Diploma - Max Planck Institute for Solid State Research

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8 2 Theoretical foundation such a system anyway (for example as a zeroth order approximation), one has to challenge additionally the convergence instability of the Fermi level determination process, since the dispersion of the localized states is weak and therefore a small rearrangement of the Fermi level changes the occupation number strongly. A possible solution is to modify the functional so that it contains some correction to the correlation energy (e.g. L(S)DA+U). To circumvent this issue the localized electrons have been treated as “core” electrons 1 (so-called open core approximation) neglecting the overlap from different sites. This approximation is legitimate, because their magnitude of localization is comparable to orbitals treated as “core” electrons, but their single-particle energy is considerably higher. Moreover, in ch. 4.1 it is shown, that the Fermi level obtained by this method is comparable to L(S)DA+U results with respect to the valence band structure. 2.1.3 Codes There are a lot of DFT codes available with various approximations depending on the implemented basis set (a few implementations of the respecting methods are given at the end of each block). In general, a distinction[17, p. 233–235] can be drawn between (a) Plane wave methods present the most general way for solving differential equations. It is easy to implement them for computation and since being the solution of the Schrödinger equation with constant potential, they are an effective basis for the nearly-free electron model (covering the crystal potential as a small perturbation) therefore one gets a valuable insight to the bandstructure of sp-metals and semiconductors. The disadvantage is enclosed in the potential representation because plane waves demand a smooth potential whereas the Coulomb potential has a singularity. Hence, those methods are often accompanied by pseudopotentials (smoothed potentials, nucleus and core electrons are combined) or grids. (e.g. Abinit [http://www.abinit.org], VASP [http://cms.mpi.univie.ac.at/vasp], Quantum- Expresso [http://www.quantum-espresso.org], CPMD [http://www.cpmd.org], ...) (b) Choosing localized (atomic-like) orbitals as a basis set respects automatically the symmetry in the vicinity of the atomic sites. As basis functions of the Bloch states are usually selected Gaussians, or numerically adjusted atomic-like orbitals (demands Bloch Theorem, cf. (2.11), (2.12)). The advantage of being able to use the bare Coulomb potential (superposition of the atomic Coulomb potentials) is gaining high accuracy for heavier elements as well as having a smaller basis set compared to plane wave methods. Since atomic orbitals from different sites are not 1 Further on, this approximation will be called “open core approximation”, in literature also frozen core or quasi-core, because we deal with not fully-occupied “core” electrons. Since it is not a stable noble gas configuration, the occupation number should in principle be determined variationally. Given that the overlap of the localized orbitals is small, one can set a fixed occupation number as an initial parameter according to the experiment
2.1 Band Structure Theory 9 orthogonal, all multicenter-integrals of the basis set have to be computed. There are also various approximate, non-DFT solutions possible e.g. tight-binding[22] which are mainly used to estimate parameters of model Hamiltonians (e.g. model, Hubbard model) for comparison with real compounds. Anderson (e.g. FPLO [http://www.fplo.de], Gaussian [http://www.gaussian.com], Siesta [http://www. icmab.es/siesta/], Crystal [http://www.cse.scitech.ac.uk/cmg/CRYSTAL/], ...) (c) Atomic sphere methods are the natural approach to adopt the basis set to the given problem dividing the arrangement of atoms into atomic sphere-like (centered around the sites) and interstitial parts. The potential in the former is similar to the atomic potential, whereat in the latter case it is smooth suggesting an augmented basis set consisting of localized functions with boundary conditions satisfying smoothly varying functions in the interstitial region (so-called APWs - Augmented Plane Waves). Adversely, this results in non-linear equations 2 which solutions are demanding. Therefore one introduced a linerization[23] around fixed energy values (eg. LAPW, ...) receiving the most accurate method today. (e.g. fleur [http://www.flapw.de], Wien2k [http://www.wien2k.at/], elk/exciting [http://exciting. sourceforge.net], Stuttgart LMTO [http://www.fkf.mpg.de/andersen/docs/manual.html], ...) The following codes have already been used successfully in our group for LDA calculations of Heavy Fermion (HF) and mixed-valent compounds in the past (and were used for all calculations in this diploma thesis) but this does not imply that they are the most suitable ones. 1. Full Potential Local Orbital code (FPLO) [24] FPLO uses a nonorthogonal local-orbital basis set | Bsµ 〉 (cf. 2.11) whose orbitals are the solution of a Schrödinger Equation with a spherically-averaged crystal potential and a limiting potential part v lim = ( r r 0 ) 4. The latter ensures a minimized basis set 3 since otherwise the amount of atomic-like basis functions needed for a sufficient expansion of weakly-bound extended states would be severely larger. Another option is a basis set extension by plane waves but the complexity and thus the additional computational effort is in no relation to the gained accuracy. The basis orbitals for which the differences between the real crystal and the sphericalaveraged potential is not perceptible, are called core orbitals – and their overlap is defined as zero. All remaining orbitals are treated as valence orbitals. Nevertheless, the overlap between core orbitals and valence orbitals from different sites is regarded. Due to the distinction between core and valence orbitals (or even 2 this does not influence the basic linearity of the SE, but due to the energy-dependence of the basis functions one is not able to solve the equations for all eigenenergies at one time 3 basis functions in the vicinity of a weak potential influence get compressed compared to the same eigenfunctions in an unmodified potential, therefore the compressed ones are more suitable to describe weakly bound / unbound states
Page 1 and 2: - Photoemission on EuRh 2 Si 2 - di
Page 3: Abstract A thorough knowledge of co
Page 7: vii Nomenclature AF APW ARPES BO BZ
Page 10 and 11: 2 1 Introduction conflicting limits
Page 12 and 13: 4 2 Theoretical foundation fore the
Page 14 and 15: 6 2 Theoretical foundation system a
Page 18 and 19: 10 2 Theoretical foundation Figure
Page 20 and 21: 12 2 Theoretical foundation Figure
Page 22 and 23: 14 2 Theoretical foundation Fi
Page 25 and 26: 17 3 Experimental foundations 3.1 P
Page 27 and 28: 3.2 General setup 19 Figure 3.2: ch
Page 29 and 30: 3.4 SLS: SIS-HRPES 21 1 3 ARPES the
Page 31 and 32: 23 4 EuRh 2 Si 2 - semi-localized e
Page 33 and 34: 4.1 Overview - properties and class
Page 49 and 50: 4.2 Hybridization: localized versus
Page 61 and 62: 4.3 Perspective: quasi-linear dispe
Page 63: 4.3 Perspective: quasi-linear dispe
Page 66 and 67:
58 5 Summary to the f-d interaction
Page 68 and 69:
60 Bibliography [13] B. Kramer. A P
Page 70 and 71:
62 Bibliography [44] T. M. Rice and
Page 72 and 73:
64 Bibliography [75] Hari C. Manoha
Page 75:
List of Tables 67 List of Tables 2.
Page 79:
Erklärung Hiermit versichere ich,
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Diploma - Max Planck Institute for Solid State Research

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