Transmission Electron Microscopic Investigation of ...

Transmission Electron Microscopic Investigation of Transmission Electron Microscopic Investigation of Intergranular Glassy FilmsIntergranular Glassy Films

ANANDH SUBRAMANIAMDepartment of Applied Mechanics

INDIAN INSTITUTE OF TECHNOLOGY DELHINew Delhi-

110016Ph: (+91) (11) 2659 1340, Fax: (+91) (11) 2658 1119

[email protected], [email protected]://web.iitd.ac.in/~anandh

http://web.iitd.ac.in/~anandh/downloads/IGF.ppt

August 2007

This talk is dedicated to the fond memory of

Prof. Rowland M Cannon 1943-2006

Last at Lawrence Berkeley National Laboratory

University of California, Berkeley

OVERVIEW OF INTERGRANULAR GLASSY FILMS (OVERVIEW OF INTERGRANULAR GLASSY FILMS (IGFsIGFs))

SystemsSystems

Formation of an Formation of an IGFIGF

Effect of Various StimuliEffect of Various Stimuli

Theoretical & Computational MethodsTheoretical & Computational Methods

EXPERIMENTAL METHODS TO STUDY THE STRUCTUREEXPERIMENTAL METHODS TO STUDY THE STRUCTURE

TRANSMISSION ELECTRON MICROSCOPY (TEM)TRANSMISSION ELECTRON MICROSCOPY (TEM)

Diffuse Dark Field Imaging Diffuse Dark Field Imaging

High Resolution Electron Microscopy High Resolution Electron Microscopy

FresnelFresnel

Fringe ImagingFringe Imaging

OUTSTANDING ISSUESOUTSTANDING ISSUES

FINITE ELEMENT ANALYSIS AT THE NANOSCALEFINITE ELEMENT ANALYSIS AT THE NANOSCALE

OUTLINEOUTLINE

Keywords: Intergranular glassy film, amorphous structure, equilibrium thickness,

partial ordering, diffuse interface, Transmission Electron Microscopy.

Development of new techniques

INTERFACES

Based on misorientationof the bounding grains

Base

d on

che

mis

try

Base

d on

ord

er

Based on phase of the bounding grains Pure

With segregation

Ordered

Disordered

Special Random

Homophase Heterophase

Low angle CSL

Monolayer Multilayer

Possible ways of looking at internal interfaces in materials. The descriptions that fit Intergranular Glassy Films have been shaded.

[1] "Intergranular glassy films: an overview" Anandh Subramaniam, C.T. Koch and M. Rühle, Materials Science and Engineering A, 422 (2006) 3-18.

[1]

0.66 nm

1.4 nm

IGF

Grain-1

Grain-2

0.66 nm

1.4 nm

IGF

Grain-1

Grain-2

High-resolution micrograph from a Lu-Mg doped Si3

N4

sample showing the presence of an Intergranular Glassy Film (IGF).

IGF

Nearly constant thickness

Basically independent of the orientation of the bounding grains

Amorphous grain boundary film ~1-2 nm thick

Dependent on the composition of the ceramic

Independent of amount of glass excess glass goes to n-pocket

Resistant to crystallization

Composition and atomic structure different from the bulk glass

Basics

Systems in which IGFs

form

Ceramic-ceramic interface:

Si3

N4

(Undoped and doped with MgO, Y2 O3 , Rare Earth oxides, CaO, F, Cl)

SiC

Al2

O3

SrTiO3

ZnO

(Bi2

O3

doped)

Si2

N2

O

SiAlON

TiO2

-SiO2

Mullite

Ceramic-ceramic heterointerface

in composites:

Ruthenate

(Ru2 O, Pb2 Ru2 O7 , Bi2 Ru2 O7 ) - silicate glass (PbO-Al2 O3 - TiO2 -

SiO2 ) composites

Si3

N4

-SiC

Metal-ceramic interface:

Al2

O3

-Cu

Al2

O3

-Ni

Si3

N4

-Al.

In addition to the grain boundaries IGFs

are also found between the crystallized triple pocket phase and the grains in Si3

N4

and SiC.

Amorphous surface films

also show equilibrium thickness:

Si1-x

Zrx

O2

films on Si

Bi2

O3

films on ZnO

Epitaxial crystallization(ABC-SiC, 1100ºC, 500h)

Desegregation(Bi2O3 doped ZnO, 700ºC, 1 GPa, 2 h)

Thinning of IGF()

Stable films with altered thickness & wide films

(Si3N4 + MgO, quenched from 1350-1650ºC)

Pressure

Provides a weak path for intergranular fracture

Bimodal film size distribution(Si3N4 + Y2O3, 1430ºC, 40 MPa, 690 h)

Thinning of IGF(Si3N4 + Yb2O3 + Al2O3, 1400ºC, 168 h, air)

FractureCreep

Oxidation

Hea

ting

Liquid phase sinteringSi3N4, SiC, Al2O3, SrTiO3

Solid state activated sinteringBi2O3 doped ZnO

Crystallization of glassSiO2-TiO2

(+ quenching)

(annealing)

Principal methods of IGF formation Effect of various stimuli and treatments on IGF

Surface films:Heating 700ºC, 780ºCBi2O3 doped ZnO powders

Surface films:Heating 700ºC, PO2<10-50atm

Si1-xZrxO2 films on Si

IGF

Principal methods for the formation of IGF

& effect of various stimuli

Properties influenced by the structure of

the IGF

Fracture toughness

Oxidation resistance

Creep resistance

Electrical properties (ZnO, SrTiO3 )

Effect of composition of the ceramic

IGF

thickness in Si3

N4

Ca Content

Addition of

Oxides

C.M. Wang, X. Pan, M.J. Hoffmann, R.M. Cannon, M. Rühle, J. Am. Ceram. Soc. 79 (1996) 788-92.

I. Tanaka, H.-J. Kleebe, M.K. Cinibulk, J. Bruley, D.R. Clarke, M.Rühle, J. Am. Ceram. Soc. 77 (1994) 911-14.

Crystal-1

Crystal-2

PocketIGF

Graded ‘equilibrium’ IGF

Transition regionfrom PocketIGF to

Bulk

IGF-POCKET CONFIGURATIONS

TP-IGF

region

Segregation of Ca

IGF-2Crystal-1

Crystal-2

IGF-1t1

t2

Different CrystalStructure

Liquid-1

Liquid-2

IGF

Heterophase

IGF

Phase separation of glass in the pocket

Theoretical and computational methods to understand IGFs

Disordered GB

Seggregation

Phase separation

Force Balance

Wetting/Adsorbtion

Phase Field Model

Diffuse Interface Model

Near wetting precursor model

IGF

Ab-initiomethods

Molecular Dynamics/ Monte Carlo

Ways of looking at IGFs Atomistic simulation Methods Continuum theoretical Models

Understanding IGF

as an multilayer segregation

‘Glassy’ region having finite thickness

Progressive addition of a species preferentially segregating to a grain boundary

Monolayer Multilayer

DispersionForces

StearicForces

Space ChargeForces

Repulsive

Attractive

Attractive orRepulsive

Force balance model for equilibrium thickness of IGFs

Schematic of two possible energy-

displacement curves leading to equilibrium thickness of film

Clarke’s model: Forces leading to equilibrium

(a)

(b)

Minima

Minimum at zero displacement

DisplacementEn

ergy

Curves (a) and (b) are displaced verticallyfor visibility

Ackler

and Chiang’s model

Clarke’s model

Clarke’s model:•

Repulsive forces balance the attractive forces to give an equilibrium thickness for the film.•

Orientation effects of the grains are ignored in the model.•

Dry boundaries are explained in terms of an interfacial energy criterion

GB (Clean)

SL

GB (a = a*)

Dry boundaries

Cusps corresponding to special boundaries

Sum of the two crystal-glass interfacial energies

Grain misorientation

Inte

rfac

ial E

nerg

y

Explanation of dry boundaries based on interface energy versus misorientation plot. The hatched dry boundaries are based on GB (clean).

The GB (a = a*) corresponds to the GB energy with adsorbate coverage at an equilibrium concentration, appropriate to the two phase co-existence condition

Dry Boundaries

In materials like

Si3

N4

it is observed

that not all boundaries

have an IGF

i.e. some boundaries

are dry

Experimental methods to understand IGFs

Experimental methods

Scanning Electron Microscopy (SEM)

Auger Electron Spectroscopy

Transmission Electron Microscopy (TEM)

Impedance Spectroscopy

Mechanical Spectroscopy

Thickness & Structure of IGFs

by

Transmission Electron Microscopy

High-resolutionLattice fringe imaging

Fresnel

fringe techniquesDiffuse dark field imaging

Diffraction analysis Electron holography

High-angle annular dark field imaging

Energy dispersive X-ray analysis

Electron energy loss spectroscopy

"Aspects regarding measurement of thickness of Intergranular Glassy Films"

S. Bhattacharyya, Anandh Subramaniam,

C.T. Koch and M. Rühle

Journal of Microscopy, 221, p. 46, 2006.

Important Advances

Technology of microscopes:

(i) Spherical aberration correctors for illumination as well as imaging lens systems, (ii) Electron beam monochromators

reducing the energy spread of the electron beam

to <0.1eV, (iii) High-energy resolution spectrometers, (iv) Imaging energy filters, (v) Advanced specimen holders for in-situ experiments, (vi) High-sensitivity and high-dynamic-range detectors for images, diffraction

patterns, electron energy loss spectra (EELS), and energy

dispersive X-ray

spectra (EDXS).

New methods, such as focal series reconstruction has added to its utility. Thus, it has

become possible to record phase contrast images with point resolutions of well below

0.1nm, or alternatively, scan electron probes of size <0.1nm

across the specimen,

recording high-angle scattering or high-energy-resolution spectra from areas as small

as a single atomic column.

In some materials (e.g. stoichiometric

SrTiO3

), the frequency of GBs

with an

IGF

is very small.

In samples with large grain-size (~tens of microns), this problem is even more

severe

The grain boundaries in stoichiometric

SrTiO3

are usually not planar and only a

portion of the GB can be made approximately edge-on. This implies that tilt

along an axis perpendicular to the GB (i.e., keeping boundary

edge-on) is severely limited.

Sample and Microscope imposed constraints in HREM

of IGFs

Material related

Microscope related

Related to resolution and available tilt angles.

For SrTiO3

only five planes fall into the resolution range of a standard

microscope like JEOL

4000EX [(001), (101), (111), (002) and (102)].

In this microscope, with a top entry holder, tilt available is 10º

and sharp lattice

fringe contrast sometimes cannot be obtained on both sides of

the IGF

within this

tilt regime.

Direction of the electon beam

Two planes in the path of the electon beam with differing order at the GB

Direction of the electon beam

True disorder versus other cases (as seen in TEM)

True disorder at the boundary

Formation of an IGF

Roughening at the GB

Strain at the grain boundary due

to presence of dislocations etc.Roughening at the GB

TEM

JEOL

400 FX

400 kV( ~ 0.2 nm)

EXPERIMENTAL DETAILS

JEOL

400 EX

400 kV( ~ 0.18 nm)

Zeiss

912

120 kV( ~ 0.8 nm, c = 1 mrad)15 eV energy window for

energy filtered images ( filter)

SAMPLES*

Lu2

O3

-MgO doped Si3

N4 SrTiO3

Stoichiometric: TiO2 / SrCO3 = 1/1

Non-stoichiometric: TiO2 / SrCO3 = 1/0.98

* Courtesy: Prof. M.J. Hoffmann, Dr. Raphaelle Satet, Univ. of Karlsuhe

No visible Grain Boundary

2.761 Å

Fourier filtered image

Dislocation structures at the Grain boundary

counts

eV

10000

11000

12000

13000

14000

15000

16000

17000

18000

19000

counts

1850 1900 1950 2000 2050 2100 2150 2200 2250eV

counts

eV

11000

12000

13000

14000

15000

16000

17000

18000

19000

20000

counts

1850 1900 1950 2000 2050 2100 2150 2200 2250eV

Si

peak at 1839 eV Sr

L2,3

peaks

Grain Boundary

Grain

eV1900 2000

EELS

2100 2200

~8º

TILT BOUNDARY IN THE SrTiO3

POLYCRYSTALGB23x4

MX23x4

Dist 5 nm

VG microscope

Fourier Filtering

FFT from the GB region

~7º

8º

6.5º

Unmasked regions

2.761 Å

SITI-3AUG-G1-30

S1T1-3AUG-G1-22-P2-math S1T1-3AUG-G1-30-P1-mathS1T1-3AUG-G1-22-P1B-math

SAD Patterns from the boundary~8º

TILT BOUNDARY IN THE SrTiO3

POLYCRYSTAL

Position

Stru

ctur

al p

aram

eter

lattice fringesHRTEM

Fourier filteredHRTEM image

(to removelattice fringes)

Position

Chem

ical

par

amet

er

Core

FWHM

ELNES width

Thickness of an IGF!!

Fourier Filtering Intensity profiles

Window for intensityprofile

40 nm

16 nm

22 n

mA

A

A A

aFWHM ~ 3.2 nm


40 nm

16 nm

22 n

mA

A

A A

aFWHM ~ 3.2 nm

40 nm


16 nm

22 n

mA

’A’

A’ A’

aZZ = 3.1 nm

40 nm


16 nm

22 n

mA

’A’

A’ A’

aZZ = 3.1 nm

(a)

Distance (Å)

a = 11 Å

aFWHM = 20.4 Å

a0 = 32.4 Å P (a)

Distance (Å)

a = 11 Å

aFWHM = 20.4 Å

a0 = 32.4 Å P

(b)Masked area (0.026 Å1)

Reciprocal distance (Å1)

(b)Masked area (0.026 Å1)

Reciprocal distance (Å1)

(c)

Distance (Å)

a’ = 11 Å

aZZ = 20.9 Å

aPP = 31.9 Å (c)

Distance (Å)

a’ = 11 Å

aZZ = 20.9 Å

aPP = 31.9 Å

Diffuse Dark Field image from a grain boundary in SrTiO3

Fourier filtered image using a mask of 0.073 nm-1 diameter

Fresnel

Contrast Images (FCI)

20 nm20 nm A

A

20 nm20 nm A

A

20 nm20 nm B

B

20 nm20 nm B

B

10 15Distance (nm)50

Inte

nsity FWHM

P P


Inte

nsity FWHM

P PP P

Inte

nsity FWHM

V V


Inte

nsity FWHM

V V


FCI from Lu-Mg doped Si3

N4

Defocus value of +1.2 m

Defocus value of 1.2 m

Ideal profiles assumed in literature

Overfocus

Underfocus

The Broken Symmetries

Ideal profiles assumed in literature

Underfocus

Overfocus

This plot shows two features: (i) Left-right mirror symmetry (LR

symmetry), (ii) inversion symmetry across the x-axis between overfocus

and underfocus

profiles (OU

symmetry)

These symmetries are broken in real Fresnel

contrast images

Fresnel

fringe extrapolation method to measure the thickness of an IGF

In the standard Fresnel

fringe extrapolation method

The primary Fresnel

fringe spacing is measured at various values of underfocus

and overfocus The measured spacing values are plotted vs

the square-root of defocus

Two separate straight line fits are passed through the points measured

for +ve

and ve

defocus values The lines are extrapolated to zero defocus (in-focus) The average of the two extrapolated values is taken as the width of the IGF

Frin

ge sp

acin

g

(Defocus)

1 20

/s s c f

Average of two values gives the IGF

thickness

Fourier Filtering Fresnel

Contrast Images

00 -80 -60 -40 -20 0 20 40 60 80 100

23.3 Å

42.2 Å

Distance (Å)00 -80 -60 -40 -20 0 20 40 60 80 100

23.3 Å

42.2 Å

Distance (Å)00 -80 -60 -40 -20 0 20 40 60 80 100

5

4

3

2

0

2

3

4

5

Distance (Å)

23.3 Å

42.2 Å

00 -80 -60 -40 -20 0 20 40 60 80 10000 -80 -60 -40 -20 0 20 40 60 80 1005

4

3

2

0

2

3

4

5

Distance (Å)

23.3 Å

42.2 Å

00 -80 -60 -40 -20 0 20 40 60 80 100

20 60Distance (Å)

-20 0-60

20.4 Å

-40 40-80 8020 60Distance (Å)

-20 0-60

20.4 Å

-40 40-80 80 20 60Distance (Å)

-20 0-60

20.4 Å

-40 40-80 80

Zero intensity line

20 60Distance (Å)

-20 0-60

20.4 Å

-40 40-80 80

Zero intensity line

(a)

Distance (Å)

a = 11 Å

aFWHM = 20.4 Å

a0 = 32.4 Å P (a)

Distance (Å)

a = 11 Å

aFWHM = 20.4 Å

a0 = 32.4 Å P


Zero-defocus (in-focus)


Defocus value of 1.2 m

20 nm20 nmC

C

20 nm20 nmC

C

20 nm20 nm D

D

20 nm20 nm D

D

Inte

nsity

P P

Z Z


Inte

nsity

P PP P

Z Z


Inte

nsity

V V

Z Z

128Distance (nm)60

Inte

nsity

V V

Z Z

128Distance (nm)60

Measurements from Fourier Filtered (FF) FCI from Lu-Mg doped Si3

N4

7

6

5

4

3

2

0

2

3

-20 0 20-40 40

20.4 Å0 V

6.44 V

(a)

4

6

8

4

6

8

1.0

1.2

1.4

(b)

(c)

(d)

m1 m2

7

6

5

4

3

2

0

2

3

-20 0 20-40 40

20.4 Å0 V

6.44 V

(a)

4

6

8

4

6

8

1.0

1.2

1.4

(b)

(c)

(d)

7

6

5

4

3

2

0

2

3

-20 0 20-40 40

20.4 Å0 V

6.44 V

(a)

4

6

8

4

6

8

1.0

4

6

8

4

6

8

1.0

1.2 1.2

1.4 1.4

(b)

(c)

(d)

m1 m2

1.0

1.2

1.4

1.6

1.8

2.0

-10 0 10-20 20 30-30

Overfocus (+) Underfocus ()

2.2

(e)

(f)

(g)

(h)

1.6 1.6

1.8 1.8

2.0 2.0

-10 0 10-20 20 30-30


2.2

(e)

(f)

(g)

(h)2.2

2.0

1.8

1.6 Simulation of FCIs

y = 37.952x - 13.456R2 = 0.9979

y = -37.952x - 13.456R2 = 0.9979

20

25

30

35

40

45

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2(f)1/2 ()1/2

Frin

ge sp

acin

g (s

) (Å

)

y = 37.952x - 13.456R2 = 0.9979

y = -37.952x - 13.456R2 = 0.9979

20

25

30

35

40

45

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2(f)1/2 ()1/2

Frin

ge sp

acin

g (s

) (Å

)

2.0

-10 0 10-20 20 30-30


2.2

(g)

(h)

2.0 2.0

-10 0 10-20 20 30-30


2.2

(g)

(h)2.2

2.0

1 20

/s s c f s → Fringe spacing, s0 → IGF thickness, f →: Defocus

The thickness of the IGF is not obtained even in the simulation !!

Plot of the fringe spacing data obtained from FCI

and FF-FCI. Lines AA and A’A’

are the extrapolation of the data of FCI

PP/VV. Lines CC and DD are the extrapolation of the overfocus

data in FF-FCI

PP/VV and underfocus

data in FF-FCI

ZZ.

-1

0

1

2

3

4

5

6

7

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

(Defocus)1/2 (nm)1/2

Frin

ge s

paci

ng (n

m)

FCI-FWHM

FCI-PP/VV

FF-ZZ

FF-PP/VV

A

A'

A'

C

CD

D

A

Lu-Mg doped Si3

N4

sample

Fresnel

fringes spacing extrapolation

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-20 -15 -10 -5 0 5 10 15 20 25(Defocus)1/2 (nm)1/2

spac

ing

(s) (

nm)

Region of approximate linearity (R2=0.99)

Region of approximatelinearity(R2=0.9968)

Spacing at zero defocus

15 nm

(a)

(b)

1 nm 1.1 nm1 nm 1.1 nm(c)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-20 -15 -10 -5 0 5 10 15 20 25(Defocus)1/2 (nm)1/2

spac

ing

(s) (

nm)

Region of approximate linearity (R2=0.99)

Region of approximatelinearity(R2=0.9968)

Spacing at zero defocus

15 nm

(a)

(b)

1 nm 1.1 nm1 nm 1.1 nm(c)

Fresnel

fringes spacing extrapolation

FF-FCI

ZZ data obtained from a GB containing an IGF

from the non-stoichiometric SrTiO3

(STO2) sample: (a) plot of the spacing (s) versus the root of defocus ((f)1/2), (b) profile across a image taken at an Overfocus

of 10nm, integrated over a width of 12.4 nm.

Fresnel

fringes hidden in high-resolution micrographs

Lu-Mg doped Si3

N4

sample

1.3 nm

0.66 nm

1.3 nm

0.66 nm

(a) HRM

10 nm10 nm10 nm10 nm10 nm10 nm

(b) FF-HRM

0 4 8Distance (nm)

FWHM

V V

(Two ways of measuring V V)

Inte

nsity

0 4 8Distance (nm)

FWHM

V V

(Two ways of measuring V V)

Inte

nsity

(c) profile across the GB

2 4Distance (nm) 60

Z Z

V V

Inte

nsity

2 4Distance (nm) 60

Z Z

V V

Inte

nsity

(d) profile across the GB in the FF-HRM

Stoichiometric

SrTiO3

sample

0.75 nm

(a) HRM

10 nm10 nm

(b) FF-HRM

2 3Distance (nm)

10

P P

Inte

nsity FWHM

2 3Distance (nm)

10

P P

Inte

nsity FWHM

(c) profile across the GB in the HRM

2 Distance (nm) 4

Z Z

P P

Inte

nsity

0 2 Distance (nm) 4

Z Z

P P

Inte

nsity

0

(d) profile across the GB in the FF-HRM

Thickness from high-resolution

micrograph (HRt)

Ft-

FWHM

Thickness after Fourier filtering the HRM

(FF-HRt)

(MacLaren thickness)

FF-Ft-

Zero to zero (ZZ)

From High-Resolution Micrograph (HRM)

Fourier Filtered

Raw Image

From profile across grain boundary

Ft-

Peak to peak (PP)/

Valley to valley (VV)

FF-Ft-

Peak to peak (PP)/

Valley to valley (VV)

Schematic showing the measurement of thickness from HRM. Thickness can be calculated directly from the HRM

(HRt) or by Fourier filtering the lattice fringes from the micrograph (FF-HRt). An easier alternative is to use the plot of profiles across these images. Values which correlate with each other are shown either by same shading

or with the same border.

Zero defocus imaging

2 nm2 nm2 nm

A

A

(b)(a)

1.2 nm

2 nm2 nm2 nm

A

A

(b)(a)

2 nm2 nm2 nm

A

A

(b)(a)

1.2 nm

Fourier Filtered-FCI from Si3

N4

–

grain boundary having an IGF

2.76 Å2.76 Å

Zero thickness

-5 0 5Distance (nm)

Zero thickness

-5 0 5Distance (nm)

Dry GB in SrTiO3

(a) HRM, (b) profile across the FF-FCI

showing zero thickness.

Sample(Lattice fringe orientation)

HRt(nm)

FF-HRt(Maclaren

thickness)

(nm)

From intensity (Fresnel) profiles

Ft

(nm)

FF-Ft (nm)

FWHM PP/VV Z-Z PP/VV

Lu doped Si3

N4(parallel to GB on one

side)

1.18 1.95 1.18 1.98-2.6* 1.18 1.98

StoichiometricSrTiO3

(STO1)(not parallel to GB on

either side)

0.95 1.51 0.95 1.53 0.94 1.51

* Depending on the choice of the VV distance

A comparison of the values of thickness obtained from:

High-Resolution Micrograph (HRt)

Plot of intensity profiles across the IGF

revealing Fresnel

fringes (Ft)

Fourier filtered (FF) high-resolution image (MacLaren

thickness (FF-HRt))

FF Fresnel

thickness (FF-Ft)

In-situ Heating Experiments

"The evolution of amorphous grain boundaries during In-situ heating experiments in Lu doped Si3

N4

"

S. Bhattacharyya, Anandh Subramaniam,

C.T. Koch and M. Rühle

Materials Science and Engineering A, 422, p.92, 2006.

In-situ heating experiments in Lu-Mg doped Si3 N4

400 kVJEOL 4000 FX

Lattice fringe imaging

120 kVZEISS-912

Fresnel

fringe techniques

0

100

200

300

400

500

600

700

800

900

1000

0 50 100 150

Time (minutes)

Tem

pera

ture

(ºC

)

JEOL 4000 FXZeiss-912

950ºC, 30 min

950ºC, 1hr 20 min

1.3 nm

0.66 nm

1.3 nm

0.66 nm

0.66 nm

1.3 nm

0.66 nm

1.3 nm

6.6 Å

31 Å

26 Å

6.6 Å6.6 Å

31 Å

26 Å2.2 nm

3.3 nm6.6 Å

22 Å

24 Å

2.2 nm

3.3 nm6.6 Å

22 Å

24 Å

400 kVJEOL 4000 FX

24oC 650oC

950oC24oC

0.8

1.3

1.8

2.3

2.8

3.3

Heating cycle

Thi

ckne

ss o

f the

IGF

(nm

)

650ºC, t=1 hrRT, t=0

Cooled to RT, t=2 hr 25 min

950ºC, t=2 hr 20 min

Wedge shaped IGF with a range of thicknesses

Crystal-1 Crystal-2Crystal-1 Crystal-2

Increased thickness t > tHT RT

Electron beam direction

Crystal-1

tCrystal-2

t1t

Crystal-1 Crystal-2

Wedge Shapedt2

t1

t > t2 1

Crystal-1

hickdary

Crystal-2

ness

Crystal-1 Crystal-2

Non-uniform thickness along the grain boundary

20 nm20 nm20 nm

A

A2.3 nm

20 nm20 nm20 nm

A

A2.3 nm

20 nm20 nm20 nm

B

B

2.5 nm

20 nm20 nm20 nm

B

B

20 nm20 nm20 nm

B

B

2.5 nm

20 nm20 nm20 nm

C

C 3.2 nm

20 nm20 nm20 nm

C

C 3.2 nm

Thickness during the heating cycle measured from Fourier filtered Fresnel

contrast images

Room Temperature

Cooled to Room Temperature

9500C

0

1

2

3

4

5

6

7

8

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2(Defocus)1/2 ()1/2

Spac

ing

(nm

)

RTHTCooled to RT

Extrapolation of the Fresnel

through focal series fringe spacing data to obtain the

thickness of the IGF: () RT, () 950ºC & () cooled to RT.

-40 0 40-20 20Distance (Å)

RTHTCooled to RT

-40 0 40-20 20Distance (Å)

RTHTCooled to RT

Comparison of exit wave reconstructed profiles, of the

imaginary part of the inner potential, using zero-defocus Fresnel

fringe images: () RT, (─) 950ºC, (--)

cooled to RT. The FWHM

measured from the profiles gives a measure of

the thickness of the IGF.

Thickness of IGF

using alternate methods

High-resolution (400 kV

microscope)

Fresnel fringes (120 kV microscope)

Temperature

HRt (nm)

ExtrapolationFt

(nm)

From zero-defocus image

(Fourier filtered)

(nm)

By zero-defocus Fresnel

reconstruction (FWHM, nm)

Room Temperature

(24ºC)

1.3

0.8 0.5

2.3 1.8

650ºC 1.3 - - - 950ºC 2.6 - 3.1* 1.0 0.1

2.5** 2.1**

Cooled to 24ºC

2.2 – 2.4 2.1 1.2

3.2 3.0

In-situ heating experiments: Summary of results

* micrograph was taken 1 hr 10 min after reaching 950ºC; ** micrograph was taken 15 min after reaching 950ºC.

In-situ heating experiments:Comparison of high-resolution vs

Fresnel

thicknesses

RT

HT

Cooled to RT

High-resolution

Fresnel

SUMMARY & CONCLUSIONSSUMMARY & CONCLUSIONS

Four new techniques have been developed for the measurement of the thickness

of IGFs:

Fresnel

Fringes hidden in high-resolution micrographs

Zero-defocus imaging

Fourier filtered Fresnel

congrast

imaging

Reconstruction of potential profile based on zero-defocus images

Limitations of the standard Fresnel

fringe extrapolation method are brought out

New finding during in-situ heating: fast changes to IGF thickness (in Lu-Mg doped Si3

N4

) can occur at

comparatively low temperatures (<1000ºC) under low irradiation doses

Low T Superdiffusion?

FINITE ELEMENT ANALYSIS FINITE ELEMENT ANALYSIS AT THE NANOSCALEAT THE NANOSCALE

Growth of Epitaxial Films

Stress Fields and Energetics

of Dislocations

EPITAXIAL THIN FILMSEPITAXIAL THIN FILMS

FILM

SUBSTRATE

INRTERFACIAL EDGE DISLOCATION

~100 Å

~100

EXAMPLES

GeSiSi

GaAsPGaAs

InGaAsGaP

AuAg

CoNi

Semiconductor

Metallic

FINITE ELEMENT ANALYSISFINITE ELEMENT ANALYSIS( Stress free strain)

1.

Constructing a strain-free layer of GeSi

on the Si

substrate

2.

Imposing the coherency at the interface through a lattice misfit strain

3.

Simulation is repeated for successive build-up of the layers to model the growth of the film

Elastic constants for the GeSi

alloy calculated by linear interpolation of values

Anisotropic conditions

Lattice constants at 550 0C –

the growth temperature

FILM

DISLOCATION

Edge dislocation is modelled by feeding the strain (Tdl

) corresponding to the introduction of an extra plane of atoms

b = as

/2 [110]

Tdl

= ((as

[110] + bs

) -

as

[110]) / (as

[110] + bs

) = bs

/3bs

=1/3

yx

S y m m e t r y l i n e(symmetric half of the domain taken for analyses)

Region of the domain (B)where Eshelby strain is imposed to simulate the dislocation

Region of the domain (A)where Eshelby strain is imposed to simulate the strained film

99 Elements

68 E

lem

ents

GeGe

0.50.5

SiSi

0.50.5

FILM ON FILM ON SiSi

SUBSTRATESUBSTRATE(MISFIT STRAIN = 0.0204)

x

AFTER THE GROWTH OF ONE LAYER ( ~5 Å)

50 Å

230 Å

(MPa)

Zoomed region near the edge

SUBSTRATE

FILM

EDGE

SYMMETRY LINE

GeGe

0.50.5

SiSi

0.50.5


SUBSTRATESUBSTRATE(MISFIT STRAIN = 0.0204)

x

AFTER THE GROWTH OF FIVE LAYERS14

5 Å

190 Å

(MPa)

SYMMETRY LINE

SUBSTRATE

FILM

EDGE

-5.00 -3.00 -1.00 1.00 3.00 5.00

x (Angstroms)

-5.00-4.00-3.00-2.00-1.000.001.002.003.004.005.00

y (A

ngst

rom

s)

-10.00-9.00-8.00-7.00-6.00-5.00-4.00-3.00-2.00-1.000.001.002.003.004.005.006.007.008.009.0010.00

(Contour values x 104 Mpa)

x = 222

22

)()3(

)1(2 yxyxyGb

-5.00 -3.00 -1.00 1.00 3.00 5.00

x (Angstroms)

-5.00-4.00-3.00-2.00-1.000.001.002.003.004.005.00

y (A

ngst

rom

s)

-10.00-9.00-8.00-7.00-6.00-5.00-4.00-3.00-2.00-1.000.001.002.003.004.005.006.007.008.009.0010.00

(Contour values x 104 Mpa)

x = 222

22

)()3(

)1(2 yxyxyGb

x PLOT OF THEORETICAL EQUATION

All contour values are in GPa272Å

272

Å

8.50

3.80

2.86

1.93

1.00

0.05

-9.90

-1.81

-2.75

-3.69

-8.39

x & y original grid size = b/2 = 2.72 Å

All contour values are in GPa272Å

272

Å

8.50

3.80

2.86

1.93

1.00

0.05

-9.90

-1.81

-2.75

-3.69

-8.39

8.50

3.80

2.86

1.93

1.00

0.05

-9.90

-1.81

-2.75

-3.69

-8.39

x & y original grid size = b/2 = 2.72 Å

Sim

ulat

ed

xco

ntou

rs

GeGe

0.50.5

SiSi

0.50.5


SUBSTRATE WITH EDGE DISLOCATIONSUBSTRATE WITH EDGE DISLOCATIONx

AFTER THE GROWTH OF FIVE LAYERS

80 Å

240 Å

MPa

Zoomed region near the edge

FILM

SUBSTRATE

SYMMETRY LINE EDGE

Critical thickness by considering total energy

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Number of layers of b/2 thickness (b = 3.84 Å)

Ene

rgy

(x 1

0-9J/

m)

Total energy of the growing film

Total energy of film with dislocaton

Mesh size = (1.92 x 1.92) Å

Domain size = (192 x 134.4) Å

Critical thickness of 20 Å= four film layers

GeGe

0.50.5

SiSi

0.50.5


SUBSTRATESUBSTRATE

Comparison with Comparison with experimental resultsexperimental results

0123456789

0.65 0.7 0.75 0.8 0.85 0.9

x in CuxAu(1-x)

h c (Å

)

Simulation

Theory

Comparison between theory and finite element simulation of the critical thickness for the onset of misfit dislocations in the CuxAu(1-x) /Ni system as a function of copper content x in the film.

8711hc (FEM simulated) (Å)<101013hc (experimental) (Å)

Cr/NiPt/AuCo/CuFilm/Substrate

8711hc (FEM simulated) (Å)<101013hc (experimental) (Å)

Cr/NiPt/AuCo/CuFilm/Substrate

Comparison between FEM simulation and experimental results of critical thickness (hc) for the nucleation of a dislocation in a coherently strained epitaxial film.

Metallic FilmsMetallic Films

5.00

5.50

6.00

6.50

7.00

7.50

8.00

8.50

0 5 10 15 20

Distance from centre (in b spacings)

Tot

al e

nerg

y (x

10-9

J/m

)

Domain Size = (272 x 272) Å

Mesh Size = (2.72 x 2.72) Å

Energy of an edge dislocation as a function of its distance from centre of the domain

All contour values are in GPa

Edge

8.4

3.4

2.4

1.4

0.5

-0.5

-1.4

-2.4

-3.4

-8.4

CENTRE LINE

20b = 109 Å

All contour values are in GPa

Edge

8.4

3.4

2.4

1.4

0.5

-0.5

-1.4

-2.4

-3.4

-8.4

CENTRE LINE

20b = 109 Å

Simulated x contours

Limitations of the current theoriesLimitations of the current theories

The energy of a growing film is assumed to be a linear function of the thickness

The substrate is assumed to be rigid and only the energy of the film is taken into account

Even though the energetics

of the substrate is ignored the energy of the

whole dislocation is taken into account

CONCLUSIONSCONCLUSIONS

Misfit strain is fed as the stress-free Eshelby

strain in the finite

element model to effectively simulate lattice mismatch strain in

an epitaxial layer

Feeding the stress-free strain corresponding to the introduction of an extra plane of atoms can simulate an edge dislocation

The equilibrium critical thickness for epitaxial films can be determined by a combined simulation of a growing film with a edge dislocation (The results obtained show a close correspondence with the standard theoretical expressions and experimental results for epitaxial metallic films)

For the GeSi/Si

system, the experimental values match satisfactorily

with that of the 'threshold approach', when the energy per unit area of the simulated dislocation is taken at a characteristic distance (xch

) of 5b.

Detailed Summary and Conclusions: Microscopy

(i) The utility of Fresnel contrast and high-resolution images can be enhanced by

Fourier filtering the images. This methodology, applied to high-resolution

micrographs, allows the use of images in which lattice fringes are weak (or even

absent) on one side of the IGF and in the case of Fresnel contrast images gives

better interpretability from noisy images.

(ii) A simple new technique is put forth, wherein Fresnel contrast hidden in high-

resolution images are used to objectively demarcate the glass-crystal interface and

to measure the thickness of an IGF (by Fourier filtering the lattice fringe spots).

The optimum mask size was determined by simulations on one-dimensional

scattering potential profiles.

(iii) Experimental results (on Si3N4 doped with Lu-Mg and stoichiometric & non-

stoichiometric SrTiO3) and simulations show that, the plot of fringe spacing (s)

versus root of defocus ((f)1/2) is not linear over the entire defocus range and is

found true at intermediate defocus values. Other limitations of the Fresnel

extrapolation technique are also enumerated; which includes the important

observation that the correct thickness of the IGF is not obtained by the method.

(i) A new methodology is developed, wherein spacing is measured based on Fourier

filtered Fresnel images. Appropriate choice of spacing values from over and

underfocus images, is seen to give a better symmetry in the plots Fresnel fringe

spacing data. It is also seen that this technique gives a better estimate of the

thickness of the IGF from the same set of Fresnel contrast images by extrapolation

to zero-defocus. The approximate validity of the extrapolation method could not be

understood in the framework of the simulations performed.

(ii) Simulations of Fresnel contrast, using one-dimensional potential profiles, are used

for understanding the different sources of left-right (LR) and underfocus-overfocus

(OU) symmetry breaking; which is observed in the experimental Fresnel contrast

images (FCI).

(iv)

(v)

(i) In the light of the limitations of the Fresnel extrapolation method; a new technique

for the Measurement of the thickness of IGFs is illustrated; which is based on

Fourier filtered zero-defocus Fresnel contrast images (tested on Si3N4 doped with

Lu-Mg and non-stoichiometric SrTiO3 samples). This is seen as the simplest

technique for this purpose; wherein, the high-resolution thickness can be estimated,

without actually performing lattice fringe imaging. Simulations are used to validate

the method.

(ii) Thickness measured from lattice fringe imaging (HRt) is equal to the zero-to-zero

distance in the zero-defocus FF-FCI (hidden in HRM or otherwise), which in turn

is a measure of the FWHM of the potential profile (as seen in simulations). This

implies: (a) the HRt corresponds to the FWHM of the potential profile and not the

inner width ('a') or (b) the IGF is fully diffuse. The current work is not able to

differentiate between the two cases.

(vii)

(vi)

(i) Fast changes to IGF thickness (in Lu-Mg doped Si3N4) can occur at comparatively low

temperatures (<1000ºC) under low irradiation doses, with electron beam energies as

low as 120 keV.

(ii) The tendency of the IGFs to regain the width they had before annealing has been

observed. However, the relaxation times were not long enough to observe complete

reversibility of the changes in IGF width.

iii) A tensile-compressive straining cycle may be realized by heating and cooling the

sample due to thermal gradients in the TEM sample, as a result of its special geometry.

(iv) Although local stress vectors, induced by thermal gradients in the sample, may offset

the forces determining the IGF equilibrium width in the force balance model, dynamic

observations (e.g. video recording) of the kinetics of the relaxation process may give

insight into details of diffusion processes and the viscosity of the intergranular glassy

phase.

Detailed Conclusions: In-situ heating

Outstanding issues and open questions

IGFs

represent equilibrium at which temperature?

What is the precise dependence of orientation of the bounding grains on the film thickness?

What is the exact structure of the glass-crystal interface? How can we perform an even better characterization of the amorphous material in the IGF? (3-D atomic structure of the IGF

would be the ideal goal).

How are IGFs

in various systems different from each other and what

features of the IGF

are system independent and can be studied by a universal model? And, if there are system dependent features can

there

be sub-classifications based electronic structure, bonding characteristics etc?

How to develop a unified theory which would describe the thickness of IGFs

as a function of temperature, activity of the chemical species and

crystallography and also account for the internal structure within the IGF?

How to predict the formation of IGF

in new systems?

I. MacLaren, Imaging and thickness measurement of amorphous intergranular films using TEM, Ultramicr. 99 (2004) 103-113.

ReferencesReferences

1. "Intergranular glassy films: an overview" Anandh Subramaniam, Christoph T. Koch, Rowland M. Cannon, and Manfred Rühle Materials Science and Engineering A, 422, p.3, 2006.

2. "Aspects regarding measurement of thickness of Intergranular Glassy Films" S. Bhattacharyya, Anandh Subramaniam, C.T. Koch and M. Rühle Journal of Microscopy, 221, p.46, 2006.

3. "The evolution of amorphous grain boundaries during In-situ heating experiments in Lu doped Si3N4" S. Bhattacharyya, Anandh Subramaniam, C.T. Koch and M. Rühle Materials Science and Engineering A, 422, p.92, 2006.

4. "Assessing thermodynamic properties of amorphous nanostructures by energy filtered electron diffraction" C.T. Koch, S. Bhattacharyya, Anandh Subramaniam, and M. Rühle, Microscopy and Microanalysis, 10, Suppl. 2, p.254, 2004.

Simulation of Fresnel

Contrast

Am

orph

icity

0

1

Direction of the electon beam Crystal Crystal

GlassRough interface

Projected potentialprofile

Inclined IGF

Projected potentialprofile

Glass

Limitations of a one-dimensional projected potential profile, with respect to its ability to distinguish between various structural features in the GB region having an IGF

Diffuse interface between glass and crystal

Rough interface

Inclined interface

Fresnel

fringe profile is obtained as follows: (i) the potential profile V(r) is constructed and multiplied by

and t (ii) the exponential term generated from this is Fourier transformed and multiplied

by the aperture function and MTF in reciprocal space(iii) the product is inverse Fourier transformed to real space

(the square of the absolute value is the Fresnel

intensity profile)

(r)21 iσV t iχ(k)

(r) apI = FT [FT(e )f (k)e ]

I(r)

→ image intensity V(r)

→ potential profile across the interface

→ interaction constant

t → specimen thickness fap

(k) → objective aperture function in reciprocal space ei(k)

→ microscope contrast transfer function (MTF) k → reciprocal lattice vector.

2 3 4s

1χ(k) = πΔfλk + πC λ k2

yxieyxAyxf , ),(),(

f(x,y) → specimen function

→ phase (function of V(x,y,z))

V(x,y,z) → specimen potential Vt

(x,y) → projected potential

→ Interaction constant (=/E)

Phase Object approximation

yxieyxf ,),(

Weak Phase Object approx.

),( 1),( yxVyxf t

dzzyxVyxVt ),,(),(

),( ),( yxVyxVE

d tt

d

V(x,y,z) Specimen

Phase change = f(Vt

)

yxVi teyxf , ),(

WPOA For a very thin specimen the amplitude of the transmitted wave function will be linearly related to the projected potential

Parameter Designation Value / Description

Weak Phase Object

Approximation

WPOA . t = 0.01 V1

( interaction constant, t – specimen thickness)

Phase Object

Approximation

POA . t = 0.1 V1

Potential Profile P FWHM = 20.4 Å, Diffuse interface on both sides

P1 a = a0 = aFWHM = 20.4 Å, square potential well

P2 aFWHM = 20.4 Å, left-right (LR) asymmetric profile

P3 aFWHM (of well) = 20.4 Å, aFWHM (of bump) = 4.5 Å

to simulate segregation

Wavelength () L1 0.0335 Å

Spherical aberration

(Cs)

C1 2.7 mm

Accelerating voltage - 120 kV

Objective aperture Size A0 Diffuse aperture with dFWHM =0.2496 Å1

Mask Size M1 0.026 Å1 (diameter)

M2 0.032 Å1 (diameter)

M3 Mask with diffuse edges dFWHM = 0.026 Å1

Beam sampling size - 0.23 Å

Simulation cell size - 126.73 Å

Parameters and assumptions used in the simulations of FCI and FF-FCI.

Potential Profile (scale for x-axis is in Å)

(comments)

Fresnel contrast plots (comments)

20.4 Å

0 V

6.44 V

20.4 Å

0 V

6.44 V

Standard square potential well) P1

(WPOA)

overfocus 2.2 micrometerunderfocus 2.2 micrometer

20 60Distance (Å)

-20 0-60 -40 40-80 80

Underfocus

Overfocus

A

A


20 60Distance (Å)

-20 0-60 -40 40-80 8020 60Distance (Å)

-20 0-60 -40 40-80 80

Underfocus

Overfocus

A

A

(LR symmetry present and OU symmetry is approximate)

20.4 Å

0 V

6.44 V

20.4 Å

0 V

6.44 V

P1 (POA)


20 60Distance (Å)

-20 0-60 -40 40-80 80

Underfocus

Overfocus

A

A

Shift in positionOf the peak


20 60Distance (Å)

-20 0-60 -40 40-80 8020 60Distance (Å)

-20 0-60 -40 40-80 80

Underfocus

Overfocus

A

A

A

A

Shift in positionOf the peak

(OU symmetry broken)

Effect of various IGF potential profiles and assumptions on the Fresnel

contrast image (FCI)

WPOA-Weak Phase Object Approximation

Defocus of

2.2 mCs = 2.7 mm

0 V

FWHM= 20.4 Å

6.44 V

-20 0 20-40 40

0 V

FWHM= 20.4 Å

6.44 V

-20 0 20-40 40 (Profile with Left-Right

asymmetry) P2

(WPOA) 4

6

8

2

4

6overfocus 2.2 micrometerunderfocus 2.2 micrometer

20 60Distance (Å)

-20 0-60 -40 40-80 80

Underfocus

Overfocus

AA

BB

4

6

8

2

4

6overfocus 2.2 micrometerunderfocus 2.2 micrometer

20 60Distance (Å)

-20 0-60 -40 40-80 8020 60Distance (Å)

-20 0-60 -40 40-80 80

Underfocus

OverfocusOverfocus

AA

AA

BB

(LR symmetry broken)

4.5 Å

0 V

6.44 V

FWHM= 20.4 Å

-20 0 20 40

4.5 Å

0 V

6.44 V

FWHM= 20.4 Å

-20 0 20 40

(Profile reflecting segregation) P3

(WPOA) 20 60Distance (Å)

-20 0-60 -40 40-80 80

Underfocus

Overfocus

A A

B B

20 60Distance (Å)

-20 0-60 -40 40-80 8020 60Distance (Å)

-20 0-60 -40 40-80 80

Underfocus

OverfocusOverfocus

A A

B B

(LR symmetry broken)

(i)

Under the Weak Phase Object Approximation (WPOA), LR

symmetry and approximate OU

symmetry are present (for profile P1)

(ii)

In the Fresnel

profiles generated using Phase Object Approximation (POA), the OU

symmetry is broken (P1)

(iii)

LR

asymmetry in the potential profile leads to a LR

asymmetry in the Fresnel

profiles (profile P2 & P3), however, approximate OU

symmetry is maintained

Observations

High Resolution Microscopy

)( ),( rfyxf

)( ),( rgyxg

)(rh

Point spread function )(uHContrast transfer function

Specimen

Image

Diffraction patternBack focal plane

( )F u

( )G u

Rea

l spa

ceR

eciprocal space

• The Point Spread Function (PSF) is also called the Impulse Response Function (IRF)• The Contrast Transfer Function (CTF) is also called the Modulation Transfer Function (MTF)

Fourier transform

Every imaging device is characterized by its Transfer Function (CTF) (Band filter)

This describes the magnitude with which a spatial frequency g

is

transferred through the device

Image formation in an TEM is a coherent process Object and Transfer functions are complex functions with amplitude and phase

1/

→

1

<N> Noise

→ resolution

PSF→ )(rh

CTF

→ )(uH

IDEAL

2 ) ()(

432 uCufu s

)()()( rhrfrg

)()()( uFuHuG

Point spread function

Convolution in real space gives multiplication in reciprocal space

Contrast transfer function

Aperture function

Envelope function

Aberration function

)()()()( uBuEuAuH

Property of lens

) , , ,()( sCuffu

)](exp[)( uiuB

Specimen transmission function

Phase distortion function

Point spread function

Specimen transmission function

Image

Effect of point spread function on image

Reference: Electron Microscopy- Principles and Fundamentals, Eds.: S. Amelinckx, D. van Dyck, J. van Landuyt, G. van Tendeloo, VCH, Weinheim, 1997

A measure of the resolution

High resolution implies ability to see two closely spaced features in the sample (real ‘r’

space) as distinct

This corresponds to high spatial frequencies large distances from the optic axis in the diffraction pattern (reciprocal (u) space)

Rays which pass through the lens at such large distances are bent through a larger angle by the objective lens

Due to an imperfect lens (spherical aberration) these rays are not focused at the same point by the lens

Point in the sample disc in the image (spreading of a point in the image)

Objective lens magnifies the image but confuses the detail (hence the resolution is limited)

Each point in the final image has contributions from many points

in the specimen (no linear map between the specimen and the image)

Is a linear relationship possible?

( ) ( ) i rf r A r e

f(r) → specimen function

→ phase (function of V(x,y,z))

V(x,y,z) → specimen potential Vt

(x,y) = Vt

(r) → projected potential

→ Interaction constant (=/E)

Phase Object approximation

( ) i rf r e

Weak Phase Object approx.

( ) 1 ( )tf r i V r

d

V(x,y,z) Specimen

Phase change = f(Vt

)

( ) ti V rf r e

Specimen and approximations

[ ( )]( ) ti V r rf r e

dzzyxVyxVt ),,(),(

( ) ( )t td V r V rE

Including absorption

2 3 4

11! 2! 3! 4!

x x x x xe

( ) 1 ( )tf r i V r

WPOA For a very thin specimen the amplitude of the transmitted wave function will be linearly related to the projected potential

Phase object has become

and amplitude object

The TEM operated in the phase contrast mode at optimum focus directly reveals the projected potential (under WPOA

= very thin specimen)

( ) 1 ( )r i r

Central beamDiffracted beam

If the phase of the diffracted beam is shifted by /2 w.r.t the central beam the amplitudes of the diffracted beam are multiplied by exp(i/2) = i

( ) 1 ( )r r

1 2 ( )I r The phase of the object is directly imaged

meEh

2In vacuum

),,((2'

zyxVEmeh

In the specimen

dzdzd

22'

Phase change dzzyxVE

d ),,( dzzyxVd ),,(

( , ) ( , , ) ( )t tV x y V x y z dz V r ( ) ( )t td V r V rE

→ Interaction constant (=/E) tends to a constant value as V increases(energy of electron proportional to E & 1, variables tend to compensate)

POA

holds good only for thin specimens

If specimen is very thin {Vt

(r) << 1} WPOA

In WPOA

{f(r) = 1

i

Vt

(r)} the amplitude of the transmitted wave will be linearly related to the projected potential of the specimen

(the TEM in the phase contrast mode at optimum focus directly reveals the projected potential)

Hence if object is very thin, optimum focus imaging would directly reveal atoms as dark areas and empty spaces as light

For Ti2 Nb10 O27 WPOA is valid if specimen thickness < 6Å [Fejes.P.L., Acta Cryst. A33, 109, 1977]

For thick specimens, there may not be a one to one correspondence between the projected structure of the object and exit face wavefunction

)()()( rhrfrg

( ) [1 ( )] ( )tr i V r h r

WPOA

( ) [1 ( )] [ ( ) ( )]tr i V r Cos r i Sin r

1 2 ( )] ( )tI V r Sin r

I =

.* Neglecting 2

Only imaginary part of B(u) contributes

to intensity

)](exp[)( uiuB )]([2)( uSinuB

22 ( )( ) 1i rI r e

In an ideal microscope the image wavefunction

= object wavefunctionImage intensity for a pure phase object

The image would show no contrast

This is like imaging a variable thickness glass plate in an ideal light microscope

( ) 1 ( ) ( ) ( ) ( )t tr V r Sin r i V r Cos r

)]([2 )()()( uSinuEuAuT

Objective lens transfer function Incoherent illumination T(u) = H(u)

T(u) ve

+ve

phase contrast (atoms would appear dark)

In WPOA

T(u) is sometimes called CTFApproximately

)]([)(2)( uSinuAuT

Or

coherencespatialchromaticeffective EEuTuT )()(

Envelope is like a virtual aperture at the back focal plane

u (nm1)

→

T(u)

= 2

Sin

()→

+2

2

v

2 4 6

Cs

= 1 mm, E0

= 200 keV, f = 58 nm)

(g) = /2

In this region all spatial frequencies have same phase shift

Information is transferred forward keeping a point to point relation to the object

Information still contributing to the image-

but with a wrong phase

It is scattered outside the peak of the PSF

and is distributed over a larger area in the image plane

)]([)(2)( uSinuAuT

u (nm1)

→

T(u)

= 2

Sin

()→

Cs

= 2.2 mm, E0

= 200 keV, f = -100 nm)

)]([)(2)( uSinuAuT

T eff

ectiv

e(u)→

u (nm1)

→

coherencespatialchromaticeffective EEuTuT )()(

Scherzer

defocus and Information Limits 124

3Scherzer sf C

1 34 4

Scherzer su =1.51 C

1 34 4

Scherzer s= 0.66 C r

Instrumental resolution limit can use nearly intuitive arguments to interpret the contrast

Information limit

Damping envelope

Instrumental resolution limit (Point resolution / Structural resolution) finest detail that can be interpreted in terms of the structure

can use nearly intuitive arguments to interpret the contrast

The point resolution can be improved by

↓

Cs

↓

(= ↑

accelerating voltage)

Information limit finest detail that can be resolved by the instrument

(irrespective of an possible interpretation) corresponds to the maximal diffracted beam angle that is still transmitted

with appreciable intensity the TEM (transfer function) is a low pass filter (in g) which cuts off all

information beyond the information limit

The information limit can be improved by

Improving beam coherence

↓

(= ↑

accelerating voltage)

Convolution

A convolution is an integral that expresses the amount of overlap of one function as it is shifted over another function. It therefore "blends" one function with another.

For example, in synthesis imaging, the measured dirty map is a convolution of the "true" CLEAN map with the dirty beam (the Fourier transform of the sampling distribution).

Reference: http://mathworld.wolfram.com/Convolution.html

0

( ) ( )t

f g f g t d

( ) ( )f g f g t d

1[ [ ] . [ ]]f g FT FT f FT g

Zero-Defocus Fresnel

Reconstruction

The intensity of an image is given as:

2( ) MTF(r)I r r

(r) → Complex electron exit-face wave function I (r) → Intensity of the imageMTF(r) → Microscopic transfer function. V(r) → Complex projected potential

→ Interaction constant (=/E) t

→ Specimen thickness

Using phase object approximation

Ψ( ) exp i tV(r) exp (́ ) ´́ ( )r i t V r iV r

The real part of the scattering potential (V'(r)) is due to elastic interaction of the electron with the specimen and the imaginary part (V''(r)) describes the loss of electrons from the elastic channel (and thus the zero energy filtered image), due to inelastic scattering events.

For zero-defocus imaging (and the assumed ideal aberration free imaging conditions; which is expected to hold good for the low-resolution conditions in a microscope like Zeiss-912) the following can be set:

aperture

aperture

qqqq

qmtfrMTFFT

,0,1

)(

qaperture

→ Reciprocal-space radius of the top had aperture

function

Ignoring coherent aberrations and envelope functions related to chromatic aberration, partial spatial coherence and microscope instabilities; the following equation is obtained under zero-defocus conditions:

2

2I exp ´´ 1 exp ´ exp ´´ ´́ ´

apertureq

r t V r i tV r t V r V r

´́ ´ , ln 1 exp ´aperture

apertureq

V r q i t V r

V'''(r,qaperture

)

→ a specimen and objective aperture dependent ‘pseudo-absorptive’

scattering potential

Both the inelastic scattering described by V''(r) and the pseudo-absorptive by V'''(r) increase with the atomic number (Z) of the scattering element.

In fact, the high-angle annular dark-field scanning transmission electron microscopy (HAADF

STEM) technique, being mainly based on V'''(r), is also called Z-

contrast STEM for this reason.

Vabs

(r) = V''(r) + V'''(r)

The Fresnel

contrast due to variations in the electrostatic potential V'(r, q<qaperture

) across the IGF

vanishes at zero-defocus so that:

1´´ ´´´ ln ln2

t V r V r I r I r

t and

are taken to be constant The LHS is representative of the inelastic and high-angle scattering and thus

the concentration of heavy elements at position. The FWHM

of a profile of absorptive potential across the IGF

gives a measure of its chemical thickness.

The resolution of this method is limited by the size of objective aperture (in Zeiss-912 microscope this is equal to 8 Å).

MATLAB

R12 software was used for implementing the algorithm to determine the imaginary part of the potential profile across the interface

from zero-defocus images.

-40 0 40-20 20Distance (Å)

RTHTCooled to RT

20 nm20 nm20 nm

A

A2.3 nm

20 nm20 nm20 nm

B

B

2.5 nm

20 nm20 nm20 nm

C

C 3.2 nm

RT

HT

Cooled to RT

FRESNELFRESNEL

RECONSTRUCTIONRECONSTRUCTION

Exit face wave reconstructed profiles, of the imaginary part of the inner potential, using

zero-defocus Fresnel

fringe images

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

-30 -20 -10 0 10 20 30(Defocus)1/2 (nm)1/2

spac

ing

(s) (

nm)

Region of approximatelinearity

Replot

of data fromOn the Fresnel-fringe technique for the analysis of interfacial films, D.R. Rasmussen, C.B. Carter, Ultramicr. 32 (1990) 337-348.Model A in the paper

1/length scale

Dep

ende

nce

on a

ssum

ptio

ns

Force balance Adsorption/Wetting

Diffuse Interface

Phase Field Method

Classical Density Functional Methods

First Principles Methods

Molecular Dynamics/Monte Carlo

1/length scale

Dep

ende

nce

on a

ssum

ptio

ns

Force balance Adsorption/Wetting

Diffuse Interface

Phase Field Method

Classical Density Functional Methods

First Principles Methods

Molecular Dynamics/Monte Carlo

Atomic-resolution scanning transmission electron microscope (STEM) images ofan intergranular

glassy film (IGF) in La-doped -Si3

N4

:a) High-angle annular dark-field (HAADF-STEM)b) Bright-field (BF-STEM)

N. Shibata, S.J. Pennycook, T.R. Gosnell, G.S. Painter, W.A. Shelton, P.F. Becher

“Observation of rare-earth segregation in silicon nitride ceramics at subnanometre

dimensions”

Nature 428 (2004) 730-33.

(a) (b)[0001]

HAADF-STEM images of the interface between the IGF

and the prismatic surface of an -

Si3

N4

grain. The -Si3

N4

lattice structure is superimposed onthe images. a) La atoms are observed as the bright spots (denoted by red arrows) at theedge of the IGF. The positions of La atoms are shifted from that of Si

atoms based on theextension of the -Si3

N4

lattice structure; these expected positions are shown by opengreen circles. b) reconstructed image of a, showing the La segregation sites more

clearly. The predicted La segregation sites obtained by the first-principles calculations are shown by the open white circles.

N. Shibata, S.J. Pennycook, T.R. Gosnell, G.S. Painter, W.A. Shelton, P.F. Becher

“Observation of rare-earth segregation in silicon nitride ceramics at subnanometre

dimensions”

Nature 428 (2004) 730-33.

HAADF-STEM images of La-, Sm-, Er-, Yb-, and Lu-doped Si3

N4

:The attachment of heavy atoms in the form of atomic columns oriented normal to the image plane

A. Ziegler, J.C. Idrobo, M.K. Cinibulk, C. Kisielowski, N.D. Browning, R.O. Ritchie, “Interface Structure and Atomic Bonding Characteristics in Silicon Nitride Ceramics”

Science 306 (2004) 1768-70

Coincident Site Lattice

Centre of Rotation

Rotation by 36 52'

The Coincident Site Lattice

Variation of film and substrate energy.

Varition of film & substrate energy with growth of layers

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Number of layers of b/2 thickness (b=3.84Å)

Ener

gy(X

10-1

0 J/m

)

Film energy Substrate energy

Limitations of the current theoriesLimitations of the current theories

The energy of a growing film is assumed to be a linear function of the thickness

The substrate is assumed to be rigid and only the energy of the film is taken into account

Even though the energetics of the substrate is ignored the energy of the whole dislocation is taken into account

Advantages of the current simulationAdvantages of the current simulation

(i) As growth progresses, the upper layers are expected to be more relaxed energetically as compared to the layers closer to the substrate and this aspect is captured in the simulation

(ii) The simulation calculates the energy of the interfacial dislocation in a film/substrate system (with separate material properties for the film and substrate), wherein there is considerable asymmetry between the tensile and compressive stress fields of the dislocation and hence the energy of a interfacial dislocation is different from that of a dislocation in a bulk crystal

(iii) The methodology adopted automatically takes into account the interaction between the film coherency and dislocation strain fields,

(iv) Equilibrium critical thickness is calculated taking into account the energy of the entire system and not just the film as in many models.

Limitations of the current simulationLimitations of the current simulation

(i)

E and

values calculated from single crystal data (bulk values) have been used for the thin films

(ii)

Linear interpolation is used to calculate the lattice parameter and the material properties of the alloy films

(iii)

For computational convenience the thickness and width of the substrate considered is small as compared to the real physical dimensions

(iv)

For computational convenience and for comparison with available experimental data highly strained films have been considered in the current analysis and the model will have to be tested for low strain systems wherein the critical thickness values are very large

(v)

Core structure & energy of the dislocation are ignored in the simulation.

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