997SM - FISICA DELLA MATERIA CONDENSATA I 2024
Section outline
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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 the Condensed Matter Physics training track.
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 are planned in these slots (starting from Sept. 23, 2024):
Mon 16:00-18:00
Wed 11:00-13:00in Room B, Building F (via Valerio, 2)
https://orari.units.it/agendaweb/index.php?view=easycourse&_lang=&include=attivita&anno=2024&attivita[]=EC465476&date=23-09-2024
Students are kindly invited to check the announcements for possible changes/updates
Lectures are recorded and collected on TEAMS.
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
Six dates for the final written test (2 dates in Jan/Feb 2025; 2 in June/July 2025; 2 in August/September 2025) will be scheduled and visible on esse3.The oral exam must be done within few days after the written test (depending on the number of candidates).
Typically 2.5 hours are available for the final written tests. Books and lecture notes can be used during the written exams.
Lectures will be given in Italian, but textbooks, lecture notes and other materials are in English.
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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. (Ashcroft-Mermin, Ch. 1)
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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]
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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]
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Opened: Monday, 21 October 2024, 1:54 AMCloses: Wednesday, 3 September 2025, 12:53 PM
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4. Independent electrons in a periodic potential: general aspects (21 (1h), 23 (1h), 28 (1h), 30 Oct.; 4, 6 Nov. 2024)
Periodic potential: Bloch theorem, I [21 Oct.] and II [23 Oct.] proof (Ch. 8).
Consequences of the Block theorem: crystalline momentum; velocity; energy bands. (Ch. 8) [28 Oct.]
Fermi surfaces. Density of states (DoS): different approaches. Derivation of the DoS using the properties of the delta-function . Van Hove singularities. (Ch. 8) [30 Oct.]
Brillouin zones, band folding and band indices, band plots in reduced zone / periodic / extended representation Fermi surfaces and their folding into the first Brillouin zone (end of Ch. 8]. [30 Oct.]Divergences in the DOS integrals - Van Hove singularities in 1D, 2D, 3D. Exercise n. 2 Ch. 8 of A&M [4 Nov.]"Empty lattice" band structure in 1D and FCC [4 Nov.]
A nice applet for plotting the "empty lattice" bands: PHY.K02UF Molecular and Solid State Physicshttp://lampx.tugraz.at/~hadley/ss1/emfield/empty/empty.phpComments on selected slides on: band structures, Fermi surfaces, DOS [6 Nov.]Calculation of k_F in the "empty lattice" model and comparison of the Fermi surface with the Brillouin zone [6 Nov.] -
5. Approximate treatment of electrons in a periodic potential: effects of a weak potential (11, 18(1h) Nov. 2024)
Effects of a weak perturbing potential (nearly free electrons): non degenerate case; degenerate case (two-levels system) (A&M, ch. 9)
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6. Approximate treatment of electrons in a periodic potential: tight-binding approach (18 (1h), 20 Nov. 2024)
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).
Bloch sums and their periodicity in direct space depending to k.
Exercises:
- weak potential in 1D
- tight-binding: s-band arising from a 1D linear chain of atoms, density of states; half filling of band.
- tight-binding: s-band in 2D square lattice: band dispersion along some high symmetry directions, energy isosurfaces in the Brillouin zone.