Cross-scale high-bandwidth atomic force microscopy with a stick-slip nanopositioner

Domain: physics
Published: 2025-11-26
Editorial outcome: minor_revision

Abstract
The core of atomic force microscopy (AFM) lies in the ultra-precise scanning between the tip and sample, which is enabled by nanopositioners. State-of-the-art AFMs generate the scanning motion using direct-drive nanopositioners, possessing either long range or high bandwidth, but not both. Here we show that with a triple-phase controller, the high-bandwidth (up to 363 Hz) nano-precision scanning can also be performed with a typical stick-slip nanopositioner. More importantly, by leveraging the displacement accumulation in the stepping mode, the same system achieved a 3 mm × 3 mm XY working range, 1-2 orders of magnitude larger than those direct-drive nanopositioners with a similar bandwidth. We further developed a versatile stick-slip AFM and demonstrated high-line-rate AFM imaging at 40 Hz over millimeter-scale areas. This work expands the functional scope of stick-slip nanopositioners, traditionally limited to static nanopositioning or long-range coarse positioning, and offers a cross-scale, high-bandwidth solution for next-generation AFMs.

Introduction
Atomic force microscopy (AFM) is a powerful multifunctional platform that enables the exploration of electron spin1, mechanobiology2, and more by nanometer-scale measurement3. With the rapid advances in materials science4, biophysics5, and the semiconductor industry6, observing large objects with high efficiency is now demanding. In particular, next-generation AFMs require simultaneous nanometer precision, long motion range, and high bandwidth. Such challenging requirements could only be resolved by enhancing nanopositioning performance, which is enabled by the core of AFM: the nanopositioners7.

The nanopositioners generate ultra-precise scanning motion between the AFM probe tip and the sample7. Currently, direct-drive nanopositioners have become the standard in state-of-the-art AFMs, owing to their friction-free transmission characteristics8. However, according to Hooke’s law and Lagrangian mechanics, direct-drive nanopositioners face a fundamental trade-off between the scanning range and the resonance frequency (the resonance frequency dominates the operation bandwidth)8. High-stiffness direct-drive systems perform a high resonance frequency but are limited in range, and vice versa (Fig. 1a). To fulfill different performance demands, a range of actuation technologies has been explored. Among them, piezoelectric-actuated systems are the most widely used. They leverage the high stiffness and fast response of the piezoelectric actuator to achieve a high bandwidth8,9. However, their working range is limited (typically no more than 100 μm) by the low strain (about 0.1%) of piezoelectric ceramics10,11. Voice coil-actuated nanopositioners offer a long range (typically longer than 1 mm), but suffer from low structural stiffness, resulting in resonances at tens of Hertz and restricting scanning frequencies to below 2 Hz12,13. Normal-stressed electromagnetic actuators have been proposed as a compromise, yet their performance remains intermediate14,15. Achieving both long-range and high-bandwidth in a single nanopositioning system remains an unresolved challenge.

Stick-slip nanopositioners can offer macro-scale long motion ranges (typically >1 mm) by incrementally accumulating displacements in the stepping mode16, and fast response by the piezoelectric actuation17. Despite these advantages, they have long been considered unsuitable for nano-precision applications such as AFM scanning, mainly due to the frictional relationship between the compliant mechanism and the slider. Kinetic friction happens between successive steps, leading to backward motion effect18. This effect introduces uncertainties in the step size Δstep on the order of tens of nanometers (Fig. 1b)19. Although some AFM systems incorporate stick-slip stages, they are typically used only for coarse positioning in dual-stage architectures, while the fine scanning motion is still delegated to a direct-drive nanopositioner20. Over the past decade, both industrial and academic communities have explored the feasibility of operating stick-slip nanopositioners in a mode free from kinetic friction, i.e., the scanning mode, to enhance resolution18. However, such demonstrations have so far been restricted to quasi-static conditions, such as holding position or scanning at very low frequencies21,22. Hitherto, it is still unclear whether stick-slip nanopositioners can support long-range, nano-precision, dynamic scenarios such as cross-scale high-bandwidth AFM.

In this work, we demonstrate an AFM system that simultaneously achieves nanometer-level repeatability, long-range motion (>3 mm), and high-bandwidth scanning (up to 363 Hz), all powered by a single stick-slip piezoelectric nanopositioner. This is enabled by a triple-phase control scheme. In phase-1, the system is excited at the optimal frequency to achieve high-velocity stepping (26.8 mm/s along X-axis, 23.1 mm/s along Y-axis) over a 3 mm × 3 mm working area, which is 1-2 orders of magnitude larger than the range of typical piezoelectric direct-drive nanopositioners. Phase-2 employs a wrap-state integral controller to enable seamless transitions between the scanning and stepping modes. Phase-3 introduces a damping-tracking feedback controller, allowing a high control bandwidth near the mechanical resonance frequency, delivering excellent raster scanning capability comparable to that of direct-drive piezoelectric stages. The integration of these control phases bridged the gap between high-velocity stepping and high-bandwidth scanning, enabling cross-scale-level (from nano to macro) versatility. We further integrated the stick-slip nanopositioning system with a commercial AFM head, and carried out cross-scale high-bandwidth sampling imaging and stitching imaging over millimeter-scale areas. These results show that, with appropriate control, a typical stick-slip nanopositioner can meet the precision and bandwidth demands of AFM while offering an extended range — challenging the long-standing view that AFM scanning can only be achieved by direct-drive nanopositioners. This work not only expands the functional scope of stick-slip nanopositioners, but also offers a cross-scale, high-bandwidth solution for next-generation AFMs.

Results
Structure and working principle of the stick-slip AFM nanopositioner

In three degrees of freedom nanopositioning applications, such as AFMs, long-range scanning along the X- and Y-axis is always desirable, ideally at the macro scale23,24. While the scanning along the Z-axis is generally at the micro scale, considering the height of the surface topography. Here, we present a three degrees of freedom system to move the sample relative to an AFM probe (Fig. 1c).

In particular, the system consists of Z-, X-, and Y-stages arranged in series from top to bottom. To enable the positioning from nano to macro scale, the X- and Y-stages adopt the stick-slip principle, with structural designs modified from our previously reported configuration25. Each stage comprises a piezoelectric stack, a compliant mechanism with a lever (Supplementary Fig. 1), a slider, and the connection parts (Fig. 1c). The slider is connected to the compliant mechanism via a frictional interface. The compliant mechanism incorporates a long, low-stiffness cantilever at the output to mitigate the influence of assembly variations and mechanical wear on performance25. Titanium alloy TC4 is used for the compliant mechanism to enhance the wear resistance compared to commonly-used aluminum-based mechanisms. The X- and Y- stages are identical in structural configuration and parameters (Supplementary Fig. 1), except for minor differences between their bases. The Z-axial direct-drive stage consists of a piezoelectric stack, a compliant mechanism (Supplementary Fig. 2), and a wedge for the provision of preload, allowing precise tracking of the micro surface topography (height variation) by keeping the tip-sample interaction constant, which is monitored by an embedded AFM sensing system with a laser beam, a probe, and a photodiode. The effectiveness of the compliant mechanisms was demonstrated via finite element simulation (Supplementary Figs. 1 and 2).

Stick-slip nanopositioners operate in two modes: stepping and scanning. In the stepping mode, periodical sawtooth signals are employed to excite the piezoelectric stack, enabling a long-range coarse motion (Fig. 1d)26. Each period T of the sawtooth signal comprises a loading stage and an unloading stage. The moments of the start, the maximum value, and the end of a period were labeled as moment-1, moment-2, and moment-3, respectively. From moment-1 to moment-2 (the stick stage), the piezoelectric stack gradually elongates in response to the increasing voltage, thereby exerting force on the compliant mechanism and driving its deformation. Concurrently, the slider travels a distance Δ as a consequence of the static friction between the slider and the compliant mechanism. Between moment-2 and moment-3 (the slip stage), the excitation voltage drops suddenly, causing the compliant mechanism to return rapidly to its initial position as the piezoelectric stack retracts abruptly. Owing to the inertia, the slider tends to maintain its position through relative slipping between the slider and the compliant mechanism19,26. However, kinetic friction during this slipping motion induces a small backward displacement, i.e., the so-called backward motion effect (Fig. 1b), resulting in a net step size Δstep that is slightly smaller than Δ (Fig. 1d)27. By repeating the sawtooth excitation, a macro-scale motion range can be achieved via displacement accumulation9,16. The stepping velocity can be optimized by tuning the excitation amplitude and frequency, and the motion direction can also be reversed by inverting the sawtooth signal28,29. To note, friction-induced backward motion between successive steps introduces uncertainties on the order of several tens of nanometers per step, which accumulate progressively with each step. As a result, nanometer-level precision is fundamentally unachievable in the stepping mode alone.

In the scanning mode, the piezoelectric stack is excited by waveforms with relatively low acceleration—such as triangular or sinusoidal signals—insufficient to overcome the static friction. In this regime, the slider remains fully stuck with the compliant mechanism throughout the cycle, eliminating kinetic friction and thus enabling nanometer-scale resolution (Fig. 1e)22,30. As the slider always sticks with the compliant mechanism, the stick-slip nanopositioner in the scanning mode behaves like a direct-drive system, and the pure scanning range is at the micro scale.

These two modes offer complementary capabilities: Stepping mode provides a macro-scale range but suffers from friction-induced uncertainty. While scanning mode offers nanometer precision but has a small pure scanning range. To bridge these regimes and achieve cross-scale versatility, we introduce a triple-phase control strategy (Supplementary Fig. 3). It includes phase-1 for maximizing the velocity in the stepping mode, phase-2 for the seamless transition between the two modes, and phase-3 for enhancing the control bandwidth in the scanning mode. The following two sections will present details and advanced functions of phase-1 and -2, followed by those of phase-3.

Seamless transition between stepping and scanning enables cross-scale nanopositioning and tracking

Integrating the stepping and scanning motion effectively is the key for stick-slip nanopositioners to achieve both long-range displacement and nanometer-level precision. When the position error is large, i.e., the current position is far from the target, the system should approach the target via stepping as fast as possible. Once the error falls below a critical threshold (in this paper, the threshold is 20 μm), an appropriate transition mechanism to switch from the stepping mode to the scanning mode is necessary for the system to reach the target with nanometer precision. Control phases-1 and 2 were designed to fulfill these requirements.

In Phase-1, the stick-slip nanopositioner is excited by the optimal sawtooth signal to perform long-range stepping at the highest velocity. Therefore, the effects of the excitation parameters were explored first. We fixed the excitation frequency at 10 Hz and varied the voltage amplitude to examine its effect on stepping velocity. As shown in Fig. 2a, b, for both the X- and Y- stages, the stepping velocity increased monotonically with the voltage amplitude (also see Supplementary Fig. 4). At the maximum amplitude of 120 V, forward and reverse stepping velocities reached 121.6 μm/s and -123.2 μm/s for the X-stage, and 211.0 μm/s and -237.4 μm/s for the Y-stage. The stepping velocity of the Y-stage was observed to be higher than that of the X-stage because the backward motion effect that occurred at the end of each excitation cycle is more obvious for the X-stage27,31 (Supplementary Fig. 4). Next, we selected 120 V as the optimum amplitude, and altered the excitation frequency to examine its impact on stepping velocity. From the statistical velocity curves in Fig. 2c, d (also see Supplementary Fig. 4 for displacement curves), the maximum stepping velocities along the X+, X-, Y+, Y- directions were identified as 26.8 mm/s, −23.9 mm/s, 23.1 mm/s, and −25.1 mm/s, corresponding to optimum exciting frequencies of 1800 Hz, 1800 Hz, 1400 Hz, 1200 Hz, respectively. Also, it was found that the cross-coupling effect between the axes was very small, which was less than 1% (Supplementary Fig. 5). Therefore, sawtooth signals with 120 V amplitude and these frequencies were utilized to excite the piezoelectric stacks of the nanopositioner in phase-1.

Once the macro-scale position error is reduced to below the threshold by phase-1, switching from stepping to scanning to seek a fine precision is the next problem. A widely implemented method is to control the stick-slip nanopositioner as a direct-drive one and employ an integral feedback controller (or similar)21,22. The integral feedback controller can be expressed as

C(t) = ki ∫ e(t)dt (1)

where e(t) denotes the position error, ki denotes the integral gain, C(t) is the state of the integral controller. However, it should be noted that the real-time generated C(t) can easily exceed the maximum feasible input voltage of the piezoelectric stack (or be a negative value, which is also not feasible). The required control voltage is usually not applicable, leading to a failed control. Furthermore, under this control, the system is very difficult to track large trajectories that exceed the nanopositioner’s pure scanning range. Here, in phase-2, instead of standard integral controllers, we introduce a wrap-state integral controller to govern the transition between stepping and scanning. Its output voltage signal to excite the piezoelectric stack has the expression of

V = C(t) - (CU - CL) * floor((C(t) - CL) / (CU - CL)) (2)

where CU = 40 V and CL = 0 V denote the upper and lower values of the wrapped state, respectively (see Methods for detailed parameter determination). Conceptually, the wrap-state integral controller discretizes the otherwise unfeasible real-time control voltage into implementable voltage signals bounded by CU and CL. When C reaches CU, the output voltage V would be reset to CL (or to CU when it reaches CL). This will cause an instantaneous voltage shift, akin to the sudden drops in sawtooth signals, prompting the nanopositioner to step toward the target position (wrap-induced stepping, see Supplementary Fig. 6). After several wrap-induced steps, as the error is sufficiently small, C no longer reaches the upper or lower boundaries. In this situation, the wrapping behavior ceases and the output transitions to a continuous form, and the system seamlessly enters the scanning mode—free from kinetic friction (see Supplementary Fig. 6). Notably, during the wrap-induced-stepping process, backward motion effects also occur between successive steps. However, the associated uncertainties are compensated in the final scanning process, ensuring nanometer-level positioning precision.

The combination of control phases-1 and 2 enables a series of cross-scale nanopositioning tasks. A typical example is point-to-point motion. We sent commands to move from the 0 mm position to the 3 mm position, or vice versa, at 0.1 s, to one of the stick-slip axes. The displacements of the main motions and parasitic motions of the stages were recorded simultaneously. The main motion curves (Fig. 2e and g) reveal that both X and Y axes successfully reached the target macro position in less than 0.2 seconds, with the root-mean-square steady-state positioning errors of approximately 3.2 nm, demonstrating a nanometer-level repeatability. The major error source was the noise of the displacement sensor. The transition between the two phases, as well as the transition between wrap-induced stepping and scanning in phase-2, can be identified from the control voltage curves (Supplementary Fig. 7), and were also highlighted in the enlarged views of the main motions (Fig. 2e and g). Cross-coupling-induced parasitic motions (Fig. 2f and h) were treated as disturbances and were successfully constrained to a few microns during the stepping of the main motions. Following the transition of the main motions from stepping to scanning, the errors induced by parasitic motions ultimately reached the nanometer-level. These results demonstrate the system’s ability to rapidly reduce macro-scale positioning errors to the nanometer level.

Beyond point-to-point positioning, the controller also enables cross-scale real-time tracking of arbitrary trajectories. Traditionally, the stepping velocity of stick-slip systems was controlled by changing the amplitude or frequency of the sawtooth excitation voltage19, which lacks responsiveness to real-time changes. In contrast, the control voltage of the wrap-state integral controller (in phase-2) is dynamically generated according to the real-time position error, enabling highly flexible stepping velocity dynamically and ensuring an nanometer-level precision when the tracking ends. To demonstrate this capability, we conducted experiments of circular, square, and triangular trajectory tracking (range: 1 mm; frequency: 0.05–5 Hz; Supplementary Videos 1 to 3). In the 2-dimensional XY space, the measured trajectory curves closely matched the reference curves (Fig. 2i, k, and m), demonstrating the excellent long-range tracking capability of the stick-slip nanopositioner. Under all frequencies, the maximum and root-mean-square tracking errors for these trajectories were less than 6.2% and 3.4% of the trajectory length, respectively (see Fig. 2j, l, and n, Methods, and Supplementary Table 1). In particular, under 0.05 Hz, the maximum and root-mean-square tracking errors for all trajectories were observed to be no more than 1.4% and 0.16%, respectively. Taking the tracking of 0.05 and 5 Hz circular trajectories as examples, the time-domain displacement, position error, and voltage curves were plotted in Supplementary Fig. 8. In particular, both the X- and Y-trajectories tracked their trajectories well (Supplementary Fig. 8a, d, g, and j). At 0.05 Hz, the position error was below 20 μm for both axes throughout the tracking (Supplementary Fig. 8b and e). Therefore, the system was controlled in phase-2 alone, the voltage was bounded between 0-40 V, and the stepping was wrap-induced stepping. While at high frequencies, the time delay (Supplementary Fig. 8g and j), which is an intrinsic characteristic of all feedback controllers8, had an increased impact on tracking errors. Therefore, the positioning errors at 5 Hz seemed like sinusoidal curves and the amplitude exceeded 20 μm (Supplementary Fig. 8h and k). Once the error exceeds 20 μm, the system switches to phase-1 (excited by 120 V sawtooth signals under the optimal frequency) automatically for rapid correction. The combination of high-velocity stepping (in phase-1) and wrap-induced stepping (in phase-2) enabled the system to effectively track this trajectory with both high frequency and long range. It is also important that, micron-level deviations only occurred during stepping. Once the trajectory ended and no more stepping was required, the system finally transited to the scanning mode and held the position with nanometer-level accuracy (Supplementary Fig. 8b, e, h and k).

Damping-tracking control enables high-bandwidth scanning

Traditionally, stick-slip nanopositioners were only used for coarse positioning and quasi-static fine positioning (such as point-to-point tasks), while dynamic scanning operations—such as high-bandwidth scanning—have relied on direct-drive systems. In this section, we investigate the dynamic behavior of stick-slip nanopositioners operating in scanning mode and demonstrate that, when combined with a damping-tracking control scheme (in control phase-3), these systems can also support high-bandwidth scanning.

We first characterized the open-loop performance of the stick-slip stages in the scanning mode, focusing on their range and frequency response. A 1-Hz sinusoidal signal with an amplitude of 120 V was applied to the piezoelectric stacks, resulting in the output main motions of 21.3 μm and 24.3 μm in the X- and Y- axes, respectively (Fig. 3a). These values represent the maximum scanning ranges achievable without stepping, and thus define the maximum area of a single AFM frame. The slight differences between the pure scanning ranges of the two axes might be due to the inherent performance differences between the piezoelectric stacks and the difference between preloading forces. Cross-coupling effects were very small (Supplementary Text), with parasitic motions of 0.2 μm from X to Y and 0.13 μm from Y to X (Fig. 3b), both under 1% of the main motions.

To investigate the frequency responses, chirp voltages with frequency ranging from 0.1 to 10000 Hz were fed to the piezoelectric stacks. The amplitude of the voltages was sufficiently small (2 V) to avoid inducing kinetic friction. By employing the Fourier transformation of the voltage signal and the measured displacement, sharp dominant resonance peaks at 370 Hz and 360 Hz were found in the amplitude-frequency responses of the X- and Y-stages, respectively (Fig. 3c and d). The difference in resonance frequencies between the X- and Y-axes primarily results from the difference in moving mass. Factors such as machining tolerances, variations between the two piezoelectric stacks, and preload forces also affect the resonance frequency.

These results suggest that in scanning mode, the stick-slip nanopositioner exhibits several typical control challenges, including hysteresis nonlinearities (Fig. 3a), cross-coupling, and lightly-damped resonances—similar to those found in direct-drive systems. To solve these issues, we employed a notch filter as the damping controller to suppress the sharp resonance peaks, thereby increasing the gain margin (see in Methods for parameter determination). Then, a high-gain proportional-integral controller was employed as the tracking controller (see Methods), treating the effects of hysteresis and cross-coupling as disturbances to be rejected.

Under this damping-tracking control scheme, the closed-loop frequency responses (Fig. 3e and f) revealed -3 dB control bandwidths of 363 Hz and 295 Hz for the X- and Y-stages, respectively, both of which are dozens of times higher than those of conventional millimeter-range nanopositioners driven by voice coil motors32,33. Then, 5 nm staircase reference signals were scanned by the nanopositioner to demonstrate the resolution. The measured displacement data (Fig. 3g and h) clearly resolved individual steps, confirming that both the X- and Y-axes had a nanometer-level fine resolution. It should be noted that, the resolution of the system was basically limited by the resolution and noise of the displacement sensor. When the resolution of the sensor can be improved (at a cost of a smaller measurement range) and/or the data can be smoothed (at a cost of a lower response speed), the positioning resolution can be refined to the sub-nanometer level (see in Methods and Supplementary Fig. 9)34. Next, raster scanning tests were performed over a 12.5 μm × 12.5 μm area. The X-axis acted as the fast axis and scanned the periodical triangular trajectories, while the Y-axis acted as the slow axis and scanned a ramp signal. The number of scanning periods was 250. Considering the control bandwidth of the system, the highest scanning rate for the X-axial triangular signal was set as 40 Hz. Once the impact of the time delay had been eliminated by data post-processing, the root-mean-square errors (see Methods and Supplementary Table 2) for each axis were found to be only 0.044% and 0.023% of the 12.5 μm working range at 1 Hz. As the frequency increased, the error also grew (Supplementary Table 2). Figure 3i and j illustrate the displacements recorded at a scanning line rate of 40 Hz. At 40 Hz, the root-mean-square scanning error for the X-axis remained below 1.27%, while that for the Y-axis was only 0.386%. Given the high scanning rate, these minor errors are deemed acceptable for most application scenarios. These results can ensure a minimal XY distortion in the following AFM imaging experiments.

Cross-scale high-bandwidth AFM imaging

The above experiments demonstrate the excellent nanopositioning performance from the nano to the macro scale, laying a solid foundation for the construction of cross-scale high-bandwidth application systems. In particular, the high resolution and repeatability ensure the quality of the data, the high-bandwidth scanning ability supports a high line rate during the AFM raster scanning, while the cross-scale positioning and tracking functionality enables rapid and precise switching between multiple areas of interest. As a comprehensive demonstration, this section investigates the application of the proposed stick-slip nanopositioning system, equipped with the triple-phase controller, in cross-scale high-bandwidth AFM imaging.

An integrated AFM system was constructed (Fig. 4a), in which the stick-slip nanopositioner carried the sample to achieve three degrees of freedom motion. Meanwhile, the probe-sample interaction was monitored via the deflection signal from a commercial AFM head, which served as the feedback signal for constant-force control of the Z-stage (see in Methods and Supplementary Fig. 10). The approaching and retracting processes for the probe were performed by a vertical motor in the AFM head.

A cross-scale high-bandwidth AFM sampling imaging experiment was carried out on a commercial height calibration standard (Supplementary Video 4). As illustrated in the schematic (Fig. 4b), the sample featured six typical structure arrays. Among them, the square holes and pillars with a 10 μm pitch were arranged in a 1 mm × 1 mm square; the circular pillars and holes, and the X- and Y-lines with a 5 μm pitch were arranged in a 500 μm × 500 μm square. The integrated AFM system was programmed to automatically image four areas on each array, yielding a total of 24 images (with the sequence labeled in Fig. 4b). Each image was acquired through a 250 × 250-pixel scan over a 12.5 μm × 12.5 μm area, yielding a planar resolution of 50 nm. The scanning rate was set as 40 Hz. Supplementary Table 3 summarizes the coordinates of the starting points for each scan. To prevent the significant variation of the tip-sample interaction force during the stepping process (Supplementary Fig. 11), which may be due to the cross-coupling from XY to Z and the potential sample tilt (which may exceed the Z-axial range and damage the tip or sample), the approaching and retracting processes were inserted between the acquisition of successive AFM images, ensuring stepping occurred only out of contact (Fig. 4c). At the beginning of the experiment, the nanopositioner stepped to move the 1st area to the position under the probe. Then, the system approached the probe to the sample. Thereafter, the XY raster scanning commenced, while the Z-stage tracked the surface topography. Once the scanning of this area was complete, the AFM head retracted the probe from the sample, and the system prepared for the scanning of the next area. By repeating these steps, the system can automatically complete the AFM experiment with 24 images (Fig. 4c). The approaching, retracting, and stepping processes can be observed in the photos in Fig. 4d. Figure 4e depicts the top views of the system when imaging one area of each of the six arrays, showing the relative planar positions between the probe and the sample. The imaging results of the 24 areas illustrate the imaging quality of this system (Fig. 4f), demonstrating that clear and high-resolution images can be successfully captured at any position within a millimeter-level area. To better evaluate the imaging quality, Fig. 4g depicts a cross-section of the line marked in the 1st area of Fig. 4f, wherein the circular pillars’ height (100 nm) can be clearly identified. The deflection signal (Fig. 4g) varied only around the edges of the pillars, while it was mostly constant (at the setpoint of 10 mV) on the flat area, indicating that the Z-stage tracked the surface topography well, demonstrating a satisfactory imaging quality35. The complete AFM imaging process, including approaching, scanning, retracting, and stepping, was finished in 12 min 19 s (Supplementary Fig. 12 and Supplementary Video 4), demonstrating a substantial increase in throughput while maintaining a high imaging quality. It should be noted that, for the most common conventional cross-scale AFM sampling measurement method (typically has a scanning rate of less than 5 Hz, and moves the sample from one position to another using a manual or motorized coarse stage)36, they may take 1 h to image less than ten 250 × 250-pixel AFM images. Therefore, this approach achieves a 1–2 order of magnitude efficiency improvement over conventional methods.

The sampling imaging experiments demonstrated the system’s capability to acquire multiple micron-scale images across a millimeter-scale range with high efficiency. In fact, such sampling already satisfies the demands of most applications. Nevertheless, certain scenarios may require larger-area images. The following experiments show that the proposed stick-slip system can support the acquisition of millimeter-scale large AFM images via stitching. Figure 5a shows the schematics of stitching, the overall image was generated by stitching several individual AFM frames, each covering an area of 12.5 μm × 12.5 μm with a resolution of 100 × 100 pixels. To facilitate stitching, adjacent frames were spaced 10 μm apart in the X- or Y-direction, ensuring a 20% overlap. The overall image acquisition procedure was similar to that used in the above AFM sampling imaging experiments (Fig. 4c), with the only difference being the positions of the areas of interest. The scan line rate was set to be 20 Hz.

The AFM stitching imaging was carried out on the commercial height calibration standard first. A linear array of 100 × 1 frames was acquired in series (Fig. 5b and Supplementary Fig. 13), and a composite image covering 1025 μm × 12.5 μm was obtained using commercial stitching software (Fig. 5c, also see in Supplementary Data 1 for a full-size image). As shown in Fig. 5c, the stitched image clearly reflects the surface morphology with very small distortion. The same procedure was then applied to a microchip sample featuring 350 nm-high structures with an approximate pitch of 900 nm. A total of 10 × 10 AFM frames were acquired (Fig. 5d and Supplementary Fig. 14), and stitched into a large image covering 102.5 μm × 102.5 μm (Fig. 5e). The feature structures, scratches, and stains were clearly visible, demonstrating the high image quality.

Methods
Prototypedesignandfabrication
Thecomputer-aided-designmodelfortheproposednanopositionerwasestablishedusingSolidworks2021(Fig.1c).Byintroducingthematerialpropertiesintothecomputer-aided-designmodel,finiteelementsimulationswerecarriedoutusingANSYSWorkbench2021.Thepiezoelectricstacks(PSt3.5×3.5/7fortheX-andY-stages,PSt2×3/5fortheZ-stage,Piezomechanik,Germany),sliders(BWU8-10,IKO,Japan),andscrewswerecommerciallyavailable.ThecompliantmechanismsoftheX-andY-stagesweremadeofTC4titaniumalloy,andthatoftheZ-stagewasmadeofstainlesssteelS136.Theywerefabricatedbywireelectricaldischargemachiningtool.Otherpartswerefabricatedwith7075aluminumalloybycomputernumericalcontrolmachiningtool.StructuralparametersofthecompliantmechanismsareillustratedinSupplementaryFigs.1and2.

Experimentalsetup
ThecontrolalgorithmswerewritteninMATLAB/Simulink2023b,whichweredownloadedtothedSPACEboard(MicroLabBox,dSPACEGmbH,Germany).The16-bitdigital-to-analogconvertersofthedSPACEboardoutputtheanalogvoltagesignals,whichweresuppliedtotwovoltageamplifiers(E01.A2,CoreMorrow,China,amplificationratio:20,fortheX-andY-stages;HVA200,THORLABS,USA,amplificationratio:20,fortheZ-stage)togeneratetheexcitationvoltagesforthepiezoelectricstacks.TheX-andY-axialdisplacementsweremeasuredbytheinterferometricdisplacementsensors(IDS3010withprobeD4/F17,Attocube,Germany)andrecordedbythedigitalinput/outputmoduleofthedSPACE.Thedisplacementsensorsallowedonlineadjustmentofthebalancebetweenresolutionandmeasurementrange.Forsub-nanometer-levelpositioningresolutiondemonstrations(SupplementaryFig.9),wesettheresolutionas0.05nm,insuchacase,thesensorsprovidedanapproximately0.5mmmeasurementrange.Allotherexperimentsused0.5nmresolution,correspondingtoarangeofapproximately5mm.Thesamplingfrequencyoftheexperimentalsetupwassetto20kHz.

TheAFMimagingexperimentswerecarriedoutinconstant-forcecontactmode.AcommercialAFMhead(Nanite,Nanosurf,Switzerland)equippedwithacontact-modeprobe(ContAl-G,BudgetSensors,Bulgaria,forceconstant:0.2N/m)wasused.Thedeflectionsignalwascollectedbytheanalog-to-digitalconvertersofthedSPACEandwasusedasthefeedbackcontrolsignaloftheZ-stage.Theheightcalibrationstandardsample(HS-100MG,BudgetSensors,Bulgaria)hasa100nmstepheight.ThemicrochipsamplewasprovidedbyShanghaiInstituteofOpticsandFineMechanics,ChineseAcademyofSciences.ThephotographsinFig.4dandeweretakenwiththebuilt-incamerasintheAFMhead.ThestitchingprocesswasconductedusingthecommercialsoftwareMountainsMap.

AphotographofthemainexperimentalsetupcanbefoundinSupplementaryFig.15.Allexperimentswereconductedunderambientconditions,wherenoisemayhaveoriginatedfromfloorvibrations,airflow,andelectricalinterference.

Implementationandparametersofthecontroller
Thetriple-phasecontrolalgorithmwaspresentedforcontrollingtheX-andY-stick-slipstages(SupplementaryFig.3).Inparticular,phases-1and3canbeviewedastargetedenhancementsofsteppingandscanningperformance,respectively.Inapplicationswhereneitherhigh-steppingvelocitynorwidecontrolbandwidthiscritical,thecontrollercanbesimplifiedtoawrap-stateintegralcontroller(i.e.,phase-2only),omittingphases-1and3,whichwouldstillsatisfythecross-scalefunctiondemandsofmostscenarios.Itshouldbenotedthat,theproposedcontrollerwasdesignedforerrorsuppressionandisthuswellsuitedfortrackingorscanningarbitrarytrajectories.WhenThetrajectoriesarepredefined,learning-basedcontrolalgorithmssuchasrepetitivecontrol44anditerativelearningcontrol45canbefurtheradoptedtoenhancethescanningaccuracyandfrequency.

Fromadesignperspective,thecurrentprototypewasdevelopedtomeetthegrowingperformancedemandsofAFMimaging.Therefore,itincludestwostick-slipaxesandonedirect-driveaxis.Forapplicationsthatneedtri-axialcross-scalehigh-bandwidthmotion,theZ-axiscouldalsobedesignedbyusingthestick-slipprinciple.Infact,stick-slippiezoelectricsystemscanrealizenotonlylinearmotions,butalsorotationalmotions17,28.Giventheirabilitytoprovidelong-range,high-frequency,andhigh-precisionmotion,futuredevicesbasedonthestick-slipprincipleholdpromiseforimprovingthroughputinmanycutting-edgescenarios,suchasroboticsurgery46,microassembly47,biologicalpuncture48,andscanningprobelithography49.

Incontrolphase-2,thevalueoftheintegralgainki ofthewrap-stateintegralcontrollerwasselectedas20forboththeXandYaxestoensureefficientsteppingandrapidsettling.CU=40VandCL=0Vwerechosentheupperandlowervaluesofthewrappedstate,respectively.Therefore,thecontrolvoltageinphase-2isconsistentlymaintainedbetween0Vand40V,therebyensuringasufficientrangeofatleast80V(withthemaximuminputvoltagebeing120V)tosupporttheforthcominghigh-bandwidthnano-precisionscanninginphase-3.

High-gainproportional-integralcontrollerswereemployedincontrolphase-3forahigh-bandwidthcontroloftheX-andY-stages.Theproportional-integralcontrollercanbeexpressedas

CPI(s) = kp + ki/s (5)

wherekp andki denotetheproportionalandintegralgainsofthecontroller,respectively.Withthegoalofmaximizingthecontrolbandwidth,theparameterswereelaboratelyselectedaskp=0.01,ki=44,fortheX-stage,andkp=0.01,ki=40,fortheY-stage,respectively.Theamplitude-frequencyresponsesoftheclosed-loopsystemscanbefoundinFig.3eandf.

Inconstant-forcecontactmodeAFMimaging,thegoaloftheZ-axialcontrolisessentiallytokeepthedeflectionsignalconstantatthesetpoint.Inthiswork,thesetpointofthedeflectionsignalwasselectedas10mV(Fig.4h).Asimpleintegralcontrollerwasemployedforconductingthefeedbackcontrol(SupplementaryFig.10a),whichhastheexpressionof

CI(s) = ki/s (6)

Thegainwaschosenaski=160000inordertoachieveabalancebetweenbandwidthandstability.Theclosed-loopamplitude-frequencyresponsedemonstratesthataZ-axialbandwidthof1790Hzwasattained(SupplementaryFig.10b).

Positionerrorevaluation
Thecriteriaforevaluatingthepositionerrorsoftrackingandscanningoperationsaredifferent.Thetrackingtasksrequirethenanopositionertofollowthearbitrarypositionoftheobjectinreal-time.Conversely,thescanningtasksletthenanopositionerscanapredefinedtrajectory,itispreferabletoachieveaperfectlydelayedtrackingratherthananimperfecttimelytracking.Therefore,errorsresultingfromthetimedelaycanbetoleratedforthescanningtasks.

Forevaluatingthesingle-axialpositionerrorofatrackingtask,themaximumerror

eimax = maxt∈(0,T] |ri(t) - xi(t)| / (max(ri(t)) - min(ri(t))) (7)

androot-mean-squaretrackingerror

eiRMS = sqrt(1/T ∫t∈(0,T] [ri(t) - xi(t)]^2 dt) / (max[ri(t)] - min[ri(t)]) (8)

werechosenastheindicators,wherei=X,Y,ri(t)andxi(t)denotetheX-orY-axialreferencetrajectoryandoutputdisplacement,respectively.Foratwodegreesoffreedomtracking,theerrorsintheXYplanecanbefurthercalculatedby

emax = sqrt(eXmax^2 + eYmax^2) (9)

and

eRMS = sqrt(eXRMS^2 + eYRMS^2) (10)

Forevaluatingthesingle-axialpositionerrorofascanningtask,theimpactofthetimedelayhadbeeneliminated50,51,andthemaximumscanningerrorandtheroot-mean-squarescanningerrorcanbeexpressedas

^eimax = maxt∈(0,T] |^ri(t) - xi(t)| / (max(^ri(t)) - min(^ri(t))) (11)

and

^eiRMS = sqrt(1/T ∫t∈(0,T] [^ri(t) - xi(t)]^2 dt) / (max[^ri(t)] - min[^ri(t)]) (12)

respectively.InEqs.(11)and(12),^ri(t)=ri(t-k^Ts)denotestheshiftedreference,whereTsisthesamplingperiodoftheexperimentalsetup,andk^denotesthenumberofthedelayedsamplingperiods.Thevalueofk^isdeterminedbysolving

k^ = argmink maxt∈[0,T] |ri(t-kTs) - xi(t)| (13)

Foratwodegreesoffreedomscanning,theerrorsintheXYplanecanbeexpressedas

^emax = sqrt(^eXmax^2 + ^eYmax^2) (14)

and

^eRMS = sqrt(^eXRMS^2 + ^eYRMS^2) (15)

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This paper presents the work on the atomic force microscopy imaging with a stick-slip nanomanipulator to cross-scale high-throughput nano scanning measurement. The topic is interesting. The comments are summarized as follows:\n(1) In this work, a two-DoF stick-slip stage is tried to apply in an AFM to achieve cross-scale high-throughput nanoscanning test, and it is necessary to clearly indicate which performance indicator (positioning accuracy, repeat positioning accuracy, or resolution) of the stage should be utilized during the scanning test process.\n(2) The advantages of the stick-slip stage lie in its small size and good output motion resolution. However, there are also many issues, such as backward motion control and structural design problems. It is recommended to provide more detailed on the design and control of the stage. How about the speed of the stick-slip stage compared with the VCM driven stage?\n(3) It is suggested to provide the experimental test results on the motion resolution or accuracy of the stage during the large stroke motion process and present the noise level of the environment. Figure 5 （c），（d） shows that the environmental noise is relatively high.\n(4) A more detailed performance comparison with commercial AFM should be given.\n(5) More extensive testing on large-area micro and nano structures is suggested to be conducted.", "reviewer_2": "The manuscript by Wang et al describes the development and testing of a novel atomic force microscope stage that is capable of obtaining nanoscopic control over macroscopic distances through the combination of piezoelectric motion and stick-slip actions to move the stage. The manuscript has an impressive amount of characterization and the ability to move large distances using a relatively simple system. My main complaint with this manuscript is that it does not provide a very substantial improvement over the state of the art. Thus, the authors should address the following comments before I can support publication.\n1) The specific application where this system actually provides value over commercial products is not clear. This criticism is based on two factors. First, the stated resolution of 5 nm is not competitive with commercial AFMs which feature sub-nm resolution. Second, the accuracy upon millimeter-scale motion is several to tens of micrometers. There exist commercially available AFMs that features a piezo combined with a motorized stage (e.g. a Dimension Icon) and boast 2-3 micrometer repeatability, obtaining a much greater total range and better repeatability than the present work. Thus, from an accuracy or precision perspective, the proposed system falls short of existing products. This leaves the combination of speed and large amplitude motion as the main potential virtue. However, all single-probe AFMs are inherently limited by the need to collect data from a single probe which is bandwidth limited, a fact which fundamentally limits the speed at which data can be collected and therefore the attainable resolution/throughput tradeoff. With these considerations in mind, it is not clear that the reported system provides any advantage over what is commercially available from a practical standpoint.\n2) The interferometric displacement sensor used to assess the position of the stage is a crucial element that is barely discussed in the manuscript. This should be given further details along with its range and precision.\n3) Stick-slip and the presence of dynamic friction raises the question of mechanical wear. Was there any consideration given to the wear of this system?\n4) Figure 1 contains text in the color bars of d-g that are illegible.", "reviewer_3": "This article describes the design and control of a long-range stick-slip nanopositioner, with 30um linear scanning range. The difference between this and existing designs seems to be the addition of a flexure; however, this is not clear. I believe the topic is worthy of investigation; however, there are several significant issues that would need to be addressed before this could be considered for publication. These are listed below, and arrange by the section they appear in.\n*Abstract*\n1) The abstract (and other parts of the article) switch between discussing nanomanipulation and nanopositioning. Nanomanipulation usually means micro-grippers etc. The article seems to be focussed on positioning, so it would be clear if nanomanipulation is not used.\n2) In general. the abstract could be shorter and more concise. I found it difficult to determine the exact contribution and novelty.\n*Introduction*\n3) Remove the additional hyphen from 3-degrees-of-freedom, and remove the unnecessary acronym 3-DoF, which is only used four times.\n4) Resonance frequency, instead of resonant frequency\n5) Nanopositioner, instead of nanomanipulator\n6) The grammar, conciseness, and clarity of the writing need significant improvement. This makes the article difficult to understand in places.\n7) The authors claim AFM positioners are only capable of dozens of microns; actually, the most common range is 100um.\n8) Revise the statement about infinity stroke.\n9) Avoid using confusing grammar like “high speed/frequency” – does this mean the ratio? Or? And?\n10) The differences between the proposed work and existing devices and methods is not clear. For example, is the mechanical design identical to a standard stick-slip nanopositioner with a new controller? Or new driving signals? Or is the mechanical design novel? This is not clear. Maybe a figure would help that shows the differences between the proposed design and existing methods?\n*Results*\n11) Similar issues with grammar, and lack of preciseness in the writing.\n12) Figure 2 (j and l). The cross-coupling is around 1% which is good for this type of nanopositioner. Do the authors think that this is just due to slight misalignment of the axes? Or is the Y stage making small steps when the X stage is moving?\n13) Figure 2 (o and p). Why are the resonance frequencies so similar when the Y stage is carrying so much more mass?\n14) Thanks for the video results, these were interesting and impressive.\n15) The manuscript talks about piezoelectric actuators (PEAs) and piezoelectric stacks (PESs), are these the same thing? If so, please avoid multiple different words and acronyms for the same things.\n16) Only use an acronym if the phrase is repeated frequently, e.g. 10 times or more. Too many acronyms make it difficult to read.\n17) When moving at high-speed in X and Y, what is the Z cross-coupling? Could AFM contact be maintained during fast movement?\n18) In the AFM experiments, please list all of the imaging parameters, preferably in the figure caption.\n19) Based on the abstract an introduction, I was expecting large range AFM scanning; however, the scan range is only 12x12um @ 40Hz line-rate. Therefore, the benefit of the proposed approach is that the scanner can move quickly between different scan areas. However, if the system is spending a lot of time to scan the AFM image, it doesn’t seem to matter much if the system can reposition in a few seconds. Please comment.\n20) Overall, the proposed design is not convincing as a large range AFM scanner. It is a good milli-meter range stepping stage; but it’s not clear what novelty has been introduced, especially compared to commercial products. For example, the NearField Instruments AFM has four heads that can cover an entire 300mm wafer, with high-speed small-range scanning.\n*Supplementary*\n21) The authors claim that stick-slip nanopositioners are not used in multi-DOF stages. However, these are available commercially from multiple sources, e.g. Attocube, PI, etc.\n22) The authors discuss three phases, but only two are mentioned in Figure 5 and Figure 3 in the main manuscript. What is the difference between phase 1 and phase 2? These are not well explained."}, "author_response": "Dear reviewers:\nWe would like to submit a revised manuscript entitled \"Cross-Scale High-Bandwidth Atomic Force Microscopy with a Stick-slip Nanopositioner\", which we wish to be considered for publication in NATURE COMMUNICATIONS.\nIn our previous submission (NCOMMS-24-76163), all the reviewers showed their interests and gave constructive comments on our work. The authors thank the anonymous reviewers for the thorough review of our manuscript and constructive comments. Your input has allowed us to improve the quality of our manuscript substantially. We respond to all of your questions and concerns, indicating the changes we implemented in our manuscript accordingly. All revisions are highlighted in RED in the revised manuscript, and the point-by-point responses to the comments are provided in the following.\n[... full author response text ...]"}, {"round": 2, "reviewer_comments": {"reviewer_1": "Stick-slip Nanopositioner has application potential in atomic force microscopy testing, and I have no further comments.", "reviewer_2": "The authors have addressed my comments through substantial revisions to the manuscript. I support publication in this form.", "reviewer_3": "The majority of my concerns have been addressed in the revised manuscript. I believe it is acceptable for publication."}, "author_response": ""}]

Decision letter