by w.h. müller , c. liebold icms-workshop | edinburgh | june 17-21, 2013

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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik

Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller

Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2013 /44

LKM

by

W.H. Müller, C. Liebold

ICMS-workshop | Edinburgh | June 17-21, 2013

Are higher gradient theories of elasticityamenable to experiments?- A state-of-the-art review

Technische Universität BerlinTechnische Universität Berlin Institut für Mechanik - LKM Institut für Mechanik - LKM Einsteinufer 5Einsteinufer 5 D-10587 BerlinD-10587 Berlin

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Supported by:

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Outline Introduction and motivation:

Applications in micro-mechanics The size effect (experiments from literature)

Overview on non-local continuum theories Beam bending in couple stress theory The AFM tool for deflection measurement Performing the experiments:

Force calibration of the AFM The beams

The Raman effect for strain measurement: General remarks Discussion of the penetration depth of the laser

Results for Si and SiN Conclusions about the length scale parameter of Si and SiN Improvements / discussion

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Applications in micro-mechanics I

A MEMS application:Micro 3D - gyroscope

A MEMS application:Micro-sensors and black box in automotive technology

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Applications in micro-mechanics II Objective: Stress and deflection measurements in micro beams subject to bending

A MEMS application:An acceleration sensor inthe automotive industry

Raman peak position along the thickness of a deflected silicon beam; Srikar et al. (2003):

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The size effect (experiments from literature)

• Objective: Determination of the material length scale parameter l (additional stiffness parameter) of silicon and silicon nitride using atomic force microscopy and Raman spectroscopy.

• A material length scale parameter called l is of importance in non-local theories, such as the couple stress theory used, e.g., by Yang et al. (2002).

• Strain gradient theories can be used to model the size effect of micro-size structures, e.g., for simple beam bending or torsion of small wires.

Left: Torsion of small copper wires performed by Chong et al. (2001) revealed: l = 3µm.

Right: Simple beam bending experiments on epoxy by Gao and Park (2007) revealed: l =17.6µm.

• The term “size effect” refers a stiffer (elastic) material behavior on the sub-micronscale.

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2 3 5 (3) 7

local - non-local theory

class. theory CS Couple Stress theory

MSG Modified Strain Gradient theory

SG Strain Gradient theory

stored energy density:

stress and strain

measures:(isotropic)

Sij

Sij

ijjiSij

jkijki

Sij

Sijijij

Glµ

ue

µu

2

,,21

,21

21

21

2

kijijk

kiiiikillikk

jjkiikijkijkijkijk

ijkijkijij

ddd

dd

u

5

43

21

21

21

balance law:

E – Euler Bernoulli T – Timoshenko (allg.)

workvirtualofprin.

uuµ

xwuuxwzuSijij

Sijij

zyx

,,

)(,0,)(

workvirtualofprin.

uuµ

xwuuxzuSijij

Sijij

zyx

,,

)(,0,)(

published derivations:

analytical displacement field:

)(E xw )(CSE xw )(MSG

E xw )(T xw )(CST xw )(MSG

T xw

Non-local theories (overview)

0,, ijijkiij

nr. of parameters:

jjiiijijijij

ijjiij

ijij

G

uu

u

2

,,21

21

0

2

2

2

)1(,,,2

1,

SS22

S

)1()1(21

)1()1(

,,20,

)1()1(,2

1

ijijkikiS

ilijjlkiik

ijijijij

ijkijkijkijk

illittikki

Sij

Sijijkijkikkiijij

pµe

Glµ

Gl

Glp

µpu

in use for finite element method by various authors

0,21

, titlijls

iij µTiij ,0,

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Couple stress theory for beam bending IDerivation of the deflection curve, the strain and the bending rigidity.

• The Euler-Bernoulli displacement field: )(,0,)( xwuuxwzu zyx

• Components of the strain tensor: 0,,,21

yzxzxyzzyyx

xxijjiij wzxuuu

0,)(~2

constantsLamé211

,12

,2

zzyyyzxzxyxxxxxx

ijkkijij

xwzEG

EEGG

EEν

νννEE

~0

2111:~

,

• Components of the stress tensor:

0,)()( 221 S

yzSxz

Szz

Syy

Sxx

Sxy

Sxy xwGlµxw

• New: Components of the rotation gradient tensor:

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• The principle of virtual work:

LLL

LLLL

L

V

Sij

Sijijij

wMwFxwxqW

wwZwwZwwZxwwZU

xwGAlEIVµU

Z

000

000

IV

0

0

22

δδdδ)(δ

δδdxδdδδ

d22

1d21

:

LxxMxwZQxwZ

LxxqxwGAlEIWU

or0)()(

),0()()(δδ IV221• The differential equation:

• Integration of the differential equation:

• The governing boundary conditions:

432

2213

161)(

dddd0)()(

CxCxCxCxZw

xxxxxqxwZ

Couple stress theory for beam bending II

• Using the boundary conditions for a clamped beam the solution for deflection and strain reads:

LxGAlEI

FxLxx

GAlEI

Fxw xx

221

CS23

221

CSE )(,

26)(

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Couple stress theory for beam bending III• Derivation of the bending rigidity for x=L, ν=0:

CSE32

2 33

)(

:

wLI

hElEF

LDwF

CSD

Figure: Simple beam bending experiments on epoxy by Gao and Park (2007) revealed: l=17,6µm.

inverse quadratic function in h

• Conclusion: - If l=0 the couple stress bending rigidity results in the conventional one.- The formula of the couple stress theory predicts a size effect.

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The AFM tool I

movementpiezo...x

sxw beam

cantileverAFMtheofbending...s

• The experiments for measuring w and ε: A Raman spectroscope integrated in an AFM used for the bending of µm beams

observation variables:

AFM laser

AFM Laser

Raman laser

Raman Laser

The central equation of the AFM tool for determining the beam deflection reads:

LASERPSD

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The AFM tool II• 1st. The non-linear behavior of the PSD (Photo Sensitive Diodes):

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Wheatstone bridge

„signal=voltage“

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+2

-2

0

tip deflection

U [mV]

non-linear behavior

AFM Laser

cylindrical AFM probe

range of linearity

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The AFM tool III• 2nd. Calibration of the tip deflection Δs:

The Δx feed motion was produced by a calibrated screw thread driven by piezo-elements.

m

0 xmax

-single experimentx [nm]

- rescaled feed motion curves - slopes increase linearly!

analysis points

U [m

V]

U [m

V]

Tapping the AFM probe against a rigid surface projected the Δx-input onto the voltage signal of the PSD.

This gave rise to a quadratic relationship of the form: xnaxxxmU )()(

top view of piezosturning the feed table

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[mV

]

The AFM tool IV• 3rd. The AFM probe against a micro beam (separating both deflections):

s

axnax

an

an

xwaU

an

anxnxaxU

rig2rig

2rig

rig

rigbeam2

2

)(

2 )(4242

arig, nrig … values of the regression line of the rigid surfacea, n … values of the regression line with the micro beam in between

s…defl. of the tip wbeam … defl. of the beam

x [nm]

• Conclusion: This approach compensated the non-linearity of the AFM tool.

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PtBPtBwkF

Performing the experiments I• The force calibration of the AFM probe against a calibrated cantilever:

The AFM cantilever was calibrated at the PtB (Physikalisch Technische Bundesanstalt, Braunschweig, Germany) by a nano-compensational weight bridge for the force data in combination with a capacitive read out for the deflection data.

It revealed a spring constant of the cantilever of about: F=kPtBwPtB , kPtB=4.868 N/m

Transfer of the spring constant

skF **

mNPtBPtB*

*

48.3

swk

k

FF

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Performing the experiments II• SEM analysis for determining the dimensions of the tested micro-beams:

b=39±0.2 µm

L=200±0.2 µm

t=0.98±0.075 µm

• Pictures via the optical microscope of some specimens:

force application point

SiN SiN SiN SiN SiN cSi

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Performing the experiments III• A video recording of the loading- unloading sequence via the optical microscope:

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(0)

(1)

(2)

(3)

(4)

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The Raman effect for strain measurement I• The setup of our Raman spectrometer:

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• The optical path of the laser beam:

1: live cam2: beam splitter3: polarizer (scattered)4: aperture (slit)5: notch filter6: optical grating7: CCD camera8: filters for laser power9: shutter10: beam expander11: polarizer (incident)12: sample stage13: optical lenses14: shutter15&16: laser source

The Raman effect for strain measurement II

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A typical spectral distribution of silicon. I. DeWolf (1998)

Si Raman peak @ ω 520 cm-1

The Raman effect for strain measurement III• Inside the CCD camera (qualitatively):

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matrix"constantspringeffective"skg,0

000

2effeff

eff

eff

eff

with

KsKsM

zz

yy

xx

sKsmsKsmsKsm

• With the idea of effective force constants in the directions of the generalized coordinates of the phonons, we obtain the secular equation without presence of strain:

1effeff2eff2eff ˆ,0ˆdet0det jlojololololol MKKKMK with

Some theoretical relationships The Raman effect for strain measurement IV

Si Raman peak @ ω 98,5 THz 520 cm-

1

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• Acc. to the so-called morphic effect, the strains εkl enter the secular equation as follows:

][ˆˆ,0ˆdet4

eff0

eff2eff0, εPKK

withijklijklij PK

Δwith

with

00020

2

220

20

eff0,

2))(()(,0det

0det

ˆ

ijklijkl

ijklijklij

ijij

P

P

K

we can rewrite the secular equation:

Some theoretical relationships The Raman effect for strain measurement V

• Consider pure tension in x-direction:

0,

,,0,

xyyzxz

xxzzzxxyyy

xx

000

0000

det

xxyxxz

yzxx

yzxx

qpqpq

qqp

the secular equation under presence of strain

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ω0

Δω

0

4001,1:

xxremember

00

1

03

Δ4.1Δ2

Δ2

xxqqypzxx

yzxx qqp

• A separation of the three eigenvalues is possible by polarization of the incoming and scattered laser light (Raman selection rules), so that the strain-shift relationship reads:

Some theoretical relationships The Raman effect for strain measurement VI

Experimental procedure

-reference-

• Taking spectra in loaded and unloaded configuration at different points around the fixture:• numerical analysis for spectral position: Gauss / Lorentz mixture -loaded-

this holds for silicon in the present configuration

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Raman @ 190 µNRaman @ 81 µNEuler Bernoulli-TheorieFEM @ 190 µN

F x=0

strainεxx

The Raman effect for strain measurement VIIExperimental application to silicon

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*De Wolf, I. (1996). Micro-Raman spectroscopy to study local mechanical stress in silicon integrated circuits. Semicond. Sci. Technol. 11, pp.139–154.*Aspnes, D. E., Studna, A. A. (1983). Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV . Am. Phys. Society. 27(2), pp. 985-1009.

chEPhoton

x

I

D

dp

I0

• Theoretical penetration depth of an extended laser beam in silicon (De Wolf*, 1996):

2t

• Specification for the choice of d:

with Aspnes* (1983):

nm 1200

ε

F

The problem of penetration depth of the laser I

= absorption coefficient measuredfor Si at different wavelengths

0.1 chosen

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x

I0

2tdp2pd

εmax

ε=0

Ramanp

corr

:

f

dtt

0

The problem of penetration depth of the laser II• Because the used beam thicknesses are 7000 nm, 3000 nm and 1000 nm a correction factor f for the strains can be introduced as follows:

thickness f from experiments f from formula

t=1000 nm4,2

56,3

t=3000 nm1,71

1,71,78

t=7000 nm 1,19 1,2

increase of uncertainties

fexp.

calculated by conventional theory

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Results I• Corrected strain values for silicon micro beams measured using the Raman spectroscope:

factor: f=1.19

t=7µm, L=175µm

t=3µm, L=90µm

factor: f=1.71

t=3µm, L=130µm

factor: f=1.78

measurement at different points around the fixture

measurement at different points along the beam length

corr

corr

corr

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Results II

• No strain values for silicon nitride, due to a thin metallic layer.

• The corresponding Raman signal is smeared out.

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Results III• The deflections of silicon micro beams measured via the AFM tool:

t=3µm, L=130µm

t=7µm, L=175µm t=1µm, L=130µm

t=1µm, L=50µm

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Results IV• The deflections of silicon nitride micro beams measured via the AFM tool:

t=0,8µm, L=170µm t=0,8µm, L=80µmt=0,8µm, L=120µm

t=0,6µm, L=170µm t=0,6µm, L=80µm t=0,2µm, L=29µm

51802

2

EIFL

… otherwise large bending angle

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1 2 3 4 5 6 7 8

Conclusions about the length scale parameter I

EDwLI

tElEF

LwDF

hD

0CSE32

2

,33

)(

:

• The bending rigidities of silicon micro beams measured via the AFM tool:

• Conclusion: The material length scale parameter of single crystalline silicon is smaller than l < 200 nm.

Bending rigidity vs. thickness of Epoxy

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Conclusions about the length scale parameter II• The bending rigidities of silicon nitride micro beams measured via the AFM tool:

• Conclusion: The material length scale parameter of silicon nitride is smaller than l < 50 nm.

0,1 0,2 0,3 0,6 0,8 1

Bending rigidity vs. thickness of Silicon Nitride

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A minimization of the penetration depth of the Raman laser can be achieved either by an UV laser source in combination with a UV spectrometer or by an independently moveable AFM-TERS probe. For detecting a size effect even smaller micro beam geometries (particularly smaller thicknesses) are needed.

Improvements / Discussion

Raman / Silicon:

In order to detect a more pronounced size effect materials with a more complex micro-structure should be tested.

Raman / Silicon-Nitride: For detecting a Raman signal the metallic layer should be removed (e.g., by etching).

Size Effect:

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Thank you for your patienceand endurance when listening to

(experimental)mechanics

presentations!

by w.h. müller , c. liebold icms-workshop | edinburgh | june 17-21, 2013

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