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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

3

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

6

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

€

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)

€

σ =EsE is modulus of elasticity or Young’s modulus (intrinsic material property)

(later)

Poisson’s ratio

€

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

€

sl =ΔL

L0

€

st =ΔD

D0

€

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)

€

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

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

€

T = Cs C is the Stiffness matrix

s- Strain matrix:

€

s = ST

€

S = C−1 S is the compliance matrix:

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Example for Silicon Orientation Effects

€

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.

€

x =Fl3

3EI

€

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)

€

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

€

k = 2(Ewt 3

l3)

€

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

€

max =d

LMaximum shear stress is related to the torque

€

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

€

Φ=d

r0=Lθ

r0=

L

r0

τ max

G=LTr0

r0GJ=TL

GJ

€

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.