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

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

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)

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

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: 

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:  {} [] {}

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: 

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

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

Boundary Conditions  (-1) o o o o  (1)  (0)  (2) xx  (-1) o o o o o  (1)  (0)  (2) xx  (3) 4-Point Formulae5-Point Formulae

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)

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

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

Silicon Substrate Thin Diaphragm A Silicon Diaphragm

SiO 2 Silicon Substrate Silicon-dioxide Diaphragm

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

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

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) | |

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)|

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

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 | | | |

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