445. Investigation of Thermo-Elastic Damping of Vibrations of Rectangular and Ring-Shaped MEMS Resonators

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1 . Investigation of Thermo-Elastic Damping of Vibrations of Rectangular and Ring-Shaped MEMS Resonators R. Barauskas,a, S. Kausinis,b, H. A. C. Tilmans Kaunas Universit of Technolog, Studentu -7, LT-68, Kaunas, Lithuania Interuniversit Microelectronics Center, Kapeldreef 7, B-, Leuven, Belgium, a rimantas.barauskas@ktu.lt; b saulius.kausinis@ktu.lt; Harrie.Tilmans@imec.be (Received Januar 9; accepted March 9) Abstract. The paper deals with finite element analsis of damped modal vibrations Q-factor values determined b taking into account the thermall-elastic damping. Modal properties of square- and ring-shaped MEMS resonators have been investigated b taking into account the laered structure of MEMS and the influence of the geometr of the clamping one. The calculations have been performed b emploing the COMSOL Multiphsics finite element software. The solution method has been verified comparing numericall and analiticall obtained damped modal properties of cantilever MEMS resonator. Eperimental investigations of Q-factor values have been performed. The comparison of calculated and eperimentall obtained resonant frequencies and Q-factor values indicated good agreement of tendencies of change of the quantities against temperature. Kewords: thermall-elastic damping, finite elements, MEMS, resonator. Introduction The scope of MEMS is epanding rapidl. Besides traditional MEMS, man new frontiers of practice were opened up in recent ears, []. The most advanced solutions in terms of MEMS sensors, actuators, mechanical filters or microfluidic sstems can be identified in inertial navigation sstems, computer devices, industrial process control, electronics instrumentation, telecommunications as well as in biological and medical applications. Advanced MEMS applications are driven b processes that enable greater functionalit and operational capabilities through higher levels of electronic-mechanical integration and greater numbers of mechanical components working either alone or together to enable a comple action. To achieve these goals, the was are being eplored for reducing the power dissipation of both mechanical microstructures and electronical components b using innovative combinations of low-power circuits and micromechanical devices, []. A number of scientific and engineering challenges must be overcome in order to realie the vision for nano-micro-macro integration. Along with the development of new technologies, new device configurations and new applications for micro-sensors, micro-actuators and micro-sstems, a growing need for achievement of better MEMS performance and for novel sensing capabilities can be observed. In man applications, the benefits of using resonator microstructures relate directl to small sie, high frequenc and spectral purit. The latter quantit is defined b high values of the mechanical Q-factor. The damping rate of the vibrations can be evaluated as Q, which is the rate at which energ of resonant vibrations is being lost to various environments coupled to the 77 VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 9 MARCH, VOLUME, ISSUE, ISSN 9-876

2 model. The vibrations of real resonator structures are affected b several different energ loss mechanisms, which predetermine the overall value of the Q-factor. Unfortunatel, the phsical mechanisms of dissipation affecting the Q-factor values of micro-resonators are still not completel understood. It appears as ver important to stud different energ loss mechanisms and to perceive the dominating energ dissipation processes. Several studies, [-], have eamined the different dissipation mechanisms, as well as, the dependence of the Q-factor on various parameters in both fleural and longitudinal vibration modes of MEMS structures. In particular, the Q-factor determined b thermal-elastic damping (TED) of micro-electromechanical resonators structures is a ver important dnamic characteristic since it provides the upper limit of the Q-factor that is possible to achieve in a structure of a given geometr and materials under an assumption that no internal friction and other sources of damping are present. In [] the TED of the single-crstal C-SiC UHF nano-mechanical rod resonators at longitudinal vibration mode have been investigated and Q-factor values caused b TED at the thermal insulation boundar conditions, as well as, for the fied temperature boundar conditions determined. The results have been emploed for development and verification of FEM computational models in []. In [] an etensive stud of vibration energ loss mechanisms, which limit the highest achievable Q-factor values, has been presented. The damping mechanisms are predetermined b material properties and are highl influenced b the design of the resonator structures. The include energ losses mechanisms due to support clamping, air damping, heating, TED dissipation, anharmonic mode coupling, surface roughness, etrinsic noise, dislocations and dissipation due to two-level sstems. Dependence of dissipation on the finite sie of the structure, temperature, surface-aggregated defects, magnetic field, hsteresis and other factors has been analed. The data on dissipation measurements have been presented for micron-sied single crstal GaAs and Si resonators. In [6] intrinsic and etrinsic energ loss mechanisms have been discussed and dissipation in polcrstalline diamond (pol-c) resonators has been eplored b using electrostatic and pieoelectric actuation methods. Due to material limitations of pol-si resonators, polcrstalline diamond (pol-c) has been eplored as a new MEMS resonator material. The pol-c resonators were designed, fabricated and tested. An effort to delineate the microscopic mechanisms predominantl responsible for dissipation in micromechanical resonators has been presented in [7]. Possible mechanisms contributing to dissipation in a double-clamped beam ultra-high frequenc (UHF) nanoresonators have been analed in [8]. Estimation of the dissipation contributed b the evaporated metallic laers (Al and Ti) with internal friction has been presented in [9]. Although numerous approaches to the damping theor and eperiments have been sstematicall developed and reported over several decades, advanced simulation and modelling tools are still needed in order to provide advisor service at earl stages of MEMS design. The numerical simulation is important for two reasons. First, it can be used to verif the analtical results, which often are obtained on the base of highl simplified models and then further used in order to eplain similar phenomena in more comple MEMS configurations as well. The same model investigated numericall and analticall provides the base for the verification of the numerical model. Second, for comple geometr and multiple materials there are no analtical approimations, therefore numerical simulation is necessar. This work presents the stud of TED as a dissipation mechanism in modal damping of vibrations of rectangular- and ring-shaped MEMS resonators. Much of the work presented is numerical eperiments has been performed to determining resonant frequencies and Q-factors of MEMS and providing phsical interpretations of the results obtained. In addition, several real 78 VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 9 MARCH, VOLUME, ISSUE, ISSN 9-876

3 phsical eperiments have been carried out, which demonstrated the reasonable agreement of computationall determined features to the realit.. FEM analsis of rectangular and ring-shaped resonators A D computational model has been developed, which provides an approimation of the TED of structural vibrations of rectangular- and ring-shaped MEMS resonators. The model is based on the theoretical consideration developed in []. The laouts of two tpes of resonators are presented in Fig. and Fig.. The model in COMSOL Multiphsics environment has been created as a coupled two-field Multiphsics problem. Two application modes have been coupled: MEMS Module-> Structural Mechanics->Plane Stress->Damped eigenfrequenc analsis COMSOL Multiphsics->Heat Transfer ->Conduction->Transient analsis. The thermal-mechanical coupling has been ensured b entering the bod heat source, which described the generation of the heat in the volume at given strain rates as Tκ c uɺ vɺ wɺ + +, and the thermal epansion strain κ( T T ), which was taken into ϑ account in the mechanical equations of the model. Here u(,, ), v(,, ), w(,, ) are displacements of the MEMS resonator structure at location(,, ), c - Young s modulus, ϑ - Poisson s ratio, κ - thermal epansion coefficient, T - ambient temperature. The model is able to take into account the TED effect and the influence of the geometr of the clamping one. Smmetric modes of rectangular- and ring-shaped resonators have been calculated b using quarter-smmetr models, Fig. and Fig.. The mechanicall free surfaces of the resonator are assumed to be thermall isolated basing on the fact that the heat echange rate through the surface is too slow to be comparable with quick internal processes of acquiring and loosing heat caused b high rates of elastic strain. On the contrar, fied temperature boundar conditions as T = T have been imposed on cut boundaries, which represent the contact of the resonator with the overall MEMS structure. Several in-plane (Figs.-6 and Figs.-6) and out-of-plane (Figs.7- and Figs.7-) quarter-smmetr modal shapes are presented. Fig.. The laout of the rectangular resonator Fig.. The quarter smmetr model of the rectangular resonator VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 9 MARCH, VOLUME, ISSUE, ISSN

4 Fig.. The laout of the ring-shaped resonator Fig.. The quarter smmetr model of the ring-shaped Resonator The figures do depict the contours of the temperature distribution over the vibrating structure caused b the thermal-elastic coupling (Fig., 7, 9,,,, 7, 9) as well as, the contours of von-misses stress (Fig 6, 8,,,, 6, 8, ). It could be mentioned that different modes of the investigated resonators ma possess ver different values of the ratio Q/f (Q-factor over modal frequenc). The ratio (which has the time dimension) ma be reasonabl emploed for the evaluation of the level of TED of differentl shaped modes of resonators of different geometries, if the ecitation frequenc value is predetermined VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 9 MARCH, VOLUME, ISSUE, ISSN Fig.. Distribution of temperature at st in-plane Fig. 6. Distribution of von Misses stress at quarter-smmetr modal shape of the rectangular st in-plane quarter-smmetr mode vibration of resonator presented in Fig., the rectangular resonator presented in Fig., f= H; Q=7 66, Q/f=.7s. f= H; Q=7 66, Q/f=.7s. The necessar value of modal frequenc can be ensured b selecting proper sie of the MEMS resonator, and the ratio Q/f is independent of the resonator sie as the investigated MEMS sstem is linear in the contet of the investigated models. The ver different values of ratio Q/f of the modes, which are quite similar at the first sight, ma be eplained on the base of strained state corresponding to the modal shape. As a rule, higher TED levels possess the modes with dominating longitudinal strain components, for eample, see rectangular resonators presented in Figs. and. Shear and bending strain

5 dominated modes have considerabl smaller TED damping ratios, see Figs. and 6, where the necks supporting the rectangular resonator on the main bod of the MEMS eperience mainl bending and shear strains, and the resonator is predominantl sheared. Naturall, such eplanations do not pretend to be complete and ehaustive, as volumetric strained state in a solid alwas is quite comple and alwas possesses all strain components. The eamples of two in-plane modes of ring resonators do not epose such noticable differences in Q/f ratio values as could be observed in the case of rectangular resonators. e Fig. 7. Distribution of temperature at nd in-plane Fig. 8. Distribution of von Misses stress at nd quarter-smmetr mode vibration of the rectangular in-plane quarter-smmetr mode vibration of the resonator presented in Fig., rectangular resonator presented in Fig., f=86 6 H; Q= 9 66, Q/f=.s. f=86 6 H; Q= 9 66, Q/f=.s. Out of plane modes are dominated b bending strains and ehibit lower values of the Q/f ratio. For the mode of the rectangular resonator presented in Figs. 9 and we obtained the value such low as Q/f=.s., e-, e- -, - -, - Fig. 9. Distribution of temperature at out-of-plane Fig.. Distribution of von Mises stress at quarter-smmetr mode vibration of the rectangular out-of-plane quarter-smmetr mode vibration of resonator presented in Fig., the rectangular resonator presented in Fig., f= 7 7 H; Q= 9, Q/f=.77s. f= 7 7 H; Q= 9, Q/f=.77s. VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 9 MARCH, VOLUME, ISSUE, ISSN

6 . e e Fig.. Distribution of temperature at out-of-plane Fig.. Distribution of von Mises stress at quarter-smmetr mode vibration of the rectangular out-of-plane quarter-smmetr mode vibration of resonator presented in Fig., the rectangular resonator presented in Fig., f= 9 9 H; Q= 99, Q/f=.s. f= 9 9 H; Q= 99, Q/f=.s Fig.. Distribution of temperature at nd in Fig.. Distribution of vonmisses stress at st plane quarter-smmetr mode vibration of the in-plane quarter-smmetr mode vibration of the ring resonator presented in Fig., ring resonator presented in Fig., f= 6 7 H; Q=779, Q/f=.7s. f= 6 7 H; Q=779, Q/f=.7s VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 9 MARCH, VOLUME, ISSUE, ISSN Fig.. Distribution of temperature at nd Fig. 6. Distribution of vonmisses stress at nd in-plane quarter-smmetr mode vibration of the in-plane quarter-smmetr mode vibration of the ring resonator presented in Fig., ring resonator presented in Fig., f= H; Q= , Q/f=.6s. f= H; Q= , Q/f=.6s.

7 - -. e e-6 -. e Fig. 7. Distribution of temperature at out-of-plane Fig. 8. Distribution of temperature at out-of-plane quarter-smmetr mode vibration of the ring quarter-smmetr mode vibration of the ring resonator presented in Fig., resonator presented in Fig., f= 6 H; Q= 8, Q/f=.7s. f= 6 H; Q= 8, Q/f=.7s.. e- 6 e- e Fig. 9. Distribution of temperature at out-of-plane Fig.. Distribution of vonmises stress at outquarter-smmetr mode vibration of the ring of-plane quarter-smmetr mode vibration of resonator presented in Fig., the ring resonator presented in Fig., f= H; Q= 796, Q/f=.7s f= H; Q= 796, Q/f=.7s. Model verification and validation A significant part of an simulation is the verification and validation of the model. It is necessar to be sure that the model design (conceptual model) is transformed into a computer model with sufficient level of adequac and accurac. In this stud, a sample model of a cantilever beam resonator (Fig. ) has been investigated numericall and eperimentall. Three modifications of the sample cantilever resonator numerical model have been eplored: VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 9 MARCH, VOLUME, ISSUE, ISSN

8 a. real clamping conditions and the presence of the SiO substrate taken into account as in Fig.. Full clamping and prescribed temperature (ambient temperature T ) boundar conditions imposed for all cut-boundaries that separate the computational model form the remaining bod of the MEMS. The model houses the resonating structure and a part of the surrounding domain. The mechanicall free surfaces of the model are assumed to be thermall isolated; b. the cantilever resonator presented without the surrounding material. One end of the resonator is full clamped, and fied temperature boundar conditions imposed at this end; c. The same model as in (b), with the clamped end thermall isolated. Fig.. Geometr and finite element discretiation of the sample model of cantilever beam resonator, b= m; h=. m; H =. m Table presents the first modal frequenc and Q-factor values of the three models of the Pol-Si cantilever resonator at three different values of its length. Model modification (a) is close to the situations observed in a real MEMS. However, analtical formulae are able to evaluate the situations b and c onl, therefore the are used for comparisson of the results and model verification. Table. Modal frequenc and Q-factor values of the sample cantilever resonator Resonator length L (m) Model modification a Model modification b Model modification c L=. f = 8. MH; f =. MH; f =. MH; L= L= Q=.7 f = MH; Q=. f =.8 MH; Q=.6 Q=.7 f =.8 MH; Q=.6 f =.6 MH; Q=.76 Q=.7 f =.8 MH; Q=.6 f =.6 MH; Q=.9 8 VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 9 MARCH, VOLUME, ISSUE, ISSN 9-876

9 The thermal boundar conditions influence the TED effects (i.e., the Q-factor values). Ideall clamped resonator models as in modifications (b) and (c) produce increased natural frequenc values compared with more adequate model (a). Fig. demonstrates the comparisson of the first modal frequenc and Q-factor values obtained b using FE models against analtical evaluations. The model parameters have been matched to the design data of pol-si and pol-c resonators given in [6]. The resonators were fabricated and tested b using pieoelectric and electrostatic actuation methods in low and high vacuum. Fig., presents the dependencies of the Q-factor values determined b different vibration energ loss mechanisms against the resonant frequencies of pol-si and pol-c µm thick cantilever beam resonators. Fig.. Dependencies of Q-factor values determined b different vibration energ loss mechanisms against the resonant frequencies of pol-si and pol-c cantilever resonators, [6]. Red dots indicate the numerical results obtained in this work b FE analsis for Pol-Si cantilever resonators of length 9 m; m, and 6, m The simulation results have been validated eperimentall. Series of two tpes of structures on the wafer cantelivers and double-clamped beam resonators square- and ring-shaped ones were designed and manufactured for eperimental characterisation. The resonator structure used in this stud was a cantilever beam of length 6 µm and width 6 µm. The gap between resonators and electrodes is around 7 nm (designed). The ecitation and readout mechanism was capacitive. The dependence of the numericall obtained values of the Q-factor of st damped mode against temperature is depicted in Fig.. The measurements for the validation purposes were performed at the MEMS testing laborator at the Interuniversit Microelectronics Center (IMEC). The aim of the eperiments was obtaining information concerning the effect of temperature on both the modal frequenc and Q-factor. The resonator was tested in the vacuum set-up over the range of reference temperatures and bias voltages. The pressure in the vacuum chamber was kept constant at a VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 9 MARCH, VOLUME, ISSUE, ISSN

10 pressure of.6 e- mbar. In the eperiments conducted the temperature was varied between - o C and o C in steps of K. The resonance frequenc and the Q-factor values were etracted from the recorded data, and the Q-factor/resonance frequenc dependenc on temperature was developed. The results of the etraction procedure are presented in Fig.. It can be admitted that the qualitative agreement of numerical and eperimental results is good (about drop of the Q-factor value over temperature range of o C in both cases). However, the eperimentall obtained Q-factor value is about 7 higher over all temperature range. It seems reasonable to conclude that the difference of values is caused b other damping mechanisms not represented b purel TED model. Simultaneousl, the obtained values proove the major signifficance of TED during vibrations of MEMS resonators, as the value of the Q-factor caused b TED is approimatel 8- against -8, which is obtained if simultaneousl other energ dissipation mechanisms are taken into account. Fig.. The dependence of numericall obtained values of the Q-factor of the st damped mode against temperature Fig.. The eperimentall obtained dependence of the st damped modal frequenc and Q-factor against temperature 86 VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 9 MARCH, VOLUME, ISSUE, ISSN 9-876

11 Eperimental studies focusing on energ loss mechanisms concerned are under wa and will be the subject of subsequent publications. Conclusion The finite element modelling of thermoelastic modal damping (TED) of MEMS vibrations developed in [] has been etended and applied to the analsis of rectangular- and ring-shaped resonators. The sample models have been verified b peroforming comparissons against analtical calculations and b corresponding eperimental investigations. The comparison of calculated and eperimentall obtained resonant frequencies and Q-factor values indicated a good agreement of tendencies of change of the quantities against temperature. Both the eperimentsand calculations revealed an almost linear decrease of the Q-factor against temperature. However, all eperimental Q- factor values were lower than theoretical ones. The shift of the values could be eplained b other mechanisms of damping, which are not included into the thermal-elastic damping model. Their influence can be quantitativel evaluated b the said difference of the values. Acknowledgement The research is sponsored b NATO RTO and Lithuanian State Science and Studies Foundation. References [] Cornelius T. Leondes, Ed., MEMS/NEMS Handbook: Techniques and Applications, Vol. -, Springer Science, 6. [] Wise K. D., Najafi K., The Engineering Research Center for Wireless Integrated MicroSstems (WIMS) Annual Report 7, Universit of Michigan, Michigan State Universit, Michigan Technological Universit, [] Barauskas R., Kausinis, S., De Wolf I., Stoffels S., Tilmans H. A. C., Finite element analsis of thermo-elastical modal damping of MEMS vibrations // Journal of Vibroengineering / Vibromechanika, Lithuanian Academ of Sciences, Kaunas Universit of Technolog, Vilnius Gediminas Technical Universit. ISSN , Vol. 9, no.. p. -. [] Zhang W., Kimberl L. Turner, Thermoelastic Damping in the Longitudinal Vibration: Analsis and Simulation, Proceedings of IMECE, ASME International Mechanical Engineering Congress and Eposition, November -,, Anaheim, California USA. [] Mohant P., Harrington D. A., Ekinci K. L., Yang Y. T., Murph M. J., and Roukes M. L. Intrinsic dissipation in high-frequenc micromechanical resonators, PHYSICAL REVIEW B 66, 86,. [6] Sepulveda N., Aslam D., Sullivan J.P., Polcrstalline diamond MEMS resonator technolog for sensor applications, Diamond & Related Materials (6) 98 [7] Harrington D. A., Mohant P., Roukes M. L., Energ dissipation in suspended micromechanical resonators at low temperatures, Phsica B 8}88 () -6 [8] Ekinci K. L., Roukes M. L., Nanoelectromechanical sstems, Review of Scientific Instruments, 76, 6 () [9] Feng X. L., Zorman C. A., Mehregan M., and Roukes M. L., Dissipation in Single-Crstal C- SIC Ultra-High Frequenc Nanomechanical Resonators, Tech. Digest, Solid-State Sensors, Actuators, and Microsstems Workshop (Hilton Head 6), pp [] Clarence Zener, Phsical Review.V-,(97-98). VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 9 MARCH, VOLUME, ISSUE, ISSN