Docente
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MAURI FRANCESCO
(programma)
1 Introduction
1.1 Measuring electronic bands in solids with Arpes. Lifetime broadening and kinks in the dispersion
1.2 Predictive power of theory: band structure comparison with experiments
2 Mean field description of electrons in solids: Hartree-Fock
2.1 Band structure in presence of spin-orbit interaction: spinor and Kramer’s doublets
2.2 Ground-state of N-independent electron systems
2.3 Ground-state of N-interacting electron systems
2.4 Hartree-Fock (HF). General derivation non-collinear
2.2 HF collinear
2.3 Self-consistent HF Hamiltonian
2.3 Interaction 3D Jellium: HF total energy and band structure, ferromagnetic instability, correlation energy, pair correlation function, [impact of correlation on Kinetic energy]
3 Mean field description of electrons in solids: Density functional theory
3.1 Hohenberg-Kohn (HK) and Kohn and Sham (KS) theorems
3.2 HK and KS functionals and KS Hamiltonian
3.3 Approximated exchange and correlation functionals (LDA GGA)
3.4 [Adiabatic connection theorem for the exchange and correlation functional]
4 Linear response theory in DFT for static (and dynamic) perturbations
4.1 First and second order derivatives of total energy: physical observable and evaluation within DFT
4.2 Density-density response for a generic system with a matrix formalism (in real space and reciprocal space). DFT and RPA.
4.3 Density-density response in Jellium. Lindhard functions in different dimensions.
4.4 [Phonon perturbation in reciprocal space. Calculation of dynamical matrix, electron-phonon interaction and Kohn anomalies.]
4.5 Response to a uniform static electric field in an insulator. Susceptibility, piezoelectric tensor and effective charges
4.6 Response to an electromagnetic electric field (monochromatic light) in an insulator. Susceptibility: interband-transitions and excitons
5 Spontaneous polarization
5.1 Definition of the Berry phase
5.2 Wannier functions
5.2 Spontaneous polarization as a Berry phase: formulation with the Wannier and with the Bloch representation.
6 Exercise sessions (integral part of the program)
6.1 Graphene π-electron band structure and ARPES
6.2 Density matrix and pair correlation function in Jellium
6.3 Dissociation of the H2 molecule with a 2-site Hubbard model: exact solution and approximated restricted and unrestricted HF solutions.
6.4 Calculation of the macroscopic susceptibility as long-wave limit of a sinusoidal modulated perturbation
6.5 Spontaneous polarization, Berry phase and Wannier functions of a linear 2-atom chain
Fundamentals of Condensed Matter Physics, Cohen and Louie, Cambridge University Press, Chapters 5, 6, 7, 8, 10
Solid State Physics (Second Edition), Grosso and Pastori-Parravicini, Accademic Press, Chapters 4 and 8
Quantum Theory of the electron liquid, Giuliani and Vignale, Cambridge University Press, Chapters 1, 2, 3, 4 (selected parts).
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