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Forces, Orientation, Scaling
Diamond lattice (Si)
Picture from U of UTAH
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Directions and Planes in Crystals
lattice constant
Directions (vector components: a single direction is expressed as [a set of 3 integers], equivalent directions (family) are expressed as < a set of 3 integers >Planes: a single plane is expressed as (a set of 3 integers h k l = Miler indices) and equivalent planes are expressed as {a set of 3 integers}
Miler indices: take a,b,c (multiple of basic vectors ex. x=4a, y=3a, z=2a)
reciprocals (1/4, 1/3, 1/2)-> common denominator (3/12, 4/12, 6/12) -> the smallest numerators (3 4 6)
Plummer
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Properties of crystals are determined by their orientation
(100)
(111)
(100) (110)(111)
go to http://stokes.byu.edu/diamond.htm to rate the crystal and see lattice symmetry. You can also calculate angles between various crytallographic planes and access many “clickable” material/device/process parameters calculated here http://www.ee.byu.edu/cleanroom/EW_orientation.phtml?flag=topic_index
Diamond lattice of Si
viewed in: <100> <110> <111>
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Crystallographic orientation of Si
Charges in MOS systems (measured in C-V) and many other properties depend on crystal orientation.
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CRYSTAL
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Orientation of Silicon Wafers is Important for MEMS
Primary and Secondary Flats
After Shimura
Stress and Strain
Static and dynamic behaviors of MEMS under loading determined by Newton Laws
Orientation effects are coming soon
Basic mechanical engineering info can be found here on http://en.wikipedia.org/wiki/Main_Page using Force, Stress, Strain, Hook’s Law, Newton's laws of motion as key words, etc. as well as http://www.ami.ac.uk/courses/topics/0123_mpm/index.html
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Examples of Mechanical Structures• Force balance analysis • (Traverse direction) Force and
moment balance analysis
The wall reacts on the bar with an axial force=F (see “-” sign)
F=0 Newton 1-st Law: total force=0
A pair of forces create a torque (“couple” or “moment”)
remove the wall
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M = FL − FL'= FL"∑Liu
Newton 3rd Law
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Shear Stress and Strain• Definition of normal Stress and Strain
• Shear stress does not cause elongation or shortening of the element but deformation.
• Shear strain represents angular displacement
€
σ =F
A[N /m2]
€
s =L − L0L0
=ΔL
L0
Pa
€
=F
A[N /m2]
€
=ΔXL
€
G =τ
γ[N /m2]
G=Shear modulus of
elasticity
€
G =E
2(1+ ν )G is material not dimension property
imaginary cut
Normal Stress can be tensile or compressive
Strain refers to elongation (“normal strain” if it is to area A)
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σ =EsE is modulus of elasticity or Young’s modulus (intrinsic material property)
(later)
Poisson’s ratio
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ν =sysx
=szsx
Poisson’s Ratio
Deformations along the load (force F) and perpendicular to the load lead to an axial strain and a transverse strain:
sl is tensile and st is
compressive so they have opposite signs
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sl =ΔL
L0
€
st =ΔD
D0
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v =transverse−strain
longitudinal−strain=
−stsl
=
ΔD
D0
ΔL
L0
v (Poisson’s ratio) typically is ≈ 0.3-0.5.for metals ≈ 0.3, for rubber 0.5, for cork ≈0 (does not expand at all), polysilicon 0.22.
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Mechanical Stress and Strain• Stress causes elongation and reduction of the cross-sectional area. Poisson’s
ratio v describes this effect
• Stress and strain, for small deformations, are described by Hooke’s law E is modulus of elasticity or Young’s modulus
€
ν =sysx
=szsx
€
σ =Es
€
G =shear−stress
shear−displacement−angle=τ
γ=
F
AΔX
L
[N /m2]
€
E = 2G(1+ v) = 3K(1− 2v)
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K =hydrostatic _ stress
volum _compression=
F
AΔV
V
[N /m2]
K is bulk modulus representing volume change under pressure (water K=1E9 N/m2, Al K=7E10, steel K=14E10 N/m2
For isotropic materials:Shear modulus of elasticity G:
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Scalar Relation Between Stress and Strain Generic stress-strain curve Non-generic Stress-strain relations
Important properties of materials used in MEMS: ductility, toughness, hardness, brittleness. Sign of applied load is important - some materials will fail “at lower stresses in shear, others in tension”.
Fatique failure is also important in MEMS - materials can develop cracks or weak points specially if operating under stress and/or harsh conditions. Thin films (<100 nm) can even work trillions of cycles w/o failure.
Hooke's law=material deforms linearly with load
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Silicon and Related Thin Films
• Dimensions, crystal orientation affect mechanical properties of silicon:• Young’s modulus in c-Si:
• In {100} planes: in direction of [110] 168GPa>[100] 130GPa• In {110} planes: [111] 187 GPa
• Young’s modulus in poly-Si (120-160 GPa) depends on structure and grains therefore depends on processing
• deposition conditions and subsequent annealing are important
• Shear modulus depends on crystal orientation• Poisson’s ratio for c-Si varies from 0.055 to 0.36 and for
poly-Si it is 0.15-0.36. • See Appendix A for specific data.
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General Stress-Strain Relations• Stress and strain are tensors. Use matrix to express normal stress
components σxx σyy σzz (noted as T1-T3) and shear stress yz, xz, xy (noted as T4-T6) as vectors.
s1-s2 are three independent strains,
s4-s6 are three shear strains.
€
€
T1
T2
T3
T4
T5
T6
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
=
C11 C12 C13 C14 C15 C16
C21 C22 C23 C24 C25 C26
C31 C32 C33 C34 C35 C36
C41 C42 C43 C44 C45 C46
C51 C52 C53 C54 C55 C56
C61 C62 C63 C64 C65 C66
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
s1s2
s3
s4
s5
s6
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
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T = Cs C is the Stiffness matrix
s- Strain matrix:
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s = ST
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S = C−1 S is the compliance matrix:
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Example for Silicon Orientation Effects
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CSi,<100> =
1.66 0.64 0.64 0 0 0
0.64 1.66 0.64 0 0 0
0.64 0.64 1.66 0 0 0
0 0 0 0.8 0 0
0 0 0 0 0.8 0
0 0 0 0 0 0.8
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
1011Pa
C matrix is very much simplified for Si along <100> direction
It can be used to calculate normal stress components T1, T2, T3 and find
Young’s modulus (see Example 3.5 for E[100]=T1/s1=130GPa)
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Flexural Beam Bending Analysis Under Simple Loading Conditions
Beams are described by how they are supported and how load is applied.
degree of freedom
no movement at the support
rotation is restricted
linear movement and rotation allowed
cantilevers are basic MEMS structures
Liu
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Load and Boundary Conditions
There is always a direct relation between the deflection and the load (for any boundary conditions)
Kovacs
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Bending of Cantilevers Under Various Boundary Conditions
Liu
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Longitudinal Strain Under Pure Bending and Deflection of the Beams
under compression
under tension
equal max values of tensile and compressive stress σmax!
Maximum stress is constant through the
length of the beam
M=bending moment or total torque €
M =σ max
(t
2)
h2dAh=−
t
2
t
2∫w
∫ =σ max
(t
2)I
h
“I”=moments of inertia, which for a beam with rectangular cross section is
Strain depends on total torque M smax=Mt/2EI
A more complex case since the Moment along the beam is not constant. It is important to find the maximum displacement of a cantilever.€
I =1
12wt 3
Liu
“tu” – neutral axis
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Finding Spring Constants “k”
To see how changes in the weight of an object and spring properties (length, stiffness, stretch) affect mechanical response of the system - watch these video. They show how scaling is used in MEMSs http://streamer.cen.uiuc.edu/me_mems/Video2.wmvhttp://streamer.cen.uiuc.edu/me_mems/Video3.wmv
More can be found inhttp://www.engr.uiuc.edu/OCEE/outreach.htm.
Stiffness is characterize by the spring constant k (or force constant).
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Cantilevers
€
θ =Fl2
2EI
Bending of a cantilever (by the angle θ and deflecting by x) depends on force, geometry and Young’s modulus.
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x =Fl3
3EI
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k =F
x=
3EI
l3=
Ewt 3
4 l3
so the spring constant is:
The stiffness depends on the force direction and on the direction of bending. The beam “is said that provide compliance in one direction
and resistance to movement in another”. Spring constant decreases with length. Soft material (small E) would deform more so spring constant is small.
Liu
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Force Constants of Beams Various Applications of Mechanical Structures
Force can be of various origin
Liu
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Applications of Mechanical StructuresSelected from many:
http://www.dlp.com/Default.asp?bhcp=12. Digital Mirrors made by Texas Instruments (will come soon in Chapter 4)
1. Accelerometer is shown in this http://streamer.cen.uiuc.edu/me_mems/Video6.wmv in http://www.engr.uiuc.edu/OCEE/outreach.htm
Plates (thick) and Membranes (thin)
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k =F
x=
12EI
l3=
Ewt 3
l3
The spring constant k associated with each fixed-guided beam
Each spring shares the load so for plates supported by two fixed-guided beams it is
and by four fixed-guided beams it is
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k = 2(Ewt 3
l3)
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k = 4(Ewt 3
l3)
Liu
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Torsional Deflection
Maximum shear stress in the bar is max and it is distributed along the bar. Maximum shear strain is
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max =d
LMaximum shear stress is related to the torque
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T =τ max
r0r2dA∫
For the circular beams:
the magnitude of the maximum shear stress is
and the angular displacement of the torsional bar is
€
max =Tr0
J
Torsional moment of inertia=J for a circular bar
and square-cross section
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Φ=d
r0=Lθ
r0=
L
r0
τ max
G=LTr0
r0GJ=TL
GJ
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J =πr0
4
2J = 2.25a
Liu
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Intrinsic Stress
Internal stress can be present if layered structures are made. It is due to fabrication processes that use materials/processes that have different thermo-mechanical properties.
Intrinsic stress can be uniform within a layer or have gradients in its distribution across the film thickness.
Removal of the substrate layer releases the stress but causes deformation.
Warped film
Liu
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Intrinsic Stress Introduced by Fabrication and Process History
Liu
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Stress Compensation
Liu
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Measuring of Intrinsic Stress
Testing of planar (large) areas. Testing local deformation/stress
Horizontal beams under intrinsic stress
Liu
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Measuring of Intrinsic Stress
Uniform compressive stress causes elongation or buckling of a double supported beam.Both tension and compression stress can be measured by such structures.
Ring and beam structure
For tensile stress in the ring when relaxed (underlying film released by etching) the ring expands and makes the beam compressed - up to the point of buckling (optical microscopy)
Kovacs
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Measuring of Intrinsic Stress if There is a Stress Gradient
Nonuniform stress in the film causes the film to curl. If stress gradient is positive (larger away from the substrate) the spiral will open as a bowl. If stress gradient is negative (smaller away from the substrate) the spiral will buckle down. Each type requires anchoring either in the center or at the perimeter.
Kovacs
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Measuring of Intrinsic Stress
Piezoelectric or piezoresitive sensor respond to curling of the film.
Kovacs
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Dynamics of the Beam
Damping is due to drag forces and/or to deformation in the sprig or other parts.
Any force can act as an external force: gravity, electrostatics etc. If the force has frequency dependence it will cause mechanical oscillations where the amplitude will be affected by “damping”.
Lumped mechanical structure is described as follows:
Kovacs
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Analogy to Electrical Filters
Working as in DC
At resonance, the deflection is Q times larger than in steady state.
Damping is represented by resistance, which reduces the quality factor Q
Bending and oscillation depend on stress and mass
Multifunctional Sensors
Control of arrays’ parameters (material & dimensions spring constant oscillation frequency & deflection) is ensured by IC technology
Integration of various functions possible in array sensors
Characterization techniques known in various science disciplines will have significantly improved sensitivity if used on the “diving boards”.
•Transduce signals from almost every domain to nanomechanical motion (bend/oscillate)
• thermal• chemical
biochemical (no labeling
for biding recognitions)• biological• magnetic • electrical• optical • mechanical (stress)
• Have high sensitivity (force
1.4x10-18N/Hz1/2, pH 1nm/5x10-5,
mass 1fg/Hz) and high speed
• Use minute quantities of analytes
• Fabricated by IC processing: •Si, Si3N4 with sensitizing films
for tailored applications• Dimensions: µm range
Smart Force Sensors: Cantilevers
Physical and Chemical Functions of Cantilevers
10-5 K
1.1 Hz/fg pJ
pN (5.6x10-18N)
Bending: 0.05 Å)
AFM Origin
pW fJ
active layer (Au)
H2 & O2 on Pt
Bimetal
photo induced stress
phase transformation
SAM
mH2O≈ng in zeolite m(T)
absorptionswelling
interface chargesvoltamograms
viscositytemperature
magnetic moment: 10-12 Am2
or magnetic alloys
Cantilever Arrays for Multifunction Sensors
Operation of cantilever arrays with various coatings
• Signal Detection:• Electrical (Piezoresistors)• Optical (laser detection systems)
• static mode (bending) • dynamic mode (tracking of
resonance frequency) • Future: Wireless (magnetic)
Integration with optoelectronics for miniaturization and functionality
signal pre-processing
Position-Sensitive-Detector
IBM Zurich/Concentris/Basel University
Identification and Elimination of Artifacts
Versatility of cantilevers allows to diminish main artifact effects: flow and temperature nonuniformity
Differential measurements with a reference cantilever
Fabrication of Cantilevers Using Si Technology
ITE, Poland
ITE, Poland
Fabrication of Piezoresistors Cantilevers
ITE, Poland
Two cantilevers covered by single stranded DNA.
Bending of the first cantilever by hybridization.
Bending of the second cantilever by hybridization.
The cantilevers bend by docking of molecules
Differential Bendingalso increases with concentrations
Measurements of Differencial Deflection
baseline
Single Cell Detection
Cell dimentions:1.46 µmx730 nmx350nm
5µm
Si3N4
L=15-400 µmW=5-50 µm
Anti-E. coli antibodies
Attachment of heat killed E. coli in various colonies
15x5 µm
JVST B, J. Illic et al. 2001.
0.5 pN force removes nonspecifically bound particles sensitivity pg/l
Force Amplified Biological Sensors
NRL, Baselt et al.
Foam cell/macrophage-Foam cell/macrophage-drivendrivenFoam cell/macrophage-Foam cell/macrophage-drivendriven
Smooth muscle cell-Smooth muscle cell-drivendrivenSmooth muscle cell-Smooth muscle cell-drivendriven
Unstable Unstable plaqueplaque
Stable Stable plaqueplaque
Constrictive arterial remodelingConstrictive arterial remodeling
Outward arterial remodelingOutward arterial remodeling(Galis and Khatri, (Galis and Khatri, 2002)2002)(Galis and Khatri, (Galis and Khatri, 2002)2002)
Normal arteryNormal arteryNormal arteryNormal artery
Examples of Cantilever Operation (Vulnerable Plaque Applications)
pH Sensor - Plaque has increased acidicity Interfacial stress depends on pH and sensitizing (thiol/acid)
layers on Au due to electrostatic force (Q-) in MHA
7.8 ng
Phase transition at T=32.5øC
Thermal Effects - Inflammation
Protein Recognition
Biochemical combined with magnetic sensor - SPIO in the macrophage rich regions
(40 pN antigen-antibody)IBM Zurich
NRL
monolayers
use lipid layer
Immunocytochemistry for plaque
Application to Vulnerable Plaque
• Array of multifunctional cantilevers to monitor:
• Temperature (inflammation)
• pH
• Flow (affects endothelium)
• Selected biochemical reactions (C-RP, MMP, bacteria-Chlamydia,
etc.)
• SPIO (agglomeration within VP)
•Realization
• Integration of Si based cantilevers with
• Magnetic layers (regions)
• wireless transmitter (if magnetostrictive layer is below the
active sensing layer)
• sensor (hard or soft magnetic tips)
• Piezo-resistive, - electric or dielectric layers for sensing or
detection
•Passivation layers for biocompatibility
here
NRL
IBM Zurich/Basel Univ.
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Summary
See a video on applications of various phenomena, known from the macro world, in MEMS.