Engineering the dissipation of crystalline micromechanical resonators
Erick Romero, Victor M. Valenzuela, Atieh R. Kermany, Leo Sementilli, Francesca Iacopi, Warwick P. Bowen
EEngineering the dissipation of crystalline micromechanical resonators
Erick Romero, ∗ Victor M. Valenzuela, Atieh R. Kermany, Leo Sementilli, Francesca Iacopi, and Warwick P. Bowen Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, Australia Facultad de Ciencias F´ısico-Matem´aticas, Universidad Aut´onoma de Sinaloa, M´exico School of Electrical and Data Engineering, University of Technology Sydney, NSW, Australia
High quality micro- and nano-mechanical resonators are widely used in sensing, communica-tions and timing, and have future applications in quantum technologies and fundamental studies ofquantum physics. Crystalline thin-films are particularly attractive for such resonators due to theirprospects for high quality, intrinsic stress and yield strength, and low dissipation. However, whengrown on a silicon substrate, interfacial defects arising from lattice mismatch with the substratehave been postulated to introduce additional dissipation. Here, we develop a new backside etchingprocess for single crystal silicon carbide microresonators that allows us to quantitatively verify thisprediction. By engineering the geometry of the resonators and removing the defective interfaciallayer, we achieve quality factors exceeding a million in silicon carbide trampoline resonators at roomtemperature, a factor of five higher than without the removal of the interfacial defect layer. Wepredict that similar devices fabricated from ultrahigh purity silicon carbide and leveraging its highyield strength, could enable room temperature quality factors as high as 6 × . I. INTRODUCTION
Micro- and nano-mechanical resonators have a widerange of applications in industry and fundamental sci-ence, ranging from precision sensing of mass [1], single-molecules [2], ultrasound [3], magnetic fields [4] and iner-tia [5]; to tests of spontaneous collapse models in quan-tum mechanics [6, 7], and memories and interfaces forquantum computers [8, 9]. Achieving a high resonatorquality factor is critical for many of these applications.Recently, remarkable progress has been made in im-proving the quality factor of micro- and nano-mechanicalresonators fabricated from highly stressed thin amor-phous films – most particularly amorphous silicon nitride– on a silicon substrate [10–15]. This progress has beenachieved through a combination of dissipation engineer-ing [12], to decrease both external energy loss to the envi-ronment and internal material losses, and strain engineer-ing to approach the material yield strength and therebydilute the dissipation [15, 16]. However, these strategiesare now approaching their limits for amorphous materi-als.Crystalline materials offer a range of advantages thatcould allow them to go beyond these limits. High pu-rity crystalline materials have a lower density of de-fects than amorphous materials, allowing significantlyhigher intrinsic quality factors. For instance, intrinsicquality factors above 10 have been reported for highlypure diamond [17, 18], calcium fluoride [19] and siliconcarbide [20], which has exhibited quality factors higherthan 10 when surface losses have been eliminated [21].This compares to 25,000 in amorphous silicon nitride [22]and 1000 in amorphous silicon [23]. Furthermore, dueto crystal lattice mismatch, crystalline materials can be ∗ [email protected] grown with high intrinsic stress [24], crucial for dissipa-tion dilution. Strained single crystal string resonatorshave been reported with quality factors exceeding 10 for GaNAs and 10 for SiC [25, 26], while GaAs andIn x Ga − x P nanomembranes have reached quality fac-tors above 10 [27, 28]. Moreover, crystalline materi-als have a high yield strength, increasing their poten-tial for applications using both dissipation and strain en-gineering. For instance, crystalline silicon carbide thinfilms can exhibit intrinsic stress as high as 1.5 GPa [29]and have a yield strength of 21 GPa [30]. This com-pares to 1.3 GPa and 6 GPa for amorphous silicon ni-tride. However, even with these significant advantages,thin-film crystalline resonators have not seen the samedramatic improvements in quality as their amorphouscounterparts. This is due to in part increased complex-ity of fabrication [18, 27, 28], and – when grown on asilicon substrate – to the presence of dislocations andhigh-density of stacking faults in close proximity to theinterface, which have been postulated to degrade the me-chanical quality factor [31–34].Here, we develop a new backside etching technique forthe fabrication of crystalline thin-film resonators. We usethis to fabricate high quality single-crystal silicon car-bide trampoline resonators, and to characterise the ef-fect of interfacial defect layer on their mechanical qualityfactor. By measuring trampoline resonators of varyingthicknesses, both back- and front-side etched, we are ableto build a quantitative model to extract the volumetricintrinsic quality factors of the interfacial layer and of thehigh quality silicon carbide far from the interface, as wellas the intrinsic surface quality factor. We find that de-fects degrade the volumetric quality factor by more thanan order of magnitude near the interface, and achieve adiluted quality factor exceeding one million in devices forwhich the interfacial defect layer is removed. Our modelpredicts that diluted quality factors as high as 6 × may be possible using both dissipation and strain engi- a r X i v : . [ phy s i c s . a pp - ph ] F e b neering in high purity single crystal silicon carbide. II. DEVICE DESIGN AND FABRICATION
Our trampoline design features a square inner island of40 µ m side suspended within a 700 µ m × µ m squaredhollow by four 5 µ m wide and ∼ µ m long tethers.These are connected to the substrate by adiabaticallywidened and rounded clamping points with radius of cur-vature R , as shown in the inset of Fig. 1(a). In contrast toclamp-tapered approaches, where the dissipation dilutionis achieved through localized stress [35], the dissipationdilution present in our trampolines comes from clampingpoints with radius of curvature engineered to optimizethe fraction of elastic energy stored as elongation as op-posed to bending [12, 36]. In this approach, two coun-teracting and competing mechanisms are present; the in-creased material volume at the widened clamps storesa larger amount of bending energy, while the increasedrigidity reduces the overall bending [37, 38]. The elonga-tion to bending ratio converges to a maximum value foran optimum radius of curvature R . According to our fi-nite element simulations, the optimal radius of curvaturefor our crystalline resonators is R = 30 µ m, predictinga Q about four times larger compared to rigid-clamping( R = 0).The fabrication of crystalline trampoline resonatorsintroduces technical hurdles that are not present dur-ing the fabrication of highly stressed amorphous tram-polines [10, 11]. One major difference is that, the thin-film growth of amorphous materials is independent ofthe substrate crystal orientation, while crystalline thin-films, which are often seeded by the crystal orientationof the substrate, being dependent of the crystal orienta-tion. The crystal lattice mismatch between the thin-filmand the seeding substrate induces a high density of crys-tal defects through the first few nanometers of the film,accessible by back-side etch. Release methods used foramorphous materials on silicon such as isotropic wet etchbecome ineffective for some crystalline films as the etchrate is strongly dependent on the silicon crystal orienta-tion. For this reason, we develop an alternative back-sideetch technique for the release of crystalline resonatorscompatible with all crystal orientations of the Si sub-strate.Our trampoline resonators are fabricated from ahighly-stressed 3C-SiC single crystal thin film devel-oped at the Queensland Microtechnology Facility (QMF),within the Queensland node of the Australian NationalFabrication Facility. The SiC material of initial thick-ness h = 337 nm is grown by hetero-epitaxial depositionatop a 500 µ m thick Si substrate, as depicted in Fig. 1(b-i). Using standard photolithography, the trampolines arefirst patterned on an aluminium thin film (150 nm) whichhas been evaporated atop the SiC. The wet etched alu-minium functions as a hard mask and as a protectivelayer during handling. The pattern is transferred to the Figure 1. (a) Modeshape of the trampoline’s fundamentalvibrational out-of-plane mode with lateral length L = 700 µ mand clamping points rounded with R = 30 µ m (inset). (b)Fabrication steps for single crystal SiC trampoline resonators.(c) Image of a SiC on Si trampolines chip sitting on an alu-minium holder. The different colors of the four different chipsections are produced by the different thickness of the SiC.Each device is released on a fully etched silicon window whichcan be directly observed in the image. (d) SEM image of aSiC trampoline resonator. The trampoline is suspended abovea fully back side etched hole in a 0.5 mm thick of siliconsubstrate. (e) Resonance frequency of a trampoline calcu-lated numerically using finite element modelling (blue dots)compared with the resonance frequency of a string of length L s = √ L calculated analytically (red line) using Eq. (2) withthickness h = 337 nm. SiC by Reactive Ion Etching (RIE), see Fig. 1(b-ii). Toprotect the patterned aluminium and SiC during latermanipulation, a layer of photoresist is spin coated as il-lustrated in Fig. 1(b-iii). A 150 nm film of aluminium isevaporated on the back-side, which will be used as a hardmask during the back-side etch, and the aluminium filmis protected with positive photoresist (Fig. 1(b-iv)). Thefront- and back-side patterns, with their correspondingalignment marks, are aligned with the front and rear op-tical microscope objectives of an EVG620 mask aligner.The resist is patterned defining squared windows whichare transferred to the aluminium using wet etch (Fig. 1(b-v)). The wafer is placed facing down on a secondarycarrier wafer, with a thin coating of fomblin oil betweenthem to enhance thermal contact (Fig. 1(b-vi)). The sili-con is back-side deep-etched ∼ µ m using Deep RectiveIon Etching (DRIE) as represented in Fig. 1(b-vii). Thechip is then immersed in a heated (80 ◦ C) potassiumhydroxide solution to remove the aluminium mask andthe excess of silicon “grass” formed during the etchingprocess [39].To elucidate the effect of the interface crystal defectsof the released SiC structures on the mechanical Q , aselective front-side etch is used to vary the thickness h of the SiC film. The front-side thinning is performed bymasking the chip in sections and etching the SiC layerusing RIE dry etch. An image of the chip after etch ispresented in Fig. 1(c), where the evident change in colorfor different film thickness is produced by the thin-filminterference effect of the illumination light. The trampo-lines are released using a XeF chemical dry etch of sil-icon, see Fig. 1(b-viii). As an example, Fig. 1(d) showsa SEM image of a fully released trampoline resonator.The layer with high density of crystal defects near theSi-SiC interface ( >
50 nm) [40] becomes accessible oncethe trampolines are fully released. The chip is flippedand placed on a secondary substrate and 77 nm of SiCfrom the interface is removed in some trampolines usingRIE dry back-side etch, eliminating most of the defect-rich layer.
III. EXPERIMENTAL RESULTS
The resonance frequency of highly stressed resonatorsis strongly dependent on their internal mean stress σ and scales as √ σ . The non-uniform residual stress σ r ofhetero-epitaxially grown 3C-SiC [29] allows us to investi-gate the dependence of the mean stress on film thickness σ ( h ). To find the relation σ ( h ), we measure the funda-mental resonance frequency ω m of each fabricated tram-poline from its noise power spectral density with an op-tical heterodyne detection system operating in vacuum( P ∼ − mbar) as reported previously [37, 41]. Thequality factor Q = ω m / Γ is measured via ringdown afterapplying an impulse to the trampoline and measuring thepower decay as e − Γ t , with decay rate Γ. An example ofringdown measurement (red dots) is shown in Fig. 2(a)for the fundamental mode of a back-side etched tram-poline of frequency ω m / π = 211 kHz. The fit (blackdashed line) is done using a linear regression of the free-ringdown signal, obtaining Γ / π ≈ / (8 .
23 s). The mea-sured Q ≈ . × of SiC trampolines is comparableto the Q > achieved in SiC strings [26]. Moreover,it compares favourably with the Q ∼ measured forGaNAs crystalline string resonators at room tempera-ture [25], and is comparable to the Q ∼ × ofIn x Ga − x P and GaAs crystalline membrane resonatorsof similar dimensions ( L ∼ ω m and qualityfactor Q were measured on a total of 45 devices. Theseresults are shown in Fig. 2(b) where each data point rep-resents an individual device. Nine devices, identified asred triangles, were back-side etched by 77 nm, remov-ing the defect-rich layer and leaving a final thickness of h = 260 nm. Six devices with the original film thick-ness h = 337 nm (circles) were measured without etch-ing. The other 30 devices were front-side etched withfinal thicknesses h = 293 nm (squares), h = 221 nm (rhombuses), h = 140 nm (triangles up) and h = 75 nm(triangles down). Most of the measured devices haveresonance frequencies ω m / π (cid:38)
200 kHz. However, de-vices that were largely front-side etched to a final thick-ness h = 75 nm suffered from a significant decrease inboth resonance frequency and quality factor. This dra-matic decrease in ω m and Q is caused by a substantialreduction of the mean stress σ ( h ), and is consistent withobservations that the layer near the interface is undercompressive stress [33]. PS D ( d B ) -30-20-100 Resonance Frequency (kHz) Q f ac t o r (b) 337 nm (No etch) 293 nm221 nm140 nm75 nm75 nm 140 nm 221 nm293 nm337 nm260 nmRingdown dataFit140 160 180 200 22010 Time (s)
Figure 2. (a) Normalized ringdown measurement on the fun-damental mode of a trampoline resonator with resonancefrequency ω m / π ∼
211 kHz and mechanical quality factor Q = 1 . × . Ringdown fitting (black-dashed line) using lin-ear regression of the free-ringdown signal captured in a single-shot. (b) Measured resonance frequencies ω m / π and qualityfactors Q of various single crystal SiC trampoline resonators.The blue color code is for front-side etched devices. Eachpoint represents an individual device and different shapescorrespond to different thickness: circle h =337 nm, square h =293 nm, rhombus h =221 nm, triangle up h =140 nmand triangle down h =75 nm. The triangles in red representthe devices that were back-side etched with total thickness h =260 nm. IV. MODEL AND DISCUSSION
The front- and back-side SiC etch is expected to havedifferent effects on the Q and resonance frequency of crys-talline resonators due to the presence of crystal defectsnear the interface. In order to better understand andquantify the role these crystal defects play on the dis-sipation of crystalline resonators, we have developed ageneric model for the total Q that allows us to identifyfive different energy dissipation mechanisms grouped intwo main categories, intrinsic and external. External en-ergy dissipation is attributed to two main mechanisms,gas damping Q − and clamping losses Q − [36]. Ourresonators are characterized in a vacuum chamber thatoperates at P ∼ − mbar and room temperature, witha characteristic Knudsen number K n ∼ , K n >
1) [36]. This rar-efied gas environment allows us to neglect the dampingcaused by collisions between gas molecules and the res-onator, leaving clamping losses as the main mechanismof external dissipation. These are detailed in Sec. IV B.Meanwhile, intrinsic dissipation Q − = Q − + Q − iscaused by surface losses ( Q − ) [22] and intrinsic frictionin the volume of the material ( Q − ) [42], and is detailedin Sec. IV C. Thermoelastic losses in highly stressedtrampoline resonators were calculated ( Q TED ∼ ) [37]using existing models [43], and are neglected in the restof this work due to their small contributions in thin res-onators [44]. The total quality factor is given by Q − = D − Q − + Q − , (1)where D ( h ) is the dilution factor and is well ap-proximated by the analytic expression for a string D − ( h ) ≈ (2 λ + π λ ), where λ = ( h/L ) (cid:112) E/ σ ( h ) and L is the length of the string [36]. In Fig. 3, we plotthe mean values of the experimentally measured Q (bluesquares) and the theoretical fit following Eq. (1) (blueline). In the remainder of the paper we discuss in de-tail the model developed to fit the Q as a function ofthickness of a non-uniform stressed crystalline film. A. Stress Profile and Dissipation Dilution
Quantifying D ( h ), Q int and Q clamp in Eq. (1) requiresus to identify the resonance frequency ω m ( σ ) and thick-ness dependent mean stress σ ( h ). Front-side thinned res-onators experience a reduction of the mean stress σ ( h ),shown in Fig. 4(a) as blue squares. The monotonic decre-ment of stress as a function of h is a direct consequenceof the declining proportion of high stress material in thethinner structures. The supplier specifies the mean stressof the SiC film prior to any thinning process ( h = 337 nm)to be σ ∼
620 MPa, shown in Fig 4(a) as a dashedline, which agrees reasonably with our measurements of σ ∼
660 MPa.There is no known exact analytic solution for the res-onance frequencies of a trampoline. However, we findthat the fundamental resonance frequency can be accu-rately modeled by that of a string resonator of length L s = √ L , corresponding to the diagonal length of thetrampoline. This is given by [36], ω m ( σ ) = (cid:18) π L s (cid:19) (cid:115) Eh ρ (cid:114) σ ( h ) L s Eh π , (2)where the density and Young’s modulus of SiC are ρ = 3210 kg/m and E = 400 GPa, respectively [26]. To Figure 3. Mean values of the measured Q of SiC trampolineswith different thickness h , obtained from the raw data pre-sented in Fig. 2 (b). The blue squares (red triangles) are thefront-side (back-side) thinned resonators and the error barsare the standard error. The blue line is the theoretical fitfor Q in Eq. (1) for front-side etched resonators, the shadedregion is the uncertainty of the fit. The mean Q for the back-side etched resonators is shown as a red triangle. The green(orange) dashed line is D ( h ) × Q def ( D ( h ) × Q hq ) the limitfor trampoline resonators made solely from the SiC defect-rich (high-quality) layer. The purple dashed line is Q clamp calculated from Eq. (4) for a SiC trampoline resonator withintrinsic stress σ ( h ). The black dashed line is the diluted in-trinsic quality factor D ( h ) × Q int , when clamping losses areneglected. confirm the accuracy of this model, the analytic expres-sion (red line) is compared to finite element modelling(blue dots) for the fundamental vibration mode of thetrampoline resonators as a function of the intrinsic meanstress σ of the SiC film. As shown in Fig. 1(e), we findvery good agreement.To build an approximate analytical model of σ ( h ) forfront-side thinned resonators we attempt to fit it to sev-eral basic growth functional forms including the Logis-tic model [45], Gompertz’s model [45], Bridgess growthmodel [46] and Solow’s model[45]. Of these, only aBridgess growth model [46] agrees reasonably with themean stress. The final form of the model is σ ( h ) = σ max (cid:16) − e − [ c ( h − h )] c (cid:17) , (3)where h = (68 ±
12) nm is the transition thickness atwhich the mean stress goes from compressive to tensile( i.e. σ ( h ) = 0). The exponential growth constant is c = 0 .
015 nm − and c = 0 .
54 is known as the ki-netic order. Using this model, the predicted maximumstress of the high-quality SiC layer would be enhanced to σ max = 740 MPa when there is no defect-rich layer.From the mean stress σ ( h ), it is also possible to es-timate the functional form of the layer by layer resid-
50 100 150 200 250 300 350 400
Thickness (nm) M ea n S t r e ss ( G P a ) (a) ■■■■■ ■ ■ ■ ■ Fit ■ Experimental ■ ■ ■ ■ ■ ■ D il u ti on F ac t o r FitExperimental+Model (b)
Figure 4. (a) Mean stress σ ( h ) as a function of the total thick-ness of the SiC thin film for front-side etched resonators. Theexperimental data are the mean values obtained using Eq. (3)from the measured resonance frequency. The error bars rep-resent the standard error. The black line is the theoreticalfit described in Eq. (3) and the gray band is the associateduncertainty of the fit. (b) Dilution factor D ( h ) for a thin filmtrampoline with mean stress σ ( h ). The blue circles are deter-mined from experimental data shown in Fig. 4(a). The lightblue band is the uncertainty of the theoretical fit. ual stress σ r ( z ). The residual stress is produced duringthe hetero-epitaxial growth of 3C-SiC thin-films and isrelated to the mean stress of the film through the rela-tion σ ( h ) = h (cid:82) h σ r ( z ) dz . So far, its functional formhas not been well known because existing spectroscopicmethods to measure σ r ( z ) are incompatible with thin filmanalysis. For example Raman spectroscopy suffers fromlimited resolution ( ∼ µ m) [47], while other mesure-ment methods such as bulge testing, are incompatiblewith micro- and nano-mechanical systems as they arelimited to centimeter scale sealed membranes [33]. Ourmethod to characterize the mean stress for released mi-crostructures could in future provide a precise determina-tion of the residual stress profile σ r ( z ) of hetero-epitaxialthin films beyond the resolution of existing non-invasivemethods [33, 47].The experimental D ( h ) shown in Fig. 4(b) (bluepoints) is determined by combining the measurements ofthe stress σ ( h ) (see Fig. 4(a)) and the analytical model ofthe thickness dependent dilution factor D ( h ) for a stringof length L s [36]. The good agreement provides experi-mental validation that it is appropriate to model tram-polines as strings relevant both to our work and previousresearch [11, 35, 38]. The combined results from Fig. 4suggest that the highest enhancement to the quality fac-tor should occur at a film thickness of h ∼
70 nm. How-ever, the total Q is affected not just by the dilution factorbut also by intrinsic loss mechanisms and clamping losses.In highly stressed resonators, clamping losses are oftenneglected as the largest contribution to the loss comesfrom intrinsic mechanisms. In this limit, the intrinsicquality factor Q int is estimated dividing Q ≈ D ( h ) × Q int (black dashed line in Fig. 3), by the dilution factor D ( h ). However, for our trampolines we find the experimentalresults deviate from this model at large thicknesses as itis shown in Fig. 3 (blue line), indicating that clampinglosses should be included. We further expand our modelfor clamping losses in Sec. IV B and intrinsic losses inSec. IV C. B. Clamping Loss
Clamping losses are caused by phonons tunneling fromthe resonator into the substrate [48]. The elastic energyleaks out of the resonator through the clamping pointsin the form of acoustic radiation. The amount of leak-age depends on the impedance mismatch between theresonator and the substrate [36]. To estimate the clamp-ing losses produced in our system, we use finite elementmodelling (FEM) to calculate the total elastic energy U stored in the resonator and the power P acou carried by theacoustic radiation. The clamping loss dominated qualityfactor Q clamp can be estimated as the ratio of energystored versus energy lost during one oscillation cycle as Q clamp = 2 πω m U (cid:104) P acou (cid:105) . In Fig. 5(a-i), we show the calcu-lation of P acou for a trampoline attached to a substrateof thickness h s = 500 µ m, where symmetries are used toreduce the computational demand calculating over onequarter of the domain. The trampoline design is repre-sented in Fig. 5(a-ii), where the blue shaded region is thequarter of the domain used during the calculations. Thepower crossing the interface S leaves the substrate wherea perfectly matched layer attenuates it and prevents itfrom reflecting. The parameters used in the FEM calcu-lation included the Young’s modulus of the Si substrate E s = 170 GPa, and the density ρ s = 2650 kg/m [26].The numerical results obtained from the FEM simula-tions for Q clamp require substantial computational timeand fail to predict the clamping losses at small thickness,where an analytic model represents an advantage. Tram-poline resonators are two-dimensional structures thatshare similarities with membrane resonators, for whichthe analytic expression Q clamp = α ρ s ρ (cid:115) E s ρ σ ( h ) ρ s Lh , (4)exists for the fundamental mode. Comparing our numer-ical solution to this analytic expression, we find an agree-ment for the prefactor α = 200, which is a fitting param-eter correcting for substrate imperfections and mount-ing conditions [22]. Consequently, the analytic model isused henceforth. With α fitted, the results obtained fromFEM simulations and analytic expressions agree within0.01 % for h ≥
200 nm and within ∼
5% for h ∼
100 nm,diverging for thinner thickness due to limitations on themeshing of the domain of the FEM simulation. In Fig. 3we plot Q clamp (purple dashed line) as a function of thick-ness showing it is not the primary limitation of the per-formance of our resonators, but cannot be entirely ne-glected. TheoryExperimental+Model
Figure 5. (a-i) FEM simulation of the acoustic power P acou propagating through the Si substrate, with perfectly matchedlayer at the surface S , calculated in a quarter of the totaldomain using symmetries. Zoom in of the quarter of thetrampoline used for the calculation. (a-ii) Top-view of thetrampoline design where the blue shaded region representsthe domain used during the FEM simulation. The clampingboundaries with color lines are mapped into the substrate forreference. (b) Schematic of the lateral view of the SiC singlecrystal atop Si, based on TEM studies and depictions [33, 49].The region h represents the thickness of the defect-rich layerlocalized near the interface. The lines near the interface il-lustrate stacking defects and dislocations. (c) Thickness de-pendent fitting parameters of the bi-layer system. Intrinsic Q def of the defect-rich layer (green line) and the high-qualitysingle crystal Q hq (orange line). Thickness dependent surfaceloss Q surf ( h ) (purple dashed line) of the bi-layer system. Eachlayer is limited by their respective volume losses Q hq,defvol (graydashed lines). Theoretical fit for Q int from Eq. (5) for thefront-side etched resonators compared to experimental data,with the blue-shaded region being the uncertainty of the fit. C. Intrinsic Dissipation
Intrinsic dissipation ( Q − ) in micro-resonators origi-nates from two main sources, surface ( Q − ) and volume losses ( Q − ). The dominant contribution to surface lossis expected to occur on the top and the bottom surfacesof the device, since these have much larger area comparedto lateral surfaces [22]. Volume losses in crystalline res-onators are caused by defect motion within the resonatorvolume [34]. In materials with a non-uniform verticaldensity of defects, such as hetero-epitaxially grown 3C-SiC, the dissipation profile is expected to be verticallynon-uniform.As a simple model of the non-uniform distribution ofdefects we consider the SiC film as a bi-layer system.The first layer is a defect-rich layer (def) near the Si-SiCinterface with thickness h , as represented in Fig. 5(b).The second layer is a high-quality single crystal (hq) withthickness h − h , above the defect-rich layer expectedto have a significantly reduced but not eliminated den-sity of defects (black lines) [40]. In order to identify thedifferent contributions to intrinsic dissipation in this bi-layer system, two main assumptions are made: first thateach layer has independent volume and surface dissipa-tion components; and second, that the surface dissipationfollows an inverse linear relation with thickness, as hasbeen shown for silicon nitride membranes and microcan-tilevers [22, 44]. For the high quality crystal ( h > h )the surface loss is then given by Q surfhq ( h ) = β hq ( h − h )for the high quality layer, and Q surfdef = β def × h for thedefect-rich layer.The total intrinsic dissipation Q − ( h ) is given by theweighted sum of the dissipation of each layer Q − ( h ) = (cid:18) h − h h (cid:19) (cid:2) ( Q surfhq ( h )) − + ( Q volhq ) − (cid:3) + (cid:18) h h (cid:19) (cid:2) ( Q surfdef ) − + ( Q voldef ) − (cid:3) , (5)where Q volhq and Q voldef are the volume quality factors ofthe high quality crystal and defect-rich layer, respec-tively. The experimentally determined values of Q int (blue dots) shown in Fig. 5(c) are deduced from Eq. (1),by combining the experimental Q presented in Fig. 3with the analytic models for Q clamp and D ( h ). Mean-while, the theory fit is obtained from simultaneously fit-ting the five parameters of the bi-layer SiC model Q voldef , Q volhq , β hq , β def and α , to the six data points in Fig. 5(c).The total surface loss in the bi-layer model is givenby Q surf ( h ) = (cid:104)(cid:0) h − h h (cid:1) ( Q surfhq ( h )) − + (cid:0) h h (cid:1) ( Q surfdef ) − (cid:105) − , and is shown as a function of film thickness (blue dashedline). The fitting parameters are summarized in Table I.Volumetric dissipation is associated to friction amongcrystal dislocations, stacking faults, or defects in thebulk SiC. Accordingly, the dissipation is expected to behigher in regions of the film with high defect densitythan in regions with low defect density. Our fitting pa-rameters Q volhq = (8 . ± . × and Q voldef = 750 ± Parameter Value Q voldef (0.75 ± × Q surfdef (10 ± × m − × h Q volhq (8.0 ± × Q surfhq (12 ± × m − × ( h − h ) h (68 ± × − mTable I. Volume and surface quality factors of the defect-richlayer and the high-quality layer. and high quality layer respectively. The fact that Q volhq ismore than an order of magnitude higher than Q voldef con-firms that the defect layer does indeed have significantlydegraded quality factor due to crystalline interface de-fects and these defects increase intrinsic dissipation.The surface loss of the defect and high-quality layers, β def = (10 ± × m − and β hq = (12 ± × m − , respectively, are identi-cal within the uncertainty of the fit. This suggests thatsurface losses are not drastically affected by the etchingprocess, and that interfacial defects have an impact onthe volume component of the dissipation rather thanthe surface component; consistent with the chemicalstability of the surface composition of SiC [50, 51].The highest quality factor of Q = 1 . × was ob-tained on a back-side etched resonator. The average Q ofthe back-side etched resonators is about five times higherthan the predicted average value for front-side etched res-onators of the same thickness, and more than an order ofmagnitude higher than the Q measured of the thinnestfront-side etched resonators. This results from an almostone order of magnitude increase in the intrinsic qualityfactor Q int of back-side etched resonators is enhancedby almost an order of magnitude compared to front-sideetched resonators of similar thickness. The Q of front-and back-side etched resonators are limited primarily bytheir corresponding volume losses.While already comparable with previous crystallineresonators [25, 27, 28], our results show that SiC res-onators have significant possibilities for further improve-ment. Even though the high-quality single crystal layerhas a low density of defects, these are not eliminated [40].The volumetric part of the intrinsic quality factor, whileremains well below the single crystal limit for SiC Q volhq ∼ [42] positively compares to LPCVD silicon nitrideresonators that have already reached the volume losslimit for their amorphous composition [14, 22, 38]. Thecomplete removal of interfacial crystal defects in SiCcould therefore potentially lead to exceptional enhance-ments of the quality factor as high as D ( h ) × Q int ∼ for a trampoline of thickness h = 70 nm. Moreover, im-plementing dissipation engineered designs exploiting the high yield strength of SiC could allow quality factors ashigh as Q ∼ × [15, 16]. V. CONCLUSION
This paper has explored the possibility of achieving ul-trahigh quality factors in crystalline thin-film microres-onators. Our results verify the prediction that interfa-cial defects within the thin films can severely degradethe intrinsic quality factor of crystalline resonators. Wedevelop a crystalline-material-compatible back-side etchprocedure to remove this layer, which can be applied toenhance the instrinsic quality factor of other crystallinestressed resonators made from epitaxially grown materi-als [24, 25, 27, 28].Our method allows a factor of five improvement indiluted quality factor for single-crystal silicon carbideresonators, by increasing the intrinsic quality factor,achieving values of
Q > . By developing a detailedmodel of the dissipation in bi-layer films, we are ableto precisely determine the material properties of SiCepitaxial films with higher resolution than spectroscopictechniques. Our model predicts that diluted qualityfactors as high as 6 × may be possible using bothdissipation and strain engineering if high quality single-crystal silicon carbide was used. ACKNOWLEDGEMENT
This research was funded by the Australian Re-search Council and Lockheed Martin Corporationthrough the Australian Research Council Linkage GrantLP140100595. This research is partially supported by theCommonwealth of Australia as represented by the De-fence Science and Technology Group of the Departmentof Defence. Support was also provided by the AustralianResearch Council Centre of Research Excellence for En-gineered Quantum Systems (CE110001013). E. R. andV. M. V acknowledge CONACYT (381542 and 234733respectively). V. M. V acknowledges LN-293471 and LN-299057. W.P.B. acknowledges a Future Fellowship fromthe Australian Research Council (FT140100650). The3C-SiC material was developed and supplied by LeonieHold and Alan Iacopi of the Queensland Microtechnol-ogy Facility (QMF), within the Queensland node of theAustralian National Fabrication Facility. 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