SUPPORTING INFORMATION

Transcription

1 SUPPOING INFOMAION Stress-Iduced Variatios i the Stiffess of Micro- ad Nao-Catilever Beams. B. Karabali 1, L. G. Villaueva 1, M. H. Mathey 1, J. E. Sader 2, M. L. oukes 1 1 Kavli Naosciece Istitute, Califoria Istitute of echology, Pasadea, Califoria 91125, 2 Departmet of Mathematics ad Statistics, he Uiversity of Melboure, Victoria 3010, Australia I. Effect of Surface Stress o Doubly-Clamped Beams A. Axial force model (Stress effect i able I) I this sectio, we derive the required formulas for the effect of surface stress o the resoat frequecy of doubly-clamped beams due to the axial tesio built up alog the beam (stress effect i able I). I the mai mauscript we discussed that a et axial force is iduced i doubly-clamped beams, alog their major axis. his et axial force is give by the itegral of the resultig axial stress over the beam cross-sectio. For the resoators used here, this coicides with the itegral of the axial stress i the active piezoelectric layer oly; other parts of the beam do ot cotribute to the et axial force because the beam-eds are restraied from movig. he et axial force determies the resultig effect o resoace frequecy ad/or stiffess of the device. a. z L σ + x y b h b. L σ - x z y b h Figure S1 Graphical represetatio of a doubly-clamped (a) ad a catilever (b) beam with the dimesios used i the paper, as well as the axis defiitio. I (a) a compressive stress (positive stress by covetio) is show, whereas i (b) a tesile (egative by covetio) stress is show. SI-1

2 he effect of surface stress o the resoace frequecy of a doubly-clamped beam is calculated to leadig order for small surface stress loads, ad thus gives a liear relatioship betwee the resoat frequecy shift ad the applied surface stress chage. It is assumed that the beam structure possesses a rectagular cross-sectio whose width b greatly exceeds its thickess h, i.e., the beam is formally a thi plate S1. he legth of the structure is L (see Fig. S1). We use the geeral theoretical formalism preseted elsewhere S2 surface stress, ad thus decompose the problem ito two subproblems: to calculate the effect of Subproblem (1): Deformatio of a urestraied plate uder applicatio of a et surface stress load. Subproblem (2): Beam structure with o surface stress load ad a specified i-plae displacemet to satisfy the required clamped boudary coditios at the eds. Superpositio of these two subproblems gives the required i-plae deformatio of the origial problem, with exact satisfactio of free edge ad clamped boudary coditios; see ref. S2 for details. Applicatio of a et surface stress chage to the surface of the structure, with the clamped boudary coditios removed, results i a isotropic strai whose displacemet field is: ( uv) ( 1 ) ν σs, =, Eh ( xy), (S1) where u ad v are displacemets i the x ad y directios, respectively; ( xy, ) are the Cartesia coordiates i the plae (see Fig. S1); ν ad E are the Poisso ratio ad Youg s modulus of beams material respectively; h is the device thickess; ad σ s is the total applied surface stress S2. his is the required solutio to Subproblem (1). o accout for the clamped displacemet coditios at the ed of the structure, i accord with Subproblem (2), a displacemet load i the x-directio must be applied to its eds; displacemets i the y-directio are ot importat sice the beam legth greatly exceeds its width, i accord with Sait- Veat's priciple S3. From Eq. (S1), this axial displacemet is: u = ( 1 ) ν σ Eh L s, (S2) where L is the beam legth. Such a axial displacemet iduces a axial tesile load: F axial ( 1 ) = ν σ b s, (S3) SI-2

3 where b is the beam width. he axial load i Eq. (S3) will lead to a chage i stiffess ad hece resoat frequecy. Sice the beam legth greatly exceeds width ad thickess, this effect is calculated from Euler-Beroulli beam theory. he goverig equatio for the deflectio fuctio w, for a beam of liear mass desity µ ad areal momet of iertia I, is: EI w w w x x t F 0 4 axial + µ = 2 2, (S4) which is solved with the usual clamped boudary coditios at the beam eds, w= w' = 0. We assume a explicit time depedece of exp( iωt), where ω is the agular frequecy ad t is time, i.e., wxt (, ) = W( x)exp( iωt). Multiplyig both sides of Eq. (S4) by the deflectio fuctio w, scalig x by the beam legth L, ad itegratig over the beam legth yields the followig exact result for square of the radial resoat frequecy: ω 2 = EI L 4 µ ( W (x)) 2 d x ( W (x)) 2 dx F axiall 2 EI ( W (x)) 2 dx ( W (x)) 2 dx , (S5) where x = x L is the scaled axial distace. o calculate the leadig order effect of surface stress chage o the frequecy shift, we use the deflectio fuctio for a doubly-clamped beam i the absece of surface stress. Solvig Eq. (S4) uder this coditio the gives: cosh D cos D W x Dx Dx Dx Dx sih D si D ( ) = cosh cos + ( si sih ), (S6) where D is the -th positive root of: with =1 correspodig to the fudametal mode. cosh D cos D = 1, (S7) Substitutig Eq. (S6) with =1 ito Eq. (S5), ad usig Eq. (S3), the yields the required result: ω ω ( )σ s = ν E h L h 2, (S8) SI-3

4 2 where the origial radial resoace frequecy ω = 2π f = / L E/ ρ, the beam mass desity is ρ, ad the radial frequecy shift is defied. We emphasize that Eq. (S8) is valid i the asymptotic limit of small stress loads. A alterate derivatio of this formula is give i ef. S4. B. Chage i dimesios (Geometric Effect i able I) Due to the boudary coditios i a doubly-clamped beam, after a isotropic i-plae stress is applied to the beam the strai map that develops is: h ω = ω ω ( ) ( 1 ) ( 1 ) ( 1 ) s + ν σ ν ν ν σs εxx = 0; εyy = 1 + ν ; εzz = Eh 1 ν Eh (S9) Assumig that the Youg s modulus of the material remais uchaged durig the applicatio of stress, the relative chage i the resoat frequecy would be: Δω 1Δρ Δt ΔL 1ΔV Δt ΔL = + 2 = + 2 εxx + εyy + εzz ω 2 ρ t L 2 V t L (S10) hat, usig (S9) with (S10), yields: Δω ω ( 1+ )( 1+ 2 ) ( 1 ) ν ν ν σ 1 ν Eh s (S11) which is the expressio foud i able I i the mai mauscript. I the case of doubly-clamped beams ad the typical geometries that are used tha this geometric effect. L h >> 1, the stress effect described previously is much larger SI-4

5 II. (Uphysical) Axial Force Model for Catilever Beams It has bee widely assumed that applicatio of surface stress to a catilever beam iduces a axial force alog the beam legth. his so-called axial force model has bee show to be uphysical ad i violatio of Newto s 3 rd law S2,S5,S6. For completeess, however, we reproduce the resultig formula derived from this model S7. his allows compariso ad assessmet with measuremets performed i this study. I the limit of small surface stress loads, the axial force model has bee reported uder various forms S7-12. Sice the uderlyig model is uphysical, we refrai from ay discussio o the merits of each form ad simply report the most commo formula that has bee claimed to yield good agreemet with measuremets S7 :. (S12) Comparig this result with the (physically correct) model for doubly-clamped beams i Eq. (S8), we observe that the axial force model predicts a larger shift i frequecy i catilever beams tha i doubly-clamped beams, by a factor of: Sice L/ b>> σs ω 12 σs L L L L = ω π Eh b h Eh b h ω ω 8 = ω ω ν cat beam ( 1 ). (S13), the axial force model predicts that catilever beams are much more sesitive to surface stress chages tha doubly-clamped beams. his is ot observed i the cotrolled measuremets of doubly-clamped ad catilever beams reported i this study. L b SI-5

6 III. Piezoelectric loads i doubly-clamped beams he piezoelectric effect couples the mechaical ad electrical degrees of freedom of a material by the followig relatio ( ε) = [ C]( σ) + [ d]( E ) (S14) where (ε) ad (σ) are the mechaical strai ad stress vectors, respectively; [C] is the compliace matrix of the material, (E) is the electric field vector ad [d] is the piezoelectric matrix, which is directly resposible for the mechaical/electrical couplig. For the piezoelectric material cosidered i this study, the piezoelectric matrix is of the form (with axis 1, 2 ad 3 poitig alog the beam, trasversal i-plae ad out-of-plae respectively): [ d ] 0 0 d d 31 d31 d31 d 33 =. (S15) 0 d45 0 d he applicatio of a exteral voltage betwee two electrodes cotactig the piezoelectric material (as i our experimetal case) leads to the geeratio of a electric field ad, cosequetly, deformatio of the material. Followig a similar approach to that take i Sectio I of the SI, we first calculate the deformatio of the urestraied piezoelectric material. his allows the material to expad or cotract freely ad therefore the i-plae strais ε xx alog the beam ad ε yy perpedicular to the beam ad out-of-plae strai ε zz are give by: V ε = ε = d ; ε = d xx yy 31 zz 33 hpze V h PZE, (S16) where V is the applied voltage ad h PZE is the thickess of the piezoelectric layer. However, the composite beam structure used i the preset devices cotais other materials that are ot piezoelectric. hose materials impose some restrictios o the total deformatio, which will be a combiatio of et elogatio ad bedig. As a first approximatio, bedig does ot cotribute to the stiffess of the beam S2 ad therefore we ca limit our aalysis to the et (average) strai. his et strai is give by Eq. (S17) whe the Poisso ratios of the differet materials are idetical (which is a excellet approximatio i the preset case): SI-6

7 E ε = ε = d V; ε = d PZE xx yy 31 zz 33 Eh i i htot V (S17) where E PZE is the Youg s modulus of the piezoelectric material, h tot is the total thickess of the beam, ad the summatio i the deomiator exteds to all four layers of the composite structure, with E i ad h i beig the Youg s modulus ad thickess of every layer, respectively. he brackets i equatio (S17), ε ii, mea average over the cross sectio of the beam (y-z plae). Followig the approach of Sectio I of the SI, we iitially cosider the case of a doublyclamped beam. he solutio above correspods to Subproblem (2) i Sectio I. Subproblem (2) the requires a displacemet load i the x-directio to match the required clamped boudary coditios at the beam eds, as discussed above. his axial displacemet iduces the followig axial load: F = d E Vb axial 31 PZE. (S18) Equatio (S18) ca ow be directly compared to Eq. (S3) ad the derivatio performed i Sectio I is valid here. We ca thus write a expressio equivalet to (S8), i this case for a piezoelectric beam: ω E = PZE d31v bl, (S19) ω EI where <EI> is the effective flexural rigidity of the beam. If the Youg s moduli of the differet materials i the composite structure are approximately the same, Eq. (S19) immediately leads to: 2 ω d31v L = ω htot htot, (S20) which is oly depedet o total thickess h tot. Comparig Equatios (S8) ad (S20), it is evidet that the effects of a applied voltage are equivalet to the effects of a applied surface stress, as required. his leads to the followig relatio coectig a equivalet surface stress to a give applied voltage: σ = d s. (S21) Note that i the calculatio of the chage i frequecy for a doubly-clamped beam, we have eglected ay differetial chage i dimesios, sice this is much smaller tha the cotributio give by Eq. (S20) (we assume that ay beam holds the coditio L>h tot ). ( ν ) 31 1 E V SI-7

8 IV. Catilever beams A. Isotropic material surface stress load We begi by cosiderig a catilever plate composed of a isotropic material uder a surface stress load this effect is due to surface stress chage; see ref. S2. he i-plae stress effect has bee aalyzed previously ad yields S2 : ( 1- ) ω ν σs b b = ( ) φν ω Eh L h 2, (S22) where umerical coefficiet φ( ν) ν is estimated via FEM ad accouts for 3D effects of the stress distributio close to the clamp. A detail expositio of the derivatio of this formula ad its physical features is give i ef. S4. he geometric effect is give by the differetial chage i dimesios of the catilever beam, due to the surface stress load. his tues the stiffess ad the resoat frequecy of the catilever, ad yields (cosiderig ε = ε ): xx yy ω 1 ρ h L 3 (1+ 2 ν ) σ s = + 2 = ε zz ε xx = ω 2 ρ h L 2 Eh, (S23) I the limit as h/ b 0 (commesurate with the assumptios of thi plate theory S2 ), the iplae stress cotributio i Eq. (S22) domiates the geometric effect i Eq. (S23), for large ad fixed aspect ratio L/ b. he i-plae stress effect remais prevalet for thi beams uder the coditio: b h < b 5 1+2ν L h, critical ν 1 ν b ( ) (S24) whereas for thicker beams, the geometric effect will domiate. hese results are gathered i able I of the mai mauscript. SI-8

9 B. Piezoelectric material voltage load For the piezoelectric beams used i this study, the geometric effect ca also be calculated: ω 1 ρ h L 3 3 d33 d 31 = + 2 = εzz εxx = + V ω ρ h L htot htot (S27) he chages i the Youg s modulus ad Poisso ratio of the materials are eglected. For the piezoelectric material used here, d = 2d 33 31, ad hece Eq. (S27) becomes: ω d = 4 ω h 31 tot V. (S28) Calculatio of the stress effect i the devices cosidered is complicated by the multilayer structure of the piezoelectric catilevers, ad their o-isotropic material properties. hree-dimesioal FEM aalysis is thus used to obtai the combied cotributios of stress ad geometric effects, complemetig Eq. (S28). his total effect is plotted i Fig. 3. We fid that the geometric effect domiates the results. SI-9

10 V. Measuremet of NEMS Device espose We employ a optical iterferometric detectio scheme to sese the mechaical motio of our NEMS devices S13. his detectio mechaism is used due to ease of implemetatio, high mechaical resposivity to out-of-plae beam displacemets ad egligible F backgroud oise. Figure S2 Schematic of the optical iterferometric detectio setup used for sesig the motio of NEMS devices A schematic of the experimetal setup is show i Fig. S2. he devices are mouted i a custom-built room temperature vacuum chamber fitted with a quartz optical widow for iterrogatio. A optical cavity is created betwee a two-mirror system formed by the bottom of the substrate ad the top surface of the NEMS device. he light source cosists of a regular helium eo laser (λ = 632 m) followed by a beam expader ad a atteuator. After the origial laser beam passes the beam splitter orieted at 45 o, oe compoet is focused oto the NEMS device with a les of 0.15 umerical aperture, resultig i a spot size ~10 20 μm. Iterferece betwee laser beams reflected from the top surface of the mechaical resoator ad bottom substrate is detected usig a high-badwidth low-oise photodetector. As a result, the resoator motio modulates the light itesity i proportio to the magitude of the mechaical displacemet. SI-10

11 A AC voltage betwee top ad bottom electrodes of the NEMS device actuates the out-ofplae motio, while a DC bias produces the required exteral stress studied i this work. Both voltages are combied usig a bias-tee. A vector etwork aalyzer is used to measure the resoace spectral respose of the resultig sigal. o perform the frequecy shift experimets, the etwork aalyzer operates i a cotiuous wave (CW) regime ad is cotrolled by a exteral computer that is used to provide phase locked loop (PLL) operatio. A slight asymmetry betwee tuability slopes is observed i doubly clamped beams. We attribute this to a electrostatic effect from the substrate that affects the resoat frequecy as V 2. Whe fittig the experimetal data to a secod-degree polyomial, the fit is more accurate but the liear slope, evertheless, remais the same (relative chage smaller tha 1%). I subsequet experimets, we used a wafer with a thick sacrificial silico oxide layer udereath the seed alumium itride layer ad this reduced by two orders of magitude the coefficiet of secod order i the polyomial. I this case, agai, the liear slope remais the same. We therefore believe that the coclusios draw i the curret paper are ot affected by the observed small symmetry-breakig. Fig. 3 i the mai mauscript also shows some apparet asymmetry ad, i additio, a apparet discrepacy of the scalig with 1/L 2. I order to clarify this issue, we replot Fig. 3 from the mai mauscript as Fig. S3, i this case with error bars. he error bars for the experimetal data are calculated based o the measured Alla Deviatio of , represetig the 95% cofidece bouds. he error bars for the FEM data correspod to the typical expected accuracy of FEM simulatios, accoutig for geometrical ad material ucertaities ad covergece of the frequecy shift; see Sectio VI-A. he discrepacy betwee theoretical predictio ad measuremet slopes is o the order of ~5-15%. his is a reasoable agreemet cosiderig the show experimetal error of ~10%, especially visible o scaled plot show i Fig S3(a). herefore, the slight asymmetry ad discrepacy with the scalig appear both to be egligible, whe careful aalysis of measuremet errors is take ito accout. SI-11

12 Figure S3 (a) elative Δf/f ad (b) absolute Δf frequecy shift for three catilevers (Fig. 3 i the mai mauscript) with error bars. he experimetal ucertaities show 95% cofidece bouds ad they are calculated based o the observed Alla Deviatio of a measure of catilevers frequecy fluctuatios. he error bars for FEM data are based o estimated geometrical ad material ucertaities ad covergece of the frequecy shift (see Sectio VI-A). SI-12

13 VI. Fiite Elemet Simulatios Extesive Fiite Elemet Method (FEM) simulatios have bee performed to (i) compare with experimetal results o piezoelectric beams, ad (ii) assess theoretical predictios from the preseted models for uiform beams. I all the cases, the objective of the simulatios was to compute the frequecy shift caused by a applied load, which could be either stress (i a uiform beam) or trasverse applied electric field i the piezoelectric material. A commercial software was used to perform these simulatios. he methodology used cosists of performig first a static aalysis of the system with the applied load, allowig the system to respod to this load. Subsequetly, a modal aalysis is performed usig the resultig stresses/strais from the static aalysis. his modal aalysis ca accout idividually for the effect of structural stress o the resultig stiffess, ad/or for the effect of the resultig strai o the geometric chage i dimesios. A. Mesh covergece he utilized mesh is refied util 99% covergece i the iterestig magitudes is achieved. Covergece is defied, for the static aalysis, by moitorig the maximum strai, stress ad deformatio, observig that all of them coverge at about the same rate. I the case of modal aalysis, covergece is defied by moitorig the frequecy of the first out of plae flexural vibratioal mode. he mesh refiemet coverges at the same size for both modal ad static aalysis. he covergece of the simulated frequecy shift was also studied. I the case of doublyclamped beams, the frequecy shift due to stress ca be observed to coverge at the same rate as the frequecy. However, whe cosiderig the stress effect i catilever beams or the geometric effect for either of both types of beams, the frequecy shift caot be coverged better tha 95%. his is most likely due to the property that the frequecy shift is much smaller i these cases ad the umber is so small that, as soo as the mesh refies, it also itroduces umerical ucertaity that is of the order of 5% of the frequecy shift. We therefore make all the simulatios cosiderig a 99% covergece i frequecy shift for the stress effect i doubly-clamped beams ad a 95% covergece i frequecy shift for remaiig cases. SI-13

14 B. Piezoelectric devices Catilever ad doubly-clamped beams are simulated i accord with the geometry ad clampig coditios of the fabricated devices. his icludes a small ledge i the achorig regio, which is foud to be ecessary for better compariso betwee experimets ad simulatios (brigig them about 2-4% closer). he used material properties are: E AlN = 345 GPa, ν AlN = 0.3, ρ AlN = 3230 kg/m 3, d 31,AlN = 2.5 pm/v, E Mo = 329 GPa, ν Mo = 0.31, ρ Mo = kg/m 3. By chagig the applied exteral voltage, simulatios of the experimetal measuremets are produced, as show i Fig. 2 ad Fig. 3 of the mai article. No chage i Youg s modulus ad Poisso s ratio was cosidered durig these simulatios. As itroduced before, we perform idepedet simulatios to accout for each of both effects (stress ad geometrical chages). I the latter case, we fid a remarkable quatitative agreemet for the frequecy shift whe comparig FEM results with the aalytical predictios of Eq. (S28) (5% differece). he simulated/theoretically predicted frequecy shift accouts for as much as 75% of the experimetally observed frequecy shift. Stress iduced frequecy shifts i piezoelectric catilevers are larger tha expected for uiform beams S2. Equatio (S22) (with φν ( ) 0.042ν ) predicts aroud 2% of the observed frequecy shift, while FEM simulatios represet aroud 20% of the experimetally observed frequecy shift (which added to the geometrical effect described before correspods to 95% of it). We attribute this divergece betwee FEM (ad experimetal results) ad theory to the actual o-uiform geometry of the multilayer resoators used experimetally. A possibility recetly brought up i the literature S14 could be the fact that our structure is ot completely symmetric, but has some asymmetry due to the bottom AlN layer. FEM simulatios disprove this explaatio whe we remove the asymmetry, the results (relative frequecy shifts) are ot affected. he simulatios diverge from the experimetal results by aroud 5%. I the case of the catilever beams we attribute this to the limit i the accuracy of our techique. I the case of the doubly-clamped beams, the divergece is mostly caused by a deviatio from liear behavior of the experimetal results. he origi of this effect has ot yet bee elucidated but a plausible explaatio ca lie i variatios of the Youg s modulus due to the axial stress geerated alog the beam. SI-14

15 C. Uiform beam with surface stress load o assess the validity of the theoretical model developed, simulatios of catilever ad doublyclamped beams composed of a sigle material were also performed. I this case, ideal achorig was utilized, i.e. o ledge was icluded i the model. We use the followig typical material properties: E = 200 GPa, ρ = 2000 kg/m 3. he Poisso s ratio of the material, ν, assumed the values ν = 0, 0.25, Followig the procedure described above, several simulatios were performed by varyig the applied load (i this case surface stress). he ormalized frequecy shift, Ω, was the determied: Δω σ s b Ω= ( 1 ν ) ω Eh L 0. (S29) I Fig. S4, the ormalized frequecy shifts are plotted for differet Poisso s ratios as a fuctio of the ratio b/h, for a give L/b = 10. Cotributios due to stress i the catilever material ad chage i catilever dimesios are show separately i Fig. S4. hey correspod precisely to the predictios of Eq. (S25) ad Eq. (S26). Differet crossover poits of the cotributios are foud for differet Poisso s ratios, also as predicted by Eq. (S24). Figure S4 FEM simulatio results of relative cotributio of stress (filled markers, solid lie) ad geometry (hollow markers, dashed lie) o Ω. esults give as a fuctio of ratio b/h for ν = 0, 0.25 ad he crossover regios are highlighted by large rectagles ad are located at b/h ~ SI-15

16 VII. efereces S1. imosheko, S. & Woiowsky-Krieger, S., heory of plates ad shells. 2d ed. Egieerig societies moographs. 1959, New York: McGraw-Hill S2. Lachut, M.J. & Sader, J.E. Effect of surface stress o the stiffess of catilever plates. Phys. ev. Lett. 99, (2007). S3. imosheko, S. & Goodier, J.N., heory of elasticity. 3d ed. Egieerig societies moographs. 1969, New York,: McGraw-Hill. xxiv, 567 p. S4. Lachut, M.J. & Sader, J.E. Effect of surface stress o the stiffess of thi elastic plates ad beams. Phys. ev. B 85, (2012). S5. Lu, P., Lee, H.P., Lu, C. & O'Shea, S.J. Surface stress effects o the resoace properties of catilever sesors. Phys. ev. B 72, (2005). S6. Gurti, M.E., Markescoff, X. & hursto,.n. Effect of Surface Stress o Natural Frequecy of hi Crystals. Appl. Phys. Lett. 29, (1976). S7. McFarlad, A.W., Poggi, M.A., Doyle, M.J., Bottomley, L.A. & Colto, J.S. Ifluece of surface stress o the resoace behavior of microcatilevers. Appl. Phys. Lett. 87, (2005). S8. Lagowski, J., Gatos, H.C. & Sproles, E.S. Surface Stress ad Normal Mode of Vibratio of hi Crystals - Gaas. Appl. Phys. Lett. 26, (1975). S9. Hwag, K.S., Eom, K., Lee, J.H., Chu, D.W., Cha, B.H., Yoo, D.S., Kim,.S. & Park, J.H. Domiat surface stress drive by biomolecular iteractios i the dyamical respose of aomechaical microcatilevers. Appl. Phys. Lett. 89, (2006). S10. Cheria, S. & hudat,. Determiatio of adsorptio-iduced variatio i the sprig costat of a microcatilever. Appl. Phys. Lett. 80, (2002). S11. Che, G.Y., hudat,., Wachter, E.A. & Warmack,.J. Adsorptio-Iduced Surface Stress ad Its Effects o esoace Frequecy of Microcatilevers. J. Appl. Phys. 77, (1995). S12. Lee, J.H., Kim,.S. & Yoo, K.H. Effect of mass ad stress o resoat frequecy shift of fuctioalized Pb(Zr0.52i0.48)O-3 thi film microcatilever for the detectio of C-reactive protei. Appl. Phys. Lett. 84, (2004). S13. Carr, D.W. & Craighead, H.G. Fabricatio of aoelectromechaical systems i sigle crystal silico usig silico o isulator substrates ad electro beam lithography. J. Vac. Sci. echol. B 15, (1997). S14. amayo, J., Pii, V., Gil-Satos, E., amos, D., Kosaka, P., og, H.D., va ij, C. & Calleja, M. Sheddig Light o Axial Stress Effect o esoace Frequecies of Naocatilevers. Acs Nao 5, (2011). SI-16