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.

    Lectures planned from Oct. 4, 2017 - Dec. 22, 2017:
    Wed., 9:00-11:00, Room V, Building A
    Thu., 11:00-13:00, Room A "Costruzioni", Building C9

    Additional few lectures are planned in other days (see detailed calendar;  pay attention to the announcements)

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



    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, with dates: Nov. 8, 2017 and Jan. 15, 2018 (FINAL DATES).

    Alternatively (if preferred, or if partial written tests not passed): final written test (2 dates in Jan/Feb 2018; 2 in June/July; 2 in September - see detailed calendar)

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

    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. 4 and 5)

    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. 9, 11 and 12)

    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. Comparison between predictions of Drude and Sommerfeld model.

    [Ashcroft-Mermin, Ch. 2]


  • Lattice and crystalline structures (Oct. 18, 19, 25)

    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 (Oct. 26, Nov. 15, 16, and 17)

    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 resources).

    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. 6, 7)

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

    - 23/11/10 (2.6 + 1.2, 1.3) (see also 18/11/14)

    - 24/02/12 (1.5, 1.6)

    - 11/6/13 (2.3)

    - 14/11/13 (1 , 2)

    - 17/02/14 (1)

    - 14/07/14 (1)

    - 20/11/15 (2)

    - 8/2/16 (1.4, 1.5)

  • Approximated treatment of electrons in a periodic potential: effects of a weak potential; tight binding approximation (Nov. 22 and 23)

    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; Importance of spin-orbit splitting.)

    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.


    1) weak potential: simple cases in 1D

    2) tight-binding: s-band arising from a 1D linear chain of atoms, density of states; s-band in 2D square lattice: band dispersion along some high symmetry directions, energy isosurfaces in the Brillouin zone, half filling of bands. (A&M, ch. 10)

  • The semiclassical model of electron dynamics (Nov. 29 and 30)

    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

  • The Boltzmann equation (Dec. 1)

    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)

  • Exercises on tight-binding model: bands, density of states, semiclassical model of electron dynamics (Dec. 6)

  • Semiconductors (Dec. 7, 13 and 14)

    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)

    The p-n junction in equilibrium   (Ch. 29, first part)

    Exercises (see details below)

  • Magnetism (Dec. 18 and 19)

    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.

    ("Magnetism" is not part of the written test, but only of the oral exam)

    • Exercises (Dec 21, Jan 10 and 11)

      Dec. 21, 2017:

      Ex. 3 from the written test of 25/1/2010

      Ex. 2 and 3 from the written test of 13/1/2015

      Jan. 10, 2018:

      Ex. 1 from the written test of 12/1/2011

      Ex. 3 from the written test of 15/2/2013

      Ex. 3 from the written test of 16/2/2015

      Ex. 3 from the written test of 25/2/2015

      Jan. 11, 2018:

      Ex. 3 from the written test of 15/12/2009

      Ex. 3 from the written test of 16/1/2012

      Ex. 2 from the written test of CMPII of 23/2/2011