CTN 2/23/16. EE 247B/ME 218: Introduction to MEMS Design Lecture 11m2: Mechanics of Materials. Copyright 2016 Regents of the University of California

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1 Vlume Change fr a Unaxal Stress Istrpc lastcty n 3D Istrpc = same n all drectns The cmplete stress-stran relatns fr an strpc elastc Stresses actng n a dfferental vlume element sld n 3D: (.e., a generalzed Hke s Law) x y z y y z x z z x y x xy yz zx xy yz zx Bascally, add n ff-axs strans frm nrmal stresses n ther drectns C245: Intrductn t MMS Desgn 2 Imprtant Case: Plane Stress C245: Intrductn t MMS Desgn 3 Imprtant Case: Plane Stress (cnt.) Cmmn case: very thn flm catng a thn, relatvely rgd Symmetry n the xy-plane x = y = Thus, the n-plane stran cmpnents are: x = y = substrate (e.g., a slcn wafer) where x ( )[ ] [ ( )] and where At regns mre than 3 thcknesses frm edges, the tp Baxal Mdulus = surface s stress-free z = 0 et tw cmpnents f n-plane stress: x ( )[ x ( y 0)] y ( )[ y ( x 0)] C245: Intrductn t MMS Desgn 4 Cpyrght 206 Regents f the Unversty f Calfrna C245: Intrductn t MMS Desgn 5

2 dge Regn f a Tensle ( >0) Flm Net nn-zer nplane frce (that we just analyzed) At free edge, n-plane frce must be zer Lnear Thermal xpansn As temperature ncreases, mst slds expand n vlume Defntn: lnear thermal expansn ceffcent Flm must be bent back, here There s n Pssn cntractn, s the flm s slghtly thcker, here Dscntnuty f stress at the attached crner stress cncentratn C245: Intrductn t MMS Desgn = T d x dt [Kelvn-] Remarks: T values tend t be n the 0-6 t 0-7 range Can capture the 0-6 by usng dmensns f stran/k, where 0-6 K- = stran/k In 3D, get vlume thermal V expansn ceffcent 3 T T V Peel frces that can peel the flm ff the surface Lnear thermal expansn ceffcent Fr mderate temperature excursns, T can be treated as a cnstant f the materal, but n actualty, t s a functn f temperature 6 T As a Functn f Temperature C245: Intrductn t MMS Desgn 7 Thn-Flm Thermal Stress Thn Flm ( Tf) Substrate much thcker than thn flm Slcn Substrate ( Ts = 2.8 x 0-6 K-) Assume flm s depsted stress-free at a temperature Tr, then the whle thng s cled t rm temperature Tr Substrate much thcker than thn flm substrate dctates the amunt f cntractn fr bth t and the thn flm [Madu, Fundamentals f Mcrfabrcatn, CRC Press, 998] Cubc symmetry mples that s ndependent f drectn C245: Intrductn t MMS Desgn 8 Cpyrght 206 Regents f the Unversty f Calfrna C245: Intrductn t MMS Desgn 9 2

3 Lnear Thermal xpansn MMS Materal Prpertes C245: Intrductn t MMS Desgn 20 C245: Intrductn t MMS Desgn 2 Materal Prpertes fr MMS Yung s Mdulus Versus Densty (/) s acustc velcty Lnes f cnstant acustc velcty [Mark Spearng, MIT] C245: Intrductn t MMS Desgn 22 [Ashby, Mechancs f Materals, Pergamn, 992] C245: Intrductn t MMS Desgn 23 Cpyrght 206 Regents f the Unversty f Calfrna 3

4 Yeld Strength Yeld Strength (cnt.) Defntn: the stress at whch a materal experences sgnfcant plastc defrmatn (defned at 0.2% ffset pt.) Belw the yeld pnt: materal defrms elastcally returns t ts rgnal shape when the appled stress s remved Beynd the yeld pnt: sme fractn f the defrmatn s permanent and nn-reversble Yeld Strength: defned at 0.2% ffset pt. Belw: typcal stress vs. stran curves fr brttle (e.g., S) and ductle (e.g. steel) materals Tensle Strength Stress Fracture T=30C) lastc Lmt: stress at whch permanent defrmatn begns Brttle (S) Prprtnal Lmt Ductle (Mld Steel) (r T>900C) Prprtnalty Lmt: pnt at whch curve ges nnlnear True lastc Lmt: lwest stress at whch dslcatns mve C245: Intrductn t MMS Desgn [Maluf] Stran 24 Yung s Mdulus and Useful Strength C245: Intrductn t MMS Desgn 25 Yung s Mdulus Versus Strength Lnes f cnstant maxmum stran [Ashby, Mechancs f Materals, Pergamn, 992] C245: Intrductn t MMS Desgn 26 Cpyrght 206 Regents f the Unversty f Calfrna C245: Intrductn t MMS Desgn 27 4

5 v Clamped-Clamped Beam Resnatr Qualty Factr (r Q) C245: Intrductn t MMS Desgn 28 f W r Frequency: Stffness Resnatr Beam lectrde 2 Mass v kr m r L r Yung s Mdulus.03 Densty h L 2 r (e.g., m r = r kg) kg) C245: Intrductn t MMS Desgn 29 V P V P h C(t) v Q ~0,000 V dc dt Smaller mass hgher freq. freq. range and and lwer seres R x x P Nte: If If V P = P 0V 0V devce ff ff Measure f the frequency selectvty f a tuned crcut Defntn: xample: seres LCR crcut xample: parallel LCR crcut Qualty Factr (r Q) Ttal nergy Per Cycle f Q nergy Lst Per Cycle BW C245: Intrductn t MMS Desgn 30 3dB Im Q Re Im Q Re Z L Z R CR Y Y f C L BW -3dB f Selectve Lw-Lss Flters: Need Q In resnatr-based flters: hgh tank Q lw nsertn lss At rght: a 0.% bandwdth, 3- res Hz (smulated) heavy nsertn lss fr Frequency [MHz] C245: Intrductn t MMS Desgn 3 Transmssn [db] resnatr Q < 0, Increasng Insertn Lss Tank Q 30,000 20,000 0,000 5,000 4, Cpyrght 206 Regents f the Unversty f Calfrna 5