MEMS Mechanical Fundamentals
ROCHESTER INSTITUTE OF TECHNOLOGY
MICROELECTRONIC ENGINEERING
MEMS Mechanical Fundamentals
Dr. Lynn Fuller
webpage: http://people.rit.edu/lffeee
Electrical and Microelectronic Engineering
Rochester Institute of Technology
82 Lomb Memorial Drive
Rochester, NY 14623-5604
email: Lynn.Fuller@rit.edu
microE program webpage: http://www.microe.rit.edu
5-5-2015 MEMSOutline.ppt
© May 5, 2015 Dr. Lynn Fuller
Page 1
MEMS Mechanical Fundamentals
OUTLINE
Force
Pressure
Stress, Strain and Hooke’s Law
Poisson’s Ratio
Simple Cantilever
Springs
Thermal Strain
Diaphragms
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
FORCE
F = ma (newton)
where m is mass in Kg
a is acceleration in m/sec2
gravitational acceleration is 9.8 m/sec2
newton (N) = 1 Kg 1 meter/s2
dyne = 1 gm 1 cm/s2
How do you convert dynes to newtons?
N
1000gm 100cm / s2
--- = ---------------------so 1 newton = 105 dynes
dyne
1gm 1 cm /s2
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
PRESSURE
Pressure is used to describe force per unit area.
P = F/A
Pressure due to the weight of liquid
P=hdg
where h is the height
d is the density of the liquid
g is acceleration due to gravity
980 cm/sec2
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
PRESSURE UNITS
Table of Pressure Conversions
1 atm = 14.696 lbs/in2 = 760.00 mmHg
1 atm = 101.32 kPa = 1.013 x 106 dynes/cm2
1Pascal = 1.4504 x 10-4 lbs/in2 =1 N/m2
1 mmHg = 1 Torr (at 0oC )
1SPL (Sound Pressure Levels) = 0.0002 dynes/cm2
Average speech = 70 dBSPL = 0.645 dynes/cm2
Pain = 130 dBSPL = 645 dyne/cm2
Whisper = 18 dBSPL = 1.62 x 10-3 dyne/cm2
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
GAUGE PRESSURE – ABSOLUTE PRESSURE
Gauge pressure is a differential pressure measured relative to an
ambient pressure. The output of a gauge pressure sensor is not
sensitive to a changing barometric pressure.
Absolute pressure is differential pressure measured relative to an
absolute zero pressure (perfect vacuum). The output of the
absolute sensor will change as a result of barometric pressure
change and thus can be used as a barometer. An absolute
pressure sensor can determine if you are at sea level or on top of
a mountain.
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
HOOKE’S LAW
Stress is the applied force per crossectional area, = F/A
Strain is the elongation per unit length,  = L/L

 y = yield strength

 u = ultimate or maximum strength

 b = rupture strength ( for brittle material  u =  b)
Stress, 
u
y
x
Hooke’s Law
= E 
where
E is Young’s modulus
Strain, 
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
POISSON’S RATIO
Poisson’s ratio () gives the relationship between lateral strain
and axial strain. In all engineering materials the elongation
produced by an axial tensile force in the direction of the force is
accompanied by a contraction in any transverse direction.
 = -y /x ~ 0.3
y
X, Axial
Deformed
Force
x
Original
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
PROPERTIES OF SOME MATERIALS
y u
E

d
Yield Ultimate Knoop
Youngs Density Thermal Thermal
Poisso’s
Strength
Hardness Modulus (gr/cm3) Conductivity Expansion Ratio
(1010 dyne/cm2) (Kg/mm2) (1012dyne/cm2)
(w/cm°C) (10-6/°C)
Si3N4
SiO2
Si
Al
14 28
8.4 16
12 14
0.17
3486
820
850
130
3.85
0.73
1.9
0.68
3.1
2.5
2.3
2.7
0.19
0.014
1.57
2.36
0.8
0.55
2.33
25
10 dyne/cm2 = 1 newton/m2
© May 5, 2015 Dr. Lynn Fuller
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0.3
0.3
0.32
0.334
MEMS Mechanical Fundamentals
STRESS - COMPRESSIVE, TENSILE
Stress is the force per unit area applied to a mechanical member
(Pressure)
A positive sign will be used to indicate a tensile stress (member in
tension) and a negative sign to indicate a compressive stress (member
in compression)
These forces can be axial, transverse, oblique, shearing, torsional,
thermal, etc
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
STRESS IN SPUTTERED FILMS
Compressively stressed films would like to expand parallel to the
substrate surface, and in the extreme, films in compressive stress will
buckle up on the substrate. Films in tensile stress, on the other hand,
would like to contract parallel to the substrate, and may crack if their
elastic limits are exceeded. In general stresses in films range from
1E8 to 5E10 dynes/cm2.
tensile
compressive
For AVT sputtered oxide films
Dr. Grande found Compressive
18MPa stress, 1-29-2000
10 dyne/cm2 = 1 newton/m2 = 1 Pascal
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
LOW STRESS SILICON RICH Si3N4
ADE Measured stress for various Ammonia:Dichlorosilane Flow
Ratios
10 dyne/cm2 = 1 newton/m2 = 1 Pascal
Flow Stress x E 9 dynes/cm2
10:1 + 14.63 Nitrogen Rich
5:1
+ 14.81
2.5:1 + 12.47
Stress; = (E/(6(1-v)))*(D2/(rt))
1:1
+ 10.13
where E is Youngs modulus,
1:2.5 + 7.79* Stociometric
v is Poissons ratio,
1:5
+3
D and t are substrate and film thickness
1:10 0
Silicon Rich r is radius of curvature (+ for tensile)
*standard recipe
T.H Wu, “Stress in PSG and Nitride Films as Related to Film
Properties and Annealing”, Solid State Technology, p 65-71,May ‘92
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
STRESS IN SILICON NITRIDE FILMS
Stress in an 8000 A
Nitride Film
Tensile Stress
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
STRESS IN SPUTTERED TUNGSTEN FILMS
Tungsten
CVC 601
4” Target
500 Watts
50 minutes
5 mTorr Argon
Thickness ~ 0.8 µm
Compressive Stress
Picture from scanner in gowning
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
SIMPLE CANTILEVER
F
Cantilever:
L
h
b
Ymax
Ymax = F L3/3EI
where E = Youngs Modulus
and I = bh3/12, moment of inertia
Mechanics of Materials, by Ferdinand P. Beer,
E. Russell Johnston, Jr., McGraw-Hill Book Co.1981
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
FORCE TO BEND CANTILEVER EXAMPLE
Example: Find the force needed to bend a simple
polysilicon cantilever 1 µm, with the following
parameters: b=4 µm, h=2µm, L=100 µm
F=
F=
Ymax 3 E bh3
12L3
(1e-6) 3 ( 1.9e11)(4e-6)(2e-6)3
12 (100e-6)3
F = 1.5e-6 newtons
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
FINITE ELEMENT ANALYSIS
Displacement
2.28um
IDEAS
Length = 100 um
Width = 10 um
Thickness = 2 um Polysilicon
Force applied = 7.2 uN
Maximum Deflection = 2.28 um
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
STRESS IN A CANTILEVER BEAM
x
F
x=0
h
b
L
Ymax
The maximum stress is at the top surface of the
cantilever beam at the anchor where x=0

x=0 = F Lh/2I
where I = bh3/12, moment of inertia
Mechanics of Materials, by Ferdinand P. Beer,
E. Russell Johnston, Jr., McGraw-Hill Book Co.1981
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
STRESS IN CANTILEVER BEAM EXAMPLE
Example: Find the maximum stress in a simple
polysilicon cantilever with the following parameters.
Ymax = 1 µm, b=4 µm, h=2µm, L=100 µm
12 F L
x=0 =
2b h2
12 (1.4e-6) (100e-6)
x=0 =
2 (4e-6) (2e-6)2
x=0 = 5.6e7 newton/m2 = 5.6e8 dyne/cm2
Compare to the given table value for yield strength of
7e10 dyne/cm2
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
FINITE ELEMENT ANALYSIS (FEA) OF CANTILEVER
F=1N
Ymax = 10 um
Stress = 3.8E5 N/m2
SolidWorks
Length = 1500 µm
Width = 600 µm
Thickness = 20 µm
Window ~ 300 x 300 µm
F=1N
Ymax = 1.6 um
Stress = 7.2E3 N/m2
Burak Baylav
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
ACCELEROMETER - EXAMPLE
x
x=0
Proof Mass
h
b
L
Ymax
The proof mass will create a force in the presence of
an acceleration, the strain can be measured with a
piezoresistive device or the position can be
measured with a capacitive measurement device
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
ACCELEROMETER-EXAMPLE
Example: from previous cantilever example
Ymax 3 E bh3
F=
12L3
F = 1.5e-6 newtons
m=dV
m
X µm
25 µm
25 µm
Find height X for a proof mass of volume (V) of nickel
(density (d)=8.91 gm/cm3) and maximum acceleration of
50G’s
F = mA = m 50 (9.8m/s2) = 1.5e-6 newtons
and find m = 3.1e-9 Kgm = 3.1e-6 gm
m = d V =3.1e-6 = (8.91)(25e-4)(25e-4)(X) cm
X = 0.055cm = 550µm
Approximately the thickness of a 4” wafer
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
SPRINGS
F=ky
Analogy: I = G V
k is like conductance (G)
I is Force, V is potential
where k is the spring constant
and y is the displacement
F
Cantilever:
L
h
b
Ymax
Ymax = F L3/3EI
F=ky
F = (3E(bh3/12)/ L3) y
where E = Youngs Modulus
and I = bh3/12, moment of inertia
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
FOLDED SPRING
k total =k
k total =k/2
40 µm
40 µm
I
100 x100 µm
pads
2nd layer poly
I
100 x100 µm
pads
2nd layer poly
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
BEAM ANCHORED AT BOTH ENDS
F
h
b
L/2
Ymax
Ymax = F L3/48EI
where E = Youngs Modulus
and I = bh3/12, moment of inertia
Mechanics of Materials, by Ferdinand P. Beer,
E. Russell Johnston, Jr., McGraw-Hill Book Co.1981
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
SPRING ANCHORED AT BOTH ENDS
I
I
40 µm
40 µm
2nd layer poly
100 x100 µm
pads
2nd layer poly
100 x100 µm
pads
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
THERMAL STRAIN
The fractional change in length due to a change in
temperature is given by:

L/L =  (T)
where  is the
coefficient of thermal
expansion
This is also the thermal strain T. In this case the
strain does not cause a stress unless the material is
confined in some way
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
BIMETALIC CANTILEVER BEAM
ANSYS
10μm
Aluminum
100μm
Silicon
ANSYS – Rob Manley
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
RAISE TEMPERATURE FROM 20 TO 200 C
ANSYS
ANSYS – Rob Manley
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
MOVIE OF TOTAL DEFORMATION
ANSYS – Rob Manley
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
FINITE ELEMENT ANALYSIS OF THERMAL BENDING
Small arm 300 C, 10um X 100 um
Large arm 0 C, 30 um x 100 um
Maximum Displacement = 6 um
IDEAS – Andrew Randall
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
EXCELL SPREADSHEET FOR DIAPHRAGM
CALCULATIONS
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
FINITE ELEMENT ANALYSIS
Points of
Maximum Stress
IDEAS – Rob Manley
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
ANSYS FINITE ELEMENT ANALYSIS
Regular Si Diaphragm
Corrugated Diaphragm
Layer 2: 1.5mm x 1.5mm Polysilicon 1μm thick
2mm x 2mm diaphragm 30µm thick, 50 psi applied
Rob Manley, 2005
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
DIAPHRAGM DEFORMATION MOVIE
200µm
Rob Manley, 2005
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
DIAPHRAGM STRESS MOVIE
Rob Manley, 2005
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
DIAPHRAGM
diaphragm
thickness ()
Diaphragm:
Displacement
Radius (R)
Uniform Pressure (P)
Displacement (y)
Equation for deflection at center of diaphragm
4[(1/)2-1]
3PR
y=
16E(1/)23
4
2-1]
= (249.979)PR2 [(1/)
3
 = Poisson’s Ratio
for Aluminum  =0.35
E = Young’s Modulus,
© May 5, 2015 Dr. Lynn Fuller
E(1/) 
*The second equation corrects all units
assuming that pressure is mmHg,
radius and diaphragm is m, Young’s
Modulus is dynes/cm2, and the
calculated displacement found is m.
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MEMS Mechanical Fundamentals
CIRCULAR DIAPHRAGM FINITE ELEMENT ANALYSIS0
ANSYS – Rob Manley
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
CAPTURED VOLUME
PV = nRT
F1 = force on diaphragm = external pressure times area of diaphragm
F2 = force due to captured volume of air under the diaphragm
F3 = force to mechanically deform the diaphragm
F1
rs
Ad
h
h
d
rd
F2 + F3
y
F1= F2 + F3 F1 = P x Ad F2 = nRT Ad / (Vd + Vs)
where Vd = Ad (d-y) and Vs = G1 Pi (rs2 – rd2)(d) where
G1 is the % of spacer that is not oxide
F3 = (16 E (1/ )2 h3 y)/(3 rd4[(1/ )2-1])
© May 5, 2015 Dr. Lynn Fuller
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MEMS Mechanical Fundamentals
DIAPHRAGM WITH CAPTURED VOLUME
y (µm)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
P (N/m2)
0
5103.44
10748.09
17025.03
24047
31955.23
40929.06
51199.69
63070.47
76947.31
93386.12
113169.1
137433.1
167895.7
207281.3
260184.8
335009.7
448945.4
643513.4
1051199
2446196
P (N/m2) vs displacement y (µm)
3000000
2500000
2000000
1500000
1000000
500000
0
0
0.5
© May 5, 2015 Dr. Lynn Fuller
1
1.5
2
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2.5
MEMS Mechanical Fundamentals
REFERENCES
1. Mechanics of Materials, by Ferdinand P. Beer, E. Russell Johnston, Jr.,
McGraw-Hill Book Co.1981, ISBN 0-07-004284-5
2. Electromagnetics, by John D Kraus, Keith R. Carver, McGraw-Hill Book
Co.1981, ISBN 0-07-035396-4
3. “Etch Rates for Micromachining Processing”, Journal of
Microelectromechanical Systems, Vol.5, No.4, December 1996.
4. “Design, Fabrication, and Operation of Submicron Gap Comb-Drive
Microactuators”, Hirano, et.al. , Journal of Microelectromechanical Systems,
Vol.1, No.1, March 1992, p52.
5. “There’s Plenty of Room at the Bottom”, Richard P. Feynman, Journal of
Microelectromechanical Systems, Vol.1, No.1, March 1992, p60.
6. “Piezoelectric Cantilever Microphone and Microspeaker”, Lee, Ried, White,
Journal of Microelectromechanical Systems, Vol.5, No.4, December 1996.
7. “Crystalline Semiconductor Micromachine”, Seidel, Proceedings of the 4th
Int. Conf. on Solid State Sensors and Actuators 1987, p 104
8. Fundamentals of Microfabrication, M. Madou, CRC Press, New York, 1997
© May 5, 2015 Dr. Lynn Fuller
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