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, schee@mpi-halle.de 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