Topic outline

  • Introduction

    The "Condensed Matter Physics I" Course is given in the first term, for a total number of 48 hours (6 credits). It is compulsory for all the training tracks.
    The course will provide theoretical concepts fundamental to understanding the behaviour of electrons in crystals and the basic tools to treat them, both in problems solvable with classical methods and those requiring a quantum treatment. Main topics: models for non-interacting electrons; crystalline lattices and structures; independent electrons in a periodic potential (Bloch electrons) and energy bands; semiconductors; magnetism.

    Timetable:
    Lectures are generally planned in these slots:
    Wed., 9:00-11:00
    Thu., 9:00-11:00

    in Room Z (temporarily in Room V, until Fri 4 Oct), Building A (Piazzale Europa, 1)

    Students are kindly invited to check the announcements for possible changes/updates


    Textbooks:
    N. Ashcroft, N D. Mermin, Solid State Physics, Saunders College (1976) (main text).
    G. Grosso and G. Pastori Parravicini, Solid State Physics, Academic Press (2000).
    C. Kittel, Introduction to Solid State Physics, Wiley (1996).
    L. Mihaly e M.C. Martin, Solid State Physics: Problems and Solutions, Wiley (1996).

     

    Exams:

    The exam includes written & oral parts

    Written part: Two partial written tests are planned during the Course, on the first and on the second half of the program. Dates: 15  November 2019; 20 December 2019 (see Esse3).

    Alternatively (if preferred, or if partial written tests not passed): final written test (2 dates in Jan/Feb 2020; 2 in June/July; 2 in September)

    The oral exam must be done immediately after the written test. Who pass successfully the partial written tests can afford the oral exam during the Jan/Feb session.

    Typically 2 - 3 hours available for the written tests. Books and lecture notes can be used during the written exams.



  • Transport of noninteracting electrons: Drude model (Oct. 2 and 3)

    Introduction to the Course; references. Basic assumptions of the Drude model for metals (noninteracting and free electrons; collisions, damping term, relaxation time; application of the kinetic theory of gases); DC electrical conductivity; Hall effect.

    AC electrical conductivity; dielectric function and plasma frequency. Thermal conductivity and Wiedemann-Franz law. Seebeck effect (Ashcroft-Mermin, Ch. 1) 

  • Transport of noninteracting electrons: Sommerfeld model (Oct. 10, 11, 16, 17)

    Fermi-Dirac distribution. Ground state of free and indep. electron gas; Fermi momentum; energy; temperature; prediction for the pressure exerted by electrons, bulk modulus and comparison with experiments.

    Integrals in energy and k space: density of states (see also these notes). Chemical potential. Use of Sommerfeld expansion; electronic contribution to the specific heat.

    Exercises.

    [Ashcroft-Mermin, Ch. 2 - Exercises n. 1, 3, 4]



  • Lattice and crystalline structures (Oct. 23, 24, 30, 31)

    Introduction to lattice structures: Bravais lattices and crystalline structures in real space. Lattices with basis (generalities; examples about the conventional cells of the cubic lattices; other relevant examples: diamond, graphene, graphite). Packing fraction.

    Other examples of Bravais lattices with basis: zincblende, rocksalt, wurzite (one slide). Wigner-Seitz cells. Reciprocal lattices. Families of lattice planes

    Miller indices. Brillouin zone. X-ray diffraction: Bragg and von Laue.  Structure factor.

    [Ashcroft & Mermin, Ch. 4, 5, 6]


  • Independent electrons in a periodic potential: general aspects (Nov. 6, 7, 14 (1h), 20)

    Periodic potential: Bloch theorem, I and II proof (Ch. 8).

    Consequences of the Block theorem: quasi-crystalline momentum; velocity; energy bands. (Ch. 8)

    Fermi surfaces. Density of states (DoS): different approaches. Derivation of the DoS using the properties of the delta-function (see notes).

    Band index and folding. Van Hove singularities in 1D, 2D, 3D. (Ch. 8)

    Brillouin zones, band folding and band indices, band plots in reduced zone / periodic / extended representation. Fermi surfaces [Ch. 9].

  • Exercises on free electrons models and crystalline structures (Nov. 13, 14 (1h))

    For the Ist partial written test, review of exercises of previous exams, including:

    Ex 4 of 22/11/2011; 1 of 11/4/2007; 1.2 of 18/11/2014; 1.4 of 14/11/2013; 5 of 8/11/2017; 2 of 20/6/2012; 1 of 19/1/2016; 2.5 of 14/11/2013; 1 of 18/11/2014; 1.4 of  20/11/2015; 2.6 of 14/11/2013; 1.2 of 18/11/2014
  • Approximated treatment of electrons in a periodic potential: effects of a weak potential (Nov. 21)

    Effects of a weak perturbing potential (nearly free electrons): non degenerate case; degenerate case (two-levels system) (A&M, ch. 9; excluding The geometrical structure factor in monoatomic lattices with bases)

    Exercises on weak potential: simple cases in 1D

  • Approximated treatment of electrons in a periodic potential: tight-binding approach (Nov. 27)

    The tight-binding approach: introduction, general formulation; the simplified case of s-band arising from a single atomic s-level. Tight-binding in crystals with inversion symmetry; band dispersion. (A&M, ch. 10).

    Exercise: tight-binding: s-band arising from a 1D linear chain of atoms, density of states; half filling of band.

    Suggested as homework: s-band in 2D square lattice: band dispersion along some high symmetry directions, energy isosurfaces in the Brillouin zone.

  • The semiclassical model of electron dynamics (Nov. 28, Dec. 4)

    Validity of semicl. dynamics. Equations of motions. Filled bands. Holes. [first part of Ch. 12]

    Orbits in r and k space. Motion of electrons in uniform and static electric fields. Motion of electrons in uniform and static magnetic fields; electron orbits, hole orbits, open and closed orbits. Period of closed orbits. Fermi surfaces of real metals (examples from www.phys.ufl.edu/fermisurface)

  • The Boltzmann equation (Dec. 5)

    Issues discussed in class and requested for the exam:

    Boltzmann eq.: Ch. 13 only Introduction; Ch. 16: Sect. IV (The Boltzmann eq.); Sect. I (Source of el. scattering); Ch. 16: Sect. II (Scattering prob. and relaxation time); Sect. III (Rate of change of the distribution function due to collisions). ( lecture notes, see parts 1-4). Ch. 13 Sect. IV (DC Electrical conductivity) ( lecture notes, see part 5)

    Issues not discussed in class, optional, not requested for the exam:

    AC Electric conductivity (Ch. 13 Sect. IV); transport in anisotropic materials (lecture notes, see parts 5-6)

  • Semiconductors (Dec. 12)

    Homogeneous semiconductors: materials (elemental and compounds), typical band structures, intrinsic and extrinsic semiconductors.  Intrinsic case: number of carriers in thermal equilibrium.  Extrinsic semiconductors: donor and acceptor levels.  (Ch. 28; excluding: population of impurity levels in thermal equilibrium, eqs. 28.30-28.34)


  • Exercises (Dec. 11, 18, 19)

  • Magnetism (Jan. 8,9)

    NOTE: this year the subject is NOT part of the exam.


    Few concepts about Magnetism in solids; in particular:

    Basic concepts: Magnetization density, Susceptibility; ferromagnetism, paramagnetism, diamagnetism.

    Atomic hamiltonian in a magnetic field. Composition of angular momenta, filling of the electronic shells and spectral terms; Hund's rules. Larmoor diamagnetism. Van Vleck paramagnetism.

    Susceptibility of metals: Pauli paramagnetism. 

    Heseimber hamiltonian. Ground state of a Heisemberg ferromagnet. Mean field equation and critical temperature.