Simulation of momentum resolved Electron Energy Loss Spectroscopy in the low loss region using model band structures

Master thesis English OPEN
Bilden, Sindre Rannem;
(2018)
  • Subject: spectroscopy | Electron Energy Loss Spectroscopy | EELS | phyics | nanotechnology | material science

One of the key limiting factors to progress within nano science is the ability to measure properties on the relevant length scale. The probe size provided by optical measurements is often larger than the individual nanoscale structures, and the resulting measurement is ... View more
  • References (30)
    30 references, page 1 of 3

    1 Introduction 1 1.1 Measurements on nanoscopic structures . . . . . . . . . . . . . . . . . 1 1.2 Transmission Electron Microscope . . . . . . . . . . . . . . . . . . . . 1 1.3 Electron Energy Loss Spectroscopy . . . . . . . . . . . . . . . . . . . 2 1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    2 Background 7 2.1 EELS in TEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Electrons in a periodic potential . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Crystal lattice in real space . . . . . . . . . . . . . . . . . . . 8 2.2.2 Lattice in reciprocal space . . . . . . . . . . . . . . . . . . . . 9 2.2.3 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.4 Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.5 Bloch waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.6 Wannier functions . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.7 Electron band structure of periodic potentials . . . . . . . . . 13 2.3 Crystal energy models . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 Empty lattice model . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Density functional theory and pseudo-potentials . . . . . . . . 16 2.3.3 Tight binding model . . . . . . . . . . . . . . . . . . . . . . . 16

    3 Theoretical foundation of inelastic electron scattering 19 3.1 Scattering cross section . . . . . . . . . . . . . . . . . . . . . . . . . . 19

    4 Implemented crystal model 33 4.1 Parabolic band class . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Tight binding class . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2.1 Wurtzite sp3 model with atomic orbitals . . . . . . . . . . . . 36 4.2.2 Fitting procedure . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2.3 Wannier functions . . . . . . . . . . . . . . . . . . . . . . . . . 37

    5 Implemented scattering cross section 39 5.1 Data structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2 The dielectric permittivity . . . . . . . . . . . . . . . . . . . . . . . . 41 5.2.1 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.3 Lorentzian angular distribution . . . . . . . . . . . . . . . . . . . . . 47

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    [10] P. Hohenberg and W. Kohn. Inhomogeneous electron gas. Phys. Rev., 136: B864-B871, Nov 1964. doi: 10.1103/PhysRev.136.B864. URL https://link. aps.org/doi/10.1103/PhysRev.136.B864.

    [11] W. Kohn and L. J. Sham. Self-consistent equations including exchange and correlation effects. Phys. Rev., 140:A1133-A1138, Nov 1965. doi: 10.1103/PhysRev. 140.A1133. URL https://link.aps.org/doi/10.1103/PhysRev.140.A1133.

    [12] T Koopmans. Über die zuordnung von wellenfunktionen und eigenwerten zu den einzelnen elektronen eines atoms. Physica, 1(1):104 - 113, 1934. ISSN 0031-8914. doi: https://doi.org/10.1016/S0031-8914(34)90011-2. URL http: //www.sciencedirect.com/science/article/pii/S0031891434900112.

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