Electronic Structure
A.Y. 2024/2025
Learning objectives
The Electronic Structure course provides a thorough knowledge of the theoretical and computational modeling of basic properties of solids and nanostructures, based on their electronic structure. In particular, it focuses on state-of-the-art methods for the first-principles numerical evaluation of the electronic, structural, and spectroscopic properties of crystals as well as of nanometer-sized systems, molecules, and interfaces.
Expected learning outcomes
The student is expected to: 1) learn the most diffuse methods for the state-of-the art research in electronic structure of materials, based on density functional theory (DFT). 2) learn the main numerical and practical issues related to such methods. 3) learn the field of applications, and possible extensions that could bypass the limitations. 4) learn how to handle aperiodic systems such as interfaces, defects, molecules, and other
nanometer-sized systems with methods developed for solid crystals. 5) learn some examples of evaluation by DFT of quantities that constitute important parameters for the development of more advanced theories, where the electron-electron correlation is more important. 6) become capable to compute the electronic band structure of solid crystals, as well as of related properties, by using DFT.
nanometer-sized systems with methods developed for solid crystals. 5) learn some examples of evaluation by DFT of quantities that constitute important parameters for the development of more advanced theories, where the electron-electron correlation is more important. 6) become capable to compute the electronic band structure of solid crystals, as well as of related properties, by using DFT.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1) Introduction. Historical notes on the development of the modern electronic structure calculation of materials. The density functional theory (DFT), the perturbation theory in DFT and their main successes. Brief references to Physics of crystalline solids: direct and reciprocal space, Bloch's theorem and electronic band structure.
2) The DFT. Hohenberg-Kohn and Kohn-Sham theorems, approximation of local density and density gradient. Extensions beyond the gradient approximation. In particular, DFT+U, exact exchange and hybrid functionals.
3) Common issues in DFT. Brillouin zone integration (special points, Monkhorst-Pack, tetrahedrons). Smearing of occupations. Pseudopotentials. The atomic pseudo-potential with conservation of the norm. The ultra-soft pseudo-potential. The Projector Augmented Wave method.
4) Plane waves basis sets. Energy of the Kohn-Sham system with plane waves. Schrodinger equation. Hellmann-Feynman forces. Computational aspects: augmentation charges and cutoffs, double grid in real space, self-consistency. Convergence parameters.
5) Non-periodic systems in the three dimensions. Slab method and supercell method. Corrections for charged systems or systems with dipole moments: Makov-Payne and Bengtsson. Notes on the treatment of aperiodic systems through open boundary conditions and Green's function methods.
6) Localized basis sets. Tight binding method (outline). Basis sets consisting of localized, analytical and numerical orbitals. Energy and forces. Corrections for non-orthogonality of the basis set and Pulay forces. Augmented plane waves.
7) Applications. Calculation of phonons: frozen phonon; the DFPT. Quantum molecular dynamics. Car-Parrinello method. Examples of application of DFT to surfaces, interfaces, defects.
8) Extensions and potential insights. Spectroscopy, the band gap problem in DFT and alternatives in many electron methods. Wannier's function method. Electronic transport in nanostructures.
9) Numerical sessions for the calculation of the band structure of insulating, metallic, and magnetic crystals. The compressibility of a crystal. Evaluation of vibrational modes. Comparison between plane waves and localized basis sets. Other examples to be defined also according to the prevailing interests of the students.
2) The DFT. Hohenberg-Kohn and Kohn-Sham theorems, approximation of local density and density gradient. Extensions beyond the gradient approximation. In particular, DFT+U, exact exchange and hybrid functionals.
3) Common issues in DFT. Brillouin zone integration (special points, Monkhorst-Pack, tetrahedrons). Smearing of occupations. Pseudopotentials. The atomic pseudo-potential with conservation of the norm. The ultra-soft pseudo-potential. The Projector Augmented Wave method.
4) Plane waves basis sets. Energy of the Kohn-Sham system with plane waves. Schrodinger equation. Hellmann-Feynman forces. Computational aspects: augmentation charges and cutoffs, double grid in real space, self-consistency. Convergence parameters.
5) Non-periodic systems in the three dimensions. Slab method and supercell method. Corrections for charged systems or systems with dipole moments: Makov-Payne and Bengtsson. Notes on the treatment of aperiodic systems through open boundary conditions and Green's function methods.
6) Localized basis sets. Tight binding method (outline). Basis sets consisting of localized, analytical and numerical orbitals. Energy and forces. Corrections for non-orthogonality of the basis set and Pulay forces. Augmented plane waves.
7) Applications. Calculation of phonons: frozen phonon; the DFPT. Quantum molecular dynamics. Car-Parrinello method. Examples of application of DFT to surfaces, interfaces, defects.
8) Extensions and potential insights. Spectroscopy, the band gap problem in DFT and alternatives in many electron methods. Wannier's function method. Electronic transport in nanostructures.
9) Numerical sessions for the calculation of the band structure of insulating, metallic, and magnetic crystals. The compressibility of a crystal. Evaluation of vibrational modes. Comparison between plane waves and localized basis sets. Other examples to be defined also according to the prevailing interests of the students.
Prerequisites for admission
Basic mechanics, thermodynamics, statistics, electromagnetism, quantum mechanics, and structure of matter. Especially:
Fermi-Boltzmann statistics;
Plane waves and Fourier transform;
Schrodinger's equation;
Basic concepts in the band structure and vibrations of a solid;
Basic use of the LINUX operating system.
The attendance of Physics of Solids 1 does not constitute a prerequisite for admission.
Fermi-Boltzmann statistics;
Plane waves and Fourier transform;
Schrodinger's equation;
Basic concepts in the band structure and vibrations of a solid;
Basic use of the LINUX operating system.
The attendance of Physics of Solids 1 does not constitute a prerequisite for admission.
Teaching methods
Teaching is provided in a traditional way, with frontal lectures. The topics are discussed verbally and through illustrations and equations at the blackboard, sometimes with the projection of slides to support. The teaching includes numerical exercises (hands-on sessions). We will use the dedicated myAriel website for sharing the teaching material.
Teaching Resources
R.M. Martin, Electronic Structure - Basic Theory and Practical Methods (Cambridge University Press, 2004 or 2020)
N.W. Ashcroft, N.D. Mermin, Solid State Physics (Harcourt NY 1976)
N.W. Ashcroft, N.D. Mermin, Solid State Physics (Harcourt NY 1976)
Assessment methods and Criteria
The exam consists in a colloquium of variable length typically in the 45-60 minutes range. It will focus on the topics presented during the teaching and on the discussion of an individually assigned numerical exercise, evaluating both the acquired knowledge and the competence of applying it to new cases.
Educational website(s)
Professor(s)