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Tunable Strong Coupling of Mechanical Resonance between Spatially Separated FePS3 Nanodrums

Coupled nanomechanical resonators made of two-dimensional materials are promising for processing information with mechanical modes. However, the challenge for these systems is to control the coupling. Here, we demonstrate strong coupling of motion between two suspended membranes of the magnetic 2D material FePS3. We describe a tunable electromechanical mechanism for control over both the resonance frequency and the coupling strength using a gate voltage electrode under each membrane. We show that the coupling can be utilized for transferring data between drums by amplitude modulation. Finally, we also study the temperature dependence of the coupling and how it is affected by the antiferromagnetic phase transition characteristic of this material. The presented electrical coupling of resonant magnetic 2D membranes holds the promise of transferring mechanical energy over a distance at low electrical power, thus enabling novel data readout and information processing technologies.


■ INTRODUCTION
Nanoelectromechanical systems (NEMS) are attracting the attention of the scientific community for their potential to study novel quantum and electromagnetic effects at the nanoscale. 1−3 Along with that, micro-and nanoresonators have been studied for various applications, including sensitive mass detection, 4,5 band-pass filters with variable properties, 6 logic gates, 7−9 and signal amplifiers. 10 For instance, arrays of coupled highly cooperative NEMS and oscillators are already utilized for the coherent manipulation of phonon populations 11−13 and for data processing and storage. 3,14 Recently, NEMS made out of two-dimensional (2D) materials have also gained interest due to the prospect of realizing highperformance oscillators 15,16 and novel sensor concepts. 17 This is not only due to the atomic thinness of these devices but also because the fabrication methodology allows the integration of a range of materials and their heterostructures, with a wide range of magnetic, optical, and electrical properties, on the same chip. 18 Hence, coupling NEMS resonators made of 2D materials and heterostructures promises even more interesting implementation possibilities.
Various studies have reported mechanical coupling between different resonance modes of the same 2D membrane by mechanical, optical, and electronic means. 19−22 However, the realization of coupling between resonances of spatially separated 2D membranes has appeared to be more difficult and was only recently achieved via a mechanical phononic transduction mechanism. 23,24 A mechanically mediated coupling mechanism via a phonon bath was demonstrated, 23,24 but although the coupling could be adjusted via the individual resonance frequencies of the resonators, the coupling strength itself is fixed by the mechanical geometry of the structure that determined the phonon bath. In order to achieve full control over the coupling, a tunable transduction mechanism is needed that not only adjusts the degree of coupling between the resonators but also regulates their resonance frequencies.
Here, we demonstrate an electrical transduction mechanism for coupling mechanical resonances of two spatially separated membranes made of a van der Waals material, allowing control over both the resonance frequency and the coupling strength via gate electrodes. We use the mechanism to strongly couple the fundamental mechanical modes of two suspended circular antiferromagnetic FePS 3 membranes that are separated by an edge-to-edge distance of 2 μm. We show that the coupling mechanism can be utilized for transferring data from one drum to the other, a feature that is useful in data processing and storage systems. Coupling of magnetic materials such as FePS 3 is of interest, since their mechanical resonances can be sensitive to both the magnetic phase 25 and the magnetization of that phase. 26 To investigate this, we study the temperature dependence of the coupling strength and in particular how it is affected by the antiferromagnetic phase transition at the Neél temperature, T N ≈ 114 K. 27,28 ■ RESULTS AND DISCUSSION Laser Interferometry of FePS 3 Resonators. We have fabricated two circular suspended FePS 3 resonators on top of a Si/SiO 2 substrate on which we define an array of Au bottom gate electrodes. A layer of spin-on glass (SOG) 29 is used to electrically insulate the bottom electrodes from the top electrode, as indicated in Figure 1 (see also Methods in the Supporting Information). Then, two spatially separated circular cavities of r = 3 μm in radius are etched in the SOG/Au top layer, 29 such that the local circular gate electrode with a radius of r g = 2.5 μm is located at the bottom of the cavity. We transfer a flake of few-layer FePS 3 exfoliated from a synthetically grown bulk crystal 25 over these cavities by a dry transfer technique (see Methods in the Supporting Information) to create two separated circular membranes of the same flake, as depicted in Figure 1a−c. As shown in Figure 1b, we focus red and blue lasers on these drums to excite the motion of one membrane and measure the displacement of the other using a laser interferometry technique (see Methods in the Supporting Information), thus probing the coupling between fundamental vibration modes of these membranes. To realize the study as a function of gate voltage and the corresponding electrostatically induced strain for both drums, we use local electrodes that allow us to individually adjust the gate voltage, V g , for each of the resonators (see Figure 1c). The single FePS 3 flake, out of which the resonators are formed, is contacted via a top metal electrode to ground. A false-color scanning electron microscopy (SEM) image in Figure 1a shows a 25.6 ± 0.4 nm FePS 3 flake suspended over the electrodes forming two separated membrane resonators. The resonators are placed in a dry cryostat with optical access that is connected to a laser interferometry setup, as shown in Figure 1d (see Methods in the Supporting Information).
When operating the laser interferomery technique with both lasers focused at the same position on the same membrane, we independently characterize the resonance spectra of the fundamental membrane modes of drums 1 and 2 at a temperature of 4 K, as shown in Figure 1e. We fit these spectra to a harmonic oscillator model and extract the resonance frequencies ω 1,2 as a function of V g , which are displayed with filled blue and orange dots in Figure 1f. The resonances ω 1,2 (V g ) of both drums closely follow the continuum mechanics model, 30,31 as shown by the solid blue and orange lines in Figure 1f (see section SI 1). At certain values of V g,1 and V g,2 , the frequencies ω 1 and ω 2 of the corresponding resonance peaks match at ω 1 = ω 2 , as indicated by the light blue region in Figure 1f. In this regime we can expect an avoided frequency crossing if the exchange of the excitation energy between the drums is sufficiently large and the drums are strongly coupled. 3 The coupling strength is also related to the dissipation Q 1 1,2 of the resonators involved. 11,19 Hence, we plot the corresponding mechanical energy dissipation rates  Figure   1g. Measured γ 1,2 (V g ) values follow a parabolic behavior, in accordance with a Joule dissipation model (solid blue and orange lines; see section SI 2), which can be attributed to where m 1,2 is the effective mass, is responsible for the transfer of energy between the two resonators and thus coupling of the mechanical motion. Several coupling mechanisms can contribute to J (see section SI 3). In this work, we present evidence for an electromechanical coupling mechanism for adjusting the coupling strength between two 2D material resonators. Figure 2b shows a schematic of the electrical circuit that mediates the coupling (see section SI 4). The suspended part of the thin FePS 3 flake, which covers the two cavities, is both resistively and capacitively connected to ground via the interface between the flake and the Au top electrode. We assume that the voltage V m,DC that is established between drum 1 and drum 2 is zero since the Au top electrode effectively shunts potential differences between the drums. However, since the capacitance C m between the FePS 3 flake and Au top electrode is large, it dominates the electrical coupling of the flake to ground where R m is the resistance to ground). As is outlined in section SI 4, the optothermal drive of the first drum at nonzero V g,1 then results in a nonzero flake voltage V m,AC that causes an electrostatic force on the second drum, , where x 10 is the amplitude of periodic displacement and the electrical coupling parameter J el is given by where ε 0 is the dielectric permittivity of vacuum, x g (V g ) the static deflection at the center of membrane (see section SI 1), and x c the separation between the membrane and the bottom electrode. When eqs 1 and 2 are combined, it is seen that J el results in a transfer of mechanical energy via electromechanical coupling between the spatially separated FePS 3 drums. It is notable that the driving force on the second drum F 2,AC is proportional to the product of the individual gate voltages applied to each of the drums, which originates from the quadratic dependence of electrostatic force with respect to total voltage. This means that at V g = 0 V on either drum, the electrical coupling parameter J el = 0 even if ω 1 = ω 2 and f d > 0. This property distinguishes the expected behavior of this mechanism from phonon-or tension-mediated coupling, 19,21,23,24 where frequency matching and a nonzero driving force acting on one of the drums are sufficient conditions for coupling and thus splitting of the resonance frequency to occur. We use this characteristic to provide evidence for the proposed mechanism, by first matching the resonance frequencies of the drums by tuning ω 1,2 such that ω 1 = ω 2 using electrostatic pulling, as shown in Figure 1f. Then we alter V g,1 of the drum that we drive with the modulated blue laser, while measuring the amplitude of motion of the other drum that is probed with the red laser at a constant V g,2 . Figure 2c shows the resonance peak splitting for different V g values applied to both membranes. Two distinct regimes are visible: one that corresponds to weak coupling at V g,2 = 0 V (and J el = 0) and another that corresponds to strong coupling with the avoided frequency crossing visible at V g,2 = 30 V (and J el ≠ 0). We did not observe any change to the avoided crossing related Nano Letters pubs.acs.org/NanoLett Letter to the change of laser intensity or its modulation amplitude (see section SI 5). Also, resonance peaks disappear when the blue laser drive is focused on the unsuspended region of FePS 3 (see section SI 6). The observed strong coupling between the drums, and its gate voltage dependence, can therefore not be attributed to periodic heating from the laser beam or to other parasitic electrical actuation mechanisms. Moreover, we observed the same behavior in a test sample without a suspended channel connecting the two membranes, as shown in Figure 2d (see section SI 7), thus providing additional evidence ruling out the possibility of strong tension-mediated direct mechanical coupling. We note that the nonzero amplitudes in the weak coupling regime at V g,2 = 0 V in Figure 2c,d indicate the presence of some other, much less pronounced, mechanisms of weak coupling (see section SI 3). However, the evidence above suggests that the contribution of these mechanisms, which can couple the motion of two spatially separated FePS 3 resonators under our experimental conditions, is negligible in comparison to the dominant mechanism that we propose in Figure 2b and eq 2 with J ≈ J el . Strong Coupling between Spatially Separated Nanodrums. We now study the coupling mechanism in more detail by comparing the two laser configurations using the setup described above, in which we either focus red and blue lasers on separate drums to excite one membrane and measure the motion of the other or focus them on the same drum to excite and measure a single drum (see Figure 3a). We apply the corresponding V g to the drums in order to match their ω 1,2 values and measure the avoided crossing of the resonance frequencies in both configurations of lasers. By solving eq 1, we find amplitudes for the two configurations of lasers as 35 A f A f j where A 1 is the oscillation amplitude with lasers on the same drum, A 2 the amplitude with lasers on different drums, We now investigate the gate voltage dependence of strong coupling between the separated FePS 3 membrane resonators. In Figure 3c we show the resonance peak splitting 2g with increasing V g,2 . We extract 2g from peak maxima of the measured data in Figure 3c, which we plot together with the cooperativity calculated 19,23 as g (2 ) 2 1 2 γγ . A strong coupling regime and an avoided crossing is reached when the figure of merit of the coupling, the cooperativity, is above 1, which is achieved for V g,2 > 16 V. We also fit A 2 of the same data set in Figure 3c to eq 3 to extract j. Figure 3e displays the measured coupling constant J(V g ) (filled blue dots), in comparison to the electrical coupling model of eq 2 (solid magenta line). The model follows the experiment closely for C m = 1.9 pF (as supported by finite element method simulations; see section SI 8), reproducing both the quasi-linear part of the data at V g,2 < 24 V and the nonlinear part at V g,2 > 24 V that appears due to the deflection of the membrane x g at larger V g . This result is also reproducible for different samples (see section SI 9).
Amplitude-Modulated Transmission of Information. In the strong coupling regime the excitation energy is transferred between the resonators. In Figure 4 we demonstrate that this channel of energy exchange can be amplitude-modulated to transfer binary data from one drum to another. We lock the gate voltage V g of both drums and lock the excitation frequency at ω d , as indicated with dashed lines in Figure 2c. Then, we modulate the drive power of the blue laser between 2.5 and 5 dBm with a step function and thus the amplitude of excitation force f d of drum 1, while measuring the Nano Letters pubs.acs.org/NanoLett Letter motion of drum 2 using the red laser. The peak value in the measured spectral density corresponds to the resonant motion of drum 2 at the excitation frequency ω d , as shown in Figure  4a. The lower maximum of the measured spectral peak density corresponds to a bit with a value of 0, while the larger maximum corresponds to a bit with a value of 1. Using this approach, we send a binary image to drum 1 and read it out on drum 2 (see section SI 10). The result is plotted in Figure 4b as a map of the maximum spectral density of the detected resonance peak on drum 2. The received picture is clearly distinguishable with no bits lost during the transfer.
Coupling near the Antiferromagnetic Neél Temperature. FePS 3 is an antiferromagnetic semiconductor at low temperature 36,37 with a Neél temperature T N ≈ 114 K, 27,28 where it exhibits a phase transition to a paramagnetic phase.
The phase change in FePS 3 is accompanied by a large anomaly in the thermal expansion coefficient that produces an accumulation of substantial tensile strain in the membrane 25 as it is cooled from room temperature to 4 K. As a consequence, at cryogenic temperatures membranes of FePS 3 , even those tens of nanometers thick, 25 have large quality factors of (2−6) × 10 4 that are comparable to those of high-Q membranes made of strained monolayers of WSe 2 and MoSe 2 . 34 In earlier works it was shown that the mechanical resonances of magnetic membranes can be sensitive to both the magnetic phase 25 and the magnetization of that phase. 26 Therefore, when the membranes are strongly coupled, small differences in magnetization can result in large differences in the resonance frequencies and the mechanical damping of the membranes and thus the coupling strength, making such coupled resonators very sensitive to small changes in the magnetic state of the material.
In Figure 5a−d, we study a sample of FePS 3 to assess the temperature dependence of the coupling strength near the T N value. Following the experimental organization and analysis from above, we fix V g,2 of the drum 2 at 29 V and measure the resonance frequency, coupling parameter, and cooperativity as a function of temperature. As shown in Figure 5a, when the sample is heated from 4 to 135 K, ω 1,2 values soften near the T N value of 107 K. This also appears as a characteristic peak in  Figure 5c and originates from the anomaly in the specific heat of the material at T N . 25,27 Interestingly, with the temperature approaching T N , the splitting of the resonance peak disappears, as shown in Figure 5b. However, as follows from eq 2, J el is not expected to have a strong temperature dependence or abruptly drop to zero near T N , which is also notable from Figure 5c, where we plot the experimentally obtained J(T) value. Instead, this switch from a strong to weak coupling regime is related to a continuous decrease of the cooperativity due to increasing γ 1,2 values as it approaches the transition temperature, as shown in Figure 5d. This behavior of  Nano Letters pubs.acs.org/NanoLett Letter γ 1,2 (T) can be attributed to the increasing contribution of thermoelastic dissipation to the nanomechanical motion of drums near phase transitions. 25 To support the hypothesis that the temperature-dependent spectral changes are related to the temperature dependence of the dissipative terms in the equation of motion, we fabricated a sample of MnPS 3 that exhibits an antiferromagnetic to paramagnetic phase transition at T N ≈ 78 K. 27 In Figure 5eh, we show the experimental data for MnPS 3 that revealed a behavior similar to FePS 3 . As shown in Figure 5e, ω 2 softens near T = 77 K, which is close to T N . We observe the splitting disappearing next to 75 ± 10 K, as displayed in Figure 5f. As expected, J does not show any systematic change near T N , which is depicted in Figure 5g. However, in Figure 5h the cooperativity has a sharper drop in value as the sample goes from the strong to the weak coupling regime with increasing temperature. This coincides with a broad kink in γ 1,2 that is visible near the T N value of MnPS 3 , providing evidence for the hypothesis.

■ CONCLUSIONS
In conclusion, we have demonstrated a mechanism that mediates strong coupling between spatially separated membranes made of the antiferromagnetic materials FePS 3 and MnPS 3 . This coupling mechanism can be switched on and tuned by an electrostatic gate. In addition, the electromechanical transfer of energy can be amplitude-modulated and is shown to be capable of performing bit-by-bit communication. This provides control advantages that can find use in the development of new device concepts, such as nanomechanical logic gates 7−9 and hybrid systems combining magnetic mechanical oscillators and qubits. 38 We have further shown that the magneto-mechanical properties of antiferromagnetic materials also can affect the coupling strength and cooperativity between the membranes next to the phase transition. For example, we have shown that the increasing mechanical dissipation 25 near the T N values of FePS 3 and MnPS 3 diminishes the cooperativity of such coupled membrane systems as T approaches T N . Therefore, coupled NEMS made of magnetic membrane resonators can provide a deeper insight into the coupling of magnetic properties with the nanomechanical motion. We also anticipate that in the future antiferromagnetic NEMS of this type can be useful to study more intricate magnetic phenomena, such as a magnetostriction in ultrathin layers 26  The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.1c03010.
Methods, continuum mechanics model of the circular membrane resonators of FePS 3 , Joule dissipation model, linear coupled oscillators model, electromechanical coupling model, laser intensity and driving power dependence, laser position dependence, test sample without a microchannel connecting two suspended drums, finite element method simulations, reproducibility of measurements, and transfer of information (experimental details) (PDF)