amplified bythe piezo-amplifier. The total noise of the electronic sys-tem is reduced by a low pass filter in the connector board.The critical frequency of the low pass filter is set to 100Hz, i.e. far above the excitation frequency of 25 Hz usedin this work. Any additional phase shift introduced bythe filter does not play a significant role, as the piezo-voltage is directly read using the PIEZO AMP IN con-nector. The electronics is controlled by an indigenouslydeveloped LabView program and all data are collected via a computer interface.
B. The interferometer
The detection of a mechanical displacement ∼ .In this work, a customized Michelson interferometer hasbeen designed and fabricated and the schematic of the set-up is reported in Fig. 6. The coherent light sourceused for this experiment is a He-Ne laser with a wave-length of 632.8 nm, which is coupled to a single modeoptical fiber. All optical components used in the devel-opment of this experimental set-up have been obtainedfrom Thorlabs . As shown in Fig. 6, the single modefiber collimates the beam towards the beamsplitter BS1,which splits the incoming laser beam into two beams,each with half the intensity of the original beam. One ofthe resulting beams is directed downwards into the cryo-stat through the window W1, fabricated from a N-BK7glass with anti-reflective coating. This beam is the pri-mary measurement beam. The reference beam is the onewhich passes through BS1. The primary beam is thenreflected by the mirror M1, mounted on the piezo-stage.The reflected primary beam passes through W1 and BS1and mirror M6 is used to reflect and guide the beam tothe photodiode. The reference beam follows the pathM5 → M4 → M3 → M2 and it is retroreflected by M2 backto BS1, as shown in Fig. 6. The beamsplitter BS1 thensplits the retroreflected reference beam, which is super-imposed with the primary beam. An intereference of thetwo beams occurs if all the necessary conditions for opti-cal interference are satisfied, resulting in alternate brightand dark circular fringes observed at the photodiode. Acondition for intereference is that the path difference be-tween the measurement and the reference beam is shorterthan the coherence length of the laser. In order to en-sure the conditions for interefence, the mounting heightof the mirror M2 – which is located outside the cryostat– must be adjustable. The coherence length of the He-Ne laser used here is estimated to be ∼
150 mm. In thisparticular experiment, the laser beams travel distances ofseveral meters. Thus, angular errors of even few tenthsof a degree in the optical arrangement can lead to large9eviations of the beams, to misalignment and eventuallyto the disappearence of the interference patterns. For ef-ficient manipulation and alignment of both the primaryand reference beams along with the reflected ones, eachoptical component used in this set-up is designed to beadjustable. The adjustments of the respective opticalcomponents are achieved using suitable screws markedby SN with N=1,2,...,11, as reported in Fig. 5. The colli-mator, BS1, M5 and M6 are mounted on the same stagewhich in turn can be rotated freely around a pivot point.The screws S4, S1 and S2 are used to rotate the assem-bly around the x − , y − and z − axes, respectively, whilethe screw S3 is used exclusively for a linear translation ofthe stage along the x − axis. The details of this assemblyare also shown in Fig. 6. On the other hand, the modulecontaining the photodiode and the photodiode-amplifiercan be manipulated and moved along the y − and the z − axes with S5 and S6, respectively. The screws S7, S8 andS9 are employed to tilt the mirror M2 used for retroreflec-tion of the reference beam, while the screws S10 and S11serve to manipulate M2 along the x − and the y − axes,respectively. While the mirrors M3 and M4 are fixed, themirror M1 is moved by the piezo-stage along the z − axis.The alignment of the interferometer is performed intwo steps. First, the primary beam is aligned by ad-justing S1, S2, S3 and S4 in such a way that the beamis aligned with the photodiode active area. During thisprocedure, the reference beam is screened out using awhite paper. The mirror M1 is attached to the piezo-stage, placed inside the cryostat, and it is manipulatedby rotating the piezo-stage. Once the primary beam isaligned, it is then blocked and the reference beam is un-blocked. An iterative approach is taken to tilt and ad-just M3 using S10 and S11 until the beam impinges ontoM2. Once the beam hits the center of M2, the mirror istilted and adjusted using S7, S8 and S9 until the referencebeam is visible at the photodiode. The primary beam isthen unblocked and an interference pattern is observedat the photodiode. The resulting intereference patternis focused in such a way, that the fringes are broad anddefined by fine tuning S7, S8, and S9. It is noted, thatthe adjustment of M2 does not affect the primary beamalignment. C. The piezo-stage
The piezo-stage serves to transmit the length varia-tion of the PEA stack to the mechanical system whichhouses M1. The piezo-stage also protects the PEA stackfrom mechanical failure and ensures that only compres-sive stress is applied. A computer-aided design (CAD)schematic of the piezo-stage is shown in Fig. 7. As shownin Fig. 7(a), the base (1) is the backbone of the wholepiezo-stage on which the slide (2) is allowed to move upand down along the z − direction. The whole system ispreloaded with four aluminum springs denoted by (3) inFig. 7(a). The springs provide lateral and longitudinal Air springsDamping frameCryostat
Figure 8. Damping stage for the cryostat equipped with airsprings to compensate ground vibrations. forces between the base and the slide. The mirror M1is mounted via a holder (4). A Ni-Cr cartridge heater(5) is placed at the bottom of the stage which is used forthe temperature regulation via a Model-332 Lakeshoretemperature controller. The design of the piezo-stagewithout the slide is shown in Fig. 7(b). The PEA stack(6) is positioned at the center of the stage and is sur-rounded by a ceramic case (7), which isolates the piezo-actuator from the conducting base. On the lower side ofthe piezo-actuator, the tightening block (9) is screwed tothe slide prividing tension to the springs while on the top-side, the piezo-actuator is fixed against the base througha ceramic hemispherical element (8) that minimizes thebending and the shearing stress. In order to minimizethe friction, three ceramic spheres (11) with the abilityto roll on molybdenum tracks are placed between theslide and the base. The upper tracks (10) are attachedto the slide, while the lower tracks are fixed to the baseusing molybdenum screws. Two cernox temperature sen-sors are mounted at the backside of the piezo-stage tomeasure the temperature at the top and bottom of thepiezo-actuator. These two cernox sensors in combinationwith a permanent cernox sensor located at the bottomof the sample chamber of the Janis Super Variable Tem-perature 7TM-SVM cryostat are used to set and controlthe temperature at the piezo-stage. The piezo-stage ismounted at the bottom of the SHR and cables are sol-dered with a cryo-compatible soldering paste to the PEAstack, cernox sensors and the cartridge heater. The ca-bles are led upward through the hollow rod, as alreadymentioned before.During dynamic operation, a PEA stack can be sub-jected to high accelerations and therby tensile stress maybe produced. Since common piezoelectric materials, forinstance PZT, are prone to tensile stress, for any dynamicoperation they must be protected through preloading.The magnitude of the required preload is estimated from10 p i e z o V pp ( V ) V ( V ) pho t oa m p S (x 10 arb. units) n (b)(a) S (x 10 arb. units) n 3 S (x 10 arb. units) n (c)(d) V (V) ppPiezo d ( n m ) P i e z o d i s p d ( n m ) P i e z od i s p Figure 9. (a) Output voltage of the photodiode-amplifier V Photoamp as a function of the samplenumber S n .(b) Voltage appliedto the PEA stack V Piezo as a function of S n . (c) Position of the PEA stack as a function of S n . (d) Position of the PEA stackas a function of the applied V Piezo exhibiting a hysteresis loop. the dynamic force F dyn , given by: F dyn = 2 π mf ∆ L (2)where m is the mass of the PEA stack and the mountedload, ∆ L is the peak-to-peak displacement of the piezo-actuator and f is the frequency of the applied sinusoidalvoltage. In the case considered here, since the piezo-actuator is neither fixed to the base nor to the slide, itsown weight (5 g) is the mass m . Considering a maxi-mum displacement of ∼ µ m at T ∼ . f = 100 Hz, a dynamic force F dyn = 3 mNis calculated for the piezo-actuator considered here. Theestimated value of F dyn is within the range of the charac-teristics of the particular PEA stack used in this experi-ment and thus no preloading is required. The piezo-stageis also kept mechanically backlash-free by eliminating thepreloading of the PEA stack. D. Damping
The ground vibrations for this experimental set-uphave been measured to be ∼ µ m, i.e. of the same or-der of magnitude as the displacement of the PEA stackin cryogenic conditions. In order to achieve an efficient damping it is required that the entire experimental as-sembly including the cryostat is stiff and decoupled fromthe ground via soft springs. The stiffness of the set-upis limited by the length of the cryostat and by the di-mensions of the sample rod. The damping system usedhere consists of a heavy triangular frame manufacturedfrom industry grade aluminium profiles. At the three ver-tices of the triangular frame, air springs are placed, whichproduce an air suspension for the 300 kg cryostat, as re-ported in Fig. 8. The cryostat is placed at the center ofthe frame and is suspended a few mm above ground. Theair pressure in the springs is adjusted through a pressuregauge. By inflating the air springs so that the cryostatcan be lifted a few millimeters above ground, a noise of ∼
150 nm due to the ground vibrations is measured, whichis less than one-sixth of the vibration measured for anundamped system. Thus, the damping against groundvibrations ensures reliable measurements of the piezo-actuator displacement from the interferometric fringes.
E. Data evaluation
The estimation of the maximum displacement of thePEA stack and its hysteresis as a function of T and ap-plied voltage is accomplished from the data collected at11
50 100 150 200 250255075100125 d ( n m / V ) R P i e z o (a) T (K) T (K) d ( n m / V ) R P i e z o (b)(c)
10 30 50 704080120160200
50 K73 K103 K123 K148 K174 K199 K220 K246 K d ( n m / V ) R P i e z o V (V) ppPiezo Figure 10. (a) Displacement rate of the PEA stack d P as afunction of T for 2 K < T <
250 K. (b) d P as a function of T for 2 K < T <
25 K. (c) d P as a function of the appliedpeak-to-peak voltage V piezopp measured at different T . the photodiode-amplifier by measuring the change in in-tensity of the interference fringes during a dynamic op-eration. The output voltage of the photodiode-amplifier V Photoamp , as a function of the sample number, S n (de-fined as the number of data points accumulated withina defined integration time) and the voltage V piezopp ap-plied to the PEA stack are recorded as a function of S n and are reported in Fig. 9(a) and Fig. 9(b), respectively.Over the upward slope of the triangular V piezopp , a sinu-soidal V Photoamp signal with a constant frequency is col-lected until the turning point at S n = 4500. The signal ∼ vice versa indicates that the piezo-actuator has travelleda distance of 158.2 nm, i.e. a quarter of the wavelengthof the He-Ne laser used here. The position of the PEAstack d PosPiezo at the peaks and valleys of V Photoamp is re-ported in Fig. 9(c). For a cyclic operation of the PEAstack, where V piezopp is ramped up from +1 V to +9 V andthen down to +1 V, a hysteresis in the estimated valuesof d PosPiezo is observed, and reported in Fig. 9(d).The systematic errors that arise due to disturbancesfrom residual ground vibrations, thermal gradients in thecryostat and turbulances in the liquid nitrogen and he-lium, are taken into account for the evaluation of thedata. One approach is to evaluate multiple samples andto employ statistical methods to reduce the error. The al-ternative approach follows a method of data fitting lead-ing to an approximation of the real hysteresis loop withan ellipse. This approximation is valid for low T and forlow amplitudes of the input voltages due to a decrease ofthe non-linearity of the PEA stack at these conditions.An ellipse is generally described as a parametric plot oftwo sinusiodal functions with a phase shift between them.For simplicity, it is assumed that the applied voltage tothe PEA stack is a simple sinusoidal function V ( t ), whilethe position of the PEA stack is represented by a time de-pendent function d ( t ). The excitation frequency of V ( t )is ω . The application of V ( t ) results in a sinusoidal d ( t )which is phase shifted by φ with respect to V ( t ), due tothe hysteresis: V ( t ) = V sin ( ωt ) (3) d ( t ) = d sin ( ωt ) + φ (4)When φ of d ( t ) with respect to V ( t ) is known, theabsolute hysteresis H abs and the relative hysteresis H rel are calculated according to: H abs = ∆ d = d sin ( φ ) = d φ (5) H rel = φ (6)In order to estimate d ( t ), a representation of the signalsin the frequency domain is adopted. When a sinusoidalvoltage is applied to the PEA stack, a related voltage canbe measured across the photoamplifier. The higher thefrequency of the photoamplifier voltage, the higher theabsolute value of the speed of the piezo-actuator trans-lation. The mathematical modelling of the data is per-formed according to the following equation: v − | v | = v − (cid:12)(cid:12)(cid:12)(cid:12) ∂d∂t (cid:12)(cid:12)(cid:12)(cid:12) = v − v | cos ( ωt ) + φ | (7)12
50 100 150 200 2500126 T (K) P ( % ) h ys t e r e s i s T (K) P ( % ) h ys t e r e s i s T (K) P ( % ) h ys t e r e s i s P ( % ) h ys t e r e s i s
50 K73 K103 K123 K 2 4 65101520 P ( % ) h ys t e r e s i s (a) (b)(c) (d) d ( µ m) ppPiezo d ( µ m) ppPiezo Figure 11. (a-b) Relative hysteresis of the PEA stack, P hysteresis as a function of T estimated for the T range: (a) 2K < T < < T < P hysteresis as a function of V piezopp measured for temperatures: (c) 50K, 73K, 103K and123K. and (d) 148K, 174K, 199K, 220K, and 246K. where v is an offset, v is the velocity of the PEA stackand ω = 2 πf is the driving frequency of the system.According to the model employed here, φ is also thephase shift of the piezo position d ( t ) with respect to U ( t ),which in turn is equal to H rel . Each time a measurementis recorded, the V piezopp and the V Piezoamp are acquired for10 s with a sample rate of 10000/s with an excitation fre-quency of 25 Hz. Therefore, the fitting method averagesover 250 oscillations.
IV. RESULTS AND DISCUSSIONSA. Displacement of the PEA stack
The displacement rate d piezoR of a PEA stack is definedas the change in dimension per unit applied voltage andthe knowledge of d piezoR for a given piezo-actuator at anyarbitrary T is essential for an efficient control of the PEAstack or of the mechanical stage to which it is attached.The d piezoR has been measured for 2 K ≤ T ≤
250 K andfor a constant displacement of 1.2 µ m and is reported inFig. 10(a). The displacement rate is found to increaselinearly as a function of T . For this work, the tempera-ture range 2 K ≤ T ≤
10 K is of particular interest. The piezo-actuator displacement measured in this tempera-ture range is shown in Fig. 10(b). For T ≥ d piezoR increases as a function of T , with the exception of thevalue collected at T = 8 . T ≤ µ m measured for an applied peak-to-peak voltage V piezopp = 75 V. In Fig. 10(c), the behavior of d piezoR asa function of different applied V piezopp is reported. FromFig. 10(c), it can be concluded that d piezoR increases withincreasing T . B. Hysteresis
The hysteresis of the PEA stack has been estimatedaccording to the method described above. The rela-tive hystereses P Hysteresis of the PEA stack for the tem-perature ranges 2 K < T <
250 K and 2 K < T <
10 K are reported in Figs. 11(a) and 11(b), respec-tively. In particular, multiple measurements have beenrecorded at T = 2 K and an absolute hysteresis of(9 . ± .
3) nm for a peak-to-peak maximum displace-ment of 1 . µ m is estimated. With increasing T , abroadening of the hysteresis is observed, as shown inFig. 11(b). The solid symbols in Fig. 11(b) represent the13ultiple measurements of P Hysteresis at T = 2K, whilethe hollow ones are the estimated P Hysteresis measuredat T ≥ P Hysteresis as a function of T . The P Hysteresis dependsalso the peak-to-peak displacement d piezopp of the PEAstack. The behavior of P Hysteresis as a function of d piezopp and measured at T = 50 K ,
73 K ,
103 K ,
123 K and at T = 148 K ,
174 K ,
199 K ,
220 K , and 246 K is reported inFigs. 11(c) and 11(d) respectively. For the temperaturerange 50 K ≤ T ≤
246 K, P Hysteresis increases as a func-tion of increasing d piezopp and saturates for d piezopp ≥ µ m.
500 1000 1500 2000 2500 30000.51.01.52.0 T = 10 KT = 260 KFrequency (Hz) A m p li t ude ( a r b . un i t s ) Figure 12. Amplitude of the piezo-stage as a function offrequency at 10 K and 260 K.
C. Resonant frequency of the piezo-stage
The piezo-stage is a mass-spring system in which themass is a constant quantity. The resonant frequency ofthe stage is determined by the stiffness of the system dur-ing a change of T . The frequency response of the piezo-stage has been measured at T = 10 K and T = 260 Kand is reported in Fig. 12, where the amplitude of thephotodiode-amplifier in arbitrary units is plotted as afunction of the driving frequency of the PEA stack. Thedriving voltage of the PEA stack is kept at ≤ ∼ T = 260 K to ∼
800 Hzat T = 10 K. D. Effect of magnetic field on thepiezo-displacement
The behavior of the PEA stack in the presence of amagnetic field has been studied in real time by moni-toring the interference pattern as a function of the mag-netic field µ H , which is ramped from 0 T to +6 T. Theramping rate of the magnet is chosen to be 0.01 T/s.The applied piezo-voltage is kept constant during theentire measurement cycle. Any force exerted on thePEA stack or on the piezo-stage due to µ H leads toa change in the alignment of the optical path of themeasurement and reference beams. Such a misalignmentproduces changes in the interference pattern or affectsthe interference conditions. No relevant variation of thediffraction pattern is observed during the sweeping ofthe magnetic field from 0 T to +6 T. A video of the en-tire measurement cycle is reported in the SupplementaryMaterial . Screenshots of the interference pattern at µ H = 0 T , , , , , , and 6 T are shown inFig. 13. The slight change in the position of the brightfringes is due to the thermal drift of the set-up. The d piezopp and P Hysteresis of the PEA stack for a fixed appliedvoltage and for a stable T is independent of an applied µ H . E. Capacity and equivalent series resistance
The capacity C Piezo and the equivalent series resistance(ESR) of a PEA stack are crucial parameters for the de-sign of control circuits. A PEA stack can be approxi-mated to an ideal capacitor with an ohmic resistance inseries, which in the case of a PEA is the ESR . The esti-mated C Piezo for a driving f P = 100 Hz and f P = 100 kHzas a function of T are given in Figs. 14(a) and 14(b),respectively. The corresponding ESR of the PEA stackmeasured for f P = 100 Hz and f P = 100 kHz are reportedin Figs. 14(c) and 14(d) respectively. For f P = 100 Hz,the C Piezo increases with increasing T , while the ESR hasa maximum at T = 50K. However, for f P = 100 kHz,the calculated ESR is lower than the one estimated for f P = 100 Hz for all T . Thus, the PEA stack studied hereis suitable to work at low driving frequencies for applica-tions as high precision nanopositioners and nanoscannersin low-T-high- µ H experimental set-ups. V. CONCLUSIONS
An interferometry based experimental set-up for mea-suring the displacement of a commercial PEA stack in thenm range, at temperatures down to 2 K and under ap-plied magnetic fields up to +6 T has been designed andrealized. The PEA stack is mounted on a piezo-stageand placed inside a cryostat equipped with a supercon-ducting magnet. The displacement of the PEA stack istransferred to the mechanical system, via the piezo-stage14
T0 T0 T 1 T 2 T 3 T6 T5 T4 T
Figure 13. Screenshots of the interference recorded at applied magnetic fields of 0 T, 1 T, 2 T, 3 T, 4 T, 5 T, and 6 T. T (K) C ( n F ) p i e z o T (K) ES R ( W ) p i e z o (a) (c)(b) (d) Figure 14. PEA stack capacity, C piezo for applied frequencies of (a) 100 Hz and (b) 100 kHz and ESR measured for appliedfrequencies of (c) 100 Hz, and (d) 100 kHz over the range 2K < T < equipped with a mirror, which forms the dynamic partof the designed interferometer. The d piezopp and P Hysteresis of the PEA stack have been measured as a function of T .A monotonous increase of d piezopp as a function of increas-ing T for 2 K ≤ T ≤
250 K is observed. For T ≤ d piezopp = 1 . µ m is measured for V piezopp = 75 V.For T ≤
50 K, a constant d P as a function of V piezopp fa-cilitates an open-loop control of the PEA stack position.At RT, for V piezopp = 75 V, the measured d piezopp = 25 . µ mmeasured agrees well with the datasheet provided by themanufacturer. With the decrease of T , a reduction in theabsolute value of the P Hysteresis is observed. At T = 2 K and for d piezopp = 1 . µ m, a residual maximum absolutehysteresis of (9 . ± .
3) nm is measured for the PEAstack. It is also demonstrated, that the P Hysteresis de-pends on the d piezopp and saturates for d piezopp ≥ µ m. Inline with the frequency response of the stage, it is con-cluded that the operating frequency should be kept farbelow the resonant frequency. Further, an external ap-plied µ H = +6 T is found to have no effects on neitherthe piezo-stage nor the PEA stack. Thus, the laser inter-ferometric technique described here can be used for thecharacterization, over a large range of temperature andmagnetic fields, of standard PEA stacks for applications15s nanoscanners and nanopositioners in scanning probetechniques and also in astronomy based-, aerodynamicsand space technologies. SUPPLEMENTARY MATERIAL
A video of the behavior of the interference patterns fora magnetic field sweep from 0 T to +6 T is reported inthe Supplementary Material DATA AVAILABILITY
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest and within the article and its Supplementary Ma-terial.
ACKNOWLEDGEMENTS
The work was funded by the Austrian Science Fund(FWF) through Projects No. P26830 and No. P31423.The authors thank Prof. David Stifter and Dr. BettinaHeise for the help with the He-Ne laser and the opticalfibers used in this work. The authors also thank Prof.Gzregorz Grabecki, Institute of Physics, Polish Academyof Sciences, Warsaw, Poland for the calibration of the cer-nox sensors used in this work. The authors also acknowl-edge the technical assistance of Ing. Ekkehard Nusko.
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