Ge-CdTe, 300kV Sample: D. Smith Holo: H. Lichte, M.Lehmann 10nm object wave amplitude object wave...
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Transcript of Ge-CdTe, 300kV Sample: D. Smith Holo: H. Lichte, M.Lehmann 10nm object wave amplitude object wave...
Ge-CdTe, 300kVSample: D. SmithHolo: H. Lichte,
M.Lehmann
10nm
object waveamplitude
object wavephase
FT
A000
P000
A1-11
P1-11
A1-1-1
A-111
P-111
P1-1-1
A-11-1
P-11-1
A-220
P-220
Kx(i,j)/a*
Ky(i,j)/a*
t(i,j)/Å
set 1: Ge set 2: CdTe dVo/Vo = 0.02% dV’o/V’o = 0.8%
Object Parameter Retrieval using Inverse Electron Diffraction including Potential
DifferencesKurt Scheerschmidt, Max Planck Institute of Microstructure Physics, Halle/Saale, Germany, [email protected] http://www.mpi-halle.de
trial-and-errorimage analysis
direct objectreconstruction
1. objectmodeling
2. wave simulation
3. image process
4. likelihoodmeasure
repetition
parameter &potential
reconstruction
wavereconstruction
?
image
?
no iteration same ambiguities
additional instabilities
parameter& potential
atomicdisplacementsexit object
wave
image
direct interpretation :Fourier filteringQUANTITEM
Fuzzy & Neuro-NetSrain analysishowever:
Information lossdue to data reduction
deviations fromreference structures:
displacement field (Head)algebraic discretizationNo succesful test yet
reference beam (holography)(cf. step 1)
defocus series (Kirkland, van Dyck …)Gerchberg-Saxton (Jansson)tilt-series, voltage variation
multi-slice inversion(van Dyck, Griblyuk, Lentzen,
Allen, Spargo, Koch)Pade-inversion (Spence) non-Convex sets (Spence)local linearization
cf. step 2
Inversion?
= M(X) 0
= M(X0) 0 + M(X0)(X-X0) 0
Assumptions:
- object: weakly distorted crystal
- described by unknown parameter set X={t, K,Vg, u}
- approximations of t0, K0 a priori known
M needs analytic solutions for inversion
Perturbation: eigensolution , C for K, V yields analytic solution of and its derivatives
for K+K, V+V with tr() + {1/(i-j)}
= C-1(1+)-1 {exp(2i(t+t)} (1+)C
The inversion needs generalized matrices due to different numbersof unknowns in X and measured reflexes in disturbed by noise
Generalized Inverse (Penrose-Moore):
X= X0+(MTM)-1MT.[exp- X]
A0 Ag1 Ag2 Ag3
P0 Pg1 Pg2 Pg3
...
...exp
X= X0+(MTM)-1MT.[exp- X]
i i i
j j jX X X...
t(i,j) Kx(i,j) Ky(i,j)
-lg()
lg()Regularization parameter test
Kx(i,j)/a*
Ky(i,j)/a*
t(i,j)/Å
Retrieval with iterative fit of the confidence region
lg()
step
step
< t > / Å
relative beamincidence to zone axis [110]
[-1,1,0]
[002]
iii
iii
iiiiii
(i-iii increasing smoothing)
Ky(i,j)/a*
Kx(i,j)/a*
K(i,j)/a*
t(i,j)/ Å
model/reco input 7 / 7 15 / 15 15 / 9 15 / 7beams used Influence of Modeling Errors
Replacement of trial & error image matching by direct object parameter retrieval without data information loss is partially solved by linearizing and regularizing the dynamical
scattering theory – Problems: Stabilization and including further parameter as e.g. potential and atomic displacements
Step 1: exit wave reconstructione.g. by electron holographyStep 2a: Linerizing dynamical theory
Step2b: Generalized Inverse
Step 2c: Single reflex reconstruction
Example 1: Tilted and twisted grains in Au
Step 2d: Regularization
Replacing the Penrose-Moore inverse by a regularized and generalized matrix
( regularization, C1 reflex weights, C2 pixels smoothing)
X=X0+(MTC1M + C2)-1MT
Regularizatiom Maximum-Likelihood error distribution:||exp-th||2 + ||X||2 = Min
Example 2: Grains in GeCdTe with different Composition and scattering potential
1
1
2
2
3
3
4
4
5
5
Conclusion: Stability increased & potential differences recoverable
Unsolved: Modeling errors & retrieval of complete potentials
Argand plots: selected regions ofthe reconstructedGeCdTe exit wave
1
2
3
4
5
whole wave