Slide vật liệu nano và màng mỏng mertials review part 2

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3. Technical Review – Materials

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Stress: N/m2, Pa or psi
σ = P/A0
P: load on the sample
A0: original(zero-stress) cross sectional area
Strain: unitless

ε = (l-l0)/l0 = Δl /l0
l : sample length
l0: original(zero-stress) length

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Basics – Mechanics of Materials

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σ

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Hooke’s law
σ = Eε
E: Young’s modulus or
elastic coefficient

E

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Yield strength
The stress at which a material exceeds its
elastic limits and the material begins to deform
permanently

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Maximum tensile stress that a
material withstand without rupture

ε
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Shear stress and strain

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τ= F/A

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Shear modulus: G
G = shear stress/shear displacement angle
= τ/γ = (F/A)/(ΔX/L)
ΔX Area, A

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F

L

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F
Poisson’s ratio
Under axial load,
ε(axial) = Δl /l0
ε(transverse) = Δd/d0
Poisson’s ratio;
ν = transverse strain/longitudinal strain
= -ε(transverse) / ε(axial)
Typical values are 0.2 to 0.5
‰ Relation of E and G
E=2G(1+ ν)

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SCS(Single Crystal Silicon)

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Anisotropic : crystal
Elastic : catastrophic failure
Young’s modulus = 190GPa < 200(SS)
Hardness = 850 kg/mm2 > 660(SS)
Yield strength = 7x 109 N/m2 > 2.1x109(SS)
Mightier than we think !

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From Kovacs
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Limiting factors
- Crystallographic defects and planes
- Residual stress from high temperature
process and film structure

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How to overcome?

- Stress consideration from design stage
- Minimize defects during dicing, grinding and
polishing
- Tribological measure: coating and lubrication
- Low temperature process

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Because of anisotropy of cubic system,
elastic coefficient is 6x6 matrix in the form of,
C11 C12 C12 0
0 0
Cij =
0 0
C12 C11 C12 0
0 0
C12 C12 C11 0
0 0 0 C44 0 0
0 0 0 0
C44 0
0 0 0 0
0 C44

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for [100] axis loading

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Young’s modulus vs crystallographic orientation in SCS

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(111) planes can be
considered as isotropic

From J. Kim, D. Cho and R. S.
Muller, “Why is a (111) silicon
better mechanical material for
MEMS”, Transducers ’01, 11th Int’l
Conf. Solid State Sensors and
Actuators, Munich, Germany, June
10-14, 2001

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Doping effect
Tensile or compressive stress is induced on the

doped area from local contraction or expansion of
lattice, which becomes critical in thin membrane or
beam structure.
‰ p-type Si with Boron
- tensile residual stress
‰ n-type Si with Phosphorus
- compressive residual stress
‰ p++ etch stop: heavily doped Si ( with Boron >7
x1019cm-3) is frequently used for membrane or
beam fabrication

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Piezoresistivity
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Change of bulk resistivity by the mechanical stress
applied to the material
Stress Ỉ Strain Ỉ Volume change Ỉ Energy gap
change Ỉ Number of charge carriers change Ỉ
Resistivity change
SCS has a high pzr with excellent mechanical
properties, therefore is widely used for
electromechanical transducers

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Principle

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R0 = ρ0 l/wt

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ΔR/ R0 = Δ l/l - Δ w/w - Δ t/t + Δ ρ/ρ0

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ΔR/ R0

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Using Poission’s ratio, ν
Δ w/w = Δ t/t = -ν Δ l/l
Gauge factor GF (strain sensitivity) is,
Δ ρ/ρ0

ε

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GF = --------- = 1 + 2 ν + -----------

Dimensional change

ε
Resistivity change

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Advantages
‰ GF(SCS) = 80 to 200 compared to GF(metal) = 1
to 5
‰ High sensitivity and good linearity of pzr elements
‰ Robustness of Si
‰ Ease of integration with IC
‰ Resistors can be located where the stress is
maximum (on the surface)
‰ Relatively simple calibration and compensation of
pzr elements
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Origin of PZR: Many-valley energy theory

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Mobility is lowest along <100>
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Depending on the direction, an electron has
different combination of k1, k2, and k3 in SCS.
Silicon has three pairs of valleys. Valleys are
are identical except for orientation.
Constant energy surface with unequal lengths
(along principal axes) Ỉ different effective
masses and mobilities Ỉ electrons make
anisotropic contribution to conductivity
Uncompressed : all valleys are equally
populated Ỉ isotropic conductivity
Under anisotropic stress: relative energies

change Ỉ electrons transfer from one valley to
another Ỉ populations change, mobilities
change Ỉ anisotropic conduction

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Longitudinal and transverse
piezoresitance coefficient

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Longitudinal pzr

Transverse pzr

ΔR/R = πl σl + πt σt
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Measurement

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T

[-110]

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[010]

[100]
1 dR
T R

= πl1

[110]

πl2

½(πl1+ πl2+ π44)

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For SCS (cubic system)

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‰ General expression of pzr for SCS

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‰ PZR coefficients for cubic crystal

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‰ PZR coefficients for SCS at RT (unit: 10-11 Pa-1)
p-Si
n-Si

ρ (Ohm-cm) π11
7.8
6.6
11.7
-102.2

π12
-1.1
53.4

π44
138.1
-13.6

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<110>, p-Si

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ΔR/R = πl σl + πt σt

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‰ Resistance changes as a function of stress

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ΔR/R = (π44 /2) * (σl - σt )
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<110>, n-Si
ΔR/R = (π11 + π12)/2 * (σl + σt )
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πl

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[-100]

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‰ Piezoresistance coefficient vs. orientation:
(001) p-Si

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[010]

πt
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