Polarimetric analysis of stress anisotropy in nanomechanical silicon nitride resonators
Thibault Capelle, Yeghishe Tsaturyan, Andreas Barg, Albert Schliesser
PPolarimetric analysis of stress anisotropy in nanomechanical silicon nitride resonators
T. Capelle, ∗ Y. Tsaturyan, A. Barg, and A. Schliesser † Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark (Dated: September 12, 2018)We realise a circular gray-field polariscope to image stress-induced birefringence in thin (sub-micron thick) silicon nitride (SiN) membranes and strings. This enables quantitative mappingof the orientation of principal stresses and stress anisotropy, complementary to, and in agree-ment with, finite element modeling (FEM). Furthermore, using a sample with a well known stressanisotropy, we extract a new value for the photoelastic (Brewster) coefficient of silicon nitride, C ≈ (3 . ± . × − MPa − . We explore possible applications of the method to analyse andquality-control stressed membranes with phononic crystal patterns. Silicon nitride membranes and strings under high ten-sile stress have excellent mechanical and optical prop-erties [1, 2], making them a widely used platformto study the behaviour of mechanical systems in thequantum regime [3–10]. Recently, further enhance-ment of these properties through in-plane patterninghas been explored. Examples include one-[11] and two-dimensional[12–14] subwavelength optical grating andphotonic crystal structures which can boost the reflec-tivity of SiN beyond 99 . Q · f -products be-yond 10 Hz.In all instances of patterning, the stress relaxes to anew equilibrium distribution according to the patternboundary conditions. This changes dramatically the me-chanical, and potentially, via photoelastic coupling, op-tical properties of the structure. In absence of a labo-ratory diagnostic instrument, it has so far been neces-sary to rely on FEM to simulate the stress redistribu-tion. In addition, little is known about the photoelasticcoupling in silicon nitride [21, 22]. To address these defi-ciencies, we have realised a highly sensitive polarimetricsetup which allows quantitative imaging of stress-inducedbirefringence.Among the numerous possibilities to implement animaging polarimeter (or polariscope)[23], we have chosento build a circular gray-field polariscope. Its basic idea isto illuminate the sample with circularly polarised light,and analyse the ellipticity of the transmitted beam’s po-larisation in a spatially resolved manner. This approachhas a decisive advantage over plane polariscopes work-ing with linear polarisation when it comes to measuringsmall optical retardation δ : as we will demonstrate, inthe circular polariscope the optical signal is ∝ δ , whereas ∗ Also at Laboratoire Kastler Brossel, Paris. † E-mail: [email protected] in the plane polariscope it is only ∝ δ . In our setup(Fig. 1 and Tab. I), we use a light emitting diode (Thor-labs M780LP1) as a light source, followed by a bandpassfilter that eases the requirements on achromaticity of thesubsequent optical elements. The source is imaged on adiffuser which, together with an aspheric condenser lens,provides a K¨ohler-like illumination of the sample. Be-fore reaching the sample, the circular polarisation stateis defined by a high-contrast polariser and a quarter-waveplate( λ/ LED central wavelength (nm) 780Bandpass filter bandwidth (nm) 10Working distance of the microscope objective (mm) 37Focal length of the imaging lens (mm) 400Pixel size ( µ m) 3.75 × × It is straightforward to compute the expected signalvia Jones calculus. Each area element of the sample canbe treated as a general retarder described by the Jonesmatrix S δ,θ = R − θ · (cid:32) e − iδ/ e + iδ/ (cid:33) · R θ , (1)where θ is the azimuthal angle of the polarisation eigen-state basis with respect to a fixed laboratory reference, δ is the retardation phase, and R θ = (cid:32) cos( θ ) − sin( θ )sin( θ ) cos( θ ) (cid:33) (2) a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b FIG. 1. Gray field polariscope setup. A stepper motor rotates the half wave plate. See text for more details. is the canonical rotation matrix. Specifically, in the caseof stress birefringence of a SiN membrane, θ denotes thedirection of the principal stress in the membrane plane,and δ is the retardation induced by stress anisotropy, δ = 2 πλ C l · ∆ σ, (3)where λ is the wavelength of the light source, ∆ σ thestress anisotropy, l the thickness of the sample and C the Brewster coefficient, which is a material constant.The other waveplates can easily be represented by similarmatrices, setting δ = π/ δ = π for half waveplates. The analyzer at the end projectsthe polarisation state, as described by the matrix P = (cid:32) (cid:33) . (4)If we introduce α as the angle of the output half wave-plate with respect to the output polariser, the intensityat each camera pixel is given by I gray ∝ (cid:126)J † out · (cid:126)J out , with (cid:126)J out = P · S π,α · S δ,θ · (cid:126)J in , (5)assuming, for simplicity, a perfect input polarisation (cid:126)J in = (1 , − i ) T / √ I gray the Fourier transform of theoutput signal with respect to time t , during which thehalf-wave plate rotates with constant angular velocity ω ≡ ∂α∂t . Then it is easy to show that˜ I gray (4 ω )˜ I gray (0) = 12 ie − iθ sin( δ ) . (6)We can thus extract the entities of interest by calculating δ = arcsin (cid:32) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ I gray (4 ω )˜ I gray (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) (7)and θ = −
12 arg (cid:32) ˜ I gray (4 ω )˜ I gray (0) (cid:33) + θ . (8)Here, θ is an offset angle, given by the angle of the firstpolariser plus π/
4, which corresponds to the position of the quarter waveplate. Indeed, there is no phase shiftwhen the axis of the quarter waveplate is aligned withthe principal axis of the sample.The data are acquired and processed as follows: im-ages are taken at 10 angular positions for one full ro-tation. At each position we acquire a large number offrames (typically 150), from which a frame acquired withthe shutter closed is subtracted. This is done in order toremove the dark current, which corresponds to an offsetdue to the thermal electrons detected by the CCD cam-era. The procedure is repeated without the sample toacquire background images. For each image, treated asa two-dimensional matrix, we calculate the normalizedquantity ˜ I gray (4 ω ) / ˜ I gray (0). Finally, the background issubtracted from the measurements involving the sample.This yields a complex matrix, which is translated into theorientation θ and retardation δ following eqs. (7) and (8).For initial validation, we performed a measurement ona sample with a particularly simple geometry, namelya 210 nm thick silicon nitride ribbon. We used stan-dard nanofabrication techniques to realise this sample,starting with low-pressure chemical vapor deposition(LPCVD) of 210 nm stoichiometric silicon nitride on a500 µ m double-side polished silicon wafer. The chosendeposition parameters create a film with an isotropic ten-sile stress of ca. 1 . FIG. 2. Analysis of a SiN ribbon. a) CCD photograph ofthe sample, ribbon shows dark gray. b) Measured retardation δ c) Simulated stress anisotropy ∆ σ d) Measured angle θ ofthe optical axis e) Simulated angle of principal stress. ribbon in its center, while close to the clamp it rotatesby π/
4, preserving the symmetry of the structure.The good agreement prompts us to apply this ap-proach to measure the Brewster coefficient, relating stressanisotropy with birefringence, of SiN, for which quanti-tative data is scarce [21, 22]. To that end we measureda second ribbon, whose initial stress was carefully de-termined to be σ = 1190 MPa before releasing of theribbon[24]. In order to take into account the change ofthe boundary conditions during the release of the ribbon,we have to correct this value by a factor 1 − ν , where ν is the Poisson ratio of the material, assumed here to beequal to 0 .
27. The result is presented in Figure 3. Weobtain a retardation of 6 . ± . ∼ × µ m -area in the central region of the ribbon.For a quantitative comparison, we correct this value forthe multiple reflections inside the film, which are not en-tirely negligible due to the relatively high ( n ≈ .
0) re-fractive index of SiN. To do so, we used a transfer matrixmodel for the complex transmittance of this sample[25],expanding it to first order in a small variation of re-fractive index. This yielded a correction parameter of η = 1 .
26 for the effective thickness of the sample. TheBrewster coefficient of SiN can then be evaluated as C = δ π ληl σ (1 − ν ) ≈ (3 . ± . × − MPa − , (9) FIG. 3. Retardation measurement on a sample with cal-ibrated stress. Shown is an average over all horizontal linecuts between the two red lines shown in the inset, where stressis approximately homogeneous. Inset shows full retardationimage. where l = 210 nm is the sample thickness. Remarkably,this is two orders of magnitude lower than the value pro-posed by Campillo et al. [26], but close to the value ofsilica ( C ≈ × − MPa − ) [27], another amorphoustransparent dielectric. This mistake arises from the factthat this previous work measured the refractive indexand the stress for different SiH Cl : NH gas flow ratiosduring the LPCVD process, and not the variation of therefractive index due to mechanical stress.With such calibration at hand, we proceed to apply-ing polarimetric stress analysis to more complicated res-onator structures. Membrane resonators with phononicbandgap shield are of particular interest. In the con-text of silicon nitride membranes it has been shown thatphononic crystal structures can suppress transmission ofvibrations [28, 29], resulting in suppression of dissipationthrough phonon tunneling [29, 30], whose avoidance hadpreviously required delicate and often unreliable clamp-ing techniques[2]. Patterning a phononic crystal struc-ture directly onto the membrane not only suppressesphonon tunneling losses, but also enhances the dilutionof internal losses dramatically, enabling an increase inthe quality factor by more than an order of magnitude[20], as compared to conventional membrane resonatorsembedded in silicon phononic crystal structures. Stressanalysis using polarimetry in such complex geometriescan be a reliable and simple tool to assess the periodic-ity of stress anisotropy as required for the formation of abandgap.To demonstrate this potential, we realised membranes( l ≈
210 nm) with a honeycomb pattern of ∼ µ m-diameter holes, fabricated using the same techniques asfor the ribbons. Such a patterned membrane exhibits abandgap as previously described[20]. Once again, we per-formed a measurement of the retardation, as describedabove, with the result shown in Figure 4. As expected,we observe an enhanced stress anisotropy in the tethers FIG. 4. Analysis of a patterned membrane. a) Retardationimage. b) Measured retardation along the dashed line in a),following a tether between two holes. Values are converted tostress anisotropy via the previously extracted Brewster coef-ficient, and corrected for multiple reflections in the film. (i.e. the narrow regions between the circular holes), seeFig. 4a. In particular, a cut along a tether reveals apeak, symmetric with respect to the center of the tether,and a maximum stress anisotropy of ∆ σ ∼ . (cid:38)
15 mrad) appears as dark blue rings around theholes (see Fig 4a). We attribute these to diffraction-related imaging artifacts. This is supported by the pres-ence of several weak concentric rings around the holes,likely due to higher diffraction orders. In addition, theobservation of these features already in single-shot im-ages rules out excentric rotation as their cause.In spite of such artifacts, it is straightforward to recog-nise defective membranes. Figure 5 shows retardationimages of an undamaged membrane and one with a bro-ken tether, as a comparison. Defects in the perforatedmembrane structure cause a redistribution of the stress.One could envision using this approach in assessing theoverall performance of the device (e.g. the quality fac-tor), by comparing the stress profile of the defective de-vice with simulations of an undamaged membrane res-onator.
FIG. 5. Comparison between an undamaged membrane (a)and one with a broken tether (b). Stress released in (b),and the broken periodicity can be clearly recognised, allowingidentification and localisation of the defect.
In conclusion, applied to 210 nm-thick membrane res-onators, our experimental setup achieves a ∼ . ∼ (200 µ m) area. Usingthe measured Brewster coefficient, this corresponds to aresolution in stress anisotropy of ∼
80 MPa. Thus weexpect the method to deliver relevant results also for ul-trathin ( l (cid:46)
50 nm) membranes of smaller dimensions.As an imaging modality, the present, very basic mi-croscopy setup achieves moderate transverse resolutionon the order of 10 µ m, impaired also by edge artifacts(likely diffraction-caused). It is nonetheless more thansufficient to assess the overall quality of patterned mem-brane resonators and localise defects. We have imple-mented a gray field polariscope optimised for extractingstress anisotropy in membrane resonators. We have val-idated the technique by comparing our measurement re-sults with FEM simulations, and performed an indepen-dent measurement of the Brewster coefficient, correctingan older value found in the litterature[26]. In itself, thisresult is of interest for optomechanical systems relyingon photoelastic coupling in silicon nitride [21, 22]. [1] B. M. Zwickl, W. E. Shanks, M. Jayich, C. Yang,C. Bleszynski Jayich, J. D. Thompson, and J. G. E.Harris, Applied Physics Letters , 103125 (2008).[2] D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble,Physical Review Letters , 207204 (2009).[3] T. P. Purdy, P.-L. Yu, R. W. Peterson, N. S. Kampel,and C. A. Regal, Physical Review X , 031012 (2013).[4] T. P. Purdy, R. W. Peterson, and C. Regal, , 1(2013).[5] X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. . Huebl,R. Gross, and T. J. Kippenberg, Nature Physics , 179(2013). [6] M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B.Shkarin, K. Borkje, S. M. Girvin, and J. G. E. Harris,Physical Review A , 061801(R) (2015).[7] D. J. Wilson, V. Sudhir, N. Piro, R. Schilling,A. Ghadimi, and T. J. Kippenberg, Nature , 325(2015).[8] R. Peterson, T. P. Purdy, N. Kampel, R. Andrews, P.-L.Yu, K. Lehnert, and C. Regal, Physical Review Letters , 063601 (2016), 1510.03911.[9] W. H. P. Nielsen, Y. Tsaturyan, C. B. Møller, E. S.Polzik, and A. Schliesser, PNAS , 62 (2016).[10] A. Noguchi, R. Yamazaki, M. Ataka, H. Fujita, Y. Tabuchi, T. Ishikawa, K. Usami, and Y. Nakamura,New journal of Physics (2016).[11] C. Stambaugh, H. Xu, U. Kemiktarak, J. Taylor, andJ. Lawall, Annalen der Physik , 81 (2015), 1407.1709.[12] X. Chen, C. Chardin, K. Makles, C. Ca¨er, S. Chua,R. Braive, I. Robert-Philip, T. Briant, P.-F. Co-hadon, A. Heidmann, T. Jacqmin, and S. Del´eglise,arXiv:1603.07200 (2016).[13] R. Norte, J. P. Moura, and S. Gr¨oblacher, Physical Re-view Letters , 147202 (2016).[14] S. Bernard, C. Reinhardt, V. Dumont, Y.-A. Peter, andJ. C. Sankey, Optics Letter , 5624 (2016).[15] D. Kleckner, B. Pepper, E. Jeffrey, P. Sonin, S. M. Thon,and D. Bouwmeester, Optics Express , 19708 (2011).[16] C. Reinhardt, T. M¨uller, A. Bourassa, and J. C. Sankey,Physical Review X , 021001 (2016).[17] M. J. Weaver, B. Pepper, F. Luna, F. M. Buters, H. J.Eerkens, G. Welker, B. Perock, K. Heeck, S. de Man, andD. Bouwmeester, Applied Physics Letters , 033501(2016).[18] A. H. Ghadimi, D. J. Wilson, and T. J. Kippenberg,arXiv:1603.01605 (2016).[19] A. Z. Barasheed, T. M¨uller, and J. C. Sankey,arXiv:1511.06193 (2015).[20] Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, arXiv:1603.07200 (2016).[21] K. Børkje and S. M. Girvin, New J. Phys. , 085016(2012).[22] K. E. Grutter, M. Davanco, and K. Srinivasan, IEEEJournal of Selected Topics in Quantum Electronics ,61 (2015).[23] A. Ajovalasit, G. Petrucci, and M. Scafidi, ExperimentalTechniques , 11 (2015).[24] This measurement was conducted on a separate samplewith a 238 . d and re-fractive index n is given by t = ine − ikd in cos( kdn )+(1+ n ) sin( nkd ) .[26] A. L. Campillo and J. W. P. Hsu, Journal of AppliedPhysics , 646 (2002).[27] R. Priestley, Proceedings of SPIE (2001).[28] Y. Tsaturyan, A. Barg, A. Simonsen, L. G. Villanueva,S. Schmid, A. Schliesser, and E. S. Polzik, Optics Ex-press , 6810 (2014).[29] P. L. Yu, K. Cicak, N. S. Kampel, Y. Tsaturyan, T. P.Purdy, R. W. Simmonds, and C. A. Regal, AppliedPhysics Letters (2014), 10.1063/1.4862031.[30] I. Wilson-Rae, Physical Review B - Condensed Mat-ter and Materials Physics77