tunable phonon-cavity coupling in graphene...

Tunable phonon-cavity coupling ingraphene membranesR. De Alba1*, F. Massel2, I. R. Storch1, T. S. Abhilash1, A. Hui3, P. L. McEuen1,4, H. G. Craighead3

and J. M. Parpia1

A major achievement of the past decade has been the realizationof macroscopic quantum systems by exploiting the interactionsbetween optical cavities and mechanical resonators1–3. In thesesystems, phonons are coherently annihilated or created inexchange for photons. Similar phenomena have recently beenobserved through phonon-cavity coupling—energy exchangebetween the modes of a single system mediated by intrinsicmaterial nonlinearity4,5. This has so far been demonstratedprimarily for bulk crystalline, high-quality-factor (Q > 105) mech-anical systems operated at cryogenic temperatures. Here, wepropose graphene as an ideal candidate for the study of such non-linear mechanics. The large elastic modulus of this material andcapability for spatial symmetry breaking via electrostatic forcesis expected to generate awealth of nonlinear phenomena6, includ-ing tunable intermodal coupling. We have fabricated circular gra-phene membranes and report strong phonon-cavity effects atroom temperature, despite the modest Q factor (∼100) of thissystem. We observe both amplification into parametric instability(mechanical lasing) and the cooling of Brownian motion in thefundamental mode through excitation of cavity sidebands.Furthermore, we characterize the quenching of these parametriceffects at large vibrational amplitudes, offering a window on theall-mechanical analogue of cavity optomechanics, where theobservation of such effects has proven elusive.

Mechanical resonators composed of atomically thin membraneshave been widely studied in recent years7–16. In the case of graphene,its low mass (ρg ≈ 0.75mgm−2) electrical integrability and strongoptical interaction11,17 make it a rich and versatile system that hasbeen studied largely for force and mass sensing. At room tempera-ture the moderateQ factor, extreme frequency tunability and low in-line resistance of graphene resonators make them promising asintermediate-frequency (1–50 MHz) electromechanical elements,including passive filters and oscillators. At cryogenic temperatures(T < 4 K) graphene is becoming an attractive system for the studyquantum motion, as it exhibits both large zero-point motion anda drastically enhanced Q-factor; progress towards this end hasalready been made18–20, with coupling to on-chip microwave cavitiesand significant optomechanical cooling recently demonstrated. Themechanical nonlinearity studied here represents a complementarymethod for the parametric control of these membranes based onthe intrinsic interactions of their vibrational modes. This effectcan be utilized to enhance the Q-factor (and hence sensitivity) ofgraphene-based sensors, provide multimode readout through thedetection of a single mode21 and ultimately enable informationexchange between optically cooled quantum modes. Moreover,this coupling makes graphene viable as low-power, tunable,electromechanical frequency mixers.

The primary source of nonlinearity in graphene membranes ismotion-induced tension modulation. Similar to mode coupling inother mechanical systems4,22,23, one vibrational mode (hereassumed to be the fundamental mode at frequency ω1) can be para-metrically manipulated through its interaction with a second mode,which is deemed the phonon cavity (at ωc). Exciting the coupledsystem at the cavity’s red sideband (ωc − ω1) results in energy flowfrom the fundamental to the cavity mode, whereas pumping theblue sideband (ωc + ω1) generates amplification of both the funda-mental and cavity modes; these processes are depicted in Fig. 1c.The efficiency of this intermodal energy exchange is dictated bythe coupling rate G = dωc/dx1, where x1 is the amplitude ofmotion at ω1. This coupling rate is reminiscent of cavity optomecha-nics, and an identical formalism can be used to derive the resultingequations of motion (Supplementary Section 1).

The advantages of graphene over other membrane materials(such as SiN or MoS2) in generating this effect are twofold. Aswill be shown below, G increases linearly with the static membranedeflection x0. In typical graphene devices this value can be tunedelectrostatically via a d.c. gate voltage. Moreover, graphene can with-stand exceptionally large out-of-plane stretching as a result ofits atomic thinness (h ∼ 0.3 nm) and low in-plane stiffnessC = Eh/(1 − ν2), where E and ν are the elastic modulus (160 GPa(ref. 24) to 1.0 TPa (ref. 25)) and Poisson’s ratio respectively. Previousstudies of suspended graphene have shown that x0 can exceed 3% ofthe membrane width without rupturing26. As the tension in grapheneis highly tunable, the frequency spectrum can be adjusted to obtainthree-mode alignment, ωc ±ω1 ≈ωsb. Here ωsb signifies the resonanceof a third mode that overlaps the cavity sideband and enhancespumping by a factor of Qsb; this arrangement is also depicted inFig. 1c. Under these conditions, it is thus possible to generate largephonon-cavity effects in the room-temperature graphene system.

Alternative intermodal coupling mechanisms for tensioned mem-branes do exist—most notably, mutual coupling to a resonance of thesurrounding substrate23. Such systems enable parametric membranecontrol in a manner qualitatively similar to the coupling studiedhere, but also necessitate the three-mode alignment describedabove, which can be a challenge if the spectrum is not experimentallytunable. Moreover, a unique feature of the graphene system is the tun-ability of the coupling rate itself, G ∝ x0, which is present neither inthe substrate-coupled case nor in standard optomechanical systems.

We have fabricated circular graphene drums with a diameter dranging from 5–20 μm; we report measurements of two suchdrums—Device 1 (d = 8 μm) and Device 2 (d = 20 μm)—althoughthe effects reported have been observed across a wide number ofsamples. A diagram of the experimental set-up and a micrographof Device 1 are shown in Fig. 1a,b. Motion is driven electrostatically

1Department of Physics, Cornell University, Ithaca, New York 14853, USA. 2Department of Physics, Nanoscience Center, University of Jyväskylä,Jyväskylä FI-40014, Finland. 3School of Applied & Engineering Physics, Cornell University, Ithaca, New York 14853, USA. 4Kavli Institute at Cornell forNanoscale Science, Ithaca, New York 14853, USA. *e-mail: [email protected]

LETTERSPUBLISHED ONLINE: 13 JUNE 2016 | DOI: 10.1038/NNANO.2016.86

NATURE NANOTECHNOLOGY | VOL 11 | SEPTEMBER 2016 | www.nature.com/naturenanotechnology 741

© 2016 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

mailto:[email protected]

http://dx.doi.org/10.1038/nnano.2016.86

http://www.nature.com/naturenanotechnology

Stokes

pumping

(amplification)

Anti-stokespumping

(deamplification)

c

180°

ω1 ωsb ωc

ωp

ω1

Δ = ω1 ωp

ωc ωsb

Phononcavity, ωc

Sidebandmode, ωsb

Mode ofinterest, ω1

Sidebandmode, ωsb

Phononcavity, ωc

Mode ofinterest, ω1

Vdc

HeNelaser To

photodiode

ba

vac sin ωt

Figure 1 | The nonlinear system under test. a, A schematic of the experimental set-up. Graphene motion is driven electrostatically by two metallicback-gates and detected through optical interferometry. The gates can be driven in various configurations to favour excitation of the fundamental mode,higher-frequency modes or both. b, False-colour electron micrograph of Device 1. Scale bar, 2 μm. c, Schematic of the three modes necessary forefficient sideband pumping and their relative positions in frequency space. The curved arrows indicate the direction of energy flow when the system ispumped at ωp.

Vdc (V)

Vdc (V)ω/2π (MHz)

ω/2π

(MH

z)

20

22

0 5 10 15 20 25 30

12

14

5 10 15 20 25 30 350

50

100

|x| (

pm)

ω1 ω2 ω3

ω4 ω5 ω6

Rex (a.u.)

b

c

d

ω/2π

(MH

z)

0 562

−20 −10 0 10 20

5

10

15

20

25

30

35

|x| (pm)

a

ω1

ω2

ω3

ω4

ω5

ω6

ω 2 + ω 1

ω6 − ω1

ω2

ω6

+1−1

Figure 2 | Multimode membrane characterization. a, Frequency dispersion with Vdc for the lowest six modes in Device 1. b, Mechanical mode shapesat Vdc = 5 V measured by scanning the detection laser across the membrane surface while driving on resonance; colour denotes the real part of the complexamplitude x, that is, the quadrature of x that is 90° out of phase with the applied a.c. voltage. The electron micrograph is given as a reference for orientation.c, Frequency spectrum at Vdc = 5 V. d, Resonant frequencies of mode 2 and mode 6 extracted from a in comparison with their sidebands with mode 1.Appreciable overlap between these frequencies occurs for Vdc = 0−7.5 V and strong phonon-cavity effects are thus expected in this range.

LETTERS NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2016.86

NATURE NANOTECHNOLOGY | VOL 11 | SEPTEMBER 2016 | www.nature.com/naturenanotechnology742




via an applied gate voltage Vdc + vac sin ωt (where ω, t denote thedrive frequency and time, respectively), and detected opticallythrough laser interferometry11. Unlike previous generations ofgraphene resonators, our structures feature two independent back-gates that enable efficient actuation of several modes. The gate–graphene separation is 1.7 μm. Most measurements were performedwith one gate grounded and a drive voltage applied to the other,although other configurations (shown in Fig. 1a) can be used tofavour either the fundamental or higher modes.

Device 1 has six modes that can be readily excited (Fig. 2a,b). Thefrequency dispersion of this spectrum with Vdc is shown in Fig. 2a.Between Vdc = 0−7.5 V, there is significant overlap of modes 1, 2, 6 andtheir respective sidebands (Fig. 2d); therefore this is where we expectthe strongest phonon-cavity effect. At Vdc = 5 V the graphene has naturalfrequencies and Q-factors of: ω1/2π = 8.6 MHz, ω2/2π = 12.4 MHz,ω6/2π = 21.0 MHz, Q1 =ω1/γ1 = 57, Q2 = 48 and Q6 = 37.

The general Hamiltonian for two coupled modes in a uniformlytensioned membrane is

H =∑n=i,j

p2n2m

+12mω2

nx2n + Lnxn + Snx

2n + Tnx

3n + Fnx

4n

( )

+ Tijxix2j + Tjixjx

2i + Fijx

2i x

2j (1)

wherem and pn are the membranemass andmomentum ofmode n; aderivation of equation (1) can be found in Supplementary Section 1.The fourth-order nonlinearity Fi is the stretching-induced Duffingterm and is proportional to the membrane stiffness C. The remain-ing terms Li , Si and Ti originate from the same geometric nonlinear-ity as Fi, combined with a static displacement x0. Although the exactderivation requires the full solution of the elastic problem for themembrane, Li, Si and Ti can be viewed roughly as resulting from abinomial expansion of Fi(x0 + xi cosωit)

4. The coupling terms Fijand Tij have similar origins, resulting in Tij∝ x0. In the SupplementaryInformation we present a calculation of these nonlinear coefficientsfor circular membranes and for a general membrane geometry.

To understand how each term in equation (1) influences mem-brane mechanics, it is useful to consider the forces acting onmode i (−∂H/∂xi) and examine each term in isolation. Using thisapproach we see that Li has no effect other than to exert a constantforce on mode i, whereas Si modifies the linear spring constant mω2

iand Ti and Fi contribute to a nonlinear spring constant. Thecoupling term Fij alters the mode i spring constant by an amountproportional to |xj|

2. Only the third-order term Tij generates aphonon-cavity effect on mode i. This is a result of its combinedinfluence on i and j: if mode i is driven at ωi while mode j isdriven at ωj ± ωi, mode j experiences a force Tijxixj ∝ cos ωjt, whichproduces motion at the cavity resonance ωj. The two frequency

ω/2π

(MH

z)

ωp/2π (MHz)

243 pm

ωp/2π (MHz)

ω/2π

(MH

z)

33 368

12 14 16 18 20 228.0

8.2

8.4

8.6

8.8

9.0

|x| (pm)

135 pm 289 pma

32 268

12 14 16 18 20 22 24

8.6

8.8

9.0

9.2

9.4

|x| (pm)

166 pmb

ωp/2π (MHz)

ω/2π

(MH

z)

12 14 16 18 20 228.4

8.6

8.8c

ω2+ωω1+ωω6−ω ω6ω4ω2

ω/2π (MHz)8.3 8.5 8.7 8.9

100

200

300

400

|x| (

pm)

8.3 8.5 8.7 8.9

d ωp = 12.5 MHz

ωp =21.0 MHz

ωp/2π (MHz)

ω/2π

(MH

z)

154 272

12 13

8.2

8.5

8.8

154 2888.2

8.5

8.8

66 154

eExperiment

72 154

20.5 21.5

γ1,eff /2π

Model

Figure 3 | Phonon pumping in Device 1. a,b, Mode 1 amplitude versus ωp and ω at Vdc = 5 V (a) and 10 V (b). Right panels: Vertical slices through the dataat the highest ωp value. Upper panels: Motion in the membrane at ωp measured simultaneously with the main panel. Measurements for both Vdc valueswere performed with equal excitation forces (F∝Vdcvac) at the pump frequency; probe frequency forces were also equal. Cavity amplification anddeamplification of mode 1 are stronger in a, where there is better mode–sideband alignment. c, Modelled behaviour in a based on equations (2)–(4). Solidlines denote the relevant frequencies for sideband effects. d, Measured response at the cavity sidebands for Vdc = 5 V with linearly increasing pump strength(darkening lines). e, Effective mode 1 damping as measured in a (top) and modelled by equation (3) (bottom) expressed in kHz (colour scales). The coloursin the upper panel are truncated to the intrinsic damping γ1/2π = 154 kHz. Quenching of the cavity effect near ω =ω1 is due to the large mode 1 amplitudeand a non-zero Tsb,c coupling. Only two free parameters (T1c and Tsb,c) were used to produce each of the lower panels.

NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2016.86 LETTERS





components of j then mix to exert a back-action force Tijx2j ∝ cosωit

onmode i, which will amplify or dampen its motion depending on thephase of this force (that is, whether the + or − sideband was driven).The remaining term Tji has no appreciable impact on mode i, butenables cavity effects on mode j.

Referring again to i and j as modes 1 and c, the cavity couplingrate (in a linearized approximation) is G = dωc/dx1 ≈ T1c/mωc. Tomeasure this coupling we drive our graphene membrane with aprobe signal at frequency ω ≈ ω1 and a pump signal at ωp. Interms of the cavity detuning Δ = ωp − ωc and the pumped vibrationamplitude xp, the effective resonant frequency and damping ofmode 1 are:

ω1,eff = Ω1 +2G2

∣∣xp∣∣2Δ[γ2c /4 − ω2 + Δ2]

[γ2c /4 + (ω − Δ)2][γ2c /4 + (ω + Δ)2](2)

γ1,eff = γ1 −4G2

∣∣xp∣∣2γcΔΩ1

[γ2c /4 + (ω − Δ)2][γ2c /4 + (ω + Δ)2](3)

mω1Ω1 =mω21 + 2S1 − 12

T1(T1c

∣∣xp∣∣2 + L1/2)

mω21 + 4S1

+ 4F1c∣∣xp∣∣2 (4)

where Ω1 describes the combined effects of L1, S1, T1 and F1c on themode 1 spring constant.

The vibration amplitude of Device 1, mode 1 is shown in Fig. 3a,bas ωp is swept from ω2 to ω6. The |xp|

2 terms in equation (4) generatea downward frequency shift when any mode is pumped directly onresonance; this is most visible at ωp/2π ≈ 16 MHz. Sideband ampli-fication and deamplification are also seen and occur when pumpingthe blue sideband of mode 2 and red sideband of mode 6, respect-ively (Fig. 3a). Amplification also occurs at ωp = 2ω1 and is mostnotable at Vdc = 10 V, where 2ω1 ≈ ω4; this effect is studied infurther detail in Supplementary Section 5.

The amplitude of mode 1 on sideband pumping, shown inFig. 3d, is nearly linear with pump amplitude—in contrast to the|xp|

2 dependence predicted by equation (3). Analysing the effectivedamping γ1,eff at the cavity sidebands reveals the source of this disagree-ment (Fig. 3e). Suppression of the sideband effects is observed aroundω =ω1, indicating a broadening of the sideband mode due to the probeamplitude x1. For the case of ωc =ω2, ωsb =ω6, motion at ω1 and a non-zero coupling T62 result in increased mode 6 damping γ6,eff, hinderingits ability to amplify mode 1. This quenching of the cavity effects can beavoided by probing mode 1 with lower amplitudes and speaks to thedynamic range of a micromechanical filter/amplifier based onphonon-cavity coupling. Careful engineering of the device modes

S xx (p

m2 H

z−1)

(ωp +ω)/2π (MHz)

3.74 3.76 3.78 3.80 3.82 3.84

0.0

0.2

0.4

0.6

0.8

1.0

|x| (

nm) 3.77 3.78

0.0

0.2

0.4

0.6

0.8

(MHz)

(nm

)

9.68 9.70 9.72 9.74 9.76

0

20

40

60

80

100

120

140

|x| (

pm) 9.73 9.74

020406080

(MHz)

(pm

)

2.90 2.95 3.00 3.05

0.00

0.05

0.10

0.15

0.20

0.25

2.92 2.94 2.96 2.98 3.00 3.02

0.0

0.5

1.0

1.5

2.0

2.5

3.0

|x| (

nm) 2.97 2.98 2.99

0

1

2

(MHz)

(nm

)

a b

c d

v ac,p = 0

vac,p = 46 mV

(ωp − ω)/2π (MHz)ω/2π (MHz)

ω/2π (MHz)

0 20 400.7

0.8

0.9

1.0

T/T 0

vac,p (mV)

Figure 4 | Parametric self-oscillation and cooling in Device 2. a, Amplification of mode 1 (ω1/2π = 3.0 MHz, γ1/2π = 45 kHz) and the transition tomechanical lasing (γ1,eff≤0) via mode coupling. Mode 1 is probed with a weak drive (vac = 0.4 mV) as mode 2 is pumped at its Stokes sideband(ωp = 6.8MHz) with increasing pump strength (vac,p = 0−400 mV). The curves are vertically offset for clarity. Inset: Saturation of the vibrational amplitudeand the flat-top response of the self-oscillating mode; no vertical offset is applied. b,c, Frequency mixing via mechanics. Measured membrane motionat ωp −ω (b) and ωp +ω (c) recorded simultaneously with a. d, Measured spectral noise density near ω1 on pumping the anti-Stokes sideband of mode 5(ωp = 3.8MHz). The curves are vertically offset for clarity. Inset: the effective temperature of mode 1 (normalized by T0 = 293 K), corresponding to the areaunder the Sxx fits. The frequency spectrum of Device 2 is given in Supplementary Section 2.






such that Tsb,c≈ 0 would counteract this effect, and can potentially beachieved by using a more sophisticated membrane clamping scheme27

or altering themembrane shape28. A detailed analysis of this quenchingis presented in Supplementary Section 6. Correcting for this effect(Supplementary Fig. 8) shows that the coupling rates in this deviceare G = 6 MHz nm−1 for blue sideband pumping (amplification)and G = 8 MHz nm−1 for red sideband pumping (deamplification).At the single quantum level, these correspond tog0 = G|x1,zpm| = 300Hz and g0 = 400 Hz respectively, wherex1,zpm = √(h− /2mω1) is the zero-point motion of mode 1.

Stronger phonon-cavity effects have been measured in Device 2,where the larger device diameter permits the use of muchweaker probe signals while maintaining comparable signal/noise ratios. Measurements were performed with Vdc = 4 V, sothat ω1 + ω2 ≈ ω5. Figure 4a shows the membrane response onpumping at ωp =ω1 + ω2 = 2π × 6.76 MHz with an a.c. voltage vac,pramped linearly from 0−400 mV. Mode 1 is probed with vac = 0.4 mVand its motion undergoes amplification by a factor of 8.5 (19 dB)before entering instability (γ1,eff ≤ 0) at vac,p = 300 mV. Above thispump strength, mode 1 undergoes self-oscillation and locks ontothe probe signal with a flat frequency response. The width of thisflat region is 4 kHz, significantly narrower than the unpumped line-width γ1/2π = 45 kHz. Amplification of mode 1 continues to rise forhigher pump strengths, reaching a factor of 18 (25 dB) at the highestvalue tested.

In this configuration the graphene membrane also acts as a fre-quency mixer, generating motion at ωp + ω and ωp − ω (Fig. 4b–c).Motion at ωp − ω ≈ ω2 signifies occupation of the cavity mode as aresult of down-scattered pump phonons, and so is significantlylarger (10× ) than motion at ωp + ω, where there is no mechanicalresonance. Both of these mixed tones inherit the flat-top spectrumof mode 1 when it is in the self-oscillating regime.

Similar to deamplification in Device 1, red sideband pumpingin Device 2 has been used to cool the thermal motion of mode 1to 208 K (Fig. 4d). As in previous phonon-cavity studies4, the lowcavity frequency ωc ∼ ω1 (and high thermal phonon occupation)limits cooling in the all-mechanical system. Cooling motiontowards the quantum ground state thus remains a task bestsuited for optical/microwave cavities, where ωc≫ ω1. However,interesting prospects arise if optical cavities and phononcavities are utilized simultaneously to control graphenemotion. For instance, optically cooling the phonon cavityenhances its capacity to mechanically cool the fundamentalmode—in such a case cooling is limited only by the cooperativitiesG2|xp|

2/γ1γc of the two cavities. Moreover, the mechanical pumpgrants experimental control over the interaction strength of thetwo modes. Microwave-cavity-coupled graphene systems18–20 aretherefore ideal testbeds for quantum entanglement, squeezing,thermalization and information exchange between modes neartheir ground state. The greatly enhanced Q-factors of graphene atdilution refrigerator temperatures8,29 will only serve to strengthenthese effects.

We have demonstrated tension-mediated coupling betweenmechanical modes in suspended graphene and its potential forparametric control of this system. Sideband cooling and theamplification of membrane motion up to self-oscillation havebeen observed within a single device. The potential for graphenemembranes to be used as frequency mixers with intrinsically flatpass-bands has also been shown. As graphene, transition metaldichalcogenides and related two-dimensional materials continueto be developed and exploited for their unique mechanicalproperties, these inherent membrane nonlinearities can ultimatelybe used to artificially enhance the Q-factors of future sensors andelectronics, to facilitate bitwise logic operations between coupledmembrane modes30 and to open new possibilities in the study ofcoupled quantum systems.

MethodsMethods and any associated references are available in the onlineversion of the paper.

Received 18 November 2015; accepted 26 April 2016;published online 13 June 2016

References1. O’Connell, A. D. et al. Quantum ground state and single-phonon control of a

mechanical resonator. Nature 464, 697–703 (2010).2. Teufel, J. D. et al. Sideband cooling of micromechanical motion to the quantum

ground state. Nature 475, 359–363 (2011).3. Andrews, R. W. et al. Bidirectional and efficient conversion between microwave

and optical light. Nature Phys. 10, 321–326 (2014).4. Mahboob, I., Nishiguchi, K., Okamoto, H. & Yamaguchi, H. Phonon-cavity

electromechanics. Nature Phys. 8, 387–392 (2012).5. Mahboob, I., Nishiguchi, K., Fujiwara, A. & Yamaguchi, H. Phonon lasing in an

electromechanical resonator. Phys. Rev. Lett. 110, 127202 (2013).6. Khan, R., Massel, F. & Heikkilä, T. T. Tension-induced nonlinearities of flexural

modes in nanomechanical resonators. Phys. Rev. B 87, 235406 (2013).7. Bunch, J. S. et al. Electromechanical resonators from graphene sheets. Science

315, 490–493 (2007).8. Chen, C. et al. Performance of monolayer graphene nanomechanical resonators

with electrical readout. Nature Nanotech. 4, 861–867 (2009).9. Singh, V. et al. Probing thermal expansion of graphene and modal dispersion at

low-temperature using graphene nanoelectromechanical systems resonators.Nanotechnology 21, 165204 (2010).

10. Chaste, J. et al. A nanomechanical mass sensor with yoctogram resolution.Nature Nanotech. 7, 301–304 (2012).

11. Barton, R. A. et al. Photothermal self-oscillation and laser cooling of grapheneoptomechanical systems. Nano Lett. 12, 4681–4686 (2012).

12. Song, X. et al. Stamp transferred suspended graphene mechanical resonators forradio frequency electrical readout. Nano Lett. 12, 198–202 (2012).

13. Eichler, A. et al. Nonlinear damping in mechanical resonators made fromcarbon nanotubes and graphene. Nature Nanotech. 6, 339–342 (2011).

14. Eichler, A., del Álamo Ruiz, M., Plaza, J. A. & Bachtold, A. Strong coupling betweenmechanical modes in a nanotube resonator. Phys. Rev. Lett. 109, 025503 (2012).

15. Lee, J., Wang, Z., He, K., Shan, J. & Feng, P. X.-L. High frequency MoS2nanomechanical resonators. ACS Nano 7, 6086–6091 (2013).

16. Liu, C. H., Kim, I. S. & Lauhon, L. J. Optical control of mechanical mode-coupling within a MoS2 resonator in the strong-coupling regime. Nano Lett. 15,6727–6731 (2015).

17. Cole, R. M. et al. Evanescent-field optical readout of graphene mechanicalmotion at room temperature. Phys. Rev. Appl. 3, 1–7 (2015).

18. Weber, P., Güttinger, J., Tsioutsios, I., Chang, D. E. & Bachtold, A. Couplinggraphene mechanical resonators to superconducting microwave cavities. NanoLett. 14, 2854–2860 (2014).

19. Song, X., Oksanen, M., Li, J., Hakonen, P. J. & Sillanpää, M. A. Grapheneoptomechanics realized at microwave frequencies. Phys. Rev. Lett. 113, 1–5 (2014).

20. Singh, V. et al. Optomechanical coupling between a graphenemechanical resonatorand a superconducting microwave cavity. Nature Nanotech. 9, 1–5 (2014).

21. Atalaya, J., Kinaret, J. M. & Isacsson, A. Narrowband nanomechanical massmeasurement using nonlinear response of a graphene membrane. Europhys. Lett.91, 48001 (2009).

22. Westra, H. J. R., Poot, M., van der Zant, H. S. J. & Venstra, W. J. Nonlinear modalinteractions in clamped-clamped mechanical resonators. Phys. Rev. Lett. 105,117205 (2010).

23. Patil, Y. S., Chakram, S., Chang, L. & Vengalattore, M. Thermomechanical two-mode squeezing in an ultrahigh Q membrane resonator. Phys. Rev. Lett. 115,017202 (2015).

24. Ruiz-Vargas, C. S. et al. Softened elastic response and unzipping in chemicalvapor deposition graphene membranes. Nano Lett. 11, 2259–2263 (2011).

25. Lee, C., Wei, X., Kysar, J. W. & Hone, J. Measurement of the elastic propertiesand Intrinsic strength of monolayer graphene. Science 321, 385–388 (2008).

26. Bunch, J. S. et al. Impermeable atomic membranes from graphene sheets. NanoLett. 8, 2458–2462 (2008).

27. Lee, S. et al. Electrically integrated SU-8 clamped graphene drum resonators forstrain engineering. Appl. Phys. Lett. 102, 153101 (2013).

28. Wang, Z., Lee, J., He, K., Shan, J. & Feng, P. X.-L. Embracing structuralnonidealities and asymmetries in two-dimensional nanomechanical resonators.Sci. Rep. 4, 3919 (2014).

29. Van Der Zande, A. M. et al. Large-scale arrays of single-layer grapheneresonators. Nano Lett. 10, 4869–4873 (2010).

30. Mahboob, I. & Yamaguchi, H. Bit storage and bit flip operations in anelectromechanical oscillator. Nature Nanotech. 3, 275–279 (2008).

AcknowledgementsThe authors are grateful to P. Rose for assistance in growing the CVD graphene and toD. MacNeill for insightful discussions and comments. This work was supported by the








Cornell Center for Materials Research with funding from the NSF MRSEC program(grant no. DMR-1120296) and by Nanoelectronics Research Initiative (NRI) through theInstitute for Nanoelectronics Discovery and Exploration (INDEX). Support was alsoprovided by the Academy of Finland (through the project ‘Quantum properties ofoptomechanical systems’). Fabrication was performed in part at the Cornell NanoScaleFacility, a member of the National Nanotechnology Coordinated Infrastructure, which issupported by the NSF (grant no. ECCS-15420819).

Author contributionsJ.M.P., H.G.C., P.L.M. and R.D.A. designed the experiment; F.M. developed the supportingtheory. R.D.A. and T.S.A. fabricated the samples; I.R.S. contributed to the design of the

samples and the experimental set-up. R.D.A. and A.H. carried out the measurements.F.M. and R.D.A. analysed the data. All authors discussed the results and commented onthe manuscript.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints. Correspondence andrequests for materials should be addressed to R.D.A.

Competing financial interestsThe authors declare no competing financial interests.





http://www.nature.com/reprints



MethodsMechanical resonators were fabricated by the growth of monolayer graphenethrough chemical vapour deposition and transfer to pre-patterned substrates. Beforetransfer, a supporting layer of 150 nm poly-methyl-methacrylate (PMMA) was spin-coated on the graphene surface and cured at 170 °C. During the transfer the Cugrowth substrate was etched using FeCl3 and the PMMA/graphene film was cleanedby soaking in a series of deionized water baths. After transfer, the film was coatedwith photoresist and patterned via optical lithography; the resist and PMMA wereremoved by submersion in N-methyl-2-pyrollidine at 80 °C, releasing the suspendedgraphene membranes.

All measurements were performed at room temperature in a vacuum ofP < 10−6 mbar. The detection of mechanical motion was performed through opticalinterferometry, as detailed in previous work11, using 190 μW and 130 μW of incidentlaser power for Devices 1 and 2, respectively. The light source used was an HeNe633 nm laser, focused to a spot of diameter ∼1 μm. Reflected light was monitored by ahigh-frequency photodetector (New Focus 1811-FS) and recorded using amultichannel lock-in amplifier (Zurich Instruments HF2LI). The same lock-inamplifier was used to supply excitation voltages at the pump and probe frequencies.Graphene motion was inferred from the modulated laser power using the opticalcalibration scheme detailed in Supplementary Section 3.


NATURE NANOTECHNOLOGY | www.nature.com/naturenanotechnology




NATURE NANOTECHNOLOGY | www.nature.com/naturenanotechnology 1

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NNANO.2016.86

Tunable phonon cavity coupling in graphene membranesSUPPLEMENTARY INFORMATION

Roberto De Alba,1 Francesco Massel,2 Isaac R. Storch,1 T. S. Abhilash,1

Aaron Hui,3 Paul L. McEuen,1, 4 Harold G. Craighead,3 and Jeevak M. Parpia1

1Department of Physics, Cornell University, Ithaca, New York 14853, USA2Department of Physics, Nanoscience Center, University of Jyvaskyla, F1-40014, Finland

3School of Applied & Engineering Physics, Cornell University, Ithaca, New York 14853, USA4Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA

CONTENTS

S1. Mode coupling in a 2D circular membrane with electrostatic drive – Theory 2

S2. Characterization of Devices 1 and 2 7

S3. Calibration of optical detection system 8

S4. Measurement of effective damping and frequency 10

S5. Additional mode coupling effects in Device 1 12

S6. Large-amplitude quenching of phonon cavity and sideband mode 13

S7. Mode coupling in a third device 14

S8. Duffing response & nonlinear damping 15

References 16

Tunable phonon-cavity coupling ingraphene membranes

© 2016 Macmillan Publishers Limited. All rights reserved.


2 NATURE NANOTECHNOLOGY | www.nature.com/naturenanotechnology

SUPPLEMENTARY INFORMATION DOI: 10.1038/NNANO.2016.86

SUPPLEMENTARY INFORMATION De Alba et al. 2016

S1. MODE COUPLING IN A 2D CIRCULAR MEMBRANE WITH ELECTROSTATIC DRIVE –THEORY

The static deformation and the dynamics of the circular membrane are described in terms of the following La-grangian, for which we have assumed that the induced tension can be averaged over the membrane geometry, effectivelyleading to a description in terms of a mean-field model:

L =ρ

2

∫dA x2 − D

2

∫dA(∆x)2 − C

16

[1

A

∫dA(∇x)2

]·∫

dA(∇x)2

− T0

2

∫dA(∇x)2 −

ε0V2g

2

∫dA

d− x

(S1)

where ρ is the surface density, D is the flexural rigidity, C the in-plane stiffness, T0 the built-in tension, Vg the gatevoltage, d the gate-sheet separation, x the deformation. If the description of the system can be performed in termsof a continuum model, we have

C =Eh

1− ν2(S2)

D =Eh3

12(1− ν2). (S3)

The first term in Equation S1 represents the kinetic energy associated with the dynamics of the membrane, thesecond its flexural elastic energy, the third the energy associated with the deformation-induced tension (treated on amean-field level), the fourth the energy due to built-in tension and last term corresponds to the capacitive couplingwith the back gate.

By expanding x into static and resonant components x(r, t) = x0(r)+∑

i xi(t)ξi(r) – where ξi(r) is the dimensionlessprofile of mode i, normalized such that 1

A

∫dA|ξi(r)|2 = 1 – it is straightforward to show that Equation S1 leads to

the Hamiltonian given in Equation 1 of the main text, with nonlinear coefficients

Li =

[T0 +

C

4A

∫dA(∇x0)

2

]·∫

dA(∇x0∇ξi) (S4)

Si =C

4A

[∫dA(∇x0∇ξi)

]2(S5)

Ti =C

4A

[∫dA(∇x0∇ξi)

]·∫

dA(∇ξi)2 (S6)

Fi =C

16A

[∫dA(∇ξi)

2

]2(S7)

Sij =C

2A

[∫dA(∇x0∇ξi)

]·∫

dA(∇x0∇ξj) (S8)

Tij =C

4A

[∫dA(∇x0∇ξi)

]·∫

dA(∇ξj)2 (S9)

Fij =C

8A

[∫dA(∇ξi)

2

]·∫

dA(∇ξj)2 (S10)

Above we have assumed displacements are large such that the flexural rigidity is negligible: x h.It is important to note that the terms listed in Equations S4 - S10 (and included terms in Equation 1 of the main

text) are the only nonlinear terms expected from the Lagrangian S1. Terms of the type Fijxix3j , Tijkxixjxk, and

Fijkxixjx2k do not appear in this calculation due to mode orthogonality and are therefore excluded from Equation 1.

From this perspective, our analysis differs from that performed in Reference [1], where the full Foppl-von Karmanequations are solved in the limit of vanishing flexural rigidity. In our case, while taking into account the effect offinite flexural rigidity, owing to the smallness of the membrane displacement, we neglect spatial inhomogeneities ofthe tension.

We now proceed to calculate the nonlinear coefficients L, S, T , F explicitly for a circular membrane geometry.From Equation S1 it is possible to derive an equation for the static displacement of the membrane

∆2ζ0 − τ∆ζ0 = κ0V2g (S11)

2






where ζ0 represents the static displacement (expressed in units of r0) of the membrane in presence of a time-independent external voltage Vg, κ0 = ε0r

30/

[2(d− x)2D

]and τ(= Tr20/D) is the (dimensionless) membrane tension,

whose value has to be determined from the solution of the following equation

τ = τ0 +2ε

π

∫dΩ(∇ζ0)

2. (S12)

The solution of Equation S11 is given by

ζ0 =κ0V

2g

4τ

[1− r2 + 2

I0(√τr)− I0(

√τ)

τI1(√τ)

]. (S13)

In order to determine τ , the result given in Equation S13 is substituted into Equation S12, yielding the followingself-consistent equation for τ

τ = τ0 +εκ2

0V4g

4τ2

(16

τ− 2R0(τ)√

τ−R2

0(τ) + 3

)(S14)

with R0(τ) = I0(√τ)/I1(

√τ) (In(x) is the modified Bessel function of the first kind of order n). From the adimen-

sionalized version of the Lagrangian given in Equation S1, it is possible to obtain the Hamiltonian H describing the

dynamics of small oscillations around ζ0 in terms of the operators ai, a†i , X

.= ai + a†i

H = ωaa†aaa + ωba

†bab

+ La Xa + SaX2a + Ta X3

a + Fa X4a

+ Lb Xb + SbX2b + Tb X3

b + Fb X4b

+ Tab X2b Xa + Tba X2

a Xb + Fab X2a X

2b

(S15)

where

Li =r20T

DBi(T )xi (S16)

Si =2ε

πB2

i (T )x2i (S17)

Ti =2ε

πα2i Bi(T )x

3i (S18)

Fi =ε

2πα4i x

4i (S19)

Tij =2ε

πα2j Bi(T )x

2j xi (S20)

Fij =ε

πα2i α

2j x

2i x

2j , (S21)

with i, j ∈ a, b, xi =√

12µωi

, and αi being the Bessel function zero associated with the ith circular membrane mode

(≈ 2.4048 for the fundamental). Moreover we have

Bi(T ) = Bi(τ) =

√πκ0V

2g

ταi

[2

α2i

−√τ

τ + α2i

R0(τ)

](S22)

and

ε =3r202h2

(S23)

ωi =αir0D

√T

ρ= ωi

D

(S24)

µ =ρr40D

2. (S25)

3






The dimensionless coefficients L,S,T ,F can be re-dimensionalized according to: Li = LiD/(r0xi), Si = SiD/(r0xi)2,

Tij = TijD/(r30xix2j ), Fij = FijD/(r40x

2i x

2j ). Moreover, in Equation S15 and the discussion that follows, the second order

coupling Sij has been excluded as only the fundamental mode has appreciable overlap with the static deformation x0.For large values of the induced tension the values of T and Bi(T ) are given by

T =1

4

[r20ε

20C

(d− x)4

]1/3V 4/3g (S26)

Bi(T ) =4√π

αi

[ε0r0

(d− x)2C

]1/3V 2/3g . (S27)

In Equation S15 Li is a term that can be trivially “displaced” away, Tij, Ti and Fi, Fij are the terms relevant for theradiation-pressure and Duffing physics, while Si represents a shift in frequency of the mode considered.

As an example, we focus our attention on 3 different modes of a circular membrane: (0, 1) (hereafter mode a) , (1, 1)(mode b for the red-detuned case), (3, 1) (mode b for the blue-detuned case). This choice is related to the necessityof having three modes for which ω1 ω2 + ω3. This condition is optimal in terms of radiation pressure-like couplingbetween modes. The (0, 1) mode plays the role of the mechanical mode, while modes (1, 1), (3, 1) play the role of thedriving tone (cavity) and cavity (driving tone) for red- (blue-) sideband detuning respectively. Due to the large densityof states at the cavity resonance, driving the system close to one of its resonances (which has non-negligible overlapwith the cavity resonance) allows for an efficient excitation of the sideband, leading to a stronger optomechanicalcoupling, for a given input drive, as compared to the case for which the resonance close to the pump frequency isabsent.

The physics leading to the frequency and damping shift of the fundamental mode can be essentially explained withthe same analysis performed for optomechanical systems. The driving around ωp can be interpreted as the “opticalpump”, detuned away from a “cavity” by the mechanical resonant frequency.

The calculation goes as follows: the strong drive field βin around ωp is determined by the solution of the I/Oequations for a free (i.e. uncoupled to other modes) mode, this mode will then be considered as the sideband(with respect to ωc) drive in the I/O equations for the coupled system whose unitary dynamics are described by theHamiltonian S15. The analysis of these quantum Langevin equations (QLEs) will be performed in terms of a standardlinearisation procedure, in complete analogy to what is done in the context of optomechanical systems.

With the approximations mentioned above, the cavity field around ωp can be written as

β =

√γpβin

γp

2 − i(ω − ωp)(S28)

The value of β represents thus, on one hand, the oscillation amplitude when the resonator is driven close to theresonance ωp, and, on the other the amplitude of oscillations at a frequency which is detuned by ωp − ωc ωf

(ωf = ωa). The relative values of ωa, ωb1 and ωb2 allow us therefore to have a strong field β, since we are driving thesystem on resonance, and at the same time, exploit the optomechanical-like sideband physics.

In order to describe the nonlinear sideband physics, we write the QLEs associated with the Hamiltonian S15

a =− iωaa− iLa − i2Sa(a† + a)− i3Ta(a† + a)2 − i4Fa(a

† + a)3

− iTab(b† + b)2 − i2Fab(a† + a)(b† + b)2

− γa2a+

√γaain

(S29)

b =− iωbb− i4Fb(b† + b)3

− iTab(a† + a)(b† + b)− i2Fab(a† + a)2(b† + b)− γb

2b+

√γbbin.

(S30)

We can solve Equations S29, S30 perturbatively, assuming that we can expand a and b as a → α+ a and b → β + b,where α represents the coherent oscillation amplitude of the fundamental mode induced by β whose value is givenby Equation S28. The value of α can be obtained as the solution of the zeroth-order term in the expansion ofEquation S29, which can be written as

α =− iωaα− iLa − i2Sa(α∗ + α)− i3Ta(α∗ + α)2 − i4Fa(α

∗ + α)3

− iTab(β∗ + β)2 − i2Fab(α∗ + α)(β∗ + β)2 − γa

2α+

√γaαin.

(S31)

4






In the substitution a → α+a we have assumed that α = 0. This assumption is justified when the conditions ωa < 2ωb,and γa < (ωa − ωb) are fulfilled, (rotating wave approximation), leading to

α = −2Tab|β|2 + La/2

(ωa + 4Sa)(S32)

where higher-order terms have been neglected, and we have assumed, without loss of generality α∗ = α.The first-order term in the expansion of Equations S29, S30 can be written as

a =− iωaa− i2Sa(a† + a)− i6Ta(α∗ + α)(a† + a)− i12Fa(α

∗ + α)2(a† + a)

− i2Tab(β∗b+ βb†)− i2Fab

[(β∗ + β)2(a† + a) + (α∗ + α)(β∗ + β)(b† + b)

]

− γa2a+

√γaain

(S33)

b =− iωbb− i3Fb(β∗ + β)2(b† + b)− iTab

[(β∗ + β)2(a† + a) + (α∗ + α)(β∗ + β)(a† + a)

]

− iFab

[2(α∗ + α)2(b† + b) + (α∗ + α)(β∗ + β)(a† + a)

]− γb

2b+

√γbbin.

(S34)

Neglecting again higher-order terms, Equations S33, S34 can be written as

a =− iωaa− i

[2Sa − 12

Ta(Tab|β|2 + La/2)

ωa + 4Sa+ 4Fab|β|2

]a

− i2Tab(β∗b+ βb†)− γa2a+

√γaain

(S35)

b =− iωbb− i2Tabβ(a† + a)− γb2b+

√γbbin (S36)

where RWA has been used for a, b, and β. Equations S35, S36 can be written more compactly as

a = −iΩaa− i(G∗b+Gb†)− γa2a+

√γaain (S37)

b = −iωbb− iG(a† + a)− γb2b+

√γbbin (S38)

where

Ωa = ωa + 2Sa − 12Ta(Tab|β|2 + La/2)

ωa + 4Sa+ 4Fab|β|2 (S39)

G = 2Tabβ. (S40)

Equations S37, S38 are the equation of motion of two linearly coupled harmonic oscillators, and are equivalent to thelinearised equation of motion for an optomechanical system. It can be shown that in this setup the mode a undergoesa frequency shift and a damping shift given by

ωeff =

√Ω2

a +4|G|2∆Ωa [γ2

b/4− ω2 +∆2]

[γ2b/4 + (ω −∆)2] [γ2

b/4 + (ω +∆)2](S41)

γeff = γa −4γb|G|2∆Ωa

[γ2b/4 + (ω −∆)2] [γ2

b/4 + (ω +∆)2]. (S42)

From Equations S37 and S41, it is clear how the observed frequency shifts have 2 different sources. On the onehand, it is determined by “geometric” nonlinearities, i.e. effects which are essentially determined by the eigenmodeshapes,dictating the value of Ωa in Equation S39, on the other it depends on the mechanical analogue of optomechanicaleffects ωeff .As a final note on the nonlinear parameters L,S,T ,F , we provide a numerical comparison of these quantities (based

on Equations S16 - S21) for a graphene drum with dimensions equal to Device 1. The values quoted in Table S1have been obtained by fitting the ω vs. Vg dependence for the fundamental mode, assuming the (dimensionless)built-in tension τ0 and the surface density of the drum as free parameters. From the fit, we have obtained ω|V g=0 =2π · 8.52MHz, T0 = 0.051N/m. The value of L1 has been expressed in nN since it represents the static forced inducedby Vdc on the resonator. Values shown are calculated at the single phonon level.

5






TABLE S1. Size comparison of the competing nonlinear terms

L1D

xzpmS1

D T1

D F1

D T12

D F12

D

0.86nN 8.18kHz 0.3018Hz 1.441µHz 0.4814Hz 2.29µHz

Perhaps more significantly we can show that the effective coupling (S40) takes the following values for differentdriving amplitudes:

TABLE S2. G = 2Tαββ, when expressed in Hz.

xβ = xzpm xβ = 0.1nm xβ = 10nm

GD

xβ

xzpm0.97Hz 2.15kHz 215kHz

Moreover, the correction to the frequency ωa on the first line of Equation S35 is given by:

TABLE S3. Overall frequency shift δω from other nonlinear terms.

xβ = xzpm xβ = 0.1nm xβ = 10nm

δω = 2Sa − 12... 1.3027kHz 1.3098kHz 71.4kHz

6






S2. CHARACTERIZATION OF DEVICES 1 AND 2

The graphene devices studied in this work are shown in Figure S1. Their physical properties are given in Table S4.The mass density of these membranes is ≈ 10× that of bare graphene due to surface contaminants (most likelyPMMA from fabrication2). Mass density here has been measured by fits to the AC amplitude of motion as Vdc

is varied, described in Section S3 and presented in Figure S3. These values can also be obtained from fits to theresonant frequency dispersion f(Vdc), as has been described in numerous works previously.3–6 The intrinsic tension

T0 is calculated from f = (α/2πr0)√T/ρ, where α ≈ 2.404.

A spectrum for Device 2 at Vdc = 4V is given in Figure S2. The pumping conditions used in Figure 4 of the maintext are also shown.

a b

FIG. S1. a,b, Scanning electron micrograph of Device 1 and 2, respectively. Scale bars are 5µm. In both cases graphene issuspended above a 1.7µm-deep circular trench in SiO2. Linear trenches (6 in a and 10 in b) allow fluid to drain from underthe graphene during device fabrication. All but two trenches terminate in a thin SiO2 bridge so as not to affect the membraneboundary conditions. The remaining two trenches carry 50nm-thick platinum leads to the split back-gates. Platinum sourceand drain leads contact the graphene bottom surface.

3 4 5 6 7 8 90

1

2

3

4

|x| (

mV)

ω/2π (MHz)3 4 5 6 7 8 9

0

0.3

0.6

0.9

1.2

ωp

FIG. S2. Spectrum of Device 2.The pump configuration used to ob-tain Figure 4a-c is shown. Verti-cal bands denote the three frequencyranges in which motion was measuredwhile pumping ωp.

Device # diameter (µm) ρ/ρgraphene f1(Vdc = 0)(MHz) T0(N/m)

1 7.8 11± 2 8.35 0.060

2 19.9 9.5± 1 2.9 0.040

TABLE S4. Graphene device properties

7





S3. CALIBRATION OF OPTICAL DETECTION SYSTEM

Calibration of the electrical and optical components of our setup were performed by pulling on Device 1 with avarying DC gate voltage (and fixed AC voltage) while measuring both the AC and DC components of our reflectedlaser power. As described below, this process allows us to determine the absolute deflection of our graphene membrane(both the static and resonant components), as well as the effective AC gate voltage that is experienced by the graphene,vac. This latter value is substantially smaller than the applied AC voltage, Vac, due to parasitic losses of our cablesand wire bonds, contact resistance of the graphene, and other unavoidable losses. The DC gate voltage, Vdc, does notsuffer this effect, as the graphene-gate capacitor (C) will reach the experimentally applied voltage within a few RCtime constants, where R encompasses all series resistances.

Following an approach reported previously,5 the graphene is considered to be situated in an optical standing wavegenerated by the incident laser light and reflection from the metallic back-gate. Because of the graphene’s 2.3% opticalabsorption,7 the overall reflected power out of the system is sensitive to the graphene position within this standingwave; this sensitivity enables us to detect graphene motion. The DC component of our reflected laser power dependson graphene position, x, as

Pdc = P0 +∆P sin

(4π

λx+ θ

)(S43)

where λ is the wavelength of light used (633nm) and P0, ∆P , θ are the average power, modulation depth, and phaseof the standing wave at the graphene, respectively. The position x can be altered by pulling the graphene towardsthe back-gate with a bias voltage, V = Vdc + vac sinωt. If vac Vdc, and ω is far below any mechanical resonance ofthe graphene, the ac portion of our reflected laser power is:

Pac = ∆P cos

(4π

λx+ θ

)· 4πλ

dx

dVdcvac. (S44)

In principle P0, ∆P , and θ can be calculated from the incident power, refractive index of graphene and back-gate,and the graphene-gate separation. However, this calculation is complicated by: (1) the quality and size of our laserspot (diameter ≈ 1µm), (2) the thickness, roughness, and refractive index of surface contaminants on the graphene,2

and (3) graphene adherence to the vertical walls of the trench,8 among other uncertainties. These parameters musttherefore be measured experimentally – in this case by pulling the graphene a significant fraction of the distance λ/4.To simplify calculations, the membrane deflection profile is assumed to be a paraboloid x(r) = x0(1− r2/r20), wherer, r0 are radial position and membrane radius, and x0 is the height of the membrane center. The membrane positioncan then be calculated from the balance of tension and electrostatic forces:

4πT0x0 +4πE

r20x30 =

1

2C ′V 2

dc. (S45)

Above, T0 and E are the 2D membrane tension and 2D Young’s modulus (in N/m), respectively, and C ′ = dC/dx. Thisis a modified version of force equations reported elsewhere,5,8 adjusted for the paraboloid approximation. The tensionT0 can be recast as the membrane mass density ρ by knowledge of the resonant frequency: ρ = T0(2.404/(ω1r0))

2.Equations S43 - S45 thus provide a means to model our optical system as the bias voltage Vdc is varied. Figure S3

shows a representative data set from which P0, ∆P , θ, ρ, and vac/Vac are measured for Device 1. Here, Vdc is swept(0− 35V) while a constant Vac (200mVpk, ω = 2π× 100kHz) is applied to both back-gates. AC data is fitted first, bynumerically finding the roots of Equation S45 and applying them to Equation S44; results are shown in Figure S3a.This fit provides values for ρ, ∆P · vac, and θ. With these parameters determined, Pdc is fitted to Equation S44to obtain P0 and ∆P ; this is shown in Figure S3b. The excellent agreement of this second, more constrained fitverifies the validity of this model. With all the parameters of Equations S43 - S45 determined, we can plot the DCand AC membrane deflection for this data set, as shown in Figure S3c-d. Interesting features in these curves are: 1)In Figure S3d, the transition from quadratic to sub-linear DC deflection above Vdc ≈ 25V caused by the E term inEquation S45. 2) The resulting maximum in AC deflection that this transition produces, as shown in Figure S3c.

In performing these fits, the measured modulus of similarly produced CVD graphene9 is used, E = 55N/m.Moreover, for the purposes of Equations S43 and S44, it is assumed that our laser spot performs a “point-like”measurement of x at the membrane’s center of mass, rcm = r0/

√2. Equation S45 was corrected for this off-center

measurement, x(rcm) = x0/2. The resulting mass density of Device 1 is ρ/ρg = 11 ± 2, where ρg = 0.75mg/m2is

the density of monolayer graphene; the extra mass is attributed to polymer contaminants from fabrication. The ACgate voltage “felt” by the graphene is vac = 11mVpk, or vac/Vac = 5.5%. From similar fits, Device 2 is found to haveρ/ρg = 9.5 and vac/Vac = 3.8%.

8






In the main text, resonant motion is converted from µV (generated by our photodiode) to pm using the abovemeasured values of ρ and vac/Vac to calculate the applied force during any measurement, and the resulting motion ofthe fundamental mode. This value is then compared to the measured amplitude (in µV) on resonance. It should benoted that because this calibration uses only mode 1 as a reference, the relative amplitude measured for each of thehigher modes depends upon the position of our laser spot. AC voltages given in the main text represent the voltageexperienced by the graphene, vac, so as to be independent of the particular electrical losses of the experiment.

0 10 20 300

5

10

15

20

DC Gate Voltage (V)

AC M

embr

ane

Def

lect

ion

(pm

)

0 10 20 300

20

40

60

80

DC Gate Voltage (V)

AC P

hoto

diod

e Vo

ltage

( μV)

0 10 20 300

10

20

30

40

50

60

70

DC Gate Voltage (V)

DC

Mem

bran

e D

efle

ctio

n (n

m)

0 10 20 300.48

0.49

0.5

0.51

0.52

0.53

0.54

0.55

DC Gate Voltage (V)

DC

Pho

todi

ode

Volta

ge ( μ

V)

a b

dc

FIG. S3. a, Measured ac reflected laser power as graphene is driven far below resonance at fixed Vac and varying Vdc. Redline: Three-parameter fit, as described in the text. b, Measured dc reflected laser power collected in synchrony with a (blackpoints). Red line: two-parameter fit, using values taken from fit to a. c,d, Calculated membrane deflection at ω = 2π×100kHzand Vac = 200mVpk resulting from the fits in a and b. The responsivity and transimpedance gain specified for our photodiode(New Focus 1801-fs-ac) are used to convert between measured voltage and input laser power.

9






S4. MEASUREMENT OF EFFECTIVE DAMPING AND FREQUENCY

In order to fully characterize the effects of mode coupling on the frequency and damping of mode 1, its responseamplitude |x1| and phase φ1 must both be measured. Below we show that when compared to the amplitude andphase of a “reference state” of the same resonator, these two numbers can be used to infer the effective damping andresonant frequency of the mode.

The equation of motion for mode 1 in the absence of nonlinearities is

x+ γx+ ω20x =

F

meiωt (S46)

where the ‘1’ subscripts have been excluded for brevity and ω0, γ are the natural frequency and damping. F , m arethe applied electrostatic force and membrane mass. In the presence of mode coupling and Stokes (or anti-Stokes)pumping, only ω0 and γ are altered:

x+ γeff x+ ω20,effx =

F

meiωt. (S47)

Assuming F is real, the driven response of mode 1 then becomes

x(t) =F/m

ω20,eff − ω2 + iγeffω

eiωt. (S48)

Expressing x(t) by its quadratures Rex = Xcosωt and Imx = Y sinωt, and amplitude |x| = R, the effectiveresonant frequency and damping can be determined from:

ω20,eff = ω2 +

FX

mR2(S49)

γeff = − FY

mωR2. (S50)

In generating Figure 3e of the main text, motion was calibrated based on a “reference” region of (ω, ωp) space wheremode coupling effects are negligible: ωp/2π = 22 − 22.5MHz. Data from this region was fitted to Equation S48 tocalibrate the amplitude of motion R (according to Section S2), as well as adjust the measured phase such that X = 0on resonance. Equations S49 - S50 were then used to convert X, Y to ω1,eff , γ1,eff for each point in (ω, ωp) space, asshown in Figure S4.

It should be noted that Equations S49 - S50 can easily be modified to account for a Duffing nonlinearity in the“reference” region. Slight variations in the optical detection efficiency can also be modeled quite effectively, as wasnecessary for Figures 3e & S4c,d. A slow drift in the position of our laser spot resulted in a roughly linearly decreasingdetection efficiency as ω/2π was ramped from 8MHz to 9MHz. Figure S5 compares the signal in our “reference” regionwith and without renormalizing to correct for the slowly evolving detection efficiency. This renormalization is usedonly in computing ω1,eff and γ1,eff , and all other figures here and in the main text depict raw data.

10






Pump Frequency (MHz)

Freq

uenc

y (M

Hz)

12 14 16 18 20 228

8.2

8.4

8.6

8.8

9

Dam

ping

Shi

ft (k

Hz)

−100

−50

0

50

100


Freq

uenc

y (M

Hz)

12 14 16 18 20 228

8.2

8.4

8.6

8.8

9

Freq

uenc

y Sh

ift (k

Hz)

−150

−100

−50

0

50

100

150


Freq

uenc

y (M

Hz)

12 14 16 18 20 228

8.2

8.4

8.6

8.8

9

Y qu

adra

ture

(pm

)

−400

−300

−200

−100

0


Freq

uenc

y (M

Hz)

12 14 16 18 20 228

8.2

8.4

8.6

8.8

9

X qu

adra

ture

(pm

)

−200

−100

0

100

200a b

c d

FIG. S4. a,b, X & Y quadrature of Device 1 motion corresponding to Figure 3a of the main text. c,d, ∆ω1 and ∆γ1 calculatedfrom from X & Y according to Equations S49 - S50. Note that these two are ≈ 0 (by definition) in the “reference” regionωp/2π = 22− 22.5MHz, as well as most other regions. In these lower panels, the color scales are symmetric about 0kHz so thatzero shift appears white. The intrinsic parameter values for mode 1 are ω1/2π = 8.62MHz and γ1/2π = 150kHz.

8 8.2 8.4 8.6 8.8 9

−300

−200

−100

0

100

200

Frequency (MHz)

Mot

ion

(pm

)

Raw Data

X dataY dataX fitY fit

8 8.2 8.4 8.6 8.8 9

−300

−200

−100

0

100

200

Frequency (MHz)

Mot

ion

(pm

)

Renormalized Data

X dataY dataX fitY fit

8 8.2 8.4 8.6 8.8 90.8

0.9

1

1.1

1.2

1.3

1.4

Frequency (MHz)

Det

ectio

n Ef

ficie

ncy

(AU

)

a b c

FIG. S5. a, Raw data (X & Y quadratures) compared to a fit of Equation S48. Discrepancies are caused by a slowly decreasingdetection efficiency over time (the frequency sweep shown was performed over ∼ 1 hour). b, The same data, corrected for thechanging detection efficiency. The near-perfect agreement between data and model is needed to ensure ∆ω1,∆γ1 = 0 in this“reference” region of (ω, ωp) space (see Figure S4). c, Detection efficiency used to renormalize the data and produce FiguresS4c,d & 3e.

11



SUPPLEMENTARY INFORMATION DOI: 10.1038/NNANO.2016.86SUPPLEMENTARY INFORMATION De Alba et al. 2016

S5. ADDITIONAL MODE COUPLING EFFECTS IN DEVICE 1

Measurements for Device 1 were taken at Vdc = 2.5V, 5V, 7.5V, 10V, and 15V. As seen in Figure 2d of the maintext, significant overlap between the frequencies of modes 2,6 and their respective sidebands occurs in the rangeVdc = 0 − 7.5V. Therefore coupling rates between the modes should be roughly constant in this range, which isverified in Figure S6. Here care was taken to ensure equivalent drive forces (F ∝ VdcVac) were applied to the pump(as well as the probe) for each Vdc value. At Vdc = 10V and above, the sideband overlap diminishes and enhancementat the pump frequency no longer occurs.

ωp/2π (MHz)12 14 16 18 20 22

8.4

8.6

ω/2π

(MH

z)

8.5

8.7

8.7

8.9

Vdc=5V

Vdc=7.5V

Vdc=2.5V

FIG. S6. Mode coupling in Device 1 at3 different Vdc values. Drive forces ap-plied at the pump frequency (and probefrequency) are equal across the threedata sets. Apart from a steady increasein ω1 with Vdc, the coupling rates lead-ing to amplification and cooling remainroughly constant.

As the graphene devices studied here have two independent back-gates, several driving conditions can be realized(depicted in Figure 1a). For the results shown in Figures 3 & S6, one back-gate is driven while the other is grounded.This configuration enables efficient actuation of all 6 membrane modes, as opposed to driving both back-gates in phase(which benefits the fundamental mode) or 180 out of phase (which benefits the higher modes). Figure S7 shows theresults of driving the back-gates out of phase in Device 1 at Vdc = 10V. Interestingly, this strong driving of mode 4(as well as its overlap with 2×ω1) results in strong amplification of mode 1. This coupling between modes 1 and 4 isdue to the T14x1x

24 term in the membrane Hamiltonian, Equation 1. Strong coupling between two mechanical modes

i, j where ωj = 2× ωi has been studied previously in carbon nanotube systems.10

ωp/2π (MHz)

ω/2π

(MH

z)

12 14 16 18 20 22

8.6

8.8

9

9.2

9.4

473 pm 126pm

11 193|x| (pm)

ωp/2π (MHz)

ω/2π

(MH

z)

12 14 16 18 20 22

8.6

8.8

9

9.2

9.4

393 pm 130pm

18 187|x| (pm)

a b

FIG. S7. Mode coupling measurements with back-gates driven 180 out of phase. The pump voltage vac is doubled between aand b. Both plots show parametric amplification at ωp = 2×ω1. For very strong pumping (b), there is also increased dampingat ωp = ω3 ≈ 16MHz. This is a separate effect from sideband cooling, and not yet fully understood. This feature is also seenin Figure S4d.

12



SUPPLEMENTARY INFORMATIONDOI: 10.1038/NNANO.2016.86SUPPLEMENTARY INFORMATION De Alba et al. 2016

S6. LARGE-AMPLITUDE QUENCHING OF PHONON CAVITY AND SIDEBAND MODE

As discussed in the main text and seen in Figure 3, Device 1 exhibits quenching of the sideband amplification andcooling effects due to the large vibration amplitudes of mode 1. This is caused by nonzero couplings T16 and T61

which lead to effective cooling of either sideband mode due to motion of mode 1. The experimental data was modeledby (for the case of ωc = ω2, ωsb = ω6) using Equations 2 & 3 to solve for the effective frequency and damping ofmode 1, ω1,eff and γ1,eff , while concurrently solving identical equations for ω6,eff and γ6,eff . In this case, mode 2acts like a phonon cavity for mode 6, and mode 1 is pumping its red sideband ω2 − ω6; mode 6 is thus cooled whilemode 1 is being amplified. If this process is initiated with a sufficiently small mode 1 vibration amplitude, significantamplification can occur such that γ1,eff → 0 before the sideband mode experiences much cooling (as is the case in ourmeasurements of Device 2).

Figure S8 demonstrates our method for fitting the measured γ1,eff , and demonstrates predicted behavior with andwithout quenching.

ωp/2π (MHz)

ω/2π

(MH

z)

0.1 4.820.5 21 21.5

8.4

8.6

8.8

|x| (AU)

0.1 1.38.4

8.6

8.8

|x| (AU)

8 8.25 8.5 8.75 9

40

60

80

100

120

140

160

ω/2π (MHz)

γ 1,ef

f/2π

(MH

z)

ωp/2π (MHz)

ω/2π

(MH

z)

66 154

20.5 21 21.58

8.2

8.4

8.6

8.8

9

γ1,eff (kHz)

ωp/2π (MHz)ω

/2π

(MH

z)

0.1 1.0

12 12.5 13

8.4

8.6

8.8

|x| (AU)

0.1 1.08.4

8.6

8.8

|x| (AU)

8 8.25 8.5 8.75 9

150

200

250

300

350

ω/2π (MHz)

γ 1,ef

f/2π

(kH

z)

ωp/2π (MHz)

ω/2π

(MH

z)

154 288

12 12.5 138

8.2

8.4

8.6

8.8

9

γ1,eff (kHz)

a b c

with quenching

without quenching

d e f

with quenching

without quenching

γ1/2π

γ1/2π

FIG. S8. Modeling the effective damping. a, The measured mode 1 damping (obtained by the analysis described in Section S3)during sideband cooling. b, Fit to the data in a. Black points represent a slice through the data in a at the solid black line.The fit (solid red line) has two free parameters, T16 and T26, where the latter signifies quenching of mode 2. Dashed linesdenote the same model for decreasing mode 1 amplitude (or equivalently, decreasing T26) at 75%, 50%, 25%, and 0% of theexperimental value. c, Simulated data with the fit parameters from b. Amplitudes are normalized to 1 when no cavity effectsare present. d-f, Similar results for the sideband amplification effect.

13





S7. MODE COUPLING IN A THIRD DEVICE

The parametric cooling and amplification effects described here and in the main text have been observed in severalgraphene membranes of various diameters. These effects are shown in Figure S9 for a third device (for which opticalcalibration has not been performed, and so motional amplitude is reported in photodetector µV).

ωp/2π (MHz)

ω/2π

(MH

z)

8.5 9 9.5 103.25

3.3

3.35

3.4

3.45

3.5

1.0 uV 5.3uV

ωp/2π (MHz)

ω/2π

(MH

z)

5 6 7 8 9 103.5

3.6

3.7

3.8

3.9

12.9 uV 5.8uV

3.3 3.35 3.4 3.450

1

2

3

4

5

6

ω/2π (MHz)Am

plitu

de (u

V)

a b c

linearlyincreasing

Vac(ωp)

0.4 5.9|x| (µV)

0.9 5.8|x| (µV)

FIG. S9. Mode coupling in a 3rd device, diameter 16µm. a, Mode coupling in this device at Vdc = 10V. Note the nontrivialspectrum of higher modes in the upper panel. Each mode coincides with increased damping of mode 1, suggesting sidebandcooling via coupling to cavity modes in the ωp/2π = 9 − 10MHz range. Modes in this range are not clearly visible (upperpanel), possibly due to poor capacitive actuation to these modes. b, Mode coupling in the same device at Vdc = 5V. Somesideband amplification is visible. c, Amplification of mode 1 at Vdc = 5V upon pumping the sideband at ωp/2π ≈ 9.25MHzshown in b.

14





S8. DUFFING RESPONSE & NONLINEAR DAMPING

As suggested by Equation 1, motion-induced tension modulation results in other types of mechanical nonlinearityaside from inter-modal coupling. To demonstrate this, the response of Device 1, mode 1 was measured for strongdrive amplitudes at Vdc = 5V and Vdc = 10V, as shown in Figure S10. Interestingly, for Vdc = 5V, the resonancepeak transitions from left-leaning at intermediate drive values to right-leaning at the highest drives. The intermediateamplitude behavior can be explained by a negative Duffing coefficient generated by Ta and Fa in Equation 1, whilethe transition to right-leaning is indicative of a higher order Duffing-like term (possibly H ∝ x5

a or x6a). In Figure S10a

and S10c, data has been modeled by a resonant frequency ω21,eff = ω2

1 +D1|x1|2+D2|x1|3+D3|x1|4, which reproducesthe curved spine observed in the data of Figure S10a. Data in Figures S10d and S10f were modeled using onlyω21,eff = ω2

1 +D|x1|2.Figure S10 also suggests that Device 1 undergoes nonlinear damping, as has been observed previously in graphene

and carbon nanotube resonators.11 The source of this nonlinear damping has thus far not been determined, andwarrants further study. The fits shown in Figure S10c and S10f also include a nonlinear damping term γ1,eff =γ1 +N |x1|2, which reproduces the data well. However, the decreasing amplitude observed in Figures S10b and S10emay be partly due to a decreasing detection efficiency over time (the data was acquired over a span of 15 minutes),as described in Section S4.

8.2 8.4 8.6 8.8 90

0.2

0.4

0.6

0.8

1

Frequency (MHz)

|x| (

nm)

8.2 8.4 8.6 8.8 9

0.2

0.4

0.6

0.8

1

Frequency (MHz)

|x|/V

ac (a

.u.)

8.2 8.4 8.6 8.8 90.2

0.4

0.6

0.8

1

Frequency (MHz)

|x| (

nm)

linearlyincreasing

Vac

a b c

8.5 9 9.50

0.2

0.4

0.6

0.8

1

Frequency (MHz)

|x| (

nm)

8.5 9 9.5

0.2

0.4

0.6

0.8

1

Frequency (MHz)

|x|/V

ac (a

.u.)

8.5 9 9.5

0.2

0.4

0.6

0.8

1

Frequency (MHz)

|x| (

nm)

d e flinearlyincreasing

Vac

FIG. S10. Duffing response of Device 1. a, Mode 1 response as drive amplitude is ramped from vac = 4mVrms to 56mVrms

(colored lines) at Vdc = 5V. The black central line is a spine extracted from a fit to the highest curve (shown in c). b, Thesame data from a, normalized by ac drive voltage. The decreasing peak height is indicative of nonlinear damping. c, A fit thehighest curve in a, with Duffing terms and nonlinear damping included. d-f, Similar data at Vdc = 10V and vac = 3mVrms to30mVrms.

15






REFERENCES

1 A. M. Eriksson, D. Midtvedt, A. Croy, and A. Isacsson, Nanotechnology 24, 395702 (2013)2 Y. C. Lin, C. C. Lu, C. H. Yeh, C. Jin, K. Suenaga, and P. W. Chiu, Nano Lett. 12, 414 (2012).3 C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stormer, T. F. Heinz, and J. Hone, Nat.Nanotechnol. 4, 861 (2009).

4 V. Singh, S. Sengupta, H. S. Solanki, R. Dhall, A. Allain, S. Dhara, P. Pant, and M. M. Deshmukh, Nanotechnology 21,165204 (2010).

5 R. A. Barton, I. R. Storch, V. P. Adiga, R. Sakakibara, B. R. Cipriany, B. Ilic, S. P. Wang, P. Ong, P. L. McEuen, J. M.Parpia, and H. G. Craighead, Nano Lett. 12, 4681 (2012).

6 S. Lee, C. Chen, V. V. Deshpande, G.-H. Lee, I. Lee, M. Lekas, A. Gondarenko, Y.-J. Yu, K. Shepard, P. Kim, and J. Hone,Appl. Phys. Lett. 102, 153101 (2013).

7 R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, Science320, 1308 (2008).

8 C. Lee, X. Wei, J. W. Kysar, and J. Hone, Science 321, 385 (2008).9 C. S. Ruiz-Vargas, H. L. Zhuang, P. Y. Huang, A. M. van der Zande, S. Garg, P. L. McEuen, D. A. Muller, R. G. Hennig,and J. Park, Nano Lett. 11, 2259 (2011).

10 A. Eichler, M. del Alamo Ruiz, J. A. Plaza, and A. Bachtold, Phys. Rev. Lett. 109, 025503 (2012).11 A. Eichler, J. Moser, J. Chaste, M. Zdrojek, I. Wilson-Rae, and A. Bachtold, Nat. Nanotechnol. 6, 339 (2011).

16



tunable phonon-cavity coupling in graphene...

Documents