Syllabus - a.a. 2017/2018
Here below the detailed syllabus of the Solid State Physics lessons is presented; the topics written in italic will not be discussed during the oral examination.
Introduction to the Solid State Physics Course. Drude model as a classical approach to describe the properties of solid state metal systems: hypotheses of the model, relaxation time. Modification of the second law of dynamics so as to include the collisions effect. DC conductivity, Hall coefficient. AC conductivity; calculation of the relative dielectric constant value. Plasma frequency; Plasma frequency as the frequency of the electronic cloud oscillation. [AM, Chapter 1]
Sommerfeld model: introduction and base hypotheses. Born-Von Karman periodical conditions and electronic wavefunctions. Energy vs. wavevector dependence; Fermi energy and Fermi surface, average value of the electronic energy, electronic energy density. Specific heat of the free electron gas. [AM, Chapter 2]
Failures of the free and independent electron Sommerfeld model [AM, Chapter 3].
Removal of the free electron approximation: the effect of the periodic lattice. Bloch theorem; electronic wavefunction, Born-Von Karman periodic boundary conditions. Demonstration of the Bloch theorem (just the second proof); electronic wavefunction periodicity in the reciprocal space, electron velocity. [AM, Chapter 8]
Application of the theorem to the null potential case. The effect of a small lattice potential on the electronic levels energy values; degenerate and non-degenerate case. Gap opening close to Bragg planes; Fermi surface deformation. Electronic bands, reduced zone and repeated zone scheme. General comparison between the properties of free electrons and Bloch electrons. [AM, Chapter 9]
Core electronic levels: the tight-binding method for the determination of the energy values and of the wavefunctions of the electrons that are more strongly affected by the single atom potential. General result; application of the general result to the case of s levels. Energy vs. wavevector dependence; physical interpretation of the result. [AM, Chapter 10]
Combination of core level and free electron character: the OPW method. The Schrodinger equation applied to OPW wavefunctions; pseudo-potential. Justification of the perturbative approach used to evaluate the effect of the periodic potential on the energy of the electronic levels [AM, Chapter 11].
The effect of external fields on Bloch electrons: semiclassical equations of motion. Hypotheses at the base of the semiclassical approach. Current density for Bloch electrons; the contribution of filled, empty and partially filled bands. Effect of a static electric field; quadratic approximation of the E(k) dependence, electrons and holes. Effective mass tensor; effect of the lattice on the effective mass of the electrons. Effect of a static magnetic field; closed and open orbits. Time required to travel along the electron trajectory in the reciprocal space; new definition of the electron effective mass. Electron motion in the Hall geometry, when a magnetic field is applied perpendicular to an electric field: effect on the electronic energy and on the orbit shape. Electron trajectory for mutually perpendicular electric and magnetic fields, the case of closed orbits and that of open orbits, effect of a strong magnetic field. Hall coefficient for Bloch electrons; resistance in presence of a static magnetic field: magnetoresistance. [AM, Chapter 12]
Calculation of the conductivity tensor for Bloch electrons in presence of a static electric field. [AM, Chapter 13, just the "DC Electrical conductivity" paragraph]
Comparison between the size of the first Brillouin zone and the size of the Fermi sphere: evaluation of the degree of deformation of the Fermi surface with respect to the free electron spherical case. The case of bcc alkali metals and fcc noble metals. [AM, Chapter 15]
Comparison between Fermi surfaces of different metals; E(k) dependence for Cu along high symmetry directions . Effect of the electronic band structure on electron-photon interaction. De Haas-Van Alphen effect: energy levels quantization in the plane perpendicular to the magnetic field (Landau levels), evaluation of the degree of degeneracy. Change from tangency to non tangency condition between the Fermi Surface and the cylindrical structure corresponding to Landau levels: the role of the extremal areas. [AM, Chapter 14]
Effects of electron-electron interaction: Hartree method; Hartree-Fock method and the Slater determinant. Calculation of the expectation value of a one-body operator on the N-electron state represented by the Slater determinant. Calculation of the expectation value of a two-body operator on the N-electron state represented by the Slater determinant. Direct term and exchange term. [AM, Chapter 17] [“slater.pdf” document]
Calculation of the dielectric constant of a free electron gas; Yukawa potential. [AM, Chapter 17]
Introduction to the second quantization approach; creation and destruction operators. Commutation rules. Change of base; field operators. Representation of the action of a one-body operator in terms of construction and destruction operators. Representation of a two-body operator using the second quantization approach; evaluation of the energy contribution of the electron-electron interaction: direct and exchange term. Evaluation of the total energy of the electronic system. [1, Chapter 2] [2, Chapter 1]
Failures of the static lattice model [Chapter 21].
Introduction to the harmonic theory of lattice oscillations. Normal modes of oscillation of the 3D lattice; the dynamical matrix, normal modes polarization vectors. Frequency vs. wavevector dependence for acoustical and optical normal modes. [AM, Chapter 22]
The normal modes picture vs. the phonon picture of lattice oscillations. Calculation of the specific heat of the 3D lattice using low temperature and high temperature approximations; the Debye approach for the specific heat calculation. Debye wavevector and Debye temperature. Comparison between lattice and electrons specific heat. The Einsten model. [Chapter 23]
Interactions with phonons: second quantization approach [2, page 63], crystal momentum conservation. Phonons as a source for electronic collisions; resistivity dependence on temperature [AM, Chapter 26]. Comparison between photons and neutrons as probes to interact with phonons. Detection of phonons and of their frequency vs. wavevector dependence using neutrons. [AM, Chapter 24]
Phonon contribution to the solid dielectric constant; effective electron-electron interaction. [AM, Chapter 26]
Superconductivity: Cooper Pairs; calculation of the binding energy of a Cooper Pair [1, page 133].
[AM] N. W. Ashcroft e N. D. Mermin, Solid State Physics, Saunders College, Philadelphia.
[1] Appunti di Teoria Quantistica della Materia, F. Ortolani
[2] Introduction To Quantum Field Theory In Condensed Matter Physics, H. Bruus and K. Flensberg
