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2-D Simulation of Laminating Stresses and Strains in MEMS Structures Prakash R. Apte Solid State Electronics Group Tata Institute of Fundamental Research Homi Bhabha Road, Colaba BOMBAY , India Web-page

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OUTLINE Laminated Structures in MEMS 2-D Stress-Strain Analysis for Laminae Equilibrium and Compatibility Equations 4 th Order Derivatives and 13-Point Finite Differences Boundary Conditions : Fixed, Simply-Supported Program LAMINA Simulation of Laminated Diaphragms : having Si, SiO 2 and Si 3 N 4 Layers Conclusions

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Si Tensile µn = 950 Fused Silica Si µn = 500 Silicon Si Compressive µn = 250 Sapphire 2-Layer Laminated Structures From : Fan, Tsaur, Geis, APL, 40, pp 322 (1982)

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Si3N4 SiO2 Silicon Substrate Thin Diaphragm Multi-Layer Laminated Structures in MEMS

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Equilibrium Equilibrium and Compatibility Equations M xx + 2 M xy + M yy = -q eff Equi. Eqn. where M xy are the bending moments q eff = Transverse Loads + In-plane Loads + Packaging Loads = q a + N x xx + 2N xy xy + N y yy + q pack where is the deflection in the transverse direction N are the in-plane stresses Dr. P.R. Apte:

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Compatibility Equilibrium and Compatibility Equations 10,yy - 120,xy + 20,xx = Comp. Eqn. where 20,xx a b N = M b T d where [a], [b], [d] are 3 x 3 matrices of functions of elastic, thermal and lattice constants of lamina materials Dr. P.R. Apte: {} [] {}

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Compatibility Equilibrium and Compatibility Equations Equilibrium Equations b 21 U xxxx + 2 (b 11 - b 33 )U xxyy + b 12 U yyyy + d 11 xxxx + 2 (d d 33 ) xxyy + d 22 yyyy = - [ q + U yy xx + U xx yy – 2 U xy xy ] Compatibility Equations a 22 U xxxx + (2a 12 - a 33 )U xxyy + a 11 U yyyy + b 21 xxxx + 2 (b 11 - b 33 ) xxyy + b 12 yyyy = 0 Dr. P.R. Apte:

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Stress- Strain Analysis of Laminated structures LAMINA with 8 layers Diaphragm with Mesh or Grid i j oo oo oo o o o o o oo i, j 13 - point Formulation of Finite Differences for 4th order derivatives

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4 th order derivatives in U, Airy Stress function and , deflection 13 - point Formulation of Finite Differences for 4th order derivatives xx xxxx xxyy Entries in box/circles are weights

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Boundary Conditions (-1) o o o o (1) (0) (2) xx (-1) o o o o o (1) (0) (2) xx (3) 4-Point Formulae5-Point Formulae

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Boundary Conditions: 4-Point Formulae 1 st order derivative x = [ -1/3 (-1) - 1/2 (0) + (1) - 1/6 (2)] / x which gives (-1) = [ -3/2 (0) + 3 (1) - 1/2 (2) - 3 x x (0) Similarly, 2 nd order derivative (-1) = [ 2 (0) - (1) + 3 2 x xx (0) Fixed or Built-in Edge : at x = a, then (a) = 0 and x (a) = 0 (-1) = 3 (1) - 1/2 (2) Simply-Supported Edge : at x = a, then x (a) = 0 and M x = 0 xx (a) = 0 (-1) = (1)

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INIT READIN SUMMAR ZQMAT SOLVE OUTPUT Initialization Read Input Data Print Input Summary Setup [ Z ] and [ Q ] matrices Solve for U and simultaneously Print deflection w and stress at each mesh point PROGRAM LAMINA

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Typical multi-layer Diaphragm Silicon Substrate Thin Diaphragm Si 3 N 4 SiO 2

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Silicon Substrate Thin Diaphragm A Silicon Diaphragm

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SiO 2 Silicon Substrate Silicon-dioxide Diaphragm

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A multi-layer Diaphragm bent concave by point or distributed loads Silicon Substrate Thin Diaphragm Si 3 N 4 SiO 2 Point or distributed load

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A multi-layer Diaphragm bent convex by temperature stresses Si 3 N 4 SiO 2 Silicon Substrate Thin Diaphragm Compressive stress due to cooling

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Deflection at center ‘ ’ for lateral loads ‘q’ Table 1 Deflection at center `w ' for lateral loads `q' (Reference for known results is Timoshenko [1]) | serial | Boundary conditions | known results | LAMINA | | no. | | | simulation | | 1 | X-edges fixed | 0.260E-2 | 0.261E-2 | | | Y-edges free | (TIM pp 202) | | | | | | | | 2 | X-edges simply supp.| 0.130E-1 | 0.130E-1 | | | Y-edges free | (TIM pp 120) | | | | | | | | 3 | All edges fixed | 0.126E-2 | 0.126E-2 | | | | (TIM pp 202) | | | | | | | | 4 | All edges simply | 0.406E-2 | 0.406E-2 | | | supported | (TIM pp 120) | |

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Deflection at center for lateral and in-plane loads P Table 2 Deflection at center for lateral and in-plane loads (Reference is Timoshenko [1], pp 381) |serial | Normalized | known results | LAMINA | | no. | in-plane load | | simulation | | 1 | Tensile N X =N Y =N O =1 | 0.386E-2 | 0.384E-2 | | | | | | | 2 | Tensile N O = 19 | 0.203E-2 | 0.202E-2 | | | | | | | 3 | Compressive N O =-1 | 0.427E-2 | 0.426E-2 | | | | | | | 4 | Compressive N O =-10 | 0.822E-2 | 0.830E-2 | | | | | | | 5 | Compressive N O =-20 | (indefinite) | (-0.246E+0)|

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Deflection at center for uniform bending moment Table 3 Deflection at center for uniform bending moment (Reference is Timoshenko [1], pp 183) | serial | condition | known results | LAMINA | | no. | | | simulation | | 1 | M O = 1 | 0.736E-1 | 0.732E-1 | | | on all edges | | | MM

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Diaphragm consisting of Si, SiO 2 and Si 3 N 4 layers Table 4 Diaphragm consisting of Si, SiO 2 and Si 3 N 4 layers |serial| lamina | Deflection |In-plane | stress | | no. | layers | at center |resultant | couple | | (a) | SiO 2 + Si + SiO 2 | 0 | E+2 | 0 | | | | | (b) | Si + SiO 2 | E-2 | E+1 | E-1 | | | | | (c) | SiO 2 + Si | E-2 | E+1 | E-1 | | | | E-3 | (d) | Si + SiO 2 + Si 3 N 4 | E-3 | E+2 | E-1 | | | | E+0 | (e) | SiO 2 + Si + SiO 2 | E-2 | E+0 | E-1 | | | + Si 3 N 4 | | | |

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Conclusions LAMINA takes into account Single Crystal anisotropy in elastic constants Temperature effects – growth and ambient Heteroepitaxy, lattice constant mismatch induced strains LAMINA accepts Square or rectangular diaphragm/beam/cantilever non-uniform grid spacing in x- and y-directions non-uniform thickness at each grid point – taper LAMINA can be used as design tool to minimize deflections (at the center) minimize in-plane stresses in Si, SiO 2 and Si 3 N 4 make the diaphragm insensitive to temperature or packaging parameters

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Thank You Web-page

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13 - point Formulation of Finite Differences for 4th order derivatives slide5

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