Theory and Practice of Electron Diffraction - Zuo Research...

$: Theory and Practice of Electron Diffraction - Zuo Research …cbed.matse.illinois.edu/download/ElectronDifraction... · · 2012-12-31Theory and Practice of Electron ... 2sinK θ=nG$
DOE BES/DMS Materials Science and Engineering/Frederick Seitz Materials Research Laboratory

J. M. Zuo Dept. of Materials and Engineering and Materials

Research LaboratoryUniversity of Illinois at Urbana and Champaign

1304 W Green Street, Urbana, [email protected]

http://cbed.mse.uiuc.edu

Theory and Practice of Electron Diffraction


Covered Topics

• Electron diffraction• Crystal diffraction patterns • Kikuchi Lines• Convergent beam electron diffraction• Electron nanobeam diffraction and examples


Why diffraction:

-Use diffraction patterns to orient crystals for high resolution electron imaging

-Use diffraction patterns to setup diffraction contrast imaging

-Use diffraction patterns for phase identification, crystal orientation determination and structural analysis


I. Electron imaging and diffraction


Direct beamDiffracted beam

Back focal plane

Objective Lens

Sample

Formation of Diffraction Patterns -The Objective Lens Action

Illumination Diffraction Lens


SA A perture

VirtualAperture

Specimen

Lower ObjectiveLens B ack FocalPlan

a) Selected Area Electron Diffraction

Specimen

nBack Focal Pla

ICondenser I

UpperObjective

LowerObjective

c) Convergent Beam Electron Diffraction (CBED)

Upper Objective

Lower ObjectiveSpecimen

Condenser Lens III

C2 Aperture10µm

Back Focal Plane

b) Nanobeam electron diffraction

PP’

II. Choose the electron illumination-Diffraction Techniques


10 nm

Example of nanobeam illumination

Direct beam

Au nanocrystal


CBED


III. Crystal Diffraction

The diffraction condition: diffracted waves from different planes must be in phase

SQ QT nλ+ =

sin sin 2 sinhkl hkl hkln d d dλ θ θ θ= + = (Bragg’s law)


Inside TEM

(Actual diffraction angle is much smaller)

Rewrite the Bragg’s law in 3-D (Laue equation)

12 sinhkl

nd

θλ

=

2 sinK nGθ =r

1/oK λr

Kr

oK K=r r

Incident wave

diffract wave

oK K g− =r r r

oKr

oKr

oKr K

rθ


Crystal lattice

=

LatticeCrystals-with repeated,

ordered, atomic arrangements

Unit cell (repeating

unit)Unit cell vectors, a, b, c


Lattice planesAny plane that pass through 3 lattice points (nonsystematic) repeats indefinitely in the crystal lattice (family of planes)

The smallest planar spacing is dhkl, hkl Miller indicies


Miller indices-Take the reciprocal of the intercepts

-Find the smallest multiplication factor-Enclose three integers in () with no

comma in between


The reciprocal lattice-Draw lines from O

perpendicular to lattice planes; in general (hkl) is different from [hkl], except in cubic crystals.

-The reciprocal lattice is defined such the distance from O to a

point is the inverse of the planar distance

-Refer a*= d*(100), b*=d*(010) and c*=d*(001)

( ) 1*( )

d hkld hkl

=

( )* * * *d hkl ha kb lc= + +r rr r

* * * 1a a b b c c⋅ = ⋅ = ⋅ =r rr r r r (otherwise zero)


Reciprocal Lattice Vectors

* , * , *Cell Cell Cell

b c c a a ba b cV V V× × ×

= = =r rr r r rrr r

( )CellV a b c= × ⋅rr r

sina b ab γ× =rr

Useful formula

Reciprocal space lattice vectors

cosa b ab γ⋅ =rr

Relationship between real and reciprocal space* 1, * 0, * 0

* 0, * 1, * 0

* 0, * 0, * 1

a a a b a c

b a b b b c

c a c b c c

⋅ = ⋅ = ⋅ =

⋅ = ⋅ = ⋅ =

⋅ = ⋅ = ⋅ =

rr r r r r

r r r rr r

rr r r r r


Crystal Diffraction Pattern and The Reciprocal Lattice

a*

b*

KK+g

L

2θ

2λθ gSin B =

Bragg’s Law

~ 1/ , ~ 0.025 , ~12.5o o

g A A mradλ θ


Indexing Diffraction PatternsThe basic equations:

θ

D

L

( )/ 1/ /g D Lλ =

/g D Lλ=

For cubic crystals:

2 2 2 /g h k l a= + +

h,k,l integers

uh vk lw n+ + = n = 0 for zero order Laue zone

1 2 1 2 1 22 2 2 2 2 2

1 1 1 2 2 2

cos h h k k l lh k l h k l

φ + +=

+ + + +


Find the Crystal Orientation-The zone axis

- Find two shortest reflections, measure and index to obtain (h1,k1,l1) and (h2,k2,l2)

-Check the angle between these two and make sure hkl indexing is consistent with the lattice

-Calculate the zone axis using

1 2 1 1 1 1 2 1 2 1 2 1 2 1 2 1 2

2 2 2

[ , , ]a b c

Z g g h k l k l l k l h h l h k k hh k l

= × = = − − −

rr r

r r rU V W


IV. Kikuchi LinesWe see Kikuchi lines because:1. The specimen is thick enough (~50

nm or above, 200 kV)2. Electrons are scattered off Bragg

angles in all directions (because of inelastic scattering)

3. These electrons are then Bragg diffracted by lattice planes

Some facts:1. The electron energy loss is small

(most less a few tens of eV) and electrons have similar wavelength as the incident beam

2. Most electrons travels in directions close to the incident beam (forward direction)

3. Kikuchi described this in 1928 before invention of TEM

SrTiO3, 200kV


Construct Kikuchi Maps-Select a zone axis (e.g. [001], cubic)-Draw the zone axis diffraction pattern-For each allowed reflection, draw a line perpendicular to g and at half of g-Select a second zone axis (e.g. [011] cubic) (use the stereo-projection)-Draw the Kikuchi lines using the same procedure-Orient the second map to line up the common pair of Kikuchi lines


The effect of sample rotation Incident beam || optical axis

Diffracted beam fixed to incident beam

Bragg cone fixed to crystal

Kikuchi lines fixed to crystal


CBED


Specimen

Back Focal Plan

Condenser II

Upper Objective

Lower Objective

• Crystal diffraction• Accurate thickness, unit cell and orientation measurement• Small probe • Used in STEM

Why CBED?


GaAs

InGaAs

10 nm

H. Lakner, HB501 100 kV

Real Space: 1) Probe size 2) Probe propagationReciprocal Space: Beam divergence

Resolution

Channeling

De-channeling

PP'


Probe SizeNot important

Important

Thickness changes across CBED disk

Small probe -> ~constant thickness


P'

Sg∝gθθ

Excitation Error Changes Across CBED Disk


0

0.2

0.4

0.6

0.8

1

-6 -4 -2 0 2 4 6W

Inte

nsity

0

0.2

0.4

0.6

0.8

1

-6 -4 -2 0 2 4 6W

Inte

nsity

0

0.2

0.4

0.6

0.8

1

-6 -4 -2 0 2 4 6W

Inte

nsity

0

0.2

0.4

0.6

0.8

1

-6 -4 -2 0 2 4 6W

Inte

nsity

0

0.2

0.4

0.6

0.8

1

-6 -4 -2 0 2 4 6W

Inte

nsity

0

0.2

0.4

0.6

0.8

1

-6 -4 -2 0 2 4 6W

Inte

nsity

0

0.2

0.4

0.6

0.8

1

-6 -4 -2 0 2 4 6W

Inte

nsity

0

0.2

0.4

0.6

0.8

1

-6 -4 -2 0 2 4 6W

Inte

nsity

0

0.2

0.4

0.6

0.8

1

-6 -4 -2 0 2 4 6W

Inte

nsity

t /ξG = 0.1 t /ξG = 0.25 t /ξG = 0.5

t /ξG = 0.75 t /ξG =1 t /ξG =1.5

t /ξG = 2 t /ξG = 2.5 t /ξG = 3

Dynamic Diffraction: Two Beam Case

g gw S ξ=


Electron Nanobeam DiffractionField Emission Electron Source

Semi-angle ~ 0.05 mrad

Magnetic Lens

10 micron aperture

40 nmJ~105 e/nm2/s

Exposure time ~10 s


Si(220)

Ag(220)

Where is the strain? Epitaxial Relationship

aSi/aAg≈4/3Ag(001)||Si(001)Ag[220]||Si[220]OrAg[2-20]||Si[2-20]

Si(331)

~13 degree off [001]

At RT: Δa/a = -0.3%At 450˚C: Δa/a = -1%


Determination of Individual CNT Structure

M. Gao, J.M. Zuo, R.D. Twesten, I. Petrov, L.A. Nagahara & R. Zhang, Appl. Phys. Lett. 82, 2703 (2003)

6,20~108 e/st~10 sL~50 nmI~10 e

d=1.4 nm


Ag/H-Si(111) 2 ML

Si(2-20)Si(4-2-2)/3

Ag(2-20)

(111)

0

5

10

15

20

0 10 20 30 40 50 60 70

Azimuthal Angle

Coverage (monolayer)

As deposited


Nano-crystalline or amorphous diffraction

AgCu filmAs deposited

AgCu filmAnnealed at

260 ºC


References

D.B. Williams and C. B. Carter, Transmission Electron Microscopy, Plenum, New York (1996)

J.W. Edington, Practical Electron Microscopy in Materials Science, Monograph 2, Electron Diffraction in the Electron Microscope, Philips Technical Library (1975)

J.M. Cowley, Diffraction Physics, North-Holland, New York (1981)

J.C.H. Spence and J.M. Zuo, Electron Microdiffraction, Plenum, New York (1992)

Theory and Practice of Electron Diffraction - Zuo Research...

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