Lecture 06 Materials Science for MEMS

Lecture 06 Materials Science for MEMS - Fundamentals of...

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Unformatted text preview: Fundamentals of Micromachining Dr. Bruce K. Gale BIOEN 6421 EL EN 5221 and 6221 ME EN 5960 and 6960 Basic Materials Science for MEMS • • • • • • • Basic material interactions Silicon as a material Crystallography Crystal defects and impurities Wafer manufacture Stress and strain Review of class projects Sensors and Actuators • “Sensor” (Latin sentire meaning “to perceive”) • “Transducer” (Latin transducere meaning “to lead across”) • A sensor performs a transducing action and the transducers must necessarily sense some physical or chemical signals • Types of signals: chemical, electrical, magnetic, mechanical, radiant, thermal Materials Overview • Metals – Characterized by metallic bonds • Polymers – Long chain molecules of repeating units • Ceramics – Inorganic compounds with ionic and covalent bonding • Others- glass (non-crystalline solids) and carbon Basic Atomic Interactions • Ionic • Crystals – Electrostatic bonding • Covalent – Electron sharing • Metallic – Organized, repeating 3-D pattern of molecules or atoms – Closely packed structure – Electron fluid or gas • Hydrogen – Ionic interactions between covalently bonded atoms • Van Der Waals – Shifting interactions between atoms Surface Properties • • • • Surfaces are uniquely reactive Surfaces are different from the bulk Surfaces are readily contaminated Surface material/ structure is mobile – Can change depending on environment • Surface structures or properties – – – – – Roughness Chemistry or molecules Inhomogenous surfaces Crystalline or disordered Hydrophobicity (wettability) • Contact angle Material Properties • Material failure – – – – – • Material properties Yield stress Ductile and brittle failure Plastic deformation Ultimate stress and strength Some fail in shear, compression, tension – Fatigue- failure under cyclic conditions though well below yield stress – Creep- time dependent extension • Stress relaxation – Toughness • Energy absorption to failure – Consistent numbers not always available – Variation in runs, machines, locations – Structures generally a laminate composite – Properties may be function of fabrication process or postprocessing – Measurement of key properties such as stress, Young’s modulus, strength, and Poisson’s ratio Surface Measurements • Contact angle • ESCA - Electron Spectroscopy for Chemical Analysis – Element identification and bonding state (XPS) • Auger Electron Spectroscopy • SIMS(Secondary Ion Mass Spectrometry) – Element ID, Low concentrations, Proteins • FTIR- ATR (Fourier Transform Infra Red) – Chemistry and Structure Orientation • STM- Scanning Tunneling Microscopy • SEM (Scanning Electron Microscopy) • AFM (Atomic Force Microscopy) Why Silicon? • Available technology (IC circuits) • Inexpensive • Compatible with existing semiconductor technology (easy integration) • Suitable for hybrid structures • Types: amorphous, polycrystalline, crystalline Wafer Flats Silicon Wafer Characteristics • • • • • • • • • • • Orientation (cleavage or fracture) Si cleaves between (111) planes, III-V separate on (110) Roughness Flatness Orientation of primary and secondary flat Type n or p Surface misorientation Si resistance Thickness Backside damage is induced if required Rounded wafer edge (significantly reduces edge chipping, wafer breakage, photoresist build up) Stress and Strain • • • • • • • • • Stresses are forces applied over areas Strain is a dimensional change due to an applied stress Axial stress and strain σ = F ε = ∆L A L0 Tension +, compression Hooke’s law- stress and strain proportional σ E= Young’s modulusε F τ Shear stress and strain τ= G= A γ Shear modulus- G, γ is an angle Poisson’s ratio- lateral distension for axial load ν= ε transverse =− t longitudinal εa Stress and Strain Relationships Silicon as a mechanical material for MEMS fabrication Miller Indices in Crystals • For a plane with: – – – – • classic reference in the field: x-axis intercept xo y-axis intercept yo z-axis intercept zo the Miller indices (hkl) for this plane are given by finding the inverses of xo, yo, & zo and reducing them to the smallest set of integers h: k: l having the same ratio (xo)-1: (yo)-1: (zo)-1. – K.E. Petersen "Silicon as a Mechanical Material", Proceedings of the IEEE, Vol. 70, No.5, May 1982. • http://robotics.eecs.berkeley.edu/~tahhan/MEMS/petersen/mems_ petersen.htm – tenants: • silicon is abundant, inexpensive, and can be produced in extremely high purity and perfection; • silicon processing based on very thin deposited films which are highly amenable to miniaturization; • definition and reproduction of the devices, shapes, and patterns, are performed using photographic techniques that have already proved capable of high precision; • silicon microelectronic (and therefore also mems) devices are batch-fabricated. • Conventions: – (hkl): single plane or set of all parallel planes. – ( h kl): for a plane that intercepts the x axis on the negative side of the origin. – {hkl}: for all planes of equivalent symmetry, such as {100} for (100), (010), (001), (100), (010), and (001) in cubic symmetry. – – [hkl]: for the direction perpendicular to the (hkl) plane. <hkl>: for a full set of equivalent directions. Dean P. Neikirk © 2001, last update January 29, 2001 Dept. of ECE, Univ. of Texas at Austin 11 Low Index Directions In Silicon (Cubic, Diamond Structure) • (100) Silicon crystal structure • valence 4 structure – each atom bonds to four neighbors in a tetragonal configuration • (110) • crystal lattice is face centered cubic (FCC), with two atom basis [at (0,0,0) and (1/4, 1/4, 1/4) ]: Zincblende • two “interpenetrating” FCC lattices – lattice constant “a”: cube side length 2a surface atomic density a ρ surface 100 2 silicon: ρ100 = 6.8 x 1014 atoms / cm2 Dean P. Neikirk © 2001, last update January 29, 2001 ρ surface = 110 12 3 a 4 – atomic density: (1atom+4 ⋅ [14atom]) = a • silicon (rm temp): 5.43 Å • nearest neighbor distance dn = (2atoms + 4 ⋅ [1 4 atom] + 2 ⋅ [1 2 atom]) 2 a ⋅a Dept. of ECE, Univ. of Texas at Austin • • • • 4 atoms inside cube 6 atoms “half” inside at face centers 8 atoms 1/8 inside at corners total of 8 atoms per cube: atomic density 8 / a3 Dean P. Neikirk © 2001, last update January 29, 2001 10 a ρSi = 5x1022 atoms/cm3 Dept. of ECE, Univ. of Texas at Austin Dopants and impurities in semiconductors (111) planes WebElements: the periodic table on the world-wide web http://www.shef.ac.uk/chemistry/web-elements/ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 hydrogen helium 1 3 11 element name 4 boron atomic number 9.012182(3) magnesium 1995 atomic weight (mean relative mass) K 37 Rb 85.4678(3) caesium 55 Ca Sc 40.078(4) strontium 38 39 Sr Cs Ba Y 87 88 vanadium chromium manganese 23 24 25 Ti V 40 Cr Mn 71 41 42 Zr Nb Mo * 72 73 Lu Hf Ta 74 103 104 cobalt 26 27 Fe Co nickel W 43 105 106 44 45 copper zinc 29 28 30 7 8 9 F 63.546(3) silver 46 47 65.39(2) cadmium 75 77 Re Os 107 78 79 48 80 Ir Pt Au Hg 190.23(3) hassium 108 12.0107(8) 14.00674(7) 15.9994(3)18.9984032( 20.1797(6) silicon phosphorus sulfur chlorine argon 14 15 16 P S 26.981538(2) 28.0855(3) 30.973761(2) 32.066(6) gallium germanium arsenic selenium 31 69.723(1) indium 32 33 34 49 72.61(2) tin 74.92160(2) 78.96(3) antimony tellurium 50 51 110 111 52 In Sn Sb Te 81 82 83 Tl Pb 192.217(3) 195.078(2) 196.96655(2) 200.59(2) 204.3833(2) meitnerium ununnilium unununium ununbium 109 (111) planes etch the slowest, tend to be cleavage planes Ne 101.07(2) 102.90550(2) 106.42(1) 107.8682(2) 112.411(8) 114.818(3) 118.710(7) 121.760(1) osmium iridium platinum gold mercury thallium lead bismuth 76 • 10 O Ni Cu Zn Ga Ge As Se Tc Ru Rh Pd Ag Cd [98.9063] rhenium 174.967(1) 178.49(2) 180.9479(1) 183.84(1) 186.207(1) lawrencium rutherfordiumdubnium seaborgium bohrium 89-102 iron 50.9415(1) 51.9961(6) 54.938049(9) 55.845(2) 58.933200(9) 58.6934(2) niobium molybdenum technetium ruthenium rhodium palladium 88.90585(2) 91.224(2) 92.90638(2) 95.94(1) lutetium hafnium tantalum tungsten 57-70 132.90545(2) 137.327(7) francium radium Fr Ra 22 44.955910(8) 47.867(1) yttrium zirconium 87.62(1) barium 56 titanium 4.002602(2) neon N 13 21 fluorine 6 Al Si scandium oxygen C 10.811(7) aluminium 12 20 nitrogen 5 Na Mg 39.0983(1) rubidium carbon B element symbol 22.989770(2) 24.3050(6) potassium calcium 19 He Key: beryllium Li Be 6.941(2) sodium 2 IV H 1.00794(7) lithium 18 17 18 Cl Ar 35.4527(9) bromine 35 39.948(1) krypton 36 Br Kr 79.904(1) iodine 53 I 83.80(1) xenon 84 Bi Po 85 (1 1 1) a tom 54 Xe atom (111) 127.60(3) 126.90447(3) 131.29(2) polonium astatine radon 86 e ic plan At Rn ic p lane 207.2(1) 208.98038(2) [208.9824] [209.9871] [222.0176] 112 ** Lr Rf Db Sg Bh Hs Mt Uun Uuu Uub [223.0197] [226.0254] [262.110] lanthanum 57 *lanthanides La cerium 58 Ce [261.1089] [262.1144] [263.1186] praseodymiu neodymium 59 60 **actinides Ac 90 61 samarium 62 Pr Nd Pm Sm 138.9055(2) 140.116(1) 140.90765(2) 144.24(3) actinium thorium protactinium uranium 89 promethium [264.12] 91 Th Pa 92 U [144.9127] 150.36(3) neptunium plutonium 93 94 [265.1306] 63 [269] [272] [277] gadolinium europium [268] terbium dysprosium holmium 64 65 66 67 Eu Gd Tb Dy Ho 151.964(1) americium 95 erbium 68 thulium ytterbium 69 70 Er Tm an (100) pl Yb e 157.25(3) 158.92534(2) 162.50(3) 164.93032(2) 167.26(3) 168.93421(2) 173.04(3) curium berkelium californiumeinsteinium fermium mendelevium nobelium 96 Np Pu Am Cm 97 98 99 100 101 102 Bk Cf Es Fm Md No [227.0277] 232.0381(1)231.03588(2)238.0289(1) [237.0482] [244.0642] [243.0614] [247.0703] [247.0703] [251.0796] [252.0830] [257.0951] [258.0984] [259.1011] • Symbols and names the symbols of the elements, their names, and their spellings are those recommended by IUPAC. After some controversy, the names of elements 101-109 are 69, 2471–2473. NamesPure & not been proposed as yet for the most r : now confirmed: see have Appl. Chem., 1997, elements 110–112 so those used here are IUPAC’s temporary systematic names: see Pure 51, Appl. Chem., the USA and some other countries, the spellings aluminum and cesium are normal while in the UK and elsewhere the usual spelling is sul & 381–384. In 1979, Periodic table organisation a justification of the positions of the elements La, Ac, Lu, and Lr in the WebElements periodic table see W.B. Jensen, “The positions of lanthanum (actinium) and lutetium (lawrencium) in the periodic table”, : for Group labels :the numeric system (1–18) used here is the current IUPAC convention. For a discussion of this and other common systems see: W.C. Fernelius and W.H. Powell, “Confusion in the periodic table of the elements”, J. Chem. Ed., 19 59, 504–508. Atomic weights (mean relative masses) : see Pure & Appl. Chem.,68, 2339–2359. These are the IUPAC 1995 values. Elements for which the atomic weight is contained within square brackets have no stable nuclides and are represented by one of the 1996, elements thorium, protactinium, and uranium do have characteristic terrestrial abundances and these are the values quoted. The last significant figure of each value is considered reliable to ±1 except where a larger uncertainty is giv ©1998 Dr Mark J Winter [University of Sheffield, [email protected]]. For updates to this table see http://www.shef.ac.uk/chemistry/web-elements/pdf/periodic-table.html1 March 1998. .Version date: For column IV elements (e.g., silicon) – valence III: one electron “short”, acceptor ρ surface = 111 • boron, aluminum, gallium • silicon: 7.8 x 1014 atoms / cm2 (3 ⋅ [12 atom] + 3 ⋅ [16 atom ]) – valence V: one “extra” electron, donor 3 2 ⋅ a2 • • nitrogen, phosphorus, arsenic, antimony Dean P. Neikirk © 2001, last update January 29, 2001 15 Dept. of ECE, Univ. of Texas at Austin Dean P. Neikirk © 2001, last update January 29, 2001 Impurities in Silicon 13 surface atomic density somewhat higher than (100) Dept. of ECE, Univ. of Texas at Austin (111) orientation in silicon • [111] is the “natural” orientation for zincblende crystals • Oxygen: (column VI) – common unintentional impurity from silica crucibles [111] direction • 1016 - 1018 cm-3 • usually clumps with silicon into large (~1µm) SiO2 complexes; sensitive to processing history • Carbon: (column IV) • high solid solubility (4x1018) • mainly substitutional, electrically inactive (111) plane • Gold: • deep donor or acceptor • rapid diffuser • minority carrier lifetime "killer" Dean P. Neikirk © 2001, last update January 29, 2001 18 Dept. of ECE, Univ. of Texas at Austin Dean P. Neikirk © 2001, last update January 29, 2001 14 Dept. of ECE, Univ. of Texas at Austin Point Defects in Crystals Solid solubility limits in Si • vacancy , interstitial , substitutional vacancy – isolated vacancy: Schottky defect n ≅ Natomic e substitutional impurity − E kT element Nsolid sol (cm-3) – @ 1000Û& interstitial impurity • Eformation ~ 2 eV – T = 300K: n ~ 0 • solid solubility: maximum equilibrium concentration of impurity (solute) in host material (solvent) – temperature dependent – generally lower at lower temp C N Si Pb P 1021 As 2 x 1021 Sb 4 x 1019 Bi Cu Zn Ag Cd Al 2 x 1019 Ga 3 x 1019 In Au 1016 Hg Tl – T = 1300K: n ~ 1013 – vacancy-interstitial pair: Frenkel defect n ≅ Natomic e B 1.5 x 1020 − E 2 kT • Eformation ~ 1 eV – T = 300K: n ~ 1013 – T = 1300K: n ~ 1020 Dean P. Neikirk © 2001, last update January 29, 2001 self interstitial Frenkel defect Dept. of ECE, Univ. of Texas at Austin 21 Dean P. Neikirk © 2001, last update January 29, 2001 Edge Dislocations 19 Ge Sn Dept. of ECE, Univ. of Texas at Austin Defects in crystals • point defects – “zero” dimensional – most common, lowest energy of formation EF is dislocation line • dislocations or line defects – one dimensional – collection of continuous point defects • area (planar) defects • one dimensional defect – formation energy is high, concentration usually low – two dimensional – gross change in crystal “orientation” across a surface “extra” plane of atoms dislocation from Kittel, p. 591 Dean P. Neikirk © 2001, last update January 29, 2001 22 Dept. of ECE, Univ. of Texas at Austin Dean P. Neikirk © 2001, last update January 29, 2001 20 Dept. of ECE, Univ. of Texas at Austin Bulk crystal growth Screw Dislocation (1-d defect) • melting points – silicon: 1420Û & – quartz: 1732Û & • starting material: metallurgical-grade silicon – by mixing with carbon, SiO2 reduced in arc furnace • T > 1780Û& SiC + SiO2 → Si + SiO + CO – common impurities • • • • Al: 1600 ppm (1 ppm = 5 x 1016 cm-3) B: 40 ppm Fe: 2000 ppm P: 30 ppm • – used mostly as an additive in steel Dean P. Neikirk © 2001, last update January 29, 2001 25 Dept. of ECE, Univ. of Texas at Austin Preparation of electronic-grade silicon • gas phase purification used to produce high purity silicon – ~ 600Û& – crud + Si + HCl → • • • • • layer ordering: B C A A B C – stacking fault: missing or extra (111) plane Al: below detection B: < 1 ppb (1 ppb = 5 x 1013 cm-3) Fe: 4 ppm P: < 2 ppb Sb: 1 ppb Au: 0.1 ppb 26 B • A-B-C-C-A-B-C • A-B-A-B-C – 2SiHCl3 + 2H2 (heat) → 2Si + 6HCl – after purification get Dean P. Neikirk © 2001, last update January 29, 2001 C • planar (2-d) defect now reverse reaction • • • • • • Dept. of ECE, Univ. of Texas at Austin 23 Stacking arrangement and stacking faults A SiCl4 (silicon tetrachloride) SiCl3H (trichlorosilane) SiCl2H2 (dichlorosilane) chlorides of impurities – trichlorosilane (liquid at rm temp), further purification via fractional distillation • Dean P. Neikirk © 2001, last update January 29, 2001 screw dislocations are most commonly formed during crystal growth A Dept. of ECE, Univ. of Texas at Austin Dean P. Neikirk © 2001, last update January 29, 2001 24 Dept. of ECE, Univ. of Texas at Austin Wafer preparation • boule forming, orientation measurement – old standard: “flat”perpendicular to <110> direction; – on large diameter “notch” used instead Czochralski crystal growth • silicon expands upon freezing (just like water) – if solidify in a container will induce large stress • CZ growth is “container-less” images from Mitsubishi Materials Silicon http://www.egg.or.jp/MSIL/english/ msilhist0-e.html inner diameter wafer saw • wafer slicing – <100> typically within ± 0.5Û – <111>, 2Û - 5Û RII D[LV images from Mitsubishi Materials Silicon http://www.egg.or.jp/MSIL/english/msilhist0-e.html Dean P. Neikirk © 2001, last update January 29, 2001 29 Dept. of ECE, Univ. of Texas at Austin Wafer prep (cont.) • lapping Dean P. Neikirk © 2001, last update January 29, 2001 Dept. of ECE, Univ. of Texas at Austin 27 Diameter control during CZ growth • critical factor is heat flow from liquid to solid – grind both sides, flatness ~2-3 µm – interface between liquid and solid is an isotherm • ~20 µm per side removed • temperature fluctuations cause problems! – already grown crystal is the heat sink • edge profiling • etching – chemical etch to remove surface damaged layer • ~20 µm per side removed • balance latent heat of fusion, solidification rate, pull rate, diameter, temperature gradient, heat flow • diameter inversely proportional to pull rate (typically ~ mm/min) • polishing pull direction seed – chemi-mechanical polish, SiO2 / NaOH slurry • ~25 µm per polished side removed rotation – gives wafers a “mirror” finish • cleaning and inspection images from Mitsubishi Materials Silicon http://www.egg.or.jp/MSIL/english/msilhist0-e.html Dean P. Neikirk © 2001, last update January 29, 2001 30 Dept. of ECE, Univ. of Texas at Austin Dean P. Neikirk © 2001, last update January 29, 2001 28 Dept. of ECE, Univ. of Texas at Austin Poisson’s ratio Wafer specifications • consider a bar under longitudinal tension or compression wafer diam. – Poisson’s ratio ν = transverse strain / longitudinal strain δW δ L ν= W L 300 mm ± 0.2mm δW ν = ⋅ (longitudinal stress ) W E ) L ⋅ W2 13 2 ∝ unstrained volume 50µ m 60µ m 775µ m ± 25µ m = 10µ m warp = 100µ m Dept. of ECE, Univ. of Texas at Austin 31 Mechanical properties assume bar subjected to longitudinal tensile stress 2 2 V ∝ (L + δL ) ⋅ (W − δW ) = (L + δL ) ⋅ W 2 − 2W ⋅ δW + [δ2] W 13 2 nd order ≈ 0 ( 675µ m ± 25µ m Dean P. Neikirk © 2001, last update January 29, 2001 overall volume change ≈ (L + δL ) ⋅ W 2 − 2W ⋅ δW = bow – warp: distance between highest and lowest points relative to reference plane – bow: concave or convex deformation Dept. of ECE, Univ. of Texas at Austin 19 thickness variation 200 mm ± • dimensionless (since both strains are dimensionless) Dean P. Neikirk © 2001, last update February 21, 2001 thickness 150 mm ± 0.5mm • under tension – length increases: Young’s modulus – ALSO: cross sectional area decreases – this constitutes a transverse strain δW/W • consider elastic media: “Hooke’s law” applies • consider a bar under longitudinal tension or compression – restoring force is proportional to displacement • under tension – length increases – cross sectional area decreases – note TOTAL volume can increase or decrease, depending on material constants! − 2 W ⋅ δW ⋅ L + δL ⋅ W 2 − 2 W ⋅ δW ⋅ δL 123 44 2 nd order ≈ 0 δW L ≈ L ⋅ W 2 − 2 W ⋅ δW ⋅ L + δL ⋅ W 2 = L ⋅ W 2 + W 2 ⋅ δL ⋅ 1 − 2 ⋅ ⋅ 13 2 W δL ∝ unstrained volume volume change 644 744 8 4 4 = L ⋅ W 2 + W 2 ⋅ δL ⋅ (1 − 2 ⋅ ν ) 1 24 43 • relation between stress and strain – stress (longitudinal) = force per unit area (units of pressure!) – strain: fractional change in length δL/L (dimensionless) – Young’s modulus E = stress / strain (units of force per area) • i.e., < 0 if ν > 0.5 > 0 if ν < 0.5 • ν > 0.5: total volume DECREASES under longitudinal tensile stress • ν < 0.5: volume INCREASES stress = E ⋅ δL L Young’s modulus is the stress you would have to apply to double the length of the bar (I.e., δL = L) http://www.britannica.com/seo/y/youngs-modulus/ Dean P. Neikirk © 2001, last update February 21, 2001 20 Dept. of ECE, Univ. of Texas at Austin Dean P. Neikirk © 2001, last update February 21, 2001 18 Dept. of ECE, Univ. of Texas at Austin Young’s modulus and Poisson’s ratio of “common” materials Bending of a simple cantilever beam • • for a uniformly distributed force units – 106 pounds per square inch (psi) = mega-psi = 6.89x109 Newton/m2 = 6.89 gigaPascal – W = (total force) / a • is temperature dependent L a material diamond Dean P. Neikirk © 2001, last update February 21, 2001 ) x >a Dept. of ECE, Univ. of Texas at Austin 23 Beam fixed both ends 70 0.17 Al2O3 (sapphire) Iron x<a 0.21 500 0.23 Aluminum ( x 2 ⋅ 6 ⋅ a 2 − 4 ⋅ a ⋅ x + x 2 W y (x ) = ⋅ 3 24 ⋅ E ⋅ I a ⋅ ( 4 ⋅ x − a ) 0.067 200 SiO2 x 1000 silicon y W = F/a Young’s modulus ( @ 300K ) (GigaPascal) 70 Poisson’s ratio 200 0.34 Dean P. Neikirk © 2001, last update February 21, 2001 Dept. of ECE, Univ. of Texas at Austin 21 L a Static beam equations y • for beam fixed at both ends, point load F \[ LV WKH YHUWLFDO GHIOHFWLRQ RI WKH EHDP LI [  D LI [ ! 0[ 5[ D   \[ \[   0[   0[   [ [  5[  5[   [   ( ,] [  : [  D    ( ,] LV WKH EHQGLQJ PRPHQW UHDFWLRQ DW WKH OHIW KDQG VXSSRUW  ) D /  D   / 0[  LV WKH YHUWLFDO UHDFWLRQ IRUFH DW WKH OHIW KDQG HQG VXSSRUW ) /  D  /   D  / 5[  at load \[ D   0[  D  5[  [\PD[ LV WKH KRUL]RQWDO ORFDWLRQ PD[LPXP YHUWLFDO GHIOHFWLRQ // /D  /D LI D  /   [\PD[ /D /D LI D ! /   [\PD[ 24 simple beam L long, w wide, t thick cantilever beam: supported at one end only – beam: L >> w and t Dept. of ECE, Univ. of Texas at Austin L a – point force F at position a – displacement y at position x y (x ) = y x ⋅ ( 3 ⋅ a − x ) x < a F ⋅ 6 ⋅ E ⋅ I a 2 ⋅ ( 3 ⋅ x − a ) x > a 2 I= x F beam calculator at: http://www.ecalcx.com/beamanalysis/beamcantpoi nt_in.asp other calculators at: http://www.ecalcx.com/ – E is Young’s modulus – I is bending moment of inertia • for a rectangular cross section I is D   ( ,] \PD[ LV WKH PD[LPXP GHIOHFWLRQ RI WKH EHDP   ) D /  D    /   D   ( ,] LI D  /   \PD[ LI D ! /   \PD[  ) D /  D   /   D   ( ,] Dean P. Neikirk © 2001, last update February 21, 2001 • • x 1 ⋅ w ⋅ t3 12 – note maximum displacement is at position L max y end = ( F ⋅ a 2 ⋅ 3 ⋅ L2 − a 6⋅E⋅I ) – note deflection decreases as cube of thickness Dean P. Neikirk © 2001, last update February 21, 2001 22 Dept. of ECE, Univ. of Texas at Austin Thin film on thick substrate • if film is stressed (stress σ), overall curvature results Beam fixed both ends • for beam fixed at both ends, distributed uniform load to a 'HIOHFWLRQ \[ LV WKH YHUWLFDO GHIOHFWLRQ RI WKH EHDP DW [ – E: Young’s moduls; ν: Poisson’s ratio; tsub: substrate thickness; tfilm: film thickness; r: radius of curvature LI [  D LI [ ! D \[ \[   0[   [  5[ [  5[   [  Z [     ( ,] [  Z [    Z [  D      ( ,] 0[ LV WKH EHQGLQJ PRPHQW UHDFWLRQ DW OHIW VXSSRUW 0[  5[ / /  0[ / Z D   5[ E (t sub ) 1 ⋅ ⋅ 1 − ν t film 6 ⋅ r   LV WKH YHUWLFDO UHDFWLRQ IRUFH DW WKH OHIW VXSSRUW Z D  5[ / 5[  2 σ≈   0[ 5[ / LV WKH YHUWLFDO UHDFWLRQ IRUFH ULJKW VXSSRUW Z D  /  D   / 5[ / a y [1] A. Sinha, H. Levinstein, and T. Smith, “Thermal Stresses and Cracking Resistance of Dielectric Films on Si Substrates,” Journal of Applied Physics, vol. 49, pp. 2423-2426, 1978. [2] G. Stoney, “The Tension of Metallic Films Deposited by Electrolysis,” Proceedings of the Royal Society, vol. A82, pp. 172, 1909. Dean P. Neikirk © 2001, last update February 21, 2001 Dept. of ECE, Univ. of Texas at Austin 27 Dynamic response • response of generic structure is approximately d2x m2+ 1 dt 2 3 mass*accel NSL dx b dt { + k⋅x { = Fexternal W = F/a x Dean P. Neikirk © 2001, last update February 21, 2001 • much thinner than radius r • for pinned around circumference, uniform force per unit area (i.e., uniform pressure P), no built in stress δ center = – transfer function (Laplace domain) Dept. of ECE, Univ. of Texas at Austin 25 Deflection of a circular diaphragm elastic force Hooke ’s law force ∝ velocity viscous damping L ( 3 ⋅ P ⋅ r 4 ⋅ 1− ν 2 16 ⋅ E ⋅ t 3 ) 1 H(s ) = m k b s + ⋅s + m m – this is the same as an LRC circuit 2 H(s ) = 1 ωo LC = 1 1 ωo 2 2 2 s + s + ⋅ s + ωo ⋅s + LC Q RC Dean P. Neikirk © 2001, last update February 21, 2001 2 28 Dept. of ECE, Univ. of Texas at Austin Dean P. Neikirk © 2001, last update February 21, 2001 26 Dept. of ECE, Univ. of Texas at Austin ...
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