Converse flexoelectric two-dimensional MoS_2 actuator

Domain: neuroscience
Published: 2026-02-09
Editorial outcome: minor_revision

Abstract
Converse flexoelectric actuators offer significant potential for diverse applications due to their rapid response, resilience in extreme environments, and broad material compatibility. However, their development has been limited, primarily because flexoelectric effects are negligible in bulk materials. At the nanoscale, the converse flexoelectric effect becomes much more promising, as the electric field gradient scales quadratically with decreasing material thickness. Here, we report a converse flexoelectric actuator based on two-dimensional molybdenum disulfide. Under alternating current excitation near 20 kHz, the actuator exhibits resonant displacements up to ~45 nm, approximately two orders of magnitude larger than the thickness of the molybdenum disulfide active layer. This performance exceeds that of existing flexoelectric actuators by more than an order of magnitude when normalized by active layer thickness. Moreover, the actuator maintains strong flexoelectric responses under extreme conditions, including vacuum, cryogenic temperatures, and repeated cycling, highlighting the robustness and broad applicability of two-dimensional material-based converse flexoelectric systems.

Introduction
With the growing demand for smaller systems and precise controls for nano/micro-robotics, nanoscale metrology and manipulating instruments, and aerospace applications, various types of nanoscale actuators have been actively explored. These nano-actuators would have to satisfy a number of requirements such as force, displacement, response speed, and repeatability. Among several types of nano-actuators including piezoelectric, electrostatic, electrostrictive, magenetostrictive, and thermal expansion-based actuators, piezoelectric actuators (piezo-actuators) are favorable for many applications due to their fast response, suitability for vacuum condition, nanoscale controllability, and compact size1,2. Nevertheless, several drawbacks also exist: i) only non-centrosymmetric crystals exhibit piezoelectric effect3, ii) a majority of piezo-materials contain heavy metals such as lead which are not biocompatible3,4, iii) the stroke of piezo-actuators is significantly low relative to the actuators’ size2,5,6, and iv) piezoelectric effect decreases sharply in cryogenic environment7–9.

Flexoelectric (or converse flexoelectric) effect, a similar electromechanical effect to the piezoelectric effect, is an electric polarization under non-zero strain gradient (or a mechanical response under non-zero electric field (E-field) gradient). Flexoelectricity exists in all dielectrics including centrosymmetric materials, and consequently, flexoelectric actuators (flexo-actuators) are a promising candidate to overcome aforementioned first and second limitations of piezo-actuators. In addition, since the mechanical response from converse flexoelectricity increases quadratically as the material’s thickness decreases, there is room for improving the relative stroke with extremely thin active layer10–12. Although there have been a few recent studies related to flexo-actuators, the active materials were in micro/millimeter scale, and thus the strokes were small relative to their size13–16. In this respect, two-dimensional (2D) materials can play a critical role as active materials for the flexo-actuators due to their atomically thinness and mechanical flexibility. Although the flexoelectric effect of 2D materials has been explored in both theoretical and experimental studies, most works have focused on materials-level behavior or computational modeling12,17–19. In contrast, an actuator driven by converse flexoelectric response of 2D materials has not been demonstrated to the best of our knowledge.

Here we show a flexo-actuator operated by converse flexoelectric effect leveraging 2D materials. E-field gradient is generated in a monolayer molybdenum disulfide (MoS2) through asymmetric electrode design to actuate a 600 nm thick beam-type structure. In-plane strain and strain gradient generated by the MoS2 active layer induce ~45 nm of dynamic out-of-plane actuation under 40 V peak-to-peak (Vpp) AC voltage. This actuation level is more than an order of magnitude higher than that of reported flexo-actuators when normalized by active layer thickness and applied voltage13–15, and it is even comparable to piezo-actuators20–30. In addition, the actuation of our device can be controlled with nanoscale precision by modulating the applied voltage, since the actuation is linearly proportional to the applied voltage. Furthermore, our device shows robust performance up to 1010 cycles and maintains ~70% of its actuation under 10 K cryogenic condition in contrast to the conventional piezo-actuator maintaining only ~40% of its actuation under the same condition. Our results highlight the novelty of flexoelectricity in 2D materials, demonstrating substantial scaling advantages and strong potential for cryogenic device applications.

Results
2D material-based flexo-actuator
Figure 1a shows the design of our flexo-actuator. Our device is composed of four layers: patterned 50 nm-Au top electrode, monolayer MoS2 active layer, 500 nm-Parylene-C insulating/supporting layer, and 50 nm-Ag bottom electrode (see Methods for fabrication details). In order to generate a strong E-field gradient in the MoS2 active layer, the top electrode was patterned to have a comb-like shape with 20 μm pitch (Fig. 1b), while a metal electrode without pattern was used as the bottom electrode. The multilayered structure was then transferred on the sample holder having ~1 mm gap in the middle to form a suspended beam structure (Fig. S1). Photoluminescence (PL) mapping showing strong and uniform PL intensity at 665 nm across our device demonstrated the quality and uniformity of the MoS2 active layer (Fig. 1c)31.

To characterize the mechanical actuation of our flexo-actuator device, a function generator, an amplifier, and our device were connected in series to apply a sinusoidal AC voltage to the device (Fig. 1d). When the AC voltage induced the deformation of the flexo-actuator, the resultant motion of our device was measured by a laser vibrometer as a form of vertical velocity. Both the applied voltage and vertical velocity signals were observed by an oscilloscope in real time (Fig. 1d and Fig. S2). The displacement (s) of the device was then calculated from the vertical velocity (v) and frequency (f) by s = v/2πf.

After setting up the testing environment, we began by measuring the displacement of the flexo-actuator device under different frequencies of 40 Vpp AC voltage. This measurement provided a displacement profile and maximum displacement of the flexo-actuator device. As shown in Fig. 2a, our flexo-actuator device exhibited a sharp peak near 19 kHz and actuated up to 42 nm when 40 Vpp of AC voltage was applied.

We next measured the displacement of the flexo-actuator device under various applied AC voltages (10~40 Vpp) while keeping the frequency of the applied AC voltage at the resonant frequency (19 kHz). This measurement was used to determine the relationship between the device displacement and the applied AC voltage. As shown in Fig. 2b, the displacement of our flexo-actuator device linearly increased as the applied AC E-field increased. This linearity demonstrates that the actuation of the flexo-actuator device can be controlled by simply adjusting the applied AC voltage. Additionally, the linear relationship supports that the actuation was governed by converse flexoelectricity, as the mechanical response of the converse flexoelectricity is linearly proportional to the applied E-field gradient32.

To study the origin of our device actuation, we fabricated and characterized three different types of devices: i) devices without MoS2 layer (control samples), ii) devices with monolayer MoS2, and iii) devices with bilayer MoS2. We observed linear AC E-field versus displacement for all three types of devices with different displacement levels (Figure S3). Under 40 V of Vpp, the control samples showed less than 5 nm of actuation, whereas the devices with monolayer and bilayer MoS2 exhibited 45 nm and 37 nm of average actuation, respectively (Fig. 2c). These results suggest important implications for the actuation mechanism. First, since the average displacement of the control devices is much smaller than that of monolayer and bilayer MoS2 devices, the displacement can be attributed primarily to the MoS2 active layer. Second, the similar level of displacement between monolayer and bilayer MoS2 suggests that the main cause of actuation is likely due to converse flexoelectricity. It has been understood that only odd-layer MoS2 exhibits in-plane piezoelectricity33. If in-plane piezoelectricity dominated the actuation, we should have observed much smaller displacement from bilayer MoS2 because of diminishing in-plane piezoelectricity.

To put our results in perspective with other beam-type nano-/micro-piezoelectric and flexoelectric devices, we compared the actuation performance of our device with other reported values. As a performance factor, we used displacement normalized by active layer thickness times E-field, i.e., displacement/(active layer thickness × E-field). This factor indicates the degree of device deformation with unit thickness of active layer under the same E-field. The performance factors of various actuators are plotted in Fig. 2d. Our device showed approximately 103 nm/V of performance factor which is more than an order of magnitude larger than that of state-of-the-art flexoelectric devices (refs. 13–15 in Fig. 2d). Furthermore, our device also exhibited comparable performance factor compared to the beam type piezo-actuators having similar actuation mechanism with our device (refs. 20–24 in Fig. 2d for dynamic piezo-actuators and refs. 25–30 for static piezo-actuators)20–30.

Computational study to verify the actuation mechanism
For further investigation on the actuation mechanism, we conducted computational study on our flexo-actuator. Since the maximum vertical displacement of the flexo-actuator (>40 nm) is approximately two orders of magnitude larger than the thickness of monolayer MoS2 (~0.65 nm), a vertical expansion of MoS2 layer due to vertical stress (σ33) would be an implausible source of the actuation. In contrast, in-plane stress (σ11) generated in MoS2 layer can induce a beam bending (Fig. S4). If we assume both piezoelectric and flexoelectric effects are contributing mechanisms, in-plane stresses can be derived from Eq. (1), where ε, μ and E are converse piezoelectric coefficient, converse flexoelectric coefficient, and E-field, respectively32. Subscripts 1, 2, and 3 indicate directions of beam’s length, width, and height, respectively.

σ11 = ε111E1 + ε122E2 + ε133E3 + μ1111 ∂E1/∂x1 + μ1122 ∂E2/∂x2 + μ1133 ∂E3/∂x3 + μ1112 ∂E1/∂x2 + μ1113 ∂E1/∂x3 + μ1123 ∂E2/∂x3 + μ1121 ∂E2/∂x1 + μ1131 ∂E3/∂x1 + μ1132 ∂E3/∂x2 (1)

For monolayer MoS2 (2H phase, D3h point group), Eq. (1) can be reduced to Eq. (2)18,34.

σ11 = ε111E1 + ε122E2 + ε133E3 + μ1111 ∂E1/∂x1 + μ1122 ∂E2/∂x2 + μ1133 ∂E3/∂x3 (2)

In our case, since there is no E-field and E-field gradient in the 2-direction, the second (ε122E2) and fifth (μ1122 ∂E2/∂x2) terms in Eq. (2) become zero. In addition, monolayer MoS2 exhibits zero ε133 due to inversion symmetry in the out-of-plane direction, and thus the third term (ε133E3) also becomes zero35.

In order to estimate the magnitude of each remaining term and in-plane stress, we calculated E1, ∂E1/∂x1, and ∂E3/∂x3 with electromagnetics simulation. For the simulation, a 2D model representing the cross section of the actuator in the 13-direction was developed as illustrated in the upper schematic diagram of Fig. S5. Each yellow rectangle labelled as ‘top electrode’ represents each tooth of the comb-shaped electrode. The resultant E1, E3, ∂E1/∂x1, and ∂E3/∂x3 at the mid-plane and different locations on MoS2 film are shown in Figs. S5 and S6. Clearly, all the electric fields and field gradients are highly concentrated at the near-edge area of the top electrode where the asymmetry occurs. E1 shows the same magnitude but opposite sign at the left and right edges of the top electrode. Therefore, the tensile and compressive stresses would balance each other out, and the piezo-induced stress term ε111E1 becomes zero overall. Therefore, the Eq. (2) is further reduced to the Eq. (3), concluding that the actuation is solely originated from the converse flexoelectric effect.

σ11 = μ1111 ∂E1/∂x1 + μ1133 ∂E3/∂x3 (3)

To verify whether converse flexoelectricity can fully induce a 45 nm dynamic displacement, we further refined our model, as shown in Fig. 3a, to estimate the actuator’s harmonic response. To reduce computational complexity and cost, the model was developed in 2D to represent the actuator’s cross-section in the length-height plane, as its configuration is uniform along the width. The model includes a 50 nm-thick bottom electrode, a 500 nm-thick Parylene-C layer, a MoS2 monolayer, and a comb-shaped 50 nm-thick top electrode. As fully simulating the 1-mm-long actuator with a sub-0.65 nm mesh is not feasible because of its extremely high aspect ratio and software limitations, we obtained resonant frequency and displacement trends from shorter beam lengths up to 0.20 mm and extrapolated them to estimate the behavior of the 1 mm-long actuator.

Figure 3b shows the resonant frequency trends as the actuator length changes. The resonant frequency decreases quadratically as the length increases, consistent with the theoretical relationship for beam resonance (i.e., resonant frequency ∝ 1/(length)2). From this extrapolation, we found that the 5th-mode resonant frequency of a 1 mm beam (20.8 kHz) matches well with the actuation frequency of our device (19 kHz).

We then estimated the 5th harmonic displacement of the beam. With the assumption that the 1111-converse flexoelectric stress (σ11 = μ1111 ∂E1/∂x1, the first term in Eq. (3)) is the dominant driving mechanism of our actuator and with an estimated μ1111 of 1.41 nC/m, we were able to obtain displacement of 45 nm through our electromechanical simulation, consistent with our experimental observations. As shown in Fig. 3c, the harmonic displacement increases quadratically with the beam length, and μ1111 = 1.41 nC/m yields 45 nm displacement for a 1 mm-long beam. Although the reported values of μ1111 in literature vary significantly, μ1111 used in our simulation lies within the reported range (0.6821–174.1 nC/m)36–38.

To study the effects of MoS2 thickness on the actuation behavior, we simulated displacement and resonance frequencies of bi- and tri-layer MoS2. The resonant frequency and displacement trends remained consistent for bi- and tri-layer MoS2 actuators (Fig. S7a). In addition, increasing the MoS2 thickness from monolayer to bilayer reduces the harmonic displacement of the actuator, which aligns with our experimental observation that the bilayer device exhibits approximately 18% lower displacement than the monolayer counterpart. This reduction is primarily attributed to the increased flexural stiffness resulting from the additional MoS2 layer.

We also evaluated displacement contributions from other potential mechanisms, including piezoelectricity, 1133-converse flexoelectric stress (σ11 = μ1133 ∂E3/∂x3, the second term in Eq. (3)), Joule heating, and electromagnetic effects. We confirmed that the 1111-converse flexoelectric effect is the dominant contributor (Supporting Information and Fig. S8). Further details regarding the simulation parameters and material properties are provided in the Methods and Supporting Information.

As our measurements of the actuator displacement were acquired at the 5th resonant frequency, there is a significant potential to further enhance the device performance by utilizing lower-order harmonic resonance. Reducing the width and pitch of the electrode combs to maximize the number of combs in the device would further improve actuation performance. Furthermore, the resonant frequency and the width of the resonant peak of the actuator can be tuned by adjusting the length, thickness, elastic modulus, and viscosity of the supporting layer to meet specific application requirements, offering a wide range of applicability.

Stability and robustness of the actuator under extreme environment
With the understanding of the actuation mechanism and converse flexoelectric coefficients of 2D MoS2, we further explored the testing of flexo-actuators under cryogenic and vacuum environment. Our actuator was operated in its resonant frequency with 40 Vpp AC while decreasing temperatures from room temperature (RT) to 10 K in vacuum environment. We observed stable actuation performance where the actuator was able to maintain 70% of its RT displacement at 10 K (Fig. 4a). Furthermore, similar to RT behavior, our flexo-actuator exhibited linear displacement versus E-field relationship (Fig. 4b). Most interestingly, our flexo-actuator significantly outperformed a state-of-the-art disk-type lead zirconate titanate (PZT) piezo-actuator which experienced a 60% reduction in displacement at 10 K (Fig. 4a). This result suggests flexo-actuators might serve as a robust platform for cryogenic applications where temperature scaling of piezoelectricity might pose challenges to conventional piezo-actuators7,8,39.

To further study long-term robustness of our flexo-actuator, our monolayer MoS2 device operated for 1010 cycles at both 10 K and RT. As shown in Fig. 4c, the device survived at least 1010 actuation cycles with less than 12% of performance fluctuation, regardless of operating temperature. These results support the reliability of our flexo-actuator towards cryogenic and long-term operations.

Summary and Outlook
In conclusion, we have developed a 2D MoS2 flexo-actuator based on converse flexoelectric effects of the MoS2 active layer. Experimental and simulation studies demonstrated that the primary source of actuation was converse flexoelectric effect. With increasing flexoelectric effect at the nanoscale, our device exhibited performance enhanced by orders of magnitude compared to existing flexo-actuators. Additionally, the actuation of our device was linearly proportional to the applied AC voltage, allowing electrical control of actuation. Considering the extremely thin active layer thickness and performance under cryogenic conditions of our device, this study demonstrates the potential of 2D flexo-actuators for a wide range of applications which have been dominated by piezo-actuators.

Methods
Device fabrication
To fabricate our device, we prepared a multilayer beam structure of Au/MoS2/Parylene-C/Ag on a 300 nm SiO2/Si wafer and then transferred it onto a sample holder. A 5 wt% polyacrylic acid (PAA) solution was spin-coated to form a thin sacrificial layer on the SiO2 wafer. A 50 nm thick bottom Ag electrode was then deposited by electron beam (E-beam) evaporator. For mechanical support and electrical insulation, a 500 nm thick Parylene-C layer was coated by Parylene deposition system (PDS2010). Then, a monolayer MoS2 film grown on a SiO2/Si wafer by metal-organic chemical vapor deposition (MOCVD) was transferred onto the Parylene-C layer. We placed the MoS2/SiO2/Si wafer upside down onto the Parylene-C layer and dropped a water droplet to separate monolayer MoS2 from the SiO2/Si wafer by water penetration. After we fully dried the MoS2/Parylene-C/Ag/PAA/SiO2/Si sample at ambient condition, we coated a patterned photoresist layer onto the sample through photolithography. Then, a 50 nm thick Au top electrode was deposited by E-beam evaporator. The photoresist layer was removed after the deposition. To transfer the Au/MoS2/Parylene-C/Ag layer from the wafer to a sample holder, the PAA layer was dissolved by placing the sample on water surface. After full separation, the four-layer structure was transferred onto the sample holder. The sample holder was prepared by placing two glass slides having a 1 mm gap between them.

Simulation setup
COMSOL Multiphysics was employed to investigate the actuation mechanism of our device. The model was developed in a 2D plane and consisted of four layers: a 50 nm-thick bottom electrode, a 500 nm-thick Parylene-C layer, a MoS2 layer with thicknesses of 0.65, 1.3, or 1.95 nm depending on the number of layers, and a 50 nm-thick top electrode. The top electrode was designed with 25 individual rectangular segments to mimic a comb-shaped electrode in 2D.

Electric field and electric field gradient distributions within the MoS2 layer were computed using the Electrostatics module. Effects of flexoelectricity and piezoelectricity on the device actuation were conducted using the Electrostatics, Solid Mechanics, and Piezoelectric Effect Multiphysics modules in combination. Since the flexoelectric effect is not natively supported in COMSOL, the equations for the variables solid.PpzeX (Piezoelectric polarization, X component) and solid.Spze11 (Piezoelectric stress tensor, local coordinate system, 11 component) were modified by incorporating a μ1111- and μ1133-dependent terms into each expression, mu_1111*d(solid.eel11,x) + mu_1133*d(solid.eel22,y) and -mu_1111*d(solid.EpzeX,x) - mu_1133*d(-solid.EpzeY,y), respectively. Note that y represents 3-directional coordinate as our model was built in 2D. Furthermore, electromagnetic stress calculations were performed using the Electrostatics, Solid Mechanics, and Electromechanical Forces Multiphysics modules based on the Minkowski tensor to estimate the device deformation due to electromagnetic forces under the applied AC voltage. Joule heating effects were also simulated to calculate the device actuation by thermal expansion resulting from heat generated in the top and bottom electrodes, using the Electric Currents, Heat Transfer in Solids, Electromagnetic Heating, and Thermal Expansion Multiphysics modules.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors present a MoS2 based flexoelectric actuation device, with significantly high response and durability under different thermal and mechanical conditions. The results are important, and the reported device holds potential in nano-actuation and control. The manuscript can be considered for publication after addressing the following major issues:\n1. The authors claim to report the first 3D material-based flexoelectric actuator. However, the work by Hirakata et al (DOI: 10.1088/1361-6463/ac4367) also reports a similar actuator made if MoS2. Authors should clarify the difference between their work and Hirakata et al. and justify the novelty argument.\n2. The measurements and computations presented are mostly derived from indirect calculations. For example, it appears that the calculations of stress based on the displacement and 2D electromagnetic simulations are performed assuming only elasticity, not the coupled electromechanical physics. If a 2-way electromechanical coupling is indeed considered, the boundary conditions and assumptions should be reported accurately. It is crucial to consider the effect of fully coupled electromechanical effects since it can have a significant impact on the final results at such small scales.\n3. Line 69-71: “This actuation level is more than an order of magnitude higher than that of reported flexoelectric actuators when normalized by active layer thickness and applied voltage, and it is even comparable to piezo-actuators.” The source of comparison should be cited if the authors are comparing their results with existing literature.\n4. Figure S2 just shows a screenshot of the oscilloscope measurements without axis labels or any form of quantification. This does not contribute anything to the results and raises concern about the measurements. Please replace it with proper plots of input voltage signals, output laser vibrometer measurements, and the FFT transforms, which would be very useful for reproducibility of the results.\n5. The reported gain in actuation with MoS2 vs without MoS2 layer (control) is not conclusive enough to credit the origin of actuation to flexoelectricity. The other plausible reasons for this could be a) Maxwell stress could enhance the actuation due to local fringe effects introduced by the MoS2 layer. b) thermoelastic bending can occur due to Joule heating caused by heat dissipation in the metal combs. These effects can be eliminated by introducing an inert layer or a layer with a known flexoelectric effect. Moreover, the mechanical resonance gains should also be eliminated by comparing the control and MoS2 layered samples off-resonance.", "reviewer_2": "The work is just fine and should be published \"somewhere\" but I see no substantive novelty that would warrant the readership of Nature Communications. Flexoelectricity has been well-studied theoretically and experimentally. Including in 2D materials. It would be like for me to publish a paper on a piezoelectric crystal and claim that it acts like an actuator! Of course it can and will. I don't see what the big deal of the current paper is other than busy work. It would, for example, be a perfectly fine paper for any number of lower tier journals.", "reviewer_3": "This manuscript presents a highly novel and compelling study on the development of a converse flexoelectric actuator utilizing monolayer MoS₂ as the active material. The combination of experimental characterization, electromagnetic simulation, and mechanical modeling provides a multi-faceted and convincing argument for the converse flexoelectric actuation mechanism. I recommend acceptance after addressing the minor points outlined below.\n1. The estimated range for μ₁₁₁₁ and μ₁₁₃₃ (0.1–0.2 nC/m) is notably lower than some values reported in the literature. While the authors correctly state that reported values vary widely, a brief discussion on the potential reasons for this discrepancy would strengthen the manuscript.\n2. The actuation performance of monolayer and bilayer devices is similar. The deep reason should be reached. Is it purely flexoelectricity? If so, does the interface between layers or the slip plane play a role?\n3. The actuator demonstrates impressive performance at its resonant frequency (~19 kHz). However, for many practical applications, a broader frequency band is highly desirable. Could the authors comment on the expected performance of the device when operated off-resonance? What potential strategies could be explored to achieve a wider bandwidth?"}, "author_response": "We thank the reviewer for their thoughtful and constructive feedback, as well as for recognizing the significance and potential of our work. We greatly appreciate the positive evaluation of the device performance and its relevance to nano-actuation and control applications. We have carefully addressed each of the major issues raised and provided a detailed, point-by-point response to the comments below.\n1. The authors claim to report the first 3D material-based flexoelectric actuator. However, the work by Hirakata et al (DOI: 10.1088/1361-6463/ac4367) also reports a similar actuator made if MoS2. Authors should clarify the difference between their work and Hirakata et al. and justify the novelty argument.\nWe thank the referee for this comment. The work by Hirakata et al. reported out-of-plane deformation of MoS₂ at the materials level, not the device scale. It turns out that most recent papers on the flexoelectricity of 2D materials are either computational, limited to the materials level, or focus on out-of-plane directional strain. In contrast, our actuator uses in-plane stress to actuate the entire device. To the best of our knowledge, this is the first experimentally demonstrated 2D material-based flexoelectric actuator. We have included the paper by Hirakata et al. to the references and included additional discussion in the main text to clarify our novelty, as shown below.\nRevision to MS/SI\nIn main text, page 4:\n~. Although the flexoelectric effect of 2D materials has been studied for its fundamental properties explored in both theoretical and experimental studies, most works have focused on materials-level behavior or computational modeling12,17–19. In contrast, an actuator driven by converse-flexoelectric response of 2D materials has not been demonstrated to the best of our knowledge.\nHere we show a flexo-actuator operated by converse-flexoelectric effect leveraging 2D materials. ~(text omitted)~ Furthermore, our device shows robust performance up to 1010 cycles and maintains ~70% of its actuation under 10 K cryogenic condition in contrast to the conventional piezoelectric actuator maintaining only ~40% of its actuation under the same condition. Our results highlight the novelty of flexoelectricity in 2D materials, demonstrating substantial scaling advantages and strong potential for cryogenic device applications.\nIn References:\n1\n19. Hirakata, H., Fukuda, Y. & Shimada, T. Flexoelectric properties of multilayer two-dimensional material MoS . J Phys D Appl Phys 55, 125302 (2021).\n2\n2. The measurements and computations presented are mostly derived from indirect calculations. For example, it appears that the calculations of stress based on the displacement and 2D electromagnetic simulations are performed assuming only elasticity, not the coupled electromechanical physics. If a 2-way electromechanical coupling is indeed considered, the boundary conditions and assumptions should be reported accurately. It is crucial to consider the effect of fully coupled electromechanical effects since it can have a significant impact on the final results at such small scales.\nWe thank the referee for the valuable insights. We have modified the electromechanical model to include flexoelectricity and conducted full simulations. Also, we have included these new results in our revised manuscript as well as in the Methods and Supporting Information as shown below.\nRevision to MS/SI\nIn main text, page 8:\nTo verify whether converse flexoelectricity can fully induce a 45 nm dynamic displacement, we further refined our model, as shown in Figure 3a, to estimate the actuator’s harmonic response. To reduce computational complexity and cost, the model was developed in 2D to represent the actuator’s cross-section in the length-height plane, as its configuration is uniform along the width. The model includes a 50 nm-thick bottom electrode, a 500 nm-thick Parylene-C layer, a 0.65 nm MoS monolayer, and a comb-shaped 50 nm-thick top electrode. As\n2\nfully simulating the 1-mm-long actuator with a sub-0.65-nm mesh is not feasible because of its extremely high aspect ratio and software limitations, we obtained resonant frequency and displacement trends from shorter beam lengths up to 0.20 mm and extrapolated them to estimate the behavior of the 1 mm-long actuator.\nFigure 3b shows the resonant frequency trends as the actuator length changes. The resonant frequency decreases quadratically as the length increases, consistent with the theoretical relationship for beam resonance (i.e., resonant frequency ∝ 1/(length)2). From this extrapolation, we found that the 5th-mode resonant frequency of a 1 mm beam (20.8 kHz) matches well with the actuation frequency of our device (19 kHz).\nWe then estimated the 5th harmonic displacement of the beam. With the assumption that the 1111-converse flexoelectric stress (σ = µ ∂E /∂x , the first term in equation (3)) is the\n11 1111 1 1\ndominant driving mechanism of our actuator and with an estimated µ of 1.41 nC/m, we were\n1111\nable to obtain displacement of 45 nm through our electromechanical simulation, consistent with our experimental observations. As shown in Figure 3c, the harmonic displacement increases quadratically with the beam length, and µ = 1.41 nC/m yields 45 nm displacement for a 1\n1111\nmm-long beam. Although the reported values of µ in literature vary significantly, µ used in\n1111 1111\nour simulation lies within the reported range (0.6821-174.1 nC/m37–39).\n2\ndisplacement and resonance frequencies of bi- and tri-layer MoS . The resonant frequency and\n2\ndisplacement trends remained consistent for bi- and tri-layer MoS actuators (Figure S7a). In\n2\naddition, increasing the MoS thickness from monolayer to bilayer reduces the harmonic\n2\ndisplacement of the actuator, which aligns with our experimental observation that the bilayer device exhibits approximately 18% lower displacement than the monolayer counterpart. This reduction is primarily attributed to the increased flexural stiffness resulting from the additional\nMoS layer.\nincluding piezoelectricity, 1133-converse flexoelectric stress (σ = µ ∂E /∂x , the second term\n11 1133 3 3\nin equation (3)), Joule heating, and electromagnetic effects. We confirmed that the 1111-converse flexoelectric effect is the dominant contributor (Supporting Information and Figure S8). Further details regarding the simulation parameters and material properties are provided in the Methods and Supporting Information.\nFigure 3:\n11 1111 1 1 1111\nFollowing this trend, a 1 mm actuator is predicted to yield a displacement of 45 nm.\nIn Methods:\n(2) Simulation set-up\nCOMSOL Multiphysics was employed to investigate the actuation mechanism of our device. The model was developed in a 2D plane and consisted of four layers: a 50 nm-thick bottom electrode,\n4\na 500 nm-thick Parylene-C layer, a MoS layer with thicknesses of 0.65, 1.3, or 1.95 nm\n2\ndepending on the number of layers, and a 50 nm-thick top electrode. The top electrode was designed with 25 individual rectangular segments to mimic a comb-shaped electrode in 2D.\nElectric field and electric field gradient distributions within the MoS layer were\n2\ncomputed using the Electrostatics module. Effects of flexoelectricity and piezoelectricity on the device actuation were conducted using the Electrostatics, Solid Mechanics, and Piezoelectric Effect Multiphysics modules in combination. Since the flexoelectric effect is not natively supported in COMSOL, the equations for the variables solid.PpzeX (Piezoelectric polarization, X component) and solid.Spze11 (Piezoelectric stress tensor, local coordinate system, 11 component) were modified by incorporating a μ - and μ -dependent terms into each\n1111 1133\nexpression, mu_1111*d(solid.eel11,x) + mu_1133*d(solid.eel22,y) and -\nmu_1111*d(solid.EpzeX,x) - mu_1133*d(solid.EpzeY,y), respectively. Note that y represents 3-directional coordinate as our model was built in 2D. Furthermore, electromagnetic stress calculations were performed using the Electrostatics, Solid Mechanics, and Electromechanical Forces Multiphysics modules based on the Minkowski tensor to estimate the device deformation due to electromagnetic forces under the applied AC voltage. Joule heating effects were also simulated to calculate the device actuation by thermal expansion resulting from heat generated in the top and bottom electrodes, using the Electric Currents, Heat Transfer in Solids, Electromagnetic Heating, and Thermal Expansion Multiphysics modules.\nIn Supporting Information:\n(1) Calculation of Q factor for including damping in the simulation\nIn order to accurately calculate the harmonic displacement, we first estimated the Q factor from experimental data and apply it to the simulation. Figure S9 shows the frequency-displacement plot near the 5th resonant frequency. The Q factor was calculated by dividing resonant frequency by the width of the peak at its half height (∆𝑓), which is 20,700 Hz / 690 Hz = 30.\n(2) Estimation of harmonic displacement of mono-, bi-, and tri-layer MoS actuators\n2\nlimitations, the thicker MoS layer enables us to simulate the resonant frequencies and harmonic\n2\ndisplacement of the longer beam. Figure S7a and b show the 5th harmonic frequency and displacement of mono-, bi-, and tri-layer MoS actuators with different lengths. As with the\n2\nmonolayer case, bi- and tri-layer devices also show quadratically decreasing resonant frequency and quadratically increasing harmonic displacement as the beam length increases. In addition, bilayer MoS actuator shows lower harmonic displacement compared to the monolayer MoS ,\n2 2\nagreeing well with our experimental results.\n(3) Estimation of actuation from piezoelectric effect, 1133-converse flexoelectric stress,\nelectromagnetic stress, and Joule heating\n2\npossible mechanisms to the 5th harmonic displacement, including the piezoelectric effect, 1133-converse flexoelectric stress, electromagnetic stress, and Joule heating. In case of the 1133-converse flexoelectric stress, we applied 1.41 nC/m for μ , the same value as the estimated μ\n1133 1111\nvalue. Table S1 summarizes the parameters used in each simulation, and Figure S8 illustrates the device actuation resulting from each effect. The maximum displacements obtained from the 1111- and 1133-converse flexoelectric effects, piezoelectric effect, and electromagnetic stress are 1.64, 0.014, 0.30, and 0.29 nm, respectively. The displacement resulting from Joule heating is more than 10 orders of magnitude smaller than those from the other effects and can therefore be neglected.\nThese results strongly support that the 1111-converse flexoelectric effect is the dominant mechanism driving device actuation, unless monolayer MoS exhibits a μ value that is\n2 1133\nsignificantly higher than μ . Additionally, when these other effects are taken into consideration,\n1111\nμ is estimated to be in the range of 1.41±0.52 nC/m, under the assumption that μ = μ .\n1111 1111 1133\nTable S1. Simulation parameters\nModule Parameter Value\nSolid Mechanics Young’s modulus, Au 70 GPa\nPoisson’s ratio, Au 0.44\nYoung’s modulus, Ag 83 GPa\nPoisson’s ratio, Ag 0.37\nYoung’s modulus, Parylene-C 2.75 GPa\nPoisson’s ratio, Parylene-C 0.4\nElastic modulus, C , MoS 128.4 GPa [1]\n11 2\nElastic modulus, C , MoS 128.4 GPa [1]\n22 2\nElastic modulus, C , MoS 32.6 GPa [1]\n12 2\nElastic modulus, C , MoS 38.3 GPa [2]\n33 2\nElastic modulus, C , MoS 26.5 GPa [2]\n44 2\nElastic modulus, C , MoS 26.5 GPa [2]\n55 2\n47.9 GPa\nElastic modulus, C , MoS\n66 2 ((C -C )/2)\n11 12\nFlexoelectric coefficient, μ , MoS 1.41 nC/m\n1111 2\n1133 2\nPiezoelectric coefficient, e , MoS 0.5 C/m2 [3]\n11 2\nIsotropic structural loss factor (1/Q factor) 1/30\nElectrostatics & Terminal voltage, top electrode (Au) 20 V\nElectric Currents\nTerminal voltage, bottom electrode (Ag) 0 V\nRelative permittivity, Parylene-C 3\nRelative permittivity, MoS 4 [4]\nDensity, Au 19300 kg/m3\nHeat capacity, Au 129 J/kg∙K\nThermal conductivity, Ag 429 W/m∙K\nDensity, Ag 10500 kg/m3\nHeat capacity, Ag 235 J/kg∙K\nThermal conductivity, Parylene-C 0.084 W/m∙K\nDensity, Parylene-C 1289 kg/m3\nHeat capacity, Parylene-C 712 J/kg∙K\nThermal conductivity, MoS 23.2 W/m∙K [5]\n2\n87.5 J/kg∙K\nHeat capacity, MoS\n2 (14 J/mol∙K) [6]\nThermal Expansion Thermal expansion coefficient, Au 1.42x10-5 /K\n(Multiphysics)\nThermal expansion coefficient, Ag 1.89x10-5 /K\nThermal expansion coefficient,\n3.5x10-5 /K\nParylene-C\nThermal expansion coefficient, MoS 7.6x10-6 /K [7]\n2\nFigure S7. (a) Resonant frequency trends for actuators with different MoS layer thicknesses. A\n2\nquadratically decreasing trend is maintained as the MoS layer becomes thicker with increasing\n2\nlayer number. (b) Fifth-harmonic displacement for different MoS layer thicknesses. Although the\n2\ntrend is maintained, the bi- and tri-layer MoS actuators exhibit lower displacement compared to\n2\nthe monolayer MoS actuator.\n2\neffects, piezoelectric effect, electromagnetic stress, and Joule heating. A 200 μm-long model with\nmonolayer MoS was implemented for the calculations, for which the 5th resonant frequency is\n2\nestimated to be ~430 kHz.\nFigure S9. Fifth-harmonic displacement of the monolayer MoS actuator near the resonant\n2\nfrequency range for Q-factor calcuation.\n3. Line 69-71: “This actuation level is more than an order of magnitude higher than that of reported flexoelectric actuators when normalized by active layer thickness and applied voltage, and it is even comparable to piezo-actuators.” The source of comparison should be cited if the authors are comparing their results with existing literature.\nWe appreciate the referee’s thorough review of our manuscript. We have added citations to the sentence.\nRevision to MS/SI:\nThis actuation level is more than an order of magnitude higher than that of reported flexoelectric\nactuators when normalized by active layer thickness and applied voltage13–15, and it is even\ncomparable to piezo-actuators20–30.\n4. Figure S2 just shows a screenshot of the oscilloscope measurements without axis labels or any form of quantification. This does not contribute anything to the results and raises concern about the measurements. Please replace it with proper plots of input voltage signals, output laser vibrometer measurements, and the FFT transforms, which would be very useful for reproducibility of the results.\nWe thank the referee for this comment. We have updated Figure S2 to the following figure.\nFigure S2. (a) AC voltage applied to the device at the 5th harmonic frequency for actuation. (b)\nOutput voltage signal from the laser vibrometer after fast Fourier transform (FFT), which is\nconverted to vertical velocity and then to displacement.\n5. The reported gain in actuation with MoS2 vs without MoS2 layer (control) is not conclusive enough to credit the origin of actuation to flexoelectricity. The other plausible reasons for this could be a) Maxwell stress could enhance the actuation due to local fringe effects introduced by the MoS2 layer. b) thermoelastic bending can occur due to Joule heating caused by heat dissipation in the metal combs. These effects can be eliminated by introducing an inert layer or a layer with a known flexoelectric effect. Moreover, the mechanical resonance gains should also be eliminated by comparing the control and MoS2 layered samples off-resonance.\nWe appreciate the referee’s valuable comment. We examined other possible effects as suggested and modified the manuscript and Supporting Information as follows:\nRevision to MS/SI\nIn main text, page 10:\nWe also evaluated displacement contributions from other potential mechanisms,\nincluding piezoelectricity, 1133-converse flexoelectric stress (σ = µ ∂E /∂x , the second term\n11 1133 3 3\nin equation (3)), Joule heating, and electromagnetic effects. We confirmed that the 1111-converse\nflexoelectric effect is the dominant contributor (Supporting Information and Figure S8). Further\ndetails regarding the simulation parameters and material properties are provided in the Methods\n10\nand Supporting Information.\nIn Supporting Information:\n(3) Estimation of actuation from piezoelectric effect, 1133-converse flexoelectric stress,\nelectromagnetic stress, and Joule heating\nUsing the 200 μm-long model with monolayer MoS , we also calculated the contributions of other\n2\npossible mechanisms to the 5th harmonic displacement, including the piezoelectric effect, 1133-converse flexoelectric stress, electromagnetic stress, and Joule heating. In case of the 1133-converse flexoelectric stress, we applied 1.41 nC/m for μ , the same value as the estimated μ\n1133 1111\nvalue. Table S1 summarizes the parameters used in each simulation, and Figure S8 illustrates the\ndevice actuation resulting from each effect. The maximum displacements obtained from the 1111- and 1133-converse flexoelectric effects, piezoelectric effect, and electromagnetic stress are 1.64, 0.014, 0.30, and 0.29 nm, respectively. The displacement resulting from Joule heating is more than 10 orders of magnitude smaller than those from the other effects and can therefore be neglected.\nThese results strongly support that the 1111-converse flexoelectric effect is the dominant\nmechanism driving device actuation, unless monolayer MoS exhibits a μ value that is\n2 1133\nsignificantly higher than μ . Additionally, when these other effects are taken into consideration,\n1111\nμ is estimated to be in the range of 1.41±0.52 nC/m, under the assumption that μ = μ .\n1111 1111 1133\nFigure S8:\neffect, piezoelectric effect, electromagnetic stress, and Joule heating. A 200 μm-long model with\nmonolayer MoS was used for the calculations, for which the 5th resonant frequency is estimated\n2\nto be ~430 kHz.\nThe work is just fine and should be published \"somewhere\" but I see no substantive novelty that\nwould warrant the readership of Nature Communications. Flexoelectricity has been well-studied\ntheoretically and experimentally. Including in 2D materials. It would be like for me to publish a\npaper on a piezoelectric crystal and claim that it acts like an actuator! Of course it can and will. I\ndon't see what the big deal of the current paper is other than busy work. It would, for example, be\na perfectly fine paper for any number of lower tier journals.\nWe appreciate the referee’s time and thoughtful critique. We would like to take this opportunity\nto further clarify the specific novelty and contribution of our work in the context of existing\nliterature.\nWhile flexoelectricity in 2D materials has been explored in both theoretical and\nexperimental studies, most of prior works have focused on materials-level phenomena,\ncomputational modeling, or out-of-plane deformation modes. In contrast, our study presents what\nwe believe to be the first experimental demonstration of flexoelectricity using a 2D material at\nthe device level for out-of-plane actuation. Through this demonstration, we reveal the substantial\nscaling advantages of flexoelectric effect in 2D materials and their strong potential for cryogenic\ndevice applications. This is a distinct departure from previous work and, in our view, represents a\nmeaningful step toward translating fundamental material properties into functional actuators. By\nbridging the gap between theoretical predictions and practical implementation, we aim to provide\na proof-of-concept that 2D flexoelectric materials can serve as the foundation for next-generation\nactuation systems - a message we believe aligns with the interdisciplinary and application-\noriented scope of Nature Communications.\nWe hope these points help clarify the contribution of our work and its relevance to the\njournal’s readership, and we added the following sentences in the manuscript:\nRevision to MS/SI\nIn main text, page 4:\n~. Although the flexoelectric effect of 2D materials has been studied for its fundamental\nproperties explored in both theoretical and experimental studies, most works have focused on\nmaterials-level behavior or computational modeling12,17–19. In contrast, an actuator driven by\nconverse-flexoelectric response of 2D materials has not been demonstrated to the best of our\nknowledge.\nHere we show a flexo-actuator operated by converse-flexoelectric effect leveraging 2D\nmaterials. ~(text omitted)~ Furthermore, our device shows robust performance up to 1010 cycles\nand maintains ~70% of its actuation under 10 K cryogenic condition in contrast to the\nconventional piezoelectric actuator maintaining only ~40% of its actuation under the same\ncondition. Our results highlight the novelty of flexoelectricity in 2D materials, demonstrating\nsubstantial scaling advantages and strong potential for cryogenic device applications.\nThis manuscript presents a highly novel and compelling study on the development of a converse\nflexoelectric actuator utilizing monolayer MoS₂ as the active material. The combination of\nexperimental characterization, electromagnetic simulation, and mechanical modeling provides a\nmulti-faceted and convincing argument for the converse flexoelectric actuation mechanism. I\nrecommend acceptance after addressing the minor points outlined below.\nWe sincerely thank the reviewer for their positive and encouraging feedback on our work. We are\nglad that the novelty and multi-faceted approach of our study were appreciated. We have\ncarefully addressed all the minor points raised and have made the corresponding revisions in the\nmanuscript, as detailed in our point-by-point response below.\n1. The estimated range for μ₁₁₁₁ and μ₁₁₃₃ (0.1–0.2 nC/m) is notably lower than some values\nreported in the literature. While the authors correctly state that reported values vary widely, a\nbrief discussion on the potential reasons for this discrepancy would strengthen the manuscript.\nWe thank the referee for the valuable advice. We have thoroughly revised our model with a fully\ncoupled electromechanical model for more accurate calculations and found that the measured\ndeformation arises from a higher-order harmonic rather than from the first harmonic excitation.\nThe resulting estimated flexoelectric coefficient (1.41 nC/m) has been revised accordingly and\nnow falls within the theoretically reported range (0.6821−174.1 nC/m). Potential discrepancies\nbetween our estimation and the actual value would mainly stem from the actual elastic properties\nand permittivity of MoS₂ and Parylene-C in the high-frequency range. We have revised our\nmanuscript and Supporting Information as follows:\nRevision to MS/SI\nIn main text, page 8:\nTo verify whether converse flexoelectricity can fully induce a 45 nm dynamic\ndisplacement, we further refined our model, as shown in Figure 3a, to estimate the actuator’s\nharmonic response. To reduce computational complexity and cost, the model was developed in\n2D to represent the actuator’s cross-section in the length-height plane, as its configuration is\nuniform along the width. The model includes a 50 nm-thick bottom electrode, a 500 nm-thick\nParylene-C layer, a 0.65 nm MoS monolayer, and a comb-shaped 50 nm-thick top electrode. As\n2\nfully simulating the 1-mm-long actuator with a sub-0.65-nm mesh is not feasible because of its\nextremely high aspect ratio and software limitations, we obtained resonant frequency and\ndisplacement trends from shorter beam lengths up to 0.20 mm and extrapolated them to estimate\nthe behavior of the 1 mm-long actuator.\nFigure 3b shows the resonant frequency trends as the actuator length changes. The\nresonant frequency decreases quadratically as the length increases, consistent with the theoretical\nrelationship for beam resonance (i.e., resonant frequency ∝ 1/(length)2). From this extrapolation,\nwe found that the 5th-mode resonant frequency of a 1 mm beam (20.8 kHz) matches well with\n14\nthe actuation frequency of our device (19 kHz).\nWe then estimated the 5th harmonic displacement of the beam. With the assumption that\nthe 1111-converse flexoelectric stress (σ = µ ∂E /∂x , the first term in equation (3)) is the\n11 1111 1 1\ndominant driving mechanism of our actuator and with an estimated µ of 1.41 nC/m, we were\n1111\nable to obtain displacement of 45 nm through our electromechanical simulation, consistent with\nour experimental observations. As shown in Figure 3c, the harmonic displacement increases\nquadratically with the beam length, and µ = 1.41 nC/m yields 45 nm displacement for a 1\n1111\nmm-long beam. Although the reported values of µ in literature vary significantly, µ used in\n1111 1111\nour simulation lies within the reported range (0.6821-174.1 nC/m37–39).\nTo study the effects of MoS thickness on the actuation behavior, we simulated\n2\ndisplacement and resonance frequencies of bi- and tri-layer MoS . The resonant frequency and\n2\ndisplacement trends remained consistent for bi- and tri-layer MoS actuators (Figure S7a). In\n2\naddition, increasing the MoS thickness from monolayer to bilayer reduces the harmonic\n2\ndisplacement of the actuator, which aligns with our experimental observation that the bilayer\ndevice exhibits approximately 18% lower displacement than the monolayer counterpart. This\nreduction is primarily attributed to the increased flexural stiffness resulting from the additional\nMoS layer.\nincluding piezoelectricity, 1133-converse flexoelectric stress (σ = µ ∂E /∂x , the second term\n11 1133 3 3\nin equation (3)), Joule heating, and electromagnetic effects. We confirmed that the 1111-converse\nflexoelectric effect is the dominant contributor (Supporting Information and Figure S8). Further\ndetails regarding the simulation parameters and material properties are provided in the Methods\nand Supporting Information.\nFigure 3:\nupper-left inset shows the monolayer MoS₂ located between the top electrode and the Parylene-C\nlayer. For clarity, the x:y scale has been adjusted to 1:100. (b) Resonant frequency trends for\nactuators of different beam lengths. The 4th, 5th, and 6th resonant frequencies all decrease\nquadratically as beam length increases. Extrapolating this trend yields a 5th-mode resonance\nfrequency of 20.8 kHz for a 1 mm actuator. (c) Fifth-harmonic displacement for various actuator\nlengths under 1111-converse-flexoelectric stress (σ = µ ∂E /∂x ), using µ = 1.41 nC/m.\n11 1111 1 1 1111\nFollowing this trend, a 1 mm actuator is predicted to yield a displacement of 45 nm.\nIn Methods:\n(2) Simulation set-up\nCOMSOL Multiphysics was employed to investigate the actuation mechanism of our device. The\nmodel was developed in a 2D plane and consisted of four layers: a 50 nm-thick bottom electrode,\n16\na 500 nm-thick Parylene-C layer, a MoS layer with thicknesses of 0.65, 1.3, or 1.95 nm\n2\ndepending on the number of layers, and a 50 nm-thick top electrode. The top electrode was\ndesigned with 25 individual rectangular segments to mimic a comb-shaped electrode in 2D.\nElectric field and electric field gradient distributions within the MoS layer were\n2\ncomputed using the Electrostatics module. Effects of flexoelectricity and piezoelectricity on the\ndevice actuation were conducted using the Electrostatics, Solid Mechanics, and Piezoelectric\nEffect Multiphysics modules in combination. Since the flexoelectric effect is not natively\nsupported in COMSOL, the equations for the variables solid.PpzeX (Piezoelectric polarization,\nX component) and solid.Spze11 (Piezoelectric stress tensor, local coordinate system, 11\ncomponent) were modified by incorporating a μ - and μ -dependent terms into each\n1111 1133\nexpression, mu_1111*d(solid.eel11,x) + mu_1133*d(solid.eel22,y) and -\nmu_1111*d(solid.EpzeX,x) - mu_1133*d(solid.EpzeY,y), respectively. Note that y represents 3-\ndirectional coordinate as our model was built in 2D. Furthermore, electromagnetic stress\ncalculations were performed using the Electrostatics, Solid Mechanics, and Electromechanical\nForces Multiphysics modules based on the Minkowski tensor to estimate the device deformation\ndue to electromagnetic forces under the applied AC voltage. Joule heating effects were also\nsimulated to calculate the device actuation by thermal expansion resulting from heat generated in\nthe top and bottom electrodes, using the Electric Currents, Heat Transfer in Solids,\nElectromagnetic Heating, and Thermal Expansion Multiphysics modules.\nIn Supporting Information:\n(1) Calculation of Q factor for including damping in the simulation\nIn order to accurately calculate the harmonic displacement, we first estimated the Q factor from\nexperimental data and apply it to the simulation. Figure S9 shows the frequency-displacement plot\nnear the 5th resonant frequency. The Q factor was calculated by dividing resonant frequency by\nthe width of the peak at its half height (∆𝑓), which is 20,700 Hz / 690 Hz = 30.\n(2) Estimation of harmonic displacement of mono-, bi-, and tri-layer MoS actuators\n2\nlimitations, the thicker MoS layer enables us to simulate the resonant frequencies and harmonic\n2\ndisplacement of the longer beam. Figure S7a and b show the 5th harmonic frequency and\ndisplacement of mono-, bi-, and tri-layer MoS actuators with different lengths. As with the\n2\nmonolayer case, bi- and tri-layer devices also show quadratically decreasing resonant frequency\nand quadratically increasing harmonic displacement as the beam length increases. In addition,\nbilayer MoS actuator shows lower harmonic displacement compared to the monolayer MoS ,\n2 2\nagreeing well with our experimental results.\n(3) Estimation of actuation from piezoelectric effect, 1133-converse flexoelectric stress,\nelectromagnetic stress, and Joule heating\n2\npossible mechanisms to the 5th harmonic displacement, including the piezoelectric effect, 1133-\nconverse flexoelectric stress, electromagnetic stress, and Joule heating. In case of the 1133-\nconverse flexoelectric stress, we applied 1.41 nC/m for μ , the same value as the estimated μ\n1133 1111\nvalue. Table S1 summarizes the parameters used in each simulation, and Figure S8 illustrates the\ndevice actuation resulting from each effect. The maximum displacements obtained from the 1111-\nand 1133-converse flexoelectric effects, piezoelectric effect, and electromagnetic stress are 1.64,\n0.014, 0.30, and 0.29 nm, respectively. The displacement resulting from Joule heating is more than\n10 orders of magnitude smaller than those from the other effects and can therefore be neglected.\nThese results strongly support that the 1111-converse flexoelectric effect is the dominant\nmechanism driving device actuation, unless monolayer MoS exhibits a μ value that is\n2 1133\nsignificantly higher than μ . Additionally, when these other effects are taken into consideration,\n1111\nμ is estimated to be in the range of 1.41±0.52 nC/m, under the assumption that μ = μ .\n1111 1111 1133\nTable S1. Simulation parameters\nModule Parameter Value\nSolid Mechanics Young’s modulus, Au 70 GPa\nPoisson’s ratio, Au 0.44\nYoung’s modulus, Ag 83 GPa\nPoisson’s ratio, Ag 0.37\nYoung’s modulus, Parylene-C 2.75 GPa\nPoisson’s ratio, Parylene-C 0.4\nElastic modulus, C , MoS 128.4 GPa [1]\n11 2\nElastic modulus, C , MoS 128.4 GPa [1]\n22 2\nElastic modulus, C , MoS 32.6 GPa [1]\n12 2\nElastic modulus, C , MoS 38.3 GPa [2]\n33 2\nElastic modulus, C , MoS 26.5 GPa [2]\n44 2\nElastic modulus, C , MoS 26.5 GPa [2]\n55 2\n47.9 GPa\nElastic modulus, C , MoS\n66 2 ((C -C )/2)\n11 12\nFlexoelectric coefficient, μ , MoS 1.41 nC/m\n1111 2\n1133 2\nPiezoelectric coefficient, e , MoS 0.5 C/m2 [3]\n11 2\nIsotropic structural loss factor (1/Q factor) 1/30\nElectrostatics & Terminal voltage, top electrode (Au) 20 V\nElectric Currents\nTerminal voltage, bottom electrode (Ag) 0 V\nRelative permittivity, Parylene-C 3\nRelative permittivity, MoS 4 [4]\nDensity, Au 19300 kg/m3\nHeat capacity, Au 129 J/kg∙K\nThermal conductivity, Ag 429 W/m∙K\nDensity, Ag 10500 kg/m3\nHeat capacity, Ag 235 J/kg∙K\nThermal conductivity, Parylene-C 0.084 W/m∙K\nDensity, Parylene-C 1289 kg/m3\nHeat capacity, Parylene-C 712 J/kg∙K\nThermal conductivity, MoS 23.2 W/m∙K [5]\n2\n87.5 J/kg∙K\nHeat capacity, MoS\n2 (14 J/mol∙K) [6]\nThermal Expansion Thermal expansion coefficient, Au 1.42x10-5 /K\n(Multiphysics)\nThermal expansion coefficient, Ag 1.89x10-5 /K\nThermal expansion coefficient,\n3.5x10-5 /K\nParylene-C\nThermal expansion coefficient, MoS 7.6x10-6 /K [7]\n2\nFigure S7. (a) Resonant frequency trends for actuators with different MoS layer thicknesses. A\n2\nquadratically decreasing trend is maintained as the MoS layer becomes thicker with increasing\n2\nlayer number. (b) Fifth-harmonic displacement for different MoS layer thicknesses. Although the\n2\ntrend is maintained, the bi- and tri-layer MoS actuators exhibit lower displacement compared to\n2\nthe monolayer MoS actuator.\n2\neffects, piezoelectric effect, electromagnetic stress, and Joule heating. A 200 μm-long model with\nmonolayer MoS was implemented for the calculations, for which the 5th resonant frequency is\n2\nestimated to be ~430 kHz.\nFigure S9. Fifth-harmonic displacement of the monolayer MoS actuator near the resonant\n2\nfrequency range for Q-factor calcuation.\n2. The actuation performance of monolayer and bilayer devices is similar. The deep reason\nshould be reached. Is it purely flexoelectricity? If so, does the interface between layers or the slip\nplane play a role?\nWe appreciate the referee for this comment. While there is a possibility of slipping between the\n2D layers, our calculations indicate that the lower actuation observed in the bilayer device\ncompared to the monolayer device is mainly due to the increased flexural stiffness of the bilayer\n(vs. monolayer). We have added a brief explanation of this observation in the main text and\nincluded simulation results for thicker MoS layers in the Supporting Information.\nIn main text, page 9-10:\nTo study the effects of MoS thickness on the actuation behavior, we simulated\n2\ndisplacement and resonance frequencies of bi- and tri-layer MoS . The resonant frequency and\n2\ndisplacement trends remained consistent for bi- and tri-layer MoS actuators (Figure S7a). In\n2\naddition, increasing the MoS thickness from monolayer to bilayer reduces the harmonic\n2\n21\ndisplacement of the actuator, which aligns with our experimental observation that the bilayer\ndevice exhibits approximately 18% lower displacement than the monolayer counterpart. This\nreduction is primarily attributed to the increased flexural stiffness resulting from the additional\nMoS layer.\nFigure S7. (a) Resonant frequency trends for actuators with different MoS layer thicknesses. A\n2\nquadratically decreasing trend is maintained as the MoS layer becomes thicker with increasing\n2\nlayer number. (b) Fifth-harmonic displacement for different MoS layer thicknesses. Although the\n2\ntrend is maintained, the bi- and tri-layer MoS actuators exhibit lower displacement compared to\n2\nthe monolayer MoS actuator.\n2\n3. The actuator demonstrates impressive performance at its resonant frequency (~19 kHz).\nHowever, for many practical applications, a broader frequency band is highly desirable. Could\nthe authors comment on the expected performance of the device when operated off-resonance?\nWhat potential strategies could be explored to achieve a wider bandwidth?\nWe thank the referee for this valuable insight. The off-resonance displacement is ~5 nm, 1/10 of\nthe resonant displacement. Although our actuator has a characteristic operation frequency, the\nresonant frequency and the width of the resonant peak of the actuator can be tuned by adjusting\nthe length, thickness, elastic modulus, and viscosity of the supporting layer. Furthermore,\nmaximizing the number of electrode combs would enhance both off-resonant and resonant\nactuations. To reflect the referee’s comments, a discussion of these considerations has been\nadded to the manuscript.\nRevision to MS/SI\nAs our measurements of the actuator displacement were acquired at the 5th resonant\nfrequency, there is a significant potential to further enhance the device performance by utilizing\nlower-order harmonic resonance. Reducing the width and pitch of the electrode combs to\nmaximize the number of combs in the device would further improve actuation performance.\nFurthermore, the resonant frequency and the width of the resonant peak of the actuator can be\ntuned by adjusting the length, thickness, elastic modulus, and viscosity of the supporting layer to\nmeet specific application requirements, offering a wide range of applicability.\n23"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have performed a rigorous revision of the manuscript, addressing all of the comments carefully. Therefore, I recommend the manuscript for publication in its current form.", "reviewer_3": "The authors have addressed all my concerns. I recommend this work for publication now."}, "author_response": ""}]

Decision letter