diffraction – going further - cime | epfl · • for an fcc lattice which of the zone axes [1 0...

$: Diffraction – Going further - CIME | EPFL · • For an FCC lattice which of the zone axes [1 0 0], [1 1 0] and! [1 1 1] will show a FOLZ? ... Duncan Alexander: Diffraction –$
Duncan Alexander: Diffraction – Going Further CIME, EPFL

Diffraction –Going further

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Duncan AlexanderEPFL-CIME


Contents

• Higher order Laue zones (HOLZ)

• Convergent beam electron diffraction (CBED)

• HOLZ lines in CBED

• Thickness measurements

• Polarity measurements

• Kikuchi diffractionNanobeam diffraction mapping

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Higher order Laue zones (HOLZ)

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Duncan Alexander: Diffraction – Going Further CIME, EPFL 4

Higher-order Laue Zones

ZOLZ: hU + kV + lW = 0FOLZ: hU + kV + lW = 1SOLZ: hU + kV + lW = 2

... Figures by Jean-Paul Morniroli


Higher-order Laue Zones

Figures by Jean-Paul Morniroli


Higher-order Laue zone quiz

• ZOLZ: hU + kV + lW = 0;!FOLZ: hU + kV + lW = 1;!SOLZ: hU + kV + lW = 2;!….

• For an FCC lattice which of the zone axes [1 0 0], [1 1 0] and![1 1 1] will show a FOLZ?

• FCC planes with h, k, l mixed even and odd are absent

• [1 0 0]: FOLZ equation gives h = 1; possible with h k l all odd so see FOLZ

• [1 1 0]: h + k = 1; impossible for h k l all even or all odd so no FOLZ

• [1 1 1]: h + k + l= 1; possible with h k l all odd so see FOLZ


HOLZ for higher symmetry• HOLZ reflections introduce a third dimension of reciprocal lattice

into SADP (diffraction from planes inclined to e-beam)

• These HOLZ reflections can show higher order symmetry not present in the ZOLZ – example FCC Al on [1 1 1] zone axis. FOLZ shows 3-fold symmetry (look carefully!)

ZOLZ only ZOLZ and FOLZ

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Convergent beam electron diffraction

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Instead of parallel illumination with selected-area aperture, CBED uses!highly converged illumination to select a much smaller specimen region


Small illuminated area => !no thickness and orientation variations

There is dynamical scattering, but it is useful!

Can obtain disc and line patterns!“packed” with information:

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Figures from Williams & Carter “Transmission Electron Microscopy”



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Image plane:!see image of focused

e-beam

Back-focal plane




Exact 2-beam condition

Near 2-beam condition

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Image plane:!see image of focused

e-beam

Back-focal plane


Excess and Deficient lines

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0 0 0

kIkDkk

• Look at Ewald sphere construction – here Ewald sphere intersects a HOLZ reflection exactly.In this condition this HOLZ reflection must give 2!B scattering angle.

• If we draw in rays for excess and deficient lines at +!B and –!B relative to the planes making thereflection we see excess line intersects HOLZ reflection and deficient 0 0 0 exactly.

• If we plot full angular range of incident beam it is clear that deficient HOLZ line crosses 0 0 0 disc.

0 0 0

kIkDkk


Excess and Deficient lines

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• Look at Ewald sphere construction – here Ewald sphere intersects a HOLZ reflection exactly.In this condition this HOLZ reflection must give 2!B scattering angle.

• If we draw in rays for excess and deficient lines at +!B and –!B relative to the planes making thereflection we see excess line intersects HOLZ reflection and deficient 0 0 0 exactly.

• If we plot full angular range of incident beam it is clear that deficient HOLZ line crosses 0 0 0 disc.

0 0 0

kIkDkk

0 0 0

kIkDkk

0 0 0

kIkDkk

0 0 0

kIkDkk


HOLZ lines in CBED

• If HOLZ CBED discs at have excess lines at Bragg condition these give corresponding deficient lines crossing 0 0 0 disc

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HOLZ lines in CBEDBecause HOLZ lines contain 3D information, they also show true symmetry!

e.g. three-fold {111} symmetry for cubic Al- unlike apparent six-fold axis in SADP or from ZOLZ Kikuchi lines

JEMS simulation for 300 nm thick Al, 200 keV beam energy

Deficient lines for inclined planes: Fringes from 2D

interactions/dynamical scattering; more

thickness gives more fringes

-3 -7 115 7 -117 5 -11

hU + kV + lW =?


Thickness measurement by CBED• Within CBED discs also obtain patterns from dynamical scattering. These patterns show fringes that

are somewhat analogous to thickness fringes in the TEM image.

• Can measure thickness e.g. by comparing experimental data to simulation

• Example: Blochwave simulations for Al on [0 0 1] zone axis:

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• Easier to think about in 2-beam Bragg scattering condition

• Different rays in the scattered beam sample different excitation errors for the reflection g

• Effectively we make a map of intensity for different excitation errors s along a chord in the disc g

• As thickness increases nodes in sinc2 function move to smaller s (reciprocal relationship with thickness) => more fringes in the disc

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t = 10 nm t = 25 nm t = 50 nm

Intensity vs s for 2-beam condition, different specimen thicknesses t

Thickness measurement by CBED


• Therefore to obtain CBED discs with 1-D fringes for thickness measurement tilt to 2-beam condition

• Possible to calculate thickness analytically (e.g. see Williams & Carter)

• Example: Blochwave simulations for Al with (0 0 2) reflection excited:

Thickness measurement by CBED


Identifying polarity in CBED

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Patterns from dynamical scattering in direct and diffraction discs allow determination of polarity of non-centrosymmetric crystals because dynamical scattering patterns are sensitive to

channeling down particular atomic column

T. Mitate et al. Phys. Stat. Sol. (a)!192, 383 (2002)

Simulation vs experiment:JEMS simulation: GaN [1 -1 0 0] zone axis

t = 100 nm

t = 150 nm

t = 200 nm

t = 250 nm

000-2 0000 0002


Zone axis CBED• Instead of spot pattern, obtain disc pattern

• Larger convergence semi-angle α => larger discs

Parallel beam SADP Si [001] α = 1.6 mrad α = 3.9 mrad


Zone axis CBED• Instead of spot pattern, obtain disc pattern

• Larger convergence semi-angle α => larger discs

α = 8.6 mradα = 3.9 mrad α = 19.3 mrad

See fringes in discs, and symmetry


Convergent beam electron diffraction – practical example

ZnO thin-film sample;Conditions: convergent beam, large condenser aperture, diffraction mode

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Convergent beam electron diffraction – practical example

ZnO thin-film sample;Conditions: convergent beam, large condenser aperture, diffraction mode

[1 1 0] zone axis

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Kikuchi diffraction

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Kikuchi lines

• Formation of bright and dark lines (“excess” and “deficient” lines) by combination of inelastic scattering followed by elastic scattering in parallel incident beam geometry (e.g. SADP)

• Inelastic scattering is incoherent (no preferential scattering vectors) but is generally in forwards direction

• Cones of inelastically-scattered electrons then elastically scattered, creating arcs in the diffraction plane. Because angles are small the arcs look straight.

• Resultant lines are very similar to the excess and deficient lines of CBED. They are equally sensitive to specimen orientation, and we use them e.g. to set a 2-beam condition. However the specimen must be thick (for sufficient inelastic scattering) and flat (to have sharp lines) to see them well.

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• Treat problem in “real space”

• Treat 2-beam Bragg scattering condition

• Yellow volume represents intensity distribution of inelastic scattering event; mainly forward scattered

• origin of inelastic scattering event:

"

h k lh k l– ––

"

"

"""""""""""

"""

h k l– – –h k l

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Kikuchi line formation

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h k l– – –h k l

" "

ghkl0 0 0

kI

kD

• Treat problem in reciprocal space; Ewald sphere construction

• Forward scattering from incoherent inelastic scattering gives diffuse intensity at low scattering angles around 0 0 0

• Exact Bragg condition: Kikuchi lines coincide with Bragg spots

Deficient line: forward-scattered inelastic intensity removed from Kikuchi line ! dark line on brighter background

Excess line: forward-scattered inelastic intensity redistributed into Kikuchi line ! bright line on low background


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• Treat problem in “real space”

• Treat zone axis condition


h k lh k l– ––

""""

" ""

h k l– ––h k l31

• Treat problem in reciprocal space; Ewald sphere construction

• Remember Bragg spots only visible because of relrods and excitation error s

• Zone axis condition: (by geometry) Kikuchi lines halfway between Bragg spots

Symmetrical relationship so both Kikuchi lines should have same intensity


h k l h k l– ––

ghkl0 0 0–ghkls

kI

kD

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2


Kikuchi diffraction

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Position of the Kikuchi line pairs of (excess and deficient) very sensitive to specimen orientationCan use to identify excitation vector; in particular s = 0 when diffracted beam coincides!

exactly with excess Kikuchi line (and direct beam with deficient Kikuchi line)

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Quiz: what happens to width of Kikuchi line pairs as (h k l) indices become bigger? !Answer: larger indices (h k l) ! greater scattering angle "B ! larger width of line pairs



Kikuchi lines – “road map” to reciprocal space

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Kikuchi lines traverse reciprocal space, converging on zone axes

- use them to navigate reciprocal space as you tilt the specimen!

Examples: Si simulations using JEMS

Si [1 1 0] Si [1 1 0] tilted off zone axis Si [2 2 3]

Obviously Kikuchi lines can be useful, but can be hard to see (e.g. from insufficient thickness, diffuse lines from crystal bending, strain). Need an alternative method...


Mapping in nano-beam diffraction mode

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Nano-beam set-up• In nano-beam mode, use small C2 condenser aperture and excited 3rd condenser

lens (e.g. condenser mini-lens) to make near-parallel beam of 2–3 nm diameter on specimen surface

• Typical convergence semi-angle alpha ~0.5–1 mrad; therefore obtain spot-like diffraction patterns from very small probe

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Orientation image mapping in TEM• The NanoMEGAS ASTAR system scans the nano-

beam across sample while recording diffraction patterns to make a map with one diffraction pattern for every pixel positions x, y

• “Template” matching then used to identify phase and orientation; each pattern correlated with 100s-1000s of patterns simulated at different orientations and for different phases

• Much higher spatial resolution than EBSD in SEM, and angular resolution of 1°

• Can combine with precession for greater reliability of indexing

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[1] E.F. Rauch et al., Micros. Anal. Nanotech. Supplement, 22(6), S5-S8 ,2008


ASTAR example: nanocrystalline ZnO• Plan-view sample of textured, nanocrystalline ZnO thin-film

• 750 x 750 pixel map, 2–3 nm probe size, 2 nm step size

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A. Brian Aebersold, CIME{0001}

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diffraction – going further - cime | epfl · • for an fcc lattice which of the zone axes [1 0...

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