Optical Control of Mechanical Mode-Coupling ... - ACS Publications

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Optical control of mechanical mode-coupling within a MoS resonator in the strong-coupling regime 2

Chang-Hua Liu, In Soo Kim, and Lincoln J. Lauhon Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.5b02586 • Publication Date (Web): 10 Sep 2015 Downloaded from http://pubs.acs.org on September 10, 2015

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Optical control of mechanical mode-coupling within a MoS2 resonator in the strong-coupling regime Chang-Hua Liu,† In Soo Kim, † and Lincoln J. Lauhon*,† Department of Materials Science and Engineering, Northwestern University, Evanston, IL, 60208, United States

ABSTRACT

Two-dimensional (2-D) materials including graphene and transition metal dichalcogenides (TMDs) are an exciting platform for ultrasensitive force and displacement detection in which the strong light-matter coupling is exploited in the optical control of nanomechanical motion. Here we report the optical excitation and displacement detection of a ~3 nm thick MoS2 resonator in the strong-coupling regime, which has not previously been achieved in 2-D materials. Mechanical mode frequencies can be tuned by more than 12% by optical heating, and they exhibit avoided crossings indicative of strong inter-mode coupling. When the membrane is optically excited at the frequency difference between vibrational modes, normal mode splitting is observed, and the inter-mode energy exchange rate exceeds the mode decay rate by a factor of 15. Finite element and analytical modeling quantifies the extent of mode softening necessary to control inter-mode energy exchange in the strong coupling regime.

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KEYWORDS: nanomechanics, optomechanics, transition metal dichalcogenide, 2D materials, strong coupling, NEMS TEXT Two dimensional (2D) transition metal dichalcogenides (TMDs) materials and graphene exhibit a large elasticity and low mass, and thereby provide a means to convert weak forces into large displacements while under tension.1-3 Due to the vanishing of bending rigidity with decreasing thickness, 2D materials are naturally in the membrane-limit, a regime in which displacement induced tension produces nonlinear force-displacement relations.4,

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Membranes

can therefore exhibit signatures of strong intermodal interactions6, 7 when one vibrational mode perturbs other modes via motion-induced tension. When the rate of energy transfer between modes exceeds the rate of energy dissipation, one can observe a rich variety of classical and quantum resonance phenomena including mechanically-induced transparency,8 Rabi splitting9, 10 and the Landau-Zener transition,11 as has been reported for dielectric membranes in the “strongcoupling” regime using both electrical6, 7 and optical12 excitation and detection methods. TMD membranes provide new opportunities to engineer opto-electro-mechanical responses in the strong coupling regime due to their distinct crystal symmetries and range of band structures. Furthermore, the strong light-matter interaction in TMDs13 lead to extremely large absorption coefficients,14 enabling the aforementioned resonance phenomena to be investigated using relatively simple optical detection and excitation. Here we take advantage of the significant reflection from and absorption of ultrathin 2-D materials to probe the strong coupling regime in a membrane-like MoS2 mechanical resonator. Photothermal heating15 is employed to tune the coupling between modes; time-dependent modulation of the heating promotes rapid energy exchange between coupled modes. Notably, the resonator is driven into


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the strong coupling regime without requiring sophisticated device structures or external cavities, providing a simple scheme for manipulating TMD resonators. A multilayer MoS2 film was transferred onto a TEM grid with an array of square holes (19×19 µm2) to form a suspended membrane resonator (Fig. 1a-b), and its thermally induced transverse displacement was detected using an infrared (λ= 1550 nm) fiber interferometer16 to minimize laser heating from the detection beam (Fig. 1a and c). The infrared monitoring laser is directed through an optical fiber and focused by an objective lens (NA = 0.85) onto the MoS2 surface with a spot size of ~2.25 µm. The MoS2 resonator is mounted on a X-Y-Z piezo-electric scanning stage, enabling control of the focal point on the MoS2 resonator. The light reflected from the MoS2 membrane passes through the same objective to interfere with light reflected from the fiber endface. The optical interferometric signal is detected by a high speed photodetector, which converts the optical signal into an electrical signal that is then amplified (Femto HCA40M-100K-C) and passed to a spectrum analyzer (Agilent N9010A) to attain the displacement spectrum. The measured thermal displacement spectrum from the MoS2 resonator shows multiple peaks that can be identified by comparison with finite element simulations of both the mode frequencies and the symmetries of the spatial displacements (Fig. 1d-e). The quality factor of the fundamental mode is ~877. The ratios between the frequencies of the higher order modes and the fundamental mode indicate that the mechanical response is dominated by surface tension rather than bending rigidity.1 Whereas an ideal square membrane would exhibit degenerate modes, the higher order modes shown in Fig. 1d and 1e show frequency splitting that likely arises from an asymmetric stress profile and/or surface contamination as observed in high stress silicon nitride12 and graphene17 resonators. The small frequency mismatch between nearly degenerate mode pairs is suggestive of inter-modal mechanical coupling.12


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To further characterize this mechanical coupling, we illuminate the entire MoS2 membrane from below (Fig. 1a and c) using a second laser (λ= 639 nm, spot size ~30 µm) and examine the dependence of mode frequency on laser power (Fig. 2). The resulting heating and thermal expansion of MoS2 reduces the membrane tension, greatly decreasing the resonance

frequencies, which vary as, where Γ is the tension of the membrane, is the mass density and is the thickness.1, 18, 19 The experimental variation of the fundamental mode frequency with laser intensity (Fig. 2c, red circles) is reproduced in a finite element model that calculates the temperature profile induced by laser heating (Supporting Information, Fig. S2) and the resulting change in frequency (Fig. 2c, black line). Note that the frequency varies as the square root of tension

∝−

, and therefore temperature, but the nonlinearity is not obvious over this

very small range in temperatures. In contrast, the higher order modes show obvious nonlinear power dependencies characteristic of avoided crossings (Fig. 2d and Fig. 2e, respectively), indicating that the static laser heating changes the inter-mode coupling. Therefore, modulation of the laser intensity should generate a time-dependent inter-mode coupling with the potential to influence the energy exchange between modes. Prior to exploring the inter-mode coupling, we first demonstrate the extent to which the frequency of a single mode can be modulated by varying the intensity of the above gap illumination to create a time-dependent variation of tension within the MoS2 membrane. Consequently, a series of Stokes and anti-Stokes vibrational sidebands emerge and are resolved at frequencies: , ± , where is a positive integer, , is the unperturbed resonance frequency of the fundamental mode, and is the light modulation frequency (Fig. 3a). Significantly, at fixed modulation frequency but increasing excitation power, the amplitudes of


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the sidebands increase while the central zero order resonance peak is suppressed (Fig. 3c), indicating the redistribution of energy into sidebands. The observed vibration spectra are those of a frequency-modulated harmonic resonator whose vibration sidebands are spaced by the modulation frequency ( ) with the relative amplitudes determined by the modulation index. To confirm this interpretation, we model the frequency response of the resonator using − ,

× "#$%m '( =

*+,

, where and

-,- .

+

+

are energy dissipation rate and vibration

frequency of the fundamental mode respectively, " is the excitation power,

m .

is the light

modulation frequency, /0 represents the thermal fluctuation forces, 1 is the effective mass, and =

89

= − is the optically-mediated spring softening coefficient (2.6 × 107 : ×;< ),

extracted from Fig. 2c. Additional details can be found in the Supporting Information. The simulated resonator response under varying light modulation frequency and excitation power is shown in Fig. 3b and Fig. 3d respectively. From the simulations, one can deduce that the wide tunability of the sidebands (more than 120 kHz , or > 10% of , arises from the extraordinarily large softening coefficient A, and that this tunability can be exploited as long as the inverse of light modulation frequency is shorter than the thermal equilibrium time constant of MoS2 membrane (~3 µs, see Supporting Information for the calculation). Such a widely tunable tension is challenging to achieve in conventional macroscopic resonators,6, 7, 12 highlighting the unique attributes of 2-D material membranes that could be exploited in signal processing applications.20 Fundamentally, the wide frequency tunability enables the transfer of mechanical energy between modes with a large frequency mismatch, which is difficult to realize in, e.g., Si3N4 membranes.


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To demonstrate the control of energy exchange between mechanical degrees of freedom, we examine the response of the nearly degenerate (1,2) and (2,1) modes when the membrane tension is modulated at frequencies near their frequency difference of ~125 kHz (Figure 4). Additional examples are shown in the Supporting Information Fig. S4. In the bottom of Fig. 4a, Stokes and anti-Stokes sidebands emerge from each mode as the modulation frequency is increased. As the modulation frequency approaches the frequency difference between coupled modes, each resonance peak split into two peaks. This observation of normal-mode splitting7, 10, 21-23

in the frequency domain indicates that the rate of mechanical energy transfer between the

two modes exceeds the intrinsic dissipation rate of either mode. The oscillatory exchange of energy is generated by the modulation of the MoS2 membrane tension at the resonance condition ( = , − , ); mechanical energy is transferred from (1,2) to (2,1) via the Stokes sideband of the (1,2) mode (ℏ, + ℏ = ℏ, ). Conversely, energy transfers back via the anti-Stokes sideband of the (2,1) mode (ℏ, − ℏ = ℏ, ).7,

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For our optically

modulated membrane resonator, the rate of energy transfer between modes g should increase with increasing laser power, which increases the magnitude of the softening (the coefficient is power independent) as shown in the analysis of Fig. 2c. Below, we measure the optical power dependence of the normal mode splitting (the frequency corresponding to the rate of energy transfer) and compare it with the expected mode splitting predicted by an analytical model to confirm our interpretation that time-dependent laser excitation controls the energy exchange. Fig. 4b shows that the mode splitting increases with increasing laser power as predicted. The displacement spectra at the points of minimum mode separation are shown in figure Fig. 4c; normal-mode splitting manifests as a shallow dip at low laser power, after which the peaks separate with increasing power. The power dependence of coupling rate g, which is the


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frequency difference between split modes, is extracted from the peak separations in Fig. 4c and plotted in Fig. 4d (black circles). Significantly, the coupling rate increases with the excitation power, and the highest coupling rate is greater than 70 kHz, which is 15 times greater than the intrinsic energy dissipation rate of the mechanical modes (linewidth ∆f~ 4.4 kHz) at room temperature. We model g(P), where P is the laser power, using a simple coupled resonators model7 described in the Supporting Information S5, leading to the analytical expression E~

G

H -∗ ∗

, where J is the (power–dependent) intermodal coupling coefficient, which is

calculated from the previously determined intramodal spring softening coefficients as described in the Supporting Information. The modeled result in Fig. 4d (red line) is in good agreement with the experimental data in terms of the magnitude of the coupling rate, and both the experiments and the model show an inflection point at similar temperatures. The nonlinear dependence on power arises from the power dependencies of Λ, ∗, and ∗ , and it is clear that the large thermal tunability of both intra and inter-modal coupling enables the observation of strong coupling. The degree of nonlinearity is slightly larger in the experimentally measured coupling coefficient. However, we note that the simulations of mode coupling assume a uniform membrane temperature, whereas the membrane temperature is actually peaked in the center. Furthermore, we have assumed an instantaneous change in temperature with laser power. Hence, more sophisticated models that take into account local variations in temperature may be necessary to understand the details of the nonlinear relationship between g and the laser power. However, one can also intentionally excite selected regions of the membrane to break the symmetry and exert greater control over which modes are coupled, and how strongly. Such studies are an important extension of the work described here.


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In summary, we utilize a simple device structure and coupling scheme to demonstrate rapid energy exchange between mechanical modes in a MoS2 resonator in the membrane limit. Our observations of strong coupling suggest the feasibility of coherent control of mechanical modes in TMDs resonators, which would provide novel basis for developing phononic devices24, 25

or exploring mechanical motions mimic quantum phenomena.22, 26 Furthermore, the quality

factors and coupling rates could be significantly enhanced by straining the membrane, improving material quality, and suppressing sources of damping, and implementation in single-layer devices would enable piezoelectric control.27, 28


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ASSOCIATED CONTENT Supporting Information. Detailed device characterization, finite element simulations of mode frequencies and detunings, analytical modeling the frequency detunings of coupled modes, calculation of thermal equilibrium time constant, demonstration of normal-mode splittings for the higher vibration modes, and numerical modeling of membrane dynamics and mode coupling rate. The Supporting Information is available free of charge on the ACS Publications website at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] Notes The authors declare no competing financial interest.

ACKNOWLEDGMENT This work was supported by the Materials Research Science and Engineering Center (MRSEC) of Northwestern University (National Science Foundation Grant DMR-1121262).


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Figure 1 Characterization of a MoS2 mechanical resonator. (a) Schematic illustration of the fiber-optic interferometer measurement (not to scale). (b) Scanning electron microscopy image of a square suspended MoS2 thin film synthesized by chemical vapor deposition.29 Scale bar: 5 µm. (c) Schematic illustration of the experimental setup. AOM, acousto-optic modulator; ISO, isolator; BS, beam splitter; FC, (90:10) fiber coupler; PD, photodiode. (d) Top panel: Finite element simulation and experimental 2D mapping of the mechanical (1,1), nearly degenerate (1,2) and (2,1) mode amplitudes (left to right). Bottom panel: Measured power spectral density (PSD, round symbols) and Lorentzian fit (solid line) of the (1,1) mode, nearly degenerate (1,2) and (2,1) modes. (e) Top panel: Finite element simulation and experimental 2D mapping of the mechanical (2,2), (1,3) and (3,1) mode amplitudes (left to right). Bottom panel: Measured power spectral density (PSD, round symbols) and Lorentzian fit (solid line) of the (2,2) mode, nearly degenerate (1,3) and (3,1) modes. All measurements were conducted in vacuum (10KL torr) at room temperature.


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Figure 2 Optically-mediated vibrational spectra. (a) Measured resonance frequencies of (1,1), (1,2) and (2,1) modes versus illumination power. (b) Measured resonance frequencies of (2,2), (1,3) and (3,1) modes versus illumination power. (c) Frequency detuning of the fundamental mode as a function of excitation power and average membrane temperature. Symbols are experimental data and the solid line is from a finite element simulation (Supporting Information). (d)-(e) Frequency detuning of (d) nearly degenerate (1,2) and (2,1) modes, and (e) (2,2) and (1,3) modes, as a function of excitation power. Avoided crossings indicate mode-coupling. Symbols are experimental data and solid lines are fits to a coupled oscillator model (Supporting Information).


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Figure 3 Mechanical energy redistribution. (a) Vibrational spectrum of the fundamental mode under parametric optical excitation with power fixed at 110 µW. (b) Simulated vibrational spectrum of the fundamental mode as a function of light modulation frequency. (c) Measured vibrational spectrum of the fundamental mode as a function of excitation power. The excitation laser was modulated at 30 kHz. (d) Simulated vibrational spectrum of the fundamental mode as a function of optical excitation power with the light modulation frequency fixed at 30 kHz.


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Figure 4 Normal mode splitting under parametric optical excitation. (a) Measured vibrational spectrum of the (1,2) and (2,1) modes under parametric optical excitation. The laser power was fixed at 200 µW. (b) From left to right: signature of normal-mode splitting of (2,1) mode at excitation powers of 150 µW, 210 µW, and 270 µW, respectively. The minima and maxima of the images in dBm are (-102, -80), (-102, -80) and (-100, -82) from left to right. (c) Mode splitting of (2,1) mode at different excitation powers. Each curve was measured at the resonance condition = , − , ), where the splitting of peaks is smallest, i.e., the curves are closest. (Excitation power: 150, 180, 210, 240, 270, 310 µW, bottom to top) (d) Measured (round symbols) and simulated (solid line) coupling rate as a function of excitation power (temperature variation of membrane). ACS Paragon Plus Environment

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