Diffraction –
Going further
Duncan Alexander!
EPFL-CIME
Duncan Alexander: Diffraction – Going Further
CIME, EPFL
1
Contents
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Higher order Laue zones (HOLZ)!
Kikuchi diffraction!
Convergent beam electron diffraction (CBED)!
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HOLZ lines in CBED!
Thickness measurements!
Polarity measurements!
Analysing SAED patterns!
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Sample/image rotation!
Indexing!
Simulation!
Nanobeam diffraction mapping
Duncan Alexander: Diffraction – Going Further
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2
Higher order Laue
zones (HOLZ)
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Higher-order Laue Zones
ZOLZ: hU + kV + lW = 0!
FOLZ: hU + kV + lW = 1!
SOLZ: hU + kV + lW = 2!
...
Duncan Alexander: Diffraction – Going Further
Figures by Jean-Paul Morniroli
CIME, EPFL
4
Higher-order Laue Zones
Figures by Jean-Paul Morniroli
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Higher-order Laue zone quiz
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ZOLZ: hU + kV + lW = 0;
FOLZ: hU + kV + lW = 1;
SOLZ: hU + kV + lW = 2;
….!
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For an FCC lattice which of the zone axes [1 0 0], [1 1 0] and
[1 1 1] will show a FOLZ?!
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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
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HOLZ for higher symmetry
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HOLZ reflections introduce a third dimension of reciprocal lattice
into SADP (diffraction from planes inclined to e-beam)!
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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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Inelastic + elastic scattering: Kikuchi diffraction
Inelastic scattering event scatters electrons in all directions inside crystal
Some scattered electrons in correct orientation for Bragg scattering => cone of scattering
Figures from Williams &
Carter “Transmission
Electron Microscopy”
Cones have very large diameters => intersect diffraction plane as ~straight lines
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CIME, EPFL
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Kikuchi diffraction
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Kikuchi diffraction
Figures by Jean-Paul Morniroli
Position of the Kikuchi line pairs of (excess and deficient) very sensitive to specimen orientation
Can 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)
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
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Kikuchi lines – “road map” to reciprocal space
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...
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11
Convergent beam
electron diffraction
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Convergent beam electron diffraction
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:
Figures by Jean-Paul Morniroli
Duncan Alexander: Diffraction – Going Further
CIME, EPFL
13
Convergent beam electron diffraction
Figures by Jean-Paul Morniroli
Duncan Alexander: Diffraction – Going Further
CIME, EPFL
14
Convergent beam electron diffraction
Figures by Jean-Paul Morniroli
Duncan Alexander: Diffraction – Going Further
CIME, EPFL
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Convergent beam electron diffraction
Figures from Williams & Carter “Transmission Electron Microscopy”
Duncan Alexander: Diffraction – Going Further
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Convergent beam electron diffraction
Back-focal
plane
Image plane:
see image of focused
e-beam
Figures by Jean-Paul Morniroli
Duncan Alexander: Diffraction – Going Further
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Convergent beam electron diffraction
Exact 2-beam
condition
Back-focal
plane
Image plane:
see image of focused
e-beam
Near 2-beam
condition
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Convergent beam electron diffraction
Kikuchi from:
inelastic scattering
convergent beam
Excess and deficient lines
much less diffuse for CBED!
!
=> use CBED to orientate
sample!
Figures from Williams &
Carter “Transmission
Electron Microscopy”
Duncan Alexander: Diffraction – Going Further
CIME, EPFL
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Convergent beam electron diffraction
– practical example
ZnO thin-film sample;!
Conditions: convergent beam, large condenser aperture, diffraction mode
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20
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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HOLZ lines in CBED
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If HOLZ CBED discs at have excess lines at Bragg condition these
give corresponding deficient lines crossing 0 0 0 disc
Figures by Jean-Paul Morniroli
Duncan Alexander: Diffraction – Going Further
CIME, EPFL
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HOLZ lines in CBED
Because 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
Deficient lines for
inclined planes:
-3 -7 11
5 7 -11
7 5 -11
Fringes from 2D
interactions/dynamical
scattering; more
thickness gives more
fringes
hU + kV + lW =?
JEMS simulation for 300 nm thick Al, 200 keV beam energy
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Thickness measurement by CBED
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Within CBED discs also obtain patterns from dynamical scattering. These patterns show fringes that
are somewhat analogous to thickness fringes in the TEM image.!
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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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Thickness measurement by CBED
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Easier to think about in 2-beam Bragg scattering
condition!
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Different rays in the scattered beam sample different
excitation errors for the reflection g!
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Effectively we make a map of intensity for different
excitation errors s along a chord in the disc g
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As thickness increases nodes in sinc2 function move to
smaller s (reciprocal relationship with thickness) =>
more fringes in the disc
Intensity vs s for 2-beam condition, different specimen thicknesses t
t = 10 nm
t = 25 nm
t = 50 nm
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Thickness measurement by CBED
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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:
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Identifying polarity in CBED
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
JEMS simulation: GaN [1 -1 0 0] zone axis
Simulation vs experiment:
t = 100 nm
t = 150 nm
t = 200 nm
T. Mitate et al. Phys. Stat. Sol. (a)
192, 383 (2002)
t = 250 nm
000-2
0000
0002
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Analysing SAED
patterns
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Given an SADP, how do we analyse it?
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For indexing, choose two
potential, non-parallel planes
(h1 k1 l1) and (h2 k2 l2) then
determine zone axis [U V W] as
cross product:
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Index two reflections that have
correct plane spacings, making
sure that angle between planes is
correct!
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Go from there using vectorial
addition, choice of planes that
are consistent with [U V W]
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Calibrating rotation
Optical axis
Electron
Unless
you source
are using rotation-corrected TEM (e.g. JEOL 2200FS), you must calibrate rotation
between image and diffraction pattern if you want to correlate orientation with image
Condenser lens
Specimen
Objective lens
Back focal plane/
diffraction plane
Intermediate
image 1
Use specimen with clear shape orientation
Defocus diffraction pattern (diffraction focus/
intermediate lens) to image pattern above BFP
Diffraction spots now discs; in each disc there is an
image (BF in direct beam, DF in diffracted beams
Selected area
aperture
Intermediate lens
Projector lens
BF image (GaAs nanowire)
Duncan Alexander: Diffraction – Going Further
Diffraction
Defocus SADP
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Indexing planes example
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Indexing planes example
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Indexing planes example
For cubic systems:
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Fitting by simulation
Can use e.g. Stadelmann’s JEMS software
to simulate diffraction patterns for known
crystal structure(s), and fit to
experimental data
JEMS can also do “active” fitting from
measured reciprocal lattice spacings
and angles
Duncan Alexander: Diffraction – Going Further
CIME, EPFL
34
Mapping in nano-beam
diffraction mode
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Nano-beam set-up
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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!
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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
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The NanoMEGAS ASTAR system scans the nanobeam across sample while recording diffraction
patterns to make a map with one diffraction pattern
for every pixel positions x, y!
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“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!
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Much higher spatial resolution than EBSD in SEM,
and angular resolution of 1°!
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Can combine with precession for greater reliability
of indexing
[1] E.F. Rauch et al., Micros. Anal. Nanotech. Supplement, 22(6), S5-S8 ,2008
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ASTAR example: nanocrystalline ZnO
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Plan-view sample of textured, nanocrystalline ZnO thin-film!
750 x 750 pixel map, 2–3 nm probe size, 2 nm step size
{10-10}
Y
Z
X
{0001}
{2-1-10}
A. Brian Aebersold, CIME
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