Unit CONDENSED MATTER PHYSICS
 Course
 Physics
 Studyunit Code
 GP005477
 Location
 PERUGIA
 Curriculum
 In all curricula
 Teacher
 Alessandro Paciaroni
 Teachers

 Alessandro Paciaroni
 Hours
 56 ore  Alessandro Paciaroni
 CFU
 8
 Course Regulation
 Coorte 2022
 Offered
 2022/23
 Learning activities
 Caratterizzante
 Area
 Microfisico e della struttura della materia
 Academic discipline
 FIS/03
 Type of studyunit
 Obbligatorio (Required)
 Type of learning activities
 Attività formativa monodisciplinare
 Language of instruction
 Italian
 Contents
 Many body systems with Coulom interaction. Perturbative techniques. Second quantization and condensed matter physics. Electron gas. Density functional and applications. Introduction to magnetism in condensed matter and link to the electron states. Lattice vibrations and their properties.
 Reference texts
 General, medium level: Solid State Physics, N. W. Ashcroft e N. D. Mermin
Specific: Condensed Matter in a Nutshell, G. D. Mahan
Advanced: ManyParticle Physics, G. D. Mahan
Tecnic: Quantum Theory of Solids, C. Kittel  Educational objectives
 The student should obtain the basic knowledge on the many electron systems and the relationship between the electron states and the main phenomena in condensed matter physics. Finally the student should acquire the basis of the modern calculation techniques to calculate the properties of matter by means of the Density Functional theory.
 Prerequisites
 Good knowledge of the basis of Quantum Mechanics and elements of Statistical Physics.
 Teaching methods
 Lectures
 Other information
 none
 Learning verification modality
 Oral exam. The test, lasting 4560 minutes, is devoted to the discussion with the student to define the ability in solving problems of condensed matter physics.
 Extended program
 Introduction to the course. Exam methods. Texts followed.
Outlines of the Drude model. Notes on the Sommerfeld model. Electrons in a periodic potential. "
Motion equations for the free electron. Semiclassical approximation for the free electron. Bloch electron wave packet. Bloch electron propagation speed and equivalence with group speed of the wave packet.
Fundamentals of the semiclassical model for electrons. Validity limits of the model. "Inertia" of complete bands and Green's theorem for periodic functions. Electrical and energy conduction for a complete band. Equivalence between representation of conductivity in terms of electrons or gaps. Electron motion in the presence of a uniform electric field. "Direction" of electron velocity as a function of K. Reflection at Bragg on the edge of the zone.
Motion of electrons in the presence of a magnetic field. Motion constants. Trajectory in plane perpendicular to the magnetic field. Electronlike and holelike trajectories. Trajectory in direct space.
"Hall effect in Drude's theory.
Motion of electrons in the presence of uniform and perpendicular Electric and Magnetic fields. Case of closed orbits. Current density in the plane perpendicular to the magnetic field. Hall effect. Hall coefficient. Case of electronlike, holelike and mixed orbits. Case of open orbits. Magnetoresistance. "
Introduction to nonequilibrium distribution functions. Ergodic principle. Assumptions of the relaxation time approximation. Local relaxation time. Equation for the nonequilibrium distribution function. Integration on the trajectory of the phase space for the calculation of the nonequilibrium distribution function.
Calculation of the nonequilibrium distribution function in the relaxation time approximation. Calculation of the electron survival probability in the phase space. Local dependence of g (r, k, t).
Assumptions: weak field and temperature gradient, uniform field, constant T gradient, tau independent of r and dependent on e (k). Direct current in the approximation of relaxation time. Anisotropy. Complete bands. Electronhole equivalence in metals. Case of free electron.
Transport of electrons to timedependent fields. Drude. Dependent time approximation. Case of high frequencies.
Electronic transport theory with relaxation time dependent on the nonequilibrium distribution function. "
"In" and "out" scattering probabilities and relationships with relaxation time and scattering probability W. Particular case of the relaxation time approximation. Boltzmann equation.
Steady state Boltzmann equation. Linearized Boltzmann equation. Electron scattering from impurities. Recall on the golden rule of Fermi.
Electronic scattering from impurities. Detailed budget report. WiedemannFranz law: Drude model and Sommerfeld model. Deviations from WiedemannFranz's law due to inelastic processes.
"Matthiessen rule. Limits of validity.
Hints to the tightbinding method. Functions of Wannier. "
Representation of the electron wave equation in metal with a flat wave, using the Wannier function method. Resistivity calculation with the Ziman method.
Completion of the derivation of the Ziman formula for resistivity in isotropic systems. Correspondence principle and semiclassical equations of motion for the electron starting from eq. of Schroedinger with Wannier functions.
Screenng of charges in ThomasFermi's approximation. Dielectric function.
Scattering from impurities. Many body problem. Introduction to the adiabatic principle.
Adiabatic approximation. Vibronic wave functions. Variational principle. Gauge transformation for electronic wave functions in adiabatic approximation. Functional vibronic energy.
Adiabatic principle. Vibronic states and energy surface of adiabatic potential. Schroedinger equation for the dynamics of nuclei.
Many electron problem. Hartree method. Hartree electronic density. Hartree potential. Hartree equations. Electronic antisymmetric wave functions. Determinants of Slater. Matrix elements of 1 electron operators. Matrix elements of twoelectron operators.
Wave functions for electronic Hamiltonian in adiabatic approximation. Hartree method. Coulombian potential. Wave functions of antisymmetric electrons. Permutation operator.
Determinants of Slater. Matrix elements of electron operators on Slater determinants. Matrix elements of twoelectron operators on Slater determinants. HartreeFock energy. Variational principle on HartreeFock energy. HartreeFock equations.
HartreeFock equations on electron gas. Eigenvalue equation for Fock operator. Electron gas energy. Exchange term and its anomalies. Slater approximation.
Density functional theory: premise. Definition of the electron density of a particle. Examples of electron density for Hartree and HartreeFock wave functions. HohenbergKohn theorem.
Definition of density functionalities for kinetic energy, electronelectron interaction energy and total energy. Variational principle for electron density (according to HohenbergKohn theorem). Beginning of the derivation of the KohnSham equations. Equivalent noninteracting electron system.
KohnSham equations. Total energy. Local density approximation.
Elastic scattering experiments. Cross section. Elastic neutron scattering. Golden rule of Fermi. Pseudopotential of Fermi. Scattering from a fixed nucleus system. Bragg's law. Elastic scattering of Xrays. Thompson scattering. Diffraction. Bragg's law.