Physics

1 Physical observables as derivatives of total energy 1.1 Introduction to IR and optical spectroscopy (dielectric tensor in IR and visible range) and Raman spectroscopy 1.2 First order derivatives: pressure, forces, and polarization 1.3 Second-order derivatives: electronic dielectric tensor, vibrational (phonon) frequencies, elastic constancies, IR vibrational activity (effective charges), and piezoelectric tensor 1.4 Third order derivatives: anharmonic vibrational broadening, Raman vibrational activity 2 Mean-field theories for total energy 2.1 Quick overview of Density functional theory and Hartree-Fock, with description of the adopted notations 3 Response in the static-case: density-density response, macroscopic susceptibility, effective charges, Wannier functions, spontaneous electric polarization in solids as a Berry phase and Thouless pump 3.1 Evaluation of first and second order derivatives: general expression from perturbation theory 3.2 Static density-density perturbation theory. Interacting and non-interacting density-density susceptibility in real space and in reciprocal space for a generic system in DFT and RPA. 3.3 Static density-density response in Jellium. Lindhard functions in one two and three dimensions. Screened Coulomb potential in reciprocal and real space. 3.4 Response to a uniform static electric field in an insulator. Susceptibility, piezoelectric tensor and effective charges 3.5 Ferroelectric material: measurement of hysteresis cycles in the Polarization/electric field space. Spontaneous polarization (in zero electric field) 3.6 Polarization as average dipole per unit cell. Quantization of polarization for classical discrete charges and apparent paradox associated to the definition in a quantum mechanical context. 3.7 Definition of the Berry phase in the discrete and continuous case 3.8 Example of Berry phase: Aharonov-Bohm effect, molecular Aharonov-Bohm effect 3.9 Wannier functions. Definition and properties 3.10 Centroids of Wannier functions as a Berry phase in Bloch space 3.11 Demonstration that the spontaneous polarization is given by the centroids of Wannier functions 3.12 Thouless pump: quantized charge transport in insulators under time-periodic parametric transformation 4 Time-dependent (and frequency-dependent) response: general properties 4.1 In time-space: causality principle and retarded time-dependent response functions 4.2 In frequency space: holomorphic properties (poles and branch cuts) of the retarded response functions in frequency-space, analytical extension, conjugation relations, Kramers-Kronig relations and spectral representation 4.3 Simple examples: a classical forced and damped oscillator and Drude model, plasmon of Drude model 4.4 Time-dependent perturbation theory for monochromatic perturbations 4.5 Dynamical Polarizability and F-sum rule 4.6 Impact of electron-defect, electron-phonon and electron-electron scattering on Drude peak 5 Electron-hole excitations and plasmons in the microscopic response and in the optical conductivity of metals 5.1 Time-dependent DFT: equation of motion and perturbation theory at zero and finite temperature 5.2 Dynamical density-density response in general and specifically in the case of Jellium: dynamical Lindhard function, free electron-hole excitations and plasmon dispersion 5.3 Measurement of the of microscopic screening function epsilon-1(q,q,omega) by EELS e Inelastic-Xray scattering 5.4 Relation between the macroscopic relative dielectric tensor epsilon_r(omega) and epsilon-1(q,q,omega) 5.5 Calculation with TD-DFT of epsilon_r(omega) and of optical conductivity tensor sigma(omega): Drude and interband conductivity in metals, effective plasma frequency 6 Optical spectroscopy of insulators: Bethe Salpeter equation and Frenkel and Wannier-Mott excitons 6.1 Time-dependent HF (TD-HF) and time-dependent screened HF (TD-sHF): equation of motion and derivation from stationary principle of action 6.2 Linear response of TD-sHF: Bethe Salpeter equation 6.3 Excitonic representation of Bethe Salpeter equation: excitonic two-particle Hamiltonian and interacting electron-hole excitations 6.4 Diagrammatic representation of the Bethe Salpeter equation 6.5 Strongly interacting Frenkel exciton: strongly bound excitons and unbound excitonic resonances 6.6 Weakly bound Wannier-Mott excitons in semiconductors using effective mass theory Exercise sessions (part of the program required for the exam) e.1 Calculation of the macroscopic susceptibility as long-wave limit of a sinusoidal modulated perturbation e.2 Spontaneous polarization, Berry phase, Wannier functions and Thouless pump of a linear 2-atom chain e.3 Optical conductivity and absorption and reflection of light in neutral and doped graphene

Condensed Matter I

Fundamentals of Condensed Matter Physics Marvin L. Cohen, Steven G. Louie Editore: Cambridge University Press 2016 Solid State Physics Giuseppe Grosso, Giuseppe Pastori Parravicini Editore: Elsevier Science Publishing Co Inc 2013

Attendance to the lectures is not mandatory but strongly recommended.

oral, lasting about one hour

Fundamentals of Condensed Matter Physics Marvin L. Cohen, Steven G. Louie Editore: Cambridge University Press 2016 Solid State Physics Giuseppe Grosso, Giuseppe Pastori Parravicini Editore: Elsevier Science Publishing Co Inc 2013

1 Physical observables as derivatives of total energy 1.1 Introduction to IR and optical spectroscopy (dielectric tensor in IR and visible range) and Raman spectroscopy 1.2 First order derivatives: pressure, forces, and polarization 1.3 Second-order derivatives: electronic dielectric tensor, vibrational (phonon) frequencies, elastic constancies, IR vibrational activity (effective charges), and piezoelectric tensor 1.4 Third order derivatives: anharmonic vibrational broadening, Raman vibrational activity 2 Mean-field theories for total energy 2.1 Quick overview of Density functional theory and Hartree-Fock, with description of the adopted notations 3 Response in the static-case: density-density response, macroscopic susceptibility, effective charges, Wannier functions, spontaneous electric polarization in solids as a Berry phase and Thouless pump 3.1 Evaluation of first and second order derivatives: general expression from perturbation theory 3.2 Static density-density perturbation theory. Interacting and non-interacting density-density susceptibility in real space and in reciprocal space for a generic system in DFT and RPA. 3.3 Static density-density response in Jellium. Lindhard functions in one two and three dimensions. Screened Coulomb potential in reciprocal and real space. 3.4 Response to a uniform static electric field in an insulator. Susceptibility, piezoelectric tensor and effective charges 3.5 Ferroelectric material: measurement of hysteresis cycles in the Polarization/electric field space. Spontaneous polarization (in zero electric field) 3.6 Polarization as average dipole per unit cell. Quantization of polarization for classical discrete charges and apparent paradox associated to the definition in a quantum mechanical context. 3.7 Definition of the Berry phase in the discrete and continuous case 3.8 Example of Berry phase: Aharonov-Bohm effect, molecular Aharonov-Bohm effect 3.9 Wannier functions. Definition and properties 3.10 Centroids of Wannier functions as a Berry phase in Bloch space 3.11 Demonstration that the spontaneous polarization is given by the centroids of Wannier functions 3.12 Thouless pump: quantized charge transport in insulators under time-periodic parametric transformation 4 Time-dependent (and frequency-dependent) response: general properties 4.1 In time-space: causality principle and retarded time-dependent response functions 4.2 In frequency space: holomorphic properties (poles and branch cuts) of the retarded response functions in frequency-space, analytical extension, conjugation relations, Kramers-Kronig relations and spectral representation 4.3 Simple examples: a classical forced and damped oscillator and Drude model, plasmon of Drude model 4.4 Time-dependent perturbation theory for monochromatic perturbations 4.5 Dynamical Polarizability and F-sum rule 4.6 Impact of electron-defect, electron-phonon and electron-electron scattering on Drude peak 5 Electron-hole excitations and plasmons in the microscopic response and in the optical conductivity of metals 5.1 Time-dependent DFT: equation of motion and perturbation theory at zero and finite temperature 5.2 Dynamical density-density response in general and specifically in the case of Jellium: dynamical Lindhard function, free electron-hole excitations and plasmon dispersion 5.3 Measurement of the of microscopic screening function epsilon-1(q,q,omega) by EELS e Inelastic-Xray scattering 5.4 Relation between the macroscopic relative dielectric tensor epsilon_r(omega) and epsilon-1(q,q,omega) 5.5 Calculation with TD-DFT of epsilon_r(omega) and of optical conductivity tensor sigma(omega): Drude and interband conductivity in metals, effective plasma frequency 6 Optical spectroscopy of insulators: Bethe Salpeter equation and Frenkel and Wannier-Mott excitons 6.1 Time-dependent HF (TD-HF) and time-dependent screened HF (TD-sHF): equation of motion and derivation from stationary principle of action 6.2 Linear response of TD-sHF: Bethe Salpeter equation 6.3 Excitonic representation of Bethe Salpeter equation: excitonic two-particle Hamiltonian and interacting electron-hole excitations 6.4 Diagrammatic representation of the Bethe Salpeter equation 6.5 Strongly interacting Frenkel exciton: strongly bound excitons and unbound excitonic resonances 6.6 Weakly bound Wannier-Mott excitons in semiconductors using effective mass theory Exercise sessions (part of the program required for the exam) e.1 Calculation of the macroscopic susceptibility as long-wave limit of a sinusoidal modulated perturbation e.2 Spontaneous polarization, Berry phase, Wannier functions and Thouless pump of a linear 2-atom chain e.3 Optical conductivity and absorption and reflection of light in neutral and doped graphene

Condensed Matter I

Fundamentals of Condensed Matter Physics Marvin L. Cohen, Steven G. Louie Editore: Cambridge University Press 2016 Solid State Physics Giuseppe Grosso, Giuseppe Pastori Parravicini Editore: Elsevier Science Publishing Co Inc 2013

Attendance to the lectures is not mandatory but strongly recommended.

oral, lasting about one hour

Fundamentals of Condensed Matter Physics Marvin L. Cohen, Steven G. Louie Editore: Cambridge University Press 2016 Solid State Physics Giuseppe Grosso, Giuseppe Pastori Parravicini Editore: Elsevier Science Publishing Co Inc 2013