Spectroscopy of mechanical dissipation in micro-mechanical membranes
Andreas Jöckel, Matthew T. Rakher, Maria Korppi, Stephan Camerer, David Hunger, Matthias Mader, Philipp Treutlein
aa r X i v : . [ qu a n t - ph ] A ug Spectroscopy of mechanical dissipation in micro-mechanical membranes
Andreas J¨ockel, Matthew T. Rakher, Maria Korppi, Stephan Camerer, David Hunger, Matthias Mader, andPhilipp Treutlein a) Departement Physik, Universit¨at Basel, CH-4056 Basel, Switzerland Max-Planck-Institut f¨ur Quantenoptik and Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at, 80799 M¨unchen,Germany (Dated: 9 November 2018)
We measure the frequency dependence of the mechanical quality factor ( Q ) of SiN membrane oscillators andobserve a resonant variation of Q by more than two orders of magnitude. The frequency of the fundamentalmechanical mode is tuned reversibly by up to 40% through local heating with a laser. Several distinctresonances in Q are observed that can be explained by coupling to membrane frame modes. Away from theresonances, the background Q is independent of frequency and temperature in the measured range.PACS numbers: 62.25.-g, 85.85.+j, 42.79.-e, 42.50.wkMicro-mechanical membrane oscillators are currentlyinvestigated in many optomechanics experiments, wherelasers and optical cavities are used for cooling, control,and readout of their mechanical vibrations. Applica-tions lie in the area of precision force sensing and infundamental experiments on quantum physics at macro-scopic scales. The quality factor Q of the mechanicalmodes of the membranes is a key figure of merit in suchexperiments. However, the origin of mechanical dissipa-tion limiting the attainable Q is not completely under-stood and a subject of intense research. Here we report an experiment in which we observe avariation of Q by more than two orders of magnitude asa function of the fundamental mode frequency of a SiNmembrane. Several distinct resonances in Q are observedthat can be explained by coupling to mechanical modes ofthe membrane frame. The frequency of the membranemodes is tuned reversibly by up to 40% through localheating of the membrane with a laser. This method offrequency tuning has the advantage that the frequencydependence of Q can be studied with a single membrane in situ , resulting in a detailed spectrum of the coupling tothe environment of this particular mode. Other methodsthat compare Q between various structures of differentsizes have to rely on the assumption that the environmentof these structures is comparable. We investigate “low-stress” SiN membranes that aresupported by a Si frame. The frame is glued at oneedge to a holder inside a vacuum chamber, see Fig. 1. Theeigenfrequencies of a square membrane under tension are f m,n = 12 l s Sρ ( m + n ) , where l is the side length, ρ = 2 . / cm the density, and S the tensile stress in the membrane. The modes arelabeled by the number of anti-nodes m and n along thetwo dimensions. The stress S = E ( l − l ) /l , where E is a) Electronic address: [email protected] (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:5)(cid:6)(cid:3)(cid:2)(cid:9)(cid:10)(cid:11)(cid:12)(cid:3)(cid:13)(cid:7)(cid:12)(cid:3)(cid:14)(cid:15)(cid:16)(cid:17)(cid:11)(cid:12)(cid:2) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:1)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:12)(cid:7)(cid:16)(cid:17)(cid:18)(cid:15)(cid:12)(cid:16)(cid:19)(cid:12)(cid:16)(cid:8)(cid:20)(cid:21)(cid:22)(cid:23)(cid:21)(cid:22)(cid:23) (cid:21)(cid:24)(cid:24)(cid:25)(cid:5)(cid:26)(cid:15)(cid:12)(cid:4)(cid:27)(cid:28) (cid:29) (cid:17)(cid:15)(cid:15) (cid:12) (cid:4) (cid:27) (cid:28) (cid:21)(cid:30) (cid:15)(cid:6)(cid:18)(cid:12)(cid:12)(cid:3)(cid:9)(cid:19)(cid:20)(cid:21)(cid:22)(cid:9)(cid:9)(cid:9)(cid:3)(cid:4)(cid:6)(cid:5) (cid:5)(cid:13) (cid:31) (cid:1)(cid:8) (cid:9) (cid:10) (cid:11) (cid:12) (cid:14)!(cid:26) (cid:12) (cid:7) (cid:16) " (cid:23)(cid:2)(cid:6)(cid:24)(cid:2)(cid:5) FIG. 1. Experimental setup. The SiN membrane in a Si frameis glued at one edge to an aluminum holder inside a room-temperature vacuum chamber. The heating laser (red) at780 nm is power stabilized to 2 × − RMS in a bandwidth of12 kHz, and focused onto the membrane under an angle. Themembrane vibrations are read out with a stabilized Michelsoninterferometer (blue). The interferometer signal is also usedfor feedback driving of the membrane with a piezo (PZT).
Young’s modulus, arises in the fabrication process. TheSiN membrane is stretched from its equilibrium length l to the length l of the Si frame.To read out the membrane vibrations, a Michelson in-terferometer operating at 852 nm is used, where one endmirror consists of the membrane. The interferometer isstabilized by the DC to 20 kHz part of the photodiode(PD) signal. The incident power on the membrane is580 µ W in a diameter of 150 µ m and the position sen-sitivity is 1 × − m / √ Hz. The >
100 kHz frequencycomponents of the signal are fed into a lock-in amplifierwith integrated phase locked loop, which measures themembrane amplitude and drives its motion via a piezomounted outside of the vacuum chamber.To tune the membrane frequency, a power stabilized780 nm laser is focused onto the membrane to a diameterof 350 µ m. This laser heats the membrane locally in itscenter. A second method of heating the whole membrane
40 80 120 160200 Power P [mW]400600800 (1,1)(2,1)(2,2)0 (3,1)(3,2)(4,1) F r equen cy f[ k H z ] m , n FIG. 2. Mode spectrum f m,n of a membrane ( l = 0 . t = 50 nm) as a function of P . At P = 0, the lowest13 modes lie within 2% of the expected frequency. At higher P , anticrossings between higher order modes are visible. and frame is by a resistive heater (R) in the chamber.In a first experiment, we demonstrate the tunabilityof the membrane eigenfrequencies through laser heating.Fig. 2 shows the recorded mode spectrum as a functionof heating laser power P . The spectra are recorded byFourier transforming the PD signal. One can see a re-versible decrease of all mode frequencies f m,n with P .The decrease in frequency can be attributed to athermal expansion of the membrane ∆ l /l = α ∆ T + α ∆ T , where ∆ l is the equilibrium length change and α ( α ) the first (second) order expansion coefficientfor a temperature change ∆ T . This reduces the ten-sile stress by ∆ S = − E (∆ l /l ). In a simple modelassuming a spatially homogeneous and linear tempera-ture change with power ∆ T = χP , one can describe thepower-dependence of the stress as S = S − E (∆ l /l ) = S − E (cid:0) α χP + α χ P (cid:1) . A fit of f , ( P ) = √ a + bP + cP to the data describesthe observed dependence within ± P we observe a linear shift of ∆ f , = −
363 Hz / mW.We neglect the dependence of E on ∆ T because it issmall. As shown in Tab. I, the tunability of f , de-pends strongly on the geometry.In order to extract χ from the fit, one has to mea-sure α . This is done by heating up the whole sampleholder with the resistive heater. In this case both l and l change and the difference in the expansion coefficients∆ α = α f − α of the frame and the membrane determines S − S = E ∆ α ∆ T . Heating the setup by ∆ T = 16 Kand using E = 260 GPa and α f = 2 . α = 1 . α = 1 . × − / K and χ = 0 . / mW. This yields an average membrane tem-perature of T = 100 ◦ C for P = 160 mW.To model laser absorption in the membrane, we per-form a finite element (FEM) simulation of laser heating using a Gaussian beam profile and a heat conductivity l [ µ m] 250 500 1000 1500 500 500 t [nm] 50 50 50 50 75 100 S [MPa] 66.4 98.0 120 78.8 114 217 f , [kHz] 428 260 144 77.7 281 387∆ f , [Hz/mW] -259 -363 -68.9 -49.5 -89.6 -10.5 Q max [10 ] 3 . . ⋆
10 0 . ⋆ TABLE I. Summary of measured SiN membrane parameters. Q max refers to the maximum observed Q . Values marked by ⋆ were limited by the available tuning range. κ = 3 W/K m. From the resulting temperature distri-bution we calculate the average membrane temperaturefor a given absorbed laser power. By comparing with χ ,we find that a fraction of 1 . × − of the 780 nm laserpower is absorbed, an order of magnitude larger than theabsorption in low-stress membranes at 1064 nm. In a second experiment, we use laser tuning to recorda spectrum of the quality factor Q of the fundamentalmode as a function of f , . We measure the decay time τ of the membrane amplitude in ring-down measurementsafter driving it to ≈ . Q − = 1 /πf τ . We observe dis-tinct resonances, changing Q by more than two orders ofmagnitude. To show that the spectrum directly dependson f , , the heating laser is pointed off center such thata different dependence f , ( P ) results, see Fig. 4a. Thedependence Q − ( f , ) is unchanged, showing that Q onlyindirectly depends on P . The resonances in Q can be at-tributed to coupling of the membrane mode to modes ofthe frame. To prove this, the interferometer is pointedonto the frame next to the membrane and the amplituderesponse to a driving with the piezo is recorded, as shownin the lower plot in Fig. 3. The observed frame modesclearly overlap with the resonances in Q − . If the frameis heated with the resistive heater, we observe a shift inthe resonances in Q − ( f , ), as shown in Fig. 4b. We at-tribute this to a shift of the frame modes due to thermalexpansion and decreasing Young’s modulus.All these measurements prove that the coupling toframe modes is responsible for the observed behavior of Q . A FEM simulation of the frame modes shows roughlythe right density of modes in the frequency range of in-terest. As the eigenfrequencies depend strongly on theexact mounting, dimensions, and Young’s modulus of theframe, it is difficult to model them quantitatively.For stoichiometric Si N “high-stress” membranes( S = 980 MPa) we observe a much weaker dependenceof the mode frequencies on P . The measurements in-dicate that absorption of 780 nm light is lower by twoorders of magnitude compared to the “low-stress” mem-branes. This is of importance for experiments couplingsuch membranes to atomic systems. Using the limitedtuning range of the resistive heater, we also observe achange of Q with frequency in high-stress membranes.This shows that coupling to frame modes is also impor- / Q
170 180 190 200 210 220 230 240 250 26000.20.40.60.81 F r a m e A m p l . [ a . u .] FIG. 3. Upper plot: spectrum of membrane dissipation Q − ( f , ), showing a variation over two orders of magnitude.Lower plot: vibrations of the frame measured close to themembrane. The resonances in Q − ( f , ) coincide with theframe modes. P o w e r [ m W ]
240 245 250 25510 −6 −5 −4 / Q Frequency [kHz] (cid:1)(cid:2) (cid:3)(cid:2) (cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:3)(cid:6)(cid:2)(cid:7)(cid:4)(cid:5)(cid:8)(cid:9)(cid:2)(cid:10)(cid:4)(cid:5)
FIG. 4. a) f , ( P ) and Q − ( f , ) for different heating laserpositions (membrane center: blue, off-center: red). The spec-tra Q − ( f , ) overlap, indicating that Q is directly dependenton frequency. b) Q − ( f , ) for different sample holder temper-atures. Heating shifts the frame modes to lower frequencies. tant in this case. The highest measured Q is 4 × fora high-stress membrane with l = 1 . t = 50 nm.Besides coupling to frame modes, the frequency de-pendence of other dissipation mechanisms is of interest.Fig. 5 shows the dissipation spectrum of another low-stress membrane. Away from the resonances, we observea constant baseline Q max , indicating that other dissipa-tion mechanisms are independent of f ( S ) and T withinour tuning range. This is in contrast to what has been ob-served in SiN strings. We observe that Q max increaseswith l , see Tab. I. We also studied higher order modesup to (2 ,
2) and find approximately the same Q max , incontrast to other membrane experiments. This couldbe due to the different frame geometry and mounting. Inour case, the frame is a relatively small resonant structurewith eigenmodes at distinct frequencies. This can be ex-ploited to reduce clamping loss by tuning the membrane
300 320 340 360 380 400 420 44010 −5 −4 Frequency [kHz] / Q FIG. 5. Spectrum of membrane dissipation Q − ( f , ) for an-other membrane ( l = 250 µ m, t = 50 nm). Besides couplingto frame modes, the dissipation is independent of frequency. frequency to a gap between frame modes, analogous tothe recently demonstrated phononic bandgap shielding. In conclusion, we presented a precise method for laser-tuning of micro-mechanical membrane oscillators andused it for spectroscopy of mechanical dissipation. Res-onances in the dissipation were observed and explainedas coupling to localized frame modes. Other dissipationmechanisms were found to be independent of membranefrequency and temperature in the measured range.Our laser tuning technique could be extended to sto-ichiometric Si N membranes by using a laser withsmaller wavelength and thus higher absorption. Thiswould allow further investigation of the differences be-tween low-stress and stoichiometric membranes. More-over, it could be useful in finding optimal frame geometryand mounting conditions to circumvent clamping loss.We acknowledge helpful discussions with I. Wilson-Rae, M. Aspelmeyer, K. Hammerer, and T. W. H¨ansch.Work supported by the EU project AQUTE and theNCCR Nanoscale Science. J. D. Thompson et al. , Nature , 72 (2008). B. M. Zwickl et al. , Appl. Phys. Lett. , 103125 (2008). D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble,Phys. Rev. Lett. , 207204 (2009). S. Camerer et al. , arXiv:1107.3650 (2011). D. Friedrich et al. , arXiv:1104.3251 (2011). T. J. Kippenberg and K. J. Vahala, Science , 1172 (2008),F. Marquardt and S. M. Girvin, Physics , 40 (2009), I. Faveroand K. Karrai, Nat. Photonics , 201 (2009), D. Hunger et al. ,arXiv:1103.1820 (2011) S. S. Verbridge, D. F. Shapiro, H. G. Craighead, and J. M.Parpia, Nano Letters , 1728 (2007). I. Wilson-Rae, Phys. Rev. B , 245418 (2008). D. R. Southworth et al. , Phys. Rev. Lett. , 225503 (2009). Q. P. Unterreithmeier, T. Faust, and J. P. Kotthaus, Phys. Rev.Lett. , 027205 (2010). I. Wilson-Rae et al. , Phys. Rev. Lett. , 047205 (2011). G. D. Cole et al. , Nature Communications
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