Damping in a Superconducting Mechanical Resonator
aa r X i v : . [ c ond - m a t . s up r- c on ] M a r epl draft Damping in a Superconducting Mechanical Resonator
Oren Suchoi and Eyal Buks Andrew and Erna Viterbi Department of Electrical Engineering, Technion, Haifa 32000 Israel
PACS – Superconductivity , Mechanical and acoustical properties, elasticity, and ultra-sonic attenuation
PACS –vibrations and mechanical waves
Abstract – We study a mechanical resonator made of aluminum near the normal to super con-ductivity phase transition. A sharp drop in the rate of mechanical damping is observed belowthe critical temperature. The experimental results are compared with predictions based on theBardeen Cooper Schrieffer theory of superconductivity and a fair agreement is obtained.
Mechanical resonators having low damping rate arewidely employed for sensing and timing applications [1].At sufficiently low temperatures such devices may allowthe experimental exploration of the crossover from clas-sical to quantum mechanics [2–8]. Commonly, the ob-servation of non-classical effects in such experiments ispossible only when the damping rate [9] of the mechan-ical resonator is sufficiently low. Mechanical resonatorsmade of superconductors are widely employed in such low-temperature experiments. In this study we experimentallyinvestigate the effect of superconductivity on the dampingrate of a mechanical resonator made of aluminum near itsnormal to super phase transition.The damping rate can be measured by coupling the me-chanical resonator under study to a displacement detec-tor. In general, a variety of different mechanisms maycontribute to the total mechanical damping rate. In or-der to isolate the effect of superconductivity it is impor-tant to employ a method of displacement detection thatis unaffected by the normal to super phase transition. Inaddition, systematic errors in the measured damping ratedue to back-reaction effects originating from the couplingbetween the mechanical resonator and the displacementdetector have to be kept at a sufficiently low level.In our setup (see Fig. 1) we employ the so-called op-tomechanical cavity configuration [10–12], in which dis-placement detection is performed by coupling the me-chanical resonator to an electromagnetic cavity. Whilemany of the previous studies of superconducting mechan-ical resonators have employed such a configuration witha superconducting microwave cavity [4, 6, 7, 13–21], oursetup, which is based on a cavity in the optical band,
Fig. 1: The experimental setup. (a) A sketch of the mechan-ical resonator and the fiber-based optical cavity. (b) Electronmicrograph of the trampoline. (c) The resonance lineshape ofthe fundamental mechanical mode. allows displacement detection that is unaffected by thephase transition occurring in the mechanical resonator un-der study, which, in-turn, allows isolating the effect of su-perconductivity on the mechanical damping. Moreover,back-reaction effects are suppressed by employing a rela-tively low driving power to the optical cavity (see discus-sion below).In our setup the optomechanical cavity is formed be-tween two mirrors, a stationary fiber Bragg grating (FBG)mirror and a movable mirror made of a mechanical res-p-1. Author et al. onator in the shape of a trampoline supported by fourbeams [see Fig. 1(a)]. A graded index fiber (GIF) splicedto the end of the single mode fiber (SMF) is employed forfocusing. A cryogenic piezoelectric three-axis positioningsystem having sub-nanometer resolution is employed formanipulating the position of the optical fiber. A photo-lithography process is used to pattern a t Al = 200 nm thickaluminum layer on top of a a t SiN = 100 nm thick siliconnitride layer into the shape of a 100 × µ m trampolinewith four supporting beams [see Fig. 1(b)]. Details ofthe fabrication process can be found elsewhere [22]. Mea-surements are performed in a dilution refrigerator at apressure well below 2 × − mbar. Another device on thesame wafer, which is made in the shape of a microwavemicrostrip resonator, allows characterizing the surface re-sistance of the aluminum layer [22].A tunable laser operating near the Bragg wavelengthof the FBG together with an external attenuator are em-ployed to excite the optical cavity. The optical power re-flected off the cavity is measured by a photodetector (PD),which is connected to a network analyzer (NA). Actuationis performed by applying an alternating voltage (with adirect voltage offset) between the trampoline and a sta-tionary electrode positioned 200 µ m below it.The fundamental mechanical mode is characterized byits frequency ω m / π and damping rate γ m . Both parame-ters can be extracted from the resonance lineshape of themeasured NA signal S NA vs. angular driving frequency ω NA with a fixed driving amplitude [see Fig. 1(c)]. Inthe regime of linear response S NA ( ω NA ) is expected to begiven by S NA ( ω NA ) = S NA , R (cid:16) ω NA − ω m γ m (cid:17) , (1)where S NA , R is the value of S NA ( ω NA ) at resonance, i.e.when ω NA = ω m .The measured damping rate γ m vs. temperature T ,which is extracted from the NA data using Eq. (1), isindicated by the crosses in Fig. 2. The same procedureyields the mode’s frequency ω m / π = 432 .
318 kHz, whichis found to be almost a constant for the range of temper-atures explored in this measurement (between 0 . . γ m sharply drops as the temperature T is loweredbelow the value of 1 . . T c = 1 . P L = 3 × − W.In general, heating due to optical absorption by the alu-minum layer may give rise to a systematic error in the mea-surement of γ m . Two possible mechanisms are discussedbelow. The first one is due to back-reaction originat-ing from the bolometric optomechanical coupling [23, 24], γ m / π [ H z ] Fig. 2: The measured (crosses) and theoretically calculated(solid line) damping rate γ m vs. temperature T . The calculated γ m is obtained using Eq. (2). which gives rise to a shift in the effective value of γ m de-noted by γ m , ba . The magnitude of γ m , ba can be roughlyestimated using the relation | γ m , ba | /γ m ≃ P L /P LT , where P L is the laser power that is employed for the measure-ment of γ m and P LT is the laser power at the thresholdof self-excited oscillation [25]. For the same cavity tun-ing, for which the data seen in Fig. 2 is obtained, self-excited oscillation occurs at a threshold power given by P LT = 4 × P L , and thus the effect of back-reaction canbe safely disregarded.The other possible source of a systematic error orig-inates from temperature rise due to optical absorption.Due to the low heat conductance of both superconductingaluminum and silicon nitride, this mechanism imposes asevere upper limit upon the allowed values of laser power.The heating power is given by P H = ζβ F (1 − R C ) P L ,where ζ is the absorption coefficient (for aluminum ζ =0 . β F is the cavity finesse, R C is the cavity reflectivity,and P L is the laser power [26]. For the measurement of γ m that is presented in Fig. 2 the optical cavity is tunedto have finesse of β F = 1 . R C = 0 . P H = 1 . × − W.The temperature rise ∆ T due to optical absorption isestimated by ∆ T = P H / K b , where K b is the thermalconductance of each of the four nominally identical beamsthat support the trampoline. The thermal conductance K b is given by K b = ( t Al κ Al + t SiN κ SiN ) ( w b /l b ), where κ Al ( κ SiN ) is the thermal conductivity of aluminum (siliconnitride), and where w b /l b = 0 . . κ Al ≃ × − W K − m − foraluminum [27] and κ SiN ≃ − W K − m − for silicon ni-tride [28, 29]. For these values the estimated temperaturep-2amping in a Superconducting Mechanical Resonatorrise is ∆ T = 0 .
06 K. For temperatures below 0 . T becomes even larger since both κ Al and κ SiN rapidly drops at low temperatures, and there-fore no reliable measurements can be obtained unless P H is further reduced. However, no significant reduction of P H is possible in our setup due to noise, and consequentlyreliable data far below 0 . . γ m is compared with theory. The solid lineseen in Fig. 2 represents the calculated value of γ m ob-tained from the following expression [30, 31] γ m = γ S + 2 γ N ∆( T ) k B T , (2)where the fitting parameters γ S , γ N and T c are taken to begiven by γ S / π = 3 . γ N / π = 1 . T c = 1 . T ) is found bynumerically solving the Bardeen Cooper Schrieffer (BCS)gap equation [32] ν = Z eν δ d x tanh (cid:16) ξδ √ x τ (cid:17) √ x , (3)where ν = 1 /gD is the inverse interaction strength with g being the electron-phonon coupling coefficient and D being the density of states per unit volume, δ = ∆ / ∆ is the normalized gap with ∆ being the zero tempera-ture gap, τ = T /T c is the normalized temperature, andthe number ξ is given by ξ = π/ e C E with C E ≃ . γ N represents the contribution of normal elec-trons to the total rate of mechanical damping [see Eq.(2)]. This rate has been calculated in Ref. [35] for thecase where the electron mean-free path is greater than thewavelength of the oscillating acoustic mode. However, thisassumption is not valid for our device. When the electronmean-free path is shorter than the acoustic wavelengththe rate γ N can be roughly estimated using Stokes’ lawof sound attenuation [36, 37], which relates this rate tothe electronic viscosity [38]. For the case of an acousticwave having angular frequency ω m propagating in a bulkaluminum the rate according to this approach is given by γ N , bulk = 2 η Al ω ρ Al c , (4)where ρ Al = 2 . − and c Al = 5 . × m s − arethe mass density and the speed of sound, respectively, of aluminum, and where η Al is the electronic viscosity ofaluminum. The electronic viscosity can be expressed as η Al = (2 / n Al τ Al h ǫ Al i where n Al is the density of freeelectrons, τ Al is the scattering relaxation time, and h ǫ Al i is the averaged kinetic energy [see Eq. (43.8) in [39]].At low temperatures h ǫ Al i = 3 ǫ F , Al /
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