Course: 997SM - FISICA DELLA MATERIA CONDENSATA I 2020

Section outline

Select section Introduction

Introduction

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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 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:
Mon., 9:00-11:00
Wed., 9:00-11:00
in Room A and D, 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
Six dates for the final written test are or will be scheduled (2 dates in Jan/Feb 2021; 2 in June/July 2021; 2 in September 2021: see esse3.
The oral exam must be done within few days after the written test (depending on the number of candidates).

Typically 2 hours available for the written tests. Books and lecture notes can be used during the written exams.
Starting from the Summer session and unless other info, the exams will be done in presence. Exceptionally will be done in remote under motivated and documented request, as explained in the UniTS guide lines reported on https://www.units.it/ateneo/emergenza-covid-19-indicazioni-e-aggiornamenti. The specific form of the written test will depend on the presence or not of candidates in remote mode: in such a case, the written part will include a multiple choice test for all the candidates.
- Select activity Annunci
  
  Annunci Forum
- Select activity Examples of past written tests
  
  Examples of past written tests Folder
- Select activity Written test 15/11/2019 - assignment and solution
  
  Written test 15/11/2019 - assignment and solution File PDF
- Select activity Written test 20/12/2019 - assignment (correct text)
  
  Written test 20/12/2019 - assignment (correct text) File PDF
- Select activity Written test 20/12/2019 - solution ex. 2 and 3
  
  Written test 20/12/2019 - solution ex. 2 and 3 File PDF
- Select activity Annunci
  
  Annunci Forum
Select section Transport of noninteracting electrons: Drude model (5 and 7 Oct. 2020)

Transport of noninteracting electrons: Drude model (5 and 7 Oct. 2020)
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)
- Select activity Periodic Table
  
  Periodic Table File PNG
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  Table of constants File PDF
- Select activity Electromagnetic spectrum
  
  Electromagnetic spectrum File JPG
- Select activity Exercises
  
  Exercises File PDF
Select section Transport of noninteracting electrons: Sommerfeld model (12, 14, 19, 21 Oct. 2020)

Transport of noninteracting electrons: Sommerfeld model (12, 14, 19, 21 Oct. 2020)
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]
- Select activity Derivation of the DoS of the free electron gas
  
  Derivation of the DoS of the free electron gas File PDF
- Select activity Derivation of the Sommerfeld expansion for low T
  
  Derivation of the Sommerfeld expansion for low T File PDF
- Select activity 1D electron gas with periodic boundary conditions and with hard walls (text)
  
  1D electron gas with periodic boundary conditions and with hard walls (text) File PDF
- Select activity 1D electron gas with periodic boundary conditions and with hard walls (solution)
  
  1D electron gas with periodic boundary conditions and with hard walls (solution) File PDF
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  1D electron gas with periodic boundary conditions and with hard walls (figure) File PDF
Select section Lattice and crystalline structures (Oct. 26, 28; Nov. 2, 4)

Lattice and crystalline structures (Oct. 26, 28; Nov. 2, 4)
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]
- Select activity Bravais lattices (a few slides)
  
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- Select activity Wurzite and Zincblende structures (image)
  
  Wurzite and Zincblende structures (image) File PDF
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- Select activity X-ray diffraction and structure factor
  
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- Select activity Diffraction: equivalence Von Laue / Bragg formulations
  
  Diffraction: equivalence Von Laue / Bragg formulations File PDF
Select section Exercises on free electrons models and crystalline structures (Nov. 9, 11))

Exercises on free electrons models and crystalline structures (Nov. 9, 11))
For the Ist partial written test, review of exercises of previous exams, e.g., 7/11/2018 (1st) + those uploaded here concerning the Drude & Sommerfeld models.
- Select activity Bozza soluzione primi punti es. 1 del 18/11/2014
  
  Bozza soluzione primi punti es. 1 del 18/11/2014 File PDF
Select section Independent electrons in a periodic potential: general aspects (Nov. 18, 21, 23, 25)

Independent electrons in a periodic potential: general aspects (Nov. 18, 21, 23, 25)
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].
- Select activity Brillouin zones and high symmetry points (figure)
  
  Brillouin zones and high symmetry points (figure) File PDF
- Select activity Brillouin zones in FCC and BCC lattices and their filling with "nearly free" electrons Fermi spheres (figures)
  
  Brillouin zones in FCC and BCC lattices and their filling with "nearly free" electrons Fermi spheres (figures) File PDF
- Select activity Properties of the delta function (useful for DoS calculations)
  
  Properties of the delta function (useful for DoS calculations) File PDF
- Select activity "Nearly free" electrons (or "empty lattice") bands in FCC structure (problem with solution)
  
  "Nearly free" electrons (or "empty lattice") bands in FCC structure (problem with solution) File PDF
- Select activity Free electron bands for an FCC 3D Bravais lattice (from: Ashcroft & Mermin, fig. 9.5)
  
  Free electron bands for an FCC 3D Bravais lattice (from: Ashcroft & Mermin, fig. 9.5) File PNG
- Select activity A simple code in fortran90 for electron bands in FCC "empty" lattice
  
  A simple code in fortran90 for electron bands in FCC "empty" lattice File F90
- Select activity A simple code in fortran90 for electron bands in SC "empty" lattice
  
  A simple code in fortran90 for electron bands in SC "empty" lattice File F90
- Select activity Examples of band structures for FCC and zincblende solids (free el.; Al; Si, Ge, GaAs)
  
  Examples of band structures for FCC and zincblende solids (free el.; Al; Si, Ge, GaAs) File PDF
- Select activity Van Hove singularities (from : Grosso - Pastori Parravicini)
  
  Van Hove singularities (from : Grosso - Pastori Parravicini) File PDF
- Select activity Van Hove singularity: saddle point in 3D (solution of Ashcroft-Mermin problem 8.2)
  
  Van Hove singularity: saddle point in 3D (solution of Ashcroft-Mermin problem 8.2) File PDF
Select section Approximated treatment of electrons in a periodic potential: effects of a weak potential (Nov. 30)

Approximated treatment of electrons in a periodic potential: effects of a weak potential (Nov. 30)
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
- Select activity Fundamental steps in the derivation of the formulas in the weak potential approximation
  
  Fundamental steps in the derivation of the formulas in the weak potential approximation File PDF
Select section Approximated treatment of electrons in a periodic potential: tight-binding approach (Dec. 2, 5)

Approximated treatment of electrons in a periodic potential: tight-binding approach (Dec. 2, 5)
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.
Select section The semiclassical model of electron dynamics (Dec. 9, 14)

The semiclassical model of electron dynamics (Dec. 9, 14)
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)
- Select activity Few figures on electrons and holes
  
  Few figures on electrons and holes File PDF
- Select activity Example of an electron+hole system (Bismuth)
  
  Example of an electron+hole system (Bismuth) File PDF
Select section The Boltzmann equation (Dec. 16)

The Boltzmann equation (Dec. 16)
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)
- Select activity Derivation of the Bolzmann equation (lecture notes)
  
  Derivation of the Bolzmann equation (lecture notes) File PDF
Select section Semiconductors (Dec. 21)

Semiconductors (Dec. 21)
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)
- Select activity Exercise on 2D semiconductors: estimate of charge carrier density
  
  Exercise on 2D semiconductors: estimate of charge carrier density File PDF
Select section Exercises (Dec 23, Jan 11)

Exercises (Dec 23, Jan 11)
Jan. 11, 2021:
·      20/12/2019 Esercizio 3 (punti 3),4),5)).

·      20/09/2018 Esercizio 3 (in particolare il punto 3)

·      30/01/2017 es3 ( distorsioni di Peierls ) => vedere soluzione completa

·      16/01/2017 es2 punto 6 (ampiezza dell'oscillazione per γ ed E tipici).

·      16/01/2017 Esercizio 1 (dubbi sulle richieste);

·      8/02/2016 Esercizio 2 => vedere soluzione completa

·      16/02/2015 Esercizio 2

·      19/12/2013 Esercizio 3, punti 5 e 6 (condizione sul termine di damping).

·      11/07/2012 esercizio 3
- Select activity soluzione 8/2/2016 esercizio 2
  
  soluzione 8/2/2016 esercizio 2 File PDF
- Select activity soluzione 30/1/2017 (instabilità di Peiersl)
  
  soluzione 30/1/2017 (instabilità di Peiersl) File PDF
Select section prova parziale intermedia (16 Nov)

prova parziale intermedia (16 Nov)
- Select activity Quiz - Elettroni liberi
  
  Quiz - Elettroni liberi
  
  Available until 16 November 2020, 9:50 AM
- Select activity Problemini strutture cristalline
  
  Problemini strutture cristalline Assignment
  
  Available until 16 November 2020, 11:10 AM
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  struttura β-Sn File PNG
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  testo quiz e risposte corrette File PDF
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Select section seconda prova parziale (13 genn. 2021)

seconda prova parziale (13 genn. 2021)
- Select activity Quiz II prova parziale
  
  Quiz II prova parziale
  
  Ci sono 15 domande a risposta multipla. Una sola risposta è esatta.
  Ogni domanda vale 2 punti: +2 risposta esatta, -1 risposta errata, 0 non risposto.
  Quindi il punteggio del Quiz è normalizzato a 30. Sufficienza con almeno 18/30.
  Tempo 15 minuti.
  
  Available until 13 January 2021, 9:35 AM
- Select activity Compito della II prova parziale
  
  Compito della II prova parziale Assignment
  
  Il compito consta di 10 domande. E' richiesto di farle tutte. Il tempo di consegna potrà costituire elemento di valutazione.
  
  Available until 13 January 2021, 10:45 AM
Select section Test utilizzo strumento quiz per le prove scritte on line

Test utilizzo strumento quiz per le prove scritte on line
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  quiz di prova
  
  Available until 18 January 2021, 6:15 PM
Select section prova scritta (20 gennaio 2021)

prova scritta (20 gennaio 2021)
- Select activity Quiz 20/1/2021
  
  Quiz 20/1/2021
  
  Ci sono 15 domande a risposta multipla. Una sola risposta è esatta.
  Ogni domanda vale 2 punti: +2 risposta esatta, -1 risposta errata, 0 non risposto.
  Quindi il punteggio del Quiz è normalizzato a 30.
  Tempo 15 minuti.
  
  Available until 20 January 2021, 9:30 AM
- Select activity compito 20/1/2021
  
  compito 20/1/2021 Assignment
  
  Available until 20 January 2021, 10:55 AM
Select section prova scritta (22 febbraio 2021)

prova scritta (22 febbraio 2021)
- Select activity Quiz 22/2/2021
  
  Quiz 22/2/2021
  
  Ci sono 15 domande a risposta multipla. Una sola risposta è esatta.
  
  Ogni domanda vale 2 punti: +2 risposta esatta, -1 risposta errata, 0 non risposto.
  
  Quindi il punteggio del Quiz è normalizzato a 30.
  
  Tempo 15 minuti.
  
  Available until end of 26 February 2021
- Select activity Compito 22/2/2021
  
  Compito 22/2/2021 Assignment
  
  È possibile svolgere il compito su fogli e poi scansionarli; in alternativa, usare qualsisi editor di note, testi e documenti in uso sul proprio pc.
  In ogni caso, alla fine fare un UNICO FILE PDF nominato COGNOME.pdf e caricarlo (limite 20 Mb).
  Il tempo per lo svolgimento è 1h 15'. Altri 15' extra sono concessi SOLO per il caricamento. ATTENZIONE: NON VERRANNO CONCESSE PROROGHE (IL SISTEMA CHIUDERÀ AUTOMATICAMENTE), QUINDI NON UTILIZZARE I MINUTI EXTRA PER TENTARE DI COMPLETARE LO SVOLGIMENTO.
Select section prova scritta (15 giugno 2021)

prova scritta (15 giugno 2021)
- Select activity Quiz
  
  Quiz
  
  Ci sono 15 domande a risposta multipla. Una sola risposta è esatta.
  
  Ogni domanda vale 2 punti: +2 risposta esatta, -1 risposta errata, 0 non risposto.
  
  Quindi il punteggio del Quiz è normalizzato a 30.
  
  Tempo 15 minuti.
  
  Available until 17 June 2021, 11:55 PM
- Select activity Compito
  
  Compito Assignment
  
  Available until 15 June 2021, 11:00 AM