Abstract

Optical cavities with small mode volume are well-suited to detect the vibration of sub-wavelength sized objects. Here we employ a fiber-based, high-finesse optical microcavity to detect the Brownian motion of a freely suspended carbon nanotube at room temperature under vacuum. The optical detection resolves deflections of the oscillating tube down to . A full vibrational spectrum of the carbon nanotube is obtained and confirmed by characterization of the same device in a scanning electron microscope. Our work successfully extends the principles of high-sensitivity optomechanical detection to molecular scale nanomechanical systems.

\title

Cavity-enhanced optical detection of carbon nanotube Brownian motion

\author

S. Stapfner \affiliationCenter for NanoScience and Fakultät für Physik, Ludwig-Maximilians-Universität München, Geschwister-Scholl-Platz 1, 80539 München, Germany

\author

L. Ost \affiliationCenter for NanoScience and Fakultät für Physik, Ludwig-Maximilians-Universität München, Geschwister-Scholl-Platz 1, 80539 München, Germany

\author

D. Hunger \affiliationCenter for NanoScience and Fakultät für Physik, Ludwig-Maximilians-Universität München, Geschwister-Scholl-Platz 1, 80539 München, Germany

\author

J. Reichel \affiliationLaboratoire Kastler Brossel, Ecole Normale Supérieure, Université Pierre et Marie Curie, CNRS, 24 rue Lhomond, 75005 Paris, France

\author

I. Favero \affiliationLaboratoire Matériaux et Phénomènes Quantiques, Université Paris-Diderot, Sorbonne Paris Cité, CNRS, UMR 7162, 10 rue Alice Domon et Léonie Duquet, 75013 Paris, France

\author

E. M. Weig \email[]correspondance: weig@lmu.de \affiliationCenter for NanoScience and Fakultät für Physik, Ludwig-Maximilians-Universität München, Geschwister-Scholl-Platz 1, 80539 München, Germany

\date\today\pacs\keywords

carbon nanotube, optical microcavity, optomechanics, fiber optics, thermal motion, Brownian motion, nanomechanics Probing the vibrational motion of nano-scale objects has great potential for advancing next-generation technologies such as resonant mass or bio-sensing [Jensen et al.(2008)Jensen, Kim, and Zettl, Chaste et al.(2012)Chaste, Eichler, Moser, Ceballos, Rurali, and Bachtold, Hanay et al.(2012)Hanay, Kelber, Naik, Chi, Hentz, Bullard, Colinet, Duraffourg, and Roukes]. A key example is carbon nanotubes (CNTs), which promise to display ultimate sensitivities due to their molecular scale mass and diameter along with their outstanding mechanical properties. Recently CNTs have attracted attention for their ability to resolve and exploit the quantum nature of mechanical vibration in optomechanical experiments [Wilson-Rae et al.(2012)Wilson-Rae, Galland, Zwerger, and Imamoglu, Schneider et al.(2012)Schneider, Etaki, van der Zant, and Steele]. However, small vibrational amplitudes and dimensions impose severe challenges in the realization of CNT-based mechanical devices. One fundamental challenge of particular interest is to resolve the thermally excited Brownian motion of CNTs. Indeed this level of resolution allows operation of the undriven device in the linear regime with enough dynamic range, and limits the influence of spurious non-linear effects observed in CNTs [Eichler et al.(2011)Eichler, Chaste, Moser, and Bachtold, Eichler et al.(2011)Eichler, Moser, Chaste, Zdrojek, Wilson-Rae, and Bachtold]. Transmission and scanning electron microscopy (TEM, SEM) have both been employed successfully to visualize the thermal motion of CNTs[Treacy et al.(1996)Treacy, Ebbesen, and Gibson, Babić et al.(2003)Babić, Furer, Sahoo, Farhangfar, and Schönenberger], yielding a superposition of the small-amplitude envelopes of all oscillating modes. Yet electron beam imaging is a destructive method for observing CNT oscillation because of the unavoidable deposition of amorphous carbon on the tube. Electrical schemes provide a powerful approach for detection of vibrating CNTs, but typically require a coherent drive voltage to actuate the tube[Hüttel et al.(2009)Hüttel, Steele, Witkamp, Poot, Kouwenhoven, and van der Zant, Steele et al.(2009)Steele, Hüttel, Witkamp, Poot, Meerwaldt, Kouwenhoven, and van der Zant, Chaste et al.(2011)Chaste, Sledzinska, Zdrojek, Moser, and Bachtold]. Furthermore their sensitivity is not sufficient to resolve the Brownian motion. Optical techniques are ultra-sensitive[Arcizet et al.(2006)Arcizet, Cohadon, Briant, Pinard, Heidmann, Mackowski, Michel, Pinard, Français, and Rousseau] but the optical detection of CNT motion is hindered by the diffraction limit, with typical tube diameters much smaller than the wavelength of visible light. Dark field illumination was combined with confocal microscopy to detect the mechanical oscillation of driven CNTs with a diameter of , however, the Brownian motion amplitude could not be resolved with this approach [Fukami et al.(2009)Fukami, Arie, and Akita]. Even in the subwavelength regime, the sensitivity can be enhanced by using an optical cavity[Favero and Karrai(2008)]. In earlier work[Favero et al.(2009)Favero, Stapfner, Hunger, Paulitschke, Reichel, Lorenz, Weig, and Karrai] we apply a fiber-based optical micro-cavity of small mode volume and high finesse to measure the Brownian motion of an amorphous carbon based nanorod with a diameter of about . Following the proposal of extending cavity optomechanics experiments to CNTs[Favero and Karrai(2008)], in the present work the nanorod is replaced by a carbon nanotube with a ten times smaller diameter. Taking advantage of an improved cavity with significantly increased finesse, we present data clearly resolving the Brownian motion of the CNT oscillating within the cavity light field. Research in optomechanics [Kippenberg and Vahala(2008), Marquardt and Girvin(2009), Favero and Karrai(2009), Aspelmeyer et al.(2010)Aspelmeyer, Gröblacher, Hammerer, and Kiesel] will benefit from the in-cavity implementation of carbon nanotubes, which are the smallest solid-state mechanical resonators available to date, and have not yet been explored in this context.

\includegraphics

cavity-cnt-v2.pdf

Figure \thefigure: (COLOR ONLINE) Illustration of the experimental setup. (a) Schematic view of the cavity and sample chip with the CNT introduced into the cavity mode. (b) Illustration based on an SEM image of the CNT under investigation suspended across the gap (white) with surrounding substrate (grey). The red spot illustrates the position and extension of the optical cavity mode during the experiment.

Without exploiting an optical transition, direct optical detection of a CNT of deep subwavelength dimensions is a challenging task[Sawano et al.(2010)Sawano, Arie, and Akita]. In an optical cavity, however, the dispersive and dissipative interactions between a nanotube and the light field is enhanced. Particularly for the case of a high finesse cavity of small mode volume, this gives rise to an amplification of the signature of the CNT displacement in the cavity response. For the case of a dispersive interaction, the CNT imprints a phase shift on the photons in the cavity, which results in a resonance frequency shift of the cavity mode. On the other hand, a dissipative interaction, for example caused by absorption and scattering of photons by the nanotube, leads to a net loss of energy decreasing the circulating optical power. The strength of both types of interaction depends on the position and orientation of the nanotube within the cavity mode. Thus by measuring the frequency and intensity of the cavity resonant response, information can be gained on these interactions. In the present work both dispersive and dissipative interactions of the CNT contribute to the observed signal. To detect the CNT’s Brownian motion, we employ a fiber-based micro cavity formed by two opposing, concavely shaped fiber end facets, which are coated with a highly reflective Bragg mirror optimized for the cavity wavelength of [Steinmetz et al.(2006)Steinmetz, Colombe, Hunger, Hänsch, Balocchi, Warburton, and Reichel, Hunger et al.(2010)Hunger, Steinmetz, Colombe, Deutsch, Hänsch, and Reichel, Hunger et al.(2012)Hunger, Deutsch, Barbour, Warburton, and Reichel, Flowers-Jacobs et al.(2012)Flowers-Jacobs, Hoch, Sankey, Kashkanova, Jayich, Deutsch, Reichel, and Harris]. Fig. Documenta displays a schematic of the setup. For a mirror separation of , the cavity optical mode waist radius is and the finesse is measured to be . Alignment presents a major technical challenge of this configuration in conducting optomechanical experiments, as illustrated by Fig.Documenta. In order to enter a nanomechanical object into the cavity, the hosting substrate chip must fit into the narrow slit formed between the two fibers. This implies that the substrate thickness has to be reduced below the width of that slit for an area covering the fiber cross section corresponding to their diameter of . The center of this thinned out area hosts a gap wide enough to avoid clipping losses from its edges, which would disrupt the optical mode. According to numerical simulations, the gap has to be at least wide to fulfill this condition. For ease of alignment with the wavefronts of the cavity mode the CNT is doubly clamped and freely suspended across this gap as indicated in Fig. Documenta. Suitable substrates serving as holders for the CNTs are fabricated from a thick silicon wafer using optical lithography and wet etching processes from both the top and the back side of the wafer (see supplement). Subsequently CNTs are grown across the gap using a process adapted from Ref. \onlinecitebabic_nanolett2003. Careful SEM and TEM inspection of different samples reveal that the doubly-clamped CNTs are freely suspended across gaps of up to . Additional electron diffraction analysis in the TEM show that the resulting CNTs are multi-walled tubes or thin ropes consisting of to individual tubes. Details of chip fabrication, growth and SEM/TEM analysis are provided in the supplement. Figure Documentb shows a post-processed SEM image of the suspended CNT investigated in the present work. The suspended section is long with a diameter between and . The slightly wavy shape of the tube indicates the absence of tensile stress along the tube. Close-up micrographs (see Fig. Document) reveal a feature in the center of the CNT, which might be catalyst residue from the growth process[Song et al.(2008)Song, Holleitner, Qian, Hartschuh, Döblinger, Weig, and Kotthaus].

\includegraphics

setup-cavity_nl.pdf

Figure \thefigure: (COLOR ONLINE) (a) Schematics of the experimental setup. (b) Illustration of the cavity with the CNT resonating with amplitude around equilibrium position in the optical mode.
\includegraphics

dataplots2.pdf

Figure \thefigure: (COLOR ONLINE) (a) Power spectra with subtracted background reveal the CNT’s vibrational spectrum. The gray trace is obtained with the CNT placed in the cavity mode and of white noise is applied to the drive piezo, whereas for the red trace the drive is switched off. The remaining peak near rising more than above the noise floor evidences the Brownian motion of the CNT. Black arrows point at the peaks confirmed as mechanical resonances of the CNT by subsequent SEM experiments. (b) Zoom of the CNT’s Brownian noise peak with Lorentzian fit (black) and data points calibrated to Brownian vibrational amplitudes.

For optical measurements of CNT vibration the substrate chip is introduced into the fiber-based cavity as illustrated in Fig. Documenta (a photograph of this experimental situation is shown in the supplement). In order to circumvent gas damping effects[Fukami et al.(2009)Fukami, Arie, and Akita] on the mechanical motion, the experiments are carried-out in a custom fiber-compliant vacuum cell (VAC) at a pressure of and at room temperature (), see Fig. Documenta. A three axis XYZ-positioner allows accurate placement of the CNT inside the cavity (C)(see Fig. Documentb). Optionally a piezo transducer (PT) underneath the sample chip (S) can be used to excite the CNT’s mechanical motion via a signal generator (SG). The cavity is pumped with a stabilized diode laser (LD) at . The transmitted light is sent to a photo detector (D), monitored on an oscilloscope (DAQ) and used to lock the cavity (CL) on a slope of an optical resonance. Therefore an electronic feed-back loop (PI) acts on a piezo (PT) controlling the cavity length. At the beam splitter (BS) the light reflected from the cavity is directed to a second photo detector (D). The CNT vibration is measured by analyzing the optical noise of the reflected light (spectrum analyzer SA). More details on the cavity vacuum setup as well as laser and cavity stabilization schemes can be found elsewhere[Stapfner et al.(2010)Stapfner, Favero, Hunger, Paulitschke, Reichel, Karrai, and Weig]. In order to be sensitive to both the dissipative and dispersive component of the CNT-light interaction, the cavity was locked on the slope of its optical resonance. The CNT displacement is probed by monitoring changes in the reflected and transmitted light intensities which are read out by photodetectors (D). The optical power transmitted on the empty cavity resonance is . Bringing the CNT into the optical mode, as illustrated by the red spot in Fig. Documentb the cavity transmission drops by . This drop is predominantly caused by residual clipping losses originating from the presence of the hosting substrate edges near the CNT. When locked on the cavity resonance slope, the vibrational motion of the CNT is characterized under external actuation from the drive piezo. Subject to excitation with white noise ( with bandwidth), a series of resonances is clearly observable above the noise floor (Fig. Documenta gray trace). Spectra shown in Figure Documenta are obtained after substraction of the reference background of the empty cavity. The main peak at is the first flexural mode of the tube. This peak and other spectral features (black arrrows) have been identified as mechanical resonances of the CNT through SEM experiments complemented by beam theory calculations and signal amplitude estimations, as we will detail below. This main peak at rises up from the noise floor by more than , even if the piezo is not driven (Fig. Documenta red data). A Lorentzian fit to the data points of the undriven resonance (Fig. Documentb red line) yields a quality factor of . Interestingly, this signal peak is not stable in frequency but fluctuates slowly between and on the timescale of minutes. We suggest slack and conformational instability of the tube, which are also observed in the SEM, contribute most to this behavior. Furthermore, a strong dependence of the signal amplitude on the position of the CNT in the optical mode is observed. The resonance essentially vanishes when the CNT is positioned a few micrometers away from the optical mode axis. Other information can be gained when the sample is positioned such that the optical mode lies on a protrusion of the hosting substrate (for example at the position indicated by a black arrow in Fig. Documentb, about ten micrometers away from the CNT). At this position, with the protrusion entering the optical mode, the drop in cavity transmission is still of order but the spectrum exhibits a series of extra peaks below most probably orginating from vibrating substrate modes (see supplemtental material). Such substrate resonances are not observed above , indicating that the main resonance at does not stem from a substrate mode. The cavity resonant transmission exhibits constant values for the CNT being placed at several -positions inside the cavity. This indicates that the dissipative signal component is dominated by clipping losses such that the CNT presence does not contribute to total extra cavity loss in our experiment. This implies that the nanomechanical motion detection relies primarily on the dispersive interaction. To obtain an estimate for the resonance frequency of the fundamental flexural oscillation of the measured CNT, conventional Euler-Bernoulli beam theory is applied, which models the mechanical properties of CNTs satisfactorily[Babić et al.(2003)Babić, Furer, Sahoo, Farhangfar, and Schönenberger, Garcia-Sanchez et al.(2007)Garcia-Sanchez, San Paulo, Esplandiu, Perez-Murano, Forró, Aguasca, and Bachtold, Martin and Houston(2007)]. The model of an unstressed beam[W. Weaver, S. P. Timoshenko, D. H. Young(1990)] leads to

(\theequation)

where for the fundamental mode and is the length of the nanotube. Based on our TEM analysis we consider the observed CNT to be either a multiwall nanotube or a rope comprising five to seven individual tubes with a total outer diameter between and . The area moment of inertia and the cross-section area are used for multiwall nanotubes with the inner diameter , the number of walls and the inter wall spacing , and for ropes with . The physical mass density is adopted from the mass density of graphite along with , and by geometrical considerations for both cases. Values for the Young’s modulus in the range between and can be found in literature[Salvetat et al.(1999a)Salvetat, Kulik, Bonard, Briggs, Stöckli, Méténier, Bonnamy, Béguin, Burnham, and Forró, Salvetat et al.(1999b)Salvetat, Briggs, Bonard, Bacsa, Kulik, Stöckli, Burnham, and Forró, Ruoff et al.(2003)Ruoff, Qian, and Liu, Löffler et al.(2011)Löffler, Weissker, Mühl, Gemming, and Büchner]. Entering the above input parameters into Eq. Document yields an estimated resonance frequency between and , which is consistent with the measured frequency of the CNT. The equipartition theorem allows calibration of the cavity noise peak in Fig. Documenta to the Brownian motional amplitude of the CNT using

(\theequation)

with the Boltzmann constant , room temperature , spring constant . The mean squared amplitude is the integral of the position squared spectral density in frequency space and the effective mass for the fundamental mode[Ekinci and Roukes(2005)](see supplement) is derived from the physical mass . The large uncertainty in the mass comes from the above-mentioned uncertainty in the composition and diameter of the tube and translates into an error of for the calibration shown in Fig. Documentb. The nanotube’s Brownian motion peak amplitude of was measured with a sensitivity of . Assuming a purely dispersive optomechanical interaction, the effect of this displacement of the CNT on the cavity reflection can be estimated using the dispersive cavity frequency shift[Chang et al.(2010)Chang, Regal, Papp, Wilson, Ye, Painter, Kimble, and Zoller]

(\theequation)

known from cavity perturbation theory[Waldron(1960)], with the electric field amplitude , polarization , dielectric constant and permittivity for CNTs[de Heer et al.(1995)de Heer, Bacsa, Chatelain, Gerfin, Humphrey-Baker, Forro, and Ugarte] . For the first flexural mode of the CNT and in case of perfect alignment of the tube orthogonal to the cavity mode axis, the integral of Eq. Document leads, for our numerical parameters, to a maximal frequency-pull coupling rate of along the cavity axis. With the employed laser power, detector response and optimized dispersive detection by placing the CNT in the middle between a node and an antinode of the optical field, the cavity reflection noise power should peak at for the Brownian motion of the CNT. To explain the observed peak value of , we have to consider the CNT being tilted with respect to the wavefronts of the cavity field by an angle of , which is compatible with the tolerance of alignment by binocular inspection. This independently supports the identification of the resonance as Brownian motion of the first flexural mode.

\includegraphics

cnt_excitation_nl.pdf

Figure \thefigure: SEM images showing the CNT (a) at rest and excited through the drive piezo with (b) of white noise and (c) at . Colors are inverted such that the substrate and CNT appear black and the gap is light gray.

To verify the nature of the observed signal, we carry out in-depth SEM investigation on the suspended CNT sample. This investigation starts after completion of the optical measurements in the cavity, in order to avoid electron beam induced contamination. As shown in Fig. Documenta, the CNT is still present at its original position and appears visually unchanged after being exposed to the intense light field of the cavity of about . In order to visualize the CNT vibrational modes in the SEM, external actuation through the drive piezo is employed. Using the same white noise signal as applied in Fig. Documenta the CNT image blurred, showing the envelope of its flexural oscillation modes (see Fig. Documentb) with a large deflection in the center[Babić et al.(2003)Babić, Furer, Sahoo, Farhangfar, and Schönenberger]. Note that when driven with a broad band source, the nanotube oscillates on several modes simultaneously such that the envelope consists of the superposition of the envelopes of all excited modes. Driving the sample with a sinusoidal signal whose frequency is swept isolates individual vibrational modes in the range between and . This allows us to identify all peaks from the optical cavity spectrum marked by black arrows in Fig. Documenta and in supplemental material as CNT mechanical modes at , , , , , , and . Figure Documentc shows the CNT oscillating of the first flexural mode when driven by a power sinusoidal signal at . Measuring the vibrational amplitude and sweeping the drive frequency across this resonance at much lower powers allows to estimate the quality factor to be about which comes close to the value measured in the cavity. It has to be noted that electron beam induced deposition on the CNT affects the mechanical properties, but by exposing the tube to the electron beam only at small areas during frequency scanning and taking single shot images this effect can be minimized such that during the measurements no obvious down-shift in resonance frequency was observed. One possible effect distorting the calibration of the CNT Brownian amplitude is heating of the CNT by absorption of laser light. Neglecting optical losses by clipping on the hosting substrate and scattering losses induced by the CNT, we envision the extreme case where all optical power lost by the cavity is absorbed by the CNT and turns into heat. Using a heat transfer model for the heat flow with the thermal conductivity[Hsu et al.(2009)Hsu, Pettes, Bushmaker, Aykol, Shi, and Cronin] of a comparable CNT of length and cross-section yields an increase in temperature of . Under the above assumptions the CNT would oscillate at an elevated vibrational temperature with larger Brownian amplitude and we would have to correct the measurement calibration by a factor , which is smaller than the uncertainty coming from the CNT mass estimation. Furthermore optomechanical back-action could contribute to actuate the CNT and modify the mechanical resonance and linewidth[Favero and Karrai(2008)]. Using a formalism to analyze optomechanical effects[Metzger et al.(2008)Metzger, Favero, Ortlieb, and Karrai] in our data, we estimate that the change in effective vibrational temperature due to dynamical back-action would be at most of a few kelvins, hence negligible for our present study. Further, we extract similar Q-values from measurements in the cavity and in the SEM, strongly indicating that the motion is not driven optomechanically during measurements. The relatively large zero point amplitude of CNTs makes them an interesting candidate for optomechanical studies. For the tube investigated in the present article . With an improved yet technically feasible cavity with line width and for good alignment of the CNT to the optical mode, the zero point equivalent optomechanical coupling rate is expected. A cryogenic environment usually boosts the CNT’s mechanical quality factor[Steele et al.(2009)Steele, Hüttel, Witkamp, Poot, Meerwaldt, Kouwenhoven, and van der Zant] beyond , which would place the system deeply into the strong coupling regime[Teufel et al.(2011)Teufel, Li, Allman, Cicak, Sirois, Whittaker, and Simmonds, Safavi-Naeini et al.(2011)Safavi-Naeini, Alegre, Chan, Eichenfield, Winger, Lin, Hill, Chang, and Painter, Weis et al.(2010)Weis, Rivière, Deléglise, Gavartin, Arcizet, Schliesser, and Kippenberg]. This is apparent form the cooperativity , an important figure of merit for the application of optomechanical systems, which relates the zero point optomechanical coupling rate , the intracavity photon number , the cavity linewidth and the mechanical damping rate . For the described parameters a cooperativity per photon can be expected, which is higher than that reported in state-of-the-art optomechanical experiments[Teufel et al.(2011)Teufel, Li, Allman, Cicak, Sirois, Whittaker, and Simmonds, Safavi-Naeini et al.(2011)Safavi-Naeini, Alegre, Chan, Eichenfield, Winger, Lin, Hill, Chang, and Painter, Weis et al.(2010)Weis, Rivière, Deléglise, Gavartin, Arcizet, Schliesser, and Kippenberg].
In summary, we developed thinned silicon substrates with freely suspended carbon nanotubes suitable for measurements in fiber-based optical micro cavities. Taking advantage of the small mode volume of such a cavity we clearly resolve the Brownian motion of a CNT, a mechanical nanostructure much smaller than the wavelength of the employed light, in the optical cavity reflection. Optically measured resonance frequencies are confirmed by measurements in the scanning electron microscope. In contrast to SEM imaging techniques, the optical detection technique presented here allows much higher motional sensitivity, integration in a device based on fiber optics, and does not contaminate the CNT. Furthermore we anticipate these advances to lead to novel optics based carbon nanotube architectures that will allow probing of the quantum nature of molecular mechanical systems and improvement in their performance. {acknowledgments} We would like to thank Prof. Schönenberger and Markus Weiss who kindly introduced us into the art of growing clean carbon nanotubes, Dr. Döblinger for TEM imaging and analysis of the CNTs, Prof. Kotthaus and Prof. Karraï for fruitful discussions and Darren R. Southworth for critically reading the manuscript. We gratefully acknowledge finacial support from the German-Israeli Foundation (GIF), the German Excellence Initiative via the Nanosystems Initiative Munich (NIM), the German and French Academic Exchange Service (DAAD and EGIDE Procope program) and the Bayerisch-Französisches Hochschulzentrum (BFHZ).

  • [1]
  • [Jensen et al.(2008)Jensen, Kim, and Zettl] K. Jensen, K. Kim, and A. Zettl, Nature Nanotechnology 3, 533 (2008).
  • [Chaste et al.(2012)Chaste, Eichler, Moser, Ceballos, Rurali, and Bachtold] J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, Nature Nanotechnology 7, 301 (2012).
  • [Hanay et al.(2012)Hanay, Kelber, Naik, Chi, Hentz, Bullard, Colinet, Duraffourg, and Roukes] M. S. Hanay, S. Kelber, A. K. Naik, D. Chi, S. Hentz, E. C. Bullard, E. Colinet, L. Duraffourg, and M. L. Roukes, Nature Nanotechnology 7, 602 (2012).
  • [Wilson-Rae et al.(2012)Wilson-Rae, Galland, Zwerger, and Imamoglu] I. Wilson-Rae, C. Galland, W. Zwerger, and A. Imamoglu, New Journal of Physics 14, 115003 (2012).
  • [Schneider et al.(2012)Schneider, Etaki, van der Zant, and Steele] B. H. Schneider, S. Etaki, H. S. J. van der Zant, and G. A. Steele, arXiv:1209.1514 [cond-mat.mes-hall] (2012).
  • [Eichler et al.(2011)Eichler, Chaste, Moser, and Bachtold] A. Eichler, J. Chaste, J. Moser, and A. Bachtold, Nano Letters 11, 2699 (2011).
  • [Eichler et al.(2011)Eichler, Moser, Chaste, Zdrojek, Wilson-Rae, and Bachtold] A. Eichler, J. Moser, J. Chaste, M. Zdrojek, I. Wilson-Rae, and A. Bachtold, Nature Nanotechnology 6, 339 (2011).
  • [Treacy et al.(1996)Treacy, Ebbesen, and Gibson] M. M. J. Treacy, T. W. Ebbesen, and J. M. Gibson, Nature 381, 678 (1996).
  • [Babić et al.(2003)Babić, Furer, Sahoo, Farhangfar, and Schönenberger] B. Babić, J. Furer, S. Sahoo, S. Farhangfar, and C. Schönenberger, Nano Letters 3, 1577 (2003).
  • [Hüttel et al.(2009)Hüttel, Steele, Witkamp, Poot, Kouwenhoven, and van der Zant] A. K. Hüttel, G. A. Steele, B. Witkamp, M. Poot, L. P. Kouwenhoven, and H. S. J. van der Zant, Nano Letters 9, 2547 (2009).
  • [Steele et al.(2009)Steele, Hüttel, Witkamp, Poot, Meerwaldt, Kouwenhoven, and van der Zant] G. A. Steele, A. K. Hüttel, B. Witkamp, M. Poot, H. B. Meerwaldt, L. P. Kouwenhoven, and H. S. J. van der Zant, Science 325, 1103 (2009).
  • [Chaste et al.(2011)Chaste, Sledzinska, Zdrojek, Moser, and Bachtold] J. Chaste, M. Sledzinska, M. Zdrojek, J. Moser, and A. Bachtold, Applied Physics Letters 99, 213502 (pages 3) (2011).
  • [Arcizet et al.(2006)Arcizet, Cohadon, Briant, Pinard, Heidmann, Mackowski, Michel, Pinard, Français, and Rousseau] O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Français, and L. Rousseau, Physical Review Letters 97, 133601 (pages 4) (2006).
  • [Fukami et al.(2009)Fukami, Arie, and Akita] S. Fukami, T. Arie, and S. Akita, Japanese Journal of Applied Physics 48, 06FG04 (2009).
  • [Favero and Karrai(2008)] I. Favero and K. Karrai, New Journal of Physics 10, 095006 (2008).
  • [Favero et al.(2009)Favero, Stapfner, Hunger, Paulitschke, Reichel, Lorenz, Weig, and Karrai] I. Favero, S. Stapfner, D. Hunger, P. Paulitschke, J. Reichel, H. Lorenz, E. M. Weig, and K. Karrai, Opt. Express 17, 12813 (2009).
  • [Kippenberg and Vahala(2008)] T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008).
  • [Marquardt and Girvin(2009)] F. Marquardt and S. M. Girvin, Physics 2, 40 (2009).
  • [Favero and Karrai(2009)] I. Favero and K. Karrai, Nature Photonics 3, 201 (2009).
  • [Aspelmeyer et al.(2010)Aspelmeyer, Gröblacher, Hammerer, and Kiesel] M. Aspelmeyer, S. Gröblacher, K. Hammerer, and N. Kiesel, J. Opt. Soc. Am. B 27, A189 (2010).
  • [Sawano et al.(2010)Sawano, Arie, and Akita] S. Sawano, T. Arie, and S. Akita, Nano Letters 10, 3395 (2010).
  • [Steinmetz et al.(2006)Steinmetz, Colombe, Hunger, Hänsch, Balocchi, Warburton, and Reichel] T. Steinmetz, Y. Colombe, D. Hunger, T. W. Hänsch, A. Balocchi, R. J. Warburton, and J. Reichel, Applied Physics Letters 89, 111110 (pages 3) (2006).
  • [Hunger et al.(2010)Hunger, Steinmetz, Colombe, Deutsch, Hänsch, and Reichel] D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hänsch, and J. Reichel, New Journal of Physics 12, 065038 (2010).
  • [Hunger et al.(2012)Hunger, Deutsch, Barbour, Warburton, and Reichel] D. Hunger, C. Deutsch, R. J. Barbour, R. J. Warburton, and J. Reichel, AIP Advances 2, 012119 (pages 6) (2012).
  • [Flowers-Jacobs et al.(2012)Flowers-Jacobs, Hoch, Sankey, Kashkanova, Jayich, Deutsch, Reichel, and Harris] N. E. Flowers-Jacobs, S. W. Hoch, J. C. Sankey, A. Kashkanova, A. M. Jayich, C. Deutsch, J. Reichel, and J. G. E. Harris, arXiv:1206.3558v2 [physics.optics] (2012).
  • [Song et al.(2008)Song, Holleitner, Qian, Hartschuh, Döblinger, Weig, and Kotthaus] L. Song, A. W. Holleitner, H. Qian, A. Hartschuh, M. Döblinger, E. M. Weig, and J. P. Kotthaus, The Journal of Physical Chemistry C 112, 9644 (2008).
  • [Stapfner et al.(2010)Stapfner, Favero, Hunger, Paulitschke, Reichel, Karrai, and Weig] S. Stapfner, I. Favero, D. Hunger, P. Paulitschke, J. Reichel, K. Karrai, and E. Weig, Proc. SPIE 7727, 772706 (2010), arXiv:1110.6292v1 [physics.optics].
  • [Garcia-Sanchez et al.(2007)Garcia-Sanchez, San Paulo, Esplandiu, Perez-Murano, Forró, Aguasca, and Bachtold] D. Garcia-Sanchez, A. San Paulo, M. J. Esplandiu, F. Perez-Murano, L. Forró, A. Aguasca, and A. Bachtold, Phys. Rev. Lett. 99, 085501 (2007).
  • [Martin and Houston(2007)] M. J. Martin and B. H. Houston, Applied Physics Letters 91, 103116 (pages 3) (2007).
  • [W. Weaver, S. P. Timoshenko, D. H. Young(1990)] W. Weaver, S. P. Timoshenko, D. H. Young, Vibration problems in engineering (John Wiley & Sons, 1990), 5th ed.
  • [Salvetat et al.(1999a)Salvetat, Kulik, Bonard, Briggs, Stöckli, Méténier, Bonnamy, Béguin, Burnham, and Forró] J.-P. Salvetat, A. J. Kulik, J.-M. Bonard, G. A. D. Briggs, T. Stöckli, K. Méténier, S. Bonnamy, F. Béguin, N. A. Burnham, and L. Forró, Advanced Materials 11, 161 (1999a).
  • [Salvetat et al.(1999b)Salvetat, Briggs, Bonard, Bacsa, Kulik, Stöckli, Burnham, and Forró] J.-P. Salvetat, G. A. D. Briggs, J.-M. Bonard, R. R. Bacsa, A. J. Kulik, T. Stöckli, N. A. Burnham, and L. Forró, Phys. Rev. Lett. 82, 944 (1999b).
  • [Ruoff et al.(2003)Ruoff, Qian, and Liu] R. S. Ruoff, D. Qian, and W. K. Liu, Comptes Rendus Physique 4, 993 (2003).
  • [Löffler et al.(2011)Löffler, Weissker, Mühl, Gemming, and Büchner] M. Löffler, U. Weissker, T. Mühl, T. Gemming, and B. Büchner, Ultramicroscopy 111, 155 (2011).
  • [Ekinci and Roukes(2005)] K. L. Ekinci and M. L. Roukes, Review of Scientific Instruments 76, 061101 (pages 12) (2005).
  • [Chang et al.(2010)Chang, Regal, Papp, Wilson, Ye, Painter, Kimble, and Zoller] D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, Proceedings of the National Academy of Sciences 107, 1005 (2010), and supplementary information.
  • [Waldron(1960)] R. A. Waldron, Proc. IEE 107C, 272 (1960).
  • [de Heer et al.(1995)de Heer, Bacsa, Chatelain, Gerfin, Humphrey-Baker, Forro, and Ugarte] W. A. de Heer, W. S. Bacsa, A. Chatelain, T. Gerfin, R. Humphrey-Baker, L. Forro, and D. Ugarte, Science 268, 845 (1995).
  • [Hsu et al.(2009)Hsu, Pettes, Bushmaker, Aykol, Shi, and Cronin] I.-K. Hsu, M. T. Pettes, A. Bushmaker, M. Aykol, L. Shi, and S. B. Cronin, Nano Letters 9, 590 (2009).
  • [Metzger et al.(2008)Metzger, Favero, Ortlieb, and Karrai] C. Metzger, I. Favero, A. Ortlieb, and K. Karrai, Phys. Rev. B 78, 035309 (2008).
  • [Teufel et al.(2011)Teufel, Li, Allman, Cicak, Sirois, Whittaker, and Simmonds] J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, Nature 471, 204 (2011).
  • [Safavi-Naeini et al.(2011)Safavi-Naeini, Alegre, Chan, Eichenfield, Winger, Lin, Hill, Chang, and Painter] A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, Nature 472, 69 (2011).
  • [Weis et al.(2010)Weis, Rivière, Deléglise, Gavartin, Arcizet, Schliesser, and Kippenberg] S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010).
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
254475
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description