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 high-performance 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 FePS3 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 FePS3 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 Néel temperature, TN ≈ 114 K.[27,28]

Results and Discussion

Laser Interferometry of FePS3 Resonators

We have fabricated two circular suspended FePS3 resonators on top of a Si/SiO2 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 (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 rg = 2.5 μm is located at the bottom of the cavity. We transfer a flake of few-layer FePS3 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 a–c. As shown in Figure b, 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, Vg, for each of the resonators (see Figure c). The single FePS3 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 a shows a 25.6 ± 0.4 nm FePS3 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 d (see Methods in the Supporting Information).

Figure 1

Measurement principle and setup. (a) False-color SEM image of the sample. Flake thickness: 25.6 ± 0.4 nm. (b) Schematics of the device and optical measurement principle. j(Vg) is the voltage-dependent coupling parameter. (c) Schematics of a cross-section of the device, electrically induced force F1,2, and gate voltage Vg. (d) Laser interferometry setup: red laser, λred = 632 nm; blue laser, λblue = 405 nm. (e) Detected resonance peaks for the two suspended drums at T = 4 K: filled dots, measured data; solid lines, linear harmonic oscillator fits. (f) Resonance frequencies ω(Vg) of drums 1 and 2, extracted from fits similar to that in (e): filled dots, measured data; solid line, continuum mechanics model[30,31] (see section SI 1). The blue region indicates the parameter space where the ω1 = ω2 condition can be met. (g) Dissipation rate γ(Vg) for two drums extracted from fits similar to those shown in (e): filled dots, measured data; solid lines, Joule dissipation model[32−34] (see section SI 2).

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 e. We fit these spectra to a harmonic oscillator model and extract the resonance frequencies ω1,2 as a function of Vg, which are displayed with filled blue and orange dots in Figure f. The resonances ω1,2(Vg) of both drums closely follow the continuum mechanics model,[30,31] as shown by the solid blue and orange lines in Figure f (see section SI 1). At certain values of Vg,1 and Vg,2, the frequencies ω1 and ω2 of the corresponding resonance peaks match at ω1 = ω2, as indicated by the light blue region in Figure f. 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 of the resonators involved.[11,19] Hence, we plot the corresponding mechanical energy dissipation rates of the FePS3 membranes in Figure g. Measured γ1,2(Vg) 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 capacitive displacement currents in the suspended region of the flake.[32−34]

Electromechanical Coupling Model

The mechanical behavior of coupled membrane resonators can be modeled by two coupled resonators, schematically depicted in Figure a. The motion of coupled resonators is described bywhere x1,2 are the membrane displacements and fd is the force at a drive frequency ωd. The coupling parameter , where m1,2 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 2

Strong coupling of spatially separated FePS3 membrane resonators at T = 4 K. Schematics of coupled membrane oscillators. (a) Mechanical model: m1,2 is the effective mass, k1,2 the effective stiffness, and J(Vg) the gate voltage dependent coupling parameter. (b) Electrical model: C1,2 is the capacitance of each drum toward the gate electrode that is kept at a voltage Vg, Rm the interface resistance between the flake and the underlying ground electrode, Cm the capacitance to ground, and Vm(ωd) the voltage between the membranes. (c) Sample with a suspended channel between the drums: (left) weak coupling of motion between spatially separated drums at Vg,1 = 36.9 V and Vg,2 = 0 V and Δω = ωd – ω2; (inset) optical image of the sample, with a thickness of 25.6 ± 0.4 nm (scale bar 6 μm); (right) strong coupling of motion between spatially separated drums at Vg,1 = 37.2 V and Vg,2 = 30 V. (d) Sample without a suspended channel between the drums (see section SI 7): (left) weak coupling of motion at Vg,1 = 32.4 V and Vg,2 = 0 V; (inset) optical image of the sample with a thickness of 14.5 ± 0.3 nm (scale bar 6 μm); (right) strong coupling of motion at Vg,1 = 34.5 V and Vg,2 = 32 V.

Figure b shows a schematic of the electrical circuit that mediates the coupling (see section SI 4). The suspended part of the thin FePS3 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 Vm,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 Cm between the FePS3 flake and Au top electrode is large, it dominates the electrical coupling of the flake to ground (, where Rm is the resistance to ground). As is outlined in section SI 4, the optothermal drive of the first drum at nonzero Vg,1 then results in a nonzero flake voltage Vm,AC that causes an electrostatic force on the second drum, F2,AC = −Je1x10 sin(ωdt), where x10 is the amplitude of periodic displacement and the electrical coupling parameter Jel is given bywhere ε0 is the dielectric permittivity of vacuum, xg(Vg) the static deflection at the center of membrane (see section SI 1), and xc the separation between the membrane and the bottom electrode. When eqs and 2 are combined, it is seen that Jel results in a transfer of mechanical energy via electromechanical coupling between the spatially separated FePS3 drums. It is notable that the driving force on the second drum F2,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 Vg = 0 V on either drum, the electrical coupling parameter Jel = 0 even if ω1 = ω2 and fd > 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 ω1,2 such that ω1 = ω2 using electrostatic pulling, as shown in Figure f. Then we alter Vg,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 Vg,2. Figure c shows the resonance peak splitting for different Vg values applied to both membranes. Two distinct regimes are visible: one that corresponds to weak coupling at Vg,2 = 0 V (and Jel = 0) and another that corresponds to strong coupling with the avoided frequency crossing visible at Vg,2 = 30 V (and Jel ≠ 0). We did not observe any change to the avoided crossing related 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 FePS3 (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 d (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 Vg,2 = 0 V in Figure c,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 FePS3 resonators under our experimental conditions, is negligible in comparison to the dominant mechanism that we propose in Figure b and eq with J ≈ Jel.

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 a). We apply the corresponding Vg to the drums in order to match their ω1,2 values and measure the avoided crossing of the resonance frequencies in both configurations of lasers. By solving eq , we find amplitudes for the two configurations of lasers as[35]where A1 is the oscillation amplitude with lasers on the same drum, A2 the amplitude with lasers on different drums, the coupling strength coefficient, and the detuning. In Figure b the measured amplitudes A1,2 at Vg,2 = 30 V are compared to simulations based on the continuum mechanics model (see section SI 1) as well as eqs and 2. The model is in good agreement with the experimental data (see sections SI 1–SI 4).

Figure 3

Comparison between the coupled oscillators model and experiments at T = 4 K. (a) Schematic indication of the position of lasers for each row of data in (b). (b) Measured normalized amplitudes A1,2 of the resonance peaks at Vg,1 = 37.2 V and Vg,2 = 30 V compared with the model of eqs and 2 (see sections SI 1–SI 4). The dashed horizontal line shows the extraction of data given in (c) at Δω = ωd – ω2. (c) Amplitude A2 of the resonance peak splitting of drum 2 at different Vg,2 values: filled blue dots, measured data; solid red lines, fit to the model of eq ; dashed black line, positions of peak maxima used to extract 2g. (d) Splitting 2g and cooperativity plotted against Vg: filled blue dots, measured 2g obtained from (c); solid blue line, fit to a parabola used as a guide to the eye; filled orange dots, cooperativity calculated from 2g and the corresponding γ1,2 values from Figure g. (e) Measured and modeled coupling constant J(Vg): filled blue dots, J extracted from the fit in (c); solid magenta line, comparison to the model of eq . Error bars in (d) and (e) are indicated with vertical colored lines.

We now investigate the gate voltage dependence of strong coupling between the separated FePS3 membrane resonators. In Figure c we show the resonance peak splitting 2g with increasing Vg,2. We extract 2g from peak maxima of the measured data in Figure c, which we plot together with the cooperativity calculated[19,23] as . 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 Vg,2 > 16 V. We also fit A2 of the same data set in Figure c to eq to extract j. Figure e displays the measured coupling constant J(Vg) (filled blue dots), in comparison to the electrical coupling model of eq (solid magenta line). The model follows the experiment closely for Cm = 1.9 pF (as supported by finite element method simulations; see section SI 8), reproducing both the quasi-linear part of the data at Vg,2 < 24 V and the nonlinear part at Vg,2 > 24 V that appears due to the deflection of the membrane xg at larger Vg. 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 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 Vg of both drums and lock the excitation frequency at ωd, as indicated with dashed lines in Figure c. 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 fd of drum 1, while measuring the 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 ωd, as shown in Figure a. 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 b 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.

Figure 4

Information transfer between drums at T = 4 K. (a) Measured spectral density near ω2 at 0 and 1 bit of amplitude-modulated excitation. Inset: schematics of the experiment. (b) Map of the maximum of the spectral density showing a binary picture that was sent to drum 1 and received at drum 2 at a bit rate of 4 bits/s.

Coupling near the Antiferromagnetic Néel Temperature

FePS3 is an antiferromagnetic semiconductor at low temperature[36,37] with a Néel temperature TN ≈ 114 K,[27,28] where it exhibits a phase transition to a paramagnetic phase. The phase change in FePS3 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 FePS3, even those tens of nanometers thick,[25] have large quality factors of (2–6) × 104 that are comparable to those of high-Q membranes made of strained monolayers of WSe2 and MoSe2.[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 a–d, we study a sample of FePS3 to assess the temperature dependence of the coupling strength near the TN value. Following the experimental organization and analysis from above, we fix Vg,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 a, when the sample is heated from 4 to 135 K, ω1,2 values soften near the TN value of 107 K. This also appears as a characteristic peak in in Figure c and originates from the anomaly in the specific heat of the material at TN.[25,27] Interestingly, with the temperature approaching TN, the splitting of the resonance peak disappears, as shown in Figure b. However, as follows from eq , Jel is not expected to have a strong temperature dependence or abruptly drop to zero near TN, which is also notable from Figure c, 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 γ1,2 values as it approaches the transition temperature, as shown in Figure d. This behavior of γ1,2(T) can be attributed to the increasing contribution of thermoelastic dissipation to the nanomechanical motion of drums near phase transitions.[25]

Figure 5

Temperature dependence of the coupling between antiferromagnetic membranes. (a) Resonance frequency ω2 of the FePS3 drum as a function of temperature. Inset: optical image of the FePS3 sample with a thickness of 13.9 ± 0.3 nm (scale bar 18 μm). (b) Normalized amplitude A1 of the resonance peak splitting at Δω = ωd – ω2 plotted for three different temperatures. (c) Filled blue dots indicate the measured coupling constant J, filled orange dots indicate the value of the data in (a). (d) Filled blue dots indicate the cooperativity, and filled orange dots indicate the dissipation rate γ2. (e–h) follow the same structure as the (a–d) with the data shown for a MnPS3 sample. Inset of (e): Optical image of the MnPS3 sample with a thickness of 10.5 ± 0.4 nm (scale bar 18 μm). Vertical dashed lines in all panels indicate the detected TN value. Error bars in (c), (d), (g), and (h) are indicated with vertical blue lines.

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 MnPS3 that exhibits an antiferromagnetic to paramagnetic phase transition at TN ≈ 78 K.[27] In Figure e-h, we show the experimental data for MnPS3 that revealed a behavior similar to FePS3. As shown in Figure e, ω2 softens near T = 77 K, which is close to TN. We observe the splitting disappearing next to 75 ± 10 K, as displayed in Figure f. As expected, J does not show any systematic change near TN, which is depicted in Figure g. However, in Figure h 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 γ1,2 that is visible near the TN value of MnPS3, 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 FePS3 and MnPS3. 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 TN values of FePS3 and MnPS3 diminishes the cooperativity of such coupled membrane systems as T approaches TN. 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] and the emission of spin currents by mechanical deformations—the piezospintronic effect.[39]