Low temperature and high magnetic field performance of a commercial piezo-actuator probed via laser interferometry

The development of precise nanopositioning systems with spatial resolution ∼(1–10) nm and time constant ∼(10–100) μs is relevant for both basic and applied research, as well as for industrial applications.¹ The positioning resolution of conventional actuator systems including hydraulic and ac/dc motors is too coarse for most modern technologies, even though these actuators are able to provide large output force and working strokes. The working stroke of an actuator is defined as its linear displacement under dynamic conditions. With the recent development of actuators based on piezoelectric materials, i.e., piezoelectric actuators (PEAs), it is now possible to achieve a spatial resolution of a few nm.^1,2 The application of PEA is widespread in basic research fields and in industrial sectors, such as from high resolution scanning probe microscopy (SPM)^3,4 to optical systems for astronomy⁵ and the aerospace industry.^1,6,7 The working stroke of a single piezoelectric element is generally limited to a few μm even at room temperature (RT). Alternative approaches have improved the working strokes of hybrid PEAs to a few centimeters.⁸ Despite the limitations imposed by the low working strokes of most commercial PEA stacks, the ones with working strokes ∼(10–30) μm find wide applications in the medical industry, especially in medical implants, scanning fiber endoscopes, lab-on-a-chip for mobile analytics, and nebulizers, in laser technology, in precision mechanical applications and 3D printing, and in the printing industry.^1,9–12

A major application area for the PEA is represented by scanning probe measurement systems, such as atomic force microscopy (AFM),¹³ scanning tunneling microscopy (STM),^3,4,14,15 and scanning tunneling spectroscopy (STS).¹⁶ The recent developments in the fields of STM,^3,4 STS,¹⁶ and, particularly, point contact spectroscopy (PCS), including point contact Andreev reflection (PCAR) spectroscopy,^2,17–24 have underlined the relevance of PEA for nanopositioning applications. With the emergence of new families of quantum materials encompassing topological insulators,²⁵ topological crystalline insulators,²⁶ topological superconductors,²⁷ Weyl and Dirac semimetals,²⁸ and unconventional superconductors²⁹ including heavy fermionic systems,²⁹ PCS and PCAR have proven to be efficient spectroscopic tools to study these material systems.³⁰ Recent investigations of topological crystalline insulators such as Pb_1−xSn_xSe and Pb_1−xSn_xTe have pointed at the presence of Majorana fermion-like excitations at the atomic steps of the epitaxial layers.^31,32 Most PCS and PCAR setups reported in the literature^2,20 are static, with the sample kept fixed, while the tip is the only dynamic component.^2,20 Therefore, by introducing a dynamic mode to the static PCAR setup, a lateral degree of freedom in the sample plane is added and makes it possible to map the sample surface. This improvement is expected to open new perspectives for the characterization of quantum materials. In particular, the use of PEA-based nanopositioning systems along the sample plane is promising to facilitate the mapping of surfaces of bulk specimens, thin films, and 2D layers such as (beyond-)graphene systems.³³ The scanning PCAR (SPCAR) is an exclusive tool for mapping exotic quantum phases and phenomena, such as Majorana fermions,^34,35 weak link Josephson effect,³⁶ Meissner and mixed phases of conventional³⁷ and unconventional superconductors, odd frequency superconductivity,³⁸ and Yu–Shiba–Rusinov states in magnetically doped superconductors.³⁹ 

The scanning degree of freedom of any physical probe can be accessed by employing either an electromechanical motor or a PEA stack. For applications at cryogenic temperatures, the PEA stacks are preferred over electromechanical motors, since their compactness and physical dimensions favor the integration with state-of-the-art cryostats. In addition, PEA stacks can be controlled by using high precision electronics. However, one major challenge in employing PEA stacks as scanners is represented by their limited displacements at cryogenic temperatures.^40,41 For most commercial PEA stacks, the absolute displacement reduces with temperature T and can shrink to (5–10)% of the maximum displacement at RT. Most available PEAs have a maximum displacement (20–30) μm for an applied voltage of (75–150 V) at RT. Moreover, creep and ferroelectric hysteresis associated with a PEA can be detrimental to the application of commercial PEA stacks in the cryogenic regime. The piezo-hysteresis as a function of T determines whether a particular PEA stack can be employed as a position scanner with nm resolution, particularly in techniques for mapping surfaces such as STM and SPCAR. Minimal or zero hysteresis of the piezoscanner is desirable for both spectroscopic and microscopic mapping of surfaces. However, for most commercial PEA stacks, specifications under cryogenic conditions and in the presence of magnetic fields are generally not available.

The absolute displacements of PEAs can be measured by mechanical, electrical, or optical means. Due to the enhanced signal-to-noise ratio, optical techniques are advantageous over mechanical or electrical sensors such as flexural hinges and strain gauges. In addition, light-based measurements are compatible with cryogenic applications, which can pose challenges to strain gauge or flexural hinge-based sensors.⁴² Optical interferometric techniques have been successfully applied for the measurement of nanometer level displacements, such as those related to the detection of gravitational waves.^43,44 Here, an indigenously designed and fabricated interferometric setup for the estimation of displacement, hysteresis, and creep of a commercial PEA stack is reported. The measurements have been carried out over the temperature range 2 K < T < 260 K, both in the absence and as a function of an applied magnetic field. The absolute displacement of the PEA stack is evaluated to be 25.3 μm at RT, while for T = 2 K, an absolute displacement of 1.2 μm has been estimated for the maximum allowed voltage of 75 V. Both the displacement and hysteresis of the PEA stack are found to decrease with T and an absolute residual hysteresis of $9.1±3.3nm$ is estimated at T = 2 K.

A brief account of the design and fabrication of the interferometry setup for measuring the PEA displacement as a function of T and μ₀H is presented in this section. The experimental assembly comprises (i) the electronics and (ii) the mechanical block. The electronics segment includes the PEA stack, the piezo-amplifier, the photodiode amplifier, and the connector board. The electronics block consists of the following segments:

• The PEA stack: the commercial PEA stack used here is a low voltage element with a maximum load voltage of 75 V. The zero load response time of the PEA stack lies in the sub-millisecond range and a free stroke displacement of 30 μm at RT.⁴⁵ 

• Piezo-amplifier: the main task of the piezo-amplifier is to provide a constant amplification while keeping the noise level of the output ≤25 mV, which is the minimum voltage required for a piezo-displacement ∼1 nm. The amplifier bandwidth has been estimated to be 600 Hz, which is sufficient for operation since the PEA stack is operated at 25 Hz. The noise level measured for the piezo-amplifier is ∼16 μV_RMS.

• Photodiode amplifier: the primary function of the photodiode amplifier is to convert the optical intensity at the photodiode into an electrical signal. The estimated critical frequency of the photodiode amplifier is 10 kHz.

• Connector board: the connector board is a hub between the piezo-amplifier, the photodiode amplifier, and the myDAQ. The myDAQ is a data acquisition device from National Instruments, which is used here for measurements and analysis of the piezoamplifier and the photodiode amplifier. The electronics is controlled by using an indigenously developed LabView program, and all data are collected via a computer interface.

The components of the mechanical block are the interferometer, the piezo-stage, and the vibration damping assembly. A schematic representation of the experimental setup is shown in Fig. 1. The complete experimental assembly is designed on a 1.0 m long hollow cylindrical sample holder rod (SHR) made out of G10 fiber material. The SHR is designed to fit into a Janis Super Variable Temperature 7TM-SVM cryostat equipped with a 7 T superconducting magnet. On one end of the SHR, a mechanical stage is attached, which accommodates the PEA on a stage (piezo-stage). The piezo-stage is positioned at the center of the 7 T superconducting solenoid magnet, as shown in Fig. 1. On the other end of the SHR, the optical assembly for the Michelson interferometer and for the photodiode detector is mounted. The SHR connects the interferometer platform to the piezo-stage in a stiff manner. All wires for the electrical connections to the piezo-stage are placed within the hollow SHR. The entire cryostat assembly is placed on an indigenously designed vibration damping mechanism in order to reduce ground vibrations. A detailed description of the electronics segment and the mechanical block including the main highlights and justification of the chosen elements for the electronics and the mechanical blocks is presented in the supplementary material. The photographs of the fabricated devices, circuit diagrams, and frequency responses of the piezo-amplifier, the photodiode amplifier, and the connector board are reported in Figs. S2–S4, respectively, of the supplementary material. The fabrication details and schematic of the interferometer including the detailed layout of optical and of mechanical components are reported in Fig. S5. The computer-aided-design (CAD) of the piezo-stage and the photograph of the damping stage built for the cryostat to reduce the ground vibrations are presented in Figs. S6 and S7, respectively.

The ground vibrations for this experimental setup have been measured to be ∼1 μm, i.e., of the same order of magnitude as the displacement of the PEA stack under cryogenic conditions. In order to achieve an efficient damping, it is required that the entire experimental assembly including the cryostat is stiff and decoupled from the ground via soft springs. The stiffness of the setup is limited by the length of the cryostat and by the dimensions of the sample rod. The damping system used here consists of a heavy triangular frame manufactured from industry grade aluminum profiles. At the three vertices of the triangular frame, air springs are placed, which produce an air suspension for the 300 kg cryostat, as reported in Fig. S7 of the supplementary material. The cryostat is placed at the center of the frame and is suspended a few mm above the ground. The air pressure in the springs is adjusted through a pressure gauge. By inflating the air springs so that the cryostat can be lifted a few millimeters above ground, a noise of ∼150 nm due to the ground vibrations is measured, which is less than one-sixth of the vibration measured for an undamped system. Thus, the damping against ground vibrations ensures reliable measurements of the piezo-actuator displacement from the interferometric fringes.

An estimation of the maximum displacement of the PEA stack and its hysteresis as a function of T and applied voltage is obtained from the data collected at the photodiode amplifier by measuring the change in intensity of the interference fringes during a dynamic operation. The PEA stack is driven with f_P = 25 Hz during the measurements, and the time period for recording of the data is therefore 40 ms. The chosen value of the driving frequency is a compromise in order to avoid parasitic phase shifts introduced by the electronics for f_P ≥ 1 kHz. However, for f_P ≤ 10 Hz, the time period becomes too large and enough cycles of measurements could not be recorded for the statistical treatment of the data, as discussed in the following. The output voltage of the photodiode amplifier V_Photoamp as a function of the sample number S_n (defined as the number of data points accumulated within a defined integration time) and the voltage $Vpppiezo$ applied to the PEA stack are recorded as a function of S_n and are reported in Figs. 2(a) and 2(b), respectively. Over the upward slope of the triangular $Vpppiezo$⁠, a sinusoidal V_Photoamp signal with a constant frequency is collected until the turning point at S_n = 4500. The signal ∼0.05 V due to the ambient light is a background offset in Fig. 2(a). Each transition from a peak to a valley and vice versa indicates that the piezo-actuator has traveled a distance of 158.2 nm, i.e., a quarter of the wavelength of the He–Ne laser used here. The position of the PEA stack $dppPiezo$ at the peaks and valleys of V_Photoamp is reported in Fig. 2(c). For a cyclic operation of the PEA stack, where $Vpppiezo$ is ramped up from +1 V to +9 V and then down to +1 V, a hysteresis in the estimated values of $dppPiezo$ is observed and reported in Fig. 2(d).

The systematic errors that arise due to disturbances from residual ground vibrations, thermal gradients in the cryostat and turbulences in the liquid nitrogen and helium, are taken into account in the treatment of the data. One approach is to evaluate multiple samples and to employ statistical methods to reduce the error. The alternative approach follows a method of data fitting, leading to an approximation of the real hysteresis loop with an ellipse. This approximation is valid for low T and for low amplitudes of the input voltages due to the decrease in the non-linearity of the PEA stack under these conditions. An ellipse is generally described as a parametric plot of two sinusoidal functions with a phase shift between them. For simplicity, it is assumed that the applied voltage to the PEA stack is a simple sinusoidal function V(t), while the position of the PEA stack is represented by a time dependent function d(t). The excitation frequency of V(t) is ω. The application of V(t) results in a sinusoidal d(t) that is phase shifted by ϕ with respect to V(t) due to the hysteresis,

When ϕ of d(t) with respect to V(t) is known, the absolute hysteresis H_abs and the relative hysteresis H_rel are calculated according to

In order to estimate d(t), a representation of the signals in the frequency domain is adopted. When a sinusoidal voltage is applied to the PEA stack, a related voltage can be measured across the photoamplifier. The higher the frequency of the photoamplifier voltage, the higher the absolute value of the speed of the piezo-actuator translation. The mathematical modeling of the data is performed according to the following equation:

where v₀ is an offset, v is the velocity of the PEA stack, and ω = 2πf is the driving frequency of the system.

According to the model employed here, ϕ is also the phase shift of the piezo-position d(t) with respect to U(t), which in turn is equal to H_rel. Each time a measurement is recorded, the $Vpppiezo$ and the V_Piezoamp are acquired for 10 s with a sample rate of 10 000/s with an excitation frequency of 25 Hz. Therefore, the fitting method averages over 250 oscillations.

The displacement rate $dRpiezo$ of a PEA stack is defined as the change in dimension per unit applied voltage, and the knowledge of $dRpiezo$ for a given piezo-actuator at any arbitrary T is essential for an efficient control of the PEA stack or the mechanical stage to which it is attached. The absolute displacement of the PEA stack $dppPiezo$ for an applied peak-to-peak voltage $Vpppiezo=75V$ has been estimated for 2 K ≤ T ≤ 250 K and is reported in Fig. 3(a). The $dPiezopp$ is found to increase linearly as a function of T. For this work, the temperature range 2 K ≤ T ≤ 10 K is of particular interest. The $dppPiezo$ measured in this temperature range is shown in Fig. 3(b). For T ≥ 5 K, $dppPiezo$ increases as a function of T, with the exception of the value collected at T = 8.5 K. The value of $dPiezopp$ estimated for T = 8.5 K is ∼150 nm lower than the ones estimated for the neighboring data points. This anomaly amounts to a difference of ∼150 nm when compared to the values of $dppPiezo$ for the neighboring data points. This is attributed to a combination of ground vibration and some noise spike in the electronics. It is noted here that the noise due to ground vibrations after the damping has been estimated to be 150 nm, as discussed before. The displacement is found to be constant for T ≤ 5 K, with a peak-to-peak value of 1.2 μm measured for an applied peak-to-peak voltage $Vpppiezo=75V$ at T = 2 K. In Fig. 3(c), the behavior of $dRpiezo$ as a function of different applied $Vpppiezo$ is reported. From Fig. 3(c), it can be concluded that $dRpiezo$ increases with the increase in T.

The hysteresis of the PEA stack has been estimated according to the method described above. The relative hystereses P_hysteresis of the PEA stack for the temperature ranges 2 K < T < 250 K and 2 K < T < 10 K are reported in Figs. 4(a) and 4(b), respectively. In particular, multiple measurements have been recorded at T = 2 K and an absolute hysteresis of $9.1±3.3$ nm for a peak-to-peak maximum displacement of 1.2 μm is estimated. With the increase in T, a broadening of the hysteresis is observed, as shown in Fig. 4(b). The solid symbols in Fig. 4(b) represent the multiple measurements of P_hysteresis at T = 2 K, while the hollow ones are the estimated P_hysteresis measured at T ≥ 2 K. The solid line traces a linear fitting of P_hysteresis as a function of T. The P_hysteresis depends also on the peak-to-peak displacement $dpppiezo$ of the PEA stack. The irregular pattern of P_hysteresis observed at T = 2 K as shown in Figs. 4(a) and 4(b) is attributed to the disturbances coming from the ground vibrations, which have a peak-to-peak amplitude of 150 nm, as discussed before. By taking into account the root mean square of the ground vibrations, the standard deviation of the absolute maximum hysteresis measurements performed at 2 K amounts to ∼7 nm. The behavior of P_hysteresis as a function of $dpppiezo$ and measured at T = 50 K, 73 K, 103 K, and 123 K and at T = 148 K, 174 K, 199 K, 220 K, and 246 K is reported in Figs. 4(c) and 4(d), respectively. For the temperature range 50 K ≤ T ≤ 246 K, P_hysteresis increases as a function of increasing $dpppiezo$ and saturates for $dpppiezo≥6μm$⁠.

The piezo-stage is a mass-spring system in which the mass is a constant quantity. The resonant frequency of the stage is determined by the stiffness of the system during a change in T. The frequency response of the piezo-stage has been measured at T = 10 K and T = 260 K and is reported in Fig. 5, where the amplitude of the photodiode amplifier in arbitrary units is plotted as a function of the driving frequency of the PEA stack. The driving voltage of the PEA stack is kept at ≤5 V, ensuring that the interferometer works in the linear regime. The correction factors due to the frequency responses of the piezo-amplifier and the photodiode amplifier are already taken into account while calculating the amplitude. It is observed that the resonance frequency of the system changes from ∼2500 Hz at T = 260 K to ∼800 Hz at T = 10 K.

The behavior of the PEA stack in the presence of a magnetic field has been studied in real time by monitoring the interference pattern as a function of the magnetic field μ₀H, which is ramped from 0 T to +6 T. The ramping rate of the magnet is chosen to be 0.01 T/s. The applied piezo-voltage is kept constant during the entire measurement cycle. Any force exerted on the PEA stack or on the piezo-stage due to μ₀H leads to a change in the alignment of the optical path of the measurement and reference beams. Such a misalignment produces changes in the interference pattern or affects the interference conditions. No relevant variation of the diffraction pattern is observed during the sweeping of the magnetic field from 0 T to +6 T. A video of the entire measurement cycle is provided in the supplementary material. Screenshots of the interference pattern at μ₀H = 0 T, 1 T, 2 T, 3 T, 4 T, 5 T, and 6 T are shown in Fig. 6. The positions of the bright fringes of the interference fringes are shown by the arrows. Any dependence of the properties such as displacement and hysteresis of the PEA stack on μ₀H would lead to destructive interference due to the modified path difference of the interfering beams, leading to a collapse of the interference patterns for the increasing μ₀H. However, no such collapse of the interference pattern has been observed, as shown by the marked position of the bright fringes in Fig. 6. The negligible linear displacement of the bright fringes is due to the thermal drift of the setup. The $dpppiezo$ and P_hysteresis of the PEA stack for a fixed applied voltage and for a stable T are independent of an applied μ₀H. The $dpppiezo$ and P_hysteresis at T = 2 K for μ₀H = 6 T are estimated to be ∼1.2 μm and 0.5%, respectively, which are of the same order as the ones measured in the absence of an applied μ₀H. It is also noted here that the magnetic field has no effects also on the cryogenic solder paste used to bond the PEA stack to the coaxial cables. Additionally, since all the electronic circuits including the power supply for the PEA stack are placed outside and away from the cryostat with zero stray magnetic fields, μ₀H has no effect on the electronic circuitry too. Therefore, the mechanical properties of the studied PEA stack are found to be independent of the magnetic field down to T = 2 K.

The capacity C_piezo and the equivalent series resistance (ESR) of a PEA stack are crucial parameters for the design of control circuits. A PEA stack can be approximated to an ideal capacitor with an ohmic resistance in series, which in the case of a PEA is the ESR.⁴⁶ An ideal capacitor does not dissipate any power, but real PEA stacks exhibit deviations from the ideal behavior due to the presence of the ESR, which is an internal resistance in series with the capacitance of the PEA stack. The power dissipation of the PEA stack is defined in terms of the dielectric loss factor tan δ,

where $XCPiezo$ is the capacitive reactance given by $XC=1ωCPiezo$ and ω is the operating frequency. For soft lead zirconium titanate (PZT) materials used as nanopositioners, typical values of tan δ ∼ 0.1–0.25 for $Vpppiezo∼50−100V$ and f_P ∼ 100 Hz are measured at T = 300 K.⁴⁶ In this work, the ESR values for the PEA stack as a function of T have been measured using a LCR meter.

The estimated C_piezo for a driving f_P = 100 Hz and f_P = 100 kHz as a function of T are given in Figs. 7(a) and 7(b), respectively. The two driving frequencies have been chosen in order to compare with the literature the properties of the PEA stack used here. The corresponding ESRs of the PEA stack measured for f_P = 100 Hz and f_P = 100 kHz are reported in Figs. 7(c) and 7(d) respectively. For f_P = 100 Hz, the C_piezo increases with the increase in T, while the ESR has a maximum at T = 50 K. However, for f_P = 100 kHz, the calculated ESR is lower than the one estimated for f_P = 100 Hz for all T.

The calculated tan δ as a function of T for $Vpppiezo=75V$ and f_P = 100 Hz is shown in Fig. 8. The estimated tan δ is found to be at least one order of magnitude lower than the one reported for PZT based nanopositioners,⁴⁶ indicating that heat generation due to power dissipation of the PEA stack can be neglected even for low T applications. In fact, the Cernox sensor placed on the back of the PEA stack, as described in the supplementary material, does not measure any rise of T during operations. The tan δ estimated for f_P = 100 kHz is ∼10⁻³ for 2 K ≤ T ≤ 250 K, which is one order of magnitude lower than the one measured for f_P = 100 Hz. However, operation of the PEA stack at f_P ≥ 100 kHz is not considered in order to avoid the parasitic effects, as discussed earlier.

Thus, the PEA stack studied here is suitable to work at low driving frequencies for applications as high precision nanopositioners and nanoscanners in low-T high-μ₀H experimental setups.

An interferometry based experimental setup for measuring the displacement of a commercial PEA stack in the nm range at temperatures down to 2 K and under applied magnetic fields up to +6 T has been designed and realized. The PEA stack is mounted on a piezo-stage and placed inside a cryostat equipped with a superconducting magnet. The displacement of the PEA stack is transferred to the mechanical system via the piezo-stage equipped with a mirror, which constitutes the dynamic part of the designed interferometer. The $dpppiezo$ and P_hysteresis of the PEA stack have been measured as a function of T. A monotonous increase in $dpppiezo$ as a function of increasing T for 2 K ≤ T ≤ 250 K is observed. For T ≤ 5 K, a constant $dpppiezo=1.2μm$ is measured for $Vpppiezo=75V$⁠. For T ≤ 50 K, a constant $dppPiezo$ as a function of $Vpppiezo$ facilitates an open-loop control of the PEA stack position. At RT, for $Vpppiezo=75V$⁠, the measured $dpppiezo=25.3μm$ is in good agreement with the datasheet provided by the manufacturer. With the decrease in T, a reduction in the absolute value of the P_hysteresis is observed. At T = 2 K and for $dpppiezo=1.2μm$⁠, a residual maximum absolute hysteresis of $9.1±3.3nm$ is measured for the PEA stack. It is also demonstrated that the P_hysteresis depends on the $dpppiezo$ and saturates for $dpppiezo≥6μm$⁠. In line with the frequency response of the stage, it is concluded that the operating frequency should be kept far below the resonant frequency. Furthermore, an external applied field μ₀H = + 6 T is found to have no effects on either the piezo-stage or the PEA stack. The integration of the individual components, namely, the circuit modules of the electronic block, the mechanical block, the active vibration damping, and the statistical approach for the data analysis, offers a uniqueness to the experimental setup in combination with the options for measurements at T ∼ 2 K and in the presence of μ₀H = 6 T. Thus, the laser interferometric technique described here can be used for the characterization, over a large range of temperature and magnetic fields, of standard PEA stacks for applications as nanoscanners and nanopositioners in scanning probe techniques and also in astronomy-based technology, aerodynamics, medical technology, and space technologies.

See the supplementary material for the fundamentals of piezoelectric actuators, the detailed description of the experimental setup developed in this work, and a video of the behavior of the interference patterns for a magnetic field sweep from 0 T to +6 T.

This work was funded by the Austrian Science Fund (FWF) through Project Nos. P26830 and P31423. The authors thank David Stifter and Bettina Heise for the help with the He–Ne laser and the optical fibers used in this work. The authors also thank Gzregorz Grabecki, Institute of Physics, Polish Academy of Sciences, Warsaw, Poland, for the calibration of the Cernox sensors used in this work. The authors also acknowledge the technical assistance by Ekkehard Nusko.

R.A. and A.B. conceived and planned the work. P.L. and K.D. designed and fabricated the mechanical and electronic components and were supported by R.A. and B.F. K.D., R.A., and B.F. performed the optimization of the setup, experiments, data acquisition, and data analysis. R.A. and A.B. wrote the manuscript with the support of all authors.

The data that support the findings of this study are available from the corresponding author upon reasonable request and within the article and its supplementary material.