Fys 208, Solid State

Notes about the lectures, Spring Semester, 2002

Updated 01.06.2002
Note: By now, the distributed material is in some cases only listed,
      not all are electronic (some new added recently).
      It is planned to put here links to all electronic versions

21.01.02 Introduction. Walking through the topics, about molecular physics Interesting phenomena Quantum Mechanics, Statistical Physics 23.01.02 Crystals as huge molecules Born-Oppenheimer approximation Fix the nuclei, do quant. mech. for electrons Electron energies depend on position of nuclei (for 2 atomic nuclei just the distance) The lower level goes down (the united nuclei) LCAO Linear Combination of Atomic Orbitals (mentioned) The linear chain of masses on a spring Dispersion relation Ashcroft-Mermin chapter 22 pages 430-433 Mentioned phase velocity, group velocity (derivation using Fourier transformation) relation between omega and k - the 'wave vector' Note: Basic constants (picture gif) Note: Basic constants (PDF-format)(for printout) 28.01.02 Linear chain with 2 different masses. The calculation done in detail (only 2 people present) 30.01.02 Linear chain with 2 different masses. Review Note: Density of normal modes Concepts: Lattice constant ( a ) normal coordinate x normal modes discretization (best understood on a circle!) .... periodicity (this will return many times) normal coordinates,modes: any motion can be expressed as linear combination of normal modes Mathematics of k-numbers 04.02.02 Einstein and Debye Note: Einsteins Model, Kittel jpg 148kB) calculation details, jpg 81kB Note: Debye Model, L.K. 127kB Debye Temp. and Heat Cond. (periodic table) 80kB High resolution, same table table50.jpg 150 kB Note: Boltzman Factor (Illustration only) Note to previous lectures: Note: Normal Modes (Matrix Formulation) Note to future lectures: Note: Harmonic Oscillator Discussing the Boltzmann Factor Work with Einstein Model Work with Debey Model Einstein vs. Debye model: Ashcroft-Mermin page 462 06.02.02 1. Debye; 2. Heat Conductivity Work with Debye Model: High temperature limit (see the note) The low temperature world: calorimetry with C ~ T3 Heat conductivity Suhail: Combined Einstein and Debye Model Thermal Conductivity: Ashcroft-Mermin page 499-500 Thermal Conductivity: see also electrons, Ashcroft-Mermin page 22 11.02.02 Computer simulations: (Matlab) Simulations Chains - showing the addition of wave numbers types of vibrations - waves - demonstration Diagonalization of the coupled Harm.Osc. matrix More on Heat Conductivity mean free path T-dependence ( T3 low T, T-1 high T) Thermal Conductivity of LiF fig. 25.5, Ashcroft-Mermin, page 505 Harmonic oscillator: the algebraic method, obtaining operators a,a+ Harmonic Oscillator: Quanta Phonons Anharmonic effects: terms of type u3 ... it means terms like a+k ak' ak'' Umklapp process - see "chains" simulation above(Thermal Conductivity) Anharmonic effects: Thermal expansion 13.02.02 Harmonic oscillator, phonons revisited Anharmonic effects revisited Electronic motion Classical Electron Gas model (Drude) Heat and electrical conductivity of electrons a) HEAT CONDUCTIVITY - see phonons b) electrical conductivity Drift velocity from acceleration Conductivity (relation to resistance) Wiedemann-Franz law derived Lorenz number CLASSICAL ASSUMPTIONS: a) the gas heat capacity (related to Boltzmann) b) velocity squared replaced by kT Conductivity (sigma) 18.02.01 FERMI GAS MODEL Pauli principle Independent particles-independent probabilities Indistinguishable particles -- new form of probability P(r1,r2)=P(r2,r1) Slater Determinant Available states (density of states in k-space) NOTE (scanned) fermi-density.jpg Fermi momentum, Fermi energy, Fermi temperature Lu: No two leaves are the same (Chinese saying) (just opposite of the Indistinguishable particles) 20.02.01 Fermi gas - at nonzero T Heat capacity etc Sommerfeld integrals: Fermi Gas In Ashcroft-Mermin: Sommerfeld integrals are pages 45-47 FermiSom.ps  (postscript file, overview) Scanned: sommer-meth-1.jpg (Sommerfeld's method) sommer-meth-2.jpg sommer-eval.jpg (Evaluation details) 25.02.02 Fermi gas - at nonzero T (continue last) Heat capacity etc Note: Due to Fermi statistics vanishingly small contribution to heat capacity from electrons Wiedemann-Franz law in Fermi-gas version (Table 4, section THERMAL CONDUCTIVITY OF METALS, Kittel, 7.ed.p. 166) (accidental succes of classical, Ashcroft-Mermin p. 23, p. 52) Compare heat capacity from electrons and lattice see e.g. Ashcroft-Mermin p. 463 (to be discussed in review) Bloch theorem (Ashcroft-Mermin p. 132-135) Note that Bloch theorem is stated (Ashcroft-Mermin p. 133, eq. (8)) 27.02.02 Bloch theorem via Fourier NOTE: Bloch.ps (postscript file) Expansions in function sets (Fourier series, other sets) Dirac notation This is done e.g. in Kittel, but MUCH LESS DETAIL (also Ashcroft-Mermin p. 135-139) This is referred to as "second proof of Bloch theorem" 04.03.02 Bloch theorem via Fourier continued 06.03.02 Bloch Theorem and Tight Binding models see fig. 10.4, page 183, Ashcrot Mermin Bands - weak potential limit, no potential periodicity in the k-space Ashcroft-Mermin p.159-160 see in particular fig 9.4 Ashcroft-Mermin p.160 11.03.02 3-dim crystals electronic bands the shape of bands Notes for the model construction of 3-dim bands 3dim-1.jpg 3dim-2.jpg 3dim-3.jpg 13.03.02 Semiclassical motion (background information - see textbook - Chapter 12) Our derivation of effective mass: 227-228 also on page 220 bottom - 221 some aspects are mentioned in the appendix E p. 765 but not relevant to our discussion Our discussion follows more Kittel (to be added later) The NOTE below is a complete presentation NOTE Effective Mass 2001 NOTE in PDF format PDF of 2001 This note will be updated (small misprints found, make clearer) Selfconsistent method (postponed-remember later) 18.03.02 Population of states in the conduction band Electrons and holes 'Number of available states' in a band Band gaps: Metals, Insulators, Semiconductors 20.03.02 Population of states in the valence band Holes Semiconductor equation Position of the parameter mu between the bands Impurity states The concept of p-n junction 25.03.02 No lecture 27.03.02 No lecture 01.04.02 No lecture 03.04.02 p-n junction 02.04.01 p-n junction 7 page NOTE The placement of Fermi level in acceptor and donor doped semiconductors. The mechanisms leading to the common position of Ferni level. Diffusion current and drift current. Fick's law for diffusion. Diffusion constant. Mobility. Einstein-Nernst relation Model for charge density distribution Model for the depletion zone The rectifying function of p-n junction 08.04.02 p-n junction 10.04.02 p-n junction (From LGJ thesis) 15.04.02 Metals Fermi Surface Explaining the fermi surface in 2 dimensions Electron motion in magnetic field Cyclotron Frequency ... (new *NOTE* ... to be updated) (*** add references to Ashcroft-Mermin *** to be updated) Metals as poly-micro-crystalline 17.04.02 Magnetic Properties of Materials NOTE ("All on magnetism") Diamagnetism ( Note new 2001?? *** to be updated) 22.04.02 Paramagnetism 24.04.02 Ferromagnetism ( Note to be updated) Electromagnetic nature of the exchange interaction 29.04.02 Ferromagnetism Heisenberg model of ferromagnet Mean Field model 01.05.02 No lecture 06.05.02 Hall effect (NOTE ***) Superconductivity (NOTE ***) 08.05.02 Superconductivity The matrix explanation of coherence 13.05.02 Crystal systems, Crystal Structure 15.05.02 Crystal systems, Crystal Structure Crystal lattices .Ashcroft-Mermin chapter 4, chapter 7; parts only Determination of crystal structures Ashcroft-Mermin chapter 6, parts only 20.05.02 - 04.06.02 Colloquia, Group work 06.06.02 Oral exams Compare with the last year notes