997SM - FISICA DELLA MATERIA CONDENSATA I 2023
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 training tracks: Condensed Matter Physics; Nuclear and Subnuclear Physics, Environmental and Interdisciplinary Physics.
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:
Thu 11:00-13:00
Fri 11:00-13:00in Room B, Building F (via Valerio, 2)
(*) NO LECTURES ON:
Fri. 29 Sept.
Thu. 2 Nov.
Fri. 3 Nov.
Fri. 8 Dec.
but instead:
Wed. 4 Oct.
Wed. 25 Oct.
Wed. 8 Nov.
Wed. 6 Dec.
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 2024; 2 in June/July 2024; 2 in August/September 2024) 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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Test on the Drude and Sommerfeld models for non interacting, independent and free electrons Quiz
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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. [12 Oct]
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. [13 Oct]
X-ray diffraction: Bragg and von Laue. Structure factor. [19 Oct]
[Ashcroft & Mermin, Ch. 4, 5, 6]
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Test on lattices and crystalline structures QuizOpened: Thursday, 19 October 2023, 1:00 PMClosed: Saturday, 31 August 2024, 11:59 PM
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Periodic potential: Bloch theorem, I and II proof (Ch. 8). [25 Oct.]
Consequences of the Block theorem: crystalline momentum; velocity; energy bands. (Ch. 8) [26 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)
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 + Ch. 9]. [27 Oct.]Divergences in the DOS integrals - Van Hove singularities in 1D, 2D, 3D. Exercise n. 2 Ch. 8 of A&M [6 Nov.]"Empty lattice" band structure in 1D and 2D (square lattice) + other exercised [8, 9 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 [10 Nov., I part of the lecture]Bloch sums and their periodicity in direct space depending to k. Calculation of k_F in the "empty lattice" model and comparison of the Fermi surface with the Brillouin zone [10 Nov.- II part] -
Approximated treatment of electrons in a periodic potential: tight-binding approach (Nov. 24 (2h+2h exercises on weak pot. + tight binding), 2023)
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).
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.
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Validity of semicl. dynamics. Equations of motions. Filled bands. Holes. [first part of Ch. 12] [30 Nov. 2023]
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. [1 Dec. 2023]
Fermi surfaces of real metals (examples from www.phys.ufl.edu/fermisurface)
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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)
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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: eqs. 28.23-28.27)
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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.
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