2 d simulation of laminating stresses and strains in mems structures
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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 - 400 005, India e-mail : [email protected] Web-page http://www.tifr.res.in/~apte/LAMINA.htm. OUTLINE.

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2 d simulation of laminating stresses and strains in mems structures

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 - 400 005, India

e-mail : [email protected]

Web-pagehttp://www.tifr.res.in/~apte/LAMINA.htm


Outline
OUTLINE Structures

  • Laminated Structures in MEMS

  • 2-D Stress-Strain Analysis for Laminae

  • Equilibrium and Compatibility Equations

  • 4th Order Derivatives and 13-Point Finite Differences

  • Boundary Conditions : Fixed, Simply-Supported

  • Program LAMINA

  • Simulation of Laminated Diaphragms :

    having Si, SiO2 and Si3N4 Layers

  • Conclusions


2 layer laminated structures

Tensile µn = 950 Structures

Si

Fused Silica

µn = 500

Compressive µn = 250

Si

Si

Sapphire

Silicon

2-Layer Laminated Structures

From : Fan, Tsaur, Geis, APL, 40, pp 322 (1982)


Multi layer laminated structures in mems

Si3N4 Structures

SiO2

Silicon <100>Substrate

Thin Diaphragm

Multi-Layer Laminated Structures in MEMS


Equilibrium and compatibility equations

Dr. P.R. Apte: Structures

Equilibrium and Compatibility Equations

Mxx + 2 Mxy + Myy = -qeff - - - Equi. Eqn.

where Mxy are the bending moments

qeff = Transverse Loads + In-plane Loads + Packaging Loads

=qa + Nx xx + 2Nxyxy + Nyyy + qpack

where  is the deflection in the transverse direction

N are the in-plane stresses


Equilibrium and compatibility equations1

[ Structures

]

{

}

{

}

Dr. P.R. Apte: 

Equilibrium and Compatibility Equations

10,yy - 120,xy + 20,xx = 0 - - - Comp. Eqn.

where 20,xx a b N

=

M bT d 

where [a] , [b] , [d] are

3 x 3 matrices of functions of

elastic, thermal and lattice constants

of lamina materials


Equilibrium and compatibility equations2

Dr. P.R. Apte: Structures

Equilibrium and Compatibility Equations

Equilibrium Equations

b21 Uxxxx + 2 (b11 - b33)Uxxyy + b12 Uyyyy + d11 xxxx + 2 (d12 - 2d33) xxyy + d22 yyyy

= - [ q + Uyyxx + Uxxyy – 2 Uxyxy ]

Compatibility Equations

a22 Uxxxx + (2a12 - a33)Uxxyy + a11 Uyyyy + b21 xxxx + 2 (b11 - b33) xxyy + b12 yyyy = 0


Stress strain analysis of laminated structures

8 Structures

7

6

LAMINA with 8 layers

5

4

3

2

1

j

o

DiaphragmwithMesh or Grid

o

o

o

.

i , j

i

o

o

o

o

o

o

o

o

o

13 - point Formulationof Finite Differencesfor 4th order derivatives

Stress- Strain Analysis of Laminated structures


4 th order derivatives in u airy stress function and deflection

Structuresxx

4

1

-2

1

3

11

5

 xxxx

-4

1

1

-4

6

10

6

2

1

12

9

7

13

1

-2

1

 xxyy

8

-2

-2

4

1

1

-2

13 - point Formulationof Finite Differencesfor 4th order derivatives

4th order derivatives inU, Airy Stress function and ,deflection

Entries in box/circles are weights


Boundary conditions
Boundary Conditions Structures

(0)

(0)

(-1)

(1)

(2)

(-1)

(1)

(2)

(3)

o o o o

o o o o o

x

x

4-Point Formulae

5-Point Formulae


Boundary conditions 4 point formulae
Boundary Conditions: 4-Point Formulae Structures

1st 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,

2nd order derivative (-1)= [ 2 (0) - (1) + 3 2x  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 Mx = 0   xx (a) = 0  (-1)= (1)


Program lamina

Initialization Structures

INIT

READIN

Read Input Data

SUMMAR

Print Input Summary

Setup [ Z ] and [ Q ] matrices

ZQMAT

Solve for U and simultaneously

SOLVE

Print deflection w andstress at each mesh point

OUTPUT

PROGRAM LAMINA


Typical multi layer diaphragm

Si Structures3N4

SiO2

Silicon <100>Substrate

Thin Diaphragm

Typical multi-layer Diaphragm


A silicon diaphragm

Silicon <100> StructuresSubstrate

Thin Diaphragm

A Silicon Diaphragm


Silicon dioxide diaphragm

SiO Structures2

Silicon <100>Substrate

Silicon-dioxide Diaphragm


A multi layer diaphragm bent concave by point or distributed loads

Si Structures3N4

SiO2

Silicon <100>Substrate

Thin Diaphragm

A multi-layer Diaphragm bent concave by point or distributed loads

Point or

distributed load


A multi layer diaphragm bent convex by temperature stresses

Si Structures3N4

Compressive stress due to cooling

SiO2

Silicon <100>Substrate

Thin Diaphragm

A multi-layer Diaphragm bent convex by temperature stresses


Deflection at center for lateral loads q
Deflection at center Structures‘’ 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
Deflection at center for lateral and in-plane loads Structures

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 NX=NY=NO=1 | 0.386E-2 | 0.384E-2 |

| | | | |

| 2 | Tensile NO = 19 | 0.203E-2 | 0.202E-2 |

| | | | |

| 3 | Compressive NO=-1 | 0.427E-2 | 0.426E-2 |

| | | | |

| 4 | Compressive NO=-10 | 0.822E-2 | 0.830E-2 |

| | | | |

| 5 | Compressive NO=-20 | (indefinite) | (-0.246E+0)|

-------------------------------------------------------------


Deflection at center for uniform bending moment
Deflection at center for uniform bending moment Structures

Table 3 Deflection at center for uniform bending moment

(Reference is Timoshenko [1], pp 183)

-------------------------------------------------------------

| serial | condition | known results | LAMINA |

| no. | | | simulation |

-------------------------------------------------------------

| 1 | MO = 1 | 0.736E-1 | 0.732E-1 |

| | on all edges | | |

-------------------------------------------------------------

M

M


Diaphragm consisting of si sio 2 and si 3 n 4 layers
Diaphragm consisting of StructuresSi, SiO2 and Si3N4 layers

Table 4 Diaphragm consisting of Si, SiO2 and Si3N4 layers

-------------------------------------------------------------

|serial| lamina | Deflection |In-plane | stress |

| no. | layers | at center |resultant | couple |

-------------------------------------------------------------

| (a) | SiO2+ Si+ SiO2 | 0 | -0.216E+2 | 0 |

| | | | | |

| (b) | Si + SiO2 | -0.583E-2 | -0.381E+1 | -0.628E-1 |

| | | | | |

| (c) | SiO2+ Si | +0.583E-2 | -0.381E+1 | +0.628E-1 |

| | | | | |

| (d) | Si+SiO2+ Si 3N4 | +0.583E-3 | +0.191E+2 | +0.168E-1 |

| | | | | |

| (e) | SiO2+ Si+ SiO2 | +0.515E-2 | +0.643E+0 | +0.727E-1 |

| | + Si 3N4 | | | |

-------------------------------------------------------------


Conclusions
Conclusions Structures

  • 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, SiO2 and Si3N4

    • make the diaphragm insensitive to temperature

      or packaging parameters


  • Thank You Structures

    Web-pagehttp://www.tifr.res.in/~apte/LAMINA.htm


    4 Structures

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    13 - point Formulationof Finite Differencesfor 4th order derivatives


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