Ge-CdTe, 300kV Sample: D. Smith Holo: H. Lichte, M.Lehmann 10nm object wave amplitude object wave phase 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 K x (i,j)/a* K y (i,j)/a* t(i,j)/Å set 1: Ge set 2: CdTe dV o /V o = 0.02% dV’ o /V’ o = 0.8% Object Parameter Retrieval using Inverse Electron Diffraction including Potential Differences Kurt Scheerschmidt, Max Planck Institute of Microstructure Physics, Halle/Saale, Germany, trial-and-error image analysis direct object reconstruction 1. object modeling 2. wave simulation 3. image process 4. likelihood measure repetitionrepetition parameter & potential reconstruction wave reconstruction ? image ? no iteration same ambiguities additional instabilities parameter & potential atomic displacements exit object wave image direct interpretation : Fourier filtering QUANTITEM Fuzzy & Neuro-Net Srain analysis however: Information loss due to data reduction deviations from reference structures: displacement field (Head) algebraic discretization No 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(X 0 )  0 + M(X 0 )(X-X 0 )  0 Assumptions: - object: weakly distorted crystal - described by unknown parameter set X={t, K,V g, u} - approximations of t 0, K 0 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(2  i (t+  t)} (1+  )C The inversion needs generalized matrices due to different numbers of unknowns in X and measured reflexes in  disturbed by noise Generalized Inverse (Penrose-Moore): X= X 0 +( M T M) -1 M T.[  exp -  X   ] A0A0 A g1 A g2 A g3 P0P0 P g1 P g2 P g3...  exp X= X 0 +( M T M) -1 M T.[  exp -  X   ] i ii jjj XXX... t(i,j)K x (i,j)K y (i,j) -lg(  ) lg(  ) Regularization parameter  test K x (i,j)/a* K y (i,j)/a* t(i,j)/Å Retrieval with iterative fit of the confidence region lg(  ) step / Å relative beam incidence to zone axis [110] [-1,1,0] [002] i ii iii i ii iii (i-iii increasing smoothing) K y (i,j)/a* K x (i,j)/a* K(i,j)/a* t(i,j)/ Å model/reco input 7 / 7 15 / / 9 15 / 7 beams 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 holography Step 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, C 1 reflex weights, C 2 pixels smoothing) X=X 0 +( M T C 1 M +  C 2 ) -1 M T  Regularizatiom  Maximum-Likelihood error distribution: ||  exp-  th|| 2 +  ||  X|| 2 = Min Example 2: Grains in GeCdTe with different Composition and scattering potential Conclusion: Stability increased & potential differences recoverable Unsolved: Modeling errors & retrieval of complete potentials Argand plots: selected regions of the reconstructed GeCdTe exit wave whole wave