>>>2023 notes<<<
1 Sala Hack lunedì 11 marzo 2019 10:30-12:30
2 Sala Dulbecco lunedì 18 marzo 2019 10:30-12:30
3 Sala Hack mercoledì 27 marzo 2019 11:30-13:30
4 Sala Hack martedì 2 aprile 2019 10:30-12:30
5 Sala Hack martedì 9 aprile 2019 10:30-12:30 discussion of problems
6 Sala Dulbecco lunedì 15 aprile 2019 10:30-12:30 first set of problems due
7 Sala Dulbecco venerdì 19 aprile 2019 14:30-16:30
8 Sala Dulbecco venerdì 26 aprile 2019 14:30-16:30
9 Sala Dulbecco lunedì 29 aprile 2019 10:30-12:30
10 Sala Dulbecco venerdì 3 maggio 2019 14:30-16:30 discussion of problems
11 Sala Dulbecco lunedì 6 maggio 2019 10:30-12:30 second set of problems due
mercoledì 24 gennaio 2018 14:30-16:30 Dulbecco10:30-12:30 Montalcini
Connection between statistical mechanics and quantum mechanics. Wick rotation
venerdì 26 gennaio 2018 14:30-16:30 Montalcini
mercoledì 31 gennaio 2018 14:30-16:30 Dulbecco
mercoledì 14 febbraio 2018 14:30-16:30 Dulbecco
mercoledì 21 febbraio 2018 14:30-16:30 Dulbecco
Fermi golden rule, optical absorption, transition dipole, symmetry, dipole allowed and forbidden transitions mercoledì 28 febbraio 2018 14:30-16:30 Dulbecco
k*p theory, symmetry of the band structure
mercoledì 7 marzo 2018 10:30-12:30 Hack
mercoledì 4 aprile 2018 10:00-12:00 Sala Dulbecco
Crystal field levels for p and d electrons, octahedral and tetrahedral coordination, irreducible representations, angular momentum operator, equivalence of t2g and p electrons
lunedi 9 april 2018 10:30-12:30 Sala Montalcini
1-site, 2-site Hubbard model, Mott plateaus, Hubbard bands, "Fermi sea" vs Heitler-London state on a dimer, upper and lower Hubbard bands, Mott transition, Stoner instability
mercoledì 11 aprile 2018 10:00-12:00 Dulbecco
Magnetic exchange in Mott insulators - Anderson superexchange. Antiferromagnetic and ferromagnetic superexchange on 180-degree and 90-degree M-O-M bonds. Frustration in next-nearest neighbor model. Spin spirals. Inversion symmetry breaking by spins
mercoledì 18 aprile 2018 10:00-12:00 Dulbecco Journal articles
martedì 7 novembre 2017 14:30-16:30
giovedì 9 novembre 2017 14:30-16:30
venerdì 17 novembre 2017 10:30-12:30
martedì 21 novembre 2017 10:30-12:30
venerdì 24 novembre 2017 10:30-12:30
martedì 28 novembre 2017 10:30-12:00 <discussion of problems>
giovedì 30 novembre 2017 14:30-16:30
venerdì 1 dicembre 2017 10:30-12:30
martedì 5 dicembre 2017 15:00-17:00 <first set of problems due>
giovedì 7 dicembre 2017 14:00-16:00
mercoledì 13 dicembre 2017 16:00-17:00 <discussion of problems>, seminar on metamaterials 11:00-12:00
mercoledì 20 dicembre 2017 10:30 <second set of problems due>
part 1. Basics of solid state theory
1. Brief overview of main concepts and tools in Quantum Mechanics. Particle in a box, harmonic oscillator, hydrogen atom, dimer. Bonding and antibonding states. Symmetry and invariants. Representation theory. Perturbation theory. Semiclassical approximation. Tunneling under the barrier.
2. Crystalline solids - elastic properties. Strain. Phonons, heat capacity. Dynamical matrix. Polar instability, phase transitions, ferroelectricity, piezoelectricity, LO-TO splitting. Landau theory of phase transitions. Mean-filed approximation. Critical exponents. Domain structure.
3. Electrons in crystals. Crystal field splitting. Band structure, weak coupling, tight-binding. Band metals and insulators. Optical properties on semiconductors, Lindhard function, excitons. Fermi Liquid, Electron-phonon interactions. Superconductivity. Fermi surface. Nesting, Pierls instability, charge density wave state. Density functional theory.
part 2. electronic interactions and emergent phenomena
4. Transition metal ions, Coulomb repulsion on narrow orbitals, Hubbard model, Hund exchange. Mott transition. Dynamical mean-field theory. Photoemission spectroscopy. Heavy fermions. Kondo effect. Stoner instability. Magnetism.
5. Magnetic insulators. Heisenberg exchange. Spin-orbit coupling, weak ferromagnetism, magnetic anisotropy, anisotropic exchange. Domain structures. Magnetization switching, domain wall motion. Spin currents. Spintronics.
6. Frustrated magnetism. Spiral states and skyrmions. Spin ice. Spin liquid states, resonating valence bonds and valence bond solid states. Breaking of inversion symmetry by magnetism. Exchange striction, multiferroics.
1. martedì 7 novembre 2017 14:30-16:30
outline: particle in a box, dimer, chain of ions, Bloch theorem
problems:
(1) Sketch the absorption spectrum of an electron in a cubic box of size a. What are the frequencies of dipole-allowed transitions?
(2) An electron is in the ground state in a spherical impermeable cavity 10 Å in diameter. What's the pressure it exerts on the box?
(3) An electron is introduced into the conduction band of a cubic ZnTe nanocrystal with an edge of 5 nm. The effective mass is 0.3 m0. Estimate the splitting between the lowest size quantization levels.
2. giovedì 9 novembre 2017 14:30-16:30
notes: draft version for a moment; displayed using Acrobat Reader in View->Page Display->Two Page View mode
problems:
(1) What is the symmetry group of the triangulene molecule? summary original paper
(2) Using symmetry analysis, determine the wave functions of a trimer, with three sites at the vertices of equilateral triangle.
(3) The hopping between the sites of a trimer from the previous problem is 0.5 eV. Compute the eigenenergies and eigenstates.
3. venerdì 17 novembre 2017 10:30-12:30
notes: draft version for a moment; displayed using Acrobat Reader in View->Page Display->Two Page View mode
problems:
(1) Consider a square lattice in 2D, with the nearest-neighbor hopping t. Calculate and plot the band structure.
(2) Sketch the Fermi surface at half filling.
(2*) Among simple metals, only Polonium is known to have a cubic structure, but it's highly radioactive, so we'll consider hexagonal close-packed structures of Cobalt. Sketch the Brillouin zone and the Fermi surface.
(3*) Sketch the Fermi surface of Copper.
4. martedì 21 novembre 2017 10:30-12:30
notes: draft version for a moment; displayed using Acrobat Reader in View->Page Display->Two Page View mode
problems:
(1) NaCl has a cubic structure, with chains made of alternating Na and Cl running in three directions. Consider a single chain and compute phonon dispersion along the chain, assuming that only the nearest neighbor ions interact, and using a value of a force constant 818 N/m. Calculate the phonon velocity and Debye frequency.
(2*) derive the formula for the thermal conductivity of a metal by analogy with that for electrical conductivity
5. venerdì 24 novembre 2017 10:30-12:30
problems:
(1) Using the formula for the specific heat in Einstein model, calculate Cv at high temperatures T≫Θ (Dulong-Petit law).
(2) Same as in (1) but within the Debye model (Dulong-Petit law again?)
6. martedì 28 novembre 2017 14:30-16:30 <discussion of problems>
7. giovedì 30 novembre 2017 10:30-12:00
specific heat in Debye and Einstein models. Anharmonic effects, Gruneisen coefficient, linear and volume thermal expansion. Theory of phase transitions: order parameter, spontaneous symmetry breaking, Landau functional
problems:
(1) Within Landau theory for a vector order parameter, F=a(T-Tc)M2+bM4-MH, a>0, b>0, sketch F(M) for T>Tc and T<Tc.
(2) Compute the magnetic susceptibility dM/dH at H->0 for T>Tc and T<Tc.
8. venerdì 1 dicembre 2017 10:30-12:30
Landau theory, spontaneous symmetry breaking, anomalies in susceptibility and specific heat. Fluctuations. Ginzburg-Levanyuk criterion. Domain walls.
9. martedì 5 dicembre 2017 15:00-17:00
Mean field theory of ferromagnetic phase transitions from a microscopic Heisenberg Hamiltonian. Brillouin function. Mean-field ordering temperature dependence on exchange and site coordination number.
Instability of a ferroelectric state due to dipole-dipole interactions.
10. giovedì 7 dicembre 2017 14:00-16:00
Landau theory for multidimensional order parameters. Breaking of continuous symmetry in systems with short-range interactions. Goldstone theorem. Mermin-Wagner theorem. Fluctuations of an ion in a lattice. Thermal and quantum lattice melting.
11. mercoledì 13 dicembre 2017 16:00-17:00
problem session