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Electron image interpretation based on electron densities … better:

Electron image interpretation based on electron densities … better: Substitution of Atomic Scattering Amplitudes by Analytic Bond Order Potentials. TEM analysis : object modeling & inverse problem. Empirical molecular dynamics: macroscopic relevance vs. potential problem.

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Electron image interpretation based on electron densities … better:

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  1. Electron image interpretation based on electron densities … better: Substitution of Atomic Scattering Amplitudes by Analytic Bond Order Potentials TEM analysis: object modeling & inverse problem Empirical molecular dynamics: macroscopic relevance vs. potential problem Application: scattering amplitude & frozen lattice Kurt Scheerschmidt & Volker Kuhlmann Max Planck Institute of Microstructure Physics Halle/Saale, Germany schee@mpi-halle.de http://www.mpi-halle.de

  2. 1. object modeling r e p e t i t i o n 2. wave simulation ? 3. image process wave reconstruction ? 4. likelihood measure image trial-and-error image analysis direct object reconstruction parameter & potential reconstruction

  3. Assumptions: - dynamical theory necessary - relativistic corrections - inelastic scattering: absorption - thermic diffuse scattering: frozen lattice - forward scattering approximation - object: weakly distorted crystal - well defined parameter set X={t, K,Vg, u} y = M(X) y0

  4. Why empirical MD: macroscopic relevance continuum elasticity theory: no atomic information embedding in elastic displacement field as boundary condition M/m ~ 1000 c ~ 3AE / fs tight-binding ~ 1000 atoms wavefunction =S atomic orbitals, TB-Hamilton sss, sps, pps, ppp, .... ab initio ~ 100 atoms parameter free all electron, pseudopotentials, LDA, CP, ..

  5. Classical molecular dynamics time-integration (fs-steps) up to relaxation (< 1ns) simultaneously 3N Newtonian equations of motion with V= empirical interaction potential

  6. Relaxation of an uncapped (100)-SiGe/Si pyramid Dt = 0.1fs Tmax = 900K DT = 20K/ns up/down Dtequil = 1ns Dtframe = 0.1ns

  7. 1026.6 1/cm 444.9 1/cm 444.9 1/cm 1026.6 1/cm 1393.0 1/cm 233.7 1/cm 1395.2 1/cm

  8. Empirical Potentials Cluster: 2-, 3-, …, n-body expansions (BMH, SW, …) Environment: density modified (EAM, MEAM, FS, …) Tersoff: empirical bond order potential V ~ S [exp(-l.rij) - bij exp(-m.rij)] bonds are weighted by bij ~ F(rik, rjk, gijk) bij ~ {S 1+c2/d2+c2/[d2+(h-cosdijk)2]}-n over all neighbors kij and with all parameters fitted BOP: bond order approximation justified by TB methods

  9. BOP embedded bonds instead atoms – two-center orthogonal TB density matrix instead diagonalisation – Lanczos recursion E tot = Erep + Eprom + Eband (k)   empirical s2p2->sp3SHia,jbQia,jb  hopping integrals BOmatrix Slater-Koster Pettifor-Aoki sss, sps, ppp... ½(N+-N-) ~ Im G de GreenFkt G ~ |H-E-ih|-1 continued fraction G00=1/(E-ih-a0-b12/(E-ih-a1-…)) Ebond = -2(sss+pps) Qis,js–4 pppQia,jb BOP2 => Tersoff bij=pij+(spij+spji)/2 BOP4 => p-bonds Qia,jb = [1+(1+b22-b12) / b3(b1+b2)].[1+ b22 / (b1+b3)2 ]–1/2/b1 evaluate the momenta up to order n analyzing hops to all neighbors yielding analytic terms LCAO linear combination of hydrogen like atomic orbitals

  10. j j i k i Bond Order Potential • Slater-Koster products = electronic hoppings in closed loops Inter-Site On-Site separation angular functions

  11. l k j i k j j j i i k new On-Site hops i l + + 3 2 j j j j 1 1 3 4 k k 4 2 i i new  hops + l k k i i torsion included:  & On-Site hops + = + +

  12. l k l j k i i j increased stiffness Comparison of - and -contributions -120 120 180 l k j 0 60 i -60 Si (110) surface dimerization 0K->600K->0K

  13. TS BOP4+ [001] and [110] views of a 1.4o twist bonded interface comparing 900K annealing with Tersoff versus BOP potential [110] + 1.4o red=TS green=BOP4+

  14. PDF Distribution of pair distances, angles and torsions at the 1.4o twist bonded Si-001 interface comparing Tersoff and BOP4+ potentials after annealing at 900K angles torsion

  15. TS-III SiGe/Si QD annealed at 900K for Tersoff and BOP4+ potentials BOP4+

  16. a C2H6: DFT-optimized structure and charge density (a), selected cuts of density (b), potential (c), and sampled BOP (d) b c d

  17. H4 in [100]-Si-2x2x2 & charge density ¼ ½ ¾ e- H Si H4inSi: BOP cuts in ¼, ½, ¾ depth of unit cell; sampling by e-, H, Si test species

  18. H4 in [100]-Si-2x2x2 & charge density DFT BOP H4inSi: central slices from sampled DFT (in Ha) and BOP (in eV)

  19. 17/32 13/32 14/32 15/32 16/32 BOP -50 -3.5 -3.0 -50 13/33 14/33 15/33 16/33 17/33 BOP -600 -7 -1.6 -10 14/32 14/33 17/32 17/33 18/32 19/33 EMS Comparison of scattering potentials for sliced H4 in Si 48 48 4.9 13/33

  20. EMS BOP 14/32 14/32 17/32 15/32 18/32 16/32 Comparison of scattering functions for sliced H4 in Si

  21. A P A P EMS EMS DFT DFT BOP 9.8nm 3.3nm BOP

  22. A A A A Variation of amplitudes and phases with sample thickness (nm) P P P P 2.2 4.4 BOP BOP EMS EMS A A A P P P BOP 16.3 19.6 22.8

  23. Dislocation dipole in 110-Si: BOP optimized structure (a), charge density (b), and simulated exit waves (amplitude A, phase P) for DFT, standard EMS potentials, and sampled BOP of 16 subslices (200kV, thickness=18.4nm). A b DFT a P A EMS A BOP P P

  24. Conclusion / Outlook TEM object analysis = inverse problem needs good modeling of objects & scattering Empirical molecular dynamics: nanoscopic lengths & times needs analytic TB-BOP & elastic BCs Structural and electronic properties: frozen lattices & scattering amplitudes

  25. multi-slice inversion (van Dyck, Griblyuk, Lentzen, Allen, Spargo, Koch) Pade-inversion (Spence) non-Convex sets (Spence) local linearization deviations from reference structures: displacement field (Head) algebraic discretization Inversion ? parameter & potential atomic displacements exit object wave no iteration same ambiguities additional instabilities direct interpretation by data reduction: Fourier filtering QUANTITEM Fuzzy & Neuro-Net Srain analysis image reference beam (holography) defocus series (Kirkland, van Dyck …) Gerchberg-Saxton (Jansson) tilt-series, voltage variation

  26. Direct & Inverse: black box gedankenexperiment thickness local orientation structure & defects composition microscope g output f input operator A wave image theory, hypothesis, model of scattering and imaging if unique & stable inverse A-1 exists ill-posed & insufficient data => least square direct: g=A<f experiment, measurement invers 1.kind: f=A-1<g parameter determination invers 2.kind: A=g$f -1 identification, interpretation a priori knowledge intuition & induction additional data

  27. Problem: Slater-Koster integrals not transferable & only 2 centre Solution: assume non-orthogonal basis states relate Hamiltonian matrix elements to it non-orthogonal basis: non-orthogonality matrix: find elements of Greens function via moments expansion as in BOP itself

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