Gate Tunable Cooperativity between Vibrational Modes | Nano Letters

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Gate Tunable Cooperativity between Vibrational Modes Parmeshwar Prasad, Nishta Arora, and Akshay Naik Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.9b01219 • Publication Date (Web): 13 Aug 2019 Downloaded from pubs.acs.org on August 14, 2019

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Gate Tunable Cooperativity between Vibrational Modes Parmeshwar Prasad, Nishta Arora, and A. K. Naik* Indian Institute of Science, Bangalore, India, 560012

ABSTRACT:

Coupling between a mechanical resonator and optical cavities, microwave resonators or other mechanical resonators have been used to observe interesting effects from sideband cooling to coherent manipulation of phonons. Here we demonstrate strong coupling between different vibrational modes of MoS2 drum resonators at room temperature. We observe intermodal as well as intramodal coupling. Cooperativity, a measure of coupling between the two modes, can be tuned by more than an order of magnitude by changing the DC gate bias. The large measured cooperativity of about 900 at room temperature indicates that the phonon population can be coherently transferred between the modes for more than 500 cycles. This coherent oscillation is of great interest in studying quantum effects in macroscopic objects.

Since the first recorded interaction among pendula by C. Huygens in 1673, there has been immense interest in studying these coupled oscillations. These interactions between different oscillators form the basis of many applications that have origin in electrical and mechanical engineering. Synchronized behavior in various biological entities including fireflies and heart

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cells have been associated with these interactions.1,2 In physics, coupled oscillation are extensively found in atomic and molecular physics such as electronic transition, Rabi oscillation, lasers and Zeeman splitting. While coupling and interactions in oscillators have driven the telecommunication revolution, recent interest in these interactions have been due to the possibility of observing novel quantum effects3–6 and coherent transfer of energy between oscillators.7,8 In optomechanical experiments, these couplings have been used to demonstrate sideband cooling of the mechanical oscillator5,9–11. Such cooled systems are essential to observe quantum effects in mesoscopic systems as well as in applications such as quantum computers. These interactions between different vibrational modes have also led to interesting effects such as internal resonance which can be used to stabilize the oscillator frequency.12 While most recent studies have looked at coupling phonons to photons9,11,13,14, there is a great interest in understanding the coupling of different vibrational modes.8,15–20 Okamoto et. al have demonstrated coherent transfer of energy between strongly coupled modes of different resonators.7 Mathew et. al recently have demonstrated that two vibrational modes of a graphene drum resonator exhibit strong coupling.8 Similar experiments were also performed on MoS2 and carbon nanotube using different measurement techniques.18,19 Here, we demonstrate large voltage controlled cooperativities between different vibrational modes of the NEMS at room temperature. We have performed the experiments on two samples, hereafter labeled device 1 and device 2. In device 1, we demonstrate coupling between two modes with resonant frequencies that are closer than 5 MHz and two distant modes with resonant frequencies separated by ~ 30 MHz. In device 2, we have studied the coupling between two modes separated by more than 60 MHz. Furthermore, we have used electrostatic gate voltage to demonstrate control over the coupling. The strong coupling between the two distant modes mimics the optomechanical system. We report cooperativity of more than 900 in a MoS2 drum resonator at room temperature. The coupling of optical and mechanical degrees


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of freedom in MoS2 combined with the large cooperativity demonstrated here are important for quantum optomechanics and other applications such as delay lines and wavelength convertors.

The experiments were performed on a suspended MoS2 drum resonators at vacuum levels of 10-6 Torr and at room temperature. Figure 1a shows the false-color scanning electron micrograph of the MoS2 drum resonator (device 1). The drum is a few layers thick and and the diameter of the device is 3 μm. The drum is suspended ~300 nm above the gate electrode. The membrane is suspended using dry transfer technique (See supplementary information for detailed fabrication process).21 The suspended drum and the metal gate form a variable capacitor. We employ capacitive actuation and detection technique to transduce the mechanical motion of the device.22,23 Figure 1b shows the schematic of the set up used in the experiment. A radio frequency (RF) signal combined with a DC bias is applied at the gate of the device to actuate the membrane. The RF drive (𝑉𝑎𝑐 𝑔 ) at frequency (𝜔𝑑) actuates the membrane while the DC gate bias (𝑉𝑑𝑐 𝑔 ) manipulates its tension. The capacitance of the device is modulated due to the vibrational motion of the drum. This change in the capacitance is reflected at the drain as a change in voltage. The output voltage at the drain is amplified using a low noise amplifier and detected using a lock-in amplifier.23 Figure 1c shows the density plot of the frequency dispersion with applied dc gate bias for device 1. Several vibrational modes can be observed in the drum resonator. The resonant frequencies of these modes can be tuned using the applied electrostatic force through the gate. The intrinsic strain in this device is estimated by fitting the dispersion of the resonant frequency of the lowest mode with the DC gate voltage.24 We estimate the strain to be 1.2 ± 0.1 × 10 ―5. This low inbuilt tension in the device facilitates large frequency tuning of the modes.25 The density plot also shows that the resonant frequency of different vibrational modes are tuned by different amount. This variability in resonant frequency tuning with DC gate voltage is used to manipulate the frequency difference between


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the modes and coupling between them.26 The coupling between the modes can be further manipulated by applying an additional RF signal, called parametric pump, with frequency (𝜔𝑝) at the gate as shown in the figure 1b.8,22

(a)

(b)


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(c) Figure 1. (a) False colored scanning electron micrograph of the MoS2 drum resonator. Two thick yellow colored lines represent the source and drain. Gate is 300 nm below the suspended green colored membrane. Asymmetry in the suspended membrane gives rise to a larger number of modes. Scale bar is 2 μm (b) Schematic of the experimental set up used in the experiment. The setup uses homodyne electrical actuation and detection technique. An extra pump is applied at the frequency 𝜔𝑝 at the gate to manipulate the coupling. (c) Dispersion curve showing multiple modes. Dotted rectangle shows the region where two modes come closer to each other.

The modes shown in the figure 1c are denoted in the increasing order of the resonant frequency (e.g. mode 1 = 2π × 38.73 MHz and mode 4 = 2π × 66.9 MHz ). In the first set of experiments, we evaluate the interaction between two modes with resonance frequencies that differ by less than 5 MHz. At 𝑉𝑑𝑐 𝑔 = 20 V, resonance frequencies of the two modes are 43.2 MHz (mode 2) and 45.0 MHz (mode 3) respectively. Figure 2a shows the magnified density plot of the frequency dispersion around these two modes. The resonance frequency of the two modes come closest to each other at 𝑉𝑑𝑐 𝑔 = 18.2 V. Coupling between the modes can be introduced by applying a pump signal at a frequency 𝜔𝑝. The pump signal can either be red detuned 𝜔𝑝 = 𝜔3 ― 𝜔2 or blue detuned 𝜔𝑝 = 𝜔2 + 𝜔3. Red detuned pump enables energy exchange between the two modes while the blue detuned pump amplifies the amplitude of the two modes. The processes are analogous to cooling and heating respectively that are typically observed in optomechanical systems.27


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(a)

(b)

(c)

(d)

(e) Figure 2. (a) Magnified image of the dispersion curve within the dotted rectangle in figure


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1c. (b) No splitting of mode 2 is observed when the pump is set to zero. Here, 𝑉𝑎𝑐 𝑔 is 10 mV. (c) Intermodal and intramodal coupling with applied red tuned pump 𝑉𝑎𝑐 𝑝 = 0.6 V at 𝜔𝑝 = 1.8 × (2𝜋 MHz). Splitting of the mode 2 are observed at 𝜔𝑝, 𝜔𝑝/2(d) Simulated inter and intra mode coupling of the mode 2 (43.2 × (2π MHz)). (e) Coupling between two modes are illustrated. The two modes are coupled via tension in the membrane in our device. The two modes split as the coupling strength is increased by applying red detuned pump 𝑉𝑎𝑐 𝑝 .

When the red-detuned pump signal at 𝜔𝑝 = 1.8 × (2𝜋 MHz) is set to zero (𝑉𝑎𝑐 𝑝 = 0 𝑉), the resonant frequency of mode 2 remains unchanged over the observed range of frequency (figure 2b). This indicates that the interaction between the two modes is minimal. As the amplitude of the pump signal is increased, mode 2 is expected to be cooled and subsequently enter the strong coupling regime. Our transduction scheme is not sensitive enough to observe the thermomechanical motion of the mode and thus perform cooling (Data showing cooling-like response of driven mode 1 coupled to mode 4 has been provided in the supplementary section). As the applied pump signal is increased beyond 𝑉𝑎𝑐 𝑝 = 0.63 V, the splitting of the peak can be observed (figure 2c). Beyond the pump voltage of 𝑉𝑎𝑐 𝑝 = 0.63 V, the rate of energy exchange between the modes becomes higher than the damping rate of the modes. In this regime, phonons transferred from mode 2 to mode 3 are likely to be transferred back before losing them to the environment. This is the signature of strong coupling and analogues to the phenomena widely observed in optomechanics and electromechanically induced transparency (EIT) like systems.7,16,28–30 The coupling between the two modes, lower mode, 𝜔𝑙 and higher mode 𝜔ℎ is tension mediated and can be further modified by applying pump signal at 𝜔𝑝 = 𝜔ℎ ― 𝜔𝑙 with strength 𝑉𝑎𝑐 𝑝 . At strong coupling the separation between the split modes is denoted by 𝑔 which


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indicates the strength of the coupling between two modes. The energy between the two modes are exchanged at the rate of 𝑔. As the pumping strength is increased the splitting also increases. In addition to this intermodal coupling that leads to splitting of the peak, we also observe the effects of intramodal coupling which arises due to the coupling with the higher order harmonics of pump frequency. In figure 2c, this intermodal and intramodal coupling manifests itself as mode splitting at

𝜔𝑝

𝑛 (n=1,2,3…).

Simulation of the coupled system using the experimental

parameters is shown in figure 2d. Details of these simulations are given in supplementary information. Figure 3a shows that the nature of the coupling can be changed by applying a blue-detuned pump signal. The vibrational modes lose energy through various damping mechanisms. These losses are compensated using the applied blue pump. These measurements are performed on device 1 at 𝑉𝐷𝐶 𝑔 = 20 V and the pump frequency is at 𝜔𝑝 = 𝜔2 + 𝜔3 = 2𝜋 × 84.2 MHz. With the increasing strength of the pump signal, the amplitude of the modes increases. Figure 3b shows that the damping rate is also significantly reduced as the amplitude of the blue detuned pump is increased. The parametric down conversion of the pump compensates for the mechanical dissipation in these vibrational modes. In these measurements, the dissipation rate is reduced by a factor of 8 with the blue detuned pump signal. The saturation of the damping rate beyond the pump strength of 170 mV is likely due to higher order nonlinear effects which can significantly affect the amplification gain provided by parametric pumping31–33. This nondegenerate parametric pump mechanism has been previously used to amplify mechanical motion in optomechanical systems34. Pumping the system with blue detuned frequency is also analogous to heating the mode. This process is widely observed in several phenomena such as Raman scattering, where a blue detuned laser is down converted by giving away a phonon to the system. This leads to Stokes scattering lines.


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(a)

(b)

Figure 3. (a) Phase of mode 2 with applied blue detuned pump 𝜔𝑝 = 𝜔2 + 𝜔3. Dissipation in the mode is compensated by the applied pump. (b) Damping rate with the applied blue detuned pump. The damping rate falls off rapidly due to increased parametric pumping of 2

energy into the system. The red line is a fit using the equation 𝛾2 = 𝐴 (1 ― 𝐵 𝑉𝑎𝑐 𝑝 ), it 27 indicates quadratic decrease of damping (𝛾) with pump strength (𝑉𝑎𝑐 𝑝 ).

We have also probed the coupling strength of vibrational modes that are far apart in frequency. These measurements are performed at 𝑉𝑑𝑐 𝑔 = 29 V with mode 1 at the resonant frequency of 𝜔1 = 2𝜋 × 38.7 MHz and mode 4 with resonance frequency at 𝜔4 = 2𝜋 × 66.9 MHz. A red detuned pump with a frequency 𝜔𝑝 = 𝜔4 ― 𝜔1 is applied to the device. As shown in Figure 4a, the mode 1 is not affected in the absence of the pump signal. However, in the presence of a pump signal, there is splitting in mode 1. The splitting is prominent at the pump frequency 𝜔𝑝 = 2𝜋 × 28 MHz which corresponds to the difference between the resonance frequency of the two modes. The lower mode (mode 1 or 𝜔1) = 2π × 38.7 MHz and mode 4 (𝜔4) = 2π × 66.9 MHz at 29 V 𝑉𝑑𝑐 𝑔 are analogues of the optomechanical experiments, where the lower mode is the mechanical mode and the higher mode is the mode of the optical cavity. The coupling rate between the two modes can be manipulated by changing the strength of the pump signal. Similar to figure 2c, figure 4c also shows the effect of the pump strength on the


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observed splitting of mode 1 peak. However, the coupling rate in this case is much larger and reaches as large as 2𝜋 × 4.6 MHz. The strength of the coupling is typically defined using a 4𝑔2

unitless parameter called cooperativity and is defined as 𝐶 = 𝛾1𝛾4, where 𝛾1 = 2𝜋 × 383 kHz and 𝛾4 = 2𝜋 × 243 kHz are the damping rates of the first and fourth modes. The above coupling rate corresponds to cooperativity of close to 900 between the two modes. This is an order of magnitude higher than the previously reported values in a similar device.8 Cooperativity greater than unity indicates energy exchange between the modes before the modes decay. Similar measurements with coupled NEMS devices at cryogenic temperatures have previously demonstrated coherent exchange of phonons for a few time periods before the coherence is lost.17 With the current set of devices, it should be possible to observe coherent exchange of phonons for more than 500 cycles at room temperature before the phonon is lost to the environment.

(a)

(b)


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(c)

(d)

Figure 4. (a) No coupling is observed with 𝑉𝑎𝑐 𝑝 set to zero (b) Splitting of the mode 2π × 38.7 MHz at 0.45 V Vac p with ωp = 2π × 28.2 MHz. (c) Splitting of the mode increases with increase in the strength of red detuned pump. The coupling rate increases as high as 2π × 4.6 MHz

at 1 V pump strength. (d) Cooperativity with applied pump voltage 𝑉𝑎𝑐 𝑝 , the

cooperativity reaches up to 900 near 1 V 𝑉𝑎𝑐 𝑝 . The line shows quadratic dependence of 𝐶 on 𝑉𝑎𝑐 𝑝 .

Similar cooperativity is observed in device 2 (see SI for detail). This device is also used to investigate the origin of these high cooperativity in our experiments, we measure the variation of coupling rate between two modes in device 2 for different DC gate bias. The modes are 𝜔1 = 60 ( × 2𝜋 𝑀𝐻𝑧) and 𝜔2 = 125 ( × 2𝜋 𝑀𝐻𝑧) for 𝑉𝑑𝑐 𝑔 = 20 V and above in the device 2 (See SI). Figure 5a shows the effect of pumping strength on the coupling between the two modes. The large splitting clearly indicates large cooperativity between the two modes. Figure 5b shows the cooperativity (𝐶) between the two modes can be tuned by applied pump (𝑉𝑎𝑐 𝑝 ) as well 𝑑𝑐 as gate bias (𝑉𝑑𝑐 𝑔 ). Figure 5d shows the coupling rate between the two modes for different 𝑉𝑔

for fixed pump strength (𝑉𝑎𝑐 𝑝 = 1𝑉). The increase in coupling rate with the DC gate bias is


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easily explained using a linear coupled model. This also leads to an expression for intermodal coupling constant (𝛬)17 (see SI for derivation).

(𝛼ℎ ― 𝛼𝑙)𝐹𝑝𝑘𝑐

𝛬= 2

𝛿𝛺2 𝛺𝑙𝛿𝛺 + 2

(

2

)

#(1)

+ 𝑘2𝑐

Where 𝛼𝑙 and 𝛼ℎ represent change in spring constant corresponding to the eigenfrequencies Ω𝑙 and Ωℎ, 𝛿Ω = Ωℎ ― Ω𝑙, 𝐹𝑝 is the parametric force and 𝑘𝑐 is structural coupling constant. The spring softening depends on the modulation of the tension in the membrane with applied force ∂𝑇

∂𝑇

2

∂Ω2

35 (i.e. 𝛼(𝑙,ℎ) ∝ ∂𝐹 ). For ultra-thin materials Ω ∝ √𝑇 and force 𝐹 ∝ (𝑉𝑑𝑐 𝑔 ) .i.e. 𝛼 ∝ ∂𝐹~ ∂𝐹 . Note 26 that the relation between tension T and 𝑉𝑑𝑐 𝑔 is complex. But, we have assumed a simple linear

relation to estimate the coupling. Also, the tuning of frequencies with applied gate bias is mode dependent and cannot be solved exactly.26 As the tension is manipulated with applied DC gate bias 𝑉𝑑𝑐 𝑔 the structure coupling (𝑘𝑐) between the modes also changes (See Figure 5(c)). We 2

𝑑𝑐 approximate the dependence of 𝑘𝑐 on 𝑉𝑑𝑐 𝑔 as quadratic based on 𝐹 ∝ (𝑉𝑔 ) . We estimate the

coupling rate17 𝑔 ~ 2

Λ

(𝜔𝑙𝜔ℎ)

for different gate bias using equation 1. We use this model to

estimate the structural coupling and its dependence on the DC gate bias by fitting the experimental data in figure 5d to equation 1. The good fit to the experimental data suggests that the simple model is adequate to explain the dependence of the coupling rate on the DC gate voltage. The slight deviation of the model from the experimentally observed values at 𝑑𝑐 large 𝑉𝑑𝑐 𝑔 is likely due to non-linear dependence of tension in the membrane (𝑇) and 𝑉𝑔 . The

model also indicates that coupling can be increased by increasing the tunabililty of the frequency of the modes as well as the tuning of structural coupling in the membrane. Our data indicates that the DC gate bias is the key parameter in our experiments by which coupling can


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be manipulated to a large extent. Gate bias is the knob to control the frequency tuning and structural coupling between the modes to maximize the cooperativity. Devices with low builtin tension in our experiment allow us to manipulate the tension using a simple electrostatic gate to have large cooperativity among the mechanical modes.

(a)

(b)

(c)

(d)

Figure 5: Device 2: (a) Splitting of mode 1 with pump strength 𝑉𝑎𝑐 𝑝 at 𝜔𝑝 = 71.5 𝑎𝑐 ( × 2𝜋 MHz) and 𝑉𝑑𝑐 𝑔 = 26 V. Inset, splitting of the mode around 𝜔𝑝 at 𝑉𝑝 = 300 mV.

Damping rates of the two modes 𝛾1~𝛾2 ≈ 410 ( × 2𝜋 kHz). (b) Cooperativity (𝐶) between


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𝑑𝑐 two modes in device 2. 𝐶 vs 𝑉𝑎𝑐 𝑝 are plotted for different 𝑉𝑔 . Cooperativity reaches close to 𝑑𝑐 700 at 𝑉𝑎𝑐 𝑝 = 1 V and 𝑉𝑔 = 30 V. (c) Dependence of the spring constants corresponding to

the modes (𝜔𝑙,ℎ) and structural coupling 𝑘𝑐 on gate bias (𝑉𝑑𝑐 𝑔 ). (d) Coupling rate with varying 𝑎𝑐 𝑉𝑑𝑐 𝑔 between two distant modes at pump strength 𝑉𝑝 = 1 𝑉. These experiments are

performed on device 2 with 𝜔1 > 60 ( × 2𝜋 MHz), 𝜔2 > 125 ( × 2𝜋 MHz) and 𝜔𝑝 > 65( × 2𝜋 MHz). The tension in the membrane and thus the coupling rate can be tuned with an applied gate voltage. The blue circles represent the experimental data and the line is an estimate based on the equation 1 which is obtained from the linear coupled oscillator model (see SI for detailed derivation).

CONCLUSION In conclusion, we demonstrate strong coupling between different vibrational modes of a MoS2 drum resonator at room temperature. The coupling is controlled using both the red and blue-detuned pump signals. The effective dissipation of the vibrational modes is reduced by a factor of 8 by parametrically pumping the system. We also demonstrate large cooperativity of 900 between two mechanical modes separated by more than 25 MHz. The large cooperativity enables coherent transfer of phonons between different vibrational modes. Even at room temperature, this coherent exchange is expected to last more than 500 periods making them ideal to study these phenomena. We have also used the electrostatic gate bias to change the cooperativitiy between the modes by more than an order of magnitude. A simple model indicates that the cooperativity between the modes can indeed be manipulated using electrostatic gate voltages. These strong couplings between modes can also be used to generate dark modes using only the mechanical vibrations.36 Furthermore, the band gap of MoS2 can be manipulated by strain engineering. This coupling of optical and mechanical degrees of MoS2


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holds great promise for using these devices as optomechanical systems, quantum state transfer37 and hybrid phononic–photonic systems.38 Combined with optical cavities these systems can also be used to implement optical delay line and wavelength converter.

ASSOCIATED CONTENT “Supporting Information”, Additional information on the sample fabrication, measurement technique and simulation details. AUTHOR INFORMATION Corresponding Author *(A.K.N.) E-mail: [email protected] Author Contributions PP performed the measurements and analysis, NA fabricated the device, performed some of the measurements and AKN provided the overall guidance for the project. All authors contributed in manuscript preparation. All authors have given approval to the final version of the manuscript. Funding Sources We acknowledge funding support from Nano Mission, Department of Science and Technology (DST), India through grant number SR/NM/NS-1157/2015(G) and SR/NMITP-62/2016(G) and from Board of Research in Nuclear Sciences(BRNS), India through grant number 37(3)/14/25/2016-BRNS. P.P. acknowledges scholarship support from CSIR, India. N.A. acknowledges fellowship support under Visvesvaraya PhD Scheme, Ministry of Electronics and Information Technology (MeitY), India. We also acknowledge funding from MHRD, MeitY and DST Nano Mission for supporting the facilities at CeNSE.


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ACKNOWLEDGMENT We gratefully acknowledge the usage of National Nanofabrication Facility (NNfC) and Micro and Nano Characterization Facility (MNCF) at CeNSE, IISc, Bengaluru. ABBREVIATIONS

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Gate Tunable Cooperativity between Vibrational Modes | Nano Letters

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