(Fisica della Materia Condensata I)

Laurea Magistralis in Physics

(Corso di Laurea Magistrale Interateneo in Fisica)

University of Trieste

Academic Year 2016/2017

Department of Physics (Miramare campus) - University of Trieste - Strada Costiera 11, I-34151 TRIESTE

(e-mail: peressi@ts.infn.it; tel. +39 040 2240242)

The

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.

Staring from Oct. 18, the teacher is available for questions and doubts on Wed., 12:00-13:00, in Room 205 of Building F, via Valerio 2; for other appointments, contact by e-mail.

Lecture | Date | Argument | Details |

1-2 | 27/9/2016 | Transport of noninteracting electrons: Drude model | 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. (Ashcroft-Mermin, Ch. 1, first part) |

3-4 | 28/9/2016 | Transport of noninteracting electrons: Drude model | AC electrical conductivity; dielectric function and plasma frequency. Thermal conductivity and Wiedemann-Franz law. (Ashcroft-Mermin, Ch. 1) [Seebeck effect: not explained in class, to be done]. Homework: EXERCISES on the Drude model from the textbook (2, 3 of Ch. I); other exercises |

5-6 | 5/10/2016 | Transport of noninteracting electrons: Sommerfeld model | 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. [Ashcroft-Mermin, Ch. 2 (not section 2: derivation of the FD distribution)]; |

7-8 | 6/10/2016 | Transport of noninteracting electrons: Sommerfeld model |
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, remaining parts of Ch. 2] See also lecture notes from a course at UCSD summarizing Ch. 1 and 2 of A&M book. Note: for the derivation of the Sommerfeld expansion see Course of Statistical Mechanics by prof. Senatore (e.g., see Landau, Course of Theoretical Physics, Pergamon, volume 5, Statistical Physics, Part I (in the 3rd edition: p. 169-170). |

9-10 | 13/10/2016 | " | EXERCISES on the free independent electrons models. In particular: ex. 1.1 (first two points), 1.2, 1.3, 2.1, 2.3 from A&M book. |

11-12 | 14/10/2016 | " | EXERCISES on the free independent electrons models.
In particular: 1D electron gas in the Sommerfeld model;
1D electron gas
with periodic boundary conditions and with hard walls
(
figure of the solution); ex. 2.4 from A&M book. |

13-14 | 19/10/2016 | Lattices and crystalline structures | 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 [Ch. 4]. (some slides) |

15-16 | 20/10/2016 | " | Other examples of Bravais lattices with basis: zincblende, rocksalt, wurzite (one slide). Wigner-Seitz cells. Reciprocal lattices. Families of lattice planes (some slides) . [Ch. 5] . |

17-18 | 26/10/2016 | " | Miller indices. Brillouin zone. X-ray diffraction: Bragg and von Laue (one slide) (NO: Experimental geometries suggested by the Von Laue condition) [Ch. 6]. |

19-20 | 27/10/2016 11:20-12:45 | " | Structure factor; example of diamond as a Bravais with basis (some slides). EXERCISES on crystalline lattices. |

21-22 | 4/11/2016 | Independent electrons in a periodic potential: exact results and approximations [Ch. 8, D, F, 9, 10]. | Periodic potential: Bloch theorem, I and II proof (Ch. 8). |

23-24 | 9/11/2016 | Exercises on the first part of the course | |

10/11/2016 11:15-13:15 |
WRITTEN TEST ON THE I PART OF THE COURSE (Room 2A, Building H3) | ||

25-26 | 11/11/2016 | " | Consequences of the Block theorem: quasi-crystalline momentum; velocity; energy bands. (Ch. 8) |

27-28 | 16/11/2016 | "" | Fermi surfaces. Density of states; different approaches. Derivation of the DOS using the properties of the delta-function (one slide) Band index and folding. Van Hove singularities in 1D, 2D, 3D. EXERCISES on Bloch electrons. |

29-30 | 17/11/2016 | " | Brillouin zones, band folding and band indices, band plots in reduced zone / periodic / extended representation. Fermi surfaces [Ch. 9]. Exercises on Bloch electrons: bands of free electrons in FCC structure (problem with solution; pdf file). (figure with Brillouin zones and high symmetry points). (Homework: bands of free electrons in other 2D and 3D structures and along high symmetry directions.) |

31-32 | 23/11/2016 | " | Effects of a weak perturbing potential (nearly free electrons): non degenerate case; degenerate case (two-levels system) (summary (two slides)). [Ch 9 excluding: The geometrical structure factor in monoatomic lattices with bases; Importance of spin-orbit splittig]. |

33-34 | 24/11/2016 | " | Exercises on the weak potential. The tight-binding approach: introduction, general formulation; the simplified case of s-band arising from a single atomic s-level. |

35-36 | 30/11/2016 | " | Tight-binding in crystals with inversion symmetry; band dispersion. Exercises on 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. |

37-38 | 1/12/2016 | Semiclassical dynamics of the electrons [Ch. 12] | Validity of semicl. dynamics. Equations of motions. Filled bands. Holes. [first part of Ch. 12] |

39-40 | 2/12/2016 | " | Orbits in r and k space. Motion of electrons in
uniform and static electric fields.
Motion of electrons in uniform and static magnetic
fields (and related exercise).
[Ch. 12] |

41 | 7/12/2016 | " | More on motion of electrons in a magnetic field: electron orbits, hole orbits and open orbits. Period of close orbits. Fermi surfaces of real metals (examples from www.phys.ufl.edu/fermisurface) |

42-43 | 7/12/2016 | Transport, Boltzmann equation [Ch. 13 and 16] | 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 and AC Electric conductivity); transport in anisotropic materials ( lecture notes, see parts 5-6) ( lecture notes). |

44-45 | 14/12/2016 | Semiconductors [Ch. 28] | Homogeneous semiconductors: materials (elemental and compounds), typical bandstructures, intrinsic and extrinsic semiconductors. Intrinsic case: number of carriers in thermal equilibrium. Extrinsic semiconductors: donor and acceptor levels. |

46 | 14/12/2016 | Exercises: correction of the I written partial test | |

47-48 | 15/12/2016 | Exercises on Bloch electrons and semiclassical model | |

49 | 21/12/2016 | Magnetism [Ch. 31] | Few concepts about Magnetism in solids: Larmoor diamagnetism; Pauli paramagnetism. (this lecture is not part of the exam program) |

50 | 21/12/2016 | Exercises on Bloch electrons. | |

51-52 | 11/1/2017 | Exercises | |

16/1/2017 14:00-17:00 |
WRITTEN TEST ON THE II PART OF THE COURSE (2h) and FINAL WRITTEN TEST (3h) (Dept. Physics, via Valerio 2, Building F, Room A) | ||

30/1/2017 14:00-17:00 |
FINAL WRITTEN TEST (Dept. Physics, via Valerio 2, Building F, Room B) | ||

2/2/2017 9:00-17:00 |
ORAL EXAMS (Cond Matt curriculum) (Miramare campus, Leonardo Building, Room 204) | for 5 candidates (register on esse3 system!) | |

6/2/2017 14:00-19:00 |
ORAL EXAMS (other curricula) (Miramare campus, Leonardo Building, Room 204) | for 5 candidates (register on esse3 system!) | |

9/2/2017 9:00-14:00 |
ORAL EXAMS (other curricula) (Miramare campus, Leonardo Building, Room 204) | for 5 candidates (register on esse3 system!) | |

10/2/2017 9:00-18:00 |
ORAL EXAMS (other curricula) (Miramare campus, Leonardo Building, Room 204) | for 8 candidates (register on esse3 system!) | |

22/2/2017 14:00-17:00 |
FINAL WRITTEN TEST (Dept. Physics, via Valerio 2, Building F, Room B) | ||

23/2/2017 9:00-17:00 |
ORAL EXAMS (Miramare campus, Leonardo Building, Room 204) | ||

24/2/2017 14:00-18:00 |
ORAL EXAMS (Miramare campus, Leonardo Building, Room 204) | ||

12/6/2017 14:00-17:00 |
FINAL WRITTEN TEST (Dept. Physics, via Valerio 2, Building F, Room B) | (register on esse3!) | |

15/6/2017 9:00-12:00 |
ORAL EXAMS (Miramare campus, Leonardo Building, Room 204) | (register on esse3!) | |

7/7/2017 9:00-12:00 CORRECT! (esse3) (wrong, old schedule: 14:00-17:00) |
FINAL WRITTEN TEST (Dept. Physics, via Valerio 2, Building F, Room B) | (register on esse3!) | |

5/9/2017 9:00-12:00 |
FINAL WRITTEN TEST (Miramare Campus, Leonardo Building, Room 204) | (register on esse3! "prova parziale") | |

20/9/2017 9:00-12:00 |
FINAL WRITTEN TEST (Dept. Physics, via Valerio 2, Building F, Room D) | (register on esse3! "prova parziale") |

**Textbooks and other material:**

TEXTBOOKS:

BOOK FOR EXERCISES:

**Exams:**

Two partial written tests (partial: on the first and on the second half
of the program) or one final written exam + final oral exam.
Dates to be announced.
Typically 2 - 3 hours available
for the written tests. You can bring with you books, lecture notes...

Examples of past written intermediate tests and final exams:

Test.I 23-11-2010

Test.II 12-01-2011

Final exam 25-01-2010

Test.II 16-01-2012

Final exam 23-01-2012

Final exam 24-02-2012

Final exam 20-06-2012

Final exam 11-07-2012

Test.II 14-01-2013

Final exam 28-01-2013

Final exam 15-02-2013

Final exam 11-06-2013

Test.I 14-11-2013

Test.II 19-12-2013

Final exam 27-01-2014

Final exam 17-02-2014

Final exam 24-06-2014

Final exam 14-07-2014

Final exam 10-09-2014 (the same of 20-06-2012!)

Test.I 18-11-2014

Test.II 13-01-2015

Final exam 23-01-2015

Final exam 16-02-2015

Final exam 25-02-2015

Final exam 13-07-2015

Test.I 20-11-2015

Test.II 14-01-2016

Final exam 19-01-2016

Final exam 08-02-2016

Final exam 04-07-2016

Final exam 16-01-2017

*Last modified: August 2017*